Proposition 2.2.1 Let { Yt} be a stationary time series with mean 0 and covariance function Vy. If _ju; 2 0juk Ty ( h t k - j ). j=-00 k=-00 If { Y+} ~ WN(0, 02), then Yx (h) = > wij-no2. j=-00Let {Xt} be the AR(1) process Xt = QXt-1+ Zt where {Zt} ~ WN(0, o?). Let {Yt} be Yt = Xt + Wt where {Wt} ~ WN(0, 2). Assume E(W, Zt) = 0 for all s and t. 5. Show that {Yt} is stationary and find its ACVF. Define another times series { Xt} Ut = Yt - dyt-1. Show that {Ut} is 1-correlated, and hence, is MA(1) process. (Hint) Use Proposition 2.1.1. From the previous result, conclude that {Yt } is an ARMA(1,1) process. Express {Yt} as a form of ARMA(1,1), that is, fill ( ) in the following: Yt - QYt-1 = ( )