ECONOMETRICS I
CHAPTER 8
MULTIPLE REGRESSION ANALYSIS:
THE PROBLEM OF INFERENCE
Textbook: Damodar N. Gujarati (2004) Basic Econometrics,
4th edition, The McGraw-Hill Companies
8.1 THE NORMALITY ASSUMPTION
ONCE AGAIN
• We continue to assume that the ui follow the
normal distribution with zero mean and
constant variance σ2.
• With normality assumption we find that the
OLS estimators of the partial regression
coefficients are best linear unbiased
estimators (BLUE).
8.1 THE NORMALITY ASSUMPTION
ONCE AGAIN
8.1 THE NORMALITY ASSUMPTION
ONCE AGAIN
8.2 EXAMPLE 8.1: CHILD MORTALITY
EXAMPLE REVISITED
8.2 EXAMPLE 8.1: CHILD MORTALITY
EXAMPLE REVISITED
8.3 HYPOTHESIS TESTING IN MULTIPLE REGRESSION:
GENERAL COMMENTS
8.4 HYPOTHESIS TESTING ABOUT
INDIVIDUAL REGRESSION COEFFICIENTS
8.4 HYPOTHESIS TESTING ABOUT
INDIVIDUAL REGRESSION COEFFICIENTS
8.4 HYPOTHESIS TESTING ABOUT
INDIVIDUAL REGRESSION COEFFICIENTS
8.4 HYPOTHESIS TESTING ABOUT
INDIVIDUAL REGRESSION COEFFICIENTS
8.4 HYPOTHESIS TESTING ABOUT
INDIVIDUAL REGRESSION COEFFICIENTS
8.4 HYPOTHESIS TESTING ABOUT
INDIVIDUAL REGRESSION COEFFICIENTS
8.5 TESTING THE OVERALL SIGNIFICANCE
OF THE SAMPLE REGRESSION
The Analysis of Variance Approach to Testing the Overall
Significance of an Observed Multiple Regression: The F Test
The Analysis of Variance Approach to Testing the Overall
Significance of an Observed Multiple Regression: The F Test
The Analysis of Variance Approach to Testing the Overall
Significance of an Observed Multiple Regression: The F Test
The Analysis of Variance Approach to Testing the Overall
Significance of an Observed Multiple Regression: The F Test
Testing the Overall Significance of a
Multiple Regression: The F Test
Testing the Overall Significance of a
Multiple Regression: The F Test
An Important Relationship between R2 and F
An Important Relationship between R2 and F
An Important Relationship between R2 and F
where use is made of the
definition R2 = ESS/TSS. Equation
on the left shows how F and R2
are related. These two vary
directly. When R2 = 0, F is zero
ipso facto. The larger the R2, the
greater the F value. In the limit,
when R2 = 1, F is infinite. Thus the
F test, which is a measure of the
overall significance of the
estimated regression, is also a
test of significance of R2. In other
words, testing the null hypothesis
(8.5.9) is equivalent to testing the
null hypothesis that (the
population) R2 is zero.
An Important Relationship between R2 and F
Testing the Overall Significance of a Multiple
Regression in Terms of R2
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
This F value is highly significant, as the computed p value is 0.0008.
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
This F value is highly
significant, suggesting that
addition of FLR to the model
significantly increases ESS and
hence the R2 value. Therefore,
FLR should be added to the
model.
Again, note that if you square the t value of the FLR coefficient in the multiple
regression (8.2.1), which is (−10.6293)2, you will obtain the F value of (8.5.17).
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
The “Incremental” or “Marginal” Contribution
of an Explanatory Variable
8.6 TESTING THE EQUALITY OF TWO
REGRESSION COEFFICIENTS
8.6 TESTING THE EQUALITY OF TWO
REGRESSION COEFFICIENTS
8.6 TESTING THE EQUALITY OF TWO
REGRESSION COEFFICIENTS
8.6 TESTING THE EQUALITY OF TWO
REGRESSION COEFFICIENTS
8.6 TESTING THE EQUALITY OF TWO
REGRESSION COEFFICIENTS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
8.7 RESTRICTED LEAST SQUARES:
TESTING LINEAR EQUALITY RESTRICTIONS
General F Testing
General F Testing
General F Testing
General F Testing
General F Testing
General F Testing
General F Testing
General F Testing
General F Testing
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST
8.8 TESTING FOR STRUCTURAL OR PARAMETER
STABILITY OF REGRESSION MODELS: THE CHOW TEST