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Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses

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  • Kim, Dukpa
  • Perron, Pierre

Abstract

Perron [Perron, P., 1989. The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361-1401] introduced a variety of unit root tests that are valid when a break in the trend function of a time series is present. The motivation was to devise testing procedures that were invariant to the magnitude of the shift in level and/or slope. In particular, if a change is present it is allowed under both the null and alternative hypotheses. This analysis was carried under the assumption of a known break date. The subsequent literature aimed to devise testing procedures valid in the case of an unknown break date. However, in doing so, most of the literature and, in particular the commonly used test of Zivot and Andrews [Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil price shock and the unit root hypothesis. Journal of Business and Economic Statistics 10, 251-270], assumed that if a break occurs, it does so only under the alternative hypothesis of stationarity. This is undesirable since (a) it imposes an asymmetric treatment when allowing for a break, so that the test may reject when the noise is integrated but the trend is changing; (b) if a break is present, this information is not exploited to improve the power of the test. In this paper, we propose a testing procedure that addresses both issues. It allows a break under both the null and alternative hypotheses and, when a break is present, the limit distribution of the test is the same as in the case of a known break date, thereby allowing increased power while maintaining the correct size. Simulation experiments confirm that our procedure offers an improvement over commonly used methods in small samples.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 148 (2009)
Issue (Month): 1 (January)
Pages: 1-13

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Handle: RePEc:eee:econom:v:148:y:2009:i:1:p:1-13

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Web page: http://www.elsevier.com/locate/jeconom

Related research

Keywords: Structural change Pre-test Trend function Integrated processes Hypothesis testing;

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References

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  1. Vogelsang, Timothy J & Perron, Pierre, 1998. "Additional Tests for a Unit Root Allowing for a Break in the Trend Function at an Unknown Time," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 1073-1100, November.
  2. Christiano, Lawrence J, 1992. "Searching for a Break in GNP," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(3), pages 237-50, July.
  3. Zivot, Eric & Andrews, Donald W K, 2002. "Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 25-44, January.
  4. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November.
  5. Montanes, Antonio & Olloqui, Irene, 1999. "Misspecification of the breaking date in segmented trend variables: effect on the unit root tests," Economics Letters, Elsevier, vol. 65(3), pages 301-307, December.
  6. Lanne, Markku & Lütkepohl, Helmut & Saikkonen, Pentti, 2001. "Test procedures for unit roots in time series with level shifts at unknown time," SFB 373 Discussion Papers 2001,39, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  7. Montañés, Antonio & Reyes, Marcelo, 1999. "The asymptotic behaviour of the Dickey-Fuller tests under the crash hypothesis," Statistics & Probability Letters, Elsevier, vol. 42(1), pages 81-89, March.
  8. Pierre Perron & Tomoyoshi Yabu, 2005. "Testing for Shifts in Trend with an Integrated or Stationary Noise Component," Boston University - Department of Economics - Working Papers Series WP2005-026, Boston University - Department of Economics.
  9. Perron, Pierre & Vogelsang, Timothy J, 1992. "Nonstationarity and Level Shifts with an Application to Purchasing Power Parity," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(3), pages 301-20, July.
  10. Perron, Pierre & Zhu, Xiaokang, 2005. "Structural breaks with deterministic and stochastic trends," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 65-119.
  11. Nunes, Luis C. & Kuan, Chung-Ming & Newbold, Paul, 1995. "Spurious Break," Econometric Theory, Cambridge University Press, vol. 11(04), pages 736-749, August.
  12. Perron, Pierre, 1990. "Testing for a Unit Root in a Time Series with a Changing Mean," Journal of Business & Economic Statistics, American Statistical Association, vol. 8(2), pages 153-62, April.
  13. Vogelsang, Timothy J., 1997. "Wald-Type Tests for Detecting Breaks in the Trend Function of a Dynamic Time Series," Econometric Theory, Cambridge University Press, vol. 13(06), pages 818-848, December.
  14. Kormendi, Roger C & Meguire, Philip, 1990. "A Multicountry Characterization of the Nonstationarity of Aggregate Output," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 22(1), pages 77-93, February.
  15. Perron, P, 1988. "The Great Crash, The Oil Price Shock And The Unit Root Hypothesis," Papers 338, Princeton, Department of Economics - Econometric Research Program.
  16. Monta s, Antonio & Reyes, Marcelo, 1998. "Effect Of A Shift In The Trend Function On Dickey Fuller Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 14(03), pages 355-363, June.
  17. Hecq, Alain & Urbain, Jean-Pierre, 1993. "Misspecification tests, unit roots and level shifts," Economics Letters, Elsevier, vol. 43(2), pages 129-135.
  18. Stephen J. Leybourne And Paul Newbold, 2000. "Behaviour of the standard and symmetric Dickey-Fuller-type tests when there is a break under the null hypothesis," Econometrics Journal, Royal Economic Society, vol. 3(1), pages 1-15.
  19. Perron, Pierre & Vogelsang, Timothy J, 1992. "Testing for a Unit Root in a Time Series with a Changing Mean: Corrections and Extensions," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(4), pages 467-70, October.
  20. Perron, P., 1994. "Further Evidence on Breaking Trend Functions in Macroeconomic Variables," Cahiers de recherche 9421, Universite de Montreal, Departement de sciences economiques.
  21. Bai, Jushan, 1998. "A Note On Spurious Break," Econometric Theory, Cambridge University Press, vol. 14(05), pages 663-669, October.
  22. Lee, Junsoo & Strazicich, Mark C, 2001. " Break Point Estimation and Spurious Rejections with Endogenous Unit Root Tests," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 63(5), pages 535-58, December.
  23. Nunes, Luis C & Newbold, Paul & Kuan, Chung-Ming, 1997. "Testing for Unit Roots with Breaks: Evidence on the Great Crash and the Unit Root Hypothesis Reconsidered," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 59(4), pages 435-48, November.
  24. Montañés, Antonio & Reyes, Marcelo, 2000. "Structural breaks, unit roots and methods for removing the autocorrelation pattern," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 401-409, July.
  25. Kim, Tae-Hwan & Leybourne, Stephen J & Newbold, Paul, 2000. " Spurious Rejections by Perron Tests in the Presence of a Break," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 62(3), pages 433-44, July.
  26. Leybourne, Stephen J. & C. Mills, Terence & Newbold, Paul, 1998. "Spurious rejections by Dickey-Fuller tests in the presence of a break under the null," Journal of Econometrics, Elsevier, vol. 87(1), pages 191-203, August.
  27. Montanes, Antonio, 1997. "Level shifts, unit roots and misspecification of the breaking date," Economics Letters, Elsevier, vol. 54(1), pages 7-13, January.
  28. Junsoo Lee & Mark C. Strazicich, 2003. "Minimum Lagrange Multiplier Unit Root Test with Two Structural Breaks," The Review of Economics and Statistics, MIT Press, vol. 85(4), pages 1082-1089, November.
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