Marketing variables that are included in consumer discrete choice models are often endogenous. Extant treatments using likelihood-based estimators impose parametric distributional assumptions, such as normality, on the source of endogeneity. These assumptions are restrictive because misspecified distributions have an impact on parameter estimates and associated elasticities. The normality assumption for endogeneity can be inconsistent with some marginal cost specifications given a price-setting process, although they are consistent with other specifications. In this paper, we propose a heterogeneous Bayesian semiparametric approach for modeling choice endogeneity that offers a flexible and robust alternative to parametric methods. Specifically, we construct centered Dirichlet process mixtures (CDPM) to allow uncertainty over the distribution of endogeneity errors. In a similar vein, we also model consumer preference heterogeneity nonparametrically via a CDPM. Results on simulated data show that incorrect distributional assumptions can lead to poor recovery of model parameters and price elasticities, whereas the proposed semiparametric model is able to robustly recover the true parameters in an efficient fashion. In addition, the CDPM offers the benefits of automatically inferring the number of mixture components that are appropriate for a given data set and is able to reconstruct the shape of the underlying distributions for endogeneity and heterogeneity errors. We apply our approach to two scanner panel data sets. Model comparison statistics indicate the superiority of the semiparametric specification and the results show that parameter and elasticity estimates are sensitive to the choice of distributional forms. Moreover, the CDPM specification yields evidence of multimodality, skewness, and outlying observations in these real data sets.
Li, Yang, and Asim Ansari. "A Bayesian Semiparametric Approach for Endogeneity and Heterogeneity in Choice Models." Management Science 60, no. 5 (2014): 1161-1179.
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