Copula parameter estimation: numerical considerations and implications for risk management

Gregor N. F. Weiß

Abstract

ABSTRACT

The purpose of this paper is to present a comprehensive simulation study on the finite sample properties of minimum distance and maximum likelihood estimators for bivariate and multivariate parametric copulas. For five popular parametric copulas, classical maximum likelihood is compared to a total of nine different minimum distance estimators. In particular, Cramér-von-Mises, Kolmogorov-Smirnov and L1 variants of the Cramér- von-Mises statistic based on the empirical copula process, Kendall's dependence function and Rosenblatt's probability integral transform are considered. The results presented in this paper show that in most settings canonical maximum likelihood yields smaller estimation biases at less computational effort than any of the minimum distance estimators. There exist, however, some cases (especially when the sample size increases) where minimum distance estimators based on the empirical copula process are superior to the maximum likelihood estimator. Minimum distance estimators based on Kendall's transform, on the other hand, yield only suboptimal results in all configurations of the simulation study. The results of the simulation study are confirmed by empirical examples where the value-at-risk as well as the expected shortfall of 100 bivariate portfolios are computed. Interestingly, the estimates for these risk measures differed considerably depending on the choice of parameter estimator. This result stresses the need for carefully choosing the parameter estimator in contrast to focusing all attention on choosing the parametric copula model.