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

1. Problem Statement

In empirical economics, data is frequently clustered (e.g., students within schools, households within villages). When estimating the causal effect of an endogenous treatment using Instrumental Variables (IV), researchers typically employ Two-Stage Least Squares (2SLS).

• Canonical 2SLS: Regresses the outcome on the treatment using the IV, ignoring cluster structure in the point estimation but using cluster-robust standard errors (CRSE) for inference.
• 2SLS with Fixed Effects (2SFE): Includes cluster indicators (fixed effects) in the regression to account for cluster structure in both point estimation and inference.

While 2SFE is often preferred in observational studies to control for unobserved cluster heterogeneity, its causal interpretation and efficiency properties relative to the canonical 2SLS within the Local Average Treatment Effect (LATE) framework have been underexplored. Specifically, it is unclear when 2SFE is more efficient than canonical 2SLS and whether 2SFE provides a valid causal estimand when clusters are heterogeneous.

2. Methodology and Framework

The authors analyze these two estimators using the potential outcomes framework (Imbens and Angrist, 1994) under two distinct assumptions regarding cluster homogeneity:

1. Homogeneous Clusters (Assumption 3): All units share a common marginal distribution.
2. Heterogeneous Clusters (Assumption 4): Units within a cluster share a distribution, but distributions vary across clusters.

Key Definitions:

• Canonical 2SLS ($\hat{\tau}_{2sls}$): $\text{2sls}(Y_i \sim 1 + D_i \mid 1 + Z_i)$.
• 2SFE ($\hat{\tau}_{2sfe}$): $\text{2sls}(Y_i \sim D_i + C_i \mid Z_i + C_i)$, where $C_i$ are cluster indicators.
• Assumptions: The paper assumes standard IV conditions (exclusion restriction, relevance, monotonicity) adapted for clustered data, where the IV is randomly assigned at the cluster level (or independent within clusters).

3. Key Contributions and Results

A. Validity and Efficiency under Homogeneous Clusters

When clusters are homogeneous (Assumption 3), the authors establish:

1. Validity: Both $\hat{\tau}_{2sls}$ and $\hat{\tau}_{2sfe}$ are consistent and asymptotically normal estimators of the global LATE ($\tau_c$). Cluster-robust standard errors are valid for inference in both cases.
2. Efficiency Trade-off: Contrary to the intuition that fixed effects always improve efficiency, the paper proves that 2SFE is not universally more efficient.
   • Under a linear IV model, the ratio of asymptotic variances depends on the ratio of cluster-level variance ($\sigma^2_\alpha$) to idiosyncratic variance ($\sigma^2_\epsilon$) and the within-cluster variation of the IV ($\kappa_N$).
   • Condition for 2SFE Efficiency: 2SFE is more efficient than canonical 2SLS only if:
     1. The variation in cluster-specific effects dominates idiosyncratic variation ($\sigma^2_\alpha$ is large relative to $\sigma^2_\epsilon$).
     2. The IV has sufficient within-cluster variation ($\kappa_N$ is not too small).
   • If cluster effects are negligible ($\sigma^2_\alpha \approx 0$), canonical 2SLS is actually more efficient because 2SFE "wastes" degrees of freedom by absorbing variation that could otherwise be used for estimation.
3. Covariate Adjustment:
   • Canonical 2SLS can accommodate cluster-level covariates (e.g., village size), which can improve efficiency if these covariates explain cluster heterogeneity.
   • 2SFE absorbs all cluster-level variation via fixed effects, rendering cluster-level covariates collinear and unusable. Thus, if informative cluster-level covariates exist, canonical 2SLS may be superior.

B. Causal Interpretation under Heterogeneous Clusters

When clusters are heterogeneous (Assumption 4):

1. 2SFE Interpretation: $\hat{\tau}_{2sfe}$ consistently estimates a weighted average of cluster-specific LATEs ($\sum \kappa_{g,2sfe} \tau_{c,g}$), where weights are non-negative and sum to one. This is a well-defined causal quantity.
2. Canonical 2SLS Interpretation: $\hat{\tau}_{2sls}$ generally does not estimate a weighted average of LATEs. Its probability limit includes bias terms related to the correlation between cluster-specific IV probabilities and baseline outcomes. It is not "level-independent" (i.e., it may be non-zero even if all true LATEs are zero).
3. Implication: In settings with heterogeneous clusters, 2SFE provides a more interpretable and robust estimand than canonical 2SLS.

C. Proposed Test for Cluster Heterogeneity

To guide empirical researchers in choosing between the two methods, the authors propose a Hausman-type test:

• Statistic: $t_{diff} = (\hat{\tau}_{2sls} - \hat{\tau}_{2sfe}) / \hat{se}_{diff}$.
• Logic: Under homogeneous clusters, both estimators target the same parameter, so the difference should be zero. Under heterogeneous clusters, they target different parameters (or 2SLS is biased), leading to a significant difference.
• Distribution: The authors derive the joint asymptotic distribution of the two estimators, showing $t_{diff} \xrightarrow{d} N(0,1)$ under the null hypothesis of homogeneous clusters.

4. Empirical Application and Simulation

• Simulations: The authors validate their theoretical findings through simulations.
  • In homogeneous settings, both methods achieve correct coverage, but efficiency gains depend on the variance ratio and IV variation.
  • In heterogeneous settings, the proposed test correctly rejects the null hypothesis of homogeneity with high power as heterogeneity increases, while 2SFE remains unbiased and 2SLS becomes biased.
• Application: The methods are applied to a microcredit program evaluation in Morocco.
  • The heterogeneity test failed to reject the null (clusters appeared homogeneous).
  • Results showed that the covariate-adjusted 2SFE ($2sfe-x$) yielded the narrowest confidence intervals, demonstrating the efficiency gains from combining fixed effects with covariate adjustment in this specific context.

5. Significance and Conclusion

This paper provides a rigorous theoretical foundation for choosing between canonical 2SLS and 2SFE in clustered IV settings.

• Practical Guidance: It cautions against the routine inclusion of fixed effects in IV regressions without assessing efficiency trade-offs. If cluster heterogeneity is low or IV variation within clusters is limited, canonical 2SLS (potentially with cluster-level covariates) may be more efficient.
• Robustness: It highlights that 2SFE is essential for valid causal interpretation when clusters are heterogeneous, as canonical 2SLS may yield biased estimates in such scenarios.
• Diagnostic Tool: The proposed $t$-test offers a practical, data-driven method for researchers to detect cluster heterogeneity and select the appropriate estimator.

In summary, the paper resolves the trade-off between these two standard approaches, offering clear conditions under which one dominates the other and providing a statistical test to detect the underlying data structure.