Accommodating heterogeneity and nonlinearity in price effects for predicting brand sales and profits

We propose a hierarchical Bayesian semiparametric approach to account simultaneously for heterogeneity and functional flexibility in store sales models. To estimate own- and cross-price response flexibly, a Bayesian version of P-splines is used. Heterogeneity across stores is accommodated by embedding the semiparametric model into a hierarchical Bayesian framework that yields store-specific own- and cross-price response curves. More specifically, we propose multiplicative store-specific random effects that scale the nonlinear price curves while their overall shape is preserved. Estimation is fully Bayesian and based on novel MCMC techniques. In an empirical study, we demonstrate a higher predictive performance of our new flexible heterogeneous model over competing models that capture heterogeneity or functional flexibility only (or neither of them) for nearly all brands analyzed. In particular, allowing for heterogeneity in addition to functional flexibility can improve the predictive performance of a store sales model considerably, while incorporating heterogeneity alone only moderately improved or even decreased predictive validity. Taking into account model uncertainty, we show that the proposed model leads to higher expected profits as well as to materially different pricing recommendations.