5. Generate yt and xt as two independent I(1) processes, i.e., yt = yt?1 + ut, xt = xt?1 + vt, ut and the vt as two independent sequences so that yt and xt are independent with each other. Let ?? = (X?X)?1X?Y be the OLS estimator of ? based on yt = xt?+errort, and t?? = ??/sd(??) (the usual t-statistic for testing ? = 0).

5. Generate yt and xt as two independent I(1) processes, i.e., yt = yt-1 + Ut, at = at-1 + Ut, Ut and the ut as two independent sequences so that yt and at are independent with each other. Let 3 = (X'X)-1X'Y be the OLS estimator of B based on yt = XtB +errort, and to = B/sd(B) (the usual t-statistic for testing B = 0). (a) Using simulations to examine the asymptotic behavior of n-1/2t. Discuss whether your simulation results support (2.51) or not. (b) Discuss what is the limit of It| as n - co. 2 (c) If we do not know that yt and at are non-stationary and mistakenly thought that yt and at were stationary. We estimate S based on yt = atB + ut. We test Ho: B = 0. What conclusion do we likely to get? Can we reach the correct conclusion that yt and at are uncorrelated? Hint: For (a) you can compute the mean, variance and quantiles (say, 25-th, 50-th and 75- th percentiles) of n /2to by simulations. Show that these quantities suggest that n-1/2t; converges to well defined distribution, which implies that It| diverges at the rate vn (explain why)