Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-10T03:34:12.629Z Has data issue: false hasContentIssue false
  • English
  • Français

NONPARAMETRIC INSTRUMENTAL REGRESSION WITH ERRORS IN VARIABLES

Published online by Cambridge University Press:  14 February 2018

Karun Adusumilli  and
Taisuke Otsu
Karun Adusumilli
Affiliation:
London School of Economics and Political Science
Taisuke Otsu*
Affiliation:
London School of Economics and Political Science
*
*Address correspondence to Taisuke Otsu, Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK; e-mail: t.otsu@lse.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 6 , December 2018 , pp. 1256 - 1280
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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).

References

REFERENCES

Antoniadis, A., Gregoire, G., & McKeague, I.W. (1994) Wavelet methods for curve estimation. Journal of the American Statistical Association 89, 1340–1353.CrossRefGoogle Scholar
Ashenfelter, O. & Krueger, A.B. (1994) Estimates of the economic returns to schooling from a new sample of twins. American Economic Review 84, 1157–1173.Google Scholar
Biemer, P., Groves, R., Lyberg, L., Mathiowetz, N., & Sudman, S. (eds.) (1991) Measurement Errors in Surveys. Wiley.Google Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant Engel curve. Econometrica 75, 1613–1669.CrossRefGoogle Scholar
Borus, M.E. & Nestel, G. (1973) Response bias in reports of father’s education and socioeconomic status. Journal of the American Statistical Association 68, 816–820.CrossRefGoogle Scholar
Bowles, S. (1972) Schooling and inequality from generation to generation. Journal of Political Economy 80, S219–S251.CrossRefGoogle Scholar
Carroll, R.J., Ruppert, D., Stefanski, L.A., & Crainiceanu, C.M. (2012) Measurement Error in Nonlinear Models: A Modern Perspective. CRC Press.Google Scholar
Chen, X., Hong, H., & Nekipelov, D. (2011) Nonlinear models of measurement errors. Journal of Economic Literature 49, 901–937.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica 80, 277–321.Google Scholar
Chen, X. & Reiss, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27, 497–521.CrossRefGoogle Scholar
Comte, F. & Kappus, J. (2015) Density deconvolution from repeated measurements without symmetry assumption on the errors. Journal of Multivariate Analysis 140, 31–46.Google Scholar
Darolles, S., Fan, Y., Florens, J.P., & Renault, E. (2011) Nonparametric instrumental regression. Econometrica 79, 1541–1565.Google Scholar
Delaigle, A. & Hall, P. (2008) Using SIMEX for smoothing-parameter choice in errors-in-variables problems. Journal of the American Statistical Association 103, 280–287.CrossRefGoogle Scholar
Delaigle, A., Hall, P., & Meister, A. (2008) On deconvolution with repeated measurements. Annals of Statistics 36, 665–685.Google Scholar
Donoho, D.L. (1995) Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Applied and Computational Harmonic Analysis 2, 101–126.CrossRefGoogle Scholar
Donoho, D.L. & Johnstone, I.W. (1998) Minimax estimation via wavelet shrinkage. Annals of Statistics 26, 879–921.Google Scholar
Donoho, D.L., Johnstone, I.W., Kerkyacharian, G., & Picard, D. (1995) Wavelet shrinkage: Asymptopia? Journal of the Royal Statistical Society: Series B 57, 301–369.Google Scholar
Donoho, D.L., Johnstone, I.W., Kerkyacharian, G., & Picard, D. (1996) Density estimation by wavelet thresholding. Annals of Statistics 24, 508–539.Google Scholar
Fan, J. (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 1257–1272.CrossRefGoogle Scholar
Fan, J. & Koo, J.-Y. (2002) Wavelet deconvolution. IEEE Transactions on Information Theory 48, 734–747.Google Scholar
Fan, J. & Truong, Y.K. (1993) Nonparametric regression with errors in variables. Annals of Statistics 21, 1900–1925.CrossRefGoogle Scholar
Freeman, R.B. (1984) Longitudinal analysis of the effects of trade unions. Journal of Labor Economics 2, 1–26.CrossRefGoogle Scholar
Fuller, W.A. (1987) Measurement Error Models. John Wiley & Sons.CrossRefGoogle Scholar
Gagliardini, P. & Scaillet, O. (2012) Nonparametric instrumental variable estimation of structural quantile effect. Econometrica 80, 1533–1562.Google Scholar
Hall, P. & Horowitz, J.L. (2005) Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 33, 2904–2929.Google Scholar
Hall, P. & Meister, A. (2007) A ridge-parameter approach to deconvolution. Annals of Statistics 35, 1535–1558.CrossRefGoogle Scholar
Hausman, J., Ichimura, H., Newey, W., & Powell, J. (1991) Measurement errors in polynomial regression models. Journal of Econometrics 50, 271–295.Google Scholar
Hausman, J., Newey, W., & Powell, J. (1995) Nonlinear errors in variables estimation of some Engel curves. Journal of Econometrics 65, 205–233.CrossRefGoogle Scholar
Horowitz, J.L. (2011) Applied nonparametric instrumental variables estimation. Econometrica 79, 347–394.Google Scholar
Horowitz, J.L. (2012) Specification testing in nonparametric instrumental variable estimation. Journal of Econometrics 167, 383–396.CrossRefGoogle Scholar
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.Google Scholar
Kerkyacharian, G. & Picard, D. (1992) Density estimation in Besov spaces. Statistics & Probability Letters 13, 15–24.CrossRefGoogle Scholar
Li, T. (2002) Robust and consistent estimation of nonlinear errors-in-variables models. Journal of Econometrics 110, 1–26.CrossRefGoogle Scholar
Li, T. & Vuong, Q. (1998) Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65, 139–165.CrossRefGoogle Scholar
Meister, A. (2009) Deconvolution Problems in Nonparametric Statistics. Springer.CrossRefGoogle Scholar
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, 262–276.CrossRefGoogle Scholar
Newey, W.K. (2001) Flexible simulated moment estimation of nonlinear errors-in-variables models. Review of Economics and Statistics 83, 616–627.CrossRefGoogle Scholar
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric model. Econometrica 71, 1565–1578.Google Scholar
Pensky, M. & Vidakovic, B. (1999) Adaptive wavelet estimator for nonparametric density deconvolution. Annals of Statistics 27, 2033–2053.Google Scholar
Schennach, S.M. (2004a) Estimation of nonlinear models with measurement error. Econometrica 72, 33–75.CrossRefGoogle Scholar
Schennach, S.M. (2004b) Nonparametric regression in the presence of measurement error. Econometric Theory 20, 1046–1093.CrossRefGoogle Scholar
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. 296–337. Cambridge University Press.CrossRefGoogle Scholar
Schennach, S., White, H., & Chalak, K. (2012) Local indirect least squares and average marginal effects in nonseparable structural systems. Journal of Econometrics 166, 282–302.Google Scholar
Song, S., Schennach, S.M., & White, H. (2015) Estimating nonseparable models with mismeasured endogenous variables. Quantitative Economics 6, 749–794.CrossRefGoogle Scholar
Stefanski, L. & Carroll, R.J. (1990) Deconvoluting kernel density estimators. Statistics 21, 169–184.CrossRefGoogle Scholar
Vidakovic, B. (1999) Statistical Modeling by Wavelets. Wiley.CrossRefGoogle Scholar
Yukich, J. (1987) Some limit theorems for the empirical process indexed by functions. Probability Theory and Related Fields 74, 71–90.CrossRefGoogle Scholar