Theory for a Multivariate Markov--switching GARCH Model with an Application to Stock Markets

We consider a multivariate Markov-switching GARCH model which allows for regime-specific volatility dynamics, leverage effects, and correlation structures. Stationarity conditions are derived, and consistency of the maximum likelihood estimator (MLE) is established under the assumption of Gaussian innovations. A Lagrange Multiplier (LM) test for correct specification of the correlation dynamics is devised, and a simple recursion for computing multi-step-ahead conditional covariance matrices is provided. The theory is illustrated with an application to global stock market and real estate equity returns. The empirical analysis highlights the importance of the conditional distribution in Markov-switching time series models. Specifications with Student's t innovations dominate their Gaussian counterparts both in- and out-of-sample. The dominating specification appears to be a two-regime Student's t process with correlations which are higher in the turbulent (high-volatility) regime.