How to Generalize from a Hierarchical Model?

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

Higher Minimum Quality Standards and Redistributive Effects on Consumer Welfare

“…It is well understood in the literature that standard unconstrained hierarchical prior distributions often lack economic rationality (e.g., Reiss and Wolak, 2007;Allenby, Brazell, Howell, and Rossi, 2014;Pachali et al, 2018). As a consequence, market simulations based on the posterior of the hierarchical prior can be misleading and produce counter-factual outcomes that lack face validity.…”

“…As our model incorporates heterogeneous preferences, expected market shares are obtained by integrating over the distribution of households' preferences. We follow Pachali et al (2018) and rely on a procedure defined as lower level model non smoothed (n.s.) to estimate the preference distribution representing the relevant population of German households.…”

“…Explicit posteriors associated with these priors can be found in Pachali et al (2018). The conditionally conjugate prior implies that mean and variance-covariance matrix are a priori independent which allows it to affect µ * cs more explicitly through A −1 µ * c .…”

“…We rely on a modified version of the standard Markov Chain Monte Carlo (MCMC) algorithm in Rossi, Allenby, and McCulloch (2005) that is described in Pachali, Kurz, and Otter (2018) and implement order constraints for the egg label intercepts to explicitly model vertical product differentiation in our demand framework.…”

How Do Nutritional Warning Labels Affect Prices?

“…This is important because the empirical distribution of heterogeneity determines to a large extent how firms adjust prices in response to the introduction of warning labels. We propose a hierarchical Bayesian approach to handle the computational burden in a feasible and intuitive way, using flexible mixtures of normals as first-stage priors, coupled with economically motivated constraints (e.g., Allenby et al 2014; Pachali, Kurz, and Otter 2020).…”