  This paper deals with a situation when one is interested in the dependence
structure of a multidimensional response variable in the presence of a
multivariate covariate. It is assumed that the covariate affects only the
marginal distributions through regression models while the dependence
structure, which is described by a copula, is unaffected. A parametric
estimation of the copula function is considered with focus on the maximum
pseudo-likelihood method. It is proved that under some appropriate regularity
assumptions the estimator calculated from the residuals is asymptotically
equivalent to the estimator based on the unobserved errors. In such case one
can ignore the fact that the response is first adjusted for the effect of the
covariate. A Monte Carlo simulation study explores (among others) situations
where the regularity assumptions are not satisfied and the claimed result does
not hold. It shows that in such situations the maximum pseudo-likelihood
estimator may behave poorly and the moment estimation of the copula parameter
is of interest. Our results complement the results available for nonparametric
estimation of the copula function.
