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					                     Part 15 – Regression Models




Statistics and Data
Analysis

Professor William Greene
Stern School of Business
IOMS Department
Department of Economics
                Part 15 – Regression Models




Statistics and Data Analysis


    Part 15 – Regression
              Models
      1/49
                                                                                                                                                                                                                                                                                                                                                                          Part 15 – Regression Models


                                                                                                 Linear Regression Models
                                                                                                                   Analyzing residuals
                                                                                                                     Violations of assumptions
                                                                                                                     Unusual data points
                                                                                                                     Hints for improving the model
                                                                                                                   Model building
                                                                                                                     Linear models – cost functions
                                                                                                                     Semilog models – growth models
                                                                                                                     Logs and elasticities



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                              Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                      Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                 900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                               14



                                                                                                                                                                                                                                                                                                                        e  mc  
                                                       Pepperoni                                                                                                                                                 99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                       Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                    800000
                                                                                                                                                                                                                 95
                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                                                        800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        51
                                                                                                                                                                                                                                                                           AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                         700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                               P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                  10
                                                       Garlic                                                                                                                                                    80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                               Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Percent
                                                       Meatball                                                                                                                                                  70
                                                                                      600000
                                                                                                                      Listing




                                                                                                                                                                                                                                                                                              Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                           8
                                                                                                                                                                                                       Percent




                                                                                                                                                                                                                 60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                 50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                     40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Listing
         5.8%                                                                                                                   400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                     20                                                                                                                                                                                                                                                             20
                                                                                                                                300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                 10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                    200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                          0       00     00     00     00      00     00     00     00     00
                                                                                                                                100000                                                                                                                                                                  100000
                                                                                                                                                                                                                 1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                      15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                 IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      3/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 An Enduring Art Mystery
                                                                                                                                                                                                                                                                                                                                               Graphics show relative
                                                                                                                                                                                                                                                                                                                                               sizes of the two works.




                                        The Persistence
                                        of Statistics.
                                        Hildebrand, Ott
                                        and Gray, 2005




                                                                                                 Why do larger
                                                                                                 paintings command                                                                                                                                                                                                                                                                                                                                The Persistence of
                                                                                                                                                                                                                                                                                                                                                                                                                                                  Memory. Salvador
                                                                                                 higher prices?                                                                                                                                                                                                                                                                                                                                   Dali, 1931



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      7/49
                                                                                                                                                                                                                                                                                                                                                                                                 Part 15 – Regression Models


                                                                                                             The Data
                                                                                                                                Histogram of ln (US$)
                                                           80

                                                           70

                                                           60

                                                           50
                                               Frequency




                                                           40

                                                                                                                                                                                                                                                                                                                                                                                                                       Histogram of ln (SurfaceArea)
                                                           30

                                                           20                                                                                                                                                                                                                                                                                     90

                                                                                                                                                                                                                                                                                                                                                  80
                                                           10
                                                                                                                                                                                                                                                                                                                                                  70
                                                           0
                                                                                                    10.5                        12.0                        13.5                                     15.0                                  16.5                                                                                                   60




                                                                                                                                                                                                                                                                                                                                   Frequency
                                                                                                                                                          ln (US$)
                                                                                                                                                                                                                                                                                                                                                  50

                                                                                                                                                                                                                                                                                                                                                  40

                                                                                                                                                                                                                                                                                                                                                  30

                                                                                                                                                                                                                                                                                                                                                  20

                                                                                                                                                                                                                                                                                                                                                  10

                                                                                                                                                                                                                                                                                                                                                     0
                                                                                                                                                                                                                                                                                                                                                             3.2                              4.0                                 4.8                          5.6       6.4                                                                     7.2                                   8.0                                    8.8
                                                                                                                                                                                                                                                                                                                                                                                                                                                             ln (SurfaceArea)




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                            Boxplot of Listing                                   Scatterplot of Listing vs IncomePC                                               Probability Plot of Listing                                                                   Scatterplot of Listing vs IncomePC                                                         Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                                        Normal - 95% CI                                                                                                                                                                                                                                                                       Normal
                                                            Category                                                                             900000                                                                                                                                                                   900000
                             Meatball                                                      900000                                                                                                                                                                                                                                                                                                                 14



                                                                                                                                                                                                                                                                                                                                               e  mc  
                                                            Pepperoni                                                                                                                                                             99                                                                                                                                                                                                                                                                                                                                               Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                         Mean      369687                                                                                                                                                                                          100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                            Plain
                                                            Mushroom
                                                            Sausage
                                                                                           800000                                                800000
                                                                                                                                                                                                                                  95
                                                                                                                                                                                                                                                                                             StDev
                                                                                                                                                                                                                                                                                             N
                                                                                                                                                                                                                                                                                                       156865
                                                                                                                                                                                                                                                                                                           51
                                                                                                                                                                                                                                                                                                                          800000                                      2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               51
                                                                                                                                                                                                                                                                                             AD         0.994                                                                                                                                                                                          80
                                                            Pepper and Onion                                                                     700000                                                                           90                                                                                      700000
                                                                                           700000                                                                                                                                                                                            P-Value    0.012
                                                            Mushroom and Onion                                                                                                                                                                                                                                                                                                                                    10
                                                            Garlic                                                                                                                                                                80                                                                                                                                                                                                                                                                                                                                                                          1000000
                                                                                                                                                 600000                                                                                                                                                                   600000                                                                                                                                                                       60
                                                                                                                                                                                                                                                                                                                                                                                                      Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Percent
                                                            Meatball                                                                                                                                                              70
                                                                                           600000
                                                                                                                                       Listing




                                                                                                                                                                                                                                                                                                                Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                                  8
                                                                                                                                                                                                                        Percent




                                                                                                                                                                                                                                  60                                                                                                                                                                                                                                                                                                                                                                           800000
                                                                                                                                                 500000                                                                                                                                                                   500000
                                                                                 Listing




                                                                                                                                                                                                                                  50                                                                                                                                                                                                                                                                   40
       Sausage                                                                             500000                                                                                                                                 40                                                                                                                                                                              6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Listing
         5.8%                                                                                                                                    400000                                                                           30                                                                                      400000                                                                                                                                                                                                                                                                               600000
                                                                                           400000                                                                                                                                 20                                                                                                                                                                                                                                                                   20
                                                                                                                                                 300000                                                                                                                                                                   300000                                                                                  4
                                                                                                                                                                                                                                  10                                                                                                                                                                                                                                                                                                                                                                           400000
                                                                                           300000                                                200000                                                                           5                                                                                       200000                                                                                  2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               200000
                                            32.5%
                                                                                           200000                                                                                                                                                                                                                                                                                                                                                                                                            0       00     00     00     00      00     00     00     00     00
                                                                                                                                                 100000                                                                                                                                                                   100000
                                                                                                                                                                                                                                  1                                                                                                                                                                               0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                                       15000   17500   20000   22500    25000   27500   30000   32500                  0    200000   400000    600000     800000   1000000                                      15000     17500      20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                                  IncomePC                                                             Listing                                                                                                  IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                           100000
      4/49
                                                                                                                                                                                                                                                                                                                                                                                             Part 15 – Regression Models


                                                                                                               Monet in Large and Small
                                             Sale prices of 328 signed Monet paintings

                                                                                                                                                            Fitted Line Plot
                                                                                                                                  ln (US$) = 2.825 + 1.725 ln (SurfaceArea)
                                                               18                                                                                                                                                                                                                                  S                                             1.00645
                                                                                                                                                                                                                                                                                                   R-Sq                                           20.0%
                                                               17                                                                                                                                                                                                                                  R-Sq(adj)                                      19.8%


                                                               16

                                                               15
                                                                                                                                                                                                                                                                                                                                                                                                  The residuals do not
                                                    ln (US$)




                                                               14
                                                                                                                                                                                                                                                                                                                                                                                                  show any obvious
                                                               13
                                                                                                                                                                                                                                                                                                                                                                                                  patterns that seem
                                                               12
                                                                                                                                                                                                                                                                                                                                                                                                  inconsistent with the
                                                               11                                                                                                                                                                                                                                                                                                                                 assumptions of the
                                                                           6.0                         6.2                 6.4        6.6    6.8   7.0                                                            7.2                        7.4                      7.6                                                                                                                         model.
                                                                                                                                      ln (SurfaceArea)



                                             Log of $price = a + b log surface area + e

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                              Boxplot of Listing                                   Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                                         Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                               Category                                                                            900000                                                                                                                                                                  900000
                             Meatball                                                         900000                                                                                                                                                                                                                                                                                                          14



