## Description

Given a time series, we can fit possible ARIMA models in R using the arima command. Look

at the help page on this function before attempting the following activities.

Suppose {Zt}t∈N is a white noise with mean zero and variance 0.8. Consider stochastic

process {Xt}t∈N with

Xt = 0.8 Xt−1 −

1

3

Xt−2 +

0.6

√

3

Xt−3 + Zt

. (1)

1. Name the process defined in equation (1), specifying its order.

2. Explain how to recognize this process based on an observed time series and how to determine its order.

3. Use the command set.seed(123456) to set the random seed for reproducibility and then

use function arima.sim() to generate 500 observations from the model in (1). Plot the

simulated time series.

4. Plot the sample autocorrelation function (acf). Comment on the behaviour of the sample

acf, and explain whether it appears as you would expect given the model.

5. Now plot the sample partial autocorrelation function, using command pacf(). Comment

on the behaviour of the sample pacf, and explain whether it appears as you would expect

given the model.

6. Use the arima function to fit an ARMA model to the simulated time series. You should

specify the order (which determines the class of models to be fitted), decide whether a

non-zero mean should be included, and use the default estimation method. Provide your

parameter estimates.

7. Now re-fit the model as above, but choose conditional least squares as the estimation

method. Provide your parameter estimates, and compare them with those from question 6.