On Categorical Time Series Models With Covariates

TL;DR

This paper investigates the conditions for stationarity and ergodicity in complex categorical time series models with covariates, introducing new theoretical results and coupling techniques for analysis.

Contribution

It significantly advances the understanding of stationarity and ergodicity conditions for multinomial logistic time series models with latent processes and covariates.

Findings

Improved conditions for stationarity and ergodicity of the models.

Application of coupling techniques to analyze ergodicity.

Extension of ergodicity results to models with exogenous covariates.

Abstract

We study the problem of stationarity and ergodicity for autoregressive multinomial logistic time series models which possibly include a latent process and are defined by a GARCH-type recursive equation. We improve considerably upon the existing results related to stationarity and ergodicity conditions of such models. Proofs are based on theory developed for chains with complete connections. This approach is based on a useful coupling technique which is utilized for studying ergodicity of more general finite-state stochastic processes. Such processes generalize finite-state Markov chains by assuming infinite order models of past values. For finite order Markov chains, we also discuss ergodicity properties when some strongly exogenous covariates are considered in the dynamics of the process.