No Customer Left Behind : A Distribution-Free Bayesian Approach to Accounting for Missing Xs in Marketing Models

In marketing applications, it is common that some key covariates in a regression model are subject to missingness. Notably, in the estimation of discrete choice models using scanner data, the prices and promotion values for non-purchased products are often missing. In consumer relationship management (CRM), some important consumer profiles may be missing. The convenient method that excludes the consumers with missingness in any covariate can result in substantial loss in efficiency, and may lead to strong selection bias in the estimation of consumer preference and sensitivity. In this paper, we propose a new Bayesian distribution-free approach to handle the missing covariates problem. This approach allows for flexible modeling of a joint distribution of multi-dimensional covariates that can contain both continuous and discrete variables. At the same time it minimizes the impact of distributional assumptions involved in covariates modeling because the method does not require researchers to specify parametric distributions for covariates and can automatically generate suitable distributions for missing covariates. We develop an MCMC algorithm for inference. The MCMC procedure contains an efficient Hybrid Monte Carlo (HMC) sampler to update parameters in the covariate model. Besides robustness and flexibility, the proposed approach reduces the marketing analysts’ modeling and computational efforts associated with missing covariates, and therefore makes the missing covariate problem easier to handle. We illustrate the method in two real data examples where missing covariates occur: a mixed multinomial logit discrete choice model in a ketchup dataset and a hierarchical probit purchase incidence model in a retail store dataset. We also evaluate the performance of the proposed method in repeated samples using extensive simulation studies. These analyses demonstrate that the proposed method possesses several unique features and benefits for handling missing covariate problems, as compared with alternative approaches. The method is useful to correct for the selection bias due to covariate missingness, and can substantially improve the efficiency of analysis. Using the proposed method, researchers can make better managerial decisions with the current available marketing databases. Our method also ensures that no customer is left behind for CRM and individualized marketing. The proposed method is general and can be applied to a wide range of marketing applications. Although our applications focus on consumer-level data, the proposed method can be applied to other marketing applications where other types of marketing players are the units of analysis