Two-Sample IV: Efficient Two-Step Estimation and Tests for Overidentification and Weak-Instruments

2026-06-18

AI Summary

This study addresses efficiency losses and inferential challenges in two-sample instrumental variable estimation arising from sample heterogeneity, heteroskedasticity, weak instruments, and overidentification. The authors propose a robust two-step estimator that relies solely on summary statistics—namely, coefficient vectors and variance matrices—from the reduced-form and first-stage regressions in each sample. By relaxing conventional homoskedasticity and homogeneity assumptions, the method achieves efficient estimation under heteroskedasticity and heterogeneity. It extends the Montiel-Olea-Pflueger effective F-statistic to diagnose weak instruments in the two-sample context and develops a Hansen-type overidentification test tailored to this setting. An empirical application examining the causal effect of education on voting behavior demonstrates the method’s validity and practical utility, enabling reliable causal inference with complex data structures.

Abstract

Two-sample IV is a popular estimation method when the outcome and treatment variables are available in different samples, whereas instruments are available in both samples. The standard estimator is two-sample two-stage least squares estimator, which is efficient under homoskedasticity and homogeneity of the samples. We develop a robust two-step procedure for efficient estimation under general heteroskedasticity and heterogeneity of the samples, and propose a related two-sample Hansen overidentification test. A key feature of our approach is that only summary statistics from the linear regressions of the reduced form and first-stage in the two samples are needed. These are the six objects of the estimated coefficient vectors, and the homoskedastic and heteroskedasticity robust estimated variance matrices. We further show that the first-stage F-statistic in the treatment sample can be used as a test for weak instruments in the standard way under homoskedasticity and homogeneity, with the relative bias here a proportional bias. We propose an extension of the effective F-statistic of Montiel-Olea and Pflueger (2013) for the heteroskedastic case, following the generalization in Windmeijer (2025). We illustrate the estimators and tests in an application studying the effect of education on voting behavior from Marshall (2019), with cluster robust inference.

Problem

• Two-sample IV
• heteroskedasticity
• overidentification
• weak instruments
• sample heterogeneity

Innovation

• Two-sample IV
• heteroskedasticity robust
• efficient estimation
• overidentification test
• weak instruments