Begin with the (A R(1)) model as in equation (8.1). a. Take variances of each side of
Question:
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).
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Regression Modeling With Actuarial And Financial Applications
ISBN: 9780521135962
1st Edition
Authors: Edward W. Frees
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