Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation

TL;DR

This paper introduces a rank-based, semiparametrically efficient estimator for Gaussian copula models with unknown margins, providing explicit calculations and conditions for efficiency and adaptivity, with practical implications for structured correlation matrices.

Contribution

It develops a new rank-based estimator for Gaussian copula models, explicitly characterizes efficiency conditions, and demonstrates its effectiveness for structured correlation matrices.

Findings

Pseudo-likelihood estimator is efficient for factor-structured matrices.

Efficiency can drop to 20% for Toeplitz matrices.

Monte Carlo simulations confirm theoretical results.

Abstract

We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations. We indicate how our results can be extended to joint regression models.