Given a model for a stationary time series $r_t$, written as: $r_t=\epsilon_t$ and $\epsilon_t=\sqrt{h_t} z_t$, where $z_t$ follows $N(0,1)$ and $h_t$ is some positive random variable, which is independent of $z_t$.
Is the mean of $r_t=0$ and variance equals to $h_t$, skewness=$E(z_t^3)$, kurtosis=$E(z_t^4)$?
if $h_t=\omega+ \alpha \epsilon^2_{t-1}$, can anyone recall what model it is? is it a AR model?
Can anyone help? Many thanks!
We compute \begin{align} \mathbb E[r_t] &= \mathbb E[\sqrt{h_t}z_t]=\mathbb E[\sqrt{h_t}]\mathbb E[z_t] = 0,\\\\ \mathsf{Var}(r_t) &= \mathbb E[r_t^2] - \mathbb E[r_t]^2 = \mathbb E[h_t]\mathbb E[z_t^2] = \mathbb E[h_t]\\\\ \gamma_1(R_t) &= \mathbb E\left[\left(\frac{r_t-\mathbb E[r_t]}{\sqrt{\mathsf{Var}(r_t)}}\right)^3\right] = \mathbb E\left[h_t^{3/2}\right]\\\\ \mathsf{Kurt}(r_t) &= \frac{\mathbb E[(r_t-\mathbb E[r_t])^4]}{\mathbb E[(r_t-\mathbb E[r_t])^2]} = \frac{\mathbb E[r_t^4]}{\mathbb E[r_t^2]} = \frac{\mathbb E[h_t^2]\mathbb E[z_t^4]}{\mathbb E[h_t]\mathbb E[z_t^2]}=\frac{3\mathbb E[h_t^2]}{\mathbb E[h_t]}. \end{align} If $h_t=\omega+\alpha\varepsilon_{t-1}^2$, then $$r_t = z_t\sqrt{\omega+\alpha\varepsilon_{t-1}^2} $$ which certainly isn't an AR process.