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					Regression
    Steps in Demand Estimation
• Model Specification: Identify
    Variables
•   Collect Data
•   Specify Functional Form
•   Estimate Function
•   Test the Results
             Collect data
• Time series vs cross section data

• Problems
• lack of data, Use a proxy for variables
• Measure consumers’ expectations by
  consumer surveys
• consumers taste may have changes
          Market Research
•   Consumer Surveys
•   Observational Research
•   Consumer Clinic
•   Market Experiments
•   Virtual Shopping
 Functional Form Specifications-
                         Some Examples

Linear Function:
QX  a0  a1 PX  a2 I  a3 N  a4 PY    e


Power Function:                    Estimation Format:

QX  a ( PX1 )( PYb2 )
          b
                                    ln QX  ln a  b1 ln PX  b2 ln PY



Y= a + b1X+ b2X2  Quadratic function
   Empirical Production
       Functions

Cobb-Douglas Production Function
            Q = AKaLb

Estimated using Natural Logarithms
    ln Q = ln A + a ln K + b ln L
             Estimation
• Perform regression analysis
 Testing the econometric results

• sign of each estimated slope coefficient
  must be checked
• t tests for degree of confidence and R2
  will indicate proportion of total variation
• Pass econometric tests as
  multicollinearity etc
 Estimation and interpretation of
             results
Y^ =17.944 + 1.873 X1 + 1.915 X2
t statistic   (2.663)    (2.813)

For each 1$ million increase in
 expenditure on advertising and quality
 control, sales of the firm increase by
 $1.87 million and 1.915
The Identification Problem
            Regression Analysis
Year   X     Y
 1     10    44
                    Scatter Diagram
 2     9     40
 3     11    42
 4     12    46
 5     11    48
 6     12    52
 7     13    54
 8     13    58
 9     14    56
10     15    60
       Regression Analysis
• Regression Line: Line of Best Fit

• Regression Line: Minimizes the sum of
  the squared vertical deviations (et) of
  each point from the regression line.

• Ordinary Least Squares (OLS) Method
Regression Analysis
Ordinary Least Squares (OLS)

    Model:    Yt  a  bX t  et

 Estimated:    ˆ ˆ ˆ
              Yt  a  bX t

                         ˆ
              et  Yt  Yt
Ordinary Least Squares (OLS)

 Objective: Determine the slope and
 intercept that minimize the sum of
 the squared errors.

    n       n               n

   et2   (Yt  Yt ) 2   (Yt  a  bX t ) 2
   t 1    t 1
                   ˆ
                           t 1
                                   ˆ ˆ
Ordinary Least Squares (OLS)

         Estimation Procedure

        n

       (X           t    X )(Yt  Y )
  ˆ
  b   t 1                                       ˆ
                                          a  Y  bX
                                          ˆ
               n

              (X
              t 1
                           t    X)   2
Ordinary Least Squares (OLS)
                                 Estimation Example
 Time                    Xt                   Yt     Xt  X                 Yt  Y         ( X t  X )(Yt  Y )   ( X t  X )2
   1                     10                   44       -2                      -6                 12                   4
   2                      9                   40       -3                     -10                 30                   9
   3                     11                   42       -1                      -8                  8                   1
   4                     12                   46       0                       -4                  0                   0
   5                     11                   48       -1                      -2                  2                   1
   6                     12                   52       0                       2                   0                   0
   7                     13                   54       1                       4                   4                   1
   8                     13                   58       1                       8                   8                   1
   9                     14                   56       2                       6                  12                   4
  10                     15                   60       3                      10                  30                   9
                         120                  500                                                 106                 30
                   n                    n                   n
n  10             X t  120
                  t 1
                                        Yt  500
                                       t 1
                                                        (X
                                                         t 1
                                                                t    X ) 2  30               ˆ 106  3.533
                                                                                               b
                                                                                                  30
    n                              n                        n
          X t 120                      Yt 500
X               12          Y           50      (X     t    X )(Yt  Y )  106       a  50  (3.533)(12)  7.60
                                                                                               ˆ
   t 1   n    10                 t 1 n   10            t 1
Ordinary Least Squares (OLS)
                      Estimation Example
                                               n
                                                     X t 120
 n  10                                  X                 12
                                              t 1   n    10
                                              n
                                                  Yt 500
                                         Y 
     n                 n

X
 t 1
         t    120    Y
                      t 1
                             t    500
                                             t 1 n
                                                    
                                                      10
                                                          50

 n
                                         ˆ  106  3.533
(X
t 1
         t    X )  30
                  2
                                         b
                                              30
 n

