Stationarity and ergodic properties for some observation-driven models in random environments

TL;DR

This paper investigates the stationarity and ergodic properties of a broad class of observation-driven time series models in random environments, including non-contractive models like threshold autoregressions, using coupling methods.

Contribution

It introduces a general framework for analyzing non-discrete state space Markov chains in random environments without relying on contraction assumptions.

Findings

Established existence of limits for backward iterations in Wasserstein metric.

Proved ergodic properties for a wide class of autoregressive models.

Applied results to GARCH, count autoregressions, and categorical time series.

Abstract

The first motivation of this paper is to study stationarity and ergodic properties for a general class of time series models defined conditional on an exogenous covariates process. The dynamic of these models is given by an autoregressive latent process which forms a Markov chain in random environments. Contrarily to existing contributions in the field of Markov chains in random environments, the state space is not discrete and we do not use small set type assumptions or uniform contraction conditions for the random Markov kernels. Our assumptions are quite general and allows to deal with models that are not fully contractive, such as threshold autoregressive processes. Using a coupling approach, we study the existence of a limit, in Wasserstein metric, for the backward iterations of the chain. We also derive ergodic properties for the corresponding skew-product Markov chain. Our results are illustrated with many examples of autoregressive processes widely used in statistics or in econometrics, including GARCH type processes, count autoregressions and categorical time series.