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Estimation of Semiparametric Models when the Criterion Function is not Smooth

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  • Xiaohong Chen
  • Oliver Linton
  • Ingrid Van Keilegom

Abstract

We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories like those of Pakes and Pollard (1989), Andrews (1994a) and Newey (1994). We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a 'hit rate' and a partially linear median regression with some endogenous regressors.

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Bibliographic Info

Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2003/450.

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Date of creation: May 2003
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Handle: RePEc:cep:stiecm:/2003/450

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Web page: http://sticerd.lse.ac.uk/_new/publications/default.asp

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Keywords: Empirical processes; non-smooth criterion; semiparametric estimation; stochastic equicontinuity.;

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  1. Manski, Charles F., 1986. "Analog estimation of econometric models," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 43, pages 2559-2582 Elsevier.
  2. Oliver B. Linton & Enno Mammen & J. Nielsen & Carsten Tanggaard, 2000. "Yield Curve Estimation by Kernel Smoothing Methods," Econometric Society World Congress 2000 Contributed Papers 0235, Econometric Society.
  3. Richard Blundell & Martin Browning & Ian Crawford, 2002. "Nonparametric Engel Curves and Revealed Preference," CAM Working Papers 2002-04, University of Copenhagen. Department of Economics. Centre for Applied Microeconometrics.
  4. repec:cup:etheor:v:11:y:1995:i:3:p:560-96 is not listed on IDEAS
  5. Arthur Lewbel & Oliver Linton, 2000. "Nonparametric censored and truncated regression," LSE Research Online Documents on Economics 2060, London School of Economics and Political Science, LSE Library.
  6. Horowitz, Joel L, 1992. "A Smoothed Maximum Score Estimator for the Binary Response Model," Econometrica, Econometric Society, vol. 60(3), pages 505-31, May.
  7. Newey, W.K., 1989. "The Asymptotic Variance Of Semiparametric Estimotors," Papers 346, Princeton, Department of Economics - Econometric Research Program.
  8. Chen, Songnian & Khan, Shakeeb, 2001. "Semiparametric Estimation Of A Partially Linear Censored Regression Model," Econometric Theory, Cambridge University Press, vol. 17(03), pages 567-590, June.
  9. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245 Elsevier.
  10. Elie Tamer, 2000. "Inference in Censored Models with Endogenous Regressors," Econometric Society World Congress 2000 Contributed Papers 1815, Econometric Society.
  11. Xiaohong Chen & Xiaotong Shen, 1998. "Sieve Extremum Estimates for Weakly Dependent Data," Econometrica, Econometric Society, vol. 66(2), pages 289-314, March.
  12. Powell, James L., 1984. "Least absolute deviations estimation for the censored regression model," Journal of Econometrics, Elsevier, vol. 25(3), pages 303-325, July.
  13. Pakes, Ariel & Pollard, David, 1989. "Simulation and the Asymptotics of Optimization Estimators," Econometrica, Econometric Society, vol. 57(5), pages 1027-57, September.
  14. Robinson, Peter M, 1988. "Root- N-Consistent Semiparametric Regression," Econometrica, Econometric Society, vol. 56(4), pages 931-54, July.
  15. Ariel Pakes & Steven Olley, 1994. "A Limit Theorem for a Smooth Class of Semiparametric Estimators," Cowles Foundation Discussion Papers 1066, Cowles Foundation for Research in Economics, Yale University.
  16. Donald W.K. Andrews, 1993. "Empirical Process Methods in Econometrics," Cowles Foundation Discussion Papers 1059, Cowles Foundation for Research in Economics, Yale University.
  17. Joel L. Horowitz, 1996. "Bootstrap Methods for Median Regression Models," Econometrics 9608004, EconWPA.
  18. Lee, Sokbae, 2003. "Efficient Semiparametric Estimation Of A Partially Linear Quantile Regression Model," Econometric Theory, Cambridge University Press, vol. 19(01), pages 1-31, February.
  19. Donald J. Brown & Marten H. Wegkamp, 2001. "Weighted Minimum Mean-Square Distance from Independence Estimation," Cowles Foundation Discussion Papers 1288, Cowles Foundation for Research in Economics, Yale University.
  20. Hall, Peter & Horowitz, Joel L, 1996. "Bootstrap Critical Values for Tests Based on Generalized-Method-of-Moments Estimators," Econometrica, Econometric Society, vol. 64(4), pages 891-916, July.
  21. Andrews, Donald W K, 1994. "Asymptotics for Semiparametric Econometric Models via Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 62(1), pages 43-72, January.
  22. O. B. LINTON & R. CHEN & Wolfgang HÄRDLE, 1995. "An Analysis of Transformations for Additive Nonparanetric Regression," SFB 373 Discussion Papers 1995,68, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  23. Manski, Charles F, 1983. "Closest Empirical Distribution Estimation," Econometrica, Econometric Society, vol. 51(2), pages 305-19, March.
  24. Andrews, Donald W.K., 1995. "Nonparametric Kernel Estimation for Semiparametric Models," Econometric Theory, Cambridge University Press, vol. 11(03), pages 560-586, June.
  25. Manski, Charles F., 1975. "Maximum score estimation of the stochastic utility model of choice," Journal of Econometrics, Elsevier, vol. 3(3), pages 205-228, August.
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