IDEAS home Printed from https://ideas.repec.org/a/bpj/sndecm/v29y2025i6p727-752n1004.html

Asymptotic Properties of ReLU FFN Sieve Estimators

Author

Listed:
  • Fabozzi Frank J.

    (Carey Business School, Johns Hopkins University, 100 International Drive, Baltimore, MD, 21202, USA)

  • Fallahgoul Hasan

    (School of Mathematics and Centre for Quantitative Finance and Investment Strategies, 2541 Monash University , 9 Rainforest Walk, 3800, Melbouren, Victoria, Australia)

  • Franstianto Vincentius

    (School of Mathematics and Centre for Quantitative Finance and Investment Strategies, 2541 Monash University , 9 Rainforest Walk, 3800, Melbouren, Victoria, Australia)

  • Loeper Grégoire

    (School of Mathematics and Centre for Quantitative Finance and Investment Strategies, 2541 Monash University , 9 Rainforest Walk, 3800, Melbouren, Victoria, Australia)

Abstract

Recently, machine learning algorithms have increasing become popular tools for economic and financial forecasting. While there are several machine learning algorithms for doing so, a powerful and efficient algorithm for forecasting purposes is the multi-layer, multi-node neural network with rectified linear unit (ReLU) activation function – deep neural network (DNN). Studies have demonstrated the empirical applications of DNN but have devoted less research to investigate its statistical properties which is mainly due to its severe nonlinearity and heavy parametrization. By borrowing tools from a non-parametric regression framework, sieve estimator, we first show that there exists such a sieve estimator for a DNN. We next establish three asymptotic properties of the ReLU network: consistency, sieve-based convergence rate, and asymptotic normality, and then validate our theoretical results using Monte Carlo analysis.

