A Generalized Fiducial Framework for Partially Identified Treatment Effects

Abstract

Over the past two decades, there has been renewed interest in fiducial inference, a statistical framework originally proposed by R. A. Fisher in the 1930s. Existing contributions, however, have largely focused on point-identified models, where the parameter vector is uniquely determined by the joint distribution of the observables and the model assumptions. This paper develops a unified fiducial inference procedure for partially identified models, thereby extending the applicability of the framework to settings in which the identified set is non-degenerate. The acceptance rate of the proposed sampler naturally serves as a diagnostic: high values support the identifying assumptions, while near-zero values signal their potential violation. We establish a Bernstein-von Mises theorem for the fiducial distribution, which provides theoretical guarantees for the proposed sampler. As two leading examples, we provide uncertainty quantification for the instrumental variable model and the mediation model under a range of causal assumptions and target parameters. The proposed methodology is illustrated through extensive simulations and empirical applications.