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Testing for unit roots and the impact of quadratic trends, with an application to relative primary commodity prices

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

Abstract

In practice a degree of uncertainty will always exist concerning what specification to adopt for the deterministic trend function when running unit root tests. While most macroeconomic time series appear to display an underlying trend, it is often far from clear whether this component is best modelled as a simple linear trend (so that long-run growth rates are constant) or by a more complicated non-linear trend function which may, for instance, allow the deterministic trend component to evolve gradually over time. In this paper we consider the effects on unit root testing of allowing for a local quadratic trend, a simple yet very flexible example of the latter. Where a local quadratic trend is present but not modelled we show that the quasi-differenced detrended Dickey-Fuller-type test of Elliott et al. (1996) has both size and power which tend to zero asymptotically. An extension of the Elliott et al. (1996) approach to allow for a quadratic trend resolves this problem but is shown to result in large power losses relative to the standard detrended test when no quadratic trend is present. We consequently propose a simple and practical approach to dealing with this form of uncertainty based on a union of rejections-based decision rule whereby the unit root null is rejected whenever either of the detrended or quadratic detrended unit root tests rejects. A modification of this basic strategy is also suggested which further improves on the properties of the procedure. An application to relative primary commodity price data highlights the empirical relevance of the methods outlined in this paper. A by-product of our analysis is the development of a test for the presence of a quadratic trend which is robust to whether or not the data admit a unit root.

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Bibliographic Info

Paper provided by University of Nottingham, Granger Centre for Time Series Econometrics in its series Discussion Papers with number 08/04.

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Date of creation: Jun 2008
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Handle: RePEc:not:notgts:08/04

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Postal: School of Economics University of Nottingham University Park Nottingham NG7 2RD
Phone: (44) 0115 951 5620
Fax: (0115) 951 4159
Web page: http://www.nottingham.ac.uk/economics/grangercentre/
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Related research

Keywords: Unit root test; trend uncertainty; quadratic trends; asymptotic power; union of rejections decision rule;

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References

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  1. Perron, P, 1988. "The Great Crash, The Oil Price Shock And The Unit Root Hypothesis," Papers 338, Princeton, Department of Economics - Econometric Research Program.
  2. 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.
  3. 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.
  4. Murray, Christian J. & Nelson, Charles R., 2000. "The uncertain trend in U.S. GDP," Journal of Monetary Economics, Elsevier, vol. 46(1), pages 79-95, August.
  5. Peter C.B. Phillips & Sam Ouliaris & Joon Y. Park, 1988. "Testing for a Unit Root in the Presence of a Maintained Trend," Cowles Foundation Discussion Papers 880, Cowles Foundation for Research in Economics, Yale University.
  6. David I. Harvey & Stephen J. Leybourne & A. M. Robert Taylor, 2007. "Unit root testing in practice: dealing with uncertainty over the trend and initial condition," Discussion Papers 07/03, University of Nottingham, Granger Centre for Time Series Econometrics.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. Stephan Pfaffenzeller & Paul Newbold & Anthony Rayner, 2007. "A Short Note on Updating the Grilli and Yang Commodity Price Index," World Bank Economic Review, World Bank Group, vol. 21(1), pages 151-163.
  13. Leon, Javier & Soto, Raimundo, 1995. "Structural breaks and long-run trends in commodity prices," Policy Research Working Paper Series 1406, The World Bank.
  14. Kellard, Neil & Wohar, Mark E., 2006. "On the prevalence of trends in primary commodity prices," Journal of Development Economics, Elsevier, vol. 79(1), pages 146-167, February.
  15. 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.
  16. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
  17. Stock, James H. & Watson, Mark W., 1999. "Business cycle fluctuations in us macroeconomic time series," Handbook of Macroeconomics, in: J. B. Taylor & M. Woodford (ed.), Handbook of Macroeconomics, edition 1, volume 1, chapter 1, pages 3-64 Elsevier.
  18. Bierens, Herman J., 1997. "Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate," Journal of Econometrics, Elsevier, vol. 81(1), pages 29-64, November.
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Citations

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Cited by:
  1. Presno, María José & Landajo, Manuel & Fernández, Paula, 2012. "Non-renewable resource prices. A robust evaluation from the stationarity perspective," MPRA Paper 42523, University Library of Munich, Germany.
  2. Yamada, Hiroshi & Yoon, Gawon, 2014. "When Grilli and Yang meet Prebisch and Singer: Piecewise linear trends in primary commodity prices," Journal of International Money and Finance, Elsevier, vol. 42(C), pages 193-207.
  3. Xuguang Sheng & Lan Cheng, 2012. "Combination of "Combinations of P-values," Working Papers 2012-11, American University, Department of Economics.
  4. Westerlund, Joakim, 2013. "Simple unit root testing in generally trending data with an application to precious metal prices in Asia," Journal of Asian Economics, Elsevier, vol. 28(C), pages 12-27.
  5. Presno, María José & Landajo, Manuel & Fernández, Paula, 2014. "Non-renewable resource prices: A robust evaluation from the stationarity perspective," Resource and Energy Economics, Elsevier, vol. 36(2), pages 394-416.

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