Asymptotic distribution of the score test for detecting marks in hawkes processes

Score Test for Marks in Hawkes Processes

“…[12, Proposition 1] show that when the decay function satisfies (2), the branching coefficient ϑ < 1 and the boost function is normalized such that E φ [g(X; φ, ψ)] = 1, there exists a unique stationary point process associated with intensity (1). Clinet et al [8] generalize this result to allow serial dependence in the marks, something which is required for practical application to modeling the limit order book, for example. Specifically, the marks are assumed to be observations x i = y ti on a continuous time stochastic process (y t ) t∈R taking values in X and satisfying E[g(y Theorem 1] show that there exists a marked point process with intensity given by the stationary version of (1) and that if (y t ) t∈R is stationary, then so is the marked point process.…”

“…However, for our applications, the marks cannot be assumed to be i.i.d. In [8] we show how to construct the quasi-likelihood when the marks are not i.i.d. by noting that the integervalued measure N g (dt × dx) has predictable compensator of the form λ g (t, ν)ds × F t (dx; φ), where F t denotes the conditional distribution of y t given F y t− .…”

“…It is not clear that this can be written as an integral with respect to N g (dt × dx) corresponding to the third term in (4). Nor has it been established that a stationary Hawkes process can be defined for this case as was done in [8] for the case where marks are observations on a continuous time process. Although not a formal likelihood, assuming the marks are from a discrete-time process leads to the objective function…”

“…The theoretical aspects underpinning the models and methods of this paper are provided in the companion paper [8]. There, a new result on the existence of a marked Hawkes process with stationary marks is established, the likelihood is rigorously derived, and the proof that the score test is asymptotically chi-squared is given along with verification that the required conditions hold for the important case where the offspring decay function is exponential.…”

“…The present paper derives the method and provides the key justifications and simplifications required for efficient implementation. In Section 2, we introduce the Hawkes process with marks and define various practical approximations to the quasi-likelihood presented in [8]. Section 3 introduces the score test and derives the score with respect to the boost parameters ψ, as well as the corresponding information matrix.…”