Iterations of dependent random maps and exogeneity in nonlinear dynamics

TL;DR

This paper investigates the existence and uniqueness of stationary nonlinear autoregressive processes with exogenous variables, introducing new convergence conditions and applying results to various time series models.

Contribution

It presents a novel convergence criterion based on contraction in conditional expectation, extending the analysis of dependent random maps in nonlinear dynamics.

Findings

Established new conditions for stationary process existence

Derived explicit control of functional dependence for statistical applications

Extended results to diverse models like GARCH and categorical time series

Abstract

We discuss existence and uniqueness of stationary and ergodic nonlinear autoregressive processes when exogenous regressors are incorporated in the dynamic. To this end, we consider the convergence of the backward iterations of dependent random maps. In particular, we give a new result when the classical condition of contraction on average is replaced with a contraction in conditional expectation. Under some conditions, we also derive an explicit control of the functional dependence of Wu (2005) which guarantees a wide range of statistical applications. Our results are illustrated with CHARN models, GARCH processes, count time series, binary choice models and categorical time series for which we provide many extensions of existing results.