Fourth Moment Structure of Markov Switching Multivariate GARCH Models

We derive sufficient conditions for the existence of second and fourth moments of Markov switching multivariate generalized autoregressive conditional heteroscedastic processes in the general vector specification. We provide matrix expressions in closed form for such moments, which are obtained by using a Markov switching vector autoregressive moving-average representation of the initial process. These expressions are shown to be readily programmable in addition of greatly reducing the computational cost. As theoretical applications of the results, we derive the spectral density matrix of the squares and cross products, propose a new definition of multivariate kurtosis measure to recognize heavy-tailed features in financial real data, and provide a matrix expression in closed form of the impulse-response function for the volatility. An empirical example illustrates the results.