  We consider a general multivariate model where univariate marginal
distributions are known up to a parameter vector and we are interested in
estimating that parameter vector without specifying the joint distribution,
except for the marginals. If we assume independence between the marginals and
maximize the resulting quasi-likelihood, we obtain a consistent but inefficient
QMLE estimator. If we assume a parametric copula (other than independence) we
obtain a full MLE, which is efficient but only under a correct copula
specification and may be biased if the copula is misspecified. Instead we
propose a sieve MLE estimator (SMLE) which improves over QMLE but does not have
the drawbacks of full MLE. We model the unknown part of the joint distribution
using the Bernstein-Kantorovich polynomial copula and assess the resulting
improvement over QMLE and over misspecified FMLE in terms of relative
efficiency and robustness. We derive the asymptotic distribution of the new
estimator and show that it reaches the relevant semiparametric efficiency
bound. Simulations suggest that the sieve MLE can be almost as efficient as
FMLE relative to QMLE provided there is enough dependence between the
marginals. We demonstrate practical value of the new estimator with several
applications. First, we apply SMLE in an insurance context where we build a
flexible semi-parametric claim loss model for a scenario where one of the
variables is censored. As in simulations, the use of SMLE leads to tighter
parameter estimates. Next, we consider financial risk management examples and
show how the use of SMLE leads to superior Value-at-Risk predictions. The paper
comes with an online archive which contains all codes and datasets.
