# Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression

arXiv: https://arxiv.org/abs/1508.03365 · HTML: https://mlsanity.com/paper/1508.03365/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Aug 2015.

## Built on by (most cited first, 30 of 30)

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- [Sup-Norm Convergence of Deep Neural Network Estimator for Nonparametric Regression by Adversarial Training](https://mlsanity.com/paper/2307.04042/) (2023)
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- [Learning Decision Policies with Instrumental Variables through Double Machine Learning](https://mlsanity.com/paper/2405.08498/) (2024)
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- [Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models](https://mlsanity.com/paper/2101.12282/) (2021)
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- [Personalized Pricing with Invalid Instrumental Variables: Identification, Estimation, and Policy Learning](https://mlsanity.com/paper/2302.12670/) (2023)
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- [Conformal Prediction for Nonparametric Instrumental Regression](https://mlsanity.com/paper/2603.25509/) (2026)

## Abstract (arXiv)

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Empiricists could read our real data application of UCBs for exact CS and DL functionals of gasoline demand that reveals interesting patterns and is applicable to other markets.