How to Generalize from a Hierarchical Model?

Quantitative Marketing and Economics, Forthcoming

36 Pages Posted: 16 Aug 2017 Last revised: 21 Apr 2020

See all articles by Max J. Pachali

Max J. Pachali

Tilburg University - Tilburg University School of Economics and Management

Peter Kurz

bms marketing + strategy

Thomas Otter

Goethe University Frankfurt - Department of Marketing

Date Written: March 26, 2020

Abstract

Models of consumer heterogeneity play a pivotal role in marketing and economics, specifically in random coefficient or mixed logit models for aggregate or individual data and in hierarchical Bayesian models of heterogeneity. In applications, the inferential target often pertains to a population beyond the sample of consumers providing the data. For example, optimal prices inferred from the model are expected to be optimal in the population and not just optimal in the observed, finite sample. The population model, random coefficients distribution, or heterogeneity distribution is the natural and correct basis for generalizations from the observed sample to the market. However, in many if not most applications standard heterogeneity models such as the multivariate normal, or its finite mixture generalization lack economic rationality because they support regions of the parameter space that contradict basic economic arguments. For example, such population distributions support positive price coefficients or preferences against fuel-efficiency in cars. Likely as a consequence, it is common practice in applied research to rely on the collection of individual level mean estimates of consumers as a representation of population preferences that often substantially reduce the support for parameters in violation of economic expectations. To overcome the choice between relying on a mis-specified heterogeneity distribution and the collection of individual level means that fail to measure heterogeneity consistently, we develop an approach that facilitates the formulation of more economically faithful heterogeneity distributions based on prior constraints. In the common situation where the heterogeneity distribution comprises both constrained and unconstrained coefficients (e.g., brand and price coefficients), the choice of subjective prior parameters is an unresolved challenge. As a solution to this problem, we propose a marginal-conditional decomposition that avoids the conflict between wanting to be more informative about constrained parameters and only weakly informative about unconstrained parameters. We show how to efficiently sample from the implied posterior and illustrate the merits of our prior as well as the drawbacks of relying on means of individual level preferences for decision-making in two illustrative case studies.

Keywords: Discrete Choice, Bayesian Inference, Market Simulation, Constrained Hierarchical Prior

Suggested Citation

Pachali, Max J. and Kurz, Peter and Otter, Thomas, How to Generalize from a Hierarchical Model? (March 26, 2020). Quantitative Marketing and Economics, Forthcoming, Available at SSRN: https://ssrn.com/abstract=3018670 or http://dx.doi.org/10.2139/ssrn.3018670

Max J. Pachali (Contact Author)

Tilburg University - Tilburg University School of Economics and Management ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

HOME PAGE: http://sites.google.com/site/mjpachali/

Peter Kurz

bms marketing + strategy ( email )

Landsberger Str. 487
Munich, Bavaria 81241
Germany

Thomas Otter

Goethe University Frankfurt - Department of Marketing ( email )

Frankfurt
Germany
++49.69.798.34646 (Phone)

HOME PAGE: http://www.marketing.uni-frankfurt.de/index.php?id=97?&L=1

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