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Unit root testing under a local break in trend

  • David I. Harvey
  • Stephen J. Leybourne
  • A. M. Robert Taylor

It is well known that it is vital to account for trend breaks when testing for a unit root. In practice, uncertainty exists over whether or not a trend break is present and, if it is, where it is located. Harris et al. (2009) and Carrion-i-Silvestre et al. (2009) propose procedures which account for both of these forms of uncertainty. Each uses what amounts to a pre-test for a trend break, accounting for a trend break (the associated break fraction estimated from the data) in the unit root procedure only where the pre-test signals a break. Assuming the break magnitude is fixed (independent of sample size) these authors show that their methods achieve near asymptotically ecient unit root inference in both trend break and no trend break environments. These asymptotic results are, however, somewhat at odds with the finite sample simulations reported in both papers. These show the presence of pronounced "valleys" in the finite sample power functions (when mapped as functions of the break magnitude) of the tests such that power is initially high for very small breaks, then decreases as the break magnitude increases, before increasing again. Here we show that treating the break magnitude as local to zero (in a Pitman drift sense) allows the asymptotic analysis to very closely approximate this finite sample effect, thereby providing useful analytical insights into the observed phenomenon. In response to this problem we propose practical solutions, based either on the use of a with break unit root test but with adaptive critical values, or on a union of rejections principle taken across with break and without break unit root tests. The former is shown to eliminate power valleys but at the expense of power when no break is present, while the latter considerably mitigates the valleys while not losing all the power gains available when no break exists.

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File URL: http://www.nottingham.ac.uk/economics/grangercentre/papers/11-02.pdf
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Paper provided by University of Nottingham, Granger Centre for Time Series Econometrics in its series Discussion Papers with number 10/05.

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Date of creation: Sep 2010
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Handle: RePEc:not:notgts:10/05
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Web page: http://www.nottingham.ac.uk/economics/grangercentre/

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  1. Mohitosh Kejriwal & Pierre Perron, 2006. "Unit Root Tests Allowing for a Break in the Trend Function at an Unknown Time Under Both the Null and Alternative Hypotheses," Boston University - Department of Economics - Working Papers Series WP2006-052, Boston University - Department of Economics.
  2. Josep Lluís Carrion-i-Silvestre & Dukpa Kim & Pierre Perron, 2007. "GLS-based unit root tests with multiple structural breaks both under the null and the alternative hypotheses," Boston University - Department of Economics - Working Papers Series wp2008-019, Boston University - Department of Economics.
  3. Serena Ng & Pierre Perron, 1997. "Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power," Boston College Working Papers in Economics 369, Boston College Department of Economics, revised 01 Sep 2000.
  4. Stock, James H & Watson, Mark W, 1996. "Evidence on Structural Instability in Macroeconomic Time Series Relations," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 11-30, January.
  5. Perron, P, 1988. "The Great Crash, The Oil Price Shock And The Unit Root Hypothesis," Papers 338, Princeton, Department of Economics - Econometric Research Program.
  6. Pierre Perron & Tomoyoshi Yabu, 2005. "Estimating Deterministric Trends with an Integrated or Stationary Noise Component," Boston University - Department of Economics - Working Papers Series WP2005-037, Boston University - Department of Economics.
  7. Graham Elliott & Thomas J. Rothenberg & James H. Stock, 1992. "Efficient Tests for an Autoregressive Unit Root," NBER Technical Working Papers 0130, National Bureau of Economic Research, Inc.
  8. Denis Kwiatkowski & Peter C.B. Phillips & Peter Schmidt, 1991. "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?," Cowles Foundation Discussion Papers 979, Cowles Foundation for Research in Economics, Yale University.
  9. David I. Harvey & Stephen J. Leybourne & A.M. Robert Taylor, . "Simple, Robust and Powerful Tests of the Breaking Trend Hypothesis," Discussion Papers 06/11, University of Nottingham, School of Economics.
  10. Eric Zivot & Donald W.K. Andrews, 1990. "Further Evidence on the Great Crash, the Oil Price Shock, and the Unit Root Hypothesis," Cowles Foundation Discussion Papers 944, Cowles Foundation for Research in Economics, Yale University.
  11. Mohitosh Kejriwal & Pierre Perron, 2009. "A Sequential Procedure to Determine the Number of Breaks in Trend with an Integrated or Stationary Noise Component," Purdue University Economics Working Papers 1217, Purdue University, Department of Economics.
  12. Timothy J. Vogelsang, 1998. "Trend Function Hypothesis Testing in the Presence of Serial Correlation," Econometrica, Econometric Society, vol. 66(1), pages 123-148, January.
  13. Yoosoon Chang & Joon Park, 2002. "On The Asymptotics Of Adf Tests For Unit Roots," Econometric Reviews, Taylor & Francis Journals, vol. 21(4), pages 431-447.
  14. Carrion-i-Silvestre, Josep Lluís & Kim, Dukpa & Perron, Pierre, 2009. "Gls-Based Unit Root Tests With Multiple Structural Breaks Under Both The Null And The Alternative Hypotheses," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1754-1792, December.
  15. Harris, David & Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2009. "Testing For A Unit Root In The Presence Of A Possible Break In Trend," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1545-1588, December.
  16. Perron, P., 1994. "Further Evidence on Breaking Trend Functions in Macroeconomic Variables," Cahiers de recherche 9421, Universite de Montreal, Departement de sciences economiques.
  17. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2009. "Unit Root Testing In Practice: Dealing With Uncertainty Over The Trend And Initial Condition," Econometric Theory, Cambridge University Press, vol. 25(03), pages 587-636, June.
  18. Pierre Perron & Tomoyoshi Yabu, 2007. "Testing for Shifts in Trend with an Integrated or Stationary Noise Component," Boston University - Department of Economics - Working Papers Series WP2007-025, Boston University - Department of Economics.
  19. James H. Stock & Mark W. Watson, 2005. "Implications of Dynamic Factor Models for VAR Analysis," NBER Working Papers 11467, National Bureau of Economic Research, Inc.
  20. Pierre Perron & Zhongjun Qu, 2006. "A Simple Modification to Improve the Finite Sample Properties of Ng and Perron’s Unit Root Tests," Boston University - Department of Economics - Working Papers Series WP2006-010, Boston University - Department of Economics.
  21. Anindya Banerjee & Robin L. Lumsdaine & James H. Stock, 1990. "Recursive and Sequential Tests of the Unit Root and Trend Break Hypothesis: Theory and International Evidence," NBER Working Papers 3510, National Bureau of Economic Research, Inc.
  22. 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.
  23. Perron, Pierre & Rodriguez, Gabriel, 2003. "GLS detrending, efficient unit root tests and structural change," Journal of Econometrics, Elsevier, vol. 115(1), pages 1-27, July.
  24. Jingjing Yang, 2012. "Break point estimators for a slope shift: levels versus first differences," Econometrics Journal, Royal Economic Society, vol. 15(1), pages 154-169, 02.
  25. Perron, Pierre & Zhu, Xiaokang, 2005. "Structural breaks with deterministic and stochastic trends," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 65-119.
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