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Lee Bounds for Random Objects

Daisuke Kurisu, Yuta Okamoto, Taisuke Otsu

TL;DR

This paper extends Lee bounds to outcomes in general metric spaces by embedding random-object outcomes into a Hilbert space via a mapping $\Psi$ and analyzing Fréchet means. It shows that the control mean $\mu_{0\oplus}$ is point identified while the treated mean $\mu_{1\oplus}$ is partially identified through a Lee-type bound $\Psi^{-1}(\mathcal{S}_1)$, where $\mathcal{S}_1$ is a convex set defined by support functions and conditional quantiles. The authors provide a practical estimation and bootstrap inference procedure for finite-dimensional cases, with guidance on higher-dimensional or infinite-dimensional settings via projection. They illustrate the method with compositional data and a distributional sleep outcome, demonstrating informative identified sets and valid confidence regions even under attrition. The approach broadens causal inference in economics and related fields to non-Euclidean outcomes such as distributions, functions, intervals, and networks, enabling geometry-aware partial identification and robust treatment-effect interpretation.

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 Fréchet 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.

Lee Bounds for Random Objects

TL;DR

This paper extends Lee bounds to outcomes in general metric spaces by embedding random-object outcomes into a Hilbert space via a mapping and analyzing Fréchet means. It shows that the control mean is point identified while the treated mean is partially identified through a Lee-type bound , where is a convex set defined by support functions and conditional quantiles. The authors provide a practical estimation and bootstrap inference procedure for finite-dimensional cases, with guidance on higher-dimensional or infinite-dimensional settings via projection. They illustrate the method with compositional data and a distributional sleep outcome, demonstrating informative identified sets and valid confidence regions even under attrition. The approach broadens causal inference in economics and related fields to non-Euclidean outcomes such as distributions, functions, intervals, and networks, enabling geometry-aware partial identification and robust treatment-effect interpretation.

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 Fréchet 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.
Paper Structure (17 sections, 5 theorems, 76 equations, 3 figures, 1 table)

This paper contains 17 sections, 5 theorems, 76 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Benchmark example: Compositional data
  3. Numerical illustration
  4. Generalization: Lee bounds for random objects
  5. Setup and examples
  6. Main results
  7. Characterizing treatment effects
  8. Implementation
  9. Estimation
  10. Inference
  11. Empirical illustration
  12. Additional examples
  13. Proofs
  14. Sharpness of $\mathcal{I}_1$
  15. Proof of Theorem \ref{['thm']}
  16. ...and 2 more sections

Key Result

Proposition 2.1

Consider the setup of this section (in particular, assume random assignment and monotonicity) with some regularity conditions (Assumption assumption:metric below). Then, $\mu_{0\oplus}$ is point identified as in (eq:mu0). Moreover, $\mu_{1\oplus}$ is partially identified and its sharp identified set

Figures (3)

  • Figure 1: Comparison between the Euclidean metric and the Aitchison metric
  • Figure 2: Ternary plots
  • Figure 3: Expected quantile functions

Theorems & Definitions (16)

  • Proposition 2.1
  • Example 3.1: Interval data
  • Example 3.2: Functional data
  • Example 3.3: One-dimensional probability distributions
  • Example 3.4: Networks
  • Theorem 3.1
  • Remark 3.1: Covariates
  • Remark 3.2: Distributional outcome
  • Remark 3.3: Contaminated and corrupted data model
  • Proposition 4.1
  • ...and 6 more