Efficient estimation of semiparametric multivariate copula models

We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the "curse of dimensionality" associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve further efficiency gains. Two such cases are studied: (a) the marginal distributions are equal but otherwise unspecified, and (b) some but not all marginal distributions are parametric. Monte Carlo studies indicate that the sieve MLEs perform well in finite samples, especially when prior information on the marginal distributions is incorporated.