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ASYMPTOTIC BEHAVIOR OF THE CUSUM OF SQUARES TEST UNDER STOCHASTIC AND DETERMINISTIC TIME TRENDS

Published online by Cambridge University Press:  11 July 2011

Bent Nielsen  and
Jouni S. Sohkanen
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Abstract

We generalize the cumulative sum of squares (CUSQ) test to the case of nonstationary autoregressive distributed lag models with deterministic time trends. The test may be implemented with either ordinary least squares residuals or standardized forecast errors. In explosive cases the asymptotic theory applies more generally for the least squares residuals-based test. Preliminary simulations of the tests suggest a very modest difference between the tests and a very modest variation with nuisance parameters. This supports the use of the tests in explorative analysis.

Type
Brief Report
Information
Econometric Theory , Volume 27 , Issue 4 , August 2011 , pp. 913 - 927
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

The authors received support from the ESRC (RES-000-27-0179 and PTA-031-2006-00174), the Open Society Institute and the Oxford Martin School. Comments from Andrew Whitby are gratefully acknowledged.

References

Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676–687.CrossRefGoogle Scholar
Billingsley, P. (1999) Convergence of Probability Measures. Wiley Series in Probability and Statistics. Wiley.CrossRefGoogle Scholar
Brown, R., Durbin, J., & Evans, J. (1975) Techniques for testing the constancy of regression relationships over time. Journal of the Royal Statistical Society. Series B (Methodological) 37, 149–192.Google Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Deng, A. & Perron, P. (2008a) The limit distribution of the CUSUM of squares test under general mixing conditions. Econometric Theory 24, 809–822.CrossRefGoogle Scholar
Deng, A. & Perron, P. (2008b) A non-local perspective on the power properties of the CUSUM and CUSUM of squares tests for structural change. Journal of Econometrics 142, 212–240.CrossRefGoogle Scholar
Duflo, M., Senoussi, R., & Touati, R. (1991) Propriétés asymptotiques presque sûre de l’estimateur des moindres carrés d’un modèle autorégressif vectoriel. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 27, 1–25.Google Scholar
Hall, P. & Heyde, C. (1980) Martingale Limit Theory and Its Application (Probability and Mathematical Statistics). Academic Press.Google Scholar
Lai, T.L. & Wei, C.Z. (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Annals of Statistics 10, 154–166.CrossRefGoogle Scholar
Lai, T.L. & Wei, C.Z. (1985) Asymptotic properties of multivariate weighted sums with applications to stochastic regression in linear dynamic systems. In Krishnaiah, P. (ed.), Multivariate Analysis VI, pp. 375–393. Elsevier Science.Google Scholar
Lee, S., Na, O., & Na, S. (2003) On the CUSUM of squares test for variance change in non-stationary and non-parametric time series models. Annals of the Institute of Statistical Mathematics 55, 467–485.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
Nielsen, B. (2005) Strong consistency results for least squares estimators in general vector autoregressions with deterministic terms. Econometric Theory 21, 534–561.CrossRefGoogle Scholar
Nielsen, B. (2008) Singular Vector Autoregressions with Deterministic Terms: Strong Consistency and Lag Order Determination. Discussion paper, Nuffield College.Google Scholar
Nielsen, B. & Sohkanen, J.S. (2009) Asymptotic Behaviour of the CUSUM of Squares Test under Stochastic and Deterministic Time Trends. Discussion paper, Nuffield College.Google Scholar
Phillips, P. & Magdalinos, T. (2008) Limit theory for explosively cointegrated systems. Econometric Theory 24, 865–887.CrossRefGoogle Scholar
Ploberger, W. & Krämer, W. (1986) On studentizing a test for structural change. Economics Letters 20, 341–344.CrossRefGoogle Scholar
Schumacher, M. (1984) Two-sample tests of Cramer-von Mises and Kolmogorov-Smirnov type for randomly censored data. International Statistical Review 52, 263–281.CrossRefGoogle Scholar