  We show that univariate and symmetric multivariate Hawkes processes are only
weakly causal: the true log-likelihoods of real and reversed event time vectors
are almost equal, thus parameter estimation via maximum likelihood only weakly
depends on the direction of the arrow of time. In ideal (synthetic) conditions,
tests of goodness of parametric fit unambiguously reject backward event times,
which implies that inferring kernels from time-symmetric quantities, such as
the autocovariance of the event rate, only rarely produce statistically
significant fits. Finally, we find that fitting financial data with
many-parameter kernels may yield significant fits for both arrows of time for
the same event time vector, sometimes favouring the backward time direction.
This goes to show that a significant fit of Hawkes processes to real data with
flexible kernels does not imply a definite arrow of time unless one tests it.
