Overlap

Every unit has a strictly-between-0-and-1 probability of receiving each treatment given its covariates: $0<\mathrm{Pr}(S=t\mid X)<1$ — equivalently, in Propensity-Score notation, $0<e(X)<1$. Without overlap there are no comparable control units for some treated units (or vice versa), and the Causal-Estimand is not identified by adjustment in that region. Pairs with Ignorability as the two pillars of selection-on-observables identification.

Relied on by

PSM (no comparison without common support); regression adjustment; any covariate-conditioning design.

Referenced by

Rubin1977-AssignmentOnCovariate (overlap in $X$ required), Rubin1974-EstimatingCausalEffects (comparable units for matching/adjustment).

DiD pass (overlap / common support of covariates or outcomes): Abadie2005-SemiparametricDiD, SantAnnaZhao2020-DoublyRobustDiD, CallawaySantAnna2021-DiDMultiplePeriods, AtheyImbens2006-NonlinearDiD.

Method-buildout pass (2026-06-17) — common support is central to the PSM literature: Imbens2004-NonparametricATEReview, Imbens2015-MatchingMethodsInPractice, CaliendoKopeinig2008-PSMImplementationGuidance, SmithTodd2005-ReconcilingPSMEvidence; and to the surveys AbadieCattaneo2018-EconometricMethodsProgramEvaluation, ImbensWooldridge2009-ProgramEvaluation.

New-papers pass (2026-07-04): ArkhangelskyImbens2024-FixedEffectsGeneralizedMundlak (propensity trimming on group balancing scores).