Consider the stationary \(\operatorname{AR}(1)\) model \(Y_{t}=\beta_{0}+\beta_{1} Y_{t-1}+u_{t}\), where \(u_{t}\) is i.i.d. with mean 0 and variance \(\sigma_{u}^{2}\). The model is estimated using data from time periods \(t=1\) through \(t=T\), yielding the OLS estimators \(\hat{\beta}_{0}\) and \(\hat{\beta}_{1}\). You are interested in forecasting the value of \(Y\) at time \(T+1-\) that is, \(Y_{T+1}\). Denote the forecast by \(\hat{Y}_{T+1 \mid T}=\hat{\beta}_{0}+\hat{\beta}_{1} Y_{T}\).

a. Show that the forecast error is \(Y_{T+1}-\hat{Y}_{T+1 \mid T}=u_{T+1}-\left[\left(\hat{\beta}_{0}-\beta_{0}\right)+\right.\) \(\left.\left(\hat{\beta}_{1}-\beta_{1}\right) Y_{T}\right]\).

b. Show that \(u_{T+1}\) is independent of \(Y_{T}\).

c. Show that \(u_{T+1}\) is independent of \(\hat{\beta}_{0}\) and \(\hat{\beta}_{1}\)

d. Show that \(\operatorname{var}\left(Y_{T+1 \mid T}-\hat{Y}_{T+1 \mid T}\right)=\sigma_{u}^{2}+\operatorname{var}\left[\left(\hat{\beta}_{0}-\beta_{0}\right)+\left(\hat{\beta}_{1}-\beta_{1}\right) Y_{T}\right]\).