For the following simulation study, use the last $4$ digits of your student ID as the

set.seed. For example, if your student ID is $1234567,$ then use set.seed$(4567),$ and if

it is $1230567,$ use set.seed$(567).$

In this simulation study, we want to check the asymptotic theory for the Yule$$Walker

estimator for an AR$(2)$ series with $1=1\mathrm{.}5$ and $2=0\mathrm{.}75$ by simulating $1000$ series of

length $100.$ Let the noise process Zt be a Gaussian white noise process $($with mean zero

and variance $\backslash$sigma

$2$

Z $=4).$ You can refer to Example $3\mathrm{.}4$ and Lemma $3\mathrm{.}1$ from the lecture

notes.

$($a$)$ Simulate $1000$ time series, each with $100$ observations, as mentioned above. $[$Hint:

In each iteration, generate a random normal sample of size $100$ and then use it as

the innov process in the arima.sim function.$]$

$($b$)$ In each iteration, compute the Yule$-$Walker estimator for the parameters of the

model $1$ and $2.$ Add the estimated values from each iteration to the vectors $1$

and $2$

$.[$Hint: use the ar$.$yw function, set the order.max to $2$ and dmean to

FALSE. Keep the other parameters in the function to default.$]$

$($c$)$ Compute the mean, variance, and covariance between the vectors $1$

and $2$

$.$

$($d$)$ Compare these values with the theoretical results from Lemma $3\mathrm{.}1$ and comment.

$($No need to check the normality assumption$)$