Stationarity and ergodicity of Markov switching positive conditional mean models

A general Markov-Switching autoregressive conditional mean model, valued in the set of non-negative numbers, is considered. The conditional distribution of this model is a finite mixture of non-negative distributions whose conditional mean follows a GARCH-like dynamics with parameters depending on the state of a Markov chain. Three different variants of the model are examined depending on how the lagged-values of the mixing variable are integrated into the conditional mean equation. The model includes, in particular, Markov mixture versions of various well-known non-negative time series models such as the autoregressive conditional duration model, the integer-valued GARCH (INGARCH) model, and the Beta observation driven model. For the three variants of the model, conditions are given for the existence of a stationary and ergodic solution. The proposed conditions match those already known for Markov-switching GARCH models. We also give conditions for finite marginal moments. Applications to various mixture and Markov mixture count, duration and proportion models are provided.