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Codifference as a practical tool to measure interdependence

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  • Wyłomańska, Agnieszka
  • Chechkin, Aleksei
  • Gajda, Janusz
  • Sokolov, Igor M.

Abstract

Correlation and spectral analysis represent the standard tools to study interdependence in statistical data. However, for the stochastic processes with heavy-tailed distributions such that the variance diverges, these tools are inadequate. The heavy-tailed processes are ubiquitous in nature and finance. We here discuss codifference as a convenient measure to study statistical interdependence, and we aim to give a short introductory review of its properties. By taking different known stochastic processes as generic examples, we present explicit formulas for their codifferences. We show that for the Gaussian processes codifference is equivalent to covariance. For processes with finite variance these two measures behave similarly with time. For the processes with infinite variance the covariance does not exist, however, the codifference is relevant. We demonstrate the practical importance of the codifference by extracting this function from simulated as well as real data taken from turbulent plasma of fusion device and financial market. We conclude that the codifference serves as a convenient practical tool to study interdependence for stochastic processes with both infinite and finite variances as well.

Suggested Citation

  • Wyłomańska, Agnieszka & Chechkin, Aleksei & Gajda, Janusz & Sokolov, Igor M., 2015. "Codifference as a practical tool to measure interdependence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 412-429.
    • Handle: RePEc:eee:phsmap:v:421:y:2015:i:c:p:412-429
    DOI: 10.1016/j.physa.2014.11.049
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    References listed on IDEAS

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    1. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2005. "Statistical Tools for Finance and Insurance," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook0501, December.
    2. Szymon Borak & Adam Misiorek & Rafał Weron, 2011. "Models for heavy-tailed asset returns," Springer Books, in: Pavel Cizek & Wolfgang Karl Härdle & Rafał Weron (ed.), Statistical Tools for Finance and Insurance, chapter 1, pages 21-55, Springer.
    3. repec:hum:wpaper:sfb649dp2010-049 is not listed on IDEAS
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    Citations

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    Cited by:

    1. Gajda, J. & Kumar, A. & Wyłomańska, A., 2019. "Stable Lévy process delayed by tempered stable subordinator," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 284-292.
    2. Kumar, A. & Wyłomańska, A. & Połoczański, R. & Sundar, S., 2017. "Fractional Brownian motion time-changed by gamma and inverse gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 648-667.
    3. Vygintas Gontis, 2023. "Discrete $q$-exponential limit order cancellation time distribution," Papers 2306.00093, arXiv.org, revised Oct 2023.
    4. Marcin Pitera & Aleksei Chechkin & Agnieszka Wyłomańska, 2022. "Goodness-of-fit test for $$\alpha$$ α -stable distribution based on the quantile conditional variance statistics," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(2), pages 387-424, June.
    5. Eliazar, Iddo, 2025. "Power Levy motion: Correlations and relaxation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 674(C).
    6. Janusz Gajda & Aleksandra Grzesiek & Agnieszka Wyłomańska, 2023. "Ornstein - Uhlenbeck Process Driven By $$\alpha$$ α -stable Process and Its Gamma Subordination," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-17, March.
    7. Bielak, Łukasz & Grzesiek, Aleksandra & Janczura, Joanna & Wyłomańska, Agnieszka, 2021. "Market risk factors analysis for an international mining company. Multi-dimensional, heavy-tailed-based modelling," Resources Policy, Elsevier, vol. 74(C).
    8. Karling, Maicon J. & Lopes, Sílvia R.C. & de Souza, Roberto M., 2023. "Multivariate α-stable distributions: VAR(1) processes, measures of dependence and their estimations," Journal of Multivariate Analysis, Elsevier, vol. 195(C).

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