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ON DISTINGUISHING BETWEEN RANDOM WALK AND CHANGE IN THE MEAN ALTERNATIVES

Published online by Cambridge University Press:  01 April 2009

Alexander Aue ,
Lajos Horváth ,
Marie Hušková  and
Shiqing Ling
Alexander Aue*
Affiliation:
University of California, Davis
Lajos Horváth
Affiliation:
University of Utah
Marie Hušková
Affiliation:
Charles University
Shiqing Ling
Affiliation:
Hong Kong University of Science and Technology
*
*Address correspondence to Alexander Aue, Department of Statistics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA; e-mail: alexaue@wald.ucdavis.edu.
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Abstract

We study test procedures that detect structural breaks in underlying data sequences. In particular, we wish to discriminate between different reasons for these changes, such as (1) shifting means, (2) random walk behavior, and (3) constant means but innovations switching from stationary to difference stationary behavior. Almost all procedures presently available in the literature are simultaneously sensitive to all three types of alternatives.

The test statistics under investigation are based on functionals of the partial sums of observations. These cumulative sum–type (CUSUM-type) statistics have limit distributions if the mean remains constant and the errors satisfy the central limit theorem but tend to infinity in the case when any of the alternatives (1), (2), or (3) holds. On removing the effect of the shifting mean, however, divergence of the test statistics will only occur under the random walk behavior, which in turn enables statisticians not only to detect structural breaks but also to specify their causes.

The results are underlined by a simulation study and an application to returns of the German stock index DAX.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 2 , April 2009 , pp. 411 - 441
Copyright
Copyright © Cambridge University Press 2009

