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Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis

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  • Matthieu Garcin

Abstract

We are interested in the nonparametric estimation of the probability density of price returns, using the kernel approach. The output of the method heavily relies on the selection of a bandwidth parameter. Many selection methods have been proposed in the statistical literature. We put forward an alternative selection method based on a criterion coming from information theory and from the physics of complex systems: the bandwidth to be selected maximizes a new measure of complexity, with the aim of avoiding both overfitting and underfitting. We review existing methods of bandwidth selection and show that they lead to contradictory conclusions regarding the complexity of the probability distribution of price returns. This has also some striking consequences in the evaluation of the relevance of the efficient market hypothesis. We apply these methods to real financial data, focusing on the Bitcoin.

Suggested Citation

  • Matthieu Garcin, 2023. "Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis," Papers 2305.13123, arXiv.org.
    • Handle: RePEc:arx:papers:2305.13123
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    1. repec:cii:cepiie:2014-q4-140-60 is not listed on IDEAS
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    5. Ammy-Driss, Ayoub & Garcin, Matthieu, 2023. "Efficiency of the financial markets during the COVID-19 crisis: Time-varying parameters of fractional stable dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
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    10. Kukacka, Jiri & Kristoufek, Ladislav, 2020. "Do ‘complex’ financial models really lead to complex dynamics? Agent-based models and multifractality," Journal of Economic Dynamics and Control, Elsevier, vol. 113(C).
    11. Geoffrey Ducournau, 2021. "Symbol Dynamics, Information theory and Complexity of Economic time series," Papers 2105.04131, arXiv.org.
    12. Xavier Brouty & Matthieu Garcin, 2023. "A statistical test of market efficiency based on information theory," Quantitative Finance, Taylor & Francis Journals, vol. 23(6), pages 1003-1018, June.
    13. Park, Sangun & Rao, Murali & Shin, Dong Wan, 2012. "On cumulative residual Kullback–Leibler information," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 2025-2032.
    14. Xavier Brouty & Matthieu Garcin, 2023. "Fractal properties, information theory, and market efficiency," Papers 2306.13371, arXiv.org.
    15. Matthieu Garcin, 2022. "Forecasting with fractional Brownian motion: a financial perspective," Quantitative Finance, Taylor & Francis Journals, vol. 22(8), pages 1495-1512, August.
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    Cited by:

    1. Daniele Angelini & Matthieu Garcin, 2024. "Market information of the fractional stochastic regularity model," Papers 2409.07159, arXiv.org.
    2. Antonio Squicciarini & Elio Valero Toranzo & Alejandro Zarzo, 2024. "A Time-Series Feature-Extraction Methodology Based on Multiscale Overlapping Windows, Adaptive KDE, and Continuous Entropic and Information Functionals," Mathematics, MDPI, vol. 12(15), pages 1-21, July.
    3. Brouty, Xavier & Garcin, Matthieu, 2024. "Fractal properties, information theory, and market efficiency," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).

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