Some Aspects of Modeling Dependence in Copula-based Markov chains

TL;DR

This paper explores the dependence structure of copula-based Markov chains, establishing the equivalence of geometric ergodicity and geometric ρ-mixing for common copulas, and demonstrating that mixtures of these copulas generate chains with desirable ergodic and mixing properties.

Contribution

It clarifies the relationship between geometric ergodicity and geometric ρ-mixing in copula-based Markov chains and shows that mixtures of Clayton, Gumbel, or Student copulas produce chains with these properties.

Findings

Geometric ergodicity and geometric ρ-mixing are equivalent for many well-known copulas.

Mixtures of Clayton, Gumbel, or Student copulas generate geometrically ergodic and ρ-mixing Markov chains.

A sufficient condition for ρ-mixing implies Doeblin recurrence.

Abstract

Dependence coefficients have been widely studied for Markov processes defined by a set of transition probabilities and an initial distribution. This work clarifies some aspects of the theory of dependence structure of Markov chains generated by copulas that are useful in time series econometrics and other applied fields. The main aim of this paper is to clarify the relationship between the notions of geometric ergodicity and geometric {\rho}-mixing; namely, to point out that for a large number of well known copulas, such as Clayton, Gumbel or Student, these notions are equivalent. Some of the results published in the last years appear to be redundant if one takes into account this fact. We apply this equivalence to show that any mixture of Clayton, Gumbel or Student copulas generate both geometrically ergodic and geometric {\rho}-mixing stationary Markov chains, answering in this way an open question in the literature. We shall also point out that a sufficient condition for {\rho}-mixing, used in the literature, actually implies Doeblin recurrence.