Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models

This paper studies a class of Markov models which consist of two components. Typically, one of the components is observable and the other is unobservable or 'hidden'. Conditions under which (a form of) geometric ergodicity of the unobservable component is inherited by the joint process formed of the two components are given. This immediately implies the existence of initial values such that the joint process is strictly stationary and beta-mixing. In addition to this, conditions for beta-mixing and existence of moments for the joint process are also provided in the case of (possibly) nonstationary initial values. All these results are applied to a general model which includes as special cases various first order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated nonlinear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.