Continuing with the two-variable model and assuming the AR(1) process, we can show that the BLUE estimator of 02 is given by the following expression11:
where C is a correction factor that may be disregarded in practice. Note that the subscript t now runs from t = 2 to t = n. And its variance is given by
where D too is a correction factor that may also be disregarded in practice. (See exercise 12.18.)
The estimator 0GLS, as the superscript suggests, is obtained by the method of GLS. As noted in Chapter 11, in GLS we incorporate any additional information we have (e.g., the nature of the heteroscedasticity or of the autocorrelation) directly into the estimating procedure by transforming the variables, whereas in OLS such side information is not directly taken into consideration. As the reader can see, the GLS estimator of 02 given in (12.3.1) incorporates the autocorrelation parameter p in the estimating formula, whereas the OLS formula given in (12.2.6) simply neglects it. Intuitively, this is the reason why the GLS estimator is BLUE and not the OLS estimator—the GLS estimator makes the most use of the available information.12 It hardly needs to be added that if p = 0, there is no additional information to be considered and hence both the GLS and OLS estimators are identical.
In short, under autocorrelation, it is the GLS estimator given in (12.3.1) that is BLUE, and the minimum variance is now given by (12.3.2) and not by (12.2.8) and obviously not by (12.2.7).
A Technical Note. As we noted in the previous chapter, the Gauss-Markov theorem provides only the sufficient condition for OLS to be BLUE. The necessary and sufficient conditions for OLS to be BLUE are given by
"For proofs, see Jan Kmenta, Elements of Econometrics, Macmillan, New York, 1971, pp. 274-275. The correction factor Cpertains to the first observation, (Y1, X1). On this point see exercise 12.18.
12The formal proof that ßGLS is BLUE can be found in Kmenta, ibid. But the tedious algebraic proof can be simplified considerably using matrix notation. See J. Johnston, Econometric Methods, 3d ed., McGraw-Hill, New York, 1984, pp. 291-293.
454 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
Krushkal's theorem, mentioned in the previous chapter. Therefore, in some cases it can happen that OLS is BLUE despite autocorrelation. But such cases are infrequent in practice.
What happens if we blithely continue to work with the usual OLS procedure despite autocorrelation? The answer is provided in the following section.
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