1. Let {} be i.i.d., with mean 0 and variance o2. Consider the following stochastic process {X} derived from {at}: Xt = Bo + B1Xt-1 + Et, for t = 1,2,... where Bo and 31 are constants, and Xo is independent of Et, for all t1. (a) Show that, for any t 1 and s > 0, t-1 t-1 Xs+t = + X. + s+t-i. i=0 i=0 (1) i. Express X, in terms of ,..., Et. [Hint: use the result in part (a).] ii. Hence, derive the mean and variance of X. Is Xt stationary? (2) [If you prefer, you may derive this only for the less notationally burdensome case where s = 0.] Taking s = 0, what does this imply about the dependence between {Et+1, Et+2,...} and X? (b) Suppose [31] < 1, and that X has mean Bo/(1-3) and variance o2/(1 ). Using (2), with appropriate choices for s and/or t, compute EXt, var (Xt) and cov (Xs, Xs+t) for t Z = {...,-1,0,1,}. [Hint: use the fact that - p = (1 - p)/(1 p), if p1.] Is {Xt} weakly stationary? i=0 (c) Suppose 31 = 1, and that Xo = 0. Recall that the optimal (MSFE-minimising) h-step ahead forecast is given by Xs+h|s := E[Xs+h | Xs, Xs1, . .]. (d) Derive Xs+hs for the model in (1). [Hint: take t = hin (2).] (e) Compute MSFE(Xs+h|s) = E(Xs+h - Xs+hs), and discuss how this depends on the forecast horizon when: (i) |B| < 1, and (ii) B = 1.