Commentary—Reexamining Bayesian Model-Comparison Evidence of Cross-Brand Pass-Through

Abstract

Using the Bayes factor estimated by harmonic mean [Newton, M. A., A. E. Raftery. 1994. Approximate Bayesian inference by the weighted likelihood bootstrap. J. Roy. Statist. Soc. Ser. B. 56(1) 3–48] to compare models with and without cross-brand pass-through, Dubé and Gupta [Dubé, J.-P., S. Gupta. 2008. Cross-brand pass-through in supermarket pricing. Marketing Sci. 27(3) 324–333] found that, in the refrigerated orange juice category, a model with cross-brand pass-through was selected 68% of the time. However, Lenk [Lenk, P. J. 2009. Simulation pseudo-bias correction to the harmonic mean estimator of integrated likelihoods. J. Comput. Graph. Statist. 18(1) 941–960] has demonstrated that the infinite variance harmonic mean estimator often exhibits simulation pseudo-bias in favor of more complex models. We replicate the results of Dubé and Gupta in the refrigerated orange juice category and then show that any of three more stable finite variance estimators select the model with cross-brand pass-through less than 1% of the time. Relaxing the assumption that model errors are distributed normally eliminates all instances in which the cross-brand pass-through model is selected. In 10 additional categories, the harmonic-mean-estimated Bayes factor selects the model with cross-brand pass-through 69% of the time, whereas a finite variance estimator of the Bayes factor selects the model with cross-brand pass-through only 5% of the time. Applying arguments in McAlister [McAlister, L. 2007. Cross-brand pass-through: Fact or artifact? Marketing Sci. 26(6) 876–898], these 5% of cases can be attributed to capitalization on chance. We conclude that Dubé and Gupta should not be interpreted as providing evidence of cross-brand pass-through.