Monotonic regression based on Bayesian P-splines: an application to estimating price response functions from store-level scanner data

Generalized additive models have become a widely used instrument for flexible regression analysis. In many practical situations, however, it is desirable to restrict the flexibility of nonparametric estimation in order to accommodate a presumed monotonic relationship between a covariate and the response variable. For example, consumers usually will buy less of a brand if its price increases, and therefore one expects a brand's unit sales to be a decreasing function in own price. We follow a Bayesian approach using penalized B-splines and incorporate the assumption of monotonicity in a natural way by an appropriate specification of the respective prior distributions. We illustrate the methodology in an empirical application modeling demand for a brand of orange juice and show that imposing monotonicity constraints for own- and cross-item price effects improves the predictive validity of the estimated sales response function considerably

Semiparametric Stepwise Regression to Estimate Sales Promotion Effects

Kalyanam and Shively (1998) and van Heerde et al. (2001) have proposed semiparametric models to estimate the influence of price promotions on brand sales, and both obtained superior performance for their models compared to strictly parametric modeling. Following these researchers, we suggest another semiparametric framework which is based on penalized B-splines to analyze sales promotion effects flexibly. Unlike these researchers, we introduce a stepwise procedure with simultaneous smoothing parameter choice for variable selection. Applying this stepwise routine enables us to deal with product categories with many competitive items without imposing restrictions on the competitive market structure in advance. We illustrate the new methodology in an empirical application using weekly store-level scanner data

Semiparametric Multinomial Logit Models for Analysing Consumer Choice Behaviour

The multinomial logit model (MNL) is one of the most frequently used statistical models in marketing applications. It allows to relate an unordered categorical response variable, for example representing the choice of a brand, to a vector of covariates such as the price of the brand or variables characterising the consumer. In its classical form, all covariates enter in strictly parametric, linear form into the utility function of the MNL model. In this paper, we introduce semiparametric extensions, where smooth effects of continuous covariates are modelled by penalised splines. A mixed model representation of these penalised splines is employed to obtain estimates of the corresponding smoothing parameters, leading to a fully automated estimation procedure. To validate semiparametric models against parametric models, we utilise proper scoring rules and compare parametric and semiparametric approaches for a number of brand choice data sets

A semiparametric approach to estimating reference price effects in sales response models

It is well known that store-level brand sales may not only depend on contemporaneous influencing factors like current own and competitive prices or other marketing activities, but also on past prices representing customer response to price dynamics. On the other hand, non- or semiparametric regression models have been proposed in order to accommodate potential nonlinearities in price response, and related empirical findings for frequently purchased consumer goods indicate that price effects may show complex nonlinearities, which are difficult to capture with parametric models. In this contribution, we combine nonparametric price response modeling and behavioral pricing theory. In particular, we propose a semiparametric approach to flexibly estimating price-change or reference price effects based on store-level sales data. We compare different representations for capturing symmetric vs. asymmetric and proportional vs. disproportionate price-change effects following adaptation-level and prospect theory, and further compare our flexible autoregressive model specifications to parametric benchmark models. Functional flexibility is accommodated via P-splines, and all models are estimated within a fully Bayesian framework. In an empirical study, we demonstrate that our semiparametric dynamic models provide more accurate sales forecasts for most brands considered compared to competing benchmark models that either ignore price dynamics or just include them in a parametric way

Bayesian Geoadditive Seemingly Unrelated Regression

Parametric seemingly unrelated regression (SUR) models are a common tool for multivariate regression analysis when error variables are reasonably correlated, so that separate univariate analysis may result in inefficient estimates of covariate effects. A weakness of parametric models is that they require strong assumptions on the functional form of possibly nonlinear effects of metrical covariates. In this paper, we develop a Bayesian semiparametric SUR model, where the usual linear predictors are replaced by more flexible additive predictors allowing for simultaneous nonparametric estimation of such covariate effects and of spatial effects. The approach is based on appropriate smoothness priors which allow different forms and degrees of smoothness in a general framework. Inference is fully Bayesian and uses recent Markov chain Monte Carlo techniques

