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Inference for a Nonstationary Self-Exciting Point Process with an Application in Ultra-High Frequency Financial Data Modeling

  • Feng Chen (a1) and Peter Hall (a2)

Abstract

Self-exciting point processes (SEPPs), or Hawkes processes, have found applications in a wide range of fields, such as epidemiology, seismology, neuroscience, engineering, and more recently financial econometrics and social interactions. In the traditional SEPP models, the baseline intensity is assumed to be a constant. This has restricted the application of SEPPs to situations where there is clearly a self-exciting phenomenon, but a constant baseline intensity is inappropriate. In this paper, to model point processes with varying baseline intensity, we introduce SEPP models with time-varying background intensities (SEPPVB, for short). We show that SEPPVB models are competitive with autoregressive conditional SEPP models (Engle and Russell 1998) for modeling ultra-high frequency data. We also develop asymptotic theory for maximum likelihood estimation based inference of parametric SEPP models, including SEPPVB. We illustrate applications to ultra-high frequency financial data analysis, and we compare performance with the autoregressive conditional duration models.

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Copyright

Corresponding author

Postal address: School of Mathematics and Statistics, The University of New South Wales, Sydney, NSW 2052, Australia. Email address: feng.chen@unsw.edu.au
∗∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC 3010, Australia.

References

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Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
Bauwens, L., Giot, P., Grammig, J. and Veredas, D. (2004). A comparison of financial duration models via density forecasts. Internat. J. Forecasting 20, 589609.
Chen, F. (2011). Maximum local partial likelihood estimators for the counting process intensity function and its derivatives. Statistica Sinica 21, 107128.
Chornoboy, E., Schramm, L. and Karr, A. (1988). {Maximum likelihood identification of neural point process systems}. Biol. Cybernetics 59, 265275.
Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press.
Crane, R. and Sornette, D. (2008). Robust dynamic classes revealed by measuring the response function of a social system. Proc. Nat. Acad. Sci. 105, 1564915653.
Embrechts, P., Liniger, T. and Lin, L. (2011). Multivariate Hawkes processes: an application to financial data. J. Appl. Prob. 48, 367378.
Engle, R. F. and Lunde, A. (2003). Trades and quotes: A bivariate point process. J. Financial Econometrics 1, 159188.
Engle, R. F. and Russell, J. R. (1998). Autoregressive conditional duration: A new model for irregularly spaced transaction data. Econometrica. 66, 11271162.
Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika. 58, 8390.
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493503.
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
Koopman, S. J., Lucas, A. and Monteiro, A. (2008). The multi-state latent factor intensity model for credit. J. Econometrics 142, 399424.
Kurtz, T. G. (1981). Approximation of Population Processes. Soc. Industrial Appl. Math. Philadelphia, Pennsylvania.
Kurtz, T. G. (1983). Gaussian approximations for Markov chains and counting processes. In Proc. 44th Session Internat. Statist. Inst. (Madrid, 1983; Bull. Inst. Internat. Statist. 50), Vol. 1, pp. 361376.
Lenglart, E. (1977). Relation de domination entre deux processus. Ann. Inst. H. Poincaré (B) 13, 171179.
Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia.
Monteiro, A. A. (2009). The econometrics of randomly spaced financial data: a survey. UC3M working paper, Statistics and Econometrics 09–24, Universidad Carlos III de Madrid. Available at http://hdl.handle.net/10016/5995.
Ogata, Y. (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Ann. Inst. Statist. Math. 30, 243261.
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 927.
Rathbun, S. L. (1996). Asymptotic properties of the maximum likelihood estimator for spatio-temporal point processes. J. Statist. Planning Infer. 51, 5574.
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, New York.
Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23, 470472.
Teich, M. C. and Saleh, B. E. A. (2000). Branching processes in quantum electronics. IEEE J. Sel. Topics in Quantum Electronics 6, 14501457.
Utsu, T. (1961). A statistical study of the occurrence of aftershocks. Geophysical Magazine 30, 521605.
van der Vaart, A. (1998). Asymptotic Statistics. Cambridge University Press, New York.
Vere-Jones, D. (1995). Forecasting earthquakes and earthquake risk. Internat. J. Forecasting 11, 503538.
Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20, 595601.
Zhang, M. Y., Russell, J. R. and Tsay, R. S. (2001). A nonlinear autoregressive conditional duration model with applications to financial transaction data. J. Econometrics 104, 179207.
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