# Regression

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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
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 '

(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
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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