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Note: OLShet means OLS allowing for heteroscedasticity.

Note: OLShet means OLS allowing for heteroscedasticity.

6From (5.3.6) we know that the 100(1 — a)% confidence interval for 02 is [02 ± ta/2 se (02)]. But if se (02) cannot be estimated unbiasedly, what trust can we put in the conventionally computed confidence interval?

7Russell Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, Oxford University Press, New York, 1993, pp. 549-550.

400 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

The most striking feature of these results is that OLS, with or without correction for heteroscedasticity, consistently overestimates the true standard error obtained by the (correct) GLS procedure, especially for large values of a, thus establishing the superiority of GLS. These results also show that if we do not use GLS and rely on OLS—allowing for or not allowing for heteroscedasticity—the picture is mixed. The usual OLS standard errors are either too large (for the intercept) or generally too small (for the slope coefficient) in relation to those obtained by OLS allowing for heteroscedasticity. The message is clear: In the presence of heteroscedasticity, use GLS. However, for reasons explained later in the chapter, in practice it is not always easy to apply GLS. Also, as we discuss later, unless heteroscedasticity is very severe, one may not abandon OLS in favor of GLS or WLS.

From the preceding discussion it is clear that heteroscedasticity is potentially a serious problem and the researcher needs to know whether it is present in a given situation. If its presence is detected, then one can take corrective action, such as using the weighted least-squares regression or some other technique. Before we turn to examining the various corrective procedures, however, we must first find out whether heteroscedasticity is present or likely to be present in a given case. This topic is discussed in the following section.

Although we have stated that, in cases of heteroscedasticity, it is the GLS, not the OLS, that is BLUE, there are examples where OLS can be BLUE, despite heteroscedasticity.8 But such examples are infrequent in practice.

As with multicollinearity, the important practical question is: How does one know that heteroscedasticity is present in a specific situation? Again, as in the case of multicollinearity, there are no hard-and-fast rules for detecting heteroscedasticity, only a few rules of thumb. But this situation is inevitable because of can be known only if we have the entire Y population corresponding to the chosen X's, such as the population shown in Table 2.1 or Table 11.1. But such data are an exception rather than the rule in most

8The reason for this is that the Gauss-Markov theorem provides the sufficient (but not necessary) condition for OLS to be efficient. The necessary and sufficient condition for OLS to be BLUE is given by Kruskal's Theorem. But this topic is beyond the scope of this book. I am indebted to Michael McAleer for bringing this to my attention. For further details, see Denzil G. Fiebig, Michael McAleer, and Robert Bartels, "Properties of Ordinary Least Squares Estimators in Regression Models with Nonspherical Disturbances," Journal of Econometrics, vol. 54, No. 1-3, Oct.-Dec., 1992, pp. 321-334. For the mathematically inclined student, I discuss this topic further in App. C, using matrix algebra.

A Technical Note

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