The Case of an Unknown Covariance Matrix
In the remainder of this chapter, we shall consider Model 6 assuming that 2 is unknown and therefore must be estimated. Suppose we somehow obtain an estimator 2. Then we define the feasible generalized least squares (FGLS) estimator by
fif = (X’2-1X)-1X’2_ly,
assuming 2 is nonsingular.
For fip to be a reasonably good estimator, we should at least require it to be consistent. This means that the number of the firee parameters that characterize 2 should be either bounded or allowed to go to infinity at a slower rate than T. Thus one must impose particular structure on 2, specifying how it depends on a set of free parameters that are fewer than Tin number. In this section we shall consider five types of models in succession. For each we shall impose a particular structure on 2 and then study the properties of LS and FGLS and other estimators of fi. We shall also discuss the estimation of 2. The five models we shall consider are (1) serial correlation, (2) seemingly unrelated regression models, (3) heteroscedasticity, (4) error components models, and (5) random coefficients models.
In each of the models mentioned in the preceding paragraph, 2 is obtained from the least squares residuals & = у — Xfi, ^where fi is the LS estimator. Under general conditions we shall show that fi is consistent in these models and hence that 2 is consistent. Using this result, we shall show that fip has the same asymptotic distribution as fiG.
In some situations we may wish to use (6.2.1) as an iterative procedure; that is, given fip we can calculate the new residuals у — X&, reestimate 2, and insert it into the right-hand side of (6.2.1). The asymptotic distribution is unchanged by iterating, but in certain cases (for example, if у is normal) iterating will allow convergence to the maximum likelihood estimator (see Oberhofer and Kmenta, 1974).
In this section we shall consider mainly Model 6 where u follows AR(1) defined in (5.2.1). Most of our results hold also for more general stationary processes, as will be indicated. The covariance matrix of u, denoted 2t, is as given in (5.2.9), and its inverse is given in (5.2.14).