Abstract
We consider the multiple change-point problem for multivariate time series, including strongly dependent processes, with an unknown number of change-points. We assume that the covariance structure of the series changes abruptly at some unknown common change-point times. The proposed adaptive method is able to detect changes in multivariate i.i.d., weakly and strongly dependent series. This adaptive method outperforms the Schwarz criteria, mainly for the case of weakly dependent data. We consider applications to multivariate series of daily stock indices returns and series generated by an artificial financial market.
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M. Basseville and N. Nikiforov, The Detection of Abrupt Changes — Theory and Applications, Information and System Sciences Series, Prentice Hall (1993).
I. Berkes, E. Gombay, L. Horváth, and P. S. Kokoszka, Sequential change-point detection in GARCH(p,q) models, Econometric Theory, 20, 1140–1167 (2004).
L. Birgé and P. Massart, Gaussian model selection, J. Eur. Math. Soc., 3, 203–268 (2001).
T. Bollerslev, Modeling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model, Rev. Econom. Statist., 72, 498–505 (1990).
F. Boussama, Ergodicité, mélange, and estimation dans les modèles GARCH. Thèse de doctorat, Université Paris 7 (1998).
J. V. Braun, R. K. Braun, and H. G. Muller, Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation, Biometrika, 87, 301–314 (2000).
B. Brodsky and B. Darkhovsky, Nonparametric Methods in Change — Point Problems, Kluwer Academic Publishers, The Netherlands (1993).
J. Chen and A. K. Gupta, Parametric Statistical Change Point Analysis, Birkhäuser Verlag, Basel (2000).
J. Chen and A. K. Gupta, Statistical inference of covariance change points in Gaussian models, Statistics, 38, 17–28 (2004).
C.-S. J. Chu, Detecting parameter shift in GARCH models, Econometric Rev., 14, 241–266 (1995).
M. Csörgö and L. Horváth, Limit Theorems in Change-Point Analysis, Wiley (1997).
R. Dalhaus, Fitting time series models to nonstationary processes, Ann. Statist., 25, 1–37 (1997).
P. Doukhan, G. Teyssière, and P. Winant, A LARCH(∞) vector valued process, in: P. Bertail, P. Doukhan, and Ph. Soulier (Eds.), Dependence in Probability and Statistics, Lecture Notes in Statistics, 187, Springer (2006), pp. 245–258.
L. Giraitis, R. Leipus, and D. Surgailis, Recent advances in ARCH modelling, in: G. Teyssière and A. Kirman (Eds.), Long-Memory in Economics, Springer, Berlin (2005), pp. 3–38.
L. Giraitis and R. Leipus, Testing and estimating in the change-point problem of the spectral function, Lith. Math. J., 32(1), 20–38 (1992).
L. Giraitis and R. Leipus, Functional CLT for nonparametric estimates of the spectrum and change-point problem for a spectral function, Lith. Math. J., 30(4), 674–679 (1990).
L. Giraitis, R. Leipus, and D. Surgailis, The change-point problem for dependent observations, J. Statist. Plann. Inference, 53, 297–310 (1996).
C. W. J. Granger and N. Hyung, Occasional structural breaks and long-memory, Prépublication (1999).
D. M. Hawkins, Testing a sequence of observations for a shift in location, J. Amer. Statist. Assoc., 72, 180–186 (1977).
D. M. Hawkins, Fitting multiple change-point models to data, Comput. Statist. Data Anal., 37, 323–341 (2001).
L. Horváth, P. S. Kokoszka, and G. Teyssière, Empirical process of the squared residuals of an ARCH sequence, Ann. Statist., 29, 445–469 (2001).
S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Signal Processing Series, Prentice Hall (1998).
P. S. Kokoszka and R. Leipus, Detection and estimation of changes in regime, in: P. Doukhan, G. Oppenheim, and M. S. Taqqu (Eds.), Long-Range Dependence: Theory and Applications, Birkhauser, Boston (2003), pp. 325–337.
P. S. Kokoszka and R. Leipus, Change-point estimation in ARCH models, Bernoulli, 6, 513–539 (2000).
P. S. Kokoszka and R. Leipus, Testing for parameter changes in ARCH models, Lith. Math. J., 39(2), 182–195 (1999).
P. S. Kokoszka and G. Teyssière, Change-point detection in GARCH models: Asymptotic and bootstrap tests, Prépublication (2002).
M. Lavielle, Detection of multiple changes in a sequence of dependent variables, Stochastic Process. Appl., 83, 79–102 (1999).
M. Lavielle and C. Ludeña, Random thresholds for linear model selection, Publ. INRIA, 5572 (2005), http://www.inria.fr/rrrt/rr-5572.html.
M. Lavielle and E. Moulines, Least-squares estimation of a unknown number of shifts in a time series, J. Time Ser. Anal., 21, 33–59 (2000).
M. Lavielle and G. Teyssière, Adaptive detection of multiple change-points in asset price volatility, in: G. Teyssière and A. Kirman (Eds.), Long-Memory in Economics, Springer, Berlin (2005), pp. 129–156.
B. B. Mandelbrot and R. L. Hudson, The Misbehavior of Markets: A Fractal View of Risk, Ruin, and Reward, Basic Books, New York (2004).
B. Q. Mia and L. C. Zhao, Detection of change points using rank methods, Communications in Statistics — Theory and Methods, 17, 3207–3217 (1988).
T. Mikosch and C. Stărică, Long-range dependence effects and ARCH modeling, in: P. Doukhan, G. Oppenheim, and M.S. Taqqu (Eds.), Long-Range Dependence: Theory and Applications, Birkhauser, Boston (2003), pp. 439–459.
E. Schechtman and D. A. Wolfe, Multiple change points problem — nonparametric procedures for estimation of the points of change, Communications in Statistics — Simulation and Computation, 14, 615–631 (1985).
A. Sen and M. S. Srivastava, On tests for detecting change in the mean, Ann. Statist., 3, 96–103 (1975).
G. Teyssière, Interaction models for common long-range dependence in asset price volatility, in: G. Rangarajan and M. Ding (Eds.), Processes with Long Range Correlations: Theory and Applications, Lecture Notes in Physics, 621, Springer, Berlin (2003), pp. 251–269.
G. Teyssière, Modelling exchange rates volatility with multivariate long-memory ARCH processes, Preprint (1997).
G. Teyssière and P. Abry, Wavelet analysis of nonlinear long-range dependent processes. Applications to financial time series, in: G. Teyssière and A. Kirman (Eds.), Long Memory in Economics, Springer, Berlin (2005), pp. 173–238.
L. Ju. Vostrikova, Detection of ‘disorder’ in multidimensional random processes, Soviet Math. Dokl., 24, 55–59 (1981).
Y. C. Yao, Estimating the number of change-points via Schwarz criterion, Statist. Probab. Lett., 6, 181–189 (1988).
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Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 351–376, July–September, 2006.
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Lavielle, M., Teyssière, G. Detection of multiple change-points in multivariate time series. Lith Math J 46, 287–306 (2006). https://doi.org/10.1007/s10986-006-0028-9
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DOI: https://doi.org/10.1007/s10986-006-0028-9