Journal of the Vol. 36, pp. 307-317, 2017 Nigerian Mathematical Society c Nigerian Mathematical Society ON THE GEOMETRIC ERGODICITY OF THE MIXTURE AUTOREGRESSIVE MODEL M. I. AKINYEMI 1 AND G. N. BOSHNAKOV ABSTRACT. Geometric ergodicity is very useful in establish- ing mixing conditions and central limit results for parameter estimates of a model. It also justifies the use of laws of large numbers and forms part of the basis for exploring the asymptotic theory of a model. The class of mixture autoregressive (MAR) models provides a flexible way to model various features of time series data and is well suited for density forecasting. The MAR models are able to capture many stylised properties of real data, such as multi- modality, asymmetry and heterogeneity. We show here that the MAR model is geometrically ergodic and by implication satisfies the absolutely regular and strong mixing conditions. Keywords and phrases: mixture autoregressive model, MAR model, geometric ergodicity, mixing conditions, drift condition 2010 Mathematical Subject Classification: 37A25 1. Introduction Geometric ergodicity is very useful in establishing mixing conditions and central limit results for parameter estimators of a model. It also justifies the use of laws of large numbers and forms a basis for explor- ing the asymptotic theory of the model. This further translates into examining the consistency and asymptotic normality of the parameter estimates of the model [22]. Detailed discussions on geometric ergodic- ity and mixing conditions are given by [16], [21], [23], [5]. Furthermore, [20] provide criteria for judging the strong ergodicity of regime-switching diffusion processes. They considered processes in one dimensional space and in multidimensional space separately.[12] stated sufficient condi- tions for simultaneous geometric ergodicity of Markov chain classes. In particular, they deal with non asymptotic computable bounds for the geometric convergence rate of homogeneous ergodic Markov processes. [19] derived sufficient conditions for geometric ergodicity of a general class of asymmetric nonparametric stochastic processes with stochastic volatility models with skewness driven by the hidden Markov Chain with Received by the editors May 28, 2015; Revised: January 10, 2017; Accepted: March 16, 2017 www.nigerianmathematicalsociety.org 1 Corresponding author 307