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 identify 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 helps identify the mean of B given V. Averaging over V identifies the mean of B. This leads to an analogous localize-then-average approach to estimation. We estimate conditional means with localized smooth maximum score estimators and average to obtain a root-n-consistent and asymptotically normal estimator of a trimmed mean of the distribution of B. Under the conditional median restrictions, the procedure can be adapted to produce a root-n-consistent and asymptotically normal estimator of the nonrandom regression coefficients in the models of Manski (1975,1985) and Horowitz (1992). We explore small sample performance through simulations, and present an application.