Strong mixing properties of discrete-valued time series with exogenous covariates

TL;DR

This paper establishes strong mixing conditions for various discrete-valued time series models with exogenous covariates, providing a general framework to verify mixing properties in models used in practice.

Contribution

It introduces a unified approach using coupling and random mapping methods to derive verifiable mixing conditions for a wide class of categorical and count time series models.

Findings

Derived mixing conditions for Markov chains in random environments.

Extended results to infinite memory categorical processes.

Provided explicit, checkable mixing conditions for models like multinomial and ordinal time series.

Abstract

We derive strong mixing conditions for many existing discrete-valued time series models that include exogenous covariates in the dynamic. Our main contribution is to study how a mixing condition on the covariate process transfers to a mixing condition for the response. Using a coupling method, we first derive mixing conditions for some Markov chains in random environments, which gives a first result for some autoregressive categorical processes with strictly exogenous regressors. Our result is then extended to some infinite memory categorical processes. In the second part of the paper, we study autoregressive models for which the covariates are sequentially exogenous. Using a general random mapping approach on finite sets, we get explicit mixing conditions that can be checked for many categorical time series found in the literature, including multinomial autoregressive processes, ordinal time series and dynamic multiple choice models. We also study some autoregressive count time series using a somewhat different contraction argument. Our contribution fill an important gap for such models, presented here under a more general form, since such a strong mixing condition is often assumed in some recent works but no general approach is available to check it.