Copulas for Markovian dependence

TL;DR

This paper explores the limitations and challenges of applying copulas to model temporal dependence in univariate Markov processes, revealing incompatibilities and restrictions in common copula families.

Contribution

It identifies fundamental difficulties in using copulas for Markovian dependence, including restrictions of Fréchet copulas and incompatibility of Archimedean copulas with Markov chains.

Findings

Fréchet copulas imply restricted Markov processes

Archimedean copulas are incompatible with Markov chains

Certain Markov chains are spreadable or conditionally i.i.d.

Abstract

Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper demonstrates some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete co- and countermonotonicity and independence (Fr\'{e}chet copulas) are shown to imply quite a restricted type of Markov process and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable or, equivalently, conditionally i.i.d.