$9\mathrm{.}9$ Simulate an AR$(1)$ process with $=0\mathrm{.}8$ and $=100.$ Simulate $48$ values but set aside the last $8$ values to compare forecasts to actual values.

$($a$)$ Using the first $40$ values of the series, find the values for the maximum likelihood estimates of $$ and $.$

$($b$)$ Using the estimated model, forecast the next eight values of the series. Plot the series together with the eight forecasts. Place a horizontal line at the estimate of the process mean.

$($c$)$ Compare the eight forecasts with the actual values that you set aside.

$($d$)$ Plot the forecasts together with $95\%$ forecast limits$.$ Do the actual values fall within the forecast limits$?$

$($e$)$ Repeat parts $($a$)$ through $($d$)$ with a new simulated series using the same values of the parameters and the same sample size.