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Functional linear regression with functional response

Author

Listed:
  • David Benatia

    (ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique, HEC Montréal - HEC Montréal)

  • Marine Carrasco

    (Université de Montréal, Départment d'Economie - CIREQ - Centre interuniversitaire de recherche en économie quantitative)

  • 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)

Abstract

In this paper, we develop new estimation results for functional regressions where both the regressor Z(t) and the response Y(t) are functions of Hilbert spaces, indexed by the time or a spatial location. The model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator Pi. We propose to estimate the operator Pi by Tikhonov regularization, which amounts to apply a penalty on the L-2 norm of Pi. We derive the rate of convergence of the mean square error, the asymptotic distribution of the estimator, and develop tests on Pi. As trajectories are often not fully observed, we consider the scenario where the data become more and more frequent (infill asymptotics). We also address the case where Z is endogenous and instrumental variables are used to estimate Pi. An application to the electricity consumption completes the paper.

Suggested Citation

  • David Benatia & Marine Carrasco & Jean-Pierre Florens, 2017. "Functional linear regression with functional response," Post-Print hal-03523162, HAL.
  • Handle: RePEc:hal:journl:hal-03523162
    DOI: 10.1016/j.jeconom.2017.08.008
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. David BENATIA & Etienne BILLETTE de VILLEMEUR, 2019. "Strategic Reneging in Sequential Imperfect Markets," Working Papers 2019-19, Center for Research in Economics and Statistics.
    2. Andrii Babii & Jean-Pierre Florens, 2017. "Is completeness necessary? Estimation in nonidentified linear models," Papers 1709.03473, arXiv.org, revised Nov 2021.
    3. Babii, Andrii, 2020. "Honest Confidence Sets In Nonparametric Iv Regression And Other Ill-Posed Models," Econometric Theory, Cambridge University Press, vol. 36(4), pages 658-706, August.
    4. Imaizumi, Masaaki & Kato, Kengo, 2018. "PCA-based estimation for functional linear regression with functional responses," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 15-36.
    5. Eduardo García‐Portugués & Javier Álvarez‐Liébana & Gonzalo Álvarez‐Pérez & Wenceslao González‐Manteiga, 2021. "A goodness‐of‐fit test for the functional linear model with functional response," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 502-528, June.
    6. Yousri Slaoui, 2020. "Recursive nonparametric regression estimation for dependent strong mixing functional data," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 665-697, October.
    7. Andrii Babii, 2022. "High-Dimensional Mixed-Frequency IV Regression," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(4), pages 1470-1483, October.
    8. Zhou, Zhiyang, 2021. "Fast implementation of partial least squares for function-on-function regression," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    9. Anton Rask Lundborg & Rajen D. Shah & Jonas Peters, 2022. "Conditional independence testing in Hilbert spaces with applications to functional data analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 1821-1850, November.
    10. Yong Liu & Manting Li & Juanjuan Zhao & Haidong Yu, 2019. "Responding to environmental pollution-related online posts: behavior of Web surfers and its influencing factors," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 21(6), pages 2931-2943, December.
    11. Dakyung Seong, 2022. "Binary response model with many weak instruments," Papers 2201.04811, arXiv.org, revised Jun 2024.

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    More about this item

    Keywords

    Functional regression; Instrumental variables; Linear operator; Tikhonov regularization;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General

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