(X
t 1
         t    X )(Yt  Y )  106        a  50  (3.533)(12)  7.60
                                         ˆ
              Interpretation
•   Intercept and slope coefficients
•   R 2 (linear regression)
•   Adjusted R 2 ( multiple regression)
•   T test
•   Standard error of b
•   Confidence interval
•   Estimation and interpretation of results
 Testing regression estimates
• R2 = how well overall equation explains
  changes in dependent variable
  (measures proportion of total variation
  in dependent variable explained by
  regression equation)
• T test = measures relationship between
  independent and dependent variable
          Tests of Significance

    Standard Error of the Slope Estimate


sbˆ 
                    ˆ
              (Yt  Y ) 2
                                    
                                                 et2
        (n  k ) ( X t  X )   2
                                        (n  k ) ( X t  X ) 2
                   Tests of Significance
                              Example Calculation
Time                Xt            Yt                       ˆ
                                                          Yt               ˆ
                                                                et  Yt  Yt                      ˆ
                                                                                     et2  (Yt  Yt )2         ( X t  X )2
  1                 10            44                  42.90            1.10                  1.2100                 4
  2                 9             40                  39.37            0.63                  0.3969                 9
  3                 11            42                  46.43        -4.43                 19.6249                    1
  4                 12            46                  49.96        -3.96                 15.6816                    0
  5                 11            48                  46.43            1.57                  2.4649                 1
  6                 12            52                  49.96            2.04                  4.1616                 0
  7                 13            54                  53.49            0.51                  0.2601                 1
  8                 13            58                  53.49            4.51              20.3401                    1
  9                 14            56                  57.02        -1.02                     1.0404                 4
 10                 15            60                  60.55        -0.55                     0.3025                 9
                                                                                         65.4830                   30




  n          n                     n                                          (Y  Yˆ ) 2
                                                                                                         65.4830
  e   (Yt  Yˆt )2  65.4830   (X                          sbˆ                                                 0.52
                                                                                 t
        2
                                              X )  30
                                                 2

 t 1
        t
            t 1                  t 1
                                         t
                                                                        ( n  k ) ( X  X )
                                                                                     t
                                                                                               2
                                                                                                       (10  2)(30)
        Tests of Significance
                  Example Calculation
            n               n

            et2   (Yt  Yt ) 2  65.4830
           t 1
                            ˆ
                           t 1
                     n

                     ( X t  X ) 2  30
                    t 1




sbˆ 
           (Yt  Y
                  ˆ )2
                               
                                   65.4830
                                               0.52
      ( n  k ) ( X t  X ) 2
                                 (10  2)(30)
     Tests of Significance
      Calculation of the t Statistic
        ˆ
       b 3.53
     t         6.79
       sbˆ 0.52

Degrees of Freedom = (n-k) = (10-2) = 8
   Critical Value at 5% level =2.306
       Tests of Significance
  Decomposition of Sum of Squares

Total Variation = Explained Variation + Unexplained Variation




                ˆ  Y )2   (Y  Y )2
  (Yt  Y )   (Y
               2
                                t
                                   ˆ
                                    t
  Tests of Significance
Decomposition of Sum of Squares
Tests of Significance
Coefficient of Determination


R2 
       Explained Variation
                           
                              ˆ
                              (Y  Y )2
         TotalVariation       (Yt  Y )2


                 373.84
             R 
               2
                         0.85
                 440.00
       Confidence Interval
• Estimated b +- t value x standard error
Tests of Significance
 Coefficient of Correlation


                           ˆ
     r  R2 withthe signof b


           1  r  1

        r  0.85  0.92
  Multiple Regression Analysis

Model: Y    a  b1 X 1  b2 X 2     bk ' X k '

  Adjusted Coefficient of Determination

                      (n  1)
     R  1  (1  R )
       2                2

                      (n  k )
Multiple Regression Analysis

 Analysis of Variance and F Statistic

           Explained Variation /(k  1)
       F
          Unexplained Variation /(n  k )

                   R 2 /( k  1)
             F
                (1  R 2 ) /( n  k )
Problems in Regression Analysis

• Multicollinearity: Two or more
  explanatory variables are highly
  correlated.
• Heteroskedasticity: Variance of error
  term is not independent of the Y
  variable (cross section data)
• Autocorrelation: Consecutive error
  terms are correlated (time series )
       Doctor’ Prescription
• Multicollinearity – extend sample size,
  use priori information, transform
  functional relationship, drop one of
  highly collinear variables
• Heterocedasticity – use log of
  explanatory variables, use weighted
  least square regression
• Autocorrelation – Use time as an
  additional explanatory variable
Durbin-Watson Statistic
     Test for Autocorrelation
                n

                (et  et 1 ) 2
          d   t 2
                       n

                       et2
                      t 1



If d=2, autocorrelation is absent.

				
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