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

Abstract

We propose a generic Bayesian framework for inference in distributional regression models in which each parameter of a potentially complex response distribution and not only the mean is related to a structured additive predictor. The latter is composed additively of a variety of different functional effect types such as nonlinear effects, spatial effects, random coefficients, interaction surfaces or other (possibly nonstandard) basis function representations. To enforce specific properties of the functional effects such as smoothness, informative multivariate Gaussian priors are assigned to the basis function coefficients. Inference can then be based on computationally efficient Markov chain Monte Carlo simulation techniques where a generic procedure makes use of distribution-specific iteratively weighted least squares approximations to the full conditionals. The framework of distributional regression encompasses many special cases relevant for treating nonstandard response structures such as highly skewed nonnegative responses, overdispersed and zero-inflated counts or shares including the possibility for zero-and one-inflation. We discuss distributional regression along a study on determinants of labour incomes for full-time working males in Germany with a particular focus on regional differences after the German reunification. Controlling for age, education, work experience and local disparities, we estimate full conditional income distributions allowing us to study various distributional quantities such as moments, quantiles or inequality measures in a consistent manner in one joint model. Detailed guidance

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“…The GAMLSS (or "distributional regression") models discussed by Rigby and Stasinopoulos (2005) (and also Yee and Wild 1996;Klein et al 2014Klein et al , 2015 fall within the scope of the general method. The idea is that we have independent univariate response observations, y i , whose distributions depend on several unknown parameters, each of which is determined by its own linear predictor.…”

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“…Therefore, a backfitting algorithm for finding posterior mode estimates is commonly applied before running the MCMC simulation. Details on the implementation of Bayesian inference in distributional regression can be found in Klein et al (2015d).…”

“…Klein et al (2015d) provide details on different scores for various types of responses. Basically the goal of all specifications is to evaluate the predictive ability based on rules that favor honest predictions while simultaneously taking sharpness (i.e.…”

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