# Statistics

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```					                    Part 18 – Multiple Regression:2

Statistics and Data
Analysis

Professor William Greene
IOMS Department
Department of Economics
Part 18 – Multiple Regression:2

Statistics and Data Analysis

Part 18 – Multiple
Regression: 2
1/52
Part 17 – Multiple Regression:2
18

Multiple Regression Models

                    Using Minitab To Compute A Multiple Regression
                    Basic Multiple Regression
                    Using Binary Variables
                    Logs and Elasticities
                    Hedonic Regression and Interpretation
                    Trends in Time Series Data
                    Using Quadratic Terms to Improve the Model

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
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
2/52
Part 18 – Multiple Regression:2

Application: WHO
                      Data Used in Assignment 1: WHO data
on 191 countries in 1995-1999.
 Analysis of Disability Adjusted Life
Expectancy = DALE
 EDUC = average years of education
 PCHexp = Per capita health expenditure

                      DALE = α + β1EDUC + β2HealthExp + 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
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
3/52
Part 18 – Multiple Regression:2

The (Famous) WHO Data

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
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
4/52
Part 18 – Multiple Regression:2

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
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
5/52
Part 18 – Multiple Regression:2

Specify the Variables in the Model

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
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
6/52
Part 18 – Multiple Regression:2

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
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
7/52
Part 18 – Multiple Regression:2

Graphs? Maybe

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
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
8/52
Part 18 – Multiple Regression:2

Regression Results

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
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
9/52
Part 18 – Multiple Regression:2

Practical Model Building

 Understanding the regression: The left
out variable problem
 Using different kinds of variables
 Dummy variables
 Logs
 Time trend

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
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
10/52
Part 18 – Multiple Regression:2

A Fundamental Result
What happens when you leave a crucial
Regression Analysis: g versus GasPrice (no income)
The regression equation is
g = 3.50 + 0.0280 GasPrice
Predictor      Coef   SE Coef       T     P
Constant     3.4963    0.1678 20.84 0.000
GasPrice   0.028034 0.002809     9.98 0.000
Regression Analysis: G versus GasPrice, Income
The regression equation is
G = 0.134 - 0.00163 GasPrice + 0.000026 Income
Predictor        Coef      SE Coef      T     P
Constant      0.13449      0.02081   6.46 0.000
GasPrice   -0.0016281   0.0004152 -3.92 0.000
Income     0.00002634 0.00000231 11.43 0.000

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
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
11/52
Part 18 – Multiple Regression:2

Using Dummy Variables
 Dummy variable = binary variable
= a variable that takes values 0 and 1.
 E.g. OECD Life Expectancies
compared to the rest of the world:
                  DALE = α + β1 EDUC + β2 PCHexp
+ β3 OECD + ε
Australia, Austria, Belgium, Canada, Czech Republic, Denmark, Finland, France,
Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Korea, Luxembourg,
Mexico, The Netherlands, New Zealand, Norway, Poland, Portugal, Slovak
Republic, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States.
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
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
12/52
Part 18 – Multiple Regression:2

OECD Life Expectancy

According to these
results, after
accounting for
education and health
expenditure
differences, people in
the OECD countries
have a life expectancy
that is 1.19 years
shorter than people in
other countries.

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
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
13/52
Part 18 – Multiple Regression:2

Binary Variable in Regression

The regression
shifts down by
NonOECD: DALE =36.770 + 2.9962 EDUC
+ .005079 PCHExp                                                                                                                                                                                                                                                                                                                                       1.191 years for
the OECD
countries

OECD: DALE =36.770 + 2.9962 EDUC
+ .005079 PCHExp – 1.191

We set PCHExp to 1000, approximately the sample mean.

