Instrumental Variable Treatment of Nonclassical Measurement Error Models

While the literature on nonclassical measurement error traditionally relies on the availability of an auxiliary data set containing correctly measured observations, we establish that the availability of instruments enables the identification of a large class of nonclassical nonlinear errors-in-variables models with continuously distributed variables. Our main identifying assumption is that, conditional on the value of the true regressors, some "measure of location" of the distribution of the measurement error (e.g., its mean, mode, or median) is equal to zero. The proposed approach relies on the eigenvalue-eigenfunction decomposition of an integral operator associated with specific joint probability densities. The main identifying assumption is used to "index" the eigenfunctions so that the decomposition is unique. We propose a convenient sieve-based estimator, derive its asymptotic properties, and investigate its finite-sample behavior through Monte Carlo simulations. Copyright The Econometric Society 2008.

[1]  C. Bollinger,et al.  Measurement Error in the Current Population Survey: A Nonparametric Look , 1998, Journal of Labor Economics.

[2]  John Bound,et al.  The Extent of Measurement Error in Longitudinal Earnings Data: Do Two Wrongs Make a Right? , 1988, Journal of Labor Economics.

[3]  Andrew Chesher.  Polynomial Regression with Covariate Measurement Error , 1998 .

[4]  Arthur Lewbel.  Demand Estimation with Expenditure Measurement Errors on the Left and Right Hand Side , 1996 .

[5]  Han Hong,et al.  Measurement Error Models with Auxiliary Data , 2005 .

[6]  Arthur Lewbel,et al.  Estimation of Average Treatment Effects With Misclassification , 2007 .

[7]  Victor Chernozhukov,et al.  Instrumental variable estimation of nonseparable models , 2007 .

[8]  Whitney Newey.  Convergence Rates for Series Estimators , 1993 .

[9]  J. Florens,et al.  A SPECTRAL METHOD FOR DECONVOLVING A DENSITY , 2010, Econometric Theory.

[10]  A. Gallant,et al.  Semi-nonparametric Maximum Likelihood Estimation , 1987 .

[11]  E. Tamer,et al.  A simple estimator for nonlinear error in variable models , 2003 .

[12]  W. Newey,et al.  Convergence rates and asymptotic normality for series estimators , 1997 .

[13]  L. Mattner,et al.  Some incomplete but boundedly complete location families , 1993 .

[14]  Andrew Chesher.  Parameter approximations for quantile regressions with measurement error , 2001 .

[15]  W. J. Hall,et al.  Information and Asymptotic Efficiency in Parametric-Nonparametric Models , 1983 .

[16]  Xiaohong Chen,et al.  Semiparametric Efficiency in GMM Models with Nonclassical Measurement Error , 2006 .

[17]  J. Hausman.  Mismeasured Variables in Econometric Analysis: Problems from the Right and Problems from the Left , 2001 .

[18]  J. Florens,et al.  Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization , 2003 .

[19]  Susanne M. Schennach,et al.  Estimation of Nonlinear Models with Measurement Error , 2004 .

[20]  Xiaohong Chen,et al.  Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , 2003 .

[21]  C. Stein.  Efficient Nonparametric Testing and Estimation , 1956 .

[22]  Aprajit Mahajan,et al.  Identification and Estimation of Regression Models with Misclassification , 2005 .

[23]  G. Ridder,et al.  Estimation of Nonlinear Models with Measurement Error Using Marginal Information1 , 2004 .

[24]  Arthur Lewbel,et al.  Semiparametric Latent Variable Model Estimation with Endogenous or Mismeasured Regressors , 1998 .

[25]  V. Chernozhukov,et al.  An IV Model of Quantile Treatment Effects , 2002 .

[26]  Yingyao Hu,et al.  Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution , 2008 .

[27]  Whitney K. Newey.  Flexible Simulated Moment Estimation of Nonlinear Errors-in-Variables Models , 2001, Review of Economics and Statistics.

[28]  Xiaotong Shen,et al.  On methods of sieves and penalization , 1997 .

[29]  A. Chesher.  Identification in Nonseparable Models , 2003 .

[30]  A. Chesher.  The effect of measurement error , 1991 .

[31]  Estimation in the nonlinear errors-in-variables model , 1998 .

[32]  J. Florens,et al.  Nonparametric Instrumental Regression , 2010 .

[33]  Xavier D'Haultfoeuille,et al.  ON THE COMPLETENESS CONDITION IN NONPARAMETRIC INSTRUMENTAL PROBLEMS , 2010, Econometric Theory.

[34]  H. White,et al.  An Extended Class of Instrumental Variables for the Estimation of Causal Effects , 2011 .

[35]  Hidehiko Ichimura,et al.  Identification and estimation of polynomial errors-in-variables models , 1991 .

[36]  Susanne M. Schennach,et al.  NONPARAMETRIC REGRESSION IN THE PRESENCE OF MEASUREMENT ERROR , 2004, Econometric Theory.

[37]  Tong Li,et al.  Robust and consistent estimation of nonlinear errors-in-variables models , 2002 .

[38]  John Bound,et al.  Measurement error in survey data , 2001 .

[39]  Xiaotong Shen,et al.  Sieve extremum estimates for weakly dependent data , 1998 .

[40]  W. Newey,et al.  Instrumental variable estimation of nonparametric models , 2003 .

[41]  Testing a parametric model against a nonparametric alternative with identification through instrumental variables , 2004 .

[42]  A. Chesher,et al.  Duration response measurement error , 2002 .

[43]  Xiaohong Chen,et al.  Semi‐Nonparametric IV Estimation of Shape‐Invariant Engel Curves , 2003 .

[44]  Enno Mammen,et al.  Identification of marginal effects in nonseparable models without monotonicity , 2007 .

[45]  F. Smithies.  Linear Operators , 2019, Nature.

[46]  B. McCarl,et al.  Economics , 1870, The Indian medical gazette.

[47]  Rosa L. Matzkin,et al.  Cross Section and Panel Data Estimators for Nonseparable Models with Endogenous Regressors , 2005 .

[48]  P. Hall,et al.  Nonparametric methods for inference in the presence of instrumental variables , 2003, math/0603130.

[49]  Susanne M. Schennach,et al.  Instrumental Variable Estimation of Nonlinear Errors-in-Variables Models , 2004 .

[50]  Rosa L. Matzkin.  Nonparametric Estimation of Nonadditive Random Functions , 2003 .

[51]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .