a. Let (I_{t-1}=left{e_{t-1}, e_{t-2}, ldots ight}). Use the law of iterated iterations to show that (Eleft(e_{t} mid

a. Let \(I_{t-1}=\left\{e_{t-1}, e_{t-2}, \ldots\right\}\). Use the law of iterated iterations to show that \(E\left(e_{t} \mid I_{t-1}\right)=0\) implies \(E\left(e_{t}\right)=0\).

b. Consider the variance model \(h_{t}=E\left(e_{t}^{2} \mid I_{t-1}\right)=\alpha_{0}+\alpha_{1} e_{t-1}^{2}\). Use the law of iterated iterations to show that, for \(0<\alpha_{1}<1, E\left(e_{t}^{2}\right)=\alpha_{0} /\left(1-\alpha_{1}\right)\).

c. Consider the variance model \(h_{t}=E\left(e_{t}^{2} \mid I_{t-1}\right)=\delta+\alpha_{1} e_{t-1}^{2}+\beta_{1} h_{t-1}\). Use the law of iterated iterations to show that for \(0<\alpha_{1}+\beta_{1}<1, E\left(e_{t}^{2}\right)=\delta /\left(1-\alpha_{1}-\beta_{1}\right)\).