Suppose \(Y_{t}=\beta_{0}+u_{t}\), where \(u_{t}\) follows a stationary stationary \(\operatorname{AR}(1)\) \(u_{t}=\phi_{1} u_{t-1}+\widetilde{u}_{t}\) with \(\widetilde{u}_{t}\) i.i.d. with mean 0 and variance \(\sigma_{\tilde{u}}^{2}\) and \(\left|\phi_{1}\right|<1\).

a. Show that \(\beta_{0}=\mu_{Y}=E\left(Y_{t}\right)\).

b. Let \(\bar{Y}_{1: T}=\frac{1}{T} \sum_{t=1}^{T} Y_{t}\) denote the sample mean of \(Y_{t}\) using observations from \(t=1\) through \(t=T\). Show that the OLS estimator of \(\beta_{0}\) is \(\hat{\beta}_{0}=\bar{Y}_{1: T}\).

c. Show that \(\operatorname{var}\left[\sqrt{T}\left(\bar{Y}_{1: T}-\mu_{Y}\right)\right] \rightarrow \sigma_{\tilde{u}}^{2} /\left(1-\phi_{1}\right)^{2}\).

d. Assume that \(\bar{Y}_{1: T}\) is approximately normally distributed with mean \(\mu_{Y}\) and variance \(\sigma_{\tilde{u}}^{2} /\left[T\left(1-\phi_{1}\right)^{2}\right]\). Suppose \(T=200, \sigma_{\tilde{u}}^{2}=7.9, \phi_{1}=0.3\), and the sample mean of \(Y_{t}\) is \(\bar{Y}_{1: T}=2.8\). Construct a \(95 \%\) confidence interval for \(\mu_{Y}\).

e. Suppose you are interested in the average value of \(Y_{t}\) from \(t=T+1\) through \(T+h\); that is, \(\bar{Y}_{T+1: T+h}=\frac{1}{h} \sum_{t=T+1}^{T+h} Y_{t}\), where \(h\) is a large number. Show that \(\bar{Y}_{T+1: T+h}\) has mean \(\mu_{Y}\) and variance \(\sigma_{\tilde{u}}^{2} /\left[h\left(1-\phi_{1}\right)^{2}\right]\).

f. Assume that \(\bar{Y}_{T+1: T+h}\) is approximately normally distributed. Suppose \(h=100, \sigma_{\tilde{u}}^{2}=7.9, \phi_{1}=0.3\), and \(\mu_{Y}=2.9\). Construct a \(95 \%\) forecast interval for \(\bar{Y}_{T+1: T+h}\).

g. Let \(r=h / T\). Show that \(\operatorname{var}\left[\sqrt{T}\left(\bar{Y}_{T+1: T+h}-\bar{Y}_{1: T}\right)\right] \rightarrow\left(1+r^{-1}\right) \frac{\sigma_{u}^{2}}{\left(1-\phi_{1}\right)^{2}}\), where \(r\) is held fixed as \(T \rightarrow \infty\).

h. Show that \(\bar{Y}_{T+1: T+h}-\bar{Y}_{1: T}\) has mean 0 and variance \(\left(\frac{1}{T}+\frac{1}{h}\right) \frac{\sigma_{\tilde{u}}^{2}}{\left(1-\phi_{1}\right)^{2}}\).

i. Use the result in (i) to show that the forecast interval \(\bar{Y}_{1: T} \pm 1.96 \sqrt{\left(\frac{1}{T}+\frac{1}{h}\right) \frac{\sigma_{u}^{2}}{\left(1-\phi_{1}\right)^{2}}}\) will contain the value of \(\bar{Y}_{T+1: T+h}\) with probability \(95 \%\), approximately, when \(T\) and \(h\) are large. (Assume that \(\bar{Y}_{T+1: T+h}-\bar{Y}_{1: T}\) is approximately normally distributed.)

j. Suppose \(T=200, h=100, \sigma_{\tilde{u}}^{2}=7.9, \phi_{1}=0.3\), and \(\bar{Y}_{1: T}=2.8\). Construct a \(95 \%\) forecast interval for \(\bar{Y}_{T+1: T+h}\).