The benefit of using random matrix theory to fit high-dimensional t-copulas

Jiali Xu and Loïc Brin

Abstract

ABSTRACT

In risk management, t-copulas are used to model dependencies beyond Gaussian copulas, as they take into account tail dependencies. A t-copula has two parameters: the correlation matrix and the degree of freedom; they are usually estimated by maximizing the likelihood function of the observations. In risk modeling, the dimension of the copula is often high, making the maximization intractable, as the number of pairwise correlations to estimate is too high. McNeil, Frey and Embrechts suggested a procedure that consists of using a correlation matrix estimated through Kendall's rank correlation matrix estimate, likely transformed to be positive definite, as an input in the likelihood function to deduce a value for the degree of freedom. We show the bias of this degree of freedom's estimator due to the noise in the correlation matrix estimate. We address this problem using a random-matrix-theory-based denoising technique to improve the correlation estimate. On simulation studies, we show how this improved procedure gives an estimator of the degree of freedom of t-copulas with no bias and a smaller variance. Finally, we fit a t-copula on real operational risk data in order to illustrate the necessity and the benefit of this procedure.