# Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression

arXiv: https://arxiv.org/abs/1311.0412 · HTML: https://mlsanity.com/paper/1311.0412/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2013.

## Built on by (most cited first, 2 of 2)

- [Optimal Treatment Regimes for Proximal Causal Learning](https://mlsanity.com/paper/2212.09494/) (2022)
- [Assumption-lean falsification tests of rate double-robustness of double-machine-learning estimators](https://mlsanity.com/paper/2306.10590/) (2023)

## Abstract (arXiv)

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 matrices are also provided.