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Distributional structured additive regression provides a flexible framework for modelling each parameter of a potentially complex response distribution in dependence of covariates. Structured additive predictors allow for an additive decomposition of covariate effects with non-linear effects and time trends, unit- or cluster-specific heterogeneity, spatial heterogeneity and complex interactions between covariates of different type. Within this framework, we present a simultaneous estimation approach for multiplicative random effects that allow for cluster-specific heterogeneity with respect to the scaling of a covariate′s effect. More specifically, a possibly non-linear function f(z) of a covariate z may be scaled by a multiplicative and possibly spatially correlated cluster-specific random effect (1+αc). Inference is fully Bayesian and is based on highly efficient Markov Chain Monte Carlo (MCMC) algorithms. We investigate the statistical properties of our approach within extensive simulation experiments for different response distributions. Furthermore, we apply the methodology to German real estate data where we identify significant district-specific scaling factors. According to the deviance information criterion, the models incorporating these factors perform significantly better than standard models without (spatially correlated) random scaling factors.

Random scaling factors in Bayesian distributional regression models with an application to real estate data