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While most regression models focus on explaining aspects of one single response variable alone, interest in modern statistical applications has recently shifted towards simultaneously studying of multiple response variables as well as their dependence structure. A particularly useful tool for pursuing such an analysis are copula-based regression models since they enable the separation of the marginal response distributions and the dependence structure summarised in a specific copula model.
However, so far copula-based regression models have mostly been relying on two-step approaches where the marginal distributions are determined first whereas the copula structure is studied in a second step after plugging in the estimated marginal distributions. Moreover, the parameters of the copula are mostly treated as a constant not related to covariates and most regression specifications for the marginals are restricted to purely linear predictors. We therefore propose simultaneous Bayesian inference for both the marginal distributions and the copula using computationally efficient Markov chain Monte Carlo simulation techniques. In addition, we replace the commonly used linear predictor by a generic structured additive predictor comprising for example nonlinear effects of continuous covariates, spatial effects or random effects and also allow to make the copula parameters covariate-dependent. To facilitate Bayesian inference, we construct proposal densities for a Metropolis Hastings algorithm relying on quadratic approximations to the full conditionals of regression coefficients, thereby avoiding manual tuning and thus providing an attractive adaptive algorithm. The flexibility of Bayesian conditional copula regression models is illustrated in an application on childhood undernutrition.


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