Identification and estimation in a correlated random coefficients binary response model

We study identification and estimation in a binary response model with random coefficients B allowed to be correlated with regressors X. Our objective is to identifiy the mean of the distribution of B and estimate a trimmed mean of this distribution. Like Imbens and Newey (2009), we use instruments Z and a control vector V to make X independent of B given V. A consequent conditional median restriction identifies the mean of B given V. Averaging over V identifies the mean of B. This leads to an analogous localise-then-average approach to estimation. We estimate conditional means with localised smooth maximum score estimators and average to obtain a √n-consistent and asymptotically normal estimator of a trimmed mean of the distribution of B. The method can be adapted to models with nonrandom coefficients to produce √n-consistent and asymptotically normal estimators under the conditional median restrictions. We explore small sample performance through simulations, and present an application.