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AVERAGE DERIVATIVE ESTIMATION UNDER MEASUREMENT ERROR

Published online by Cambridge University Press:  13 November 2020

Hao Dong ,
Taisuke Otsu  and
Luke Taylor
Hao Dong
Affiliation:
Southern Methodist University
Taisuke Otsu*
Affiliation:
London School of Economics
Luke Taylor
Affiliation:
Aarhus University
*
Address correspondence to Taisuke Otsu, Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK; e-mail: [email protected].
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Abstract

In this paper, we derive the asymptotic properties of the density-weighted average derivative estimator when a regressor is contaminated with classical measurement error and the density of this error must be estimated. Average derivatives of conditional mean functions are used extensively in economics and statistics, most notably in semiparametric index models. As well as ordinary smooth measurement error, we provide results for supersmooth error distributions. This is a particularly important class of error distribution as it includes the Gaussian density. We show that under either type of measurement error, despite using nonparametric deconvolution techniques and an estimated error characteristic function, we are able to achieve a $\sqrt {n}$ -rate of convergence for the average derivative estimator. Interestingly, if the measurement error density is symmetric, the asymptotic variance of the average derivative estimator is the same irrespective of whether the error density is estimated or not. The promising finite sample performance of the estimator is shown through a Monte Carlo simulation.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 5 , October 2021 , pp. 1004 - 1033
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

Financial support from the ERC Consolidator Grant (SNP 615882) (Otsu) and the AUFF Starting Grant (26852) (Taylor) is gratefully acknowledged.

