## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 9, Number 2 (2015), 1024-1052.

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

Nadja Klein, Thomas Kneib, Stefan Lang, and Alexander Sohn

#### 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 on practical aspects of model choice including the selection of several competing distributions for labour incomes and the consideration of different covariate effects on the income distribution complete the distributional regression analysis. We find that next to a lower expected income, full-time working men in East Germany also face a more unequal income distribution than men in the West, ceteris paribus.

#### Article information

**Source**

Ann. Appl. Stat., Volume 9, Number 2 (2015), 1024-1052.

**Dates**

Received: September 2013

Revised: March 2015

First available in Project Euclid: 20 July 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1437397122

**Digital Object Identifier**

doi:10.1214/15-AOAS823

**Mathematical Reviews number (MathSciNet)**

MR3371346

**Zentralblatt MATH identifier**

06499941

**Keywords**

Generalised additive models for location scale and shape income distribution iteratively weighted least squares proposal Markov chain Monte Carlo semiparametric regression wage inequality

#### Citation

Klein, Nadja; Kneib, Thomas; Lang, Stefan; Sohn, Alexander. Bayesian structured additive distributional regression with an application to regional income inequality in Germany. Ann. Appl. Stat. 9 (2015), no. 2, 1024--1052. doi:10.1214/15-AOAS823. https://projecteuclid.org/euclid.aoas/1437397122

#### Supplemental materials

- Supplement A: Case studies. Additional material on the application to regional income inequality in Germany is provided in Section A.1. A second case study on the proportion of farm outputs achieved by cereals is treated in Section A.2.Digital Object Identifier: doi:10.1214/15-AOAS823SUPPASupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.
- Supplement B: Methodology. This supplement comprises details on Bayesian inference, derivations of required quantities for the iteratively weighted least squares proposals and simulation studies.Digital Object Identifier: doi:10.1214/15-AOAS823SUPPBSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.