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Inference for a Nonstationary Self-Exciting Point Process with an Application in Ultra-High Frequency Financial Data Modeling

Part of: Parametric inference Stochastic processes Applications Limit theorems

Published online by Cambridge University Press:  30 January 2018

Feng Chen  and
Peter Hall
Feng Chen*
Affiliation:
The University of New South Wales
Peter Hall*
Affiliation:
The University of Melbourne
*
∗ Postal address: School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia. Email address: feng.chen@unsw.edu.au
∗∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC 3010, Australia.
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Abstract

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Self-exciting point processes (SEPPs), or Hawkes processes, have found applications in a wide range of fields, such as epidemiology, seismology, neuroscience, engineering, and more recently financial econometrics and social interactions. In the traditional SEPP models, the baseline intensity is assumed to be a constant. This has restricted the application of SEPPs to situations where there is clearly a self-exciting phenomenon, but a constant baseline intensity is inappropriate. In this paper, to model point processes with varying baseline intensity, we introduce SEPP models with time-varying background intensities (SEPPVB, for short). We show that SEPPVB models are competitive with autoregressive conditional SEPP models (Engle and Russell 1998) for modeling ultra-high frequency data. We also develop asymptotic theory for maximum likelihood estimation based inference of parametric SEPP models, including SEPPVB. We illustrate applications to ultra-high frequency financial data analysis, and we compare performance with the autoregressive conditional duration models.

Keywords

Asymptotic normalityconsistencyHawkes processintensity processmartingale central limit theoremmaximum likelihood estimatornonstationarypoint processself-excitingultra-high frequency

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 62F12: Asymptotic properties of estimators 62P20: Applications to economics
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1006 - 1024
Copyright
© Applied Probability Trust 

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