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Empirical process results for exchangeable arrays

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  • Laurent Davezies

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - GENES - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - GENES - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Xavier D’haultfœuille

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - GENES - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - GENES - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Yannick Guyonvarch

    (Université Paris-Saclay)

Abstract

Exchangeable arrays are natural tools to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same population. Examples include trade flows between countries or relationships in a network. Separately exchangeable arrays are well suited to multiway clustering, where units sharing the same cluster (e.g. geographical areas or sectors of activity when considering individual wages) may be dependent in an unrestricted way. We prove uniform laws of large numbers and central limit theorems for such exchangeable arrays. We obtain these results under the same moment restrictions and conditions on the class of functions as those typically assumed with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays.

Suggested Citation

  • Laurent Davezies & Xavier D’haultfœuille & Yannick Guyonvarch, 2021. "Empirical process results for exchangeable arrays," Post-Print hal-04430851, HAL.
    • Handle: RePEc:hal:journl:hal-04430851
    DOI: 10.1214/20-AOS1981
    Note: View the original document on HAL open archive server: https://hal.inrae.fr/hal-04430851v1
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    References listed on IDEAS

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    1. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, November.
    2. Laurent Davezies & Xavier D'Haultfoeuille & Yannick Guyonvarch, 2018. "Asymptotic results under multiway clustering," Papers 1807.07925, arXiv.org, revised Aug 2018.
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    Cited by:

    1. Harold D Chiang & Yukitoshi Matsushita & Taisuke Otsu, 2021. "Multiway empirical likelihood," Papers 2108.04852, arXiv.org, revised Aug 2024.
    2. Laurent Davezies & Xavier D'Haultf{oe}uille & Yannick Guyonvarch, 2025. "Analytic inference with two-way clustering," Papers 2506.20749, arXiv.org.
    3. MacKinnon, James G. & Nielsen, Morten Ørregaard & Webb, Matthew D., 2023. "Cluster-robust inference: A guide to empirical practice," Journal of Econometrics, Elsevier, vol. 232(2), pages 272-299.
    4. Daniel Gaigall & Stefan Weber, 2025. "Jointly Exchangeable Collective Risk Models: Interaction, Structure, and Limit Theorems," Papers 2504.06287, arXiv.org.
    5. James G. MacKinnon & Morten Ørregaard Nielsen & Matthew D. Webb, 2024. "Jackknife Inference with Two-Way Clustering," Working Paper 1516, Economics Department, Queen's University.
    6. Diegert, Paul & Jochmans, Koen, 2024. "Nonparametric Identification of Models for Dyadic Data”," TSE Working Papers 24-1574, Toulouse School of Economics (TSE).
    7. Bryan S. Graham, 2019. "Network Data," Papers 1912.06346, arXiv.org.
    8. Bryan S. Graham, 2019. "Dyadic Regression," Papers 1908.09029, arXiv.org.
    9. Richard K. Crump & Nikolay Gospodinov & Ignacio Lopez Gaffney, 2024. "A Jackknife Variance Estimator for Panel Regressions," Staff Reports 1133, Federal Reserve Bank of New York.
    10. Harold D Chiang & Yukun Ma & Joel Rodrigue & Yuya Sasaki, 2021. "Dyadic double/debiased machine learning for analyzing determinants of free trade agreements," Papers 2110.04365, arXiv.org, revised Dec 2022.
    11. Nan Liu & Yanbo Liu & Yuya Sasaki, 2024. "Estimation and Inference for Causal Functions with Multiway Clustered Data," Papers 2409.06654, arXiv.org.
    12. Michael Pfaffermayr, 2023. "Cross‐sectional Gravity Models, PPML Estimation, and the Bias Correction of the Two‐Way Cluster‐Robust Standard Errors," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 85(5), pages 1111-1134, October.
    13. Bryan S. Graham, 2024. "Sparse Network Asymptotics for Logistic Regression Under Possible Misspecification," Econometrica, Econometric Society, vol. 92(6), pages 1837-1868, November.
    14. Bryan S. Graham, 2019. "Network Data," CeMMAP working papers CWP71/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    15. Chiang, Harold D. & Matsushita, Yukitoshi & Otsu, Taisuke, 2025. "Multiway empirical likelihood," Journal of Econometrics, Elsevier, vol. 249(PA).
    16. Chen, Kaicheng & Vogelsang, Timothy J., 2024. "Fixed-b asymptotics for panel models with two-way clustering," Journal of Econometrics, Elsevier, vol. 244(1).

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