Conjoint-Analyse und Marktsegmentierung

Die Marktsegmentierung zählt neben der Neuproduktplanung und Preisgestaltung zu den wesentlichen Einsatzgebieten der Conjoint-Analyse. Neben traditionell eingesetzten zweistufigen Vorgehensweisen, bei denen Conjoint-Analyse und Segmentierung in zwei getrennten Schritten erfolgen, stehen heute mit Methoden wie der Clusterwise Regression oder Mixture-Modellen neuere Entwicklungen, die eine simultane Segmentierung und Präferenzschätzung ermöglichen, zur Verfügung. Der Beitrag gibt einen Überblick über die vorliegenden methodischen Ansätze zur Verknüpfung von Conjoint-Analyse und Marktsegmentierung und zeigt die Vorzüge simultaner Conjointsegmentierungsmethoden gegenüber den in der Unternehmenspraxis noch immer weit verbreiteten zweistufigen Verfahren auf. Along with new product/concept identification and pricing, market segmentation ranks among the primary purposes in commercial conjoint applications. Traditionally, this conjoint segmentation has been accomplished by a two-step procedure, (1) either by first segmenting markets and subsequently estimating conjoint models at the segment level or (2) by first conducting conjoint analysis at the individual level and then clustering individual level part-worths. However, in recent years, some powerful techniques for simultaneously performing market segmentation and calibrating segment-level part-worths such as clusterwise regression procedures and mixture models have been proposed. This article provides an overview of existing conjoint segmentation methods and particularly focuses on the newer simultaneous approaches which offer substantial advantages compared to the traditional two-step procedures.Marktsegmentierung; Conjoint-Analyse ; Simultanverfahren;

Additive models with random scaling factors: Applications to modeling price response functions

We discuss inference for additive models with random scaling factors. The additive effects are of the form (1+g)f(z) where f is a nonlinear function of the continuous covariate z modeled by P(enalized)-splines and 1+g is a random scaling factor. Additionally, monotonicity constraints on the nonlinear functions are possible. Our work is motivated by the situation of a retailer analyzing the impact of price changes on a brand's sales in its orange juice product category. Relating sales to a brand's own price as well as to the prices of competing brands in the category, we estimate own- and cross-item price response functions flexibly to represent nonlinearities and irregular pricing effects in sales response. Monotonicity constraints are imposed so that a brand's own price is inversely related and the prices of competing brands are directly related to the number of items sold, as suggested by economic theory. Unobserved store-specific heterogeneity is accounted for by allowing the price response curves to vary between different stores.@Wir behandeln additive Modelle mit zufälligen Skalierungsfaktoren. Die additive Effekte haben die Form (1 + ã)f (z). f ist eine nichtlineare Funktion der stetigen Kovariable z, modelliert mittels P(enalized)- splines und 1 + ã ist ein zufälliger Skalierungsfaktor. Den nichtlinearen Funktionen können zusätzlich Monotonierestriktionen auferlegt werden. Den Ausgangspunkt unserer Arbeit bildet die Situation eines Einzelhändlers, der den Einfluss von Preisänderungen auf den Absatz einer Orangensaftmarke in seinem Sortiment analysieren möchte. Eine entsprechende Absatzreaktionsfunktion lässt sich schätzen, indem der Absatz der betrachteten Marke als nichtlineare Funktion des eigenen Preises sowie der Preise der Konkurrenzmarken modelliert wird. Monotonierestriktionen für die Preiseffekte gewährleisten darüber hinaus einen inversen Verlauf des Absatzes bezüglich des eigenen Preises sowie eine direkte Beziehung des Absatzes zu Konkurrenzpreisen, wie es in Anlehnung an die ökonomische Preistheorie zu erwarten ist. Unbeobachtete Heterogenität wird berücksichtigt, indem die Preiseffekte über die einzelnen Geschäfte des Händlers zufällig variieren können

Report on the First Working Group Meeting of the “AG Marketing”

This contribution reports on the first meeting of the new formed working group “Data Analysis and Classification in Marketing (AG Marketing)” of the data science society (GfKl) held at the KIT, Karlsruhe, November 14th – 15th, 2019. The abstracts of the presentations given reflect the ongoing trend to exploit a large variety of digital data sources for marketing purposes and the need for advanced and innovative analysis methods