Disentangling endogeneity and exogeneity in cryptocurrency markets using the modulated renewal Hawkes process with marks

Hawkes processes have been used to quantify activity in cryptocurrency markets that are either exogenously or endogenously driven. The renewal Hawkes (RHawkes) process, which allows the underlying process for exogenous events to be a renewal process, has been found to provide a better fit to financial activity data in some markets. Although substantially more flexible, the RHawkes process is stationary in nature and unsuitable when systematic trends in the event occurrence rate exist. Therefore, we propose an RHawkes process with an exogenous arrival intensity modulated by a multiplicative trend function. When a suitable parametric form of the trend function is unavailable, we approximate it using basis spline functions. We propose algorithms to evaluate the likelihood, assess model fit, and simulate the process. Simulation experiments and an analysis of the endogeneity of the Bitcoin cryptocurrency are presented. Supplementary materials contain code to implement the proposed methodologies.