Lee bounds are a powerful econometric tool originally developed to assess treatment effects in the presence of sample selection or attrition, typically applied in contexts where outcomes are scalar and well-defined. Extending Lee bounds to more complex data types—specifically outcomes that reside in general metric spaces such as compositional data (e.g., proportions summing to one) or distributional data (e.g., probability distributions)—is a cutting-edge area of research that addresses key challenges in causal inference when outcomes are not simple real numbers.
**Short answer:** Lee bounds can be generalized to outcomes in general metric spaces by leveraging the structure of these spaces—such as metrics defining distances between points—to construct partial identification regions that respect the geometry and constraints of compositional or distributional data, enabling robust inference under selection or attrition without requiring full parametric assumptions.
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Background: The Original Lee Bounds and Their Limitations
Lee bounds, introduced by David S. Lee in 2009, provide a nonparametric way to bound treatment effects when outcomes are only observed for a selected subpopulation, for example, when attrition or sample selection depends on treatment status. The method trims the distribution of observed outcomes to account for unobserved potential outcomes of non-respondents, yielding sharp bounds on average treatment effects.
Traditionally, Lee bounds have been applied to scalar outcomes—such as income or test scores—where ordering is straightforward and trimming can be done by removing the lowest or highest fraction of outcomes. This assumption of a total order and scalar outcome space is crucial to the original method.
However, in many modern applications, outcomes are more complex. For example, compositional data represent parts of a whole (like budget shares or microbial proportions) and live on the simplex, a constrained multidimensional space without a natural total order. Distributional data—such as income distributions or probability densities—are infinite-dimensional objects with their own intrinsic geometry. These data types challenge the direct application of Lee bounds because there is no simple way to "trim" outcomes in the absence of a natural ordering.
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Extending Lee Bounds to General Metric Spaces
To apply Lee bounds beyond scalar outcomes, researchers have developed approaches that exploit the metric structure of the outcome space. A metric space is a set equipped with a distance function that quantifies how "far apart" two points are, satisfying properties like symmetry and the triangle inequality.
In compositional data analysis, the simplex can be endowed with metrics such as the Aitchison distance, which respects the relative nature of parts and the unit-sum constraint. Similarly, for distributional data, Wasserstein metrics (also called earth mover’s distances) provide a natural way to measure distances between probability distributions, capturing differences in shape and location.
By using these metrics, it becomes possible to define trimmed sets or quantiles in a generalized sense. Instead of trimming the lowest or highest values, one can trim subsets of the outcome space that correspond to the "most extreme" points under the metric, effectively generalizing the idea of quantile trimming to complex spaces.
This generalization preserves the partial identification spirit of Lee bounds: it does not require parametric assumptions about the outcome distribution or the selection mechanism, but it uses the geometry of the space to bound the treatment effect.
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Practical Approaches and Examples
One practical approach is to use metric balls or neighborhoods around observed outcomes to define trimming. For example, in compositional data, one might trim those observations that lie furthest from a central point (like the geometric mean composition), ensuring that the trimmed sample mimics the selection process assumed in Lee bounds. For distributional outcomes, trimming can be performed by excluding distributions that lie beyond a certain quantile of the Wasserstein distance from a reference distribution.
This approach has been applied in emerging econometric and statistical literature focusing on causal inference with functional or distributional outcomes. It aligns with advances in optimal transport theory and compositional data analysis, which provide computational tools to calculate distances and medians in these spaces.
While the original Lee bounds provide sharp bounds on average treatment effects, extending to metric spaces often leads to bounds on more general functionals, such as Fréchet means (generalized means in metric spaces) or other location parameters defined via the metric structure.
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Challenges and Open Questions
Extending Lee bounds to general metric spaces is not straightforward. Unlike the scalar case, where trimming is done on a well-defined quantile scale, metric spaces may lack a total order, making the notion of "lowest" or "highest" ambiguous. Defining and computing quantiles or trimmed sets requires careful mathematical formulation and computational methods.
Moreover, the geometry of the space affects the shape and size of bounds. For example, compositional data constraints (e.g., non-negativity and unit sum) mean that trimmed sets must respect these constraints to remain valid compositions. Distributional data often require infinite-dimensional optimization, which can be computationally intensive.
Additionally, identification results depend on assumptions about the selection mechanism and how it relates to the metric structure of outcomes. Researchers must carefully specify assumptions analogous to monotonicity or rank preservation but adapted to metric spaces.
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Contextualizing with Related Research
Although the NBER working paper excerpt (Blanchflower & Bryson, 2020) focuses on labor economics and job satisfaction, it highlights the importance of longitudinal data and conditioning on covariates for causal inference, which parallels the need for careful conditioning in Lee bounds extensions.
The arXiv paper on multiferroic materials (Qian Song et al., 2021) illustrates the complexity of studying phenomena in high-dimensional and structured spaces (like 2D materials with magnetic and electric order parameters), analogous to the complexity of outcomes in metric spaces.
The mention of admissible Bayes equivariant estimation in spherically symmetric distributions (projecteuclid.org) touches on estimation problems in structured spaces, which informs how statistical methods can be adapted to non-Euclidean spaces—relevant for extending Lee bounds.
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Takeaway
Extending Lee bounds to general metric spaces like compositional and distributional data represents a significant advance in causal inference methodology, accommodating modern complex data types. By leveraging the geometry of these spaces through appropriate metrics, researchers can construct robust, assumption-light bounds on treatment effects even when outcomes lack natural ordering or scalar structure. This extension opens pathways for rigorous analysis in diverse fields—from economics to biology—where outcomes are inherently multidimensional or functional, ensuring that the powerful insights of Lee bounds remain applicable in the data-rich, complex world of today’s empirical research.
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Suggested Readings and Resources
- Econometric literature on Lee bounds and partial identification methods (NBER working papers, Econometrica). - Compositional data analysis texts explaining the Aitchison geometry (e.g., John Aitchison’s foundational work). - Optimal transport and Wasserstein distance in statistics (texts by Cédric Villani). - Recent papers on causal inference with functional and distributional outcomes (arXiv.org for preprints). - Statistical estimation in metric spaces (projecteuclid.org and related journals). - Tutorials and lectures on partial identification and causal inference (e.g., Raj Chetty & Kosuke Imai’s methods lectures). - Software packages implementing compositional data methods and optimal transport computations (e.g., R packages `compositions`, `transport`).
These resources provide the theoretical foundation and practical tools for applying Lee bounds in metric spaces, helping researchers navigate the challenges and harness the opportunities presented by complex outcome types.
