State-Dependent Hawkes Models for Limit Order Book Dynamics

Summary

This paper develops state-dependent Hawkes processes, in which event counts and a changing market state affect one another. The state determines the excitation kernels governing self- and cross-excitation, while events in the counting process can trigger state changes. The work establishes existence and uniqueness, describes simulation, and develops maximum-likelihood estimation for parametric models.

Applications to high-frequency limit order book data use bid-ask spread and queue imbalance as alternative state variables. The fitted models find that order-flow excitation varies with the book state, and that excitation-based endogeneity is stronger in disequilibrium states. These results offer a framework for representing feedback between order flow and book shape, but the document provides no quantitative performance comparison or detail about how the model performs beyond the studied data.

Key ideas

• The model couples event counts with a state process that can change when events occur.
• Self- and cross-excitation kernels vary according to the current state.
• The paper presents simulation methods and maximum-likelihood estimation.
• Bid-ask spread and queue imbalance are used as state variables in order book applications.
• Order-flow excitation is reported to be stronger in disequilibrium states.

Tags

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


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









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.

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