Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T17:32:25.041Z Has data issue: false hasContentIssue false
  • English
  • Français

ON RATE OPTIMALITY FOR ILL-POSED INVERSE PROBLEMS IN ECONOMETRICS

Published online by Cambridge University Press:  11 October 2010

Xiaohong Chen  and
Markus Reiss
Xiaohong Chen*
Affiliation:
Yale University
Markus Reiss
Affiliation:
Humboldt University Berlin
*
*Address correspondence to Xiaohong Chen, Cowles Foundation for Research in Economics, Yale University, Box 208281, New Haven, CT 06520 USA; e-mail: xiaohong.chen@yale.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and NPIV models under two basic regularity conditions: the approximation number and the link condition. We show that both a simple projection estimator for the NPIR model and a sieve minimum distance estimator for the NPIV model can achieve the minimax risk lower bounds and are rate optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.

Type
ARTICLES
Information
Econometric Theory , Volume 27 , Issue 3: SPECIAL ISSUE ON INVERSE PROBLEMS IN ECONOMETRICS , June 2011 , pp. 497 - 521
Copyright
Copyright © Cambridge University Press 2011

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.)

References

REFERENCES

Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71, 1795–1844.CrossRefGoogle Scholar
Bissantz, N., Hohage, T., Munk, A., & Ruymgaart, F. (2007) Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM Journal on Numerical Analysis 45, 2610–2636.CrossRefGoogle Scholar
Blundell, R., Chen, X., & Kristensen, D. (2007) Semi-nonparametric IV estimation of shape-invariant engel curves. Econometrica 75, 1613–1670.CrossRefGoogle Scholar
Blundell, R. & Powell, J.L. (2003) Endogeneity in semiparametric and nonparametric regression models. In Dewatripont, M., Hansen, L. P., & Turnovsky, S. J. (eds.), Advances in Economics and Econometrics: Theory and Applications, vol. 2, pp. 312–357. Cambridge University Press.CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. In Heckman, J.J. & Leamer, E.E. (eds.), Handbook of Econometrics, vol. 6, Ch. 77. North-Holland.Google Scholar
Chen, X. & Pouzo, D. (2008) Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Moments. Cowles Foundation Discussion Paper d1650, Yale University.CrossRefGoogle Scholar
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatment effects. Econometrica 73, 245–261.CrossRefGoogle Scholar
Chernozhukov, V., Imbens, G., & Newey, W. (2007) Instrumental variable estimation of nonseparable models. Journal of Econometrics 139, 4–14.CrossRefGoogle Scholar
Cohen, A., Daubechies, I., & Vial, P. (1993) Wavelets on the interval and fast wavelet transforms. Applied Computational Harmonic Analysis 1(1), 54–81.CrossRefGoogle Scholar
Cohen, A., Hoffmann, M., & Reiss, M. (2004) Adaptive wavelet Galerkin methods for linear inverse problems. SIAM Journal of Numerical Analysis 42, 1479–1501.CrossRefGoogle Scholar
Darolles, S., Florens, J.-P., & Renault, E. (2006) Nonparametric Instrumental Regression. Mimeo, GREMAQ, University of Toulouse.Google Scholar
D’Haultfoeuille, X. (2011) On the completeness condition in nonparametric instrumental problems. Econometric Theory 27 (this issue).CrossRefGoogle Scholar
Donoho, D., Liu, R., & MacGibbon, B. (1990) Minimax risk over hyperrectangles, and implications. Annals of Statistics 18, 1416–1437.CrossRefGoogle Scholar
Edmunds, D. & Evans, W. (1987) Spectral Theory and Differential Operators. Oxford University Press.Google Scholar
Efromovich, S. & Koltchinskii, V. (2001) On inverse problems with unknown operators. IEEE Transactions. on Information Theory 47, 2876–2893.CrossRefGoogle Scholar
Engl, H., Hanke, M., & Neubauer, A. (1996) Regularization of Inverse Problems. Kluwer Academic.CrossRefGoogle Scholar
Florens, J.-P. (2003) Inverse problems and structural econometrics: The example of instrumental variables. In Dewatripont, M., Hansen, L.P., & Turnovsky, S.J. (eds.), Advances in Economics and Econometrics: Theory and Applications, vol. 2, pp. 284–311. Cambridge University Press.CrossRefGoogle Scholar
Florens, J.-P., Johannes, J., & Van Bellegem, S. (2011) Identification and estimation by penalization in nonparametric instrumental regression, Econometric Theory 27 (this issue).CrossRefGoogle Scholar
Gagliardini, P. & Scaillet, O. (2009) Tikhonov Regularisation for Functional Minimum Distance Estimation. Mimeo, HEC Université de Genève.Google Scholar
Hall, P. & Horowitz, J. (2005) Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 33, 2904–2929.CrossRefGoogle Scholar
Hoffmann, M. & Reiss, M. (2008) Nonlinear estimation for linear inverse problems with error in the operator. Annals of Statistics 36(1), 310–336.CrossRefGoogle Scholar
Horowitz, J. & Lee, S. (2007) Nonparametric instrumental variables estimation of a quantile regression model. Econometrica 75, 1191–1208.CrossRefGoogle Scholar
Johannes, J., Van Belleghem, S., & Vanhems, A. (2011) Convergence rates for ill-posed inverse problems with an unknown operator. Econometric Theory 27 (this issue).CrossRefGoogle Scholar
Korostelev, A. & Tsybakov, A. (1993) Minimax Theory of Image Reconstruction, Lecture Notes in Statistics, v.82. Springer.CrossRefGoogle Scholar
Le Cam, L. & Lo Yang, G. (2000) Asymptotics in Statistics. Some Basic Concepts, 2nd ed. Springer.Google Scholar
Mair, A. & Ruymgaart, F. (1996) Statistical inverse problems in Hilbert scales. SIAM Journal on Applied Mathematics 56, 1424–1444.CrossRefGoogle Scholar
Nair, M., Pereverzev, S., & Tautenhahn, U. (2005) Regularization in Hilbert scales under general smoothing conditions. Inverse Problems 21, 1851–1869.CrossRefGoogle Scholar
Newey, W.K. & Powell, J. (2003) Instrumental variables estimation for nonparametric models. Econometrica 71, 1565–1578.CrossRefGoogle Scholar
Tautenhahn, U. (1998) Optimality for ill-posed problems under general source conditions. Numerical Functional Analysis and Optimization 19, 377–98.CrossRefGoogle Scholar
Yang, Y. & Barron, A. (1999) Information-theoretic determination of minimax rates of convergence. Annals of Statistics 27(5), 1564–1599.CrossRefGoogle Scholar