Effective measure of endogeneity for the Autoregressive Conditional Duration point processes via mapping to the self-excited Hawkes process

In order to disentangle the internal dynamics from exogenous factors within the Autoregressive Conditional Duration (ACD) model, we present an effective measure of endogeneity. Inspired from the Hawkes model, this measure is defined as the average number of direct offspring for each observed event. Having an explicit meaning allowing one to define a distance to criticality, this “branching ratio” parameter also characterizes the average fraction of events that are triggered due to internal feedback mechanisms within the total population. We provide a direct comparison of the Hawkes and ACD models based on numerical simulations and show that our effective measure of endogeneity for the ACD can be mapped onto the “branching ratio” of the Hawkes model.