Semiparametric estimation of multinomial discrete-choice models using a subset of choices

Nonlogit maximum-likelihood estimators are inconsistent when using data on a subset of the choices available to agents. I show that the semiparametric, multinomial maximum-score estimator is consistent when using data on a subset of choices. No information is required for choices outside of the subset. The required conditions about the error terms are the same conditions as for using all the choices. Estimation can proceed under additional restrictions if agents have unobserved, random consideration sets. A solution exists for instrumenting endogenous continuous variables. Monte Carlo experiments show the estimator performs well using small subsets of choices.