Consistency of the maximum likelihood estimate for Non-homogeneous Markov-switching models

Many nonlinear time series models have been proposed in the last decades. Among them, the models with regime switchings provide a class of versatile and interpretable models which have received a particular attention in the literature. In this paper, we consider a large family of such models which generalize the well known Markov-switching AutoRegressive (MS-AR) by allowing non-homogeneous switching and encompass Threshold AutoRegressive (TAR) models. We prove various theoretical results related to the stability of these models and the asymptotic properties of the Maximum Likelihood Estimates (MLE). The ability of the model to catch complex nonlinearities is then illustrated on various time series.