From the abstract:
In this paper, we present copula regression as an alternative to OLS and GLM. The major advantage of a copula regression is that there are no restrictions on the probability distributions that can be used. In this paper, we will present the formulas and algorithms necessary for conducting a copula regression analysis using the normal copula. However, the ideas presented here can be used with any copula function that can incorporate multiple variables with varying degrees of association.
In the paper they outline a 3 step process for accomplishing this:
1) Assume a model for the joint distribution of all the variables (response and covariates)
2) Estimate the parameters of the model (the parameters for the selected marginal distributions and the parameters of the copula)
3) Compute the predicted values of Y given a set of covariates by using the conditional mean of Y given the covariates.
I'd also like to point out this interesting virtual course on copula regression from Dr. Edward Frees at the University of Wisconsin: https://sites.google.com/a/wisc.edu/jed-frees/multivariate-regression-using-copulas
I have not had a chance to view these materials in detail, but absolutely think it could be valuable to anyone wanting to learn more about these methods.
Modeling Dependence with Copulas and R
Copula Based Agricultural Risk Models
Intro to Copulas using SAS
Copulas, R, and the Financial Crisis
Parsa, Rahul A, and Stuart A. Klugman, "Copula Regression," Variance 5:1, 2011, pp. 45-54.