Copula Regression
I will be adding examples of copula regression here.
Start with a tutorial by Jed, Multivariate Regression Using Copulas
See the review paper entitled "Multivariate Frequency-Severity Regression Models in Insurance" below. The supporting code and data is available at my research site under the header "Wisconsin Property Fund".
For some other approaches to multivariate regression, check out
Correlated Data Analysis (2006), Peter X.-K Song
Vector Generalized Linear and Additive Models (2015) Thomas Yee