Econometrics and Applied Micro Seminar

As the (Gale-Shapley) Deferred-Acceptance mechanism becomes more popular in school choice, many attempts have been made to estimate student preferences from rank-order school choice data that have been increasingly available. Under the assumption that students rank schools truthfully, discrete choice models as those in the product-demand estimation are usually applied. However, this approach requires truth-telling to be the unique equilibrium, which the mechanism does not guarantee. Instead, it is less restrictive to assume that the matching outcome is stable, i.e., every student is assigned to her most preferred school among those she is qualified for. We focus on the case where schools strictly rank students by scores known by students and researchers. Given the observability of school preferences, stability implies a discrete choice model with personalized choice sets. We theoretically model school choice as a game of incomplete information and show that the outcome is asymptotically stable in equilibrium -- when holding the number of schools constant and increasing the number of students, the fraction of students who have a profitable unilateral deviation after observing the realization of the game goes to zero. While both truth-telling and stability assumptions lead to maximum likelihood estimation under the usual parametric assumptions, we also provide an approach that only relies on the moment inequalities implied by undominated strategies even if stability is not satisfied. A set of tests are proposed for model selection. We illustrate the estimation and testing in Monte Carlo simulations. When applying the methods to school choice data from Paris, we strongly reject the hypothesis that students are truth-telling. Comparing with our preferred estimates from stability (with or without moment inequalities), incorrectly imposing truth-telling leads to a serious under-estimation of the qualities of popular schools.