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10.4 GARCH Models for Bivariate Returns

Since the same techniques can be used to generalize many univariate volatility models to the multivariate case, we focus our discussion on the multivariate GARCH model. Other multivariate volatility models can also be used.

For a k-dimensional return series , a multivariate GARCH model uses “exact equations” to describe the evolution of the k(k + 1)/2-dimensional vector over time. By exact equation, we mean that the equation does not contain any stochastic shock. However, the exact equation may become complicated even in the simplest case of k = 2 for which is three dimensional. To keep the model simple, some restrictions are often imposed on the equations.

10.4.1 Constant-Correlation Models

To keep the number of volatility equations low, Bollerslev (1990) considers the special case in which the correlation coefficient ρ21, t = ρ21 is time invariant, where |ρ21| < 1. Under such an assumption, ρ21 is a constant parameter and the volatility model consists of two equations for , which is defined as . A GARCH(1,1) model for becomes

10.21

where

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