On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood

TL;DR

This paper critically evaluates the continuous extension method for estimating multivariate normal copula models with discrete data, revealing biases and proposing a nearly efficient simulated likelihood approach using quasi Monte Carlo methods.

Contribution

The paper analyzes the limitations of existing continuous extension methods and introduces a maximum simulated likelihood approach with quasi Monte Carlo techniques for improved estimation accuracy.

Findings

Continuous extension methods can lead to biased estimates of latent correlations.

The proposed simulated likelihood method achieves near-ML efficiency.

The method is effective for binary, Poisson, and negative binomial regressions.

Abstract

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.