ON RATE OPTIMALITY FOR ILL-POSED INVERSE PROBLEMS IN ECONOMETRICS

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.

[1]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[2]  Victor Chernozhukov,et al.  Instrumental variable estimation of nonseparable models , 2007 .

[3]  Ulrich Tautenhahn,et al.  Optimality for ill-posed problems under general source conditions , 1998 .

[4]  P. Gagliardini,et al.  TIKHONOV REGULARISATION FOR FUNCTIONAL , 2006 .

[5]  Grace L. Yang,et al.  Asymptotics In Statistics , 1990 .

[6]  Advances in Economics and Econometrics: Endogeneity in Nonparametric and Semiparametric Regression Models , 2003 .

[7]  Bernard A. Mair,et al.  Statistical Inverse Estimation in Hilbert Scales , 1996, SIAM J. Appl. Math..

[8]  P. Gagliardini,et al.  Tikhonov Regularization for Functional Minimum Distance Estimators , 2006 .

[9]  Xiaohong Chen,et al.  Semi‐Nonparametric IV Estimation of Shape‐Invariant Engel Curves , 2003 .

[10]  Vladimir Koltchinskii,et al.  On inverse problems with unknown operators , 2001, IEEE Trans. Inf. Theory.

[11]  Xiaohong Chen,et al.  Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , 2003 .

[12]  Demian Pouzo,et al.  Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Moments , 2008 .

[13]  Marc Hoffmann,et al.  Nonlinear estimation for linear inverse problems with error in the operator , 2008, 0803.1956.

[14]  J. Florens,et al.  Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization , 2003 .

[15]  Yuhong Yang,et al.  Information-theoretic determination of minimax rates of convergence , 1999 .

[16]  P. Hall,et al.  Nonparametric methods for inference in the presence of instrumental variables , 2003, math/0603130.

[17]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[18]  Joel L. Horowitz,et al.  Nonparametric Instrumental Variables Estimation of a Quantile Regression Model , 2006 .

[19]  V. Chernozhukov,et al.  An IV Model of Quantile Treatment Effects , 2002 .

[20]  J. Florens,et al.  IDENTIFICATION AND ESTIMATION BY PENALIZATION IN NONPARAMETRIC INSTRUMENTAL REGRESSION , 2010, Econometric Theory.

[21]  W. Newey,et al.  Instrumental variable estimation of nonparametric models , 2003 .

[22]  Nicolai Bissantz,et al.  Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications , 2007, SIAM J. Numer. Anal..

[23]  J. Florens.  Advances in Economics and Econometrics: Inverse Problems and Structural Econometrics: The Example of Instrumental Variables , 2003 .

[24]  J. Florens,et al.  Nonparametric Instrumental Regression , 2010 .

[25]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[26]  T. N. Sriram.  Asymptotics in Statistics–Some Basic Concepts , 2002 .

[27]  CONVERGENCE RATES FOR ILL-POSED INVERSE PROBLEMS WITH AN UNKNOWN OPERATOR , 2010, Econometric Theory.

[28]  Sergei V. Pereverzev,et al.  Regularization in Hilbert scales under general smoothing conditions , 2005 .

[29]  D. Donoho,et al.  Minimax Risk Over Hyperrectangles, and Implications , 1990 .

[30]  Xavier D'Haultfoeuille,et al.  ON THE COMPLETENESS CONDITION IN NONPARAMETRIC INSTRUMENTAL PROBLEMS , 2010, Econometric Theory.

[31]  A. Tsybakov,et al.  Minimax theory of image reconstruction , 1993 .

[32]  Albert Cohen,et al.  Adaptive Wavelet Galerkin Methods for Linear Inverse Problems , 2004, SIAM J. Numer. Anal..