Partial Identification in Moment Models with Incomplete Data via Optimal Transport

Abstract: In this paper, we develop a unified approach to study partial identification of a finite-dimensional parameter defined by a moment equality model with incomplete data. We establish a novel characterization of the identified set for the true parameter in terms of a continuum of inequalities defined by optimal transport costs. For a special class of moment functions, we show that the identified set is convex, and its support function can be easily computed by solving an optimal transport problem. We demonstrate the generality and effectiveness of our approach through several running examples, including the linear projection model and two algorithmic fairness measures.