Consider a simple VAR2(1) model for xt = (xt1 xt2) given by xt =

Gxt 1+ t with t =( t1 t2) N(0W)Denotethe(i j) element of G by gij so the evolution equation element-by-element is xt1 = g11xt 11 +g12xt 12 + t1 xt2 = g21xt 11 +g22xt 12 + t2

(a) Write these equations in terms of backshift operators and manipulate them to eliminate xt2 in the equation for xt1; this will yield an equa tion for xt1 in terms of its lagged values and innovation terms-the implied marginal model for xt1 alone.

(b) Show that this leads to (B)xt1 = t where (B) = 1 1B 2B2 and t is a zero-mean, normal process with a lag 1 dependency.

(c) Is xt1 a Markov process?

(d) Give expressions for 1 2, and t in terms of the VAR2(1) model parameters and innovations.

(e) Verify the simple identities 1 = trace(G) and 2 = G