Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression

TL;DR

This paper derives optimal uniform convergence rates for sieve estimators in nonparametric instrumental variables regression, addressing endogeneity and ill-posedness, with implications for econometrics and statistics.

Contribution

It establishes the first general upper bounds on sup-norm convergence for sieve NPIV estimators, matching minimax lower bounds and confirming their optimality.

Findings

Optimal sup-norm convergence rates for spline and wavelet estimators.

Rates match known $L^2$-norm bounds for ill-posed problems.

Validation of sieve NPIV estimators' wide applicability.

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.