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PERFORMANCE OF EMPIRICAL RISK MINIMIZATION FOR LINEAR REGRESSION WITH DEPENDENT DATA

Published online by Cambridge University Press:  10 November 2023

Christian Brownlees Open the ORCID record for Christian Brownlees [Opens in a new window]  and
Guđmundur Stefán Guđmundsson Open the ORCID record for Guđmundur Stefán Guđmundsson [Opens in a new window]
Christian Brownlees*
Affiliation:
Universitat Pompeu Fabra and Barcelona SE
Guđmundur Stefán Guđmundsson
Affiliation:
Aarhus University
*
Address correspondence to Christian Brownlees, Department of Economics and Business, Universitat Pompeu Fabra and Barcelona SE, Barcelona, Spain; e-mail: christian.brownlees@upf.edu.
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Abstract

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This paper establishes bounds on the performance of empirical risk minimization for large-dimensional linear regression. We generalize existing results by allowing the data to be dependent and heavy-tailed. The analysis covers both the cases of identically and heterogeneously distributed observations. Our analysis is nonparametric in the sense that the relationship between the regressand and the regressors is not specified. The main results of this paper show that the empirical risk minimizer achieves the optimal performance (up to a logarithmic factor) in a dependent data setting.

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Type
ARTICLES
Information
Econometric Theory , Volume 41 , Issue 2 , April 2025 , pp. 391 - 420
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Footnotes

We have benefited from discussions with Liudas Giraitis, Emmanuel Guerre, Petra Laketa, Gabor Lugosi, Stanislav Nagy, Jordi Llorens-Terrazas, Yaping Wang, and Geert Mesters as well as seminar participants at the Granger Center, Nottingham University and School of Economics and Finance, Queen Mary University of London. We would also like to thank the Co-Editor Liangjun Su and two anonymous referees for their useful comments. Christian Brownlees acknowledges support from the Spanish Ministry of Science and Technology (Grant No. MTM2012-37195); the Severo Ochoa Programme for Centres of Excellence in R&D (Barcelona School of Economics CEX2019-000915-S) funded by MCIN/AEI/10.13039/501100011033; the Ayudas Fundación BBVA Proyectos de Investigación Cientìfica en Matemáticas 2021. Guðmundur Stefán Guðmundsson acknowledges financial support from Danish National Research Foundation (DNRF Chair Grant No. DNRF154).

References

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