Unconditional variance of MA(1)-GARCH(1,1) process

Question

Let $y_t = \Delta{p_t}$ denote a time series of asset log-returns, where $p_t$ are logarithmic prices; $y_t$ is generated by the conditionally heteroscedastic MA(1) process

$y_t = \epsilon_t + \theta \epsilon_{t-1}$, where $\epsilon_t = \sqrt{h_t}z_t$ and $z_t\sim i.i.d N (0,1) $ with $|\theta|<1$, $h_t = \omega+\alpha \epsilon_t^2+\beta h_{t-1},\quad \omega>0, \alpha>0 ,\alpha+\beta<1$.

1. Derive the expressions for the unconditional mean $E(y_t)$, unconditional variance $Var(y_t)$ and autocorrelation function of $y_t, \rho(k),k=1,2,.....$

I computed the unconditional mean being equal to 0 because $E(\sqrt{h_t} z_t)+\theta E(\sqrt{h_{t-1}}z_{t-1})=0 $ for the unconditional variance $Var(y_t)= E(y_t^2)= (h_t z_t^2+\theta^2 h_{t-1} z_t^2)$ = $E(h_t+\theta^2 h_{t-1})$ because $z_t\sim i.i.d N (0,1)$.

From now on I am not sure. Should I substitute $h_t$ to be done? Any help would be really appreciated.

Answer 1

So far so good.

To obtain an expression for the unconditional variance, you need to work out $\mathbb{E}(\epsilon_t^2) =\mathbb{E}(h_t)$. Call this $\sigma^2_t$.

Substituting in $h_t$ as you suggested gives $$ \sigma^2_t = \mathbb{E}(\omega + \alpha \epsilon_{t-1}^2 + \beta h_{t-1}) = \omega + (\alpha + \beta) \sigma^2_{t-1} $$ Now substitute in $\sigma^2_{t-1}$. Can you see what's going to happen?