Statistics 451 Assignment 5 Spring 2019
1. Consider the model Zt = ?0 ? ?1at?1 + at
. Show that the ACF of this model does not depend
2. Briefly explain why a realization of size 300 would be better than a realization of size 75 when
trying to identify p and q from an ARMA(p, q) model.
3. Briefly state the conditions needed for a stochastic process to be weakly (or covariance) sta-
4. Any MA(q) model, with q being a finite integer, is stationary. Pointing to results from Module
4, briefly explain how you could substantiate this statement.
5. There is an invertible time series process Zt that has the ACF function ?0 = 1, ?1 = 0.16, and
?k = 0 for all values of k > 1.
(a) Compute the first three PACF values for Zt? (See the hints in slide 4–16 and you can
use the RTseries function show.true.acfpacf to check your work.)
(b) It is possible to identify the values of p and q from an ARMA(p, q) model from the above
information. What are the values of p and q?
(c) For this model, is Zt stationary? Why or why not?
6. Consider the AR(1) model Zt = ?0 + ?1Zt?1 + at with ?1 < ?1 < 1.
(a) Give an expression for the root of the model-defining polynomial for this model.
(b) Is the model stationary or not? Explain why or why not.
(c) Derive an expression for the mean of Zt
(d) Derive an expression for the variance of Zt
(e) Derive expressions for ?1 and ?2, the first two ACF values for this model (note that you
first have to obtain expressions for the corresponding covariances).
7. For a MA(1) model, show why ?2 = Cov(Zt
, Zt+2) = E[(Zt ? µz)(Zt+1 ? µz)] = 0.
8. The autocorrelation of a stationary time series is defined as ?k = ?k/?0. In a sense, the
autocorrelation ?k and the autocovariance ?k provide similar information (about the linear
association between Zt and Zt+k). Briefly explain why we generally prefer to report ?k instead
9. Consider the following MA(2) time series model
Zt = 5 + (1 ? 0.2B)(1 + 0.8B)at
, at ? nid(0, ?2
(a) Find the root(s) of the model-defining polynomial (hint: there is an easy way to do this).
(b) Is the model for Zt
invertible or not? Why or why not?
(c) Is the model for Zt stationary or not? Why or why not?