Two-stage least squares with clustered data

2026-01-20

AI Summary

This study addresses the challenge of correctly accounting for cluster structure when estimating causal effects of endogenous treatment variables in clustered data. Within the local average treatment effect (LATE) framework, it systematically compares standard two-stage least squares (2SLS) with two-stage fixed effects (2SFE) that incorporate cluster-level fixed effects. The analysis shows that both estimators are consistent under homogeneous cluster effects, but under heterogeneous clusters, 2SFE identifies a weighted average of LATEs while 2SLS does not. Moreover, 2SFE is more efficient only when variation in cluster-specific effects dominates. The paper also proposes a Wald-type test to detect cluster heterogeneity and provides both theoretical guidance and statistical tools to inform method selection in empirical practice.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Clustered data -- where units of observation are nested within higher-level groups, such as repeated measurements on users, or panel data of firms, industries, or geographic regions -- are ubiquitous in business research. When the objective is to estimate the causal effect of a potentially endogenous treatment, a common approach -- which we call the canonical two-stage least squares (2sls) -- is to fit a 2sls regression of the outcome on treatment status with instrumental variables (IVs) for point estimation, and apply cluster-robust standard errors to account for clustering in inference. When both the treatment and IVs vary within clusters, a natural alternative -- which we call the two-stage least squares with fixed effects (2sfe) -- is to include cluster indicators in the 2sls specification, thereby incorporating cluster information in point estimation as well. This paper clarifies the trade-off between these two approaches within the local average treatment effect (LATE) framework, and makes three contributions. First, we establish the validity of both approaches for Wald-type inference of the LATE when clusters are homogeneous, and characterize their relative efficiency. We show that, when the true outcome model includes cluster-specific effects, 2sfe is more efficient than the canonical 2sls only when the variation in cluster-specific effects dominates that in unit-level errors. Second, we show that with heterogeneous clusters, 2sfe recovers a weighted average of cluster-specific LATEs, whereas the canonical 2sls generally does not. Third, to guide empirical choice between the two procedures, we develop a joint asymptotic theory for the two estimators under homogeneous clusters, and propose a Wald-type test for detecting cluster heterogeneity.

Problem

• clustered data
• two-stage least squares
• local average treatment effect
• instrumental variables
• cluster heterogeneity

Innovation

• two-stage least squares
• clustered data
• local average treatment effect
• fixed effects
• heterogeneity test