Consider forecasting with the \(A R(1)\) model.

a. Use the forecasting chain rule in equation (8.4) to show

\[y_{T+k}-\widehat{y}_{T+k} \approx \varepsilon_{T+k}+\beta_{1} \varepsilon_{T+k-1}+\cdots+\beta_{1}^{k-1} \varepsilon_{T+1}\]

b. From part (a), show that the approximate variance of the forecast error is \(\sigma^{2} \sum_{l=0}^{k-1} \beta_{1}^{2 l}\).