  This paper considers (partial) identification of a variety of parameters,
including counterfactual choice probabilities, in a general class of binary
response models with possibly endogenous regressors. Importantly, our framework
allows for nonseparable index functions with multi-dimensional latent
variables, and does not require parametric distributional assumptions. We
demonstrate how various functional form, independence, and monotonicity
assumptions can be imposed as constraints in our optimization procedure to
tighten the identified set, and we show how these assumptions have meaningful
interpretations in terms of restrictions on latent types. In the special case
when the index function is linear in the latent variables, we leverage results
in computational geometry to provide a tractable means of constructing the
sharp set of constraints for our optimization problems. Finally, we apply our
method to study the effects of health insurance on the decision to seek medical
treatment.
