Two-stage least squares with clustered data

TL;DR

The paper develops a rigorous framework for conducting causal inference with endogenous treatments on clustered data using two IV-based approaches: the canonical 2sls and 2sfe. It establishes Wald-type inference validity under homogeneous clusters and derives precise efficiency conditions, showing that 2sfe can be more efficient when cluster-specific effects dominate unit-level noise, while the canonical 2sls can benefit from cluster-level covariates. It extends the analysis to heterogeneous clusters, proving that 2sfe consistently targets a weighted average of cluster-specific LATEs, and it provides a Wald-type test to detect cross-cluster heterogeneity, guiding empirical choice between the methods. The paper also develops a comprehensive joint asymptotic theory for the estimators, includes covariate-adjusted variants, and demonstrates the methods with simulations and an application to a Moroccan microcredit program, offering practical guidance for applied researchers dealing with clustered IV settings.

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.

Figures (2)

• Figure 1: Distributions of the $t$-statistic we proposed in \ref{['eq:t-stat']} under homogeneous and heterogeneous clusters over 1,000 independent replications. The curve in Figure \ref{['fig:t']}(a) represents the standard normal density. The table below reports the corresponding average values of $\hat{\tau}_\textup{2sls}$ and $\hat{\tau}_{\textup{2sfe}}$.
• Figure 2: Violin plots of the cluster bootstrap distributions of the eight point estimators over 1,000 replications. The width of each violin represents the empirical probability density, with wider sections indicating higher data concentration. Circles ($\circ$) denote point estimates from the original data. Error bars indicate 95% confidence intervals computed from the cluster-robust standard errors.

Theorems & Definitions (47)

• Definition 1
• Definition 2: Canonical 2sls
• Definition 3: Two-stage least squares with fixed effects (2sfe)
• Remark 1
• Proposition 1
• Theorem 1
• Corollary 1
• Corollary 2
• Proposition 2
• Proposition 3