IDEAS home Printed from https://ideas.repec.org/p/not/notgts/19-03.html

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.nottingham.ac.uk/research/groups/grangercentre/documents/19-03.pdf
    Download Restriction: no
    ---><---

    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. Bent Nielsen, 2008. "Power of Tests for Unit Roots in the Presence of a Linear Trend," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 70(5), pages 619-644, October.
    4. Francke, Marc K. & de Vos, Aart F., 2007. "Marginal likelihood and unit roots," Journal of Econometrics, Elsevier, vol. 137(2), pages 708-728, April.
    5. Harvey, David I. & Leybourne, Stephen J., 2014. "Asymptotic behaviour of tests for a unit root against an explosive alternative," Economics Letters, Elsevier, vol. 122(1), pages 64-68.
    6. Patrick Marsh, "undated". "Saddlepoint Approximations for Optimal Unit Root Tests," Discussion Papers 09/31, Department of Economics, University of York.
    7. 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.
    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. 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.
    10. 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.
    11. 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.
    12. Leamer, Edward E, 1985. "Sensitivity Analyses Would Help," American Economic Review, American Economic Association, vol. 75(3), pages 308-313, June.
    13. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    14. 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.
    15. 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.
    16. Podivinsky, Jan M. & King, Maxwell L., 2000. "The exact power envelope of tests for a unit root," Discussion Paper Series In Economics And Econometrics 0026, Economics Division, School of Social Sciences, University of Southampton.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. Marsh, Patrick, 2011. "Saddlepoint And Estimated Saddlepoint Approximations For Optimal Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 27(5), pages 1026-1047, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Samuel Brien & Michael Jansson & Morten Ørregaard Nielsen, 2022. "Nearly Efficient Likelihood Ratio Tests of a Unit Root in an Autoregressive Model of Arbitrary Order," Working Paper 1429, Economics Department, Queen's University.
    2. Patrick Marsh, "undated". "Saddlepoint Approximations for Optimal Unit Root Tests," Discussion Papers 09/31, Department of Economics, University of York.
    3. Mehdi Hosseinkouchack & Uwe Hassler, 2016. "Powerful Unit Root Tests Free of Nuisance Parameters," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(4), pages 533-554, July.
    4. Niels Haldrup & Robinson Kruse & Timo Teräsvirta & Rasmus T. Varneskov, 2013. "Unit roots, non-linearities and structural breaks," Chapters, in: Nigar Hashimzade & Michael A. Thornton (ed.), Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 4, pages 61-94, Edward Elgar Publishing.
    5. David I. Harvey, & Stephen J. Leybourne, & A. M. Robert Taylor, 2007. "Testing for a unit root when uncertain about the trend [Revised to become 07/03 above]," Discussion Papers 06/03, University of Nottingham, Granger Centre for Time Series Econometrics.
    6. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2012. "Testing for unit roots in the presence of uncertainty over both the trend and initial condition," Journal of Econometrics, Elsevier, vol. 169(2), pages 188-195.
    7. 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.
    8. Lieberman, Offer & Phillips, Peter C.B., 2020. "Hybrid stochastic local unit roots," Journal of Econometrics, Elsevier, vol. 215(1), pages 257-285.
    9. Gabriel Zsurkis & JoÃo Nicolau & Paulo M. M. Rodrigues, 2021. "A Re‐Examination of Inflation Persistence Dynamics in OECD Countries: A New Approach," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 83(4), pages 935-959, August.
    10. M. Hashem Pesaran, 2007. "A simple panel unit root test in the presence of cross-section dependence," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(2), pages 265-312.
    11. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2012. "Unit root testing under a local break in trend," Journal of Econometrics, Elsevier, vol. 167(1), pages 140-167.
    12. Sven Otto, 2021. "Unit root testing with slowly varying trends," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(1), pages 85-106, January.
    13. Sandberg, Rickard, 2016. "Trends, unit roots, structural changes, and time-varying asymmetries in U.S. macroeconomic data: the Stock and Watson data re-examined," Economic Modelling, Elsevier, vol. 52(PB), pages 699-713.
    14. Stephan Smeekes, 2013. "Detrending Bootstrap Unit Root Tests," Econometric Reviews, Taylor & Francis Journals, vol. 32(8), pages 869-891, November.
    15. David I. Harvey & Stephen J. 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 & Francis Journals, vol. 30(5), pages 514-547, October.
    16. Shintani, Mototsugu, 2001. "A simple cointegrating rank test without vector autoregression," Journal of Econometrics, Elsevier, vol. 105(2), pages 337-362, December.
    17. Clayton Webb & Suzanna Linn & Matthew J. Lebo, 2020. "Beyond the Unit Root Question: Uncertainty and Inference," American Journal of Political Science, John Wiley & Sons, vol. 64(2), pages 275-292, April.
    18. Joakim Westerlund & Jörg Breitung, 2013. "Lessons from a Decade of IPS and LLC," Econometric Reviews, Taylor & Francis Journals, vol. 32(5-6), pages 547-591, August.
    19. Sabzikar, Farzad & Wang, Qiying & Phillips, Peter C.B., 2020. "Asymptotic theory for near integrated processes driven by tempered linear processes," Journal of Econometrics, Elsevier, vol. 216(1), pages 192-202.
    20. Pierre Perron & Eduardo Zorita & Iliyan Georgiev & Paulo M. M. Rodrigues & A. M. Robert Taylor, 2017. "Unit Root Tests and Heavy-Tailed Innovations," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(5), pages 733-768, September.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:not:notgts:19/03. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/tsnotuk.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.