Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T04:04:53.287Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATION FOR THE PREDICTION OF POINT PROCESSES WITH MANY COVARIATES

Published online by Cambridge University Press:  24 April 2017

Alessio Sancetta
Alessio Sancetta*
Affiliation:
Royal Holloway, University of London
*
*Address correspondence to Alessio Sancetta, Department of Economics, Royal Holloway University of London, Egham TW20 0EX, UK; e-mail: [email protected], URL: http://sites.google.com/site/wwwsancetta/.
Get access Rights & Permissions [Opens in a new window]

Abstract

Estimation of the intensity of a point process is considered within a nonparametric framework. The intensity measure is unknown and depends on covariates, possibly many more than the observed number of jumps. Only a single trajectory of the counting process is observed. Interest lies in estimating the intensity conditional on the covariates. The impact of the covariates is modelled by an additive model where each component can be written as a linear combination of possibly unknown functions. The focus is on prediction as opposed to variable screening. Conditions are imposed on the coefficients of this linear combination in order to control the estimation error. The rates of convergence are optimal when the number of active covariates is large. As an application, the intensity of the buy and sell trades of the New Zealand Dollar futures is estimated and a test for forecast evaluation is presented. A simulation is included to provide some finite sample intuition on the model and asymptotic properties.

Type
ARTICLES
Information
Econometric Theory , Volume 34 , Issue 3 , June 2018 , pp. 598 - 627
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

I would like to thank the Editor, the Co-Editor, the referees, and Luca Mucciante for comments that led to substantial improvements both in content and presentation.

References

REFERENCES

Andersen, P.K. & Gill, R.D. (1982) Cox’s regression model for counting processes: A large sample study. Annals of Statistics 10, 1100–1120.CrossRefGoogle Scholar
Barron, A.R., Cohen, A., Dahmen, W., & DeVore, R.A. (2008) Approximation and learning by greedy algorithms. Annals of Statistics 36, 64–94.CrossRefGoogle Scholar
Bauwens, L. & Hautsch, N. (2009) Modelling financial high frequency data using point processes. In Andersen, T.G., Davis, R.A., Kreiss, J.-P., & Mikosch, T. (eds.), Handbook of Financial Time Series, pp. 953–982. Springer.CrossRefGoogle Scholar
Bühlmann, P. & van de Geer, S. (2011) Statistics for High-Dimensional Data. Springer.CrossRefGoogle Scholar
Bradic, J., Fan, J., & Jiang, J. (2011) Regularization for Cox’s proportional hazards model with NP-dimensionality. Annals of Statistics 39, 3092–3120.Google ScholarPubMed
Brémaud, P. (1981) Point Processes and Queues: Martingales Dynamics. Springer.CrossRefGoogle Scholar
Brémaud, P. & Massoulié, L. (1996) Stability of nonlinear Hawkes processes. Annals of Probability 24, 1563–1588.Google Scholar
Brémaud, P., Nappo, G., & Torrisi, G.L. (2002) Rate of convergence to equilibrium of marked Hawkes processes. Journal of Applied Probability 39, 123–136.CrossRefGoogle Scholar
Burman, P., Chow, E., & Nolan, D. (1994) A cross-validatory method for dependent data. Biometrika 81, 351–358.CrossRefGoogle Scholar
Cont, R., Kukanov, A., & Stoikov, S. (2014) The price impact of order book events. Journal of Financial Econometrics 12, 47–88.CrossRefGoogle Scholar
Engle, R.F. & Russell, J.R. (1998) Autoregressive conditional duration: A new model for irregularly spaced transaction data. Econometrica 66, 1127–1162.CrossRefGoogle Scholar
Fan, J., Gijbels, I., & King, M. (1997) Local likelihood and local partial likelihood in hazard regression. Annals of Statistics 25, 1661–1690.Google Scholar
Gaiffas, S. & Guilloux, A. (2012) High dimensional additive Hazards models and the Lasso. Electronic Journal of Statistics 6, 522–546.Google Scholar
Hall, D. & Hautsch, N. (2007) Modelling the buy and sell intensity in a limit order book market. Journal of Financial Markets 10, 249–286.CrossRefGoogle Scholar
Hasbrouck, J. (1991) Measuring the information content of stock prices. Journal of Finance 46, 179–207.CrossRefGoogle Scholar
Jaggi, M. (2013) Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization. Journal of Machine Learning Research (Proceedings ICML 2013). Available at http://jmlr.org/proceedings/papers/v28/jaggi13-supp.pdf.Google Scholar
Katznelson, Y. (2002) An Introduction to Harmonic Analysis. Cambridge University Press.Google Scholar
Lorentz, G.G. (1986) Bernstein Polynomials. Chelsea Publishing Company.Google Scholar
Lillo, F., Farmer, J.D., & Mantegna, R.N. (2003) Master curve for the price-impact function. Nature 421, 129–130.CrossRefGoogle ScholarPubMed
Meinshausen, N. & Bühlmann, P. (2010) Stability selection (with discussion). Journal of the Royal Statistical Society B 72, 417–473.CrossRefGoogle Scholar
Nielsen, J.P. & Linton, O.B. (1995) Kernel estimation in a nonparametric marker dependent Hazard model. Annals of Statistics 5, 1735–1748.CrossRefGoogle Scholar
Ogata, Y. (1978) The asymptotic behaviour of the maximum likelihood estimator for stationary point processes. Annals of the Institute of Statistical Mathematics 30(A), 243–261.CrossRefGoogle Scholar
Sancetta, A. (2015) A nonparametric estimator for the covariance function of functional data. Econometric Theory 31, 1359–1381.CrossRefGoogle Scholar
Sancetta, A. (2016) Greedy algorithms for prediction. Bernoulli 22, 1227–1277.CrossRefGoogle Scholar
Seillier-Moiseiwitsch, F. & Dawid, A.P. (1993). On testing the validity of sequential probability forecasts. Journal of the American Statistical Association 88, 355–359.Google Scholar
Tsybakov, A.B. (2003) Optimal rates of aggregation. In Scholkopf, B. & Warmuth, M.K. (eds.), Proceedings of COLT-2003. Lecture Notes in Artificial Intelligence, vol. 2777, pp. 303–313. Springer.Google Scholar
van de, Geer (1995) Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Annals of Statistics 23, 1779–1801.CrossRefGoogle Scholar
van Dijk, D., Teräsvirta, T., & Franses, P.H. (2002) Smooth transition autoregressive models - a survey of recent developments. Econometric Reviews 21, 1–47.CrossRefGoogle Scholar
van der Vaart, A. & Wellner, J.A. (2000) Weak Convergence and Empirical Processes. Springer.Google Scholar
Yukich, J.E., Stinchcombe, M.B., & White, H. (1995) Sup-norm approximation bounds for networks through probabilistic methods. IEEE Transactions on Information Theory 41, 1021–1027.CrossRefGoogle Scholar
Supplementary material: PDF

Sancetta supplementary material

Sancetta supplementary material

Download Sancetta supplementary material(PDF)
PDF 341.4 KB