# Garch for covariance matrix?

I have seen plenty of literature about GARCH on estimation volatility. how about covariance? There are plenty of risk models depending on the covariance matrix.

I guess we can assume the correlation is constant and volatility changes. But in reality in super volatile moment correlation between stocks increases.

Or there is a separate model for estimating correlation?

I think you're looking for multivariate GARCH models of which this is an overview paper.

Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of correlations. This paper by Caporin and McAleer might be of interest in that regard.

Not sure your question is about having a process for covariance or to have multivariate GARCH.

The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for instance Philipov, A. and M. E. Glickman (2006, July) Multivariate stochastic volatility via wishart processes. Journal of Business & Economic Statistics 24 (3), 313-328. You will find all the formulas.

Just note in dimension one, it is like using a Gamma distribution for your volatility, using a "time serie" (a stochastic process) on the parameter $\beta$. I.e.

$$X_t|\sigma^2_t \sim {\cal N}(0, \sigma_t)$$

and

$$\sigma_t^{-2}|\alpha,\beta_t\sim \Gamma(\alpha,\beta_t).$$