Keywords: BEKK GARCH, CCC (Constant Conditional Correlations) GARCH, DCC GARCH, Diagonal GARCH, Factor GARCH, FF (Full-Factor) GARCH,v Multivariate ARMA, Multivariate Autocovariances and Autocorrelation, Multivariate White Noise, Multivariate Wold's decomposition, PCA (Principal Component Analysis), PC-GARCH, O-GARCH (Orthogonal-GARCH), Top Lyapounov Exponent, Vector GARCH.
Description: While the volatility of univariate series has been the focus of the previous chapters, modeling the comovements of several series is of great practical importance. When several series displaying temporal or contemporaneous dependencies are available, it is useful to analyze them jointly, by viewing them as the components of a vector-valued (multivariate) process. The standard linear modeling of real time series has a natural multivariate extension through the framework of the VARMA (Vector ARMA) models. In particular, the sub-class of the 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 cointegration). Similarly, it is important to extend the concepts and models of GARCH to the multivariate case. 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 models 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 a m × m matrix. A general extension of the univariate GARCH processes would consist in 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 number of parameters that this approach would entail, it is not feasible from a statistical point of view. An alternative approach is to introduce some specification constraints which, while preserving a certain generality, make these models operational. We start by reviewing the main concepts for the analysis of the multivariate time series.