Question 2 Consider the following panel data model: yit = B'xit + Hi + ft + eit, i = 1,..., n; t = 1, ..., I, where yit is the scalar outcome variable, it is the K x 1 vector of regressors, ui and ft are the unobserved individual effects and time effects. Define the unit- specific average over time as ai = + E -1: It=1 Tit. Define the time-specific average over the cross section as It = n Lizi it. The overall average is defined as I = nT Li=1 Et=1%it. Define the grand-difference as bit = it - Ti - It + 5. (a) Show that the fixed-effects estimator BFE can be obtained by OLS regression of yit on it. (b) Show that the fixed-effects estimator BFE can be obtained by OLS regression of yit on 1, it, Ti, It. (c) Suppose we observe a time-variant regressor z; and a common time effect wt. Show that in the OLS regression of yit on 1, it, Ti, It, Zi, Wt, the OLS estimates for the coefficients on z; and wt are both zero. (d) Consider a policy intervention that occurred in period To. Let Di be a dummy variable indicating whether unit i is subject to the policy intervention. Let Ft denote the time dummy such that Ft = 1 if t > To. Define the interaction term Xit = Di X Ft. Let BFE denote the fixed-effects estimator in the regression yit = BXit + Hit f+ + eit. Let PDD denote the difference-in-differences estimator in the regression yit = BotPXittyDi+ F+ + wit. Show that BFE = PDD