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ADAPTIVE ESTIMATION OF FUNCTIONALS IN NONPARAMETRIC INSTRUMENTAL REGRESSION

Published online by Cambridge University Press:  30 March 2015

Christoph Breunig  and
Jan Johannes
Christoph Breunig*
Affiliation:
Humboldt-Universität zu Berlin
Jan Johannes
Affiliation:
CREST-Ensai and Université catholique de Louvain
*
*Address correspondence to Christoph Breunig, Humboldt-Universität zu Berlin, Department of Economics, Spandauer Str. 1, 10178 Berlin, Germany, e-mail: christoph.breunig@hu-berlin.de.
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Abstract

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We consider the problem of estimating the value ℓ(ϕ) of a linear functional, where the structural function ϕ models a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parameter m depending on certain characteristics of the structural function ϕ and the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice of m which combines model selection and Lepski’s method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural function ϕ.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 3 , June 2016 , pp. 612 - 654
Copyright
Copyright © Cambridge University Press 2015 

References

REFERENCES

Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71, 1795–1843.Google Scholar
Ai, C. & Chen, X. (2007) Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables. Journal of Econometrics 141, 5–43.Google Scholar
Ai, C. & Chen, X. (2012) The semiparametric efficiency bound for models of sequential moment restrictions containing unknown functions. Journal of Econometrics 170(2), 442–457.Google Scholar
Barron, A., Birgé, L., & Massart, P. (1999). Risk bounds for model selection via penalization. Probability Theory and Related Fields 113(3), 301–413.Google Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant engel curves. Econometrica 75(6), 1613–1669.Google Scholar
Blundell, R. & Horowitz, J.L. (2007) A nonparametric test of exogeneity. Review of Economic Studies 74, 1035–1058.Google Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes. Springer.Google Scholar
Brown, L.D. & Low, M.G. (1996) A constrained risk inequality with applications to nonparametric functional estimation. Annals of Statistics 24(6), 2524–2535.Google Scholar
Cardot, H. & Johannes, J. (2010) Thresholding projection estimators in functional linear models. Journal of Multivariate Analysis 101, 395–408.Google Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics: Estimation based on spectral decomposition and regularization. Handbook of Econometrics, vol. 6. North Holland.Google Scholar
Chen, X. & Pouzo, D. (2013) Sieve Quasi Likelihood Ratio Inference on Semi/Nonparametric Conditional Moment Models1. Technical report, Cowles Foundation for Research in Economics, Yale University.Google Scholar
Chen, X. & Reiß, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27, 497–521.Google Scholar
Darolles, S., Fan, Y., Florens, J.P., & Renault, E. (2011) Nonparametric instrumental regression. Econometrica 79(5), 1541–1565. ISSN 1468–0262.Google Scholar
Donoho, D. (1994) Statistical estimation and optimal recovery. Annals of Statistics 22, 238–270.Google Scholar
Donoho, D. & Low, M. (1992) Renormalization exponents and optimal pointwise rates of convergence. Annals of Statistics 20, 944–970.Google Scholar
Efromovich, S. & Koltchinskii, V. (2001) On inverse problems with unknown operators. IEEE Transactions on Information Theory 47(7), 2876–2894.CrossRefGoogle Scholar
Engl, H.W., Hanke, M., & Neubauer, A. (2000) Regularization of Inverse problems. Kluwer Academic.Google Scholar
Florens, J.-P. (2003) Inverse problems and structural econometrics: The example of instrumental variables. Advances in Economics and Econometrics: Theory and Applications – Eight World Congress, Volume 36 of Econometric Society Monographs, vol. 2, pp. 284. Cambridge University Press.Google Scholar
Florens, J.P., Johannes, J., & Van Bellegem, S. (2011) Identification and estimation by penalization in nonparametric instrumental regression. Econometric Theory 27, 522–545.Google Scholar
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2012) Instrumental regression in partially linear models. The Econometrics Journal, 15(2), 304–324.CrossRefGoogle Scholar
Florens, J.-P. & Simoni, A. (2012) Nonparametric estimation of an instrumental regression: A quasi-bayesian approach based on regularized posterior. Journal of Econometrics 170(2), 458–475.CrossRefGoogle Scholar
