1. Let fYtg 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., std = st for all t)

(a) Is fXtg a stationary process? Why or Why not?
(b) Let Ut = (B)Xt where (z) = (1 zd)2. Show that fUtg is stationary.
(c) Write the autocovariance function of fUtg in terms of the autocovariance function, Y , of fYtg.

2. We have seen that P1 j=0 jZtj is the unique stationary solution to the AR(1) dierence equation: Xt Xt1 = Zt for jj < 1. But there can be many non-stationary solutions. Show that Xt = ct + P1 j=0 jZtj 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 2z2 and fZtg 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 2 1 < 1 j2j < 1:

4. Consider the AR(2) model: Xt Xt1 + 0:5Xt2 = Zt where fZtg is white noise. Show that there exists a unique causal stationary solution. Find the autocorrelation function.

5. Consider the ARMA(2, 1) model: XtXt1+0:5Xt2 = Zt+0:5Zt1 where fZtg is white noise. Show that there exists a unique causal stationary solution. Find the autocorrelation function.

6. Let fYtg be a doubly innite sequence of random variables that is stationary. Let

Xt = 0 + 1t + + qtq + Yt
where 0; : : : ; q are real numbers with q 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.

7. Let fYtg be a doubly innite mean zero sequence of random variables that is stationary. Dene Xt = Yt 0:4Yt1 and Wt = Yt 2:5Yt1.

(a) Express the autocovariance functions of fXtg and fWtg in terms of the autocovariance function of fYtg.

(b) Show that fXtg and fWtg have the same autocorrelation functions.