Estimation of the Hawkes Process with Renewal Immigration Using the EM Algorithm

37 Pages Posted: 8 Aug 2014 Last revised: 8 Nov 2014

See all articles by Spencer Wheatley

Spencer Wheatley

ETH Zurich

Vladimir Filimonov

Swiss Federal Institute of Technology Zurich (ETH Zurich)

Didier Sornette

ETH Zürich - Department of Management, Technology, and Economics (D-MTEC); Swiss Finance Institute

Date Written: July 26, 2014

Abstract

We introduce the Hawkes process with renewal immigration and make its statistical estimation possible with two Expectation Maximization (EM) algorithms. The standard Hawkes process introduces immigrant points via a Poisson process, and each immigrant has a subsequent cluster of associated offspring of multiple generations. We generalize the immigration to come from a Renewal process; introducing dependence between neighbouring clusters, and allowing for over/under dispersion in cluster locations. This complicates evaluation of the likelihood since one needs to know which subset of the observed points are immigrants. Two EM algorithms enable estimation here: The first is an extension of an existing algorithm that treats the entire branching structure - which points are immigrants, and which point is the parent of each offspring - as missing data. The second considers only if a point is an immigrant or not as missing data and can be implemented with linear time complexity. Both algorithms are found to be consistent in simulation studies. Further, we show that misspecifying the immigration process introduces significant bias into model estimation - especially the branching ratio, which quantifies the strength of self excitation. Thus, this extended model provides a valuable alternative model in practice.

Keywords: Expectation-maximization algorithm; Branching process models; Renewal Cluster process models; Point process models; non-parametric estimation; Hawkes process; immigration; branching structure.

JEL Classification: C40, C63

Suggested Citation

Wheatley, Spencer and Filimonov, Vladimir and Sornette, Didier, Estimation of the Hawkes Process with Renewal Immigration Using the EM Algorithm (July 26, 2014). Swiss Finance Institute Research Paper No. 14-53. Available at SSRN: https://ssrn.com/abstract=2477432 or http://dx.doi.org/10.2139/ssrn.2477432

Spencer Wheatley (Contact Author)

ETH Zurich ( email )

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Vladimir Filimonov

Swiss Federal Institute of Technology Zurich (ETH Zurich) ( email )

Scheuchzerstrasse 7, SEC F3
Zurich, CH-8092
Switzerland

Didier Sornette

ETH Zürich - Department of Management, Technology, and Economics (D-MTEC) ( email )

Scheuchzerstrasse 7
Zurich, ZURICH CH-8092
Switzerland
41446328917 (Phone)
41446321914 (Fax)

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