Question $3.$ Suppose that the data follows ARMA$(1,1)$ model

$x_t-=(x_{t-1}-)+lo{n}_t-lo{n}_{t-1}$

where $lo{n}_t$ are independent identical distributed normal random variables with mean $0$ and variance ${}^2.$

The model is fitted using a dataset and the result is:

Coefficients:

ar$1$ ma$1$ intercept

$0\mathrm{.}9078,0\mathrm{.}2311,3\mathrm{.}9552$

s$.$e$.,0\mathrm{.}0152,0\mathrm{.}0317,0\mathrm{.}0235$

sigma$^2$ estimated as $0\mathrm{.}002592$

$($a$)$ Write down the equation for the fitted ARMA$(1,1)$ model.

$($b$)$ Assume the following MA$($ model is another representation of the above ARMA$(1,1)$

model.

$x_t-=lo{n}_t+{}_{1}lo{n}_{t-1}+{}_{2}lo{n}_{t-2}+{}_{3}lo{n}_{t-3}+{}_{4}lo{n}_{t-4}+$cdots

Using the theorem of $MA()$ representation on page $7$ of the lecture note, express ${}_k$ in

terms of $a{b}^k$ for some constants a and $b.$ Hence, calculate the value of ${}_1,{}_2$ and ${}_3.$

$($c$)$ Suppose $x_{n-2}=3\mathrm{.}956,x_{n-1}=3\mathrm{.}941,x_n=3\mathrm{.}902$ and assume $lo{n}_{n-2}=0\mathrm{.}0948.$ Forecast

$x_{n+1},x_{n+2},x_{n+3}$ by giving a prediction interval of $95\%.$

$($Hint: when you calculate the conditional expectation, you may use the original form of

ARMA$(1,1).$ When you calculate the conditional variance, you have to use the $MA()$ form

in $($b$).)$

Question $4.$ The time series data of U$.$S$.$ retail sales and disposable income are stored in

data$\_$asg$\_1.$csv$.$

$($a$)$ Using RStudio, plot the sample autocorrelation function for the retail sales data from time lag

$0$ to lag $20.$ Hence, suggest whether $AR(1),MA(1)$ or ARMA$(1,1)$ is a good choice to fit the

data.

Fit the model to the AR$(1)$ process using the moment matching method. Show your working

step and write down the fitted equation.

$($b$)$ Consider the causal regression model with time lag:

$RE{T}_t={}_0+{}_{1}DI{S}_{t-1}+$ error

where $RE{T}_t$ is the retail sales at year $t,DI{S}_{t-1}$ is the disposable income at year $t-1.$

Using R$,$ fit the model using the data from $1970$ to $1990.$ Write down the equation and make a

one$-$year ahead prediction for the retail sales at year $1991.$