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Functional Ecological Inference

Author

Listed:
  • Christian Bontemps

    (ENAC-LAB - Laboratoire de recherche ENAC - ENAC - Ecole Nationale de l'Aviation Civile, TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Jean-Pierre Florens

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Nour Meddahi

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

Abstract

In this paper we consider the problem of ecological inference when one observes the conditional distributions of Y |W and Z|W from aggregate data and wants to infer the conditional distribution of Y |Z without observing Y and Z in the same sample. First, we show that this problem can be transformed into a linear equation involving operators for which, under suitable regularity assumptions, least squares solutions are available. Then we propose to use the least squares solution with the minimum Hilbert-Schmidt norm, which in our context can be structurally interpreted as the solution with minimum dependence between Y and Z. Interestingly, in the case where the conditioning variable W is discrete and belongs to a finite set, such as the labels of units/groups/cities, the solution of this minimal dependence has a closed form. In the more general case, we use a regularization scheme and show the convergence of our proposed estimator. A numerical evaluation of our procedure is proposed.

Suggested Citation

  • Christian Bontemps & Jean-Pierre Florens & Nour Meddahi, 2024. "Functional Ecological Inference," Working Papers hal-04709684, HAL.
    • Handle: RePEc:hal:wpaper:hal-04709684
    Note: View the original document on HAL open archive server: https://hal.science/hal-04709684v1
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