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References

Andreau, E. & Spanos, A. (2003) Statistical adequacy and the testing of trend versus difference stationarity. Econometric Reviews 22, 217–237.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Aue, A., Berkes, I., & Horváth, L. (2006) Strong approximation for the sums of squares of augmented GARCH sequences. Bernoulli 12, 583–608.CrossRefGoogle Scholar
Bai, J. (1994) Least squares estimation of a shift in linear processes. Journal of Time Series Analysis 15, 453–470.CrossRefGoogle Scholar
Banerjee, A., Lumsdaine, R., & Stock, J. (1992) Recursive and sequential tests of the unit root and trend break hypothesis: Theory and international evidence. Journal of Business & Economic Statistics 10, 271–288.CrossRefGoogle Scholar
Bartlett, M.S. (1950) Periodogram analysis of continuous spectra. Biometrika 37, 1–16.CrossRefGoogle ScholarPubMed
Belaire-Franch, J. (2005) A proof of the power of Kim's test against stationary processes with structural breaks. Econometric Theory 21, 1172–1176.CrossRefGoogle Scholar
Berkes, I., Horváth, L., Kokoszka, P., & Shao, Q.-M. (2005) Almost sure convergence of the Bartlett estimator. Periodica Mathematica Hungarica 51, 11–25.CrossRefGoogle Scholar
Berkes, I., Horváth, L., Kokoszka, P., & Shao, Q.-M. (2006) On discriminating between long-range dependence and changes in the mean. Annals of Statistics 34, 1140–1165.CrossRefGoogle Scholar
Berkes, I. & Philipp, W. (1979) Approximation theorem for independent and weakly dependent random vectors. Annals of Probability 7, 29–54.CrossRefGoogle Scholar
Busetti, F. & Taylor, A.M.R. (2004) Tests of stationarity against a change in persistence. Journal of Econometrics 123, 33–66.CrossRefGoogle Scholar
Breidt, F.J., Crato, N., & de Lima, P (1998) On the detection and estimation of long memory in stochastic volatility. Journal of Econometrics 83, 325–348.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Csörgő, M. & Horváth, L. (1997) Limit Theorems in Change-Point Analysis. Wiley.Google Scholar
Csörgő, M. & Révész, P. (1981) Strong Approximations in Probability and Statistics. Academic Press.Google Scholar
Den Haan, W.J. & Levin, A. (1996) Inference from Parametric and Non-parametric co-Variance Matrix Estimation Procedures. National Bureau of Economic Research Technical Working Paper 2000–11.CrossRefGoogle Scholar
Feller, W. (1951) The asymptotic distribution of the range of sums of independent random variables. Annals of Mathematical Statistics 22, 427–432.CrossRefGoogle Scholar
Giraitis, L., Kokoszka, P., & Leipus, R. (2001) Testing for long memory in the presence of a general trend. Journal of Applied Probability 10, 1002–1024.Google Scholar
Giraitis, L., Kokoszka, P., Leipus, R., & Teyssiére, G. (2003) Rescaled variance and related tests for long memory in volatility and levels. Journal of Econometrics 112, 265–294.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Harvey, D.I. & Mills, T.C. (2004) Tests for stationarity in a series with endogenously determined structural change. Oxford Bulletin of Economics and Statistics 66, 863–894.CrossRefGoogle Scholar
Hurst, H. (1951) Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers 116, 770–799.CrossRefGoogle Scholar
Kim, J.-Y. (2000) Detection of change in persistence of a linear time series. Journal of Economics 95, 97–116.CrossRefGoogle Scholar
Kim, J.-Y, Belaire-Franch, J., & Badillo-Amador, R. (2002) Corringendum to “Detection of change in persistence of a linear time series.” Journal of Economics 109, 389–392.CrossRefGoogle Scholar
Kuiper, N.H. (1960) Tests concerning random points on the circle. Proceedings of the Koninklijke Nederlandse Akademie Van Wettenschappen, Series A 63, 38–47.Google Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Leybourne, S., Kim, T.-H., Smith, V., & Newbold, P. (2003) Tests for a change in persistence against the null of difference-stationarity. Econometrics Journal 6, 291–311.CrossRefGoogle Scholar
Lo, A. (1991) Long-term memory in stock market prices. Econometrica 59, 1279–1313.CrossRefGoogle Scholar
Mandelbrot, B.B. & Taqqu, M.S. (1979) Robust R/S analysis of long run serial correlation. In 42nd Session of the International Statistical Institute, Manila, book 2, pp. 69–99.Google Scholar
Nelson, C.R. & Plosser, C.I. (1982) Trends and random walks in macroeconomics time series: Some evidence and implications. Journal of Monetary Economics 10, 139–162.CrossRefGoogle Scholar
Nyblom, J. & Mäkeläinen, T. (1983) Comparisons of tests for the presence of random walk coefficients in a simple linear model. Journal of the American Statistical Association 78, 856–864.CrossRefGoogle Scholar
Page, E.S. (1954) Continuous inspection schemes. Biometrika 41, 100–105.CrossRefGoogle Scholar
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.CrossRefGoogle Scholar
Perron, P. (1990) Testing for a unit root in a time series with a changing mean. Journal of Business & Economic Statistics 8, 153–162.CrossRefGoogle Scholar
Perron, P. & Vogelsang, T.J. (1992) Testing for a unit root in a time series with a changing mean: Corrections and extensions. Journal of Business & Economic Statistics 10, 467–470.CrossRefGoogle Scholar
Philipp, W. (1986) Invariance principles for independent and weakly dependent random variables. In Dependence in Probability and Statistics, Progress in Probability and Statistics, 11, pp. 225–268 Birkhäuser.CrossRefGoogle Scholar
Shao, Q.-M. (1993) Almost sure invariance principles for mixing sequences of random variables. Stochastic Processes and Their Applications 48, 319–334.CrossRefGoogle Scholar
Shorack, G.R. & Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. Wiley.Google Scholar
Taylor, A.M.R. (2005) Fluctuation tests for change in persistance. Oxford Bulletin of Economics and Statistics 67, 207–229.CrossRefGoogle Scholar
Teverovsky, V., Taqqu, M.S., & Willinger, W. (1999). Stock market prices and long-range dependence. Finance and Stochastics 3, 1–13.Google Scholar
Vostrikova, L.J. (1981) Discovery of “discord” in multidimensional random processes. Doklady Akademii Nauk SSSR 259, 270–274.Google Scholar
Watson, G.S. (1961) Goodness-of-fit tests on a circle. Biometrika 48, 109–114.CrossRefGoogle Scholar
Xiao, Z. (2001) Testing the null hypothesis of stationarity against an autoregressive unit root alternative. Journal of Time Series Analysis 22, 87–105.CrossRefGoogle Scholar
Yao, Y.-C. (1988). Estimating the number of change-points via Schwartz's criterion. Statistics and Probability Letters 6, 173–177.CrossRefGoogle Scholar
Zhang, A., Gabrys, R., & Kokoszka, P. (2007) Discriminating between long memory and volatility shifts. Austrian Journal of Statistics 36, 253–275.Google Scholar