  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.
