Feasible Generalized Least Squares Estimator

Inserting^ defined in (6.3.3) into X71 defined in (5.2.14) and then inserting the estimate of X-1 into (6.2.1), we obtain the FGLS estimator fiF. As noted earlier, <72 drops out of the formula (6.2.1) and therefore need not be estimated in the calculation of fip. (However, er2 needs to be estimated in order to estimate the covariance matrix of ftp.) The consistency of ftp can be proved easily under general conditions, but the proof of the asymptotic normality is rather involved. The reader is referred to Amemiya (1973a), where it has been proved that fip has the same asymptotic distribution as fiG under general assumptions on X when {u,} follow an autoregressive process with moving – average errors. More specifically,

yff(fip-fi)-+N[0, lim ДХ’Х-‘Х)-‘]. (6.3.7)

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In a finite sample, ftp may not be as efficient as fiQ. In fact, there is no assurance that fip is better than the LS estimator fi. Harvey (1981a, p. 191) presented a summary of several Monte Carlo studies comparing fip with fi. Taylor (1981) argued on the basis of analytic ajjproximations of the moments of the estimators that the relative efficiency offip vis-&-vis fi crucially depends on the process of the independent variables.

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