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Testing for unit roots in the presence of uncertainty over both the trend and initial condition

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

We provide a joint treatment of two major problems that surround testing for a unit root in practice, namely uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In earlier work [Harvey, Leybourne and Taylor, 2008] we proposed methods to deal with trend uncertainty when the initial condition is assumed to be (asymptotically) negligible, together with methods to deal with uncertainty over the initial condition when the form of the trend function was taken as known. In each case we recommended a simple union of rejections-based decision rule. In the first case rejecting the unit root null whenever either of the quasi-differenced (QD) detrended or QD demeaned augmented Dickey-Fuller [ADF] unit root tests yields a rejection, and in the second case if either of the QD and OLS detrended/demeaned ADF tests rejects. Both approaches were shown to work well. In this paper we extend these procedures to allow for both trend and initial condition uncertainty, proposing a four-way union of rejections decision rule based on the QD and OLS demeaned and the QD and OLS detrended ADF tests. This is shown to work well but to lack power, relative to the best available test, in some scenarios. A modification of the basic union, based on auxiliary information including linear trend pre-test statistics, is proposed and shown to deliver significant improvements. A by-product of our analysis is that the power functions of the associated trend function pre-tests are shown to be heavily dependent on the initial condition.

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Paper provided by University of Nottingham, Granger Centre for Time Series Econometrics in its series Discussion Papers with number 08/03.

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Date of creation: May 2008
Date of revision:
Handle: RePEc:not:notgts:08/03
Contact details of provider: Postal: School of Economics University of Nottingham University Park Nottingham NG7 2RD
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  1. David I. Harvey & Stephen J. Leybourne & A. M. Robert Taylor, 2008. "Testing for unit roots in the presence of uncertainty over both the trend and initial condition," Discussion Papers 08/03, University of Nottingham, Granger Centre for Time Series Econometrics.
  2. 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.
  3. Peter C.B. Phillips & Zhijie Xiao, 1998. "A Primer on Unit Root Testing," Cowles Foundation Discussion Papers 1189, Cowles Foundation for Research in Economics, Yale University.
  4. Bunzel, Helle & Vogelsang, Timothy J., 2005. "Powerful Trend Function Tests That Are Robust to Strong Serial Correlation, With an Application to the Prebisch-Singer Hypothesis," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 381-394, October.
  5. Peter C.B. Phillips, 1990. "To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends," Cowles Foundation Discussion Papers 950, Cowles Foundation for Research in Economics, Yale University.
  6. Kwiatkowski, D. & Phillips, P.C.B. & Schmidt, P., 1990. "Testing the Null Hypothesis of Stationarity Against the Alternative of Unit Root : How Sure are we that Economic Time Series have a Unit Root?," Papers 8905, Michigan State - Econometrics and Economic Theory.
  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. Peter C.B. Phillips & Hyungsik R. Moon, 1999. "Linear Regression Limit Theory for Nonstationary Panel Data," Cowles Foundation Discussion Papers 1222, Cowles Foundation for Research in Economics, Yale University.
  9. Timothy J. Vogelsang, 1998. "Trend Function Hypothesis Testing in the Presence of Serial Correlation," Econometrica, Econometric Society, vol. 66(1), pages 123-148, January.
  10. Schmidt, Peter & Phillips, C B Peter, 1992. "LM Tests for a Unit Root in the Presence of Deterministic Trends," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 54(3), pages 257-87, August.
  11. Peter C.B. Phillips, 1989. "Time Series Regression with a Unit Root and Infinite Variance Errors," Cowles Foundation Discussion Papers 897R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1989.
  12. Ulrich K. M¸ller & Graham Elliott, 2003. "Tests for Unit Roots and the Initial Condition," Econometrica, Econometric Society, vol. 71(4), pages 1269-1286, 07.
  13. Serena Ng & Pierre Perron, 2001. "LAG Length Selection and the Construction of Unit Root Tests with Good Size and Power," Econometrica, Econometric Society, vol. 69(6), pages 1519-1554, November.
  14. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
  15. Choi, In & Phillips, Peter C. B., 1993. "Testing for a unit root by frequency domain regression," Journal of Econometrics, Elsevier, vol. 59(3), pages 263-286, October.
  16. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
  17. Phillips, Peter C.B. & Ploberger, Werner, 1994. "Posterior Odds Testing for a Unit Root with Data-Based Model Selection," Econometric Theory, Cambridge University Press, vol. 10(3-4), pages 774-808, August.
  18. David I. Harvey, & Stephen J. Leybourne, & A. M. Robert Taylor, 2006. "A simple, robust and powerful test of the trend hypothesis," Discussion Papers 06/01, University of Nottingham, Granger Centre for Time Series Econometrics.
  19. 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.
  20. Elliott, Graham & Muller, Ulrich K., 2006. "Minimizing the impact of the initial condition on testing for unit roots," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 285-310.
  21. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  22. Peter C.B. Phillips, 1991. "Bayesian Routes and Unit Roots: de rebus prioribus semper est disputandum," Cowles Foundation Discussion Papers 986, Cowles Foundation for Research in Economics, Yale University.
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