                                                                                                                                                                                                                                                                                                                                           e  mc  
                                                               Pepperoni                                                                                                                                                            99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                          Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                               Plain
                                                               Mushroom
                                                               Sausage
                                                                                              800000                                               800000
                                                                                                                                                                                                                                    95
                                                                                                                                                                                                                                                                                              StDev
                                                                                                                                                                                                                                                                                              N
                                                                                                                                                                                                                                                                                                        156865
                                                                                                                                                                                                                                                                                                            51
                                                                                                                                                                                                                                                                                                                           800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           51
                                                                                                                                                                                                                                                                                              AD         0.994                                                                                                                                                                                     80
                                                               Pepper and Onion                                                                    700000                                                                           90                                                                                     700000
                                                                                              700000                                                                                                                                                                                          P-Value    0.012
                                                               Mushroom and Onion                                                                                                                                                                                                                                                                                                                             10
                                                               Garlic                                                                                                                                                               80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                                   600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                                  Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Percent
                                                               Meatball                                                                                                                                                             70
                                                                                              600000
                                                                                                                                         Listing




                                                                                                                                                                                                                                                                                                                 Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                              8
                                                                                                                                                                                                                          Percent




                                                                                                                                                                                                                                    60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                                   500000                                                                                                                                                                  500000
                                                                                    Listing




                                                                                                                                                                                                                                    50                                                                                                                                                                                                                                                             40
       Sausage                                                                                500000                                                                                                                                40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Listing
         5.8%                                                                                                                                      400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                              400000                                                                                                                                20                                                                                                                                                                                                                                                             20
                                                                                                                                                   300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                                    10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                              300000                                               200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           200000
                                            32.5%
                                                                                              200000                                                                                                                                                                                                                                                                                                                                                                                                     0       00     00     00     00      00     00     00     00     00
                                                                                                                                                   100000                                                                                                                                                                  100000
                                                                                                                                                                                                                                    1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                                         15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                                    IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                              100000
      8/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Monet Regression




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      9/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Using the Residuals

                                                           How do you know the model is “good?”
                                                           Various diagnostics to be developed over
                                                            the semester.
                                                           But, the first place to look is at the
                                                            residuals.



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      10/49
                                                                                                                                                                                                                                                                                                                                                                         Part 15 – Regression Models


                                                                                                  Residuals Can Signal a
                                                                                                  Flawed Model
                                                                                                Standard application: Cost function
                                                                                                 for output of a production process.
                                                                                                Compare linear equation to a
                                                                                                 quadratic model (in logs)
                                                                                                (124 American Electric Utilities)




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                 Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                     Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                              14



                                                                                                                                                                                                                                                                                                                       e  mc  
                                                       Pepperoni                                                                                                                                                99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                      Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                   800000
                                                                                                                                                                                                                95
                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                    156865
                                                                                                                                                                                                                                                                                        51
                                                                                                                                                                                                                                                                                                       800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                          AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                        700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                              P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                 10
                                                       Garlic                                                                                                                                                   80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                               600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                              Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Percent
                                                       Meatball                                                                                                                                                 70
                                                                                      600000
                                                                                                                     Listing




                                                                                                                                                                                                                                                                                             Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                          8
                                                                                                                                                                                                      Percent




                                                                                                                                                                                                                60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                               500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                    40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Listing
         5.8%                                                                                                                  400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                    20                                                                                                                                                                                                                                                             20
                                                                                                                               300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                   200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                         0       00     00     00     00      00     00     00     00     00
                                                                                                                               100000                                                                                                                                                                  100000
                                                                                                                                                                                                                1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                     15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Electricity Cost Function




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      11/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Candidate Model for Cost
                                                                                                                                                                                                Log c = a + b log q + e

                                                                                                                                                                                                                                     Scatterplot of logCost vs logOutput
                                                                                                                                                               7

                                                                                                                                                               6

                                                                                                                                                               5
                                                                                                                                                                                  Most of the points in
                                                                                                                                                               4
                                                                                                                                                                                  this area are above                                                                                                                                                                                                                                                                          Most of the points in
                                                                                                                                             logCost




                                                                                                                                                               3                  the regression line.                                                                                                                                                                                                                                                                         this area are above
                                                                                                                                                                                                                                                                                                                                                                                                                                                                               the regression line.
                                                                                                                                                               2

                                                                                                                                                               1
                                                                                                                                                                                                                                                                                                                            Most of the points in
                                                                                                                                                               0                                                                                                                                                            this area are below the
                                                                                                                                                                                                                                                                                                                            regression line.
                                                                                                                                                            -1
                                                                                                                                                                             0                                              2                                                               4                                     6                                                                       8                                                         10                                                            12
                                                                                                                                                                                                                                                                                                                              logOutput



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                     Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                   Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                          Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                      100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                      80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                     1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                   60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                      800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                              40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                           600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                              20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                      400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                      0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                         0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                 00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                        10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                               Listing                                                                                      IncomePC
                                                                                      100000
      12/49
                                                                                                                                                                                                                                                                                                                                                                                   Part 15 – Regression Models


                                                                                                 A Missing Variable?
                                                                                                      Residuals from the (log)linear cost model

                                                                                                                                                                                                        Residuals Versus logOutput
                                                                                                                                                                                                                                   (response is logCost)
                                                                                                                    2.0


                                                                                                                    1.5


                                                                                                                    1.0
                                                                                                     Residual




                                                                                                                    0.5


                                                                                                                    0.0


                                                                                                                    -0.5


                                                                                                                    -1.0
                                                                                                                           0                                                     2                                                 4                                   6                                                                           8                                                     10                                                     12
                                                                                                                                                                                                                                                                   logOutput




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                                       Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                               Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                          900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                        14



                                                                                                                                                                                                                                                                                                                                 e  mc  
                                                       Pepperoni                                                                                                                                                          99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                             800000
                                                                                                                                                                                                                          95
                                                                                                                                                                                                                                                                                    StDev
                                                                                                                                                                                                                                                                                    N
                                                                                                                                                                                                                                                                                              156865
                                                                                                                                                                                                                                                                                                  51
                                                                                                                                                                                                                                                                                                                 800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 51
                                                                                                                                                                                                                                                                                    AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                                  700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                        P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                           10
                                                       Garlic                                                                                                                                                             80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                         600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                        Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Percent
                                                       Meatball                                                                                                                                                           70
                                                                                      600000
                                                                                                                               Listing




                                                                                                                                                                                                                                                                                                       Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                    8
                                                                                                                                                                                                                Percent




                                                                                                                                                                                                                          60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                         500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                          50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                              40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Listing
         5.8%                                                                                                                            400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                              20                                                                                                                                                                                                                                                             20
                                                                                                                                         300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                          10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                             200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                                   0       00     00     00     00      00     00     00     00     00
                                                                                                                                         100000                                                                                                                                                                  100000
                                                                                                                                                                                                                          1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                               15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                          IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      13/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 A Better Model?




                                                                                                 Log Cost = α + β1 logOutput + β2 [logOutput]2 + ε



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      14/49
                                                                                                                                                                                                                                                                                                                                                                           Part 15 – Regression Models

                                                                                                 Candidate Models for Cost
                                                                                                                    The quadratic equation is the appropriate model.

