Mixed Marginal Copula Modeling

ABSTRACT

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model. Supplementary materials for this article are available online.

KEY WORDS:

• Bayesian analysis
• Markov chain Monte Carlo
• Mixtures of copulas
• Multivariate income dynamics.

SUPPLEMENTARY MATERIALS

There is an online supplement for our article that contains the following: Section S1 gives the density, conditional distribution function, and MCMC sampling methods for the Gaussian, Gumbel and Clayton copulas. Section S2 contains a trivariate example to illustrate the methodology. Section S3 provides the proof of Lemma 3. Section S4 provides some additional empirical results.

ACKNOWLEDGEMENTS

The authors thank two anonymous referees and the associate editor for suggestions that helped improve the clarity of the article.

Additional information

Funding

The research of David Gunawan and Robert Kohn was partially supported by an Australian Research Council Discovery Grant DP150104630.