Bayesian estimation of a simultaneous probit model using error augmentation: An application to multi-buying and churning behavior

Researchers in marketing are often interested in analyzing how an agent's discrete choice decision affects a subsequent or concurrent discrete choice decision by the same or different agent. This analysis may necessitate the use of a simultaneous equations model with discrete and continuous endogenous variables as explanatory variables. In this paper, we offer an error augmentation approach to Hierarchical Bayesian estimation of a simultaneous bivariate probit model containing both discrete and continuous endogenous variables. We accomplish the error augmentation in our MCMC algorithm using a Metropolis-Hastings step that generates the error components of the latent variables in our model. Using simulated data, we demonstrate that our error augmentation algorithm recovers closely the true parameters of the simultaneous bivariate probit model. We then apply our algorithm to customer churn data from a wireless service provider. We formulate a simultaneous bivariate probit model to study the impact of a customer's multiple product relationships with a firm (multibuying) on the likelihood of churn by that customer. The empirical results show that the act of multi-buying significantly reduces churn even though the customers who are more predisposed to multi-buy have an inherently higher predisposition to churn.