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:


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

  • $\begingroup$ You seem to have forgotten to square the $\varepsilon_t$ in your formula. $\endgroup$ Dec 18 '15 at 15:46
  • $\begingroup$ Ok thanks, but any idea of how to perform forecasts in a bayesian setting for a GARCH(1,1)? $\endgroup$
    – Eren
    Dec 18 '15 at 15:54
  • $\begingroup$ No, unfortunately I am no expert in Bayesian methods. $\endgroup$ Dec 18 '15 at 16:05
  • $\begingroup$ For those of us not familiar with a GARCH model, can you tell us what $\epsilon_t$ are and what your data are? $\endgroup$
    – jaradniemi
    Dec 18 '15 at 20:18
  • $\begingroup$ I edited my question. $\endgroup$
    – Eren
    Dec 18 '15 at 20:27

A Bayesian forecast that takes into account the parameter uncertainty is constructed as follows when we use the posterior predictive distribution shown below:

$p(\sigma^2_{t+1}|\sigma^2)= \int_{\theta}p(\sigma^2_{t+1}|\theta,\sigma^2)p(\theta|\sigma^2)d\theta$

where $\theta = [\alpha_0, \alpha_1, \beta]$ and $p(\theta|\sigma^2)$ is the posterior distribution of the parameters. The $p(\sigma^2_{t+1}|\theta,\sigma^2)$ is the distribution of the predicted volatilities.

According to Rachev et al. (2008) we should plug-in the estimated parameters in the GARCH equation to get (f.e. 1000) simulated values of $\sigma^2_{t+1}$. The thousand $\sigma^2_{t+1}$ are an estimation of this distribution of predicted volatilities: $p(\sigma^2_{t+1}|\theta,\sigma^2)$.

To approximate the integral above we use the following:

$p(\sigma^2_{t+1}|\sigma^2)= \int_{\theta}p(\sigma^2_{t+1}|\theta,\sigma^2)p(\theta|\sigma^2)d\theta = E_{\theta|\sigma^2}(p(\sigma^2_{t+1}|\theta,\sigma^2))$

The mean is approximated by taking the sample average of all the predicted volatilities.

Basically, we approximate this integral (which is an epectation) by using simulated sample averages.


Rachev, S. T., Hsu, J. S., Bagasheva, B. S., & Fabozzi, F. J. (2008). Bayesian methods in finance (Vol. 153). John Wiley & Sons


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