Vine copula regression for observational studies

If explanatory variables and a response variable of interest are simultaneously observed, then fitting a joint multivariate density to all variables would enable prediction via conditional distributions. Regular vines or vine copulas with arbitrary univariate margins provide a rich and flexible class of multivariate densities for Gaussian or non-Gaussian dependence structures. The density enables calculation of all regression functions for any subset of variables conditional on any disjoint set of variables, thereby avoiding issues of transformations, heteroscedasticity, interactions, and higher-order terms. Only the question of finding an adequate vine copula remains. Heteroscedastic prediction inferences based on vine copulas are illustrated with two data sets, including one from the National Longitudinal Study of Youth relating breastfeeding to IQ. Some usual methods based on linear and quadratic equations are shown to have some undesirable inferences.