Electronic Journal of Statistics

Multiple breaks detection in general causal time series using penalized quasi-likelihood

Jean-Marc Bardet, William Kengne, and Olivier Wintenberger

Full-text: Open access

Enhanced PDF (866 KB)

Abstract

This paper is devoted to the off-line multiple breaks detection for a general class of models. The observations are supposed to fit a parametric causal process (such as classical models AR(∞), ARCH(∞) or TARCH(∞)) with distinct parameters on multiple periods. The number and dates of breaks, and the different parameters on each period are estimated using a quasi-likelihood contrast penalized by the number of distinct periods. For a convenient choice of the regularization parameter in the penalty term, the consistency of the estimator is proved when the moment order r of the process satisfies r≥2. If r≥4, the length of each approximative segment tends to infinity at the same rate as the length of the true segment and the parameters estimators on each segment are asymptotically normal. Compared to the existing literature, we added the fact that a dependence is possible over distinct periods. To be robust to this dependence, the chosen regularization parameter in the penalty term is larger than the ones from BIC approach. We detail our results which notably improve the existing ones for the AR(∞), ARCH(∞) and TARCH(∞) models. For the practical applications (when n is not too large) we use a data-driven procedure based on the slope estimation to choose the penalty term. The procedure is implemented using the dynamic programming algorithm. It is an O(n2) complexity algorithm that we apply on AR(1), AR(2), GARCH(1,1) and TARCH(1) processes and on the FTSE index data.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 435-477.

Dates
First available in Project Euclid: 19 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1332162336

Digital Object Identifier
doi:10.1214/12-EJS680

Mathematical Reviews number (MathSciNet)
MR2988415

Zentralblatt MATH identifier
1337.62210

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F12: Asymptotic properties of estimators

Keywords
Change detection causal processes ARCH(∞) processes AR(∞) processes quasi-maximum likelihood estimator model selection by penalized likelihood

Citation

Bardet, Jean-Marc; Kengne, William; Wintenberger, Olivier. Multiple breaks detection in general causal time series using penalized quasi-likelihood. Electron. J. Statist. 6 (2012), 435--477. doi:10.1214/12-EJS680. https://projecteuclid.org/euclid.ejs/1332162336