                                                                                                                                                                                Logc = a + b1 logq + b2 log2q + e




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                               Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                       Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                  900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                14



                                                                                                                                                                                                                                                                                                                         e  mc  
                                                       Pepperoni                                                                                                                                                  99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                        Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                     800000
                                                                                                                                                                                                                  95
                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                                                         800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                            AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                          700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                   10
                                                       Garlic                                                                                                                                                     80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                 600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Percent
                                                       Meatball                                                                                                                                                   70
                                                                                      600000
                                                                                                                       Listing




                                                                                                                                                                                                                                                                                               Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                            8
                                                                                                                                                                                                        Percent




                                                                                                                                                                                                                  60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                 500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                  50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                      40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Listing
         5.8%                                                                                                                    400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                      20                                                                                                                                                                                                                                                             20
                                                                                                                                 300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                  10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                     200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                           0       00     00     00     00      00     00     00     00     00
                                                                                                                                 100000                                                                                                                                                                  100000
                                                                                                                                                                                                                  1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                       15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                  IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      15/49
                                                                                                                                                                                                                                                                                                                                                                                                  Part 15 – Regression Models


                                                                                                                     Missing Variable Included
                                                                                                                                                                                                                                             Residuals from the quadratic cost model

                                                                                                          Residuals Versus logOutput
                                                                                                                                                                                                                                                                                                                                                      Residuals Versus logOutput
                                                                                                               (response is logCost)                                                                                                                                                                                                                                           (response is logCost)
                                                   2.0
                                                                                                                                                                                                                                                            0.50
                                                   1.5


                                                   1.0

                                                                                                                                                                                                                                                            0.25
                                        Residual




                                                   0.5


                                                   0.0



                                                                                                                                                                                                                                             Residual
                                                   -0.5
                                                                                                                                                                                                                                                            0.00

                                                   -1.0
                                                               0                        2                      4                 6       8                              10                       12
                                                                                                                             logOutput                                                                                                                     -0.25


                                         Residuals from the                                                                                                                                                                                                -0.50
                                         linear cost model
                                                                                                                                                                                                                                                                                  0                                                      2                                       4                                6                                                             8                                             10                                                    12
                                                                                                                                                                                                                                                                                                                                                                                                              logOutput




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                                    Boxplot of Listing                                 Scatterplot of Listing vs IncomePC                                                Probability Plot of Listing                                                            Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                                              Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                                   Category                                                                            900000                                                                                                                                                                   900000
                             Meatball                                                                 900000                                                                                                                                                                                                                                                                                                       14



                                                                                                                                                                                                                                                                                                                                                e  mc  
                                                                   Pepperoni                                                                                                                                                            99                                                                                                                                                                                                                                                                                                                                          Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                               Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                                   Pepperoni
                                                   21.8%
                                                                   Plain
                                                                   Mushroom
                                                                   Sausage
                                                                                                      800000                                           800000
                                                                                                                                                                                                                                        95
                                                                                                                                                                                                                                                                                                   StDev
                                                                                                                                                                                                                                                                                                   N
                                                                                                                                                                                                                                                                                                             156865
                                                                                                                                                                                                                                                                                                                 51
                                                                                                                                                                                                                                                                                                                                800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                51
                                                                                                                                                                                                                                                                                                   AD         0.994                                                                                                                                                                                     80
                                                                   Pepper and Onion                                                                    700000                                                                           90                                                                                      700000
                                                                                                      700000                                                                                                                                                                                       P-Value    0.012
                                                                   Mushroom and Onion                                                                                                                                                                                                                                                                                                                              10
                                                                   Garlic                                                                                                                                                               80                                                                                                                                                                                                                                                                                                                                                                     1000000
                                                                                                                                                       600000                                                                                                                                                                   600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                                       Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Percent
                                                                   Meatball                                                                                                                                                             70
                                                                                                      600000
                                                                                                                                             Listing




                                                                                                                                                                                                                                                                                                                      Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                                   8
                                                                                                                                                                                                                              Percent




                                                                                                                                                                                                                                        60                                                                                                                                                                                                                                                                                                                                                                      800000
                                                                                                                                                       500000                                                                                                                                                                   500000
                                                                                            Listing




                                                                                                                                                                                                                                        50                                                                                                                                                                                                                                                              40
       Sausage                                                                                        500000                                                                                                                            40                                                                                                                                                                         6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Listing
         5.8%                                                                                                                                          400000                                                                           30                                                                                      400000                                                                                                                                                                                                                                                                          600000
                                                                                                      400000                                                                                                                            20                                                                                                                                                                                                                                                              20
                                                                                                                                                       300000                                                                                                                                                                   300000                                                                             4
                                                                                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                      400000
                                                                                                      300000                                           200000                                                                           5                                                                                       200000                                                                             2                                                                                     0
             Mushroom                               Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                200000
                                                    32.5%
                                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                                  0       00     00     00     00      00     00     00     00     00
                                                                                                                                                       100000                                                                                                                                                                   100000
                                                                                                                                                                                                                                        1                                                                                                                                                                          0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                                             15000   17500   20000   22500    25000   27500   30000   32500                  0    200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                                        IncomePC                                                             Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                                      100000
      16/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Heteroscedasticity

                                                                                                  Hetero - differences
                                                                                                  Scedastic - function, variation around
                                                                                                               the mean
                                                                                                  Arises when y is “proportional” to x

                                                                                                  Arises sometimes when there are
                                                                                                   natural, heterogeneous groups


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      17/49
                                                                                                                                                                                                                                                                                                                                                                            Part 15 – Regression Models


                                                                                                     Heteroscedasticity
                                                                                                                             Scatterplot of Residuals vs Predictions                                                                                                                                                                                                                                                                                           Residuals from
                                                                                                30000
                                                                                                                                                                                                                                                                                                                                                                                                                                                               a regression of
                                                                                                                                                                                                                                                                                                                                                                                                                                                               salaries on
                                                                                                20000
                                                                                                                                                                                                                                                                                                                                                                                                                                                               years of
                                                                                                10000
                                                                                                                                                                                                                                                                                                                                                                                                                                                               experience.
                                                                            Residuals




                                                                                                          0



                                                                                                -10000



                                                                                                -20000
                                                                                                      10000                         20000                                        30000                                       40000      50000                                                                                           60000                                    70000
                                                                                                                                                                                                                            Predictions



                                                                        Standard deviation of the residuals seems not to be constant.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                    Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                        Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                   900000                                                                                                                                                                  900000
                             Meatball                                                  900000                                                                                                                                                                                                                                                                                                14



                                                                                                                                                                                                                                                                                                                          e  mc  
                                                       Pepperoni                                                                                                                                                   99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                         Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                       800000                                     800000
                                                                                                                                                                                                                   95
                                                                                                                                                                                                                                                                             StDev
                                                                                                                                                                                                                                                                             N
                                                                                                                                                                                                                                                                                       156865
                                                                                                                                                                                                                                                                                           51
                                                                                                                                                                                                                                                                                                          800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                             AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                           700000                                                                           90                                                                                     700000
                                                                                       700000                                                                                                                                                                                P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                    10
                                                       Garlic                                                                                                                                                      80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                  600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                 Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Percent
                                                       Meatball                                                                                                                                                    70
                                                                                       600000
                                                                                                                        Listing




                                                                                                                                                                                                                                                                                                Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                             8
                                                                                                                                                                                                         Percent




                                                                                                                                                                                                                   60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                  500000                                                                                                                                                                  500000
                                                                             Listing




                                                                                                                                                                                                                   50                                                                                                                                                                                                                                                             40
       Sausage                                                                         500000                                                                                                                      40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Listing
         5.8%                                                                                                                     400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                       400000                                                                                                                      20                                                                                                                                                                                                                                                             20
                                                                                                                                  300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                   10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                       300000                                     200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          200000
                                            32.5%
                                                                                       200000                                                                                                                                                                                                                                                                                                                                                                                           0       00     00     00     00      00     00     00     00     00
                                                                                                                                  100000                                                                                                                                                                  100000
                                                                                                                                                                                                                   1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                        15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                   IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                       100000
      18/49
                                                                                                                                                                                                                                                                                                                                                                           Part 15 – Regression Models


                                                                                                 Problem with the Model?
                                                                                                                                                                                                                                              Residuals Versus Output
                                                                                                                                                                                                                                                                     (response is Cost)
                                                                                                                                                               150


                                                                                                                                                               100


                                                                                                                                                                 50
                                                                                                                                            Residual




                                                                                                                                                                     0


                                                                                                                                                               -50


                                                                                                                                                           -100

                                                                                                                                                                                        0                         10000                    20000                             30000                             40000                              50000                         60000                                70000                            80000
                                                                                                                                                                                                                                                                                                              Output




                                                                                                                                            This usually suggests the model should be
                                                                                                                                            defined in terms of logs of the variable.


