Let xt be an ARMA(p; q) process. Show that _2 is the one-step prediction error variance and that (0) is the in_nite steps ahead prediction error variance. Furthermore, verify that (0) 2. A...

Let xt be an ARMA(p; q) process. Show that _2 is the one-step prediction

error variance and that (0) is the in_nite steps ahead prediction error

variance. Furthermore, verify that (0) 2.

A quarterly economic time series was modeled by rzt = 0:5+(1􀀀B +

0:5B2)at with _2 a = 0:04.

(a) Given z48 = 130 , a47 = 􀀀0:3 , a48 = 0:2 , calculate and draw the

predictions ^ Z48(l) for l = 1; 2 12

(b) Include 80% prediction bands in the graph.