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

Spencer Wheatley, Vladimir Filimonov and Didier Sornette
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Spencer Wheatley: ETH Zurich
Vladimir Filimonov: ETH Zurich
Didier Sornette: ETH Zurich and Swiss Finance Institute

No 14-53, Swiss Finance Institute Research Paper Series from Swiss Finance Institute

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 (search for similar items in EconPapers)
JEL-codes: C40 C63 (search for similar items in EconPapers)
Pages: 37 pages
Date: 2014-08
New Economics Papers: this item is included in nep-ecm
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:chf:rpseri:rp1453

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