STA 451 Time Series Assignment

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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

on ?0.

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-

tionary.

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

of ?k.

9. Consider the following MA(2) time series model

Zt = 5 + (1 ? 0.2B)(1 + 0.8B)at

, at ? nid(0, ?2

a

).

(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?

Statistics 451 Assignment 5 Spring 2019 1. Consider the model Zt = θ0 − θ1 at−1 + at . Show that the ACF of this model does not depend on θ0 . 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) stationary. 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 ρ0 = 1, ρ1 = 0.16, and ρk = 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 = θ0 + φ1 Zt−1 + at with −1 < φ1 < 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 ρ1 and ρ2 , 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 γ2 = Cov(Zt , Zt+2 ) = E[(Zt − µz )(Zt+1 − µz )] = 0. 8. The autocorrelation of a stationary time series is defined as ρk = γk /γ0 . In a sense, the autocorrelation ρk and the autocovariance γk provide similar information (about the linear association between Zt and Zt+k ). Briefly explain why we generally prefer to report ρk instead of γk . 9. Consider the following MA(2) time series model Zt = 5 + (1 − 0.2B)(1 + 0.8B)at , at ∼ nid(0, σa2 ). (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? 1

Tutor Answer

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School: Duke University

Attached.

Running Head: TIME SERIES ASSIGNMENT.

Time series assignment.
Name of the course:
Name of the institution:
Name of the student:

1

TIME SERIES ASSIGNMENT.

2

Time series assignment.
Statistics 451

Spring 2019

1. Consider the model Zt = θ0 − θ1at−1 + at. Show that the ACF of this model does
not depend on θ0.
Solution
Zt is a response variable while at−1 is an explanatory variable.
θ0 and θ1 are constants while at is a white noise, therefore the model does not depend
on θ0 since it is a constant.
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.
Solution
In choosing the ARMA (p, q) models, Bayesian Information Criterion (BIC) plays a critical
role.
BIC= −2log(L)+klog(n), where n is the number of data points (realization size) in the time series.
When n is large, the resulting value of log(n) in the BIC formula will be large. This larger value
of log (n) will in turn results to a better value of BIC. Therefore, since better values of BIC
indicates the better models, then it is true to conclude that a realization of size 300 would be
better than a realization of size 75.
3. Briefly state the conditions needed for a stochastic process to be weakly (or
covariance) stationary.

TIME SERIES ASSIGNMENT.
Solution
A stochastic process is said to be weak if its mean and varian...

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Tutor went the extra mile to help me with this essay. Citations were a bit shaky but I appreciated how well he handled APA styles and how ok he was to change them even though I didnt specify. Got a B+ which is believable and acceptable.

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