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This paper deals with a situation when one is interested in the dependencestructure of a multidimensional response variable in the presence of amultivariate covariate. It is assumed that the covariate affects only themarginal distributions through regression models while the dependencestructure, which is described by a copula, is unaffected. A parametricestimation of the copula function is considered with focus on the maximumpseudo-likelihood method. It is proved that under some appropriate regularityassumptions the estimator calculated from the residuals is asymptoticallyequivalent to the estimator based on the unobserved errors. In such case onecan ignore the fact that the response is first adjusted for the effect of thecovariate. A Monte Carlo simulation study explores (among others) situationswhere the regularity assumptions are not satisfied and the claimed result doesnot hold. It shows that in such situations the maximum pseudo-likelihoodestimator may behave poorly and the moment estimation of the copula parameteris of interest. Our results complement the results available for nonparametricestimation of the copula function.

Maximum pseudo-likelihood estimation based on estimated residuals in copula semiparametric models