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SEQUENTIAL CHANGE-POINT DETECTION IN GARCH(p,q) MODELS

Published online by Cambridge University Press:  01 December 2004

István Berkes ,
Edit Gombay ,
Lajos Horváth  and
Piotr Kokoszka
István Berkes
Affiliation:
Hungarian Academy of Sciences
Edit Gombay
Affiliation:
University of Alberta
Lajos Horváth
Affiliation:
University of Utah
Piotr Kokoszka
Affiliation:
Utah State University
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Abstract

We suggest a sequential monitoring scheme to detect changes in the parameters of a GARCH(p,q) sequence. The procedure is based on quasi-likelihood scores and does not use model residuals. Unlike for linear regression models, the squared residuals of nonlinear time series models such as generalized autoregressive conditional heteroskedasticity (GARCH) do not satisfy a functional central limit theorem with a Wiener process as a limit, so its boundary crossing probabilities cannot be used. Our procedure nevertheless has an asymptotically controlled size, and, moreover, the conditions on the boundary function are very simple; it can be chosen as a constant. We establish the asymptotic properties of our monitoring scheme under both the null of no change in parameters and the alternative of a change in parameters and investigate its finite-sample behavior by means of a small simulation study.This research was partially supported by NSF grant INT-0223262 and NATO grant PST.CLG.977607. The work of the first author was supported by the Hungarian National Foundation for Scientific Research, grants T 29621, 37886; the work of the second author was supported by NSERC Canada.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 6 , December 2004 , pp. 1140 - 1167
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Andreou, E. & E. Ghysels (2002) Detecting multiple breaks in financial market volatility dynamics. Journal of Applied Econometrics 17, 579–600.Google Scholar
Berkes, I. & L. Horváth (2004) The efficiency of the estimators of the parameters in GARCH processes. Annals of Statistics 32, 633–655.Google Scholar
Berkes, I. & L. Horváth (2003) Limit results for the empirical process of squared residuals in GARCH models. Stochastic Processes and Their Applications 105, 279–298.Google Scholar
Berkes, I., L. Horváth, & P. Kokoszka (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.Google Scholar
Besseville, M. & I.V. Nikifirov (1993) Detection of Abrupt Changes: Theory and Applications. Prentice Hall.
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Boldin, M.V. (1998) On residual empirical distribution functions in ARCH models with applications to testing and estimation. Mitteilungen aus dem Mathematischen Seminar Giessen 235, 49–66.Google Scholar
Bougerol, P. & N. Picard (1992a) Strict stationary of generalized autoregressive processes. Annals of Probability 20, 1714–1730.Google Scholar
Bougerol, P. & N. Picard (1992b) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.Google Scholar
Brodsky, B.E. & B.S. Darkhovsky (1993) Nonparametric Methods in Change-Point Problems. Kluwer.
Brodsky, B.E. & B.S. Darkhovsky (2000) Non-Parametric Statistical Diagnosis. Kluwer.
Chow, Y.S. (1960) A martingale inequality and the law of large numbers. Proceedings of the American Mathematics Society 11, 107–111.Google Scholar
Chu, C.-S.J. (1995) Detecting parameter shift in GARCH models. Econometric Reviews 14, 241–266.Google Scholar
Chu, C.-S.J., M. Stinchcombe, & H. White (1996) Monitoring structural change. Econometrica 64, 1045–1065.Google Scholar
Csörgő, M. & L. Horváth (1997) Limit Theorems in Change-Point Analysis. Wiley.
Csörgő, M. & P. Révész (1981) Strong Approximations in Probability and Statistics. Academic Press.
Gombay, E. (1996) The weighted sequential likelihood ratio. Canadian Journal of Statistics 24, 229–239.Google Scholar
Gombay, E. (2003) Sequential change-point detection and estimation. Sequential Analysis 22, 203–222.Google Scholar
Gouriéroux, C. (1997) ARCH Models and Financial Applications. Springer-Verlag.
Horváth, L., P. Kokoszka, & G. Teyssière (2001) Empirical process of the squared residuals of an ARCH sequence. Annals of Statistics 29, 445–469.Google Scholar
Inclan, C. & G. Tiao (1994) Use of cumulative sums of squares for retrospective detection of change in variance. Journal of the American Statistical Association 89, 913–929.Google Scholar
Inoue, A. (2001) Testing for distributional change in time series. Econometric Theory 17, 156–187.Google Scholar
Kim, S., S. Cho, & S. Lee (2000) On the cusum test for parameter changes in GARCH(1,1) model. Communications in Statistics—Theory and Methodology 29, 445–462.Google Scholar
Kokoszka, P. & R. Leipus (1999) Testing for parameter changes in ARCH models. Lithuanian Mathematical Journal 39, 231–247.Google Scholar
Kokoszka, P. & R. Leipus (2000) Change-point estimation in ARCH models. Bernoulli 6, 513–539.Google Scholar
Kokoszka, P. & G. Teyssière (2002) Change Point Detection in GARCH Models: Asymptotic and Bootstrap Tests. Preprint available at http://stat.usu.edu/∼piotr/research.html. Utah State University.
Koul, H. (2002) Weighted Empirical Processes in Dynamic Nonlinear Models. Springer.
Kulperger, R. & H. Yu (2003) High Moment Partial Sum Processes of Residuals in GARCH Models. Preprint, University of Western Ontario.
Lamoureux, C. & W.D. Lastrapes (1990) Persistence in variance, structural change, and the GARCH model. Journal of Business and Economic Statistics 8, 225–234.Google Scholar
Lee, S.-W. & B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.Google Scholar
Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 6, 575–596.Google Scholar
Lundbergh, S. & T. Teräsvirta (2002) Evaluating GARCH models. Journal of Econometrics 110, 417–435.Google Scholar
Mikosch, T. & C. Stărică (2002) Long-range dependence effects and ARCH modeling. In P. Doukhan, G. Oppenheim, & M.S. Taqqu (eds.), Theory and Applications of Long-Range Dependence, pp. 439–459. Birkhäuser.
Nyblom, Y. (1989) Testing for the constancy of parameters over time. Journal of the American Statistical Association 6, 223–230.Google Scholar
Sowell, F. (1996) Optimal tests for parameter instability in the generalized method of moments framework. Econometrica 64, 1085–1107.Google Scholar