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                             Scatterplot of Listing vs IncomePC                                                Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                       Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                     900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                14



                                                                                                                                                                                                                                                                                                                         e  mc  
                                                       Pepperoni                                                                                                                                                  99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                        Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                                  95
                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                                                         800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                            AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                              90                                                                                     700000
                                                                                      700000                                                                                                                                                                                P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                   10
                                                       Garlic                                                                                                                                                     80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                     600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Percent
                                                       Meatball                                                                                                                                                   70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                               Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                            8
                                                                                                                                                                                                        Percent




                                                                                                                                                                                                                  60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                     500000
                                                                            Listing




                                                                                                                                                                                                                  50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                      40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Listing
         5.8%                                                                                                                 400000                                                                              30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                      20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                     300000                                                                             4
                                                                                                                                                                                                                  10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                              5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                           0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                     100000
                                                                                                                                                                                                                  1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500      20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                  IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      19/49
                                                                                                                                                                                                                                                                                                                                                                           Part 15 – Regression Models

                                                                                                 Sometimes Heteroscedasticity Can
                                                                                                 Be Cured By Taking Logs
                                                                                                                                                                                                                                                                       Scatterplot of Residual vs LogSalary
                                                                                                                                                                                              0.3


                                                                                                                                                                                              0.2

                                    Residuals from a                                                                                                                                          0.1
                                    regression of logs
                                                                                                                                                                                              0.0
                                                                                                                                                                             Residual


                                    of salaries on years
                                    of experience.                                                                                                                                            -0.1

                                    Salary = αeβteε                                                                                                                                           -0.2

                                    We will explore this                                                                                                                                      -0.3
                                    model below.
                                                                                                                                                                                              -0.4
                                                                                                                                                                                                                     9.5                                                                       10.0                                                                  10.5                                                                                       11.0                                                                                         11.5
                                                                                                                                                                                                                                                                                                                                                                   LogSalary




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                                 Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                       Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                     900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                14



                                                                                                                                                                                                                                                                                                                         e  mc  
                                                       Pepperoni                                                                                                                                                99                                                                                                                                                                                                                                                                                                                                            Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                        Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                                95
                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                                                         800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                            AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                            90                                                                                       700000
                                                                                      700000                                                                                                                                                                                P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                   10
                                                       Garlic                                                                                                                                                   80                                                                                                                                                                                                                                                                                                                                                                       1000000
                                                                                                                              600000                                                                                                                                                                     600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Percent
                                                       Meatball                                                                                                                                                 70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                               Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                            8
                                                                                                                                                                                                      Percent




                                                                                                                                                                                                                60                                                                                                                                                                                                                                                                                                                                                                        800000
                                                                                                                              500000                                                                                                                                                                     500000
                                                                            Listing




                                                                                                                                                                                                                50                                                                                                                                                                                                                                                               40
       Sausage                                                                        500000                                                                                                                    40                                                                                                                                                                          6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Listing
         5.8%                                                                                                                 400000                                                                            30                                                                                       400000                                                                                                                                                                                                                                                                           600000
                                                                                      400000                                                                                                                    20                                                                                                                                                                                                                                                               20
                                                                                                                              300000                                                                                                                                                                     300000                                                                             4
                                                                                                                                                                                                                10                                                                                                                                                                                                                                                                                                                                                                        400000
                                                                                      300000                                  200000                                                                            5                                                                                        200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                           0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                     100000
                                                                                                                                                                                                                1                                                                                                                                                                           0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000    27500   30000   32500                   0    200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                         15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                               Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                       IncomePC
                                                                                      100000
      21/49
                                                                                                                                                                                                                                                                                                                                                                            Part 15 – Regression Models

                                                                                                     Should I Worry
                                                                                                     About Heteroscedasticity?
                                                                                                Not a problem for using least squares
                                                                                                 to estimate α or β.
                                                                                                But, there is a better method than
                                                                                                 least squares.
                                                                                                Assessment of the uncertainty of the
                                                                                                 least squares estimates may be too
                                                                                                 optimistic.
                                                                                                (Not contagious)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                    Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                        Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                   900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                 14



                                                                                                                                                                                                                                                                                                                          e  mc  
                                                       Pepperoni                                                                                                                                                   99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                         Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                      800000
                                                                                                                                                                                                                   95
                                                                                                                                                                                                                                                                             StDev
                                                                                                                                                                                                                                                                             N
                                                                                                                                                                                                                                                                                       156865
                                                                                                                                                                                                                                                                                           51
                                                                                                                                                                                                                                                                                                          800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                             AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                           700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                 P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                    10
                                                       Garlic                                                                                                                                                      80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                  600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                 Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Percent
                                                       Meatball                                                                                                                                                    70
                                                                                      600000
                                                                                                                        Listing




                                                                                                                                                                                                                                                                                                Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                             8
                                                                                                                                                                                                         Percent




                                                                                                                                                                                                                   60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                  500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                   50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                       40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Listing
         5.8%                                                                                                                     400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                       20                                                                                                                                                                                                                                                             20
                                                                                                                                  300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                   10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                      200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                            0       00     00     00     00      00     00     00     00     00
                                                                                                                                  100000                                                                                                                                                                  100000
                                                                                                                                                                                                                   1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                        15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                   IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      24/49
                                                                                                                                                                                                                                                                                                                                                                                         Part 15 – Regression Models


                                                                                                       Unusual Data Points
                                                                                        Outliers have (what appear to be) very large disturbances, ε

                                                                                                                          Scatterplot of Weight vs TLength                                                                                                                                                                                                                      Regression of Foreign Box Office on Domestic
                                                                                                                                                                                                                                                                                                                                                                                                                   Overseas = 6.693 + 1.051 Domestic
                                                                     160
                                                                                                                                                                                                                                                                                                                                              1400                                                                                                                                                                                                                         S                                    73.0041
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           R-Sq                                  52.2%
                                                                                                                                                                                                                                                                                                                                              1200                                                                                                                                                                                                                         R-Sq(adj)                             52.1%
                                                                     140
                                                                                                                                                                                                                                                                                                                                              1000

                                                                     120
                                                       Weight




                                                                                                                                                                                                                                                                                                                                                 800




                                                                                                                                                                                                                                                                                                                                  Overseas
                                                                                                                                                                                                                                                                                                                                                 600
                                                                     100

                                                                                                                                                                                                                                                                                                                                                 400

                                                                         80
                                                                                                                                                                                                                                                                                                                                                 200

                                                                                                                                                                                                                                                                                                                                                       0
                                                                         60
                                                                                                10         12               14          16                        18     20                                  22                       24                   26                      28                                                                                 0                  100                             200                  300                                400                          500                              600
                                                                                                                                                                  TLength                                                                                                                                                                                                                                                                   Domestic




                                                                     Wolf weight vs. tail length                                                                                                                                                                                                                                    The 500 most successful movies

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                      Boxplot of Listing                                    Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                                Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                                  Normal - 95% CI                                                                                                                                                                                                                                                                     Normal
                                                        Category                                                                            900000                                                                                                                                                                  900000
                             Meatball                                                  900000                                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                                      e  mc  
                                                        Pepperoni                                                                                                                                                            99                                                                                                                                                                                                                                                                                                                                            Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                   Mean      369687                                                                                                                                                                                        100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                        Plain
                                                        Mushroom
                                                        Sausage
                                                                                       800000                                               800000
                                                                                                                                                                                                                             95
                                                                                                                                                                                                                                                                                       StDev
                                                                                                                                                                                                                                                                                       N
                                                                                                                                                                                                                                                                                                 156865
                                                                                                                                                                                                                                                                                                     51
                                                                                                                                                                                                                                                                                                                    800000                                    2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                       AD         0.994                                                                                                                                                                                        80
                                                        Pepper and Onion                                                                    700000                                                                           90                                                                                     700000
                                                                                       700000                                                                                                                                                                                          P-Value    0.012
                                                        Mushroom and Onion                                                                                                                                                                                                                                                                                                                                10
                                                        Garlic                                                                                                                                                               80                                                                                                                                                                                                                                                                                                                                                                       1000000
                                                                                                                                            600000                                                                                                                                                                  600000                                                                                                                                                                     60
                                                                                                                                                                                                                                                                                                                                                                                              Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Percent
                                                        Meatball                                                                                                                                                             70
                                                                                       600000
                                                                                                                                  Listing




                                                                                                                                                                                                                                                                                                          Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                          8
                                                                                                                                                                                                                   Percent




                                                                                                                                                                                                                             60                                                                                                                                                                                                                                                                                                                                                                        800000
                                                                                                                                            500000                                                                                                                                                                  500000
                                                                             Listing




                                                                                                                                                                                                                             50                                                                                                                                                                                                                                                                40
       Sausage                                                                         500000                                                                                                                                40                                                                                                                                                                           6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            Listing
         5.8%                                                                                                                               400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                             600000
                                                                                       400000                                                                                                                                20                                                                                                                                                                                                                                                                20
                                                                                                                                            300000                                                                                                                                                                  300000                                                                                4
                                                                                                                                                                                                                             10                                                                                                                                                                                                                                                                                                                                                                        400000
                                                                                       300000                                               200000                                                                           5                                                                                      200000                                                                                2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       200000
                                            32.5%
                                                                                       200000                                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                                            100000                                                                                                                                                                  100000
                                                                                                                                                                                                                             1                                                                                                                                                                            0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                                  15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000    17500     20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                             IncomePC                                                            Listing                                                                                                IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                       100000
      25/49
                                                                                                                                                                                                                                                                                                                                                                                  Part 15 – Regression Models


