Fiducial inference for partially identified parameters with applications to instrumental variable models
Yifan Cui, Jan Hannig
TL;DR
This paper develops a generalized fiducial inference framework for partially identified parameters, with instrumental variable models as the running example. It introduces an acceptance-sampling mechanism to draw fiducial samples of the observational probabilities and the underlying bounds, and proves a Bernstein–von Mises theorem establishing asymptotically valid fiducial confidence intervals for lower and upper bounds. The authors demonstrate the method via simulation studies and a real-data Vitamin A mortality example, showing nominal coverage and interpretability of the fiducial acceptance rate as a diagnostic for IV validity. The work advances both fiducial inference and causal partial identification by providing practical uncertainty quantification without priors and by connecting fiducial plausibility to IV feasibility.
Abstract
In the past two decades, there has been a fast-growing literature on fiducial inference since it was first proposed by R. A. Fisher in the 1930s. However, most of the fiducial inference based methods and related approaches have been developed for point-identified models, i.e., statistical models where the parameters of interest are uniquely determined by the observed data and the model's assumptions. In this paper, we propose a novel fiducial approach for partially identified statistical models. As a leading example, we consider the instrumental variable model with a variety of causal assumptions and estimands. The proposed methods are illustrated through extensive simulations and a data analysis evaluating the effect of consuming Vitamin A supplementation on reducing mortality rates.