10 Multivariate GARCH Processes

While the volatility of univariate series has been the focus of the previous chapters, modelling the co‐movements of several series is of great practical importance. When several series displaying temporal or contemporaneous dependencies are available, it is useful to analyse them jointly, by viewing them as the components of a vector‐valued (multivariate) process. The standard linear modelling of real‐time series has a natural multivariate extension through the framework of the vector ARMA (VARMA) models. In particular, the subclass of vector autoregressive (VAR) models has been widely studied in the econometric literature. This extension entails numerous specific problems and has given rise to new research areas (such as co‐integration).

Similarly, it is important to introduce the concept of multivariate GARCH (MGARCH) model. For instance, asset pricing and risk management crucially depend on the conditional covariance structure of the assets of a portfolio. Unlike the ARMA models, however, the GARCH model specification does not suggest a natural extension to the multivariate framework. Indeed, the (conditional) expectation of a vector of size m is a vector of size m, but the (conditional) variance is an m × m matrix. A general extension of the univariate GARCH processes would involve specifying each of the m(m + 1)/2 entries of this matrix as a function of its past values and the past values of the other entries. Given the excessive ...

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