The ARCH model is sometimes presented in the following multiplicative form:

\[\begin{aligned}& y_{t}=\beta_{0}+e_{t} \\& e_{t}=z_{t} \sqrt{h_{t}}, z_{t} \sim N(0,1) \\& h_{t}=\alpha_{0}+\alpha_{1} e_{t-1}^{2}, \quad \alpha_{0}>0, \quad 0 \leq \alpha_{1}<1\end{aligned}\]

This form describes the distribution of the standardized residuals \(e_{t} / \sqrt{h_{t}}\) as standard normal \(z_{t}\). However, the properties of \(e_{t}\) are not altered.

a. Show that the conditional mean \(E\left(e_{t} \mid I_{t-1}\right)=0\).

b. Show that the conditional variance \(E\left(e_{t}^{2} \mid I_{t-1}\right)=h_{t}\).

c. Show that \(e_{t} \mid I_{t-1} \sim N\left(0, h_{t}\right)\).