Lee Bounds for Random Objects

TL;DR

This paper extends Lee bounds to outcomes in general metric spaces, enabling causal inference with compositional and distributional data by embedding outcomes into Euclidean or Hilbert spaces.

Contribution

It introduces a novel methodology to apply Lee bounds to complex outcomes in metric spaces using embeddings and Fréchet means, with practical estimators and confidence regions.

Findings

Method successfully bounds causal effects for compositional data.

Numerical examples demonstrate applicability to distributional data.

Proposed estimator performs well in simulations.

Abstract

In applied research, Lee (2009) bounds are widely applied to bound the average treatment effect in the presence of selection bias. This paper extends the methodology of Lee bounds to accommodate outcomes in a general metric space, such as compositional and distributional data. By exploiting a representation of the Fr\'echet mean of the potential outcome via embedding in an Euclidean or Hilbert space, we present a feasible characterization of the identified set of the causal effect of interest, and then propose its analog estimator and bootstrap confidence region. The proposed method is illustrated by numerical examples on compositional and distributional data.