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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

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8.1 THE NORMALITY ASSUMPTION ONCE AGAIN We continue to assume that the u i follow the normal distribution with zero mean and constant variance σ 2. With normality assumption we find that the OLS estimators of the partial regression coefficients are best linear unbiased estimators (BLUE).

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8.1 THE NORMALITY ASSUMPTION ONCE AGAIN

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8.2 EXAMPLE 8.1: CHILD MORTALITY EXAMPLE REVISITED

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8.3 HYPOTHESIS TESTING IN MULTIPLE REGRESSION: GENERAL COMMENTS

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8.4 HYPOTHESIS TESTING ABOUT INDIVIDUAL REGRESSION COEFFICIENTS

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8.5 TESTING THE OVERALL SIGNIFICANCE OF THE SAMPLE REGRESSION

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The Analysis of Variance Approach to Testing the Overall Significance of an Observed Multiple Regression: The F Test

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Testing the Overall Significance of a Multiple Regression: The F Test

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An Important Relationship between R 2 and F

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where use is made of the definition R 2 = ESS/TSS. Equation on the left shows how F and R 2 are related. These two vary directly. When R 2 = 0, F is zero ipso facto. The larger the R 2, the greater the F value. In the limit, when R 2 = 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 R 2. In other words, testing the null hypothesis (8.5.9) is equivalent to testing the null hypothesis that (the population) R 2 is zero.

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An Important Relationship between R 2 and F

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Testing the Overall Significance of a Multiple Regression in Terms of R 2

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The “Incremental” or “Marginal” Contribution of an Explanatory Variable

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This F value is highly significant, as the computed p value is

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The “Incremental” or “Marginal” Contribution of an Explanatory Variable

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This F value is highly significant, suggesting that addition of FLR to the model significantly increases ESS and hence the R 2 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 (− ) 2, you will obtain the F value of (8.5.17).

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The “Incremental” or “Marginal” Contribution of an Explanatory Variable

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8.6 TESTING THE EQUALITY OF TWO REGRESSION COEFFICIENTS

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8.7 RESTRICTED LEAST SQUARES: TESTING LINEAR EQUALITY RESTRICTIONS

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General F Testing

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8.8 TESTING FOR STRUCTURAL OR PARAMETER STABILITY OF REGRESSION MODELS: THE CHOW TEST

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