Testing for unit roots and the impact of quadratic trends, with an application to relative primary commodity prices
AbstractIn 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 InfoPaper provided by University of Nottingham, Granger Centre for Time Series Econometrics in its series Discussion Papers with number 08/04.
Date of creation: Jun 2008
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Unit root test; trend uncertainty; quadratic trends; asymptotic power; union of rejections decision rule;
Other versions of this item:
- David Harvey & Stephen Leybourne & A.M. Robert Taylor, 2011. "Testing for Unit Roots and the Impact of Quadratic Trends, with an Application to Relative Primary Commodity Prices," Econometric Reviews, Taylor and Francis Journals, vol. 30(5), pages 514-547.
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