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1. An alternative way to deal with a higher order difference equation is to rewrite it as a rst order vector equation. For example, consider the second order equation 9: = (lm1 + (Milt2 + 5:: (51462 1 0 and dene :3; = (gt, yt_1)', v; = (61;, 0)' and F = a) Write down a rst order linear difference equation for the vector .13; so that the rst equation gives back the original higher order equation and the second equation is an identity. b) Use backward substitution to express 3:; in terms of some initial value altk, k > 0 and subsequent shocks v_k+1,...,vt. Describe how you would nd Byt iet = Byt/Bet_j for j Z 0. (Do not try to obtain a closed form expression. Just tell me which element of which matrix would you look at.) c) For Q51 = 0.9 and (:52 = 0.2, compute the numerical value 8yt+3faet using the approach in part b). d) Let 2:1 and 2:2 denote the characteristic roots of the original difference equation, and let A1 and A2 be the eigenvalues of the matrix F. Show that A1 = 1/21 and A2 = 1/z2