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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