Generalized Least Squares Theory

1. If rank (ЩХ) = K, fiis uniquely determined by (6.1.2).

2. Farebrother (1980) presented the relevant tables for the case in which there is no intercept.

3. Breusch (1978) and Godfrey (1978) showed that Durbin’s test is identical to Rao’s score test (see Section 4.3.1). See also Breusch and Pagan (1980) for the Lagrange multiplier test closely related to Durbin’s test.

4. In the special case in which N=2 and fi changes with і (so that we must estimate both /?, and fi2), statistic (6.3.9) is a simple transformation of the F statistic (1.5.44). In this case (1.5.44) is preferred because the distribution given there is exact.

5. It is not essential to use the unbiased estimators a) here. If 6} are used, the distribution of у is only trivially modified.

6. In either FGLS or MLE we can replace by Д without affecting the asymptotic distribution.

7. Hildreth and Houck suggested another estimator (Z’NfZ)r1Z/fl2, which is the instrumental variables estimator applied to (6.3.23) using Z as the instrumental vari­ables. Using inequality (6.5.29), we can show that this estimator is also asymptotically less efficient than a3. See Hsiao (1973) for an interesting derivation of the two estima­tors of Hildreth and Houck from the MINQUE principle of Rao (1970). Froehlich (1973) suggested FGLS applied to (6.5.23). By the same inequality, the estimator can be shown to be asymptotically less efficient than d3. Froehlich reported a Monte Carlo study that compared the FGLS subject to the nonnegativity of the variances with several other estimators. A further Monte Carlo study has been reported by Dent and Hildreth (1977).

8. The symbol 1 here denotes an ЛТ-vector of ones. The subscripts will be omitted whenever the size of 1 is obvious from the context. The same is true for the identity matrix.

9. For the consistency of fitv we need both N and Г to go to °°. For further discus­sion, see Anderson and Hsiao (1981, 1982).

10. The presence of the vector f in a variance term would cause a problem in the derivation of the asymptotic results if T were allowed to go to but in the Lillaid – Weiss model, as in most of econometric panel-data studies, T is small and N is large.

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