  We model the arrival of mid-price changes in the E-Mini S&P futures contract
as a self-exciting Hawkes process. Using several estimation methods, we find
that the Hawkes kernel is power-law with a decay exponent close to -1.15 at
short times, less than approximately 10^3 seconds, and crosses over to a second
power-law regime with a larger decay exponent of approximately -1.45 for longer
times scales in the range [10^3, 10^6] seconds. More importantly, we find that
the Hawkes kernel integrates to unity independently of the analysed period,
from 1998 to 2011. This suggests that markets are and have always been close to
criticality, challenging a recent study which indicates that reflexivity
(endogeneity) has increased in recent years as a result of increased automation
of trading. However, we note that the scale over which market events are
correlated has decreased steadily over time with the emergence of higher
frequency trading.
