  Inference and testing in general point process models such as the Hawkes
model is predominantly based on asymptotic approximations for likelihood-based
estimators and tests, as originally developed in Ogata (1978). As an
alternative, and to improve finite sample performance, this paper considers
bootstrap-based inference for interval estimation and testing. Specifically,
for a wide class of point process models we consider a novel bootstrap scheme
labeled 'fixed intensity bootstrap' (FIB), where the conditional intensity is
kept fixed across bootstrap repetitions. The FIB, which is very simple to
implement and fast in practice, naturally extends previous ideas from the
bootstrap literature on time series in discrete time, where the so-called
'fixed design' and 'fixed volatility' bootstrap schemes have shown to be
particularly useful and effective. We compare the FIB with the classic
recursive bootstrap, which is here labeled 'recursive intensity bootstrap'
(RIB). In RIB algorithms, the intensity is stochastic in the bootstrap world
and implementation of the bootstrap is more involved, due to its sequential
structure. For both bootstrap schemes, no asymptotic theory is available; we
therefore provide a new bootstrap (asymptotic) theory, which allows to assess
bootstrap validity. We also introduce novel 'non-parametric' FIB and RIB
schemes, which are based on resampling time-changed transformations of the
original waiting times. We show effectiveness of the different bootstrap
schemes in finite samples through a set of detailed Monte Carlo experiments. As
far as we are aware, this is the first detailed Monte Carlo study of bootstrap
implementations for Hawkes-type processes. Finally, in order to illustrate, we
provide applications of the bootstrap to both financial data and social media
data.