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
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 18 – Multiple Regression:2

Plotting  For DALE_NonOECD, remove -1.191

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
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 18 – Multiple Regression:2

Two Plots
Life Expectancy in OECD and NonOECD Countries
80                                                                                                                                                                                                                                                                                                                                                                         Variable
DALE_OECD
DALE_NonOECD

70
Y-Data

60

50

40
0                                            2                                          4                                        6                                                              8                                       10                                                     12
EDUC

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
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
14/52
Part 18 – Multiple Regression:2

Dummy Variable in Log Regression

E.g., Monet’s signature equation

Log\$Price = α + β1 logArea + β2 Signed

Unsigned: PriceU                                                                                                         = exp(α) Areaβ1
Signed: PriceS                                                                                                           = exp(α) Areaβ1 exp(β2)
Signed/Unsigned                                                                                                          = exp(β2)
%Difference                                                                                                              = 100%(Signed-Unsigned)/Unsigned
= 100%[exp(β2) – 1]

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
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
15/52
Part 18 – Multiple Regression:2

The Signature Effect: 253%

100%[exp(1.2618) – 1] = 100%[3.532 – 1] = 253.2 %

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
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
16/52
Part 18 – Multiple Regression:2

Monet Paintings in Millions
Scatterplot of Price vs Square Inches
30                                                                                                                                                                                                                                                                                                                     Signed
0
1
25

20

Price

15

10

5

0

0                        1000                               2000                     3000    4000                                                                        5000                             6000                                 7000
Square Inches

Predicted Price is exp(4.122+1.3458*logArea+1.2618*Signed) / 1000000

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
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
17/52
Part 18 – Multiple Regression:2

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
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
18/52
Part 18 – Multiple Regression:2

Dummy Variable for One
Observation
Proofs:
See p. 40.
                     Single out one observation for special attention.

                     The equation will predict that observation
perfectly.

                     For the other coefficients, it is the same as
removing that observation from the sample.

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
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
19/52
Part 18 – Multiple Regression:2

A “London Effect” on UK
Electronic Store Sales?

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
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 18 – Multiple Regression:2

20/52

Observation 2 is London
Fit = Actual, Residual=0.

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
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
21/52
Part 18 – Multiple Regression:2

Logs in 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
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
22/52
Part 18 – Multiple Regression:2

Elasticity
 The coefficient on log(Area) is 1.346
 For each 1% increase in area, price
goes up by 1.34% - even accounting
for the signature effect.
 The elasticity is +1.34

 Remarkable. Not only does price
increase with area, it increases faster
than area.
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
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
23/52
Part 18 – Multiple Regression:2

Monet: By the Square Inch
Scatterplot of Price vs Area

20000000

15000000
price

10000000

5000000

0

0                              1000                                2000                                                3000    4000                                                                      5000                                       6000                                       7000
Area

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
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
24/52
Part 18 – Multiple Regression:2

Elasticities of Demand for Gasoline

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
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
25/52
Part 18 – Multiple Regression:2

Logs and Elasticities
Theory: In the equation
y = α + β1x1 + β2x2 + … βKxK + ε
β = (change in y) / (unit change in x)
Elasticity = β * mean of x / mean of y
When the variables are in logs: change in logx = %change in x
log y = α + β1 log x1 + β2 log x2 + … βK log xK + ε
Elasticity = β
These will often give approximately the same answer.

When in doubt, use logs.

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
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
26/52
Part 18 – Multiple Regression:2

Elasticities

Price elasticity                                                                            = -0.02070                                                                                                              Income elasticity = +1.10318

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
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
27/52
Part 18 – Multiple Regression:2

A Set of Dummy Variables

 Complete set of dummy variables
divides the sample into groups.
 Fit the regression with “group” effects.

 Need to drop one (any one) of the
variables to compute the regression.
(Avoid the “dummy variable trap.”)

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
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
28/52
Nancy Burnett: Journal of Economic                                                                                                                                                                                                                         Part 18 – Multiple Regression:2
Education, 1998

Rankings of 132 U.S.Liberal Arts Colleges
Reputation=α+β1Religious + β2GenderEcon + β3EconFac +
β4North + β5South + β6Midwest + β7West + 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
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
29/52
Part 18 – Multiple Regression:2

Minitab to the Rescue

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
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
30/52
Part 18 – Multiple Regression:2

Unordered Categorical Variables

House price data (fictitious)
Type 1 = Split level
Type 2 = Ranch
Type 3 = Colonial
Type 4 = Tudor
Use 3 dummy variables for this kind
of data. (Not all 4)
Using variable STYLE in the model
makes no sense. You could change
the numbering scale any way you
like. 1,2,3,4 are just labels.

