Question: 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
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 \(\sigma^{2} \sum_{l=0}^{k-1} \beta_{1}^{2 l}\).
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