Abstract
Wang and Zhou propose an iterative estimation algorithm for the binary choice model in “Working paper no. E-180-95, the Center for Business and Economic Research, College of Business and Economics, University of Kentucky (1995).” The method is easy-to-implement, semiparametric, and free from choosing nonparametric tuning parameters such as a kernel bandwidth. In this paper, a rigorous proof for consistency of the estimator will be given.
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- 1.
Wang and Zhou estimate F by the maximum likelihood method,
$$\displaystyle{\hat{F}_{\hat{b}}^{MLE}(\,\cdot \,) =\mathop{ \mathrm{argmax}}\limits _{ S\in \mathcal{F}}\sum _{i=1}^{n}D_{ i}\log S(x_{i}^\prime \hat{b})) + (1 - D_{i})\log (1 - S(x_{i}^\prime \hat{b})).}$$However, the maximum likelihood estimator is shown numerically identical as the LSE (8). For details, see e.g. p. 43 of Groeneboom and Wellner [8] or p. 7 of Robertson et al. [20].
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Acknowledgements
The author would like to deeply appreciate the financial support by the Seimeikai Foundation at Bank of Tokyo-Mitsubishi UFJ, 2-7-1 Marunouchi, Chiyoda-ku, Tokyo 100-8388, Japan, and gratefully acknowledges helpful comments and suggestions from anonymous referees.
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Tanaka, H. (2013). Local consistency of the iterative least-squares estimator for the semiparametric binary choice model. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics Volume 17. Advances in Mathematical Economics, vol 17. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54324-4_5
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