Two-way fixed effects regression

(a) Prove that the Within estimator \(\widetilde{\beta}=\left(X^{\prime} Q Xight)^{-1} X^{\prime} Q y\) with \(Q\) defined in (3.3) can be obtained from OLS on the panel regression model (2.3) with disturbances defined in (3.2). Hint: Use the Frisch-Waugh-Lovell theorem of Davidson and MacKinnon (1993, p. 19), and also the generalized inverse matrix result given in problem 9.6. See the complete solution in Chap. 3 of the companion, Baltagi (2009).

(b) Within two-way is equivalent to two Withins one-way. This is based on problem 98.5.2 in Econometric Theory by Baltagi (1998). Show that the Within two-way estimator of \(\beta\) can be obtained by applying two Within (one-way) transformations. The first is the Within transformation ignoring the time effects followed by the Within transformation ignoring the individual effects. Show that the order of these two Within (one-way) transformations is unimportant. Give an intuitive explanation for this result. See solution 98.5.2 in Econometric Theory by Li (1999).

y=aINT + XB+u =Z8+u (2.3)