Two-stage Least Squares with Clustered Data under the Local Average Treatment Effect Framework

TL;DR

This paper compares two-stage least squares methods for clustered data within the LATE framework, analyzing their validity, efficiency, and heterogeneity detection.

Contribution

It provides theoretical insights into when each 2sls approach is valid and efficient, and introduces a test for cluster heterogeneity.

Findings

Both methods yield valid inference when clusters are homogeneous.

2sfe is more efficient when cluster effects dominate idiosyncratic variation.

2sfe estimates a weighted average of cluster-specific LATEs in heterogeneous clusters.

Abstract

To estimate the causal effect of an endogenous treatment using clustered data, the canonical two-stage least squares (2sls) estimates a linear regression of the outcome on treatment status using an instrumental variable (IV) and conducts inference with cluster-robust standard errors. When both the treatment and the IV vary within clusters, an alternative two-stage least squares with fixed effects (2sfe) additionally includes cluster indicators in the regression, thereby incorporating cluster information into point estimation as well. This paper studies the trade-off between these approaches within the local average treatment effect (LATE) framework. When clusters are homogeneous, we show that both approaches yield valid large-sample inference for the LATE, and that 2sfe is more efficient than canonical 2sls only when the variation in cluster-specific effects dominates idiosyncratic variation and the IV has sufficient within-cluster variation. When clusters are heterogeneous, we show that 2sfe identifies a weighted average of cluster-specific LATEs, whereas the canonical 2sls generally does not. We further propose a test for detecting cluster heterogeneity.