1. Let {Yt} be a doubly infinite sequence of random variables that is stationary with autocovariance function γY . Let Xt = (a + bt)st + Yt, where a and b are real numbers and st is a deterministic seasonal function with period d (i.e., st-d = st for all t) (a) Is {Xt} a stationary process? Why or Why not? (b) Let Ut = ψ(B)Xt where ψ(z) = (1 - z d ) 2 . Show that {Ut} is stationary. (c) Write the autocovariance function of {Ut} in terms of the autocovariance function, γY , of {Yt}.
2. We have seen that P∞ j=0 φ jZt-j is the unique stationary solution to the AR(1) difference equation: Xt - φXt-1 = Zt for |φ| < 1. But there can be many non-stationary solutions. Show that Xt = cφt + P∞ j=0 φ jZt-j is a solution to the difference equation for every real number c. Show that this is non-stationary for c 6= 0.
3. Consider the AR(2) model: φ(B)Xt = Zt where φ(z) = 1 - φ1z - φ2z 2 and {Zt} is white noise. Show that there exists a unique causal stationary solution if and only if the pair (φ1, φ2) satisfies all of the following three inequalities: φ2 + φ1 < 1 φ2 - φ1 < 1 |φ2| < 1.
4. Consider the AR(2) model: Xt - Xt-1 + 0.5Xt-2 = Zt where {Zt} is white noise. Show that there exists a unique causal stationary solution. Find the autocorrelation function.
5. Let {Yt} be a doubly infinite sequence of random variables that is stationary. Let Xt = β0 + β1t + · · · + βqt q + Yt where β0, . . . , βq are real numbers with βq 6= 0. (a) Show that (I - B) kYt is stationary for every k ≥ 1.(b) Show that (I - B) kXt is not stationary for k < q and that it is stationary for k ≥ q
6. Let {Yt} be a doubly infinite mean zero sequence of random variables that is stationary. Define Xt = Yt - 0.4Yt-1 and Wt = Yt - 2.5Yt-1. (a) Express the autocovariance functions of {Xt} and {Wt} in terms of the autocovariance function of {Yt}. (b) Show that {Xt} and {Wt} have the same autocorrelation functions.