The Hawkes process with renewal immigration & its estimation with an EM algorithm

In its original form, the self-excited Hawkes process is a cluster process where immigrants follow a Poisson process, and each immigrant may form a cluster of multi-generational offspring. The Hawkes process is generalized by replacing the Poisson immigration process with a renewal process. This generalization makes direct MLE impossible. Thus, two EM algorithms are introduced: The first extends the existing EM algorithm for the Hawkes process to consider renewal immigration. It treats the entire branching structure-which points are immigrants, and which point is the parent of each offspring-as missing data. The second algorithm reduces the amount of missing data, considering only if a point is an immigrant or not as missing data. This significantly reduces computational complexity and memory requirements, enabling estimation on larger datasets. Both algorithms are found to perform well in simulation studies. A case study shows that the Hawkes process with renewal immigration is superior to the standard Hawkes process for the modeling of high-frequency price fluctuations. Further, it is demonstrated that misspecification of the immigration process can bias estimation of the branching ratio, which quantifies the degree of self-excitation.

[1]  J. Rasmussen,et al.  Perfect simulation of Hawkes processes , 2005, Advances in Applied Probability.

[2]  Didier Sornette,et al.  Quantification of the High Level of Endogeneity and of Structural Regime Shifts in Commodity Markets , 2013 .

[3]  D. Sornette,et al.  Limits of declustering methods for disentangling exogenous from endogenous events in time series with foreshocks, main shocks, and aftershocks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Shy Shoham,et al.  Frontiers in Computational Neuroscience Correlation-based Analysis and Generation of Multiple Spike Trains Using Hawkes Models with an Exogenous Input , 2022 .

[5]  Rama Cont,et al.  Statistical Modeling of High-Frequency Financial Data , 2011, IEEE Signal Processing Magazine.

[6]  Yosihiko Ogata,et al.  Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes , 1988 .

[7]  Klaus Reiner Schenk-Hoppé,et al.  Handbook of Financial Markets: Dynamics and Evolution , 2009 .

[8]  T. Ozaki.  Maximum likelihood estimation of Hawkes' self-exciting point processes , 1979 .

[9]  F. Massey.  The Kolmogorov-Smirnov Test for Goodness of Fit , 1951 .

[10]  P. Reynaud-Bouret,et al.  Adaptive estimation for Hawkes processes; application to genome analysis , 2009, 0903.2919.

[11]  Yacine Ait-Sahalia,et al.  Modeling Financial Contagion Using Mutually Exciting Jump Processes , 2010 .

[12]  Erik A. Lewis,et al.  RESEARCH ARTICLE A Nonparametric EM algorithm for Multiscale Hawkes Processes , 2011 .

[13]  Rama Cont,et al.  Statistical Modeling of High Frequency Financial Data: Facts, Models and Challenges , 2011, IEEE Signal Process. Mag..

[14]  G. Shedler,et al.  Simulation of Nonhomogeneous Poisson Processes by Thinning , 1979 .

[15]  A. Hawkes.  Spectra of some self-exciting and mutually exciting point processes , 1971 .

[16]  Didier Sornette,et al.  Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data , 2013, 1308.6756.

[17]  Peter F. Halpin,et al.  A Scalable EM Algorithm for Hawkes Processes , 2013 .

[18]  D. Sornette,et al.  Quantifying Reflexivity in Financial Markets: Towards a Prediction of Flash Crashes , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Ioane Muni Toke,et al.  "Market making" behaviour in an order book model and its impact on the bid-ask spread , 2010, 1003.3796.

[20]  Stephen J. Hardiman,et al.  Critical reflexivity in financial markets: a Hawkes process analysis , 2013, 1302.1405.

[21]  Luc Bauwens,et al.  Département des Sciences Économiques de l'Université catholique de Louvain Modelling Financial High Frequency Data Using Point Processes , 2019 .

[22]  Clive G. Bowsher.  Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models , 2003 .

[23]  P. Embrechts,et al.  Multivariate Hawkes processes: an application to financial data , 2011, Journal of Applied Probability.

[24]  S. S. Wilks.  The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses , 1938 .

[25]  Y. Ogata.  The asymptotic behaviour of maximum likelihood estimators for stationary point processes , 1978 .

[26]  V. Chavez-Demoulin,et al.  High-frequency financial data modeling using Hawkes processes , 2012 .

[27]  Michael I. Jordan,et al.  On Convergence Properties of the EM Algorithm for Gaussian Mixtures , 1996, Neural Computation.

[28]  D. Sornette,et al.  2 Definitions and formalism 2 . 1 Definitions for the Hawkes self-excited conditional Poisson process , 2013 .

[29]  New York Dover,et al.  ON THE CONVERGENCE PROPERTIES OF THE EM ALGORITHM , 1983 .

[30]  Agus Salim,et al.  Extensions of the Bartlett-Lewis model for rainfall processes , 2003 .

[31]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[32]  A. Hawkes.  Point Spectra of Some Mutually Exciting Point Processes , 1971 .

[33]  Sophie Schbath,et al.  FADO: A Statistical Method to Detect Favored or Avoided Distances between Occurrences of Motifs using the Hawkes' Model , 2005, Statistical applications in genetics and molecular biology.

[34]  Yuichi Mori,et al.  Handbook of Computational Statistics , 2004 .

[35]  D. Marsan,et al.  Extending Earthquakes' Reach Through Cascading , 2008, Science.

[36]  D. Sornette,et al.  Quantifying reflexivity in financial markets: towards a prediction of flash crashes , 2012, 1201.3572.

[37]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[38]  Kay Giesecke,et al.  Affine Point Processes and Portfolio Credit Risk , 2010, SIAM J. Financial Math..

[39]  Yosihiko Ogata,et al.  A Prospect of Earthquake Prediction Research , 2013, 1312.7712.

[40]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[41]  E. Bacry,et al.  Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data , 2011, 1112.1838.

[42]  Zoubin Ghahramani,et al.  Optimization with EM and Expectation-Conjugate-Gradient , 2003, ICML.

[43]  Paul S. P. Cowpertwait.  A renewal cluster model for the inter-arrival times of rainfall events , 2001 .

[44]  A. Veen,et al.  Estimation of Space–Time Branching Process Models in Seismology Using an EM–Type Algorithm , 2006 .

[45]  H. Kile,et al.  Bandwidth Selection in Kernel Density Estimation , 2010 .

[46]  Erik A. Lewis,et al.  Self-exciting point process models of civilian deaths in Iraq , 2011, Security Journal.

[47]  David Oakes,et al.  The Markovian self-exciting process , 1975, Journal of Applied Probability.

[48]  A. C. Davison,et al.  Estimating value-at-risk: a point process approach , 2005 .

[49]  F. Papangelou.  Integrability of expected increments of point processes and a related random change of scale , 1972 .