                                                                                                 Outliers (?)
                                                                                                      Remember the empirical rule, 99.5% of observations will lie within
                                                                                                      mean ± 3 standard deviations? We show (a+bx) ± 3se below.)
                                                                                                                                                                     Regression of Foreign Box Office on Domestic
                                                                                                                                                                                                              Overseas = 6.693 + 1.051 Domestic
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Titanic is 8.1
                                                                                                                          1400                                                                                                                                                                                                                                                                                       S                                               73.0041
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 standard
                                                                                                                                                                                                                                                                                                                                                                                                                     R-Sq                                             52.2%                                                      deviations
                                                                                                                          1200                                                                                                                                                                                                                                                                                       R-Sq(adj)                                        52.1%

              These                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              from the
              observations                                                                                                1000                                                                                                                                                                                                                                                                                                                                                                                                   regression!
              might                                                                                                       800                                                                                                                                                                                                                                                                                                                                                                                                    Only 0.86% of
                                                                                                               Overseas




              deserve a                                                                                                   600
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 the 466
              close look.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        observations
                                                                                                                          400
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 lie outside
                                                                                                                          200                                                                                                                                                                                                                                                                                                                                                                                                    the bounds.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 (We will
                                                                                                                            0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 refine this
                                                                                                                                                     0                           100                                    200                 300                                                      400                                      500                              600
                                                                                                                                                                                                                                          Domestic                                                                                                                                                                                                                                                                               later.)


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                                     Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                              Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                             Normal - 95% CI                                                                                                                                                                                                                                                                   Normal
                                                       Category                                                                        900000                                                                                                                                                                   900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                       14



                                                                                                                                                                                                                                                                                                                                e  mc  
                                                       Pepperoni                                                                                                                                                        99                                                                                                                                                                                                                                                                                                                                          Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                              Mean      369687                                                                                                                                                                                      100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                           800000
                                                                                                                                                                                                                        95
                                                                                                                                                                                                                                                                                  StDev
                                                                                                                                                                                                                                                                                  N
                                                                                                                                                                                                                                                                                            156865
                                                                                                                                                                                                                                                                                                51
                                                                                                                                                                                                                                                                                                                800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                51
                                                                                                                                                                                                                                                                                  AD         0.994                                                                                                                                                                                      80
                                                       Pepper and Onion                                                                700000                                                                           90                                                                                      700000
                                                                                      700000                                                                                                                                                                                      P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                          10
                                                       Garlic                                                                                                                                                           80                                                                                                                                                                                                                                                                                                                                                                     1000000
                                                                                                                                       600000                                                                                                                                                                   600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                       Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Percent
                                                       Meatball                                                                                                                                                         70
                                                                                      600000
                                                                                                                             Listing




                                                                                                                                                                                                                                                                                                      Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                   8
                                                                                                                                                                                                              Percent




                                                                                                                                                                                                                        60                                                                                                                                                                                                                                                                                                                                                                      800000
                                                                                                                                       500000                                                                                                                                                                   500000
                                                                            Listing




                                                                                                                                                                                                                        50                                                                                                                                                                                                                                                              40
       Sausage                                                                        500000                                                                                                                            40                                                                                                                                                                         6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Listing
         5.8%                                                                                                                          400000                                                                           30                                                                                      400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                            20                                                                                                                                                                                                                                                              20
                                                                                                                                       300000                                                                                                                                                                   300000                                                                             4
                                                                                                                                                                                                                        10                                                                                                                                                                                                                                                                                                                                                                      400000
                                                                                      300000                                           200000                                                                           5                                                                                       200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                                  0       00     00     00     00      00     00     00     00     00
                                                                                                                                       100000                                                                                                                                                                   100000
                                                                                                                                                                                                                        1                                                                                                                                                                          0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                             15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                       15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                        IncomePC                                                            Listing                                                                                              IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
                                                                                                                                                                                                                                                                                                                                                                              Part 15 – Regression Models


                                                                                                        Prices paid at auction for Monet paintings vs. surface area (in logs)




                                                                                               logPrice = a + b logArea + e
                                                                                                                      Not an outlier: Monet chose to paint a small painting.
                                                                                                                      Possibly an outlier: Why was the price so low?

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                  Boxplot of Listing                                Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                          Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                     900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                   14



                                                                                                                                                                                                                                                                                                                            e  mc  
                                                       Pepperoni                                                                                                                                                     99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                           Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                        800000
                                                                                                                                                                                                                     95
                                                                                                                                                                                                                                                                               StDev
                                                                                                                                                                                                                                                                               N
                                                                                                                                                                                                                                                                                         156865
                                                                                                                                                                                                                                                                                             51
                                                                                                                                                                                                                                                                                                            800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            51
                                                                                                                                                                                                                                                                               AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                             700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                   P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                      10
                                                       Garlic                                                                                                                                                        80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                    600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                   Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Percent
                                                       Meatball                                                                                                                                                      70
                                                                                      600000
                                                                                                                          Listing




                                                                                                                                                                                                                                                                                                  Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                               8
                                                                                                                                                                                                           Percent




                                                                                                                                                                                                                     60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                    500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                     50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                         40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Listing
         5.8%                                                                                                                       400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                         20                                                                                                                                                                                                                                                             20
                                                                                                                                    300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                     10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                        200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                              0       00     00     00     00      00     00     00     00     00
                                                                                                                                    100000                                                                                                                                                                  100000
                                                                                                                                                                                                                     1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                          15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                     IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      26/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 What to Do About Outliers

                                                                        (1) Examine the data
                                                                        (2) Are they due to mismeasurement error or obvious
                                                                            “coding errors?” Delete the observations.
                                                                        (3) Are they just unusual observations? Do nothing.
                                                                        (4) Generally, resist the temptation to remove outliers.
                                                                             Especially if the sample is large. (500 movies is
                                                                             large. 10 wolves is not.)
                                                                        (5) Question why you think it is an outlier. Is it really?


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      29/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Regression Options




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      32/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models

                                                                                               Minitab’s Opinions

                                                                                                                                                                                                                                                                                                                                                                                                                Minitab uses ± 2S to flag
                                                                                                                                                                                                                                                                                                                                                                                                                “large” residuals.


                                                                                                                                                                                                                                                                                                                                                                                                                Influential observations
                                                                                                                                                                                                                                                                                                                                                                                                                have very large values
                                                                                                                                                                                                                                                                                                                                                                                                                 of | xi - x | .




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      33/49
                                                                                                                                                                                                                                                                                                                                                                          Part 15 – Regression Models


                                                                                                 On Removing Outliers

                                                                                                 Be careful about singling out particular
                                                                                                  observations this way.
                                                                                                                    The resulting model might be a product of your
                                                                                                                    opinions

                                                                                                                    Removing outliers might create new outliers
                                                                                                                    that were not outliers before.

                                                                                                                    Statistical inferences from the model will be
                                                                                                                    incorrect.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                              Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                      Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                 900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                               14



                                                                                                                                                                                                                                                                                                                        e  mc  
                                                       Pepperoni                                                                                                                                                 99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                       Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                    800000
                                                                                                                                                                                                                 95
                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                                                        800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        51
                                                                                                                                                                                                                                                                           AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                         700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                               P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                  10
                                                       Garlic                                                                                                                                                    80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                               Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Percent
                                                       Meatball                                                                                                                                                  70
                                                                                      600000
                                                                                                                      Listing




                                                                                                                                                                                                                                                                                              Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                           8
                                                                                                                                                                                                       Percent




                                                                                                                                                                                                                 60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                 50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                     40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Listing
         5.8%                                                                                                                   400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                     20                                                                                                                                                                                                                                                             20
                                                                                                                                300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                 10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                    200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                          0       00     00     00     00      00     00     00     00     00
                                                                                                                                100000                                                                                                                                                                  100000
                                                                                                                                                                                                                 1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                      15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                 IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models

                                                                                                 Mechanically Remove
                                                                                                 Outliers?




