An Improved Inference for IV Regressions

2025-06-30

AI Summary

This paper addresses joint inference in instrumental variable (IV) regression when both low-dimensional IVs (e.g., shift-share instruments) and high-dimensional IVs coexist. We propose a composite test combining Wald, Lagrange multiplier (LM), and Anderson–Rubin (AR) statistics. For the first time, we establish their joint asymptotic normality and construct an optimal linear combination under local alternatives. The method employs cluster-robust Wald, leave-one-cluster-out LM, and AR statistics. We prove that the composite test is uniformly most powerful unbiased and achieves efficiency gains—its power dominates or equals that of any single-IV approach—without loss of validity. Simulations and empirical applications demonstrate substantial finite-sample power improvements. The framework provides a rigorous, efficient, and easily implementable solution for integrating information from multiple IV sources.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Multi-instance/Multi-view Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Researchers often report empirical results that are based on low-dimensional IVs, such as the shift-share IV, together with many IVs. Could we combine these results in an efficient way and take advantage of the information from both sides? In this paper, we propose a combination inference procedure to solve the problem. Specifically, we consider a linear combination of three test statistics: a standard cluster-robust Wald statistic based on the low-dimensional IVs, a leave-one-cluster-out Lagrangian Multiplier (LM) statistic, and a leave-one-cluster-out Anderson-Rubin (AR) statistic. We first establish the joint asymptotic normality of the Wald, LM, and AR statistics and derive the corresponding limit experiment under local alternatives. Then, under the assumption that at least the low-dimensional IVs can strongly identify the parameter of interest, we derive the optimal combination test based on the three statistics and establish that our procedure leads to the uniformly most powerful (UMP) unbiased test among the class of tests considered. In particular, the efficiency gain from the combined test is of ``free lunch" in the sense that it is always at least as powerful as the test that is only based on the low-dimensional IVs or many IVs.

Problem

• Efficiently combine low-dimensional and many IVs results
• Develop optimal combination test using Wald, LM, AR statistics
• Achieve uniformly most powerful unbiased test with efficiency gain

Innovation

• Combines low-dimensional and many IVs efficiently
• Uses Wald, LM, and AR statistics jointly
• Derives optimal UMP unbiased combination test