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We present a careful analysis of possible issues on the application of theself-excited Hawkes process to high-frequency financial data. We carefullyanalyze a set of effects leading to significant biases in the estimation of the"criticality index" n that quantifies the degree of endogeneity of how muchpast events trigger future events. We report a number of model biases that areintrinsic to the estimation of brnaching ratio (n) when using power law memorykernels. We demonstrate that the calibration of the Hawkes process on mixturesof pure Poisson process with changes of regime leads to completely spuriousapparent critical values for the branching ratio (n~1) while the true value isactually n=0. More generally, regime shifts on the parameters of the Hawkesmodel and/or on the generating process itself are shown to systematically leadto a significant upward bias in the estimation of the branching ratio. We alsodemonstrate the importance of the preparation of the high-frequency financialdata and give special care to the decrease of quality of the timestamps of tickdata due to latency and grouping of messages to packets by the stock exchange.Altogether, our careful exploration of the caveats of the calibration of theHawkes process stresses the need for considering all the above issues beforeany conclusion can be sustained. In this respect, because the above effects areplaguing their analyses, the claim by Hardiman, Bercot and Bouchaud (2013) thatfinancial market have been continuously functioning at or close to criticality(n~1) cannot be supported. In contrast, our previous results on E-mini S&P 500Futures Contracts and on major commodity future contracts are upheld.

Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data