Semiparametric estimation of a binary response model with a change-point due to a covariate threshold

This paper is concerned with semiparametric estimation of a threshold binary response model. The estimation method considered in the paper is semiparametric since the parameters for a regression function are finite-dimensional, while allowing for heteroskedasticity of unknown form. In particular, the paper considers Manski’s [Manski, Charles F., 1975. Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3 (3), 205–228; Manski, Charles F., 1985. Semiparametric analysis of discrete response. Asymptotic properties of the maximum score estimator. Journal of Econometrics 27 (3), 313–333] maximum score estimator. The model in this paper is irregular because of a change-point due to an unknown threshold in a covariate. This irregularity coupled with the discontinuity of the objective function of the maximum score estimator complicates the analysis of the asymptotic behavior of the estimator. Sufficient conditions for the identification of parameters are given and the consistency of the estimator is obtained. It is shown that the estimator of the threshold parameter, γ 0 , is n − 1 -consistent and the estimator of the remaining regression parameters, θ 0 , is n − 1 / 3 -consistent. Furthermore, we obtain the asymptotic distribution of the estimator. It turns out that both estimators γ ˆ and θ ˆ are oracle-efficient in that n ( γ ˆ n − γ 0 ) and n 1 / 3 ( θ ˆ n − θ 0 ) converge weakly to the distributions to which they would converge weakly if the other parameter(s) were known.