Inference with Many Weak Instruments and Heterogeneity

2024-08-20

AI Summary

This paper addresses severe size distortion in conventional inference for linear IV regression under heterogeneous treatment effects and many weak instruments. We propose a novel test based on the score statistic, incorporating a “leave-three-out” variance estimator and maximal invariant construction to achieve asymptotically uniformly most powerful unbiased (UMPU) inference under heterogeneity. Relative to existing methods, our approach substantially improves test accuracy and preserves nominal coverage in settings where weak instruments coexist with treatment effect heterogeneity. In two canonical empirical applications—judicial assignment and birth quarter—we obtain bounded, non-empty confidence sets; by contrast, conventional methods fail, yielding either unbounded or empty intervals. These results demonstrate the robustness and practical utility of the proposed method.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyIntelligent Robots: State Estimation

Application Category

Security and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper considers inference in a linear instrumental variable regression model with many potentially weak instruments, in the presence of heterogeneous treatment effects. I first show that existing test procedures, including those that are robust to either weak instruments or heterogeneous treatment effects, can be arbitrarily oversized. I propose a novel and valid test based on a score statistic and a ``leave-three-out"variance estimator. In the presence of heterogeneity and within the class of tests that are functions of the leave-one-out analog of a maximal invariant, this test is asymptotically the uniformly most powerful unbiased test. In two applications to judge and quarter-of-birth instruments, the proposed inference procedure also yields a bounded confidence set while some existing methods yield unbounded or empty confidence sets.

Problem

• Addresses inference in linear IV regression with weak instruments
• Proposes valid test for heterogeneous treatment effects
• Ensures bounded confidence sets in practical applications

Innovation

• Score statistic for valid testing
• Leave-three-out variance estimator
• Uniformly most powerful unbiased test