Some Comments on Copula-Based Regression

Abstract

In a recent article, Noh, El Ghouch, and Bouezmarni proposed a new semiparametric estimate of a regression function with a multivariate predictor, which is based on a specification of the dependence structure between the predictor and the response by means of a parametric copula. This comment investigates the effect which occurs under misspecification of the parametric model. We demonstrate by means of several examples that even for a one or two-dimensional predictor the error caused by a "wrong" specification of the parametric family is rather severe, if the regression is not monotone in one of the components of the predictor. Moreover, we also show that these problems occur for all of the commonly used copula families and we illustrate in several examples that the copula-based regression may lead to invalid results even when flexible copula models such as vine copulas (with the common parametric families) are used in the estimation procedure.

D-vine copula based quantile regression

“…However, we observe that the nonparametrically estimated copula manages to model the non-monotonic dependence of the data quite well (for visualization purposes we added the data points transformed to have standard normal margins as well). Hence, by using a nonparametric copula to model the dependence of the pair (Y, X 1 ), a model misspecification as described in Dette et al (2014) would be avoided. Further, as discussed in Nagler and Czado (2016) in detail, by modeling only bivariate copulas nonparametrically the dreaded curse of dimensionality is evaded.…”

“…Moreover, we have seen that there is still room for future research. With the bivariate pair-copulas implemented in the package VineCopula there is currently no possibility to model non-monotonic dependencies between a response and its predictors as is already pointed out in Dette et al (2014). A remedy for the resulting issue of possible misspecification is the inclusion of nonparametric paircopulas in the construction of the D-vine used for quantile regression.…”

Copula-based piecewise regression

“…In [1] it is argued that when the regression function is non-monotone, copula-based regression estimates do not reproduce the qualitative features of the regression function under commonly used parametric copula families. This occurs because very often such parametric copulas lead to monotone regression functions, but in case there is evidence that the underlying regression function is non-monotone a piecewise regression approach may be applied in order to break up a non-monotone relationship into a piecewise monotonic one, and then seek for the best copula fit for each piece.…”

“…The ideas explained in the previous sections may be useful in tackling the concerns raised by [1] when the dependence relationship between random variables implies a non-monotone regression function, considering that the most common families of parametric copulas lead to monotone regression functions, and a possible solution might be to break up such dependence into pieces such that within each one the dependence implies a piecewise monotone regression function, and possibly one of the common families of parametric copulas may have an acceptable fit for each piece. In pursuing this objective, when dealing with data from which the dependence has to be estimated, a methodology to find break-point candidates, that is changepoint detection, becomes necessary.…”

Semiparametric copula quantile regression for complete or censored data

“…Overall, their suggested approach results in a semiparametric regression estimator that is not exposed to the curse of dimensionality. However, in order to avoid possible shortcomings highlighted by Dette et al (2014) that are induced by the misspecification of the parametric copula, we propose in this work, both for complete and censored data, an alternative semiparametric estimation strategy for the copula itself. Our resulting methodology is flexible for multidimensional data with or without censoring, easy to implement and does not require any iterative procedure in opposition to existing semiparametric alternatives.…”