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We propose a generic Bayesian framework for inference in distributionalregression models in which each parameter of a potentially complex responsedistribution and not only the mean is related to a structured additivepredictor. The latter is composed additively of a variety of differentfunctional effect types such as nonlinear effects, spatial effects, randomcoefficients, interaction surfaces or other (possibly nonstandard) basisfunction representations. To enforce specific properties of the functionaleffects such as smoothness, informative multivariate Gaussian priors areassigned to the basis function coefficients. Inference can then be based oncomputationally efficient Markov chain Monte Carlo simulation techniques wherea generic procedure makes use of distribution-specific iteratively weightedleast squares approximations to the full conditionals. The framework ofdistributional regression encompasses many special cases relevant for treatingnonstandard response structures such as highly skewed nonnegative responses,overdispersed and zero-inflated counts or shares including the possibility forzero- and one-inflation. We discuss distributional regression along a study ondeterminants of labour incomes for full-time working males in Germany with aparticular focus on regional differences after the German reunification.Controlling for age, education, work experience and local disparities, weestimate full conditional income distributions allowing us to study variousdistributional quantities such as moments, quantiles or inequality measures ina consistent manner in one joint model. Detailed guidance on practical aspectsof model choice including the selection of several competing distributions forlabour incomes and the consideration of different covariate effects on theincome distribution complete the distributional regression analysis. We findthat next to a lower expected income, full-time working men in East Germanyalso face a more unequal income distribution than men in the West, ceterisparibus.

Bayesian structured additive distributional regression with an application to regional income inequality in Germany