On stationarity and β-mixing for ARMA-ACD model

Abstract

Among the simplest and most widely-used models of stationary stochastic processes are the autoregressive moving-average or ARMA models. Engle and Russell(1995) developed a model for data that do not arrive in equal time intervals. This model is called the autoregressive conditional duration(ACD) model because the model formulation focuses on the intertemporally correlations of the duration. In this paper we will propose ARMA-ACD(p,q,r,s) models and give some evidences and proofs to show the new model is Markov geometrically ergodic, and if X_(0) is initialized from the invariant distribution, the X_(i) is strictly stationary and β mixing with exponential decay.