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

Abstract

We present a careful analysis of possible issues on the application of the self-excited Hawkes process to high-frequency financial data. We analyse a set of effects leading to significant biases in the estimation of the ‘criticality index’ n that quantifies the degree of endogeneity of how much past events trigger future events. We report the following model biases: (i) evidence of strong upward biases on the estimation of n when using power law memory kernels in the presence of outliers, (ii) strong effects on n resulting from the form of the regularization part of the power law kernel, (iii) strong edge effects on the estimated n when using power law kernels and (iv) the need for an exhaustive search of the absolute maximum of the log-likelihood function due to its complicated shape. Moreover, we demonstrate that the calibration of the Hawkes process on mixtures of pure Poisson process with changes of regime leads to completely spurious apparent critical values for the branching ratio (), while the true value is actually . More generally, regime shifts on the parameters of the Hawkes model and/or on the generating process itself are shown to systematically lead to a significant upward bias in the estimation of the branching ratio. We demonstrate the importance of the preparation of the high-frequency financial data, in particular: (a) the impact of overnight trading in the analysis of long-term trends, (b) intraday seasonality and detrending of the data and (c) vulnerability of the analysis to day-to-day non-stationarity and regime shifts. Special care is given to the decrease of quality of the timestamps of tick data due to latency and grouping of messages to packets by the stock exchange. Altogether, our careful exploration of the caveats of the calibration of the Hawkes process stresses the need for considering all the above issues before any conclusion can be sustained. In this respect, because the above effects are plaguing their analyses, the claim by Hardiman et al. [Eur. Phys. J. B – Cond. Matter Comp. Syst., 2013, 86, 442] that financial market has been continuously functioning at or close to criticality () cannot be supported. In contrast, our previous results on E-mini S&P 500 Futures Contracts and on major commodity future contracts are upheld.

Keywords:

• Hawkes process
• Poisson process
• Endogeneity
• Reflexivity
• Branching ratio
• Outliers
• Memory kernel
• High-frequency data
• Criticality
• Statistical biases
• Power laws
• Regime shifts

JEL Classification:

• C14
• C15
• C41
• C63
• G12
• G14

Acknowledgements

We are grateful to Spencer Wheatley for helpful discussions and suggestions while preparing this manuscript as well as for pointing out the impact of outliers (section 3.1). We also thank Igor Artyukhin for the help in numerical tests. We are grateful to Stephen Hardiman and Jean-Philippe Bouchaud for many long and fruitful discussions while preparing the manuscript, as well as for pointing out some inconsistencies and mistakes in preliminary versions of this article. We would like to express our deep gratitude to Professor Alexander Saichev for many fruitful discussions and collaborations on the critical regime of branching processes. Professor Saichev, who passed away on 8 June 2013, has been an outstanding contributor to the general theory and to the practical applications of self-excited processes that we have been developing at ETH Zurich.

Notes

No potential conflict of interest was reported by the authors.

1 In the sections 4.1–4.4, we discuss in details every step of this data preprocessing and show that all of them result in significant upward biases for the estimated branching ratio.

2 Note that, in contrast to figure 1, both lengths of the realizations and the initial burned period are larger, and the exponent is smaller, which results in more robust estimations.