The Normality Assumption Once Again

We know by now that if our sole objective is point estimation of the parameters of the regression models, the method of ordinary least squares (OLS), which does not make any assumption about the probability distribution of the disturbances ui, will suffice. But if our objective is estimation as well as inference, then, as argued in Chapters 4 and 5, we need to assume that the ui follow some probability distribution.

For reasons already clearly spelled out, we assumed that the ui follow the normal distribution with zero mean and constant variance a2. We continue to make the same assumption for multiple regression models. With the normality assumption and following the discussion of Chapters 4 and 7, we find that the OLS estimators of the partial regression coefficients, which are identical with the maximum likelihood (ML) estimators, are best linear unbiased estimators (BLUE).1 Moreover, the estimators fa, fa, and fa are

'With the normality assumption, the OLS estimators ft, ft, and ft are minimum-variance estimators in the entire class of unbiased estimators, whether linear or not. In short, they are BUE (best unbiased estimators). See C. R. Rao, Linear Statistical Inference and Its Applications, John Wiley & Sons, New York, 1965, p. 258.

CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 249

themselves normally distributed with means equal to true ft2, ft, and ft and the variances given in Chapter 7. Furthermore, (n — 3)a2/a2 follows the x2 distribution with n — 3 df, and the three OLS estimators are distributed independently of a2. The proofs follow the two-variable case discussed in Appendix 3. As a result and following Chapter 5, one can show that, upon replacing a2 by its unbiased estimator a2 in the computation of the standard errors, each of the following variables t =

 ft - ft se (ft) ft - ft se (ft) ft ft3 se (ft)

follows the t distribution with n — 3 df.

Note that the df are now n — 3 because in computing Y U and hence a2 we first need to estimate the three partial regression coefficients, which therefore put three restrictions on the residual sum of squares (RSS) (following this logic in the four-variable case there will be n — 4 df, and so on). Therefore, the t distribution can be used to establish confidence intervals as well as test statistical hypotheses about the true population partial regression coefficients. Similarly, the x2 distribution can be used to test hypotheses about the true a2. To demonstrate the actual mechanics, we use the following illustrative example.

8.2 EXAMPLE 8.1: CHILD MORTALITY EXAMPLE REVISITED

In Chapter 7 we regressed child mortality (CM) on per capita GNP (PGNP) and the female literacy rate (FLR) for a sample of 64 countries. The regression results given in (7.6.2) are reproduced below with some additional information:

CM* = 263.6416 - 0.0056 PGNP; - 2.2316 FLR* se = (11.5932) (0.0019) (0.2099)

where " denotes extremely low value.

250 PART ONE: SINGLE-EQUATION REGRESSION MODELS

In Eq. (8.2.1) we have followed the format first introduced in Eq. (5.11.1), where the figures in the first set of parentheses are the estimated standard errors, those in the second set are the t values under the null hypothesis that the relevant population coefficient has a value of zero, and those in the third are the estimated p values. Also given are R2 and adjusted R2 values. We have already interpreted this regression in Example 7.1.

What about the statistical significance of the observed results? Consider, for example, the coefficient of PGNP of -0.0056. Is this coefficient statistically significant, that is, statistically different from zero? Likewise, is the coefficient of FLR of -2.2316 statistically significant? Are both coefficients statistically significant? To answer this and related questions, let us first consider the kinds of hypothesis testing that one may encounter in the context of a multiple regression model.

8.3 HYPOTHESIS TESTING IN MULTIPLE REGRESSION: GENERAL COMMENTS

Once we go beyond the simple world of the two-variable linear regression model, hypothesis testing assumes several interesting forms, such as the following:

1. Testing hypotheses about an individual partial regression coefficient (Section 8.4)

2. Testing the overall significance of the estimated multiple regression model, that is, finding out if all the partial slope coefficients are simultaneously equal to zero (Section 8.5)

3. Testing that two or more coefficients are equal to one another (Section 8.6)

4. Testing that the partial regression coefficients satisfy certain restrictions (Section 8.7)

5. Testing the stability of the estimated regression model over time or in different cross-sectional units (Section 8.8)

6. Testing the functional form of regression models (Section 8.9)

Since testing of one or more of these types occurs so commonly in empirical analysis, we devote a section to each type.