Bivariate copula additive models for location, scale and shape

In generalized additive models for location, scale and shape (GAMLSS), the response distribution is not restricted to belong to the exponential family and all the model’s parameters can be made dependent on additive predictors that allow for several types of covariate effects (such as linear, non-linear, random and spatial effects). In many empirical situations, however, modeling simultaneously two or more responses conditional on some covariates can be of considerable relevance. The scope of GAMLSS is extended by introducing bivariate copula models with continuous margins for the GAMLSS class. The proposed computational tool permits the copula dependence and marginal distribution parameters to be estimated simultaneously, and each parameter to be modeled using an additive predictor. Simultaneous parameter estimation is achieved within a penalized likelihood framework using a trust region algorithm with integrated automatic multiple smoothing parameter selection. The introduced approach allows for straightforward inclusion of potentially any parametric marginal distribution and copula function. The models can be easily used via the copulaReg() function in the R package SemiParBIVProbit. The proposal is illustrated through two case studies and simulated data.