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The Properties Of Kullback–Leibler Divergence For The Unit Root Hypothesis


  • Marsh, Patrick


The fundamental contributions made by Paul Newbold have highlighted how crucial it is to detect when economic time series have unit roots. This paper explores the effects that model specification has on our ability to do that. Asymptotic power, a natural choice to quantify these effects, does not accurately predict finite-sample power. Instead, here the Kullback–Leibler divergence between the unit root null and any alternative is used and its numeric and analytic properties detailed. Numerically it behaves in a similar way to finite-sample power. However, because it is analytically available we are able to prove that it is a minimizable function of the degree of trending in any included deterministic component and of the correlation of the underlying innovations. It is explicitly confirmed, therefore, that it is approximately linear trends and negative unit root moving average innovations that minimize the efficacy of unit root inferential tools. Applied to the Nelson and Plosser macroeconomic series the effect that different types of trends included in the model have on unit root inference is clearly revealed.

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  • Marsh, Patrick, 2009. "The Properties Of Kullback–Leibler Divergence For The Unit Root Hypothesis," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1662-1681, December.
  • Handle: RePEc:cup:etheor:v:25:y:2009:i:06:p:1662-1681_99

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    References listed on IDEAS

    1. de Jong, Robert M. & Davidson, James, 2000. "The Functional Central Limit Theorem And Weak Convergence To Stochastic Integrals I," Econometric Theory, Cambridge University Press, vol. 16(05), pages 621-642, October.
    2. Davidson, James & Hashimzade, Nigar, 2008. "Alternative Frequency And Time Domain Versions Of Fractional Brownian Motion," Econometric Theory, Cambridge University Press, vol. 24(01), pages 256-293, February.
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    Cited by:

    1. Astill, Sam & Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2014. "Robust tests for a linear trend with an application to equity indices," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 168-185.
    2. Patrick Marsh, "undated". "Saddlepoint Approximations for Optimal Unit Root Tests," Discussion Papers 09/31, Department of Economics, University of York.

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