Statistics 153 Introduction to Time Series Homework 2 problem 1 part b and part c not solved

Question: #1123

Statistics 153 (Introduction to Time Series) Homework 2

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 equa-
tion: 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.