and

This paper is concerned with semiparametric estimation of a threshold binary re-sponse model. The estimation method considered in the paper is semiparametric since the parameters for a regression function are finite-dimensional, while allowing for het-eroskedasticity of unknown form. In particular, the paper considers Manski (1975, 1985)’s 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, γ0, is n−1-consistent and the estimator of the remaining regression parameters, θ0, is n−1/3-consistent. Furthermore, we obtain the asymptotic distribution of the estimator. It turns out that a suitably nor-malized estimator of γ0 converges weakly to the distribution to which it would converge weakly if θ0 were known and likewise for θ0. Key Words: Binary response model, maximum score estimation, nonlinear random utility models, semiparametric estimation, threshold regression. ∗Lee thanks the Leverhulme Trust through the funding of the Centre for Microdata Methods and Practice (http://cemmap.ifs.org.uk) and the research programme Evidence, Inference and Inquiry (http://www.evidencescience.org). Seo thanks the Economic and Social Research Council (ESRC) for fi-nancial support.