Please use this identifier to cite or link to this item: http://hdl.handle.net/10419/97421
Authors: 
Chen, Xiaohong
Christensen, Timothy
Year of Publication: 
2013
Series/Report no.: 
cemmap working paper, Centre for Microdata Methods and Practice CWP56/13
Abstract: 
We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult ill-posed inverse problem with unknown operator in statistics. We first establish a general upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal L2- norm rates for severely ill-posed problems, and are power of log (n) slower than the optimal L2- norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matricies are also provided.
Subjects: 
Nonparametric instrumental variables
Statistical ill-posed inverse problems
Optimal uniform convergence rates
Weak dependence
Random matrices
Splines
Wavelets
JEL: 
C13
C14
C32
Persistent Identifier of the first edition: 
Document Type: 
Working Paper

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