Fragility in Average Treatment Effect on the Treated under Limited Covariate Support

This paper studies the identification of the average treatment effect on the treated (ATT) under unconfoundedness when covariate overlap is partial. A formal diagnostic is proposed to characterize empirical support -- the subset of the covariate space where ATT is point-identified due to the presence of comparable untreated units. Where support is absent, standard estimators remain computable but cease to identify meaningful causal parameters. A general sensitivity framework is developed, indexing identified sets by curvature constraints on the selection mechanism. This yields a structural selection frontier tracing the trade-off between assumption strength and inferential precision. Two diagnostic statistics are introduced: the minimum assumption strength for sign identification (MAS-SI), and a fragility index that quantifies the minimal deviation from ignorability required to overturn qualitative conclusions. Applied to the LaLonde (1986) dataset, the framework reveals that nearly half the treated strata lack empirical support, rendering the ATT undefined in those regions. Simulations confirm that ATT estimates may be stable in magnitude yet fragile in epistemic content. These findings reframe overlap not as a regularity condition but as a prerequisite for identification, and recast sensitivity analysis as integral to empirical credibility rather than auxiliary robustness.