Equality Constraints in Linear Hawkes Processes

Conditional independence is often used as a testable implication of causal models of random variables. In addition, equality constraints have been proposed to distinguish between data-generating mechanisms. We show that one can also find equality constraints in linear Hawkes processes, extending this theory to a class of continuous-time stochastic processes. This is done by proving that Hawkes process models in a certain sense satisfy the equality constraints of linear structural equation models. These results allow more refined constraint-based structure learning in this class of processes. Arguing the existence of equality constraints leads us to new identification results for Hawkes processes. We also describe a causal interpretation of the linear Hawkes process which is closely related to its so-called cluster representation.