  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.
