The equations of an ARCH-in-mean model are shown below:

\[\begin{aligned}& y_{t}=\beta_{0}+\theta h_{t}+e_{t} \\& e_{t} \mid I_{t-1} \sim N\left(0, h_{t}\right) \\& h_{t}=\delta+\alpha_{1} e_{t-1}^{2}, \quad \delta>0, \quad 0 \leq \alpha_{1}<1\end{aligned}\]

Let \(y_{t}\) represent the return from a financial asset and let \(e_{t}\) represent "news" in the financial market. Now use the third equation to substitute out \(h_{t}\) in the first equation, to express the return as

\[y_{t}=\beta_{0}+\theta\left(\delta+\alpha_{1} e_{t-1}^{2}\right)+e_{t}\]

a. If \(\theta\) is zero, what is \(E_{t}\left(y_{t+1}\right)\), the conditional mean of \(y_{t+1}\) ? In other words, what do you expect next period's return to be, given information today?

b. If \(\theta\) is not zero, what is \(E_{t}\left(y_{t+1}\right)\) ? What extra information have you used here to forecast the return?