Let fytg be an ARMA time series plus a noise de_ned by yt = xt +Wt; where fWtg _ WN(0; _2w ), fxtg is the ARMA(p,q) process satisfying _(B)xt = _(B)"t, f"tg _ WN(0; _2 " ) and E(Wszt) = 0 for all s...

Let fytg be an ARMA time series plus a noise de_ned by yt = xt +Wt;

where fWtg _ WN(0; _2w

), fxtg is the ARMA(p,q) process satisfying _(B)xt =

_(B)"t, f"tg _ WN(0; _2 " ) and E(Wszt) = 0 for all s and t.

(a) Show that fxtg is stationary and _nd its autocovariance function in

terms of _2w

and the autocovariance function of fxtg.

(b) Show that Ut := _(B)yt, is r-correlated where r = max(p; q). Conclude

that fytg is an ARMA(p; r) process.