Throughout this question, we maintain the following assumptions. (a) For every t 2 0, Y, and Ut+1 are independent random variables. This implics that AY, and Utti are also independent for every t 2 0. (b) The covariance matrix of Yo is Wo and the covariance matrix of Ut+1 is V for every 120. 4. [10 marks] (Challenging) Consider the following process: Ct+1 = aidt + a2xt-1 + 03 t-2 + Et+1, for t = 2, 3, 4, .... (4) Here at and et are random variables for t 2 0 and et+1 is independent of (at, "t-1, "t-2) for every t 2 2. As Eq. (4) looks very different from Eq. (2), the theory from Parts 1 to 3 of the question do not immediately apply to the process in Eq. (4). Eq. (4) may be rewritten as the following system of equations: "t+1 = allt + andt-1+ 032t-2 + Et+1; It = at + 0x-1+ 021-2 + 0; "-1 = Out +x-1+ 0*1-2+0. The first equation is Eq. (4), and the last two equations hold trivially for every t > 2. Using this trick, convert Eq. (4) to the form in Eq. (2) with appropriately chosen A, Ye and Ut+1. Finally, find the characteristic polynomial of your A in terms of a1, a2 and a3