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Partial Identification under Stratified Randomization

Bruno Ferman, Davi Siqueira, Vitor Possebom

TL;DR

This paper addresses partial identification of treatment effects when outcomes are missing due to attrition in stratified randomized experiments. It develops a unified framework that covers both equal-share and heterogeneous-share designs, deriving design-consistent inference for Lee bounds in the equal-share case and introducing Lee-IPW bounds with a global trimming rule for heterogeneous shares. The authors cast Lee-bound estimation as a moment-based GMM problem, establish CLTs and variance decompositions that respect block structure, and provide feasible, design-consistent variance estimators applicable to a broad class of estimators. They further extend the results to label-based stratification and offer practical guidance on when to use Lee bounds versus Lee-IPW bounds, with a focus on tighter confidence intervals and robustness to small or unbalanced strata. Overall, the paper advances partial identification and inference in complex, stratified experiments by closely tying identification, estimation, and variance estimation to the experimental design.

Abstract

This paper develops a unified framework for partial identification and inference in stratified experiments with attrition, accommodating both equal and heterogeneous treatment shares across strata. For equal-share designs, we apply recent theory for finely stratified experiments to Lee bounds, yielding closed-form, design-consistent variance estimators and properly sized confidence intervals. Simulations show that the conventional formula can overstate uncertainty, while our approach delivers tighter intervals. When treatment shares differ across strata, we propose a new strategy, which combines inverse probability weighting and global trimming to construct valid bounds even when strata are small or unbalanced. We establish identification, introduce a moment estimator, and extend existing inference results to stratified designs with heterogeneous shares, covering a broad class of moment-based estimators which includes the one we formulate. We also generalize our results to designs in which strata are defined solely by observed labels.

Partial Identification under Stratified Randomization

TL;DR

This paper addresses partial identification of treatment effects when outcomes are missing due to attrition in stratified randomized experiments. It develops a unified framework that covers both equal-share and heterogeneous-share designs, deriving design-consistent inference for Lee bounds in the equal-share case and introducing Lee-IPW bounds with a global trimming rule for heterogeneous shares. The authors cast Lee-bound estimation as a moment-based GMM problem, establish CLTs and variance decompositions that respect block structure, and provide feasible, design-consistent variance estimators applicable to a broad class of estimators. They further extend the results to label-based stratification and offer practical guidance on when to use Lee bounds versus Lee-IPW bounds, with a focus on tighter confidence intervals and robustness to small or unbalanced strata. Overall, the paper advances partial identification and inference in complex, stratified experiments by closely tying identification, estimation, and variance estimation to the experimental design.

Abstract

This paper develops a unified framework for partial identification and inference in stratified experiments with attrition, accommodating both equal and heterogeneous treatment shares across strata. For equal-share designs, we apply recent theory for finely stratified experiments to Lee bounds, yielding closed-form, design-consistent variance estimators and properly sized confidence intervals. Simulations show that the conventional formula can overstate uncertainty, while our approach delivers tighter intervals. When treatment shares differ across strata, we propose a new strategy, which combines inverse probability weighting and global trimming to construct valid bounds even when strata are small or unbalanced. We establish identification, introduce a moment estimator, and extend existing inference results to stratified designs with heterogeneous shares, covering a broad class of moment-based estimators which includes the one we formulate. We also generalize our results to designs in which strata are defined solely by observed labels.
Paper Structure (63 sections, 26 theorems, 389 equations)

This paper contains 63 sections, 26 theorems, 389 equations.

Table of Contents

  1. Introduction
  2. General Setting under Heterogeneous-Shares Stratification
  3. Inference under Equal-Share Stratification
  4. Setup: Equal-Share Stratification
  5. Partial Identification and Consistency of Lee Bounds under Equal-Share Stratification
  6. Asymptotic Distribution and Variance for Lee Bounds under Equal-Share Stratification
  7. Simulation Results
  8. Lee–IPW Bounds under Heterogeneous-Shares Stratification
  9. Partial Identification Strategy with Lee-IPW Bounds
  10. Lee-IPW Estimator and Moment Conditions
  11. Inference for Moment-Based Estimators under Heterogeneous-Shares Stratification
  12. Asymptotic Distribution under Heterogeneous-Shares Stratification
  13. Variance Decomposition and Estimation under Heterogeneous-Shares Stratification
  14. Empirical Implementation of Standard Errors for Lee-IPW Bounds
  15. Conclusion
  16. ...and 48 more sections

Key Result

Lemma 1

Suppose Assumption ass:A holds. Then

Theorems & Definitions (30)

  • Remark 1: Label-based stratification
  • Lemma 1: Equal-share stratification implies unconditional independence
  • Proposition 1: Partial ID and sharpness under equal-share stratification
  • Proposition 2: Consistency under equal-share stratification
  • Proposition 3: Asymptotics of Lee bounds under equal-share stratification
  • Remark 2: Equal shares and label-based designs
  • Lemma 2: Heterogeneous-shares stratification implies stratum-level independence
  • Proposition 4: Partial Identification with Lee-IPW Bounds
  • Proposition 5: Moment Characterization of the Lee-IPW Lower Bound
  • Remark 3: Heterogeneous shares and label-based designs
  • ...and 20 more