OLS and GLS are equivalent for the Within transformed model.

(a) Using generalized inverse, show that OLS or GLS on (2.6) yields \(\widetilde{\beta}\), the Within estimator given in (2.7).

(b) Show that (2.6) satisfies the necessary and sufficient condition for OLS to be equivalent to GLS (see Baltagi (1989)). Hint: show that \(\operatorname{var}(Q v)=\sigma_{v}^{2} Q\) which is positive semi-definite and then use the fact that \(Q\) is idempotent and is its own generalized inverse.

Transcribed Image Text:

Qy = QX + Qv = + (2.6)