Consider the stationary AR1 process:
Z = a1Zt-1 + ε
where |a1| < 1 and ε is white noise.
(a) Determine the mean of Zt. [2]
(b) Show that Cov(Zt, Zt-k) = a^k * Cov(Zt, Zt) for k = 0, 1, 2, where Cov(Y, Z) represents the autocovariance of Y and Cov(Yt, Zt) represents the autocovariance of Zt. [3]
(c) In the previous question, you had to show that Cov(Yt, Yt-k) = -a^(k-1) * Cov(Yt, Zt) + 2 * Cov(Yt, Yt-k+1) - Cov(Yt, Yt-k+2), where Cov(Y, Z) represents the autocovariance of Y and Cov(Yt, Zt) represents the autocovariance of the stationary AR1 process {Z}. Using this information, answer the following questions. [3]
(d) Suppose that Yt = β0 + β1t + Zt and β0 = 0. Is {Yt} weakly stationary? [3]
(e) What do your previous two answers imply about differencing a trend-stationary series versus differencing a random walk with drift? [2]