Stationarity of generalized autoregressive moving average models

Abstract

Time series models are often constructed by combining nonstationary effects such as trends with stochastic processes that are believed to be stationary. Although stationarity of the underlying process is typically crucial to ensure desirable properties or even validity of statistical estimators, there are numerous time series models for which this stationarity is not yet proven. A major barrier is that the most commonly-used methods assume φ-irreducibility, a condition that can be violated for the important class of discrete-valued observation-driven models. We show (strict) stationarity for the class of Generalized Autoregressive Moving Average (GARMA) models, which provides a flexible analogue of ARMA models for count, binary, or other discrete-valued data. We do this from two perspectives. First, we show conditions under which GARMA models have a unique stationary distribution (so are strictly stationary when initialized in that distribution). This result potentially forms the foundation for broadly showing consistency and asymptotic normality of maximum likelihood estimators for GARMA models. Since these conclusions are not immediate, however, we also take a second approach. We show stationarity and ergodicity of a perturbed version of the GARMA model, which utilizes the fact that the perturbed model is φ-irreducible and immediately implies consistent estimation of the mean, lagged covariances, and other functionals of the perturbed process. We relate the perturbed and original processes by showing that the perturbed model yields parameter estimates that are arbitrarily close to those of the original model.