  We introduce and show the existence of a Hawkes self-exciting point process
with exponentially-decreasing kernel and where parameters are time-varying. The
quantity of interest is defined as the integrated parameter
$T^{-1}\int_0^T\theta_t^*dt$, where $\theta_t^*$ is the time-varying parameter,
and we consider the high-frequency asymptotics. To estimate it na\"ively, 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 non-na\"ive
estimator which is constructed as the na\"ive one when applying a first-order
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.