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
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
31/52
Part 18 – Multiple Regression:2

Transform Style to Types

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
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
32/52
Part 18 – Multiple Regression:2

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
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
33/52
Part 18 – Multiple Regression:2

House Price Regression

Each of these is relative to a Split
Level, since that is the omitted
category. E.g., the price of a Ranch
house is \$74,369 less than a Split
Level of the same size with the
same number of bedrooms.

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
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
34/52
Part 18 – Multiple Regression:2

Ordered Categories
             Health Satisfaction:
1=Poor, 2=So_so, 3=OK, 4=Good, 5=Great
             How to handle such a variable?
 Just use as is? No, So_so – Poor =1, but this is not
equal to Great – Good = 1 (necessarily)
 Use 4 of the indicator variables.
             Coding. It is not useful to consider modifications of the
variable, such as -2,-1,0,1,2 or 2,4,6,8,10. None make
sense as this is just a label. Could also use 1,4,8,17,26
which would also make no sense.
             This needs a special kind of model if it is the dependent
variable – not a regression equation.

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
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
35/52
Part 18 – Multiple Regression:2

Hedonic Regression
                A theory of prices
 Price = sum of prices for components
                House price
 Land size
 Rooms: Fixed amount per room
 Swimming pool
 View
 N car garage
 Etc.
                Computers
 Speed
 Screen size
 Other features…

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
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
36/52
Part 18 – Multiple Regression:2

Fumiro Computer Data

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
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
37/52
Part 18 – Multiple Regression:2

Transform Manufacturer
Names to Indicator Variables
Calc  Make Indicator Variables…

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
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
38/52
Part 18 – Multiple Regression:2

Hedonic 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
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
39/52
Part 18 – Multiple Regression:2

Time Trends in Regression

 y = α + β 1x + β 2t + ε
β2 is the year to year increase not
explained by anything else.
 log y = α + β1log x + β2t + ε
(not log t, just t)
100β2 is the year to year
% increase
not explained by anything else.
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
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
40/52
Part 18 – Multiple Regression:2

Time Trend Regression

After accounting for Income, the
price and the price of new cars,
per capita gasoline consumption
falls by 1.25% per year. I.e., if
Income and the prices were
unchanged, consumption would
fall by 1.25%. But, of course,
these other things do not remain
unchanged.

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
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
41/52
Part 18 – Multiple Regression:2

Nonlinear Equation

 Using a quadratic (like using logs)
 y = α + β1x + β2x2 + ε

 Usually β1 > 0.

 If β2 > 0              If β2 < 0
y                                                                                                                                                                                                                                         y

x                                                                                                                                                                                                                                                                     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
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
42/52
Part 18 – Multiple Regression:2

Regression
+----------------------------------------------------+
| LHS=HHNINC    Mean                  =  .3520836       |
|               Standard deviation    =  .1769083       |
| Model size    Parameters            =          3      |
|               Degrees of freedom    =     27323       |
| Residuals     Sum of squares        =  794.9667       |
|               Standard error of e =    .1705730       |
| Fit           R-squared             =  .7040754E-01 |
+----------------------------------------------------+
+--------+--------------+--+--------+
|Variable| Coefficient | Mean of X|
+--------+--------------+-----------+ Note the coefficient on
Constant| -.39266196                  Age squared is
AGE     | .02458140      43.5256898   negative. Age ranges
AGESQ   | -.00027237     2022.85549
from 25 to 65.
EDUC    | .01994416      11.3206310
+--------+--------------+-----------+
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
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
43/52
Part 18 – Multiple Regression:2

Implied By The Model

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
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
44/52
Part 18 – Multiple Regression:2

Case Study: A Huge Sports
Contract
       Alex Rodriguez hired by the Texas Rangers for
something like \$25 million per year.
       Costs – the salary plus and minus some fine
tuning of the numbers
       Benefits – more fans in the stands.
       How to determine if the benefits exceed the
costs? Use a regression model.

Marginal Plot of Listing vs IncomePC
Pie Chart of Percent vs Type                                                Boxplot of Listing                            Scatterplot of Listing vs IncomePC                                              Probability Plot of Listing      ```