The Cascade Identity: 2SLS as a Policy Parameter in Capacity-Constrained Settings

2026-03-23

AI Summary

This study addresses the interpretational ambiguity of the conventional two-stage least squares (2SLS) estimator in capacity-constrained rationing systems, where treatment effect heterogeneity and first-stage spillovers obscure its policy relevance. The authors introduce a “cascade identity” that establishes conditions under which the 2SLS coefficient precisely identifies the total societal cascade effect—i.e., the aggregate reallocation triggered by adding one marginal treatment slot—provided the instrument and policy operate on the same margin. This result holds for arbitrary heterogeneity structures and rationing mechanisms, including queuing, waitlists, or market clearing, thereby overcoming key limitations in existing causal inference frameworks. Empirical validation using Swedish university admissions lotteries and charitable giving data confirms the theoretical predictions, demonstrating broad applicability to rationed settings such as education, housing, and healthcare.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A growing literature shows that two-stage least squares (2SLS) with multiple treatments yields coefficients that are difficult to interpret under heterogeneous treatment effects and cross-effects in the first stage. We show that in capacity-constrained allocation systems, these cross-effects are not a nuisance but the source of a clean policy interpretation. When treatments are rationed and the instrument operates on the same margin as the policy of interest, the 2SLS coefficient $β_k$ equals the total societal effect of expanding treatment $k$ by one slot, including all cascading reallocations through the system. The mechanism is general: it applies whenever fixed supply constrains allocation, whether through ranked queues, waitlists, or market-clearing prices. This cascade identity $\mathbf{T} = \mathbfβ$ holds for any first-stage matrix, under arbitrary treatment effect heterogeneity, and requires only instrument relevance and that the instrument operates on the same margin as the policy. The result applies to university admissions, school choice, medical residency matching, public housing, and other rationed allocation settings. We provide an empirical application using lottery-based admission to Swedish university programs and charitable giving as the outcome.

Problem

• 2SLS
• treatment effect heterogeneity
• capacity-constrained allocation
• policy interpretation
• cross-effects

Innovation

• cascade identity
• 2SLS
• capacity-constrained allocation
• treatment effect heterogeneity
• instrumental variables