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 means of individual level preferences for decision-making in two illustrative case studies.

Omitted Budget Constraint Bias and Implications for Competitive Pricing

“…We adopt the marginal-conditional decomposition in Pachali, Kurz, and Otter (2020) to sign-constrain the budget and price coefficients in line with economic theory in the flexible S-component mixture of normals hierarchical prior/random coefficient distribution:…”

“…We adopt the marginal-conditional decomposition in Pachali, Kurz, and Otter (2020) to sign-constrain the budget and price coefficients in line with economic theory in the flexible S-component mixture of normals hierarchical prior/random coefficient distribution: where normalZ ¯ normali is row i of the demeaned matrix of financial demographic variables, such as disposable income and liquid funds; Δ* relates demographics to budget, price, and remaining demand coefficients (as suggested by, e.g., Petrin [2002]); false( μ normals * , normalV normals * false) are component-specific mean and variance-covariance parameters; and η s measures the weight of mixture component s. We need to take into account that Δ* is estimated on the log scale, and Web Appendix B shows how to arrive at the estimates of the relationship between demographics and sign-constrained demand parameters. We decide the number of mixture components (S) in the empirical application by comparing approximations to the marginal likelihood of different model specifications (similarly to the approach proposed in Dubé, Hitsch, and Rossi [2010]).…”

“…Algorithm 1 summarizes each step of our Markov chain Monte Carlo (MCMC) method, based on the DAG in Figure 1. The priors and subjective priors for the upper-level coefficients are specified as proposed in Pachali, Kurz, and Otter (2020), accounting for the change of variable to impose economic constraints. The prior for the error variance follows an inverse chi-square distribution with subjective prior parameters specified as suggested in Rossi, Allenby, and McCulloch (2005).…”

Firm Competition and Cooperation with Norm‐Based Preferences for Sustainability

“…Taken together, when now comparing the willingness to pay for the more sustainable with that for the less sustainable alternative, as perceived market share shifts from the 28 While here the coefficients 𝛽 i,k are assumed to be normally distributed, we note that for the sustainbility attribute(s) also order (sign) constraints would be appropriate, so that 𝛽 i,sust ≥ 0. We have also estimated the model by substituting the respective coefficients by e 𝛽 i,sust , albeit we have thereby used a Bayesian approach (implementing the techniques and code developed in Pachali et al [2020]). The values for the (expected) willingness-to-pay, including for the interactive term, are largely comparable.…”