Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T12:43:51.924Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LIMIT DISTRIBUTION OF THE CUSUM OF SQUARES TEST UNDER GENERAL MIXING CONDITIONS

Published online by Cambridge University Press:  26 February 2008

Ai Deng  and
Pierre Perron
Ai Deng*
Affiliation:
Bates White, LLC
Pierre Perron
Affiliation:
Boston University
*
Address correspondence to Ai Deng, Bates White, LLC, 1300 Eye Street NW, Suite 600, Washington, DC 20005, USA; e-mail: ai.deng@bateswhite.com.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the cumulative sum (CUSUM) of squares test in a linear regression model with general mixing assumptions on the regressors and the errors. We derive its limit distribution and show how it depends on the nature of the error process. We suggest a corrected version that has a limit distribution free of nuisance parameters. We also discuss how it provides an improvement over the standard approach to testing for a change in the variance in a univariate times series. Simulation evidence is presented to support this.

Type
NOTES AND PROBLEMS
Information
Econometric Theory , Volume 24 , Issue 3 , June 2008 , pp. 809 - 822
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Andrews, D.W.K. ’ Monahan, J.C. (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60, 953–966.CrossRefGoogle Scholar
Bai, J. ’ Perron, P. (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78.CrossRefGoogle Scholar
Beare, B.K. (2005) Robustifying Unit Root Tests to Permanent Changes in Innovation Variance. Manuscript, Department of Economics, Yale University.Google Scholar
Birnbaum, Z.W. ’ Marshall, A.W. (1961) Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Annals of Mathematical Statistics 32, 687–703.CrossRefGoogle Scholar
Brown, R.L., Durbin, J., ’ Evans, J.M. (1975) Techniques for testing the constancy of regression relationships over time. Journal of the Royal Statistical Society, Series B 37, 149–163.Google Scholar
Cavaliere, G. ’ Taylor, A.M.R. (2006) Testing for Unit Roots in Time Series Models with Non-Stationary Volatility. Manuscript, Department of Economics, University of Nottingham.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Inclán, C. ’ Tiao, G.C. (1994) Use of the cumulative sums of squares for retrospective detection of changes of variance. Journal of the American Statistical Association 89, 913–923.Google Scholar
Koksozka, P. ’ Leipus, R. (2000) Change-point estimation in ARCH models. Bernoulli 6, 1–28.Google Scholar
Lee, S., Ha, J., ’ Na, O. (2003) The CUSUM test for parameter change in time series models. Scandinavian Journal of Statistics 30, 781–796.CrossRefGoogle Scholar
Lee, S. ’ Park, S. (2001) The CUSUM of squares test for scale changes in infinite order moving average processes. Scandinavian Journal of Statistics 28, 625–644.CrossRefGoogle Scholar
Lee, S., Tokutsu, Y., ’ Maekawa, K. (2004) The CUSUM test for parameter change in regression models with ARCH errors. Journal of the Japanese Statistical Society 34, 173–188.CrossRefGoogle Scholar
Loretan, M. ’ Phillips, P.C.B. (1994) Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance 1, 211–248.CrossRefGoogle Scholar
McCabe, B.P.M. ’ Harrison, M.J. (1980) Testing the constancy of regression relationships over time using least squares residuals. Applied Statistics 29, 142–148.CrossRefGoogle Scholar
Pagan, A.R. ’ Schwert, G.W. (1990) Testing for covariance stationarity in stock market data. Economics Letters 33, 165–170.CrossRefGoogle Scholar
Perron, P. (2006) Dealing with structural breaks. In Patterson, K. ’ Mills, T.C. (eds.), Palgrave Handbook of Econometrics, vol. 1: Econometric Theory, pp. 278–352. Palgrave Macmillan.Google Scholar
Phillips, P.C.B. ’ Xu, K.-L. (2005) Inference in autoregression under heteroskedasticity. Journal of Time Series Analysis 27, 289–308.CrossRefGoogle Scholar
Ploberger, W. (1989) The local power of the CUSUM-SQ test against heteroskedasticity. In Hackl, P. (ed.), Statistical Analysis and Forecasting of Econometric Structural Change, pp. 127–133. International Institute for Applied Systems Analysis, Springer-Verlag.CrossRefGoogle Scholar
Ploberger, W. ’ Krämer, W. (1986) On studentizing a test for structural change. Economics Letters 20, 341–344.CrossRefGoogle Scholar
Ploberger, W. ’ Krämer, W. (1990) The local power of the CUSUM and CUSUM of squares tests. Econometric Theory 6, 335–347.CrossRefGoogle Scholar
Qu, Z. ’ Perron, P. (2007) Estimating and testing multiple structural changes in multivariate regressions. Econometrica 75, 459–502.CrossRefGoogle Scholar
Sansó, A., Aragó, V., ’ Carrion-i-Silvestre, J.L. (2004) Testing for changes in the unconditional variance of financial time series. Revista de Economía Financiera 4, 32–53.Google Scholar
Stock, J.H. ’ Watson, M.W. (2002) Has the business cycle changed and why? In Gertler, M. ’ Rogoff, K. (eds.), NBER Macroeconomics Annual 17, pp. 159–218. MIT Press.Google Scholar
Tang, S.M. ’ MacNeill, I.B. (1993) The effect of serial correlation on tests for parameter change at unknown time. Annals of Statistics 21, 552–575.CrossRefGoogle Scholar