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

## TL;DR

This paper investigates the maximum pseudo-likelihood estimation of copula models with residual-based data, demonstrating asymptotic equivalence to unobserved error-based estimators under certain conditions, and exploring limitations via simulations.

## Contribution

It establishes the asymptotic properties of residual-based pseudo-likelihood estimators in copula models and examines their performance when regularity conditions fail.

## Key findings

- Residual-based estimator is asymptotically equivalent to error-based estimator under regularity.
- Simulation shows poor behavior of pseudo-likelihood estimator when assumptions are violated.
- Moment estimation of copula parameters can be preferable in irregular cases.

## Abstract

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.