What happens to OLS estimators and their variances if we introduce heteroscedasticity by letting E(u2) = of but retain all other assumptions of the classical model? To answer this question, let us revert to the two-variable model:
Yi = 01 + 02 Xi + Ui Applying the usual formula, the OLS estimator of 02 is
Ex nTXY -VXTY
394 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
but its variance is now given by the following expression (see Appendix 11A, Section 11A.1):
which is obviously different from the usual variance formula obtained under the assumption of homoscedasticity, namely,
Of course, if o2 = a2 for each i, the two formulas will be identical. (Why?)
Recall that fa is best linear unbiased estimator (BLUE) if the assumptions of the classical model, including homoscedasticity, hold. Is it still BLUE when we drop only the homoscedasticity assumption and replace it with the assumption of heteroscedasticity? It is easy to prove that fa is still linear and unbiased. As a matter of fact, as shown in Appendix 3A, Section 3A.2, to establish the unbiasedness of fa it is not necessary that the disturbances (ui) be homoscedastic. In fact, the variance of ui, homoscedastic or het-eroscedastic, plays no part in the determination of the unbiasedness property. Recall that in Appendix 3A, Section 3A.7, we showed that fa is a consistent estimator under the assumptions of the classical linear regression model. Although we will not prove it, it can be shown that fa is a consistent estimator despite heteroscedasticity; that is, as the sample size increases indefinitely, the estimated fa converges to its true value. Furthermore, it can also be shown that under certain conditions (called regularity conditions), fa is asymptotically normally distributed. Of course, what we have said about fa also holds true of other parameters of a multiple regression model.
Granted that fa is still linear unbiased and consistent, is it "efficient" or "best"; that is, does it have minimum variance in the class of unbiased estimators? And is that minimum variance given by Eq. (11.2.2)? The answer is no to both the questions: fa is no longer best and the minimum variance is not given by (11.2.2). Then what is BLUE in the presence of heteroscedas-ticity? The answer is given in the following section.
11.3 THE METHOD OF GENERALIZED LEAST SQUARES (GLS)
Why is the usual OLS estimator of fa given in (11.2.1) not best, although it is still unbiased? Intuitively, we can see the reason from Table 11.1. As the table shows, there is considerable variability in the earnings between employment classes. If we were to regress per-employee compensation on the size of employment, we would like to make use of the knowledge that there is considerable interclass variability in earnings. Ideally, we would like to devise
CHAPTER ELEVEN: HETEROSCEDASTICITY 395
the estimating scheme in such a manner that observations coming from populations with greater variability are given less weight than those coming from populations with smaller variability. Examining Table 11.1, we would like to weight observations coming from employment classes 10-19 and 20-49 more heavily than those coming from employment classes like 5-9 and 250-499, for the former are more closely clustered around their mean values than the latter, thereby enabling us to estimate the PRF more accurately.
Unfortunately, the usual OLS method does not follow this strategy and therefore does not make use of the "information" contained in the unequal variability of the dependent variable Y, say, employee compensation of Table 11.1: It assigns equal weight or importance to each observation. But a method of estimation, known as generalized least squares (GLS), takes such information into account explicitly and is therefore capable of producing estimators that are BLUE. To see how this is accomplished, let us continue with the now-familiar two-variable model:
which for ease of algebraic manipulation we write as
where X0i = 1 for each i. The reader can see that these two formulations are identical.
Now assume that the heteroscedastic variances a2 are known. Divide (11.3.2) through by ai to obtain
ai ai ai ai which for ease of exposition we write as
where the starred, or transformed, variables are the original variables divided by (the known) ai .We use the notation faf and fa|, the parameters of the transformed model, to distinguish them from the usual OLS parameters fai and fa2.
What is the purpose of transforming the original model? To see this, notice the following feature of the transformed error term uf:
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