Statistical inference for the doubly stochastic selfexciting process
Abstract
We introduce and show the existence of a Hawkes selfexciting point process with exponentiallydecreasing kernel and where parameters are timevarying. The quantity of interest is defined as the integrated parameter $T^{1}\int_0^T\theta_t^*dt$, where $\theta_t^*$ is the timevarying parameter, and we consider the highfrequency asymptotics. To estimate it naïvely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a nonnaïve estimator which is constructed as the naïve one when applying a firstorder bias reduction to the local MLE. We show the associated central limit theorem. Monte Carlo simulations show the importance of the bias correction and that the method performs well in finite sample, whereas the empirical study discusses the implementation in practice and documents the stochastic behavior of the parameters.
 Publication:

arXiv eprints
 Pub Date:
 July 2016
 arXiv:
 arXiv:1607.05831
 Bibcode:
 2016arXiv160705831C
 Keywords:

 Quantitative Finance  Statistical Finance;
 Mathematics  Statistics Theory
 EPrint:
 47 pages, 4 figures, 4 tables. Under revision for Bernoulli Journal