Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression

37 Pages Posted: 4 Nov 2013

See all articles by Xiaohong Chen

Xiaohong Chen

Yale University - Cowles Foundation

Timothy Christensen

New York University (NYU) - Department of Economics

Date Written: August 2013

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 L^2-norm rates for severely ill-posed problems, and are power of log(n) slower than the optimal L^2-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 matrices are also provided.

Keywords: Nonparametric instrumental variables; Statistical ill-posed inverse problems; Optimal uniform convergence rates; Weak dependence; Random matrices; Splines; Wavelets

JEL Classification: C13, C14, C32

Suggested Citation

Chen, Xiaohong and Christensen, Timothy, Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression (August 2013). Cowles Foundation Discussion Paper No. 1923, Available at SSRN: https://ssrn.com/abstract=2349684 or http://dx.doi.org/10.2139/ssrn.2349684

Xiaohong Chen (Contact Author)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States

Timothy Christensen

New York University (NYU) - Department of Economics ( email )

269 Mercer Street, 7th Floor
New York, NY 10011
United States

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