Center for Social Information Sciences (CSIS) Seminar

Abstract: Consider a discrete choice problem where each alternative is associated with an unknown price distribution. A selection function maps a realized price vector to a probability distribution over the alternatives. We observe only the price of an alternative conditional on it being selected. Given the selection function and the observed selected price distributions, we nonparametrically recover the unselected price distributions. We achieve this by constructing an operator whose fixed point is the unselected price distributions, and showing that it is a functional contraction.

Building on this result, we propose a multi-step semiparametric maximum likelihood estimator for the selection function. The consistency and asymptotic normality of the estimator are established. Our estimator is applicable in various empirical settings where only a selected sample of outcomes is observed. Examples include consumer demand models where only transaction prices are available, auctions that record only winning bids, and Roy models with data on accepted wages.

Written with Yi Xin.