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Local consistency of the iterative least-squares estimator for the semiparametric binary choice model

  • Hisatoshi Tanaka
Chapter
Part of the Advances in Mathematical Economics book series (MATHECON, volume 17)

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.

Key words

Binary choice model EM algorithm Isotonic regression Iteration method Semiparametric estimation 

Notes

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.

References

  1. 1.
    Ayer, M., Brunk, H.D., Ewing, G.M., Reid, W.T., Silverman, E.: An empirical distribution function for sampling with incomplete information. Ann. Math. Stat. 26, 641–647 (1955)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bilrman, M.Š., Solomjak, M.Z.: Piece-wise polynomial approximations of functions in the classes W p α. Math. USSR Sb. 73, 295–317 (1967)CrossRefGoogle Scholar
  3. 3.
    Cavanagh, C., Sherman, R.P.: Rank estimators for monotonic index models. J. Econometrics 84, 351–381 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chamberlain, G.: Asymptotic efficiency in semi-parametric models with censoring. J. Econometrics 32, 189–218 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cosslett, S.R.: Distribution-free maximum likelihood estimator of the binary choice model. Econometrica 51, 765–782 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dominitz, J., Sherman, R.P.: Some convergence theory for iterative estimation procedures with an application to semiparametric estimation. Econom. Theor. 21, 838–863 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gao, F., Wellner, J.A.: Entropy Estimate For High Dimensional Monotonic Functions. University of Idaho, Mimeo (2008)Google Scholar
  8. 8.
    Groeneboom, P., Wellner, J.A.: Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel (1992)zbMATHCrossRefGoogle Scholar
  9. 9.
    Han, A. K.: Non-parametric analysis of a generalized regression model: The maximum rank correlation estimator. J. Econometrics 35, 303–316 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Horowitz, J.L.: Semiparametric Methods in Econometrics. Springer, New York (1998)zbMATHCrossRefGoogle Scholar
  11. 11.
    Ichimura, H.: Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58, 71–120 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kim, J., Pollard, D.: Cube root asymptotics. Ann. Stat. 18, 191–219 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Klein, R.W., Spady, R.H.: An Efficient Semiparametric Estimator for Binary Response Models. Econometrica 61, 387–421 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kosorok, M.R.: Introduction to Empirical Processes and Semiparametric Inference. Springer (2008)Google Scholar
  15. 15.
    Lewbel, A., Schennach, S.: A simple ordered data estimator for inverse density weighted functions. J. Econometrics 186, 189–211 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Maddala, G.S.: Limited-dependent and qualitative variables in econometrics. Cambridge University Press, Cambridge (1986)Google Scholar
  17. 17.
    Manski, C.F.: Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. J. Econometrics 27, 313–333 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Pollard, D.: Empirical Processes: Theory and Applications. Nsf-Cbms Regional Conference Series in Probability and Statistics 2. Inst of Mathematical Statistic (1991)Google Scholar
  19. 19.
    Robertson, T., Wright, F.T.: Consistency in generalized isotonic regression. Ann. Stat. 3, 350–362 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Robertson, T., Wright, F.T., Dykstra, R.L.: Ordered Restricted Statistical Inference. Wiley, New York (1988)Google Scholar
  21. 21.
    Sherman, R.P.: The limiting distribution of the maximum rank correlation estimator. Econometrica 61, 123–137 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    van de Geer: Empirical Processes in M-Estimation. Cambridge University Press, Cambridge (2000)Google Scholar
  23. 23.
    van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)zbMATHCrossRefGoogle Scholar
  24. 24.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York (1996)zbMATHCrossRefGoogle Scholar
  25. 25.
    Wang, W., Zhou, M.: Iterative Least Squares Estimator of Binary Choice Models: A Semi-parametric Approach, Working Paper no. E-180-95, The Center for Business and Economic Research, College of Business and Economics, University of Kentucky (1995)Google Scholar
  26. 26.
    Wang, W.: Semi-parametric estimation of the effect of health on labour-force participation of married women. Appl. Econ. 29, 325–329 (1997)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.Waseda UniversityTokyoJapan

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