  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.
