  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.
