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Unit Root Testing In Practice: Dealing With Uncertainty Over The Trend And Initial Condition

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

In this paper we focus on two major issues that surround testing for a unit root in practice, namely, (i) uncertainty as to whether or not a linear deterministic trend is present in the data and (ii) uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. In each case simple testing procedures are proposed with the aim of maintaining good power properties across such uncertainties. For the first issue, if the initial condition is negligible, quasi-differenced (QD) detrended (demeaned) Dickey–Fuller-type unit root tests are near asymptotically efficient when a deterministic trend is (is not) present in the data generating process. Consequently, we compare a variety of strategies that aim to select the detrended variant when a trend is present, and the demeaned variant otherwise. Based on asymptotic and finite-sample evidence, we recommend a simple union of rejections-based decision rule whereby the unit root null hypothesis is rejected whenever either of the detrended or demeaned unit root tests yields a rejection. Our results show that this approach generally outperforms more sophisticated strategies based on auxiliary methods of trend detection. For the second issue, we again recommend a union of rejections decision rule, rejecting the unit root null if either of the QD or ordinary least squares (OLS) detrended/demeaned Dickey–Fuller-type tests rejects. This procedure is also shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended/demeaned test for small (large) initial conditions.

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Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 25 (2009)
Issue (Month): 03 (June)
Pages: 587-636

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Handle: RePEc:cup:etheor:v:25:y:2009:i:03:p:587-636_09
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  1. 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.
  2. Cavaliere, Giuseppe & Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2011. "Testing For Unit Roots In The Presence Of A Possible Break In Trend And Nonstationary Volatility," Econometric Theory, Cambridge University Press, vol. 27(05), pages 957-991, October.
  3. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2007. "A simple, robust and powerful test of the trend hypothesis," Journal of Econometrics, Elsevier, vol. 141(2), pages 1302-1330, December.
  4. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, vol. 64(4), pages 813-36, July.
  5. Ayat, Leila & Burridge, Peter, 2000. "Unit root tests in the presence of uncertainty about the non-stochastic trend," Journal of Econometrics, Elsevier, vol. 95(1), pages 71-96, March.
  6. Kenneth D. West & Whitney K. Newey, 1995. "Automatic Lag Selection in Covariance Matrix Estimation," NBER Technical Working Papers 0144, National Bureau of Economic Research, Inc.
  7. Peter C.B. Phillips & Werner Ploberger, 1992. "Posterior Odds Testing for a Unit Root with Data-Based Model Selection," Cowles Foundation Discussion Papers 1017, Cowles Foundation for Research in Economics, Yale University.
  8. David Harris & David I. Harvey & Stephen J. Leybourne & A. M. Robert Taylor, 2007. "Testing for a unit root in the presence of a possible break in trend," Discussion Papers 07/04, University of Nottingham, Granger Centre for Time Series Econometrics.
  9. Peter C. B. Phillips & Zhijie Xiao, 1998. "A Primer on Unit Root Testing," Journal of Economic Surveys, Wiley Blackwell, vol. 12(5), pages 423-470, December.
  10. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
  11. 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.
  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. Bunzel, Helle & Vogelsang, Timothy J., 2003. "Powerful Trend Function Tests That Are Robust to Strong Serial Correlation with an Application to the Prebisch-Singer Hypothesis," Staff General Research Papers 10353, Iowa State University, Department of Economics.
  15. 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.
  16. 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.
  17. West, Kenneth D, 1988. "Asymptotic Normality, When Regressors Have a Unit Root," Econometrica, Econometric Society, vol. 56(6), pages 1397-1417, November.
  18. Timothy J. Vogelsang, 1998. "Trend Function Hypothesis Testing in the Presence of Serial Correlation," Econometrica, Econometric Society, vol. 66(1), pages 123-148, January.
  19. Marsh, Patrick, 2007. "The Available Information For Invariant Tests Of A Unit Root," Econometric Theory, Cambridge University Press, vol. 23(04), pages 686-710, August.
  20. 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.
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  22. Zhije Xiao & Peter C.B. Phillips, 1998. "An ADF coefficient test for a unit root in ARMA models of unknown order with empirical applications to the US economy," Econometrics Journal, Royal Economic Society, vol. 1(RegularPa), pages 27-43.
  23. Elliott, Graham, 1999. "Efficient Tests for a Unit Root When the Initial Observation Is Drawn from Its Unconditional Distribution," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 40(3), pages 767-83, August.
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