Begin with the \(A R(1)\) model as in equation (8.1).

a. Take variances of each side of equation (8.1) to show that \(\sigma_{y}^{2}\left(1-\beta_{1}^{2}\right)=\) \(\sigma^{2}\), where \(\sigma_{y}^{2}=\operatorname{Var} y_{t}\) and \(\sigma^{2}=\operatorname{Var} \varepsilon_{t}\).

b. Show that \(\operatorname{Cov}\left(y_{t}, y_{t-1}\right)=\beta_{1} \sigma_{y}^{2}\).

c. Show that \(\operatorname{Cov}\left(y_{t}, y_{t-k}\right)=\beta_{1}^{k} \sigma_{y}^{2}\).

d. Use part (c) to establish equation (8.2).

Transcribed Image Text:

y=Bo+Biy-1+1, t = 2,..., T. (8.1)