Geometric Ergodicity of a Class of Markov Chains with Applications to Time Series Models

Abstract

We set up general conditions for a general non-linear Markov model to be geometrically ergodic (which implies beta-mixing of the stationary solution) and existence of certain moments. The conditions are fairly general and can be applied to most known time series models. We demonstrate the usefulness of our general result by applying it to various popular time series models. For each model, we give conditions for beta-mixing and existence of certain moments. In many cases, our conditions are weaker than those found elsewhere in the literature. In particular, we derive sufficient conditions for a class of univariate GARCH models to be geometrically ergodic without having a 2nd moment. In certain cases, the conditions are also sufficient. We also consider multivariate GARCH models and give conditions for stationarity with finite 2nd moment.

Keywords: Markov chain, geometric ergodicity, mixing, moments, GARCH, bilinear model

JEL Classification: C22, C32