Autoregressive processes with data-driven regime switching

We develop a switching-regime vector autoregressive model in which changes in regimes are governed by an underlying Markov process. In contrast to the typical hidden Markov approach, we allow the transition probabilities of the underlying Markov process to depend on past values of the time series and exogenous variables. Such processes have potential applications in finance and neuroscience. In the latter, the brain activity at time t (measured by electroencephalograms) will be modelled as a function of both its past values as well as exogenous variables (such as visual or somatosensory stimuli). In this article, we establish stationarity, geometric ergodicity and existence of moments for these processes under suitable conditions on the parameters of the model. Such properties are important for understanding the stability properties of the model as well as for deriving the asymptotic behaviour of various statistics and model parameter estimators.