A Nonparametric Bayesian Approach to Copula Estimation

TL;DR

This paper introduces a new nonparametric Bayesian method using a Dirichlet-based Pólya tree prior for copula estimation, demonstrating improved flexibility, consistency, and robustness over traditional methods, especially with limited data and complex structures.

Contribution

It develops a novel D-P tree prior for copula estimation, providing a flexible, consistent, and smoothing Bayesian approach superior to existing models.

Findings

D-P tree prior ensures consistency in copula estimation.

Method detects complex copula structures better than Gaussian mixtures.

Robustness demonstrated during 2007-08 financial crisis.

Abstract

We propose a novel Dirichlet-based P\'olya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Further, the continuity of the imposed D-P tree prior leads to a more favorable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S\&P 500 index and the IBM stock prices during the 2007-08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviors.