  This paper considers (partial) identification of a variety of counterfactual
parameters in binary response models with possibly endogenous regressors. Our
framework allows for nonseparable index functions with multi-dimensional latent
variables, and does not require parametric distributional assumptions. We
leverage results on hyperplane arrangements and cell enumeration from the
literature on computational geometry in order to provide a tractable means of
computing the identified set. We demonstrate how various functional form,
independence, and monotonicity assumptions can be imposed as constraints in our
optimization procedure to tighten the identified set. Finally, we apply our
method to study the effects of health insurance on the decision to seek medical
treatment.
