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Second order statistics characterization of Hawkes processes and non-parametric estimation

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  • Emmanuel Bacry
  • Jean-Francois Muzy

Abstract

We show that the jumps correlation matrix of a multivariate Hawkes process is related to the Hawkes kernel matrix through a system of Wiener-Hopf integral equations. A Wiener-Hopf argument allows one to prove that this system (in which the kernel matrix is the unknown) possesses a unique causal solution and consequently that the second-order properties fully characterize a Hawkes process. The numerical inversion of this system of integral equations allows us to propose a fast and efficient method, which main principles were initially sketched in [Bacry and Muzy, 2013], to perform a non-parametric estimation of the Hawkes kernel matrix. In this paper, we perform a systematic study of this non-parametric estimation procedure in the general framework of marked Hawkes processes. We describe precisely this procedure step by step. We discuss the estimation error and explain how the values for the main parameters should be chosen. Various numerical examples are given in order to illustrate the broad possibilities of this estimation procedure ranging from 1-dimensional (power-law or non positive kernels) up to 3-dimensional (circular dependence) processes. A comparison to other non-parametric estimation procedures is made. Applications to high frequency trading events in financial markets and to earthquakes occurrence dynamics are finally considered.

Suggested Citation

  • Emmanuel Bacry & Jean-Francois Muzy, 2014. "Second order statistics characterization of Hawkes processes and non-parametric estimation," Papers 1401.0903, arXiv.org, revised Feb 2015.
    • Handle: RePEc:arx:papers:1401.0903
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    References listed on IDEAS

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    1. E. Bacry & J. F Muzy, 2013. "Hawkes model for price and trades high-frequency dynamics," Papers 1301.1135, arXiv.org.
    2. E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 65-77, January.
    3. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Post-Print hal-01313995, HAL.
    4. Mohler, G. O. & Short, M. B. & Brantingham, P. J. & Schoenberg, F. P. & Tita, G. E., 2011. "Self-Exciting Point Process Modeling of Crime," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 100-108.
    5. Luc Bauwens & Nikolaus Hautsch, 2009. "Modelling Financial High Frequency Data Using Point Processes," Springer Books, in: Thomas Mikosch & Jens-Peter Kreiß & Richard A. Davis & Torben Gustav Andersen (ed.), Handbook of Financial Time Series, chapter 41, pages 953-979, Springer.
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    Cited by:

    1. Matthias Kirchner, 2017. "An estimation procedure for the Hawkes process," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 571-595, April.
    2. Marcello Rambaldi & Emmanuel Bacry & Fabrizio Lillo, 2016. "The role of volume in order book dynamics: a multivariate Hawkes process analysis," Papers 1602.07663, arXiv.org.
    3. Bonnet, Anna & Martinez Herrera, Miguel & Sangnier, Maxime, 2021. "Maximum likelihood estimation for Hawkes processes with self-excitation or inhibition," Statistics & Probability Letters, Elsevier, vol. 179(C).

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