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
                                                                                                                                                                                                                                                                                                                                                                                     Part 15 – Regression Models



                                                                                                                    Removing Outliers Creates Outliers

                                                                                                                                                                                                  N umber of Observations A fter R emoving Outliers
                                                                                                                                              450



                                                                                                                                              425



                                                                                                                                              400
                                                                                                                    N U MO B S




                                                                                                                                              375



                                                                                                                                              350



                                                                                                                                              325



                                                                                                                                              300
                                                                                                                                                                  0                                                                                              10                                                                                                         20                                                                                                             30

                                                                                                                                                                                                                                                                                                                        IT ER




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                                         Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                                 Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                            900000                                                                                                                                                                  900000
                                                                                      900000                                                                                                                                                                                                                                                                                                          14




                                                                                                                                                                                          Were they really outliers?
                             Meatball




                                                                                                                                                                                                           e  mc  
                                                       Pepperoni                                                                                                                                                            99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                  Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                               800000
                                                                                                                                                                                                                            95
                                                                                                                                                                                                                                                                                      StDev
                                                                                                                                                                                                                                                                                      N
                                                                                                                                                                                                                                                                                                156865
                                                                                                                                                                                                                                                                                                    51
                                                                                                                                                                                                                                                                                                                   800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   51
                                                                                                                                                                                                                                                                                      AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                                    700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                          P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                             10
                                                       Garlic                                                                                                                                                               80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                           600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                          Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 Percent
                                                       Meatball                                                                                                                                                             70
                                                                                      600000
                                                                                                                                 Listing




                                                                                                                                                                                                                                                                                                         Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                      8
                                                                                                                                                                                                                  Percent




                                                                                                                                                                                                                            60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                           500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                            50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                                40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Listing
         5.8%                                                                                                                              400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                                20                                                                                                                                                                                                                                                             20
                                                                                                                                           300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                            10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                               200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                                     0       00     00     00     00      00     00     00     00     00
                                                                                                                                           100000                                                                                                                                                                  100000
                                                                                                                                                                                                                            1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                                 15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                            IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      34/49
                                                                                                                                                                                                                                                                                                                                                                                  Part 15 – Regression Models


                                                                                                 Normal Distribution of ei?
                                                                                                                                                                                                                                           Probability Plot of e_movies
                                                                                                                                                                                                                                                                                             Normal - 95% CI
                                                                                                                              99.9
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Mean                                       -4.89299E-14
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              StDev                                             72.93
                                                                                                                                        99
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              N                                                   466
                                                                                                                                                                                                                                                                                                                                                                                                                                                                              AD                                                4.307
                                                                                                                                        95                                                                                                                                                                                                                                                                                                                                    P-Value                                          <0.005
                                                                                                                                        90
                                                                                                                                        80
                                                                                                                                        70
                                                                                                                    Percent




                                                                                                                                        60
                                                                                                                                        50
                                                                                                                                        40
                                                                                                                                        30
                                                                                                                                        20
                                                                                                                                        10
                                                                                                                                           5

                                                                                                                                           1


                                                                                                                                 0.1
                                                                                                                                                 -400                                           -200                                                       0               200                                                                                         400                                                         600
                                                                                                                                                                                                                                                                     e_movies


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                                      Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                      Empirical CDF of Listing
                                                                                                                                                                                                                                              Normal - 95% CI                                                                                                                                                                                                                                                                    Normal
                                                       Category                                                                         900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                       14



                                                                                                                                                                                                                                                                                                                                e  mc  
                                                       Pepperoni                                                                                                                                                         99                                                                                                                                                                                                                                                                                                                                           Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                               Mean      369687                                                                                                                                                                                       100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                            800000
                                                                                                                                                                                                                         95
                                                                                                                                                                                                                                                                                   StDev
                                                                                                                                                                                                                                                                                   N
                                                                                                                                                                                                                                                                                             156865
                                                                                                                                                                                                                                                                                                 51
                                                                                                                                                                                                                                                                                                                800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  51
                                                                                                                                                                                                                                                                                   AD         0.994                                                                                                                                                                                       80
                                                       Pepper and Onion                                                                 700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                       P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                          10
                                                       Garlic                                                                                                                                                            80                                                                                                                                                                                                                                                                                                                                                                      1000000
                                                                                                                                        600000                                                                                                                                                                  600000                                                                                                                                                                    60
                                                                                                                                                                                                                                                                                                                                                                                       Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Percent
                                                       Meatball                                                                                                                                                          70
                                                                                      600000
                                                                                                                              Listing




                                                                                                                                                                                                                                                                                                      Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                   8
                                                                                                                                                                                                               Percent




                                                                                                                                                                                                                         60                                                                                                                                                                                                                                                                                                                                                                       800000
                                                                                                                                        500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                         50                                                                                                                                                                                                                                                               40
       Sausage                                                                        500000                                                                                                                             40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Listing
         5.8%                                                                                                                           400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                            600000
                                                                                      400000                                                                                                                             20                                                                                                                                                                                                                                                               20
                                                                                                                                        300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                         10                                                                                                                                                                                                                                                                                                                                                                       400000
                                                                                      300000                                            200000                                                                           5                                                                                      200000                                                                             2                                                                                       0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                                    0       00     00     00     00      00     00     00     00     00
                                                                                                                                        100000                                                                                                                                                                  100000
                                                                                                                                                                                                                         1                                                                                                                                                                         0                                                                                                  00     00     00     00      00     00     00     00     00
                                                                                                                                              15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                         10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                         IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                                Listing                                                                                      IncomePC
                                                                                      100000
      35/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Probability Plot
                                             Graph -> Probability Plots …




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      36/49
                                                                                                                                                                                                                                                                                                                                                                          Part 15 – Regression Models


                                                                                                 Using and Interpreting the Model


                                                                                                                   Interpreting the linear model

                                                                                                                   Semilog and growth models

                                                                                                                   Log-log model and elasticities


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                              Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                      Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                 900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                               14



                                                                                                                                                                                                                                                                                                                        e  mc  
                                                       Pepperoni                                                                                                                                                 99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                       Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                    800000
                                                                                                                                                                                                                 95
                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                                                        800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        51
                                                                                                                                                                                                                                                                           AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                         700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                               P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                  10
                                                       Garlic                                                                                                                                                    80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                               Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Percent
                                                       Meatball                                                                                                                                                  70
                                                                                      600000
                                                                                                                      Listing




                                                                                                                                                                                                                                                                                              Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                           8
                                                                                                                                                                                                       Percent




                                                                                                                                                                                                                 60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                 50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                     40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Listing
         5.8%                                                                                                                   400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                     20                                                                                                                                                                                                                                                             20
                                                                                                                                300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                 10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                    200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                          0       00     00     00     00      00     00     00     00     00
                                                                                                                                100000                                                                                                                                                                  100000
                                                                                                                                                                                                                 1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                      15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                 IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      37/49
                                                                                                                                                                                                                                                                                                                                                                                  Part 15 – Regression Models


                                                                                                     Statistical Cost Analysis
                                                                                                                         Fitted Line Plot
                                                                                                                    Cost = 2.444 + 0.005291 Output
                                                                                                                                                                                                                                                                                                                                      The units of the LHS and RHS must
                                                    500                                                                                                                                                                                    S                                 20.5111                                                  be the same.
                                                                                                                                                                                                                                           R-Sq                               92.4%
                                                                                                                                                                                                                                           R-Sq(adj)                          92.3%
                                                    400                                                                                                                                                                                                                                                                               $M cost = a + b MKWH
                                                    300
                                                                                                                                                                                                                                                                                                                                      Y = $ cost
                                        Cost




                                                    200
                                                                                                                                                                                                                                                                                                                                      a = $ cost                                                                                        = 2.444 $M
                                                    100
                                                                                                                                                                                                                                                                                                                                      b = $M /MKWH = 0.005291 $M/MKWH
                                                       0

                                                                      0                    10000 20000 30000 40000 50000 60000 70000 80000
                                                                                                                                                                                                                                                                                                                                      So,…..
                                                                                                            Output
                                                                                                                                                                                                                                                                                                                                      a = fixed cost = total cost if MKWH = 0

                                        Generation cost ($M) and output                                                                                                                                                                                                                                                               b = marginal cost = dCost/dMKWH
                                        (Millions of KWH) for 124 American                                                                                                                                                                                                                                                            b * MKWH = variable cost
                                        electric utilities. (1970).

