We introduce and show the existence of a continuous time-varying parameter extension model to the self-exciting point process . The kernel shape is assumed to be exponentially decreasing . The quantity of interest is defined as the integrated parameter over time T^{-1}\int_0^T\theta_t^*dt, where \theta_t^* is the time-varying parameter  . To estimate it na\"{i}vely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. Correspondingly, we give conditions on the parameter process and the block length under which we can establish the local central limit theorem, and the boundedness of moments of order 2\kappa of the local estimators, where \kappa > 1. Under those assumptions, the global estimator asymptotic bias explodes asymptotically . As a consequence, we provide a non-na\"{i}ve estimator , which is constructed as the na\"{i}ve one when applying a first-order bias reduction to the local MLE. We derive such first-order bias formula for the self-exciting process, and provide further conditions under which the non-na\"{i .
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\"{i}vely, 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\"{i}ve estimator which is constructed as the na\"{i}ve 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 .
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continuous time-varying parameter extension model to the
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. The kernel shape is assumed to be exponentially decreasing
with exponentially-decreasing kernel and where parameters are time-varying
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over time
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, and we consider the high-frequency asymptotics
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Correspondingly, we give conditions on the parameter process and the block length under which we can establish the local central limit theorem, and the boundedness of moments of order 2\kappa of the local estimators, where \kappa > 1. Under those assumptions, the global estimator
The
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. As a consequence,
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,
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derive such first-order bias formula for the self-exciting process, and provide further conditions under which the non-na\"{i
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