Chapter 10
Multivariate Volatility Models and Their Applications
In this chapter, we generalize the univariate volatility models of Chapter 3 to the multivariate case and discuss some simple methods for modeling the dynamic relationships between volatility processes of multiple asset returns. By multivariate volatility, we mean the conditional covariance matrix of multiple asset returns. Multivariate volatilities have many important financial applications. They play an important role in portfolio selection and asset allocation, and they can be used to compute the value at risk of a financial position consisting of multiple assets.
Consider a multivariate return series 
. We adopt the same approach as the univariate case by rewriting the series as
where 
is the conditional expectation of 
given the past information 
, and 
is the shock, or innovation, of the series at time t. In addition, we assume that 
follows a multivariate time series model of Chapter 8 so that 
is the 1-step-ahead prediction of the model. For most return series, it suffices to employ a simple vector ARMA structure with exogenous variables for 
—that is,
where 
denotes an m-dimensional vector of exogenous (or explanatory) variables with x1t = 1, 
is a k × m matrix, and p and q are nonnegative integers. We refer to Eq. (10.1) as the mean equation of 
.
The conditional covariance matrix of 
given 
is a k × k positive-definite matrix 
defined by 
). Multivariate volatility modeling is concerned with the time evolution of 
. We refer to a model for the 
process as a volatility model for the return series 
.
There are many ways to generalize univariate volatility models to the multivariate case, but the curse of dimensionality quickly becomes a major obstacle in applications because there are k(k + 1)/2 quantities in 
for a k-dimensional return series. To illustrate, there are 15 conditional variances and covariances in 
for a five-dimensional return series. The goal of this chapter is to introduce some relatively simple multivariate volatility models that are useful, yet remain manageable in real application. In particular, we discuss some models that allow for time-varying correlation coefficients between asset returns. Time-varying correlations are useful in finance. For example, they can be used to estimate the time-varying beta of the market model for a return series.
We begin by using an exponentially weighted approach to estimate the covariance matrix in Section 10.1. This estimated covariance matrix can serve as a benchmark for multivariate volatility estimation. Section 10.2 discusses some generalizations of univariate GARCH models that are available in the literature. We then introduce two methods to reparameterize 
for volatility modeling in Section 10.3. The reparameterization based on the Cholesky decomposition is found to be useful. We study some volatility models for bivariate returns in Section 10.4, using the GARCH model as an example. In this particular case, the volatility model can be bivariate or three dimensional. Section 10.5 is concerned with volatility models for higher dimensional returns and Section 10.6 addresses the issue of dimension reduction. We demonstrate some applications of multivariate volatility models in Section 10.7. Finally, Section 10.8 gives a multivariate Student-t distribution useful for volatility modeling.