References

REFERENCES

Ahn, H. & Powell, J. L. (1993) Semiparametric estimation of censored selection models with a nonparametric selection mechanism. Journal of Econometrics 58, 3–29.CrossRefGoogle Scholar
Bissantz, N., Dümbgen, L., Holzmann, H., & Munk, A. (2007) Non-parametric confidence bands in deconvolution density estimation. Journal of the Royal Statistical Society: Series B 69, 483–506.CrossRefGoogle Scholar
Blundell, R., Duncan, A., & Pendakur, K. (1998) Semiparametric estimation and consumer demand. Journal of Applied Econometrics 13, 435–461.3.0.CO;2-K>CrossRefGoogle Scholar
Brainard, W. C. & Tobin, J. (1968) Econometric models: Their problems and usefulness: Pitfalls in financial model building. American Economic Review 58, 99–122.Google Scholar
Butucea, C. & Comte, F. (2009) Adaptive estimation of linear functionals in the convolution model and applications. Bernoulli 15, 69–98.CrossRefGoogle Scholar
Carroll, R. J., Ruppert, D., Stefanski, L. A., & Crainiceanu, C. M. (2006) Measurement Error in Nonlinear Models, 2nd Edition. Chapman & Hall/CRC Press.CrossRefGoogle Scholar
Cattaneo, M. D., Crump, R. K., & Jansson, M. (2010) Robust data-driven inference for density-weighted average derivatives. Journal of the American Statistical Association 105, 1070–1083.CrossRefGoogle Scholar
Cattaneo, M. D., Crump, R. K., & Jansson, M. (2014) Small bandwidth asymptotics for density-weighted average derivatives. Econometric Theory 30, 176–200.CrossRefGoogle Scholar
Chen, X., Hong, H., & Nekipelov, D. (2011) Nonlinear models of measurement errors. Journal of Economic Literature 49, 901–937.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.CrossRefGoogle Scholar
Das, M., Newey, W. K., & Vella, F. (2003) Nonparametric estimation of sample selection models. Review of Economic Studies 70, 33–58.CrossRefGoogle Scholar
Dattner, I., Reiß, M., & Trabs, M. (2016) Adaptive quantile estimation in deconvolution with unknown error distribution. Bernoulli 22, 143–192.CrossRefGoogle Scholar
Delaigle, A. & Gijbels, I. (2004) Practical bandwidth selection in deconvolution kernel density estimation. Computational Statistics & Data Analysis 45, 249–267.CrossRefGoogle Scholar
Delaigle, A., Hall, P., & Meister, A. (2008) On deconvolution with repeated measurements. Annals of Statistics 36, 665–685.CrossRefGoogle Scholar
Devroye, L. (1992) A note on the usefulness of superkernels in density estimates. Annals of Statistics 20, 2037–2056.CrossRefGoogle Scholar
Dong, H. & Otsu, T. (2018) Nonparametric Estimation of Additive Model with Errors-in-Variables, Working paper.Google Scholar
Dong, Y. & Lewbel, A. (2015) A simple estimator for binary choice models with endogenous regressors. Econometric Reviews 34, 82–105.CrossRefGoogle Scholar
Erickson, T. & Whited, T. M. (2000) Measurement error and the relationship between investment and q. Journal of Political Economy 108, 1027–1057.CrossRefGoogle Scholar
Fan, J. (1991) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19, 1257–1272.CrossRefGoogle Scholar
Fan, J. & Masry, E. (1992) Multivariate regression estimation with errors-in-variables: asymptotic normality for mixing processes. Journal of Multivariate Analysis 43, 237–271.CrossRefGoogle Scholar
Fan, J. & Truong, Y. K. (1993) Nonparametric regression with errors in variables. Annals of Statistics 21, 1900–1925.CrossRefGoogle Scholar
Fan, Y. (1995) Average derivative estimation with errors-in-variables. Journal of Nonparametric Statistics 4, 395–407.CrossRefGoogle Scholar
Härdle, W., Hildenbrand, W., & Jerison, M. (1991) Empirical evidence on the law of demand. Econometrica 59 1525–1549.CrossRefGoogle Scholar
Horowitz, J. L. (2009) Semiparametric and Nonparametric Methods in Econometrics. Springer.CrossRefGoogle Scholar
Ichimura, H. (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58, 71–120.CrossRefGoogle Scholar
Kato, K. & Sasaki, Y. (2018) Uniform confidence bands in deconvolution with unknown error distribution. Journal of Econometrics 207, 129–161.CrossRefGoogle Scholar
Li, Q. & Racine, J. S. (2007) Nonparametric Econometrics. Princeton University Press.Google 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
Liang, H., Thurston, S. W., Ruppert, D., Apanasovich, T., & Hauser, R. (2008) Additive partial linear models with measurement errors. Biometrika 95, 667–678.CrossRefGoogle ScholarPubMed
Mamuneas, T. P., Savvides, A., & Stengos, T. (2006) Economic development and the return to human capital: A smooth coefficient semiparametric approach. Journal of Applied Econometrics 21, 111–132.CrossRefGoogle Scholar
McMurry, T. L. & Politis, D. N. (2004) Nonparametric regression with infinite order flat-top kernels. Journal of Nonparametric Statistics 16, 549–562.CrossRefGoogle Scholar
Meister, A (2009) Deconvolution Problems in Nonparametric Statistics. Springer.CrossRefGoogle Scholar
Otsu, T. & Taylor, L. (2020) Specification testing for errors-in-variables models. Econometric Theory, first published online 19 June 2020. https://doi.org/10.1017/S0266466620000262.CrossRefGoogle Scholar
Politis, D. N. & Romano, J. P. (1995) Bias-corrected nonparametric spectral estimation. Journal of Time Series Analysis 16, 67–103.CrossRefGoogle Scholar
Powell, J. L., Stock, J. H., & Stoker, T. M. (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.CrossRefGoogle Scholar
Powell, J.L. & Stoker, T. M. (1996) Optimal bandwidth choice for density-weighted averages. Journal of Econometrics 75, 291–316.CrossRefGoogle Scholar
Racine, J. (1997) Consistent significance testing for nonparametric regression Journal of Business & Economic Statistics 15, 369–378.Google Scholar
Schennach, S. M. (2004) Estimation of nonlinear models with measurement error. Econometrica 72, 33–75.CrossRefGoogle Scholar
Schennach, S. M. & Hu, Y. (2013) Nonparametric identification and semiparametric estimation of classical measurement error models without side information. Journal of the American Statistical Association 108, 177–186.CrossRefGoogle Scholar
van Es, B. & Gugushvili, S. (2008) Weak convergence of the supremum distance for supersmooth kernel deconvolution. Statistics & Probability Letters 78, 2932–2938.CrossRefGoogle Scholar
van Es, B. & Uh, H.-W. (2005) Asymptotic normality for kernel type deconvolution estimators. Scandinavian Journal of Statistics 32, 467–483.Google Scholar
Yatchew, A. (2003) Semiparametric Regression for the Applied Econometrician. Cambridge University Press.CrossRefGoogle Scholar
You, J. & Chen, G. (2006) Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model. Journal of Multivariate Analysis 97, 324–341.CrossRefGoogle Scholar