Suggested Citation

  • Fabozzi Frank J. & Fallahgoul Hasan & Franstianto Vincentius & Loeper Grégoire, 2025. "Asymptotic Properties of ReLU FFN Sieve Estimators," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 29(6), pages 727-752.
    • Handle: RePEc:bpj:sndecm:v:29:y:2025:i:6:p:727-752:n:1004
    DOI: 10.1515/snde-2023-0072
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/snde-2023-0072
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/snde-2023-0072?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Fallahgoul, Hasan & Franstianto, Vincentius & Lin, Xin, 2024. "Asset pricing with neural networks: Significance tests," Journal of Econometrics, Elsevier, vol. 238(1).
    2. Daniele Bianchi & Matthias Büchner & Andrea Tamoni, 2021. "Bond Risk Premiums with Machine Learning [Quadratic term structure models: Theory and evidence]," The Review of Financial Studies, Society for Financial Studies, vol. 34(2), pages 1046-1089.
    3. Shihao Gu & Bryan Kelly & Dacheng Xiu, 2020. "Empirical Asset Pricing via Machine Learning," Review of Finance, European Finance Association, vol. 33(5), pages 2223-2273.
    4. Stefan Hubner & Pavel Čížek, 2019. "Quantile-based smooth transition value at risk estimation," The Econometrics Journal, Royal Economic Society, vol. 22(3), pages 241-261.
    5. Xiaohong Chen & Xiaotong Shen, 1998. "Sieve Extremum Estimates for Weakly Dependent Data," Econometrica, Econometric Society, vol. 66(2), pages 289-314, March.
    6. Jia Chen, 2019. "Estimating latent group structure in time-varying coefficient panel data models," The Econometrics Journal, Royal Economic Society, vol. 22(3), pages 223-240.
    7. Shihao Gu & Bryan Kelly & Dacheng Xiu, 2020. "Empirical Asset Pricing via Machine Learning," The Review of Financial Studies, Society for Financial Studies, vol. 33(5), pages 2223-2273.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Fallahgoul, Hasan & Franstianto, Vincentius & Lin, Xin, 2024. "Asset pricing with neural networks: Significance tests," Journal of Econometrics, Elsevier, vol. 238(1).
    2. Yuan Liao & Xinjie Ma & Andreas Neuhierl & Linda Schilling, 2025. "The Uncertainty of Machine Learning Predictions in Asset Pricing," Papers 2503.00549, arXiv.org.
    3. Michalski, Lachlan & Low, Rand Kwong Yew, 2024. "Determinants of corporate credit ratings: Does ESG matter?," International Review of Financial Analysis, Elsevier, vol. 94(C).
    4. Chai, Bailin & Jiang, Fuwei & Lin, Yihao & You, Tian, 2025. "Predicting bond risk premiums with machine learning: Evidence from China," Pacific-Basin Finance Journal, Elsevier, vol. 93(C).
    5. Eghbal Rahimikia & Stefan Zohren & Ser-Huang Poon, 2021. "Realised Volatility Forecasting: Machine Learning via Financial Word Embedding," Papers 2108.00480, arXiv.org, revised Apr 2026.
    6. Rad, Hossein & Low, Rand Kwong Yew & Miffre, Joëlle & Faff, Robert, 2023. "The commodity risk premium and neural networks," Journal of Empirical Finance, Elsevier, vol. 74(C).
    7. Victor Duarte & Diogo Duarte & Dejanir H. Silva, 2024. "Machine Learning for Continuous-Time Finance," CESifo Working Paper Series 10909, CESifo.
    8. Borup, Daniel & Christensen, Bent Jesper & Mühlbach, Nicolaj Søndergaard & Nielsen, Mikkel Slot, 2023. "Targeting predictors in random forest regression," International Journal of Forecasting, Elsevier, vol. 39(2), pages 841-868.
    9. repec:rim:rimwps:20-27 is not listed on IDEAS
    10. Cheng, Zhuo (June) & Fang, Jing & Zhang, Yinglei, 2026. "Idiosyncratic volatility and return: A finite mixture approach," The British Accounting Review, Elsevier, vol. 58(2).
    11. Qian, Yihe & Zhang, Yang, 2025. "Long-term forecasting in asset pricing: Machine learning models’ sensitivity to macroeconomic shifts and firm-specific factors," The North American Journal of Economics and Finance, Elsevier, vol. 78(C).
    12. Adel Javanmard & Jingwei Ji & Renyuan Xu, 2024. "Multi-Task Dynamic Pricing in Credit Market with Contextual Information," Papers 2410.14839, arXiv.org, revised Dec 2025.
    13. DeMiguel, Victor & Gil-Bazo, Javier & Nogales, Francisco J. & Santos, André A.P., 2023. "Machine learning and fund characteristics help to select mutual funds with positive alpha," Journal of Financial Economics, Elsevier, vol. 150(3).
    14. Zeng, Qing & Lu, Xinjie & Xu, Jin & Lin, Yu, 2024. "Macro-Driven Stock Market Volatility Prediction: Insights from a New Hybrid Machine Learning Approach," International Review of Financial Analysis, Elsevier, vol. 96(PB).
    15. Zhao, Albert Bo & Cheng, Tingting, 2022. "Stock return prediction: Stacking a variety of models," Journal of Empirical Finance, Elsevier, vol. 67(C), pages 288-317.
    16. Yilie Huang & Yanwei Jia & Xun Yu Zhou, 2024. "Mean--Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study," Papers 2412.16175, arXiv.org, revised Mar 2026.
    17. Luo, Qin & Bu, Jinfeng & Xu, Weiju & Huang, Dengshi, 2023. "Stock market volatility prediction: Evidence from a new bagging model," International Review of Economics & Finance, Elsevier, vol. 87(C), pages 445-456.
    18. Goutte, Stéphane & Le, Hoang-Viet & Liu, Fei & von Mettenheim, Hans-Jörg, 2023. "Deep learning and technical analysis in cryptocurrency market," Finance Research Letters, Elsevier, vol. 54(C).
    19. Ricardo P. Masini & Marcelo C. Medeiros & Eduardo F. Mendes, 2023. "Machine learning advances for time series forecasting," Journal of Economic Surveys, Wiley Blackwell, vol. 37(1), pages 76-111, February.
    20. Huang, Xinyu & Newton, David P. & Platanakis, Emmanouil & Sutcliffe, Charles, 2025. "Single-stage portfolio optimization with automated machine learning for M6," International Journal of Forecasting, Elsevier, vol. 41(4), pages 1450-1460.
    21. Sergio Consoli & Luca Tiozzo Pezzoli & Elisa Tosetti, 2022. "Neural forecasting of the Italian sovereign bond market with economic news," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(S2), pages 197-224, December.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bpj:sndecm:v:29:y:2025:i:6:p:727-752:n:1004. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Peter Golla (email available below). General contact details of provider: https://www.degruyterbrill.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.