Simulate an AR$(1)$ process with $=0\mathrm{.}8$ and $\backslash$mu $=100.$ 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 $\backslash$mu $.$ Check if these estimates are same as CSS or Yule$-$Walker estimated.

$($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 actual values fall within the forecast limits$?$ in r code