State-dependent Hawkes processes and their application to limit order book modelling

TL;DR

This paper introduces state-dependent Hawkes processes, extending traditional models by coupling a counting process with a state process, and applies them to model the feedback loop in high-frequency limit order book data.

Contribution

It establishes the theoretical foundation, simulation methods, and maximum likelihood estimation for state-dependent Hawkes processes, and demonstrates their application to limit order book modeling.

Findings

Excitation effects are strongly state-dependent in order flow.

Order flow endogeneity varies with the state of the order book.

The model captures feedback between order flow and order book shape.

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.