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Quantifying Distributional Model Risk in Marginal Problems via Optimal Transport

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  • Yanqin Fan
  • Hyeonseok Park
  • Gaoqian Xu

Abstract

This paper studies distributional model risk in marginal problems, where each marginal measure is assumed to lie in a Wasserstein ball centered at a fixed reference measure with a given radius. Theoretically, we establish several fundamental results including strong duality, finiteness of the proposed Wasserstein distributional model risk, and the existence of an optimizer at each radius. In addition, we show continuity of the Wasserstein distributional model risk as a function of the radius. Using strong duality, we extend the well-known Makarov bounds for the distribution function of the sum of two random variables with given marginals to Wasserstein distributionally robust Markarov bounds. Practically, we illustrate our results on four distinct applications when the sample information comes from multiple data sources and only some marginal reference measures are identified. They are: partial identification of treatment effects; externally valid treatment choice via robust welfare functions; Wasserstein distributionally robust estimation under data combination; and evaluation of the worst aggregate risk measures.

Suggested Citation

  • Yanqin Fan & Hyeonseok Park & Gaoqian Xu, 2023. "Quantifying Distributional Model Risk in Marginal Problems via Optimal Transport," Papers 2307.00779, arXiv.org.
    • Handle: RePEc:arx:papers:2307.00779
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    References listed on IDEAS

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    1. Daido Kido, 2022. "Distributionally Robust Policy Learning with Wasserstein Distance," Papers 2205.04637, arXiv.org, revised Aug 2022.
    2. Christopher Adjaho & Timothy Christensen, 2022. "Externally Valid Policy Choice," Papers 2205.05561, arXiv.org, revised Nov 2025.
    3. Ridder, Geert & Moffitt, Robert, 2007. "The Econometrics of Data Combination," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 75, Elsevier.
    4. Bryan S. Graham & Cristine Campos de Xavier Pinto & Daniel Egel, 2016. "Efficient Estimation of Data Combination Models by the Method of Auxiliary-to-Study Tilting (AST)," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(2), pages 288-301, April.
    5. Fan, Yanqin & Guerre, Emmanuel & Zhu, Dongming, 2017. "Partial identification of functionals of the joint distribution of “potential outcomes”," Journal of Econometrics, Elsevier, vol. 197(1), pages 42-59.
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    Cited by:

    1. Yanqin Fan & Yuan Qi & Gaoqian Xu, 2025. "Policy Learning with $\alpha$-Expected Welfare," Papers 2505.00256, arXiv.org.
    2. Zhaonan Qu & Yongchan Kwon, 2024. "Distributionally Robust Instrumental Variables Estimation," Papers 2410.15634, arXiv.org, revised Dec 2024.

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