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 salesmodels. To estimate ownand 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 ownand 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. © 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved. 1. Motivation and literature review In recent years, two streams of research for estimating sales response models based on store-level data have evolved: on the one hand, researchers have proposed hierarchical Bayesian (HB) store sales models allowing for heterogeneity of marketing effects across stores (e.g., Andrews, Currim, Leeflang, & Lim, 2008; Blattberg & George, 1991; Boatwright, McCulloch, & Rossi, 1999; Hruschka, 2006b; Montgomery, 1997; Montgomery & Rossi, 1999). While some of these studies have shown that considering heterogeneity can improve model fit, the accuracy of sales forecasts, or expected profits (e.g., Hruschka, 2006b; Montgomery, 1997), recent research of Andrews et al. (2008) has demonstrated rather marginal improvements in fit and predictive performance from incorporating store ∗ Corresponding author. Tel.: +49 5323 72 7650; fax: +49 5323 72 7659. E-mail addresses: [email protected] (S. Lang), [email protected] (W.J. Steiner), [email protected] (A. Weber), [email protected] (P. Wechselberger). 1 Tel.: +43 512 507 7110; fax: +43 512 507 2851. 2 Tel.: +49 5323 72 7658. 3 Tel.: +49 89 4423400 39. heterogeneity. One possible reason for this latter finding is that the HB models mentioned above assume a strictly parametric functional form thereby limiting the scope for model calibration to an a priori fixed parametrization. Hence, although accounting for heterogeneity, a source of bias remains if the assumed parametric form differs from the true underlying function. On the other hand, researchers have proposed nonparametric regression models in order to accommodate potential nonlinearities in store sales response (e.g., Brezger & Steiner, 2008; Haupt, Kagerer, & Steiner, 2014; van Heerde, Leeflang, & Wittink, 2001; Kalyanam & Shively, 1998; Steiner, Brezger, & Belitz, 2007). The empirical results of this second stream indicate that ownand cross-price effects may show complex nonlinearities which are difficult or not at all to capture by parametric models. The main potential weakness of this second group of nonparametric approaches, however, is that heterogeneity across stores has not been considered. Consequently, bias due to potential heterogeneity across stores here remains. There is so far only one approach that has consolidated the two streams: Hruschka (2006a, 2007) proposed a hierarchical Bayesian multilayer perceptron (MLP) that allows for nonlinearity in price effects and yields store-specific coefficients. In an empirical study, his flexible heterogenous MLP turned out to be superior in terms of http://dx.doi.org/10.1016/j.ejor.2015.02.047 0377-2217/© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved. S. Lang et al. / European Journal of Operational Research 246 (2015) 232–241 233 Table 1 Descriptive statistics for weekly brand prices, market shares, and unit sales. Refrigerated orange juice category (64 oz) Brand Retail price Market share Unit sales Range ($) Mean ($) SD ($) Range (percent) Mean (percent) SD (percent) Minimum Maximum Premium brands Tropicana Pure [1.60; 3.55] 2.95 .53 [3;73] 15 15 6388 100,712 Florida Natural [1.57; 3.16] 2.86 .33 [1;53] 5 7 1138 56,037 National brands Citrus Hill [1.09; 2.82] 2.31 .31 [1;78] 8 12 2006 151,570 Minute Maid [1.29; 2.92] 2.23 .40 [3;87] 21 22 4805 243,711 Tropicana [1.49; 2.75] 2.20 .35 [2;75] 21 23 3041 102,629 Florida Gold [.99; 2.83] 2.17 .39 [1;63] 4 8 325 150,945 Tree Fresh [1.07; 2.48] 2.15 .27 [1;42] 4 6 916 39,401 Store brand Dominick’s [.99; 2.47] 1.75 .4 [1;83] 22 22 2170 189,462 a The unit sales of all eight brands amount to 96.25 percent of the total sales volume in the refrigerated orange juice category (64 oz) during the time span considered. b Reading example: For Tropicana Pure the lowest observed price across all stores and weeks was 1.60 $, its lowest market share (unit sales) in a week pooled across stores was 3 percent (6388 units), and its mean price level averaged across all stores and weeks was 2.95 $. posterior model probability (Hruschka, 2006a) and further with respect to predictive validity (Hruschka, 2007) compared to a heterogeneous parametric multiplicative model, respectively. In addition, Hruschka (2007) analyzed profit implications for the flexible heterogeneousMLP froma retailer’s point of view. Specifically, he shows that taking menu costs into account a moderately risk averse retailer may prefer a clusterwise pricing strategy to a store-specific pricing strategy. Ifmenucosts are ignored, expectedprofits however increasewith the number of clusters and reach their maximum for a store-specific pricing strategy.3 Our approach proposed here differs from that of Hruschka (2006a, 2007) in several ways: First, we will show that accounting for store heterogeneity alone might not be advantageous per se (as is assumed byHruschka), and that accounting for functional flexibility is the primary driver for model improvement (at least for our data). In particular, we find that allowing for heterogeneity in addition to functionalflexibility can improve thepredictiveperformance of a store salesmodel considerably,while incorporatingheterogeneity alone onlymoderately improves or evendecreases predictive validity. Second, we illustrate why accommodating store heterogeneity may pay off only once nonlinearity in price response is modeled appropriately. And third, we compare our flexible heterogeneous model in terms of expected profits to competing models that capture heterogeneity or functional flexibility only (or neither of them). This way, we investigate howmuch loss in expected profitsmanagement incurs by not using the model with the highest predictive performance. In the following, we develop a store sales model which accommodates both functional flexibility and heterogeneity within a unified regression framework. We propose a structured additive approach where ownand cross-price response is estimated flexibly using P-splines, while heterogeneity in price response across stores is simultaneously accommodated by multiplicative store-specific random effects that scale the nonlinear price curves while their overall shape is preserved. The rest of the paper is organized as follows. In Section 2, we introduce our hierarchical Bayesian semiparametric model. Using store-level data from Dominick’s Finer Foods we analyze in Section 3 whether, how much, and why the performance of a store sales model can be improved from considering either heterogeneity, functional flexibility, or both features. Our results indicate that the most complex model accommodating both heterogeneity and func3 The advantages of using nonparametric or seminonparametric techniques for estimating response functions were demonstrated previously by Hruschka in the context of market share modeling (Hruschka, 2002), brand choice modeling (Hruschka, Fettes, & Probst, 2004) and catalog allocation modeling (Baumgartner & Hruschka, 2005), too. tional flexibility outperforms competing models in predictive validity for most brands, and provides substantial increases in expected profits for those brands. We further show that the proposed model leads to materially different pricing implications. We conclude in Section 4 with an outlook on future research perspectives. 2. Data and model framework