Skip to main content


  • Karun Adusumilli (a1) and Taisuke Otsu (a1)

This paper considers nonparametric instrumental variable regression when the endogenous variable is contaminated with classical measurement error. Existing methods are inconsistent in the presence of measurement error. We propose a wavelet deconvolution estimator for the structural function that modifies the generalized Fourier coefficients of the orthogonal series estimator to take into account the measurement error. We establish the convergence rates of our estimator for the cases of mildly/severely ill-posed models and ordinary/super smooth measurement errors. We characterize how the presence of measurement error slows down the convergence rates of the estimator. We also study the case where the measurement error density is unknown and needs to be estimated, and show that the estimation error of the measurement error density is negligible under mild conditions as far as the measurement error density is symmetric.

Corresponding author
*Address correspondence to Taisuke Otsu, Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK; e-mail:
Hide All

The authors would like to thank three anonymous referees and a co-editor for helpful comments. Otsu gratefully acknowledges financial support from the ERC Consolidator Grant (SNP 615882).

Hide All
Antoniadis, A., Gregoire, G., & McKeague, I.W. (1994) Wavelet methods for curve estimation. Journal of the American Statistical Association 89, 13401353.
Ashenfelter, O. & Krueger, A.B. (1994) Estimates of the economic returns to schooling from a new sample of twins. American Economic Review 84, 11571173.
Biemer, P., Groves, R., Lyberg, L., Mathiowetz, N., & Sudman, S. (eds.) (1991) Measurement Errors in Surveys. Wiley.
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant Engel curve. Econometrica 75, 16131669.
Borus, M.E. & Nestel, G. (1973) Response bias in reports of father’s education and socioeconomic status. Journal of the American Statistical Association 68, 816820.
Bowles, S. (1972) Schooling and inequality from generation to generation. Journal of Political Economy 80, S219S251.
Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C.M. (2012) Measurement Error in Nonlinear Models: A Modern Perspective. CRC Press.
Chen, X., Hong, H., & Nekipelov, D. (2011) Nonlinear models of measurement errors. Journal of Economic Literature 49, 901937.
Chen, X. & Pouzo, D. (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica 80, 277321.
Chen, X. & Reiss, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27, 497521.
Comte, F. & Kappus, J. (2015) Density deconvolution from repeated measurements without symmetry assumption on the errors. Journal of Multivariate Analysis 140, 3146.
Darolles, S., Fan, Y., Florens, J.P., & Renault, E. (2011) Nonparametric instrumental regression. Econometrica 79, 15411565.
Delaigle, A. & Hall, P. (2008) Using SIMEX for smoothing-parameter choice in errors-in-variables problems. Journal of the American Statistical Association 103, 280287.
Delaigle, A., Hall, P., & Meister, A. (2008) On deconvolution with repeated measurements. Annals of Statistics 36, 665685.
Donoho, D.L. (1995) Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied and Computational Harmonic Analysis 2, 101126.
Donoho, D.L. & Johnstone, I.W. (1998) Minimax estimation via wavelet shrinkage. Annals of Statistics 26, 879921.
Donoho, D.L., Johnstone, I.W., Kerkyacharian, G., & Picard, D. (1995) Wavelet shrinkage: Asymptopia? Journal of the Royal Statistical Society: Series B 57, 301369.
Donoho, D.L., Johnstone, I.W., Kerkyacharian, G., & Picard, D. (1996) Density estimation by wavelet thresholding. Annals of Statistics 24, 508539.
Fan, J. (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 12571272.
Fan, J. & Koo, J.-Y. (2002) Wavelet deconvolution. IEEE Transactions on Information Theory 48, 734747.
Fan, J. & Truong, Y.K. (1993) Nonparametric regression with errors in variables. Annals of Statistics 21, 19001925.
Freeman, R.B. (1984) Longitudinal analysis of the effects of trade unions. Journal of Labor Economics 2, 126.
Fuller, W.A. (1987) Measurement Error Models. John Wiley & Sons.
Gagliardini, P. & Scaillet, O. (2012) Nonparametric instrumental variable estimation of structural quantile effect. Econometrica 80, 15331562.
Hall, P. & Horowitz, J.L. (2005) Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 33, 29042929.
Hall, P. & Meister, A. (2007) A ridge-parameter approach to deconvolution. Annals of Statistics 35, 15351558.
Hausman, J., Ichimura, H., Newey, W., & Powell, J. (1991) Measurement errors in polynomial regression models. Journal of Econometrics 50, 271295.
Hausman, J., Newey, W., & Powell, J. (1995) Nonlinear errors in variables estimation of some Engel curves. Journal of Econometrics 65, 205233.
Horowitz, J.L. (2011) Applied nonparametric instrumental variables estimation. Econometrica 79, 347394.
Horowitz, J.L. (2012) Specification testing in nonparametric instrumental variable estimation. Journal of Econometrics 167, 383396.
Hu, Y. (2016) Microeconomic Models with Latent Variables: Applications of Measurement Error Models in Empirical Industrial Organization and Labor Economics. Working paper, John Hopkins University.
Kerkyacharian, G. & Picard, D. (1992) Density estimation in Besov spaces. Statistics & Probability Letters 13, 1524.
Li, T. (2002) Robust and consistent estimation of nonlinear errors-in-variables models. Journal of Econometrics 110, 126.
Li, T. & Vuong, Q. (1998) Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65, 139165.
Meister, A. (2009) Deconvolution Problems in Nonparametric Statistics. Springer.
Morey, E.R. & Waldman, D.M. (1998) Measurement error in recreation demand models: The joint estimation of participation, site choice, and site characteristics. Journal of Environmental Economics and Management 35, 262276.
Newey, W.K. (2001) Flexible simulated moment estimation of nonlinear errors-in-variables models. Review of Economics and Statistics 83, 616627.
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric model. Econometrica 71, 15651578.
Pensky, M. & Vidakovic, B. (1999) Adaptive wavelet estimator for nonparametric density deconvolution. Annals of Statistics 27, 20332053.
Schennach, S.M. (2004a) Estimation of nonlinear models with measurement error. Econometrica 72, 3375.
Schennach, S.M. (2004b) Nonparametric regression in the presence of measurement error. Econometric Theory 20, 10461093.
Schennach, S.M. (2013) Measurement error in nonlinear models - A review. In Acemoglu, D., Arellano, M., & Dekel, E. (eds.), Advances in Economics and Econometrics, vol. 3, pp. 296337. Cambridge University Press.
Schennach, S., White, H., & Chalak, K. (2012) Local indirect least squares and average marginal effects in nonseparable structural systems. Journal of Econometrics 166, 282302.
Song, S., Schennach, S.M., & White, H. (2015) Estimating nonseparable models with mismeasured endogenous variables. Quantitative Economics 6, 749794.
Stefanski, L. & Carroll, R.J. (1990) Deconvoluting kernel density estimators. Statistics 21, 169184.
Vidakovic, B. (1999) Statistical Modeling by Wavelets. Wiley.
Yukich, J. (1987) Some limit theorems for the empirical process indexed by functions. Probability Theory and Related Fields 74, 7190.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed