Multivariate GARCH and Wishart Models for Time-Varying Covariance

Summary

The discussion addresses how to model a covariance matrix when asset volatilities and correlations can change over time. It identifies multivariate GARCH as a family of models for joint volatility and covariance dynamics. A central practical drawback is estimation complexity: modeling many correlations can make these systems difficult to fit as the number of assets grows.

A second approach described is stochastic covariance modeling with Wishart processes, with a cited research paper as a source for its formulation. The response offers a one-dimensional analogy involving stochastic volatility and a time-varying parameter, but does not develop a complete estimation procedure. The material is conceptual and provides references rather than model comparisons, empirical results, or guidance on selecting among specifications. It highlights that assuming fixed correlations while volatility changes may miss joint behavior in turbulent periods, but does not establish which model will forecast covariance most reliably.

Key ideas

• Multivariate GARCH models extend volatility modeling to joint covariance dynamics across multiple assets.
• Estimating multivariate GARCH can become difficult as the number of correlations increases.
• Wishart processes are presented as a stochastic-process approach to covariance modeling.
• The discussion supplies references and intuition but no empirical comparison or detailed model-selection guidance.

Tags

# Garch for covariance matrix?


# 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?

## Answer by Bob Jansen (score 4, accepted)

https://quant.stackexchange.com/a/17140

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.

## Answer by lehalle (score 3)

https://quant.stackexchange.com/a/17147

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

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

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