Gagliardini, P. & Scaillet, O. (2012) Tikhonov regularization for nonparametric instrumental variable estimators. Journal of Econometrics 167(1), 61–75.Google Scholar
Goldenshluger, A. & Lepski, O. (2011) Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. Annals of Statistics 39(3), 1608–1632.Google Scholar
Goldenshluger, A. & Pereverzev, S.V. (2000) Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations. Probability Theory and Related Fields 118, 169–186.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
Heinz, E. (1951) Beiträge zur störungstheorie der spektralzerlegung. Mathematische Annalen 123, 415–438.Google Scholar
Hoffmann, M. & Reiß, M. (2008) Nonlinear estimation for linear inverse problems with error in the operator. Annals of Statistics 36(1), 310–336.Google Scholar
Horowitz, J.L. (2014) Adaptive nonparametric instrumental variables estimation: Empirical choice of the regularization parameter. Journal of Econometrics 180, 158–173.Google Scholar
Horowitz, J.L. & Lee, S. (2007) Nonparametric instrumental variables estimation of a quantile regression model. Econometrica 75, 1191–1208.Google Scholar
Ibragimov, I. & Has’minskii, R. (1984) On nonparametric estimation of the value of a linear functional in Gaussian white noise. Theory of Probability and its Applications 29, 18–32.Google Scholar
Johannes, J. & Schwarz, M. (2010) Adaptive Nonparametric Instrumental Regression by Model Selection. Technical report, Université catholique de Louvain.Google Scholar
Johannes, J., Van Bellegem, S., & Vanhems, A. (2011) Convergence rates for ill-posed inverse problems with an unknown operator. Econometric Theory 27, 472–496.Google Scholar
Kawata, T. (1972) Fourier Analysis in Probability Theory. Academic Press.Google Scholar
Krein, S. & Petunin, Y.I. (1966) Scales of banach spaces. Russian Mathematical Surveys 21, 85–169.Google Scholar
Lepski, O.V. (1990) On a problem of adaptive estimation in Gaussian white noise. Theory of Probability and its Applications 35, 454–466.CrossRefGoogle Scholar
Li, K. (1982) Minimaxity of the method of regularization of stochastic processes. Annals of Statistics 10, 937–942.Google Scholar
Loubes, J.-M. & Marteau, C. (2009) Oracle Inequalities for Instrumental Variable Regression. Technical report, Toulouse.Google Scholar
Mair, B.A. (1994) Tikhonov regularization for finitely and infinitely smoothing operators. SIAM Journal on Mathematical Analysis 25, 135–147.Google Scholar
Massart, P. (2007) Concentration Inequalities and Model Selection. Ecole d’Eté de Probabilités de Saint-Flour XXXIII – 2003. Lecture Notes in Mathematics, vol. 1896. Springer, xiv, pp. 337.Google Scholar
Natterer, F. (1984) Error bounds for Tikhonov regularization in Hilbert scales. Applicable Analysis 18, 29–37.Google Scholar
Neubauer, A. (1988a) An a posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates. SIAM Journal of Numerical Analysis 25(6) 1313–1326.Google Scholar
Neubauer, A. (1988b) When do Sobolev spaces form a Hilbert scale? Proceedings of the American Mathematical Society 103(2), 557–562.Google Scholar
Newey, W.K. (1990) Efficient instrumental variables estimation of nonlinear models. Econometrica 58, 809–837.Google Scholar
Newey, W.K. & Powell, J.L. (2003) Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578.Google Scholar
Petrov, V.V. (1995) Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Studies in Probability, 4th ed.Clarendon Press.Google Scholar
Santos, A. (2011) Instrumental variable methods for recovering continuous linear functionals. Journal of Econometrics 161, 129–146.Google Scholar
Severini, T.A. & Tripathi, G. (2012) Efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors. Journal of Econometrics 170(2), 491–498, ISSN 0304–4076.Google Scholar
Speckman, P. (1979) Minimax estimation of linear functionals in a Hilbert space. Unpublished manuscript.Google Scholar
Wang, Q. & Phillips, P.C. (2009a) Asymptotic theory for local time density estimation and nonparametric cointegrating regression. Econometric Theory 25(03), 710–738.Google Scholar
Wang, Q. & Phillips, P.C. (2009b) Structural nonparametric cointegrating regression. Econometrica 77(6), 1901–1948.Google Scholar
Wang, Q. & Phillips, P.C. (2015) Nonparametric cointegrating regression with endogeneity and long memory. Econometric Theory. Forthcoming.Google Scholar