2. Heteroskedasticity and WLS Suppose we have a regression model with n samples of i.i.d. data yi = Bxi + Wi where E[uilxi] = 0, Var[uilxi] = 5x? (x; > 0), and ui I uj for all i / j (i.e. they are independent each other). (a) Obtain the OLS estimator BOLS for B (b) Obtain the optimal WLS estimator BwLS for B (c) Obtain the variance of the WLS estimator BwLS in part (b) (d) Explain why the WLS estimator is preferred to the OLS estimator in terms of efficiency. In other words, why does the Gauss-Markov theorem hold for the WLS? 3. Moving Average: MA(3) Suppose that yt = Et + 0.2Et-1 - 0.4Et-2 + Et-3 (1) where at is a white noise such that Elet] = 0 for all t, E[e?] = o2 for all t, and EletEv] = 0 for all t # t'. (a) Show that Elyt] = 0 for all t (b) Obtain Varlyt] (c) Obtain Coolyt, yt-1], Covlyt, yt-2], Covlyt, yt-3], and Covlyt, yt-k] with k > 3