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Asymptotic distribution of the score test for detecting marks in hawkes processes

Author

Listed:
  • Simon Clinet

    (Keio University)

  • William T. M. Dunsmuir

    (University of New South Wales)

  • Gareth W. Peters

    (Heriot-Watt University)

  • Kylie-Anne Richards

    (University of Technology Sydney)

Abstract

The score test is a computationally efficient method for determining whether marks have a significant impact on the intensity of a Hawkes process. This paper provides theoretical justification for use of this test. It is shown that the score statistic has an asymptotic chi-squared distribution under the null hypothesis that marks do not impact the intensity process. For local power, the asymptotic distribution against local alternatives is proved to be non-central chi-squared. A stationary marked Hawkes process is constructed using a thinning method when the marks are observations on a continuous time stationary process and the joint likelihood of event times and marks is developed for this case, substantially extending existing results which only cover independent and identically distributed marks. These asymptotic chi-squared distributions required for the size and local power of the score test extend existing asymptotic results for likelihood estimates of the unmarked Hawkes process model under mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for the exponential decay Hawkes process.

Suggested Citation

  • Simon Clinet & William T. M. Dunsmuir & Gareth W. Peters & Kylie-Anne Richards, 2021. "Asymptotic distribution of the score test for detecting marks in hawkes processes," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 635-668, October.
    • Handle: RePEc:spr:sistpr:v:24:y:2021:i:3:d:10.1007_s11203-021-09245-5
    DOI: 10.1007/s11203-021-09245-5
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    References listed on IDEAS

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    1. Emmanuel Bacry & Iacopo Mastromatteo & Jean-Franc{c}ois Muzy, 2015. "Hawkes processes in finance," Papers 1502.04592, arXiv.org, revised May 2015.
    2. T. S. Breusch & A. R. Pagan, 1980. "The Lagrange Multiplier Test and its Applications to Model Specification in Econometrics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(1), pages 239-253.
    3. Alan G. Hawkes, 2018. "Hawkes processes and their applications to finance: a review," Quantitative Finance, Taylor & Francis Journals, vol. 18(2), pages 193-198, February.
    4. Simon Clinet & Yoann Potiron, 2016. "Statistical inference for the doubly stochastic self-exciting process," Papers 1607.05831, arXiv.org, revised Jun 2017.
    5. Clinet, Simon & Yoshida, Nakahiro, 2017. "Statistical inference for ergodic point processes and application to Limit Order Book," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1800-1839.
    6. Kylie-Anne Richards & William T. M. Dunsmuir & Gareth W. Peters, 2019. "Score Test for Marks in Hawkes Processes," Research Paper Series 405, Quantitative Finance Research Centre, University of Technology, Sydney.
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