Advances in Mathematical Economics Volume 17 pp 139-161

Part of the Advances in Mathematical Economics book series (MATHECON, volume 17) | Cite as

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



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 


  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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chamberlain, G.: Asymptotic efficiency in semi-parametric models with censoring. J. Econometrics 32, 189–218 (1986)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cosslett, S.R.: Distribution-free maximum likelihood estimator of the binary choice model. Econometrica 51, 765–782 (1983)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHGoogle 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)MATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Horowitz, J.L.: Semiparametric Methods in Econometrics. Springer, New York (1998)MATHCrossRefGoogle Scholar
  11. 11.
    Ichimura, H.: Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58, 71–120 (1993)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kim, J., Pollard, D.: Cube root asymptotics. Ann. Stat. 18, 191–219 (1990)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Klein, R.W., Spady, R.H.: An Efficient Semiparametric Estimator for Binary Response Models. Econometrica 61, 387–421 (1993)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle 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)MATHCrossRefGoogle Scholar
  24. 24.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York (1996)MATHCrossRefGoogle 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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