or TO Suppose we have the following VAR model: Yt a10a11 Lyt + a12Lzt + et Zt=a20 + a21 Lyt + a22Lzt + e2t (1 a11 L) Yt = a10+ a12Lzt + e1t (1-a22L) + a20 + a21 Lzt + e2t zt If we use this last equation to solve for zt, it follows that Lzt is Lzt = L(a20+ a21 Lyt + e2t) / (1 - a22 L) so that (1 a11L) yt = a10+ a12L [(a20 + a21 Lyt + e2t) / (1 a22L)] + e1t - Notice that we have transformed the first-order VAR in the {y} and {z} sequences into a second-order stochastic difference equation in the {y} sequence. Explicitly solving for yt, we get Yt = a10 (1a22) + a12a20 + (1 - a22 L) e1t + a12e2t-1 (1 a11L) (1-a22L) - a12a21 L In the same manner, you should be able to demonstrate that the solution for zt is xt = a20 (1a11) + a21a10 + (1 - a11L) e2t + a21e1t1 - (1 a11L) (1a22L) - a12a21 L Using the above, answer the following questions. (a) Find the appropriate second-order stochastic difference equation for yt by considering the following expression: 0.8 0.2 0.2 0.8 = C-289-9 Yt-1 (b) Determine whether the sequence {y} is stationary. ayt+i for ayt+j and delt de2t (c) Discuss the shape of the impulse response function of yt in response to a one-unit shock in et and to a one-unit shock in e2t. That is, what are j 0, 1, 2, 3,...? (d) Denote A = 0.8 0.2 0.2 0.8 What are the eigenvalues of A? (e) Use a computer to calculate A2, A3, A10, and A20. Do your results make sense?