OLS and GLS are equivalent for the Within transformed model. (a) Using generalized inverse, show that OLS

Question:

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)). 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.