Score Test for Marks in Hawkes Processes

Asymptotic Distribution of the Score Test for Detecting Marks in Hawkes Processes

“…Also (Breusch and Pagan, 1980) the information matrix can be replaced by any matrix with the same limit in probability, for example the negative of the matrix of second derivatives of the log-likelihood, and the large sample properties of the score statistic will be the same. Under Condition 1 stated below on the functions h defining the boost functions g via (3) the information matrix is shown in Richards et al (2018) to be block diagonal which, together with ∂ ν l g (ν T ) = (0, 0, ∂ ψ l g (ν T )) T , allows simplification of (12) toQ…”

“…Increasingly marked Hawkes processes, in which marks attached to past event times influence future intensities, are being considered for a range of applications. For example, Richards et al (2018) consider the use of marked Hawkes processes for modelling millisecond recordings of activity in the limit order book for a range of assets traded on international futures markets. In these applications there are numerous potential marks that are recorded at each event and a method is required to efficiently screen out those that are not influential on future event arrival intensities before the joint models for the event times and associated marks are estimated.…”

“…An alternative approach for assessing the impact of marks on intensity is the score test as proposed in Richards et al (2018) where details of computational implementation, simulation and application to limit order book event series is presented. As is well known, this test, also known as the Lagrange Multiplier test (Breusch and Pagan, 1980), is computed using the score of the likelihood evaluated under the null hypothesis, which, in this application, is that marks do not impact the intensity, so that the event times are that of an unmarked Hawkes process.…”

“…( 3) It is no loss of generality to require h(X; 0) ≡ 1. Many examples of boost functions including the polynomial, exponential and power functions forms for h, as presented in Liniger (2009), as well as additive and multiplicative combinations of individual elements of X used in Richards et al (2018) can be formulated in this way. The null hypothesis being assessed with the score test is H 0 : ψ = 0, which is equivalent to g(x; φ, 0) = 1 so that marks do not boost intensity.…”

“…, N T }. In Richards et al (2018) relevant marks constitute a vector of correlated marks which are also serially dependent. In order to accommodate such examples we explain how to define a marked Hawkes process with serially dependent marks, give conditions for stationarity of the point process, and, define the relevant quasi-likelihood.…”