Author:
Christine Farr

Let {Yt } be a doubly inﬁnite sequence of random variables that is stationary with autocovariancefunction γY . LetXt = (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∞j =0φj Zt−j is the unique stationary solution to the AR(1) diﬀerence equa-tion: Xt − φXt−1 = Zt for |φ| < 1. But there can be many non-stationary solutions. Show thatXt = cφt +∞j =0φj Zt−j is a solution to the diﬀerence equation for every real number c. Showthat this is non-stationary for c = 0.3. Consider the AR(2) model: φ(B )Xt = Zt where φ(z ) = 1 − φ1 z − φ2 z 2 and {Zt } is white noise.Show that there exists a unique causal stationary solution if and only if the pair (φ1 , φ2 ) satisﬁesall of the following three inequalities:φ2 + φ1 < 1φ2 − φ1 < 1|φ2 | < 1.

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