Stationarity of multivariate Markov-switching ARMA models

In this article we consider multivariate ARMA models subject to Markov switching. In these models, the parameters are allowed to depend on the state of an unobserved Markov chain. A natural idea when estimating these models is to impose local stationarity conditions, i.e. stationarity within each regime. In this article we show that the local stationarity of the observed process is neither sufficient nor necessary to obtain the global stationarity. We derive stationarity conditions and we compute the autocovariance function of this nonlinear process. Interestingly, it turns out that the autocovariance structure coincides with that of a standard ARMA. Some examples are proposed to illustrate the stationarity conditions. Using Monte Carlo simulations we investigate the consequences of accounting for the stationarity conditions in statistical inference.