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                    Boxplot of Listing                                  Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                              Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                           Category                                                                     900000                                                                                                                                                                  900000
                             Meatball                                                     900000                                                                                                                                                                                                                                                                                                   14



                                                                                                                                                                                                                                                                                                                                e  mc  
                                                           Pepperoni                                                                                                                                                     99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                               Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                           Plain
                                                           Mushroom
                                                           Sausage
                                                                                          800000                                        800000
                                                                                                                                                                                                                         95
                                                                                                                                                                                                                                                                                   StDev
                                                                                                                                                                                                                                                                                   N
                                                                                                                                                                                                                                                                                             156865
                                                                                                                                                                                                                                                                                                 51
                                                                                                                                                                                                                                                                                                                800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                51
                                                                                                                                                                                                                                                                                   AD         0.994                                                                                                                                                                                     80
                                                           Pepper and Onion                                                             700000                                                                           90                                                                                     700000
                                                                                          700000                                                                                                                                                                                   P-Value    0.012
                                                           Mushroom and Onion                                                                                                                                                                                                                                                                                                                      10
                                                           Garlic                                                                                                                                                        80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                        600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                       Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Percent
                                                           Meatball                                                                                                                                                      70
                                                                                          600000
                                                                                                                              Listing




                                                                                                                                                                                                                                                                                                      Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                   8
                                                                                                                                                                                                               Percent




                                                                                                                                                                                                                         60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                        500000                                                                                                                                                                  500000
                                                                                Listing




                                                                                                                                                                                                                         50                                                                                                                                                                                                                                                             40
       Sausage                                                                            500000                                                                                                                         40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Listing
         5.8%                                                                                                                           400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                          400000                                                                                                                         20                                                                                                                                                                                                                                                             20
                                                                                                                                        300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                         10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                          300000                                        200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                200000
                                            32.5%
                                                                                          200000                                                                                                                                                                                                                                                                                                                                                                                              0       00     00     00     00      00     00     00     00     00
                                                                                                                                        100000                                                                                                                                                                  100000
                                                                                                                                                                                                                         1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                              15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                         IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                          100000
      38/49
                                                                                                                                                                                                                                                                                                                                                                                          Part 15 – Regression Models


                                                                                                             Semilog Models and Growth Rates

                                                                                                                                                  Fitted Line Plot
                                                                                                                                LogSalary = 9.841 + 0.04998 YEARS
                                                                                                                                                                                                                                                                                                                                                              Salary = e9.84+0.05Years
                                                                                                                                                                                                                                                                         S                                              0.154111
                                                                      11.5
                                                                                                                                                                                                                                                                         R-Sq
                                                                                                                                                                                                                                                                         R-Sq(adj)
                                                                                                                                                                                                                                                                                                                          86.4%
                                                                                                                                                                                                                                                                                                                          86.1%
                                                                                                                                                                                                                                                                                                                                                              Years = 0 at Starting Salary
                                                                      11.0
                                                                                                                                                                                                                                                                                                                                                             Salary = e9.84 = 18,770
                                                       LogSalary




                                                                      10.5                                                                                                                                                                                                                                                                                   Marginal change. From year
                                                                      10.0
                                                                                                                                                                                                                                                                                                                                                             t to year t+1, log Salary goes
                                                                                                                                                                                                                                                                                                                                                             up by 0.05. Salary changes
                                                                           9.5
                                                                                                  0            5                 10                  15                                  20                            25                            30
                                                                                                                                                   YEARS                                                                                                                                                                                                      from e9.84+0.05Y to e9.84+0.05(Y+1) .
                                                                             LogSalary = 9.84 + 0.05 Years + e



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                            Boxplot of Listing                                  Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                                      Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                                   Category                                                                     900000                                                                                                                                                                  900000
                             Meatball                                                             900000                                                                                                                                                                                                                                                                                                   14



                                                                                                                                                                                                                                                                                                                                        e  mc  
                                                                   Pepperoni                                                                                                                                                     99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                                       Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                                   Plain
                                                                   Mushroom
                                                                   Sausage
                                                                                                  800000                                        800000
                                                                                                                                                                                                                                 95
                                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                                                                        800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        51
                                                                                                                                                                                                                                                                                           AD         0.994                                                                                                                                                                                     80
                                                                   Pepper and Onion                                                             700000                                                                           90                                                                                     700000
                                                                                                  700000                                                                                                                                                                                   P-Value    0.012
                                                                   Mushroom and Onion                                                                                                                                                                                                                                                                                                                      10
                                                                   Garlic                                                                                                                                                        80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                                600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                               Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Percent
                                                                   Meatball                                                                                                                                                      70
                                                                                                  600000
                                                                                                                                      Listing




                                                                                                                                                                                                                                                                                                              Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                                           8
                                                                                                                                                                                                                       Percent




                                                                                                                                                                                                                                 60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                                500000                                                                                                                                                                  500000
                                                                                        Listing




                                                                                                                                                                                                                                 50                                                                                                                                                                                                                                                             40
       Sausage                                                                                    500000                                                                                                                         40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Listing
         5.8%                                                                                                                                   400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                                  400000                                                                                                                         20                                                                                                                                                                                                                                                             20
                                                                                                                                                300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                                 10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                                  300000                                        200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        200000
                                            32.5%
                                                                                                  200000                                                                                                                                                                                                                                                                                                                                                                                              0       00     00     00     00      00     00     00     00     00
                                                                                                                                                100000                                                                                                                                                                  100000
                                                                                                                                                                                                                                 1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                                      15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                                 IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                                  100000
      39/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Growth in a Semilog Model
                                                                                                       logSalary  9.84  0.05Years  e
                                                                                                      Salary = e9.84+0.05Years
                                                                                                      Years = 0 at Starting Salary = 18,770
                                                                                                      Marginal change. From year t to year t+1,
                                                                                                      log Salary goes up by 0.05. Salary changes
                                                                                                       from e9.84+0.05Y to e9.84+0.05(Y+1) . Salary change is
                                                                                                       Salary(t+1)-Salary(t) Salary(t+1)
                                                                                                                                                 1
                                                                                                             Salary(t)               Salary(t)
                                                                                                                               e9.84+0.05(Y+1)
                                                                                                                              9.84+0.05Y  1  e0.05  1  0.0513.
                                                                                                                                 e
                                                                                                       Conclude: The slope is the growth rate per period or the
                                                                                                       proportional increase for a 1 unit change in the "x."



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      40/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Using Semilog Models for Trends
                                                                                                                                                                                                                                                                                                                                                     Scatterplot of Flights vs Month
                                                                                                                                                                                                                                                                 350


                                                                                                                                                                                                                                                                 300


                                                                                                                                                                                                                                                                 250

                                                                                                                                                                                                                                                                 200




                                                                                                                                                                                                                                                    Flights
                                                                                                                                                                                                                                                                 150


                                                                                                                                                                                                                                                                 100

                                                                                                                                                                                                                                                                     50


                                                                                                                                                                                                                                                                           0
                                                                                                                                                                                                                                                                                                      0                             10                         20                                      30                      40                                   50                                            60                                      70                                   80
                                                                                                                                                                                                                                                                                                                                                                                                                             Month




                                                                                                                                                                                                                                               Frequent Flyer Flights for 72 Months.
                                                                                                                                                                                                                                               (Text, Ex. 11.1, p. 508)


                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      41/49
                                                                                                                                                                                                                                                                                                                                                                          Part 15 – Regression Models


                                                                                                 Regression Approach
                                                                                                                                                                   logFlights = α + β Months + ε
                                                                                                                                                                  a = 2.770, b = 0.03710, s = 0.06102

                                                                                                                                                                                                                                                                 Fitted Line Plot
                                                                                                                                                                                                                                   LogFlights = 2.770 + 0.03710 Month
                                                                                                                                                 6.0                                                                                                                                                                                                                                                               S                                         0.247017
                                                                                                                                                                                                                                                                                                                                                                                                                   R-Sq                                        90.9%
                                                                                                                                                 5.5                                                                                                                                                                                                                                                               R-Sq(adj)                                   90.8%



                                                                                                                                                 5.0
                                                                                                                                 LogFlights




                                                                                                                                                 4.5

                                                                                                                                                 4.0

                                                                                                                                                 3.5

                                                                                                                                                 3.0


                                                                                                                                                                     0                         10                     20                     30                    40                                            50                   60                       70                               80
                                                                                                                                                                                                                                                                 Month



                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                              Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                      Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                    900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                               14



                                                                                                                                                                                                                                                                                                                        e  mc  
                                                       Pepperoni                                                                                                                                                 99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                       Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                                 95
                                                                                                                                                                                                                                                                           StDev
                                                                                                                                                                                                                                                                           N
                                                                                                                                                                                                                                                                                     156865
                                                                                                                                                                                                                                                                                         51
                                                                                                                                                                                                                                                                                                        800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        51
                                                                                                                                                                                                                                                                           AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                             90                                                                                     700000
                                                                                      700000                                                                                                                                                                               P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                  10
                                                       Garlic                                                                                                                                                    80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                    600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                               Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Percent
                                                       Meatball                                                                                                                                                  70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                              Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                           8
                                                                                                                                                                                                       Percent




