As a beginner in Bayesian statistics, I was wondering how one can make a GARCH(1,1) volatility point forecast using a Bayesian approach in the following model:
$$\sigma^2_{t+1}=\alpha_0+\alpha_1\varepsilon^2_t+\beta\sigma^2_t$$
The $\varepsilon_t$ corresponds to the zero mean financial returns. The $\sigma^2_t$ corresponds to the conditional variance of these demeaned returns (i.e. $Var(\varepsilon_t|I_t)$ where $I_t$ denotes the information set available at time t).
Suppose you have a window of 100 observations and you simulate 1000 values from the posterior distribution of the parameters. So, you have 1000 simulated values for $\alpha_0$, $\alpha_1$ and $\beta$.
How can one make a forecast using a Bayesian approach? Is one, for example, allowed to do the following: Calculate a 1000 values for $\sigma^2_{t+1}$ using the simulated values for $\alpha_0$, $\alpha_1$ and $\beta$ and take the average over the 1000 values for $\sigma^2_{t+1}$ and use this average over the 1000 values for $\sigma^2_{t+1}$ as your point forecast of the volatility?
And if this is allowed, how is making a Bayesian point forecast in this specific case related to the posterior predictive density (which I thought was the way to go in Bayes).
So, if one assumes a Normal distribution for the residuals ($\varepsilon_t$) with mean zero and variance equal to $\sigma^2_{t}$, what is then the posterior predictive density for a GARCH(1,1) model (given that we use a truncated normal prior for our parameters $\alpha_0$ and $\alpha_1$ with hyperparameters $\mu_{\alpha}$ which is a $2\times1$ vector of zeros and $\Sigma_{\alpha}$. For the parameter $\beta$ we also use a truncated normal prior. Also, we assume prior independence between parameters $\alpha$ and $\beta$) ?
Many Thanks