Abstract
We study statistical aspects of state-dependent Hawkes processes, which are an extension of Hawkes processes where a self- and cross-exciting counting process and a state process are fully coupled, interacting with each other. The excitation kernel of the counting process depends on the state process that, reciprocally, switches state when there is an event in the counting process. We first establish the existence and uniqueness of state-dependent Hawkes processes and explain how they can be simulated. Then we develop maximum likelihood estimation methodology for parametric specifications of the process. We apply state-dependent Hawkes processes to high-frequency limit order book data, allowing us to build a novel model that captures the feedback loop between the order flow and the shape of the limit order book. We estimate two specifications of the model, using the bid–ask spread and the queue imbalance as state variables, and find that excitation effects in the order flow are strongly state-dependent. Additionally, we find that the endogeneity of the order flow, measured by the magnitude of excitation, is also state-dependent, being more pronounced in disequilibrium states of the limit order book.
Highlights
Hawkes processes are a class of self- and cross-exciting point processes where events of different types may increase the rate of new events of the same or other type (Hawkes, 1971; Laub et al, 2015), dispensing with the independence-of-increments property of Poisson processes
Besides Intel Corporation (INTC), we have studied in the same manner the stocks of Advanced Micro Devices Inc. (AMD), Micron Technology Inc. (MU), Snap Inc. (SNAP) and Twitter Inc. (TWTR) from January to April 2018, but due to space constraints we only report our results on INTC as they are largely representative of the findings on the other stocks. (Full results are available from the authors upon request.)
In the context of limit order book (LOB) modelling, they provide a probabilistic foundation for a novel class of continuous-time models that encapsulate the feedback loop between the order flow and the the shape of the LOB
Summary
Hawkes processes are a class of self- and cross-exciting point processes where events of different types may increase the rate of new events of the same or other type (Hawkes, 1971; Laub et al, 2015), dispensing with the independence-of-increments property of Poisson processes. Hawkes processes are complemented by the other main approach to LOB modelling, based on continuous-time Markov chains (Huang et al, 2015; Huang and Rosenbaum, 2017; Cartea et al, 2018a) This alternative approach extends some of the earlier zero-intelligence models (Smith et al, 2003; Cont et al, 2010; Cont and De Larrard, 2011) and postulates that the arrival rate of orders is driven by the state of the LOB alone, making the state part of the model. The process X switches state when there is an event in N , according to a Markov transition matrix that depends on the type of the event This two-way interaction between N and X makes them fully coupled, just like the order flow and the state of the LOB in an order-driven market. A Python library called mpoints that implements the models and estimation methodology of this paper is available from https://mpoints.readthedocs.io
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