Export citation

References

  • [1] Bai J. and Perron P. Estimating and testing linear models with multiple structural changes., Econometrica 66 (1998), 47–78.
    Mathematical Reviews (MathSciNet): MR1616121
    Digital Object Identifier: doi:10.2307/2998540
  • [2] Bardet, J.-M. and Wintenberger, O. Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes., Ann. Statist. 37, (2009), 2730–2759.
    Mathematical Reviews (MathSciNet): MR2541445
    Zentralblatt MATH: 1173.62063
    Digital Object Identifier: doi:10.1214/08-AOS674
    Project Euclid: euclid.aos/1247836667
  • [3] Basseville, M. and Nikiforov, I. Detection of Abrupt Changes: Theory and Applications. Prentice Hall, Englewood Cliffs, NJ, 1993.
    Mathematical Reviews (MathSciNet): MR1210954
  • [4] Baudry, J.-P., Maugis, C. and Michel, B. Slope Heuristics: overview and implementation., RR-INRIA, (2010), n○7223.
  • [5] Berkes, I., Horváth, L., and Kokoszka, P. GARCH processes: structure and estimation., Bernoulli 9 (2003), 201–227.
    Mathematical Reviews (MathSciNet): MR1997027
    Digital Object Identifier: doi:10.3150/bj/1068128975
    Project Euclid: euclid.bj/1068128975
  • [6] Davis, R. A., Huang, D. and Yao, Y.-C. Testing for a change in the parameter values and order of an autoregressive model., Ann. Statist. 23, (1995), 282–304.
    Mathematical Reviews (MathSciNet): MR1331669
    Zentralblatt MATH: 0822.62072
    Digital Object Identifier: doi:10.1214/aos/1176324468
    Project Euclid: euclid.aos/1176324468
  • [7] Davis, R. A., Lee, T. C. M. and Rodriguez-Yam, G. A. Break detection for a class of nonlinear time series models., Journal of Time Series Analysis 29, (2008), 834–867.
    Mathematical Reviews (MathSciNet): MR2450899
    Zentralblatt MATH: 1199.62006
    Digital Object Identifier: doi:10.1111/j.1467-9892.2008.00585.x
  • [8] Doukhan, P. and Wintenberger, O. Weakly dependent chains with infinite memory., Stochastic Process. Appl. 118, (2008) 1997-2013.
    Mathematical Reviews (MathSciNet): MR2462284
    Zentralblatt MATH: 1166.60031
    Digital Object Identifier: doi:10.1016/j.spa.2007.12.004
  • [9] Duflo, M. Méthodes récursives aléatoires., Masson, Paris, 1990; English Edition, Springer, 1996.
    Mathematical Reviews (MathSciNet): MR1082344
  • [10] Francq, C., and Zakoïan, J.-M. Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes., Bernoulli 10 (2004), 605–637.
    Mathematical Reviews (MathSciNet): MR2076065
    Digital Object Identifier: doi:10.3150/bj/1093265632
    Project Euclid: euclid.bj/1093265632
  • [11] Fryzlewicz, P., and Subba Rao, S. BaSTA: consistent multiscale multiple change-point detection for piecewise-stationary ARCH processes., Preprint (2010). http://stats.lse.ac.uk/fryzlewicz/basta/basta.pdf
  • [12] Hinkley, D. V. Inference about the change-point in a sequence of random variables., Biometrika 57, (1970), 1–17.
    Mathematical Reviews (MathSciNet): MR273727
    Zentralblatt MATH: 0198.51501
    Digital Object Identifier: doi:10.1093/biomet/57.1.1
  • [13] Jeantheau, T. Strong consistency of estimators for multivariate arch models., Econometric Theory 14 (1998), 70–86.
    Mathematical Reviews (MathSciNet): MR1613694
    Digital Object Identifier: doi:10.1017/S0266466698141038
  • [14] Kokoszka, P. and Leipus, R. Change-point estimation in ARCH models., Bernoulli 6, (2000), 513–539.
    Mathematical Reviews (MathSciNet): MR1762558
    Digital Object Identifier: doi:10.2307/3318673
    Project Euclid: euclid.bj/1081616703
  • [15] Kay, S. M. Fundamentals of Statistical Signal Processing, vol. 2., Prentice-Hall, Englewood Cliffs, NJ, (1998)
  • [16] Kounias, E. G. and Weng, T.-S. An inequality and almost sure convergence., Annals of Mathematical Statistics 40, (1969), 1091–1093.
    Mathematical Reviews (MathSciNet): MR245058
    Zentralblatt MATH: 0176.48003
    Digital Object Identifier: doi:10.1214/aoms/1177697615
    Project Euclid: euclid.aoms/1177697615
  • [17] Lavielle, M. and Ludena, C. The multiple change-points problem for the spectral distribution., Bernoulli 6, (2000) 845–869.
    Mathematical Reviews (MathSciNet): MR1791905
    Digital Object Identifier: doi:10.2307/3318759
    Project Euclid: euclid.bj/1081282692
  • [18] Lavielle, M. and Moulines, E. Least squares estimation of an unknown number of shifts in a time series., Journal of Time Series Analysis 21, (2000) 33–59.
    Mathematical Reviews (MathSciNet): MR1766173
    Zentralblatt MATH: 0974.62070
    Digital Object Identifier: doi:10.1111/1467-9892.00172
  • [19] Nelson, D. B. and Cao, C. Q. Inequality Constraints in the Univariate GARCH Model., Journal of Business & Economic Statistics 10 , (1992), 229–235.
  • [20] Page, E. S. A test for a change in a parameter occurring at an unknown point., Biometrika 42, (1955), 523–526.
    Mathematical Reviews (MathSciNet): MR72412
    Zentralblatt MATH: 0067.11602
  • [21] Robinson, P., and Zaffaroni, P. Pseudo-maximum likelihood estimation of, ARCH(∞) models. Ann. Statist. 34 (2006), 1049–1074.
    Mathematical Reviews (MathSciNet): MR2278351
    Digital Object Identifier: doi:10.1214/009053606000000245
    Project Euclid: euclid.aos/1152540742
  • [22] Straumann, D., and Mikosch, T. Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach., Ann. Statist. 34 (2006), 2449–2495.
    Mathematical Reviews (MathSciNet): MR2291507
    Zentralblatt MATH: 1108.62094
    Digital Object Identifier: doi:10.1214/009053606000000803
    Project Euclid: euclid.aos/1169571804
  • [23] Yao, Y.C. Estimating the number of change-points via Schwarz criterion., Statistics & Probability Letters 6, (1988), 181–189.
    Mathematical Reviews (MathSciNet): MR919373

The Institute of Mathematical Statistics and the Bernoulli Society

Electronic Journal of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more