We propose a new methodology for structural estimation of dynamic discrete choice models. We combine the Dynamic Programming (DP) solution algorithm with the Bayesian Markov Chain Monte Carlo algorithm into a single algorithm that solves the DP problem and estimates the parameters simultaneously. As a result, the computational burden of estimating a dynamic model becomes comparable to that of a static model. Another feature of our algorithm is that even though per solution-estimation iteration, the number of grid points on the state variable is small, the number of effective grid points increases with the number of estimation iterations. This is how we help ease the "Curse of Dimensionality". We simulate and estimate several versions of a simple model of entry and exit to illustrate our methodology. We also prove that under standard conditions, the parameters converge in probability to the true posterior distribution, regardless of the starting values.

By using a unique data set that contains detailed information about consumer payment choice and consumers’ attitudes toward each payment method, we estimate the effects of payment card rewards on consumer choice of payment methods. Our approach allows us to control for consumer heterogeneity. We find the effects of rewards to be statistically significant across five retail types. Our policy experiments suggest that for the sub-population who hold both credit and debit cards, removing rewards would increase their share of paper-based payment methods (i.e., cash and checks), measured in terms of in-store transactions, by no more than 4 percentage points.