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Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression

Xiaohong Chen () and Timothy Christensen
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Timothy Christensen: Dept. of Economics, Yale University

No 1923, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

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 (search for similar items in EconPapers)
JEL-codes: C13 C14 C32 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm
Date: 2013-11
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Working Paper: Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression (2013) Downloads
Working Paper: Optimal uniform convergence rates for sieve nonparametric instrumental variables regression (2013) Downloads
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