So, I have some data.. and a parameterized Hawkes process which I estimate parameters for via maximum likelihood... the residuals ( the compensator aka the dual-predictable projection) are good in that they both have nearly 1 for mean and variance and have almost no remaining autocorrelation. The odd thing is that even though the residuals fit so well, the theoretical moments are extremely far away from the empirical moments. The opposite happens if I estimate to match the moments... then the moments are closer to matching but the compensator is extremely far from Poissonian.
Is this some sort of paradox? How do I reconcile these facts? Does it imply that the parameterization needs to be modified somehow since it doesn't fit well enough?