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Properties of the power envelope for tests against both stationary and explosive alternatives: the effect of trends

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  • Patrick Marsh

Abstract

This paper details a precise analytic effect that inclusion of a linear trend has on the power of Neyman-Pearson point optimal unit root tests and thence the power envelope. Both stationary and explosive alternatives are considered. The envelope can be characterized by probabilities for two, related, sums of chi-square random variables. A stochastic expansion, in powers of the local-to-unity parameter, of the difference between these loses its leading term when a linear trend is included. This implies that the power envelope converges to size at a faster rate, which can then be exploited to prove that the power envelope must necessarily be lower. This effect is shown to be, analytically, greater asymptotically than in small samples and numerically far greater for explosive than for stationary alternatives. Only a linear trend has a specific rate effect on the power envelope, however other deterministic variables will have some effect. The methods of the paper lead to a simple direct measure of this effect which is then informative about power, in practice.

Suggested Citation

  • Patrick Marsh, 2019. "Properties of the power envelope for tests against both stationary and explosive alternatives: the effect of trends," Discussion Papers 19/03, University of Nottingham, Granger Centre for Time Series Econometrics.
    • Handle: RePEc:not:notgts:19/03
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    References listed on IDEAS

    as
    1. Patrick Marsh, 2007. "Constructing Optimal tests on a Lagged dependent variable," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(5), pages 723-743, September.
    2. Maxwell L. King & Sivagowry Sriananthakumar, 2015. "Point Optimal Testing: A Survey of the Post 1987 Literature," Monash Econometrics and Business Statistics Working Papers 5/15, Monash University, Department of Econometrics and Business Statistics.
    3. Moon, Hyungsik Roger & Perron, Benoit & Phillips, Peter C.B., 2007. "Incidental trends and the power of panel unit root tests," Journal of Econometrics, Elsevier, vol. 141(2), pages 416-459, December.
    4. Patrick Marsh, "undated". "Saddlepoint Approximations for Optimal Unit Root Tests," Discussion Papers 09/31, Department of Economics, University of York.
    5. Elliott, Graham & Rothenberg, Thomas J & Stock, James H, 1996. "Efficient Tests for an Autoregressive Unit Root," Econometrica, Econometric Society, vol. 64(4), pages 813-836, July.
    6. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    7. Anna Bykhovskaya & Peter C. B. Phillips, 2017. "Point Optimal Testing with Roots That Are Functionally Local to Unity," Cowles Foundation Discussion Papers 3007, Cowles Foundation for Research in Economics, Yale University.
    8. Anna Bykhovskaya & Peter C. B. Phillips, 2018. "Boundary Limit Theory for Functional Local to Unity Regression," Journal of Time Series Analysis, Wiley Blackwell, vol. 39(4), pages 523-562, July.
    9. Marsh, Patrick, 2007. "The Available Information For Invariant Tests Of A Unit Root," Econometric Theory, Cambridge University Press, vol. 23(4), pages 686-710, August.
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    16. 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(3), pages 587-636, June.
    17. Bykhovskaya, Anna & Phillips, Peter C.B., 2020. "Point optimal testing with roots that are functionally local to unity," Journal of Econometrics, Elsevier, vol. 219(2), pages 231-259.
    18. Leamer, Edward E, 1985. "Sensitivity Analyses Would Help," American Economic Review, American Economic Association, vol. 75(3), pages 308-313, June.
    19. Podivinsky, Jan M. & King, Maxwell L., 2000. "The exact power envelope of tests for a unit root," Discussion Paper Series In Economics And Econometrics 26, Economics Division, School of Social Sciences, University of Southampton.
    20. Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(1), pages 1-28, February.
    21. Nabeya, Seiji & Tanaka, Katsuto, 1990. "Limiting power of unit-root tests in time-series regression," Journal of Econometrics, Elsevier, vol. 46(3), pages 247-271, December.
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