SEMIPARAMETRIC ESTIMATION OF A BINARYRESPONSE MODEL WITH A CHANGE-POINTDUE TO A COVARIATE THRESHOLD

This paper is concerned with semiparametric estimation of a threshold binaryresponse model. The estimation method considered in the paper is semiparametricsince the parameters for a regression function are finite-dimensional, whileallowing for heteroskedasticity of unknown form. In particular, the paper considersManski (1975, 1985)'s maximum score estimator. The model in this paper isirregular because of a change-point due to an unknown threshold in a covariate.This irregularity coupled with the discontinuity of the objective function of themaximum score estimator complicates the analysis of the asymptotic behavior ofthe estimator. Sufficient conditions for the identification of parameters are givenand the consistency of the estimator is obtained. It is shown that the estimator ofthe threshold parameter is n-consistent and the estimator of the remainingregression parameters is cube root n-consistent. Furthermore, we obtain theasymptotic distribution of the estimators. It turns out that a suitably normalizedestimator of the regression parameters converges weakly to the distribution towhich it would converge weakly if the true threshold value were known andlikewise for the threshold estimator.