Nonparametric Instrumental Variable Inference with Many Weak Instruments

2025-05-12

AI Summary

This paper studies asymptotically efficient inference on linear functionals in nonparametric instrumental variable (NPIV) models under a “many weak instruments” asymptotic regime, where the number of discrete instruments grows with the sample size—particularly relevant for causal extrapolation across experiments, using historical experiments as instruments to identify the nonparametric effect of short-term outcomes on long-term effects. Addressing the challenge that the structural function is only identified up to moment restrictions (not point-identified), we propose npJIVE: the first extension of the Jackknife Instrumental Variables Estimator (JIVE) to nonparametric inverse problems. npJIVE integrates minimal-norm solutions, a jackknife-based IV extension, and debiased machine learning (DML) to achieve automatic bias correction and robust inference. We establish its consistency, asymptotic normality, and semiparametric efficiency. To our knowledge, npJIVE is the first method for nonparametric IV inference with many weak instruments that is both theoretically rigorous and computationally feasible.

Technology Category

• Machine Learning: Causal Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We study inference on linear functionals in the nonparametric instrumental variable (NPIV) problem with a discretely-valued instrument under a many-weak-instruments asymptotic regime, where the number of instrument values grows with the sample size. A key motivating example is estimating long-term causal effects in a new experiment with only short-term outcomes, using past experiments to instrument for the effect of short- on long-term outcomes. Here, the assignment to a past experiment serves as the instrument: we have many past experiments but only a limited number of units in each. Since the structural function is nonparametric but constrained by only finitely many moment restrictions, point identification typically fails. To address this, we consider linear functionals of the minimum-norm solution to the moment restrictions, which is always well-defined. As the number of instrument levels grows, these functionals define an approximating sequence to a target functional, replacing point identification with a weaker asymptotic notion suited to discrete instruments. Extending the Jackknife Instrumental Variable Estimator (JIVE) beyond the classical parametric setting, we propose npJIVE, a nonparametric estimator for solutions to linear inverse problems with many weak instruments. We construct automatic debiased machine learning estimators for linear functionals of both the structural function and its minimum-norm projection, and establish their efficiency in the many-weak-instruments regime.

Problem

• Inference on linear functionals in NPIV with many weak instruments
• Estimating long-term causal effects using short-term outcome instruments
• Debiased machine learning for nonparametric structural function estimation

Innovation

• Nonparametric estimator for linear inverse problems
• Debiased machine learning for structural functions
• Jackknife extension to many weak instruments