                                                                                                                                                                                                                 60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                    500000
                                                                            Listing




                                                                                                                                                                                                                 50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                     40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Listing
         5.8%                                                                                                                 400000                                                                             30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                     20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                    300000                                                                             4
                                                                                                                                                                                                                 10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                             5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                          0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                    100000
                                                                                                                                                                                                                 1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000     17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                 IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      43/49
                                                                                                                                                                                                                                                                                                                                                                             Part 15 – Regression Models


                                                                                                     Elasticity and Loglinear Models
                                                                                                            logY = α + βlogx + ε
                                                                                                            The “responsiveness” of one variable to changes
                                                                                                             in another
                                                                                                            E.g., in economics
                                                                                                             demand elasticity = (%ΔQ) / (%ΔP)
                                                                                                            Math: Ratio of percentage changes
                                                                                                                       %ΔQ / %ΔP = {100%[(ΔQ )/Q] / {100%[(ΔP)/P]}
                                                                                                                       Units of measurement and the 100% fall out of this eqn.
                                                                                                                       Elasticity = (ΔQ/ΔP)*(P/Q)
                                                                                                                       Elasticities are units free
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                    Boxplot of Listing                             Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                         Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                    900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                  14



                                                                                                                                                                                                                                                                                                                           e  mc  
                                                       Pepperoni                                                                                                                                                    99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                          Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                       800000
                                                                                                                                                                                                                    95
                                                                                                                                                                                                                                                                              StDev
                                                                                                                                                                                                                                                                              N
                                                                                                                                                                                                                                                                                        156865
                                                                                                                                                                                                                                                                                            51
                                                                                                                                                                                                                                                                                                           800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           51
                                                                                                                                                                                                                                                                              AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                            700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                  P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                     10
                                                       Garlic                                                                                                                                                       80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                   600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                  Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Percent
                                                       Meatball                                                                                                                                                     70
                                                                                      600000
                                                                                                                         Listing




                                                                                                                                                                                                                                                                                                 Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                              8
                                                                                                                                                                                                          Percent




                                                                                                                                                                                                                    60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                   500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                    50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                        40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                Listing
         5.8%                                                                                                                      400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                        20                                                                                                                                                                                                                                                             20
                                                                                                                                   300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                    10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                       200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                             0       00     00     00     00      00     00     00     00     00
                                                                                                                                   100000                                                                                                                                                                  100000
                                                                                                                                                                                                                    1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                         15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                    IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      8/49
                                                                                                                                                                                                                                                                                                                                                                        Part 15 – Regression Models


                                                                                                 Monet Regression




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                    Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                               900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                             14



                                                                                                                                                                                                                                                                                                                      e  mc  
                                                       Pepperoni                                                                                                                                               99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                     Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                  800000
                                                                                                                                                                                                               95
                                                                                                                                                                                                                                                                         StDev
                                                                                                                                                                                                                                                                         N
                                                                                                                                                                                                                                                                                   156865
                                                                                                                                                                                                                                                                                       51
                                                                                                                                                                                                                                                                                                      800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      51
                                                                                                                                                                                                                                                                         AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                       700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                             P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                10
                                                       Garlic                                                                                                                                                  80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                              600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                             Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                    Percent
                                                       Meatball                                                                                                                                                70
                                                                                      600000
                                                                                                                    Listing




                                                                                                                                                                                                                                                                                            Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                         8
                                                                                                                                                                                                     Percent




                                                                                                                                                                                                               60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                              500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                               50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                   40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Listing
         5.8%                                                                                                                 400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                   20                                                                                                                                                                                                                                                             20
                                                                                                                              300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                               10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                  200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                        0       00     00     00     00      00     00     00     00     00
                                                                                                                              100000                                                                                                                                                                  100000
                                                                                                                                                                                                               1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                    15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                               IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000
      49/49
                                                                                                                                                                                                                                                                                                                                                                            Part 15 – Regression Models


                                                                                                 Summary
                                                                                                             Residual analysis
                                                                                                                     Consistent with model assumptions?
                                                                                                                     Suggest missing elements in the
                                                                                                                      model
                                                                                                             Building the regression model
                                                                                                                     Interpreting the model – cost function
                                                                                                                     Growth model – semilog
                                                                                                                     Double log and estimating elasticities
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Marginal Plot of Listing vs IncomePC
                   Pie Chart of Percent vs Type                                                Boxplot of Listing                                Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing                                                             Scatterplot of Listing vs IncomePC                                                          Histogram of Listing                                                                    Empirical CDF of Listing
                                                                                                                                                                                                                                        Normal - 95% CI                                                                                                                                                                                                                                                                  Normal
                                                       Category                                                                   900000                                                                                                                                                                  900000
                             Meatball                                                 900000                                                                                                                                                                                                                                                                                                 14



                                                                                                                                                                                                                                                                                                                          e  mc  
                                                       Pepperoni                                                                                                                                                   99                                                                                                                                                                                                                                                                                                                                         Mean    369687
                         Garlic 5.0%                                                                                                                                                                                                                                         Mean      369687                                                                                                                                                                                     100
    Mushroom and Onion
                  9.2%
                         2.3%
                                           Pepperoni
                                           21.8%
                                                       Plain
                                                       Mushroom
                                                       Sausage
                                                                                      800000                                      800000
                                                                                                                                                                                                                   95
                                                                                                                                                                                                                                                                             StDev
                                                                                                                                                                                                                                                                             N
                                                                                                                                                                                                                                                                                       156865
                                                                                                                                                                                                                                                                                           51
                                                                                                                                                                                                                                                                                                          800000                                 2                                           12
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              StDev
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              N
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      156865
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          51
                                                                                                                                                                                                                                                                             AD         0.994                                                                                                                                                                                     80
                                                       Pepper and Onion                                                           700000                                                                           90                                                                                     700000
                                                                                      700000                                                                                                                                                                                 P-Value    0.012
                                                       Mushroom and Onion                                                                                                                                                                                                                                                                                                                    10
                                                       Garlic                                                                                                                                                      80                                                                                                                                                                                                                                                                                                                                                                    1000000
                                                                                                                                  600000                                                                                                                                                                  600000                                                                                                                                                                  60
                                                                                                                                                                                                                                                                                                                                                                                 Frequency




Pepper and Onion




                                                                                                                                                                                                                                                                                                                                                                                                                                                                        Percent
                                                       Meatball                                                                                                                                                    70
                                                                                      600000
                                                                                                                        Listing




                                                                                                                                                                                                                                                                                                Listing




            7.3%                                                                                                                                                                                                                                                                                                                                                                             8
                                                                                                                                                                                                         Percent




                                                                                                                                                                                                                   60                                                                                                                                                                                                                                                                                                                                                                     800000
                                                                                                                                  500000                                                                                                                                                                  500000
                                                                            Listing




                                                                                                                                                                                                                   50                                                                                                                                                                                                                                                             40
       Sausage                                                                        500000                                                                                                                       40                                                                                                                                                                        6




                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Listing
         5.8%                                                                                                                     400000                                                                           30                                                                                     400000                                                                                                                                                                                                                                                                          600000
                                                                                      400000                                                                                                                       20                                                                                                                                                                                                                                                             20
                                                                                                                                  300000                                                                                                                                                                  300000                                                                             4
                                                                                                                                                                                                                   10                                                                                                                                                                                                                                                                                                                                                                     400000
                                                                                      300000                                      200000                                                                           5                                                                                      200000                                                                             2                                                                                     0
             Mushroom                       Plain
                16.2%
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          200000
                                            32.5%
                                                                                      200000                                                                                                                                                                                                                                                                                                                                                                                            0       00     00     00     00      00     00     00     00     00
                                                                                                                                  100000                                                                                                                                                                  100000
                                                                                                                                                                                                                   1                                                                                                                                                                         0                                                                                                00     00     00     00      00     00     00     00     00
                                                                                                                                        15000   17500   20000   22500    25000   27500   30000   32500                  0   200000   400000    600000     800000   1000000                                      15000   17500   20000   22500    25000   27500   30000   32500                    200000   300000   400000   500000 600000   700000   800000   900000                       10     20     30     40      50     60     70     80     90                                        15000       20000       25000    30000
                                                                                                                                                                   IncomePC                                                            Listing                                                                                             IncomePC                                                                              Listing                                                                              Listing                                                                                      IncomePC
                                                                                      100000

				
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