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 that consist of two components+ Typically, one of the components is observable and the other is unobservable or “hidden+” Conditions under which geometric ergodicity of the unobservable component is inherited by the joint process formed of the two components are given+ This implies existence of initial values such that the joint process is strictly stationary and b-mixing+ In addition to this, conditions for the existence of moments are also obtained, and extensions to the case of nonstationary initial values are provided+ All these results are applied to a general model that 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+