# STAT 443 Lab 3: Properties of MA and AR Processes solution

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## Description

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• The aim of this lab is to explore properties of MA and AR processes that were discussed
in lectures, including the behaviour of their autocorrelation functions and conditions of
invertibility and stationarity.

• You will use the R command arima.sim() to simulate realizations from MA and AR time

• The lab must be completed in R Markdown. Create a pdf file or html file (via ’knit to
pdf’ or ’knit to html’) and use it as your lab submission.

1. Let {Zt}t∈N be a white noise process with mean zero and variance 0.2. Use arima.sim()
function to simulate 500 observations from the process
Xt = Zt − 4.25Zt−1 + 5.75Zt−2 − 1.80Zt−3.

(a) Create a time series plot for the simulated data. How should the autocorrelation
function behave for the process {Xt}t∈N?

(b) Use the command acf to plot the sample acf, and comment on its behaviour. Does
the sample acf look like you would expect? Repeat the simulation a few times to see
what is happening.

(c) The command ARMAacf provides the autocorrelation function for an ARMA model
(that is, a model that is an MA, an AR, or a mixture of the two). Learn about this
function, and then find the acf for {Xt}t∈N.

(d) Explain the behaviour of the sample acf here.

2. Let {Zt}t∈N be a white noise process with mean zero and variance 0.9.
(a) Use arima.sim() to simulate 1000 observations from the process
Xt = Zt + 5Zt−1.

Create a plot for the simulated data. Plot the sample acf, and comment on its behaviour.
(b) Use arima.sim() to simulate 1000 observations from the process
Xt = Zt + 0.2Zt−1.

Plot the sample acf, and comment on its behaviour.

(c) Repeat parts (a) and (b) several times. What do you notice? Explain what is happening.

3. Use arima.sim() to generate 1000 observations from the model
Xt = αXt−1 + Zt
for α = 0.3,
where {Zt}t∈N is a white noise process with Zt ∼ N (0, 0.2
2
) for all t.

(a) Create a time series plot of your simulated data. Plot the sample acf, and comment
on its behaviour.

(b) How does the acf of the process change as α → 1? Explore this in R, and explain what
you observe.

(c) What happens to the sample acf if the parameter α changes sign? Experiment with
this, changing the sign of α in your simulations. Explain what is happening.