Stacked Triple Differences

2026-04-24

AI Summary

This study addresses the challenges of forbidden comparisons and ambiguous causal interpretation that arise when applying traditional difference-in-differences-in-differences (DDD) designs in staggered treatment settings. The authors propose a “stacked DDD” framework that constructs self-contained “stacks” composed of four units—combining treated and clean control groups, each further partitioned into units with and without access to treatment—to form a unified dataset. This approach avoids forbidden comparisons while enabling transparent regression-based estimation. Built upon a fully saturated fixed-effects model, the method leverages the stack structure within event-time windows to identify causal effects and permits direct control over local parallel trends assumptions and aggregation weights. Empirical applications demonstrate that the proposed estimator yields substantively different conclusions compared to existing GMM or imputation-based approaches, underscoring its validity, robustness, and enhanced interpretability.

Technology Category

• Reasoning under Uncertainty: Causality
• Multiagent Systems: Mechanism Design
• Machine Learning: Causal Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Triple differences (DDD) is a workhorse quasi-experimental design in applied economics. But, under staggered adoption, its conventional three-way fixed-effects (3WFE) implementation inherits the forbidden-comparison and interpretation issues now well understood in the difference-in-differences literature. To resolve these issues, I introduce stacked DDD. I extend the stacked difference-in-differences approach to the DDD setting by creating self-contained stacks, each consisting of four cells over an event window: treated and clean comparison cohorts, each with treatment-eligible and treatment-ineligible units. Appending these stacks yields a unified dataset for estimating treatment effects without making forbidden comparisons. I prove that, at each post-treatment event-time, a linear regression with fully saturated fixed-effects applied to the stacked dataset identifies a strictly positive, cell-size-weighted average of stack-level conditional average treatment effects, with stack weights proportional to stack-level cell sizes. Building on this characterization, I outline alternative weighting schemes that recover distinct, transparent causal estimands with clear interpretations. Stacked DDD complements recent GMM and imputation-based frameworks by trading efficiency for regression-based transparency, pairwise (rather than global) parallel trends, and direct control over aggregation weights. I provide two empirical illustrations where stacked DDD yields substantially different quantitative conclusions compared to existing procedures.

Problem

• triple differences
• staggered adoption
• forbidden comparisons
• causal interpretation
• fixed-effects

Innovation

• Stacked Triple Differences
• Staggered Adoption
• Forbidden Comparisons
• Fixed-Effects Regression
• Causal Estimands