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Essays on the econometric theory of rank regressions

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  • Subbotin, Viktor

Abstract

Several semiparametric estimators recently developed in the econometrics literature are based on the rank correlation between the dependent and explanatory variables. Examples include the maximum rank correlation estimator (MRC) of Han [1987], the monotone rank estimator (MR) of Cavanagh and Sherman [1998], the pairwise-difference rank estimators (PDR) of Abrevaya [2003], and others. These estimators apply to various monotone semiparametric single-index models, such as the binary choice models, the censored regression models, the nonlinear regression models, and the transformation and duration models, among others, without imposing functional form restrictions on the unknown functions and distributions. This work provides several new results on the theory of rank-based estimators. In Chapter 2 we prove that the quantiles and the variances of their asymptotic distributions can be consistently estimated by the nonparametric bootstrap. In Chapter 3 we investigate the accuracy of inference based on the asymptotic normal and bootstrap approximations, and provide bounds on the associated error. In the case of MRC and MR, the bound is a function of the sample size of order close to n^(-1/6). The PDR estimators, however, belong to a special subclass of rank estimators for which the bound is vanishing with the rate close to n^(-1/2). In Chapter 4 we study the efficiency properties of rank estimators and propose weighted rank estimators that improve efficiency. We show that the optimally weighted MR attains the semiparametric efficiency bound in the nonlinear regression model and the binary choice model. Optimally weighted MRC has the asymptotic variance close to the semiparametric efficiency bound in single-index models under independence when the distribution of the errors is close to normal, and is consistent under practically relevant deviations from the single index assumption. Under moderate nonlinearities and nonsmoothness in the data, the efficiency gains from weighting are likely to be small for MRC in the transformation model and for MRC and MR in the binary choice model, and can be large for MRC and MR in the monotone regression model. Throughout, the theoretical results are illustrated with Monte-Carlo experiments and real data examples

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 14086.

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Date of creation: Dec 2008
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Handle: RePEc:pra:mprapa:14086

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Keywords: Semiparametric models; Bootstrap; Maximum rank correlation estimator; Monotone rank estimator; Efficiency; U-processes; U-statistics; Maximal Inequalities; Econometric theory; Rank regressions;

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  1. Alberto Abadie & Guido W. Imbens, 2006. "On the Failure of the Bootstrap for Matching Estimators," NBER Technical Working Papers 0325, National Bureau of Economic Research, Inc.
  2. Abrevaya, Jason, 1999. "Leapfrog estimation of a fixed-effects model with unknown transformation of the dependent variable," Journal of Econometrics, Elsevier, Elsevier, vol. 93(2), pages 203-228, December.
  3. Chamberlain, Gary, 1992. "Efficiency Bounds for Semiparametric Regression," Econometrica, Econometric Society, Econometric Society, vol. 60(3), pages 567-96, May.
  4. Y. Nishiyama & Peter Robinson, 2004. "The bootstrap and the Edgeworth correction for semiparametric averaged derivatives," CeMMAP working papers, Centre for Microdata Methods and Practice, Institute for Fiscal Studies CWP12/04, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  5. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, Econometric Society, vol. 57(6), pages 1403-30, November.
  6. Pollard, David, 1985. "New Ways to Prove Central Limit Theorems," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 1(03), pages 295-313, December.
  7. Pakes, Ariel & Pollard, David, 1989. "Simulation and the Asymptotics of Optimization Estimators," Econometrica, Econometric Society, Econometric Society, vol. 57(5), pages 1027-57, September.
  8. Sherman, Robert P, 1993. "The Limiting Distribution of the Maximum Rank Correlation Estimator," Econometrica, Econometric Society, Econometric Society, vol. 61(1), pages 123-37, January.
  9. Khan, Shakeeb & Tamer, Elie, 2007. "Partial rank estimation of duration models with general forms of censoring," Journal of Econometrics, Elsevier, Elsevier, vol. 136(1), pages 251-280, January.
  10. Jason Abrevaya, 1999. "Rank estimation of a transformation model with observed truncation," Econometrics Journal, Royal Economic Society, Royal Economic Society, vol. 2(2), pages 292-305.
  11. Han, Aaron K., 1987. "Non-parametric analysis of a generalized regression model : The maximum rank correlation estimator," Journal of Econometrics, Elsevier, Elsevier, vol. 35(2-3), pages 303-316, July.
  12. Newey, Whitney K & Stoker, Thomas M, 1993. "Efficiency of Weighted Average Derivative Estimators and Index Models," Econometrica, Econometric Society, Econometric Society, vol. 61(5), pages 1199-223, September.
  13. Hall, Peter & Horowitz, Joel L, 1996. "Bootstrap Critical Values for Tests Based on Generalized-Method-of-Moments Estimators," Econometrica, Econometric Society, Econometric Society, vol. 64(4), pages 891-916, July.
  14. Klein, Roger W & Spady, Richard H, 1993. "An Efficient Semiparametric Estimator for Binary Response Models," Econometrica, Econometric Society, Econometric Society, vol. 61(2), pages 387-421, March.
  15. Songnian Chen, 2002. "Rank Estimation of Transformation Models," Econometrica, Econometric Society, Econometric Society, vol. 70(4), pages 1683-1697, July.
  16. Abrevaya, Jason, 2003. "Pairwise-Difference Rank Estimation of the Transformation Model," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 21(3), pages 437-47, July.
  17. Chunrong Ai & Xiaohong Chen, 2003. "Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions," Econometrica, Econometric Society, Econometric Society, vol. 71(6), pages 1795-1843, November.
  18. Abrevaya, Jason, 1999. "Computation of the maximum rank correlation estimator," Economics Letters, Elsevier, Elsevier, vol. 62(3), pages 279-285, March.
  19. Han, Aaron K., 1987. "A non-parametric analysis of transformations," Journal of Econometrics, Elsevier, Elsevier, vol. 35(2-3), pages 191-209, July.
  20. Asparouhova, Elena & Golanski, Robert & Kasprzyk, Krzysztof & Sherman, Robert P. & Asparouhov, Tihomir, 2002. "Rank Estimators For A Transformation Model," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 18(05), pages 1099-1120, October.
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