Statistics 153 (Introduction to Time Series) Homework 2 Due on 27 February, 2013 16 February, 2013 1. Let {Yt } be a doubly innite sequence of random variables that is stationary with autocovariance function Y . 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 = (B)Xt where z d )2 . Show that {Ut } is stationary. (z) = (1 (c) Write the autocovariance function of {Ut } in terms of the autocovariance function, {Yt }. 2. We have seen that tion: Xt Xt = c t P1 j=0 j Zt j Y , of is the unique stationary solution to the AR(1) dierence equa- Xt 1 = Zt for | | < 1. But there can be many non-stationary solutions. Show that P1 j + j=0 Zt j is a solution to the dierence equation for every real number c. Show that this is non-stationary for c 6= 0. 3. Consider the AR(2) model: (B)Xt = Zt where (z) = 1 1z 2z 2 and {Zt } is white noise. Show that there exists a unique causal stationary solution if and only if the pair ( 1, 2) satises all of the following three inequalities: 2 + 1 <1 4. Consider the AR(2) model: Xt 2 Xt 1 + 0.5Xt 2 | <1 1 2| < 1. = 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 innite sequence of random variables that is stationary. Let Xt = where 0, . . . , q are real numbers with (a) Show that (I 0 + + + 1t q 6= 0. B)k Yt is stationary for every k 1 qt 1. q + Yt (b) Show that (I B)k Xt is not stationary for k < q and that it is stationary for k q. 6. Let {Yt } be a doubly innite mean zero sequence of random variables that is stationary. Dene 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. 2