Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution

We consider a binary response model yi=1{xi′βi+εi⩾0} with xi independent of the unobservables (βi, εi). No finite-dimensional parametric restrictions are imposed on F0, the joint distribution of (βi, εi). A nonparametric maximum likelihood estimator for F0 is shown to be consistent. We analyze some conditions under which F0 is or is not identified. In particular, we show that if the support of F0 is a subset of any half of the unit hypersphere, then F0 is identified relative to all distributions on the unit hypersphere. We also provide some Monte Carlo evidence on the small sample performance of our estimator.