# Bayesian Approaches to Copula Modelling

Damien, P. P. Dellaportas, N. G. Polson, D. A. Stephens, (2013), Bayesian Theory and Applications, OUP, pp. 336-358.

33 Pages Posted: 19 Dec 2011 Last revised: 30 Mar 2015

See all articles by Michael S. Smith

## Michael S. Smith

University of Melbourne - Melbourne Business School

Date Written: December 12, 2011

### Abstract

Copula models have become one of the most widely used tools in the applied modelling of multivariate data. Similarly, Bayesian methods are increasingly used to obtain efficient likelihood-based inference. However, to date, there has been only limited use of Bayesian approaches in the formulation and estimation of copula models. This article aims to address this shortcoming in two ways. First, to introduce copula models and aspects of copula theory that are especially relevant for a Bayesian analysis. Second, to outline Bayesian approaches to formulating and estimating copula models, and their advantages over alternative methods.

Copulas covered include Archimedean, copulas constructed by inversion, and vine copulas; along with their interpretation as transformations. A number of parameterisations of a correlation matrix of a Gaussian copula are considered, along with hierarchical priors that allow for Bayesian selection and model averaging for each parameterisation. Markov chain Monte Carlo sampling schemes for fitting Gaussian and D-vine copulas, with and without selection, are given in detail. The relationship between the prior for the parameters of a D-vine, and the prior for a correlation matrix of a Gaussian copula, is discussed. Last, it is shown how to compute Bayesian inference when the data are discrete-valued using data augmentation. This approach generalises popular Bayesian methods for the estimation of models for multivariate binary and other ordinal data to more general copula models. Bayesian data augmentation has substantial advantages over other methods of estimation for this class of models.

Keywords: Copula, Dependence Modelling, Vines, Gaussian copula, Archimedean copula, Dependence priors, Bayesian selection, Bayesian model averaging, Discrete copulas

JEL Classification: C11, C30, C58

Suggested Citation

Smith, Michael S., Bayesian Approaches to Copula Modelling (December 12, 2011). Damien, P. P. Dellaportas, N. G. Polson, D. A. Stephens, (2013), Bayesian Theory and Applications, OUP, pp. 336-358., Available at SSRN: https://ssrn.com/abstract=1974297 or http://dx.doi.org/10.2139/ssrn.1974297