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A Measure of Distance for the Unit Root Hypothesis

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  • Patrick Marsh
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    Abstract

    This paper proposes and analyses a measure of distance for the unit root hypothesis tested against stochastic stationarity. It applies over a family of distributions, for any sample size, for any specification of deterministic components and under additional autocorrelation, here parameterised by a finite order moving-average. The measure is shown to obey a set of inequalities involving the measures of distance of Gibbs and Su (2002) which are also extended to include power. It is also shown to be a convex function of both the degree of a time polynomial regressors and the moving average parameters. Thus it is minimisable with respect to either. Implicitly, therefore, we find that linear trends and innovations having a moving average negative unit root will necessarily make power small. In the context of the Nelson and Plosser (1982) data, the distance is used to measure the impact that specification of the deterministic trend has on our ability to make unit root inferences. For certain series it highlights how imposition of a linear trend can lead to estimated models indistinguishable from unit root processes while freely estimating the degree of the trend yields a model very different in character.

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    Bibliographic Info

    Paper provided by Department of Economics, University of York in its series Discussion Papers with number 05/02.

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    Handle: RePEc:yor:yorken:05/02

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    1. 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.
    2. Hillier, Grant H., 1987. "Classes of Similar Regions and Their Power Properties for Some Econometric Testing Problems," Econometric Theory, Cambridge University Press, vol. 3(01), pages 1-44, February.
    3. Peter C.B. Phillips & Werner Ploberger, 1999. "Empirical Limits for Time Series Econometric Models," Cowles Foundation Discussion Papers 1220, Cowles Foundation for Research in Economics, Yale University.
    4. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
    5. DeJong, David N & Whiteman, Charles H, 1991. "The Case for Trend-Stationarity Is Stronger Than We Thought," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 6(4), pages 413-21, Oct.-Dec..
    6. 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.
    7. Campbell, J.Y. & Perron, P., 1991. "Pitfalls and Opportunities: What Macroeconomics should know about unit roots," Papers 360, Princeton, Department of Economics - Econometric Research Program.
    8. 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.
    9. Forchini, G., 2002. "The Exact Cumulative Distribution Function Of A Ratio Of Quadratic Forms In Normal Variables, With Application To The Ar(1) Model," Econometric Theory, Cambridge University Press, vol. 18(04), pages 823-852, August.
    10. Steven N. Durlauf & Peter C.B. Phillips, 1986. "Trends Versus Random Walks in Time Series Analysis," Cowles Foundation Discussion Papers 788, Cowles Foundation for Research in Economics, Yale University.
    11. Dufour, Jean-Marie & King, Maxwell L., 1991. "Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary AR(1) errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 115-143, January.
    12. Leybourne, Stephen J. & C. Mills, Terence & Newbold, Paul, 1998. "Spurious rejections by Dickey-Fuller tests in the presence of a break under the null," Journal of Econometrics, Elsevier, vol. 87(1), pages 191-203, August.
    13. Bhargava, Alok, 1986. "On the Theory of Testing for Unit Roots in Observed Time Series," Review of Economic Studies, Wiley Blackwell, vol. 53(3), pages 369-84, July.
    14. 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.
    15. Peter C.B. Phillips, 2001. "Regression with Slowly Varying Regressors," Cowles Foundation Discussion Papers 1310, Cowles Foundation for Research in Economics, Yale University.
    16. repec:cup:etheor:v:10:y:1994:i:3-4:p:774-808 is not listed on IDEAS
    17. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
    18. DeJong, David N, et al, 1992. "Integration versus Trend Stationarity in Time Series," Econometrica, Econometric Society, vol. 60(2), pages 423-33, March.
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