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Instrumental regression in partially linear models

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  • FLORENS, Jean-Pierre
  • JOHANNES, Jan
  • VAN BELLEGEM, Sébastien

Abstract

We consider the semiparametric regression X t +(Z) where and (r and function, and where the variables (X, Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of . An incorrect parametrization of generally leads to an inconsistent estimator of , whereas consistent nonparametric estimators for have a slow rate of convergence. An additional complication is that the solution of the equation necessitates the inversion of a compact operator which can be estimated nonparametrically. In general this inversion is not stable, thus the estimation of is ill-posed. In this paper, a n-consistent estimator for is derived under mild assumptions. One of these assumptions is given by the socalled source condition which we explicit and interpret in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroskedastic.

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Bibliographic Info

Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2006025.

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Date of creation: 00 Mar 2006
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Handle: RePEc:cor:louvco:2006025

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Keywords: partially linear model; semiparametric regression; instrumental variables; endogeneity; ill-posed inverse problem; Tikhonov regularization; root-N consistent estimation; semiparametric efficiency bound;

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References

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  1. Xiaohong Chen & Oliver Linton & Ingred Van Keilegom, 2002. "Estimation of semiparametric models when the criterion function is not smooth," CeMMAP working papers CWP02/02, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  2. Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
  3. Newey, Whitney K, 1990. "Efficient Instrumental Variables Estimation of Nonlinear Models," Econometrica, Econometric Society, vol. 58(4), pages 809-37, July.
  4. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, 09.
  5. Robinson, Peter M, 1988. "Root- N-Consistent Semiparametric Regression," Econometrica, Econometric Society, vol. 56(4), pages 931-54, July.
  6. Hardle, Wolfgang & LIang, Hua & Gao, Jiti, 2000. "Partially linear models," MPRA Paper 39562, University Library of Munich, Germany, revised 01 Sep 2000.
  7. Richard Blundell & Joel L. Horowitz, 2007. "A Non-Parametric Test of Exogeneity," Review of Economic Studies, Oxford University Press, vol. 74(4), pages 1035-1058.
  8. Chunrong Ai & Xiaohong Chen, 2003. "Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions," Econometrica, Econometric Society, vol. 71(6), pages 1795-1843, November.
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Citations

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Cited by:
  1. Xiaohong Chen & Victor Chernozhukov & Sokbae 'Simon' Lee & Whitney Newey, 2012. "Local identification of nonparametric and semiparametric models," CeMMAP working papers CWP37/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  2. Johannes, Jan & Van Bellegem, Sébastien & Vanhems, Anne, 2011. "Convergence Rates For Ill-Posed Inverse Problems With An Unknown Operator," Econometric Theory, Cambridge University Press, vol. 27(03), pages 522-545, June.
  3. Kapetanios, George & Marcellino, Massimiliano, 2010. "Factor-GMM Estimation with Large Sets of Possibly Weak Instruments," CEPR Discussion Papers 7726, C.E.P.R. Discussion Papers.
  4. FLORENS, Jean-Pierre & JOHANNES, Jan & VAN BELLEGEM, Sébastien, 2007. "Identification and estimation by penalization in nonparametric instrumental regression," CORE Discussion Papers 2007085, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. Gao, Jiti & Phillips, Peter C.B., 2013. "Semiparametric estimation in triangular system equations with nonstationarity," Journal of Econometrics, Elsevier, vol. 176(1), pages 59-79.
  6. Senay Sokullu, 2012. "Nonparametric Estimation of Semiparametric Transformation Models," Bristol Economics Discussion Papers 12/625, Department of Economics, University of Bristol, UK.
  7. Jorge Manzi & Ernesto San Martín & Sébastien Van Bellegem, 2014. "School System Evaluation by Value Added Analysis Under Endogeneity," Psychometrika, Springer, vol. 79(1), pages 130-153, January.
  8. Jiti Gao & Peter C.B. Phillips, 2011. "Semiparametric Estimation in Multivariate Nonstationary Time Series Models," Monash Econometrics and Business Statistics Working Papers 17/11, Monash University, Department of Econometrics and Business Statistics.
  9. Xiaohong Chen & Yingyao Hu, 2006. "Identification and Inference of Nonlinear Models Using Two Samples with Arbitrary Measurement Errors," Cowles Foundation Discussion Papers 1590, Cowles Foundation for Research in Economics, Yale University.

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