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


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 r2. If r4, 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

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

First available in Project Euclid: 19 March 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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


  • [1] Bai J. and Perron P. Estimating and testing linear models with multiple structural changes., Econometrica 66 (1998), 47–78.
  • [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.
  • [3] Basseville, M. and Nikiforov, I. Detection of Abrupt Changes: Theory and Applications. Prentice Hall, Englewood Cliffs, NJ, 1993.
  • [4] Baudry, J.-P., Maugis, C. and Michel, B. Slope Heuristics: overview and implementation., RR-INRIA, (2010), n7223.
  • [5] Berkes, I., Horváth, L., and Kokoszka, P. GARCH processes: structure and estimation., Bernoulli 9 (2003), 201–227.
  • [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.
  • [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.
  • [8] Doukhan, P. and Wintenberger, O. Weakly dependent chains with infinite memory., Stochastic Process. Appl. 118, (2008) 1997-2013.
  • [9] Duflo, M. Méthodes récursives aléatoires., Masson, Paris, 1990; English Edition, Springer, 1996.
  • [10] Francq, C., and Zakoïan, J.-M. Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes., Bernoulli 10 (2004), 605–637.
  • [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.
  • [13] Jeantheau, T. Strong consistency of estimators for multivariate arch models., Econometric Theory 14 (1998), 70–86.
  • [14] Kokoszka, P. and Leipus, R. Change-point estimation in ARCH models., Bernoulli 6, (2000), 513–539.
  • [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.
  • [17] Lavielle, M. and Ludena, C. The multiple change-points problem for the spectral distribution., Bernoulli 6, (2000) 845–869.
  • [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.
  • [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.
  • [21] Robinson, P., and Zaffaroni, P. Pseudo-maximum likelihood estimation of, ARCH() models. Ann. Statist. 34 (2006), 1049–1074.
  • [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.
  • [23] Yao, Y.C. Estimating the number of change-points via Schwarz criterion., Statistics & Probability Letters 6, (1988), 181–189.