Stationarity and ergodicity of Markov switching positive conditional mean models

Abstract

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.