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A class of Simple Semiparametrically Efficient Rank-Based Unit Root Tests

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Author Info
Marc Hallin
Ramon van den Akker
Bas Werker

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Abstract

We propose a class of simple rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference density g, which needs not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite sample size, is guaranteed by distribution-freeness, irrespective of the value of the drift and the actual underlying f. When based on a Gaussian reference density g, our tests (of the van der Waerden form) perform uniformly better, in terms of asymptotic relative effciency, than the Dickey and Fuller test --except under Gaussian f, where they are doing equally well. Under Student t3 density f, the effciency gain is as high as 110%, meaning that Dickey-Fuller requires over twice as many observations as we do in order to achieve comparable performance. This gain is even larger in case the underlying f has fatter tails; under Cauchy f, where Dickey and Fuller is no longer valid, it can be considered infinite. The test associated with reference density g is semiparametrically e±cient when f happens to coincide with g, in the ubiquitous case that the model contains a non-zero drift. Finally, with an estimated density f(n) substituted for the reference density g, our tests achieve uniform (with respect to f) semiparametric efficiency.

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Publisher Info
Paper provided by Université Libre de Bruxelles, Ecares in its series ECARES Working Papers with number 2009_001.

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Length: 23 pages
Date of creation: 2009
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Handle: RePEc:eca:wpaper:2009_001

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Related research
Keywords: Dickey-Fuller test; Local Asymptotic Normality;

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Find related papers by JEL classification:
C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Hypothesis Testing
C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions

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  1. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June. [Downloadable!] (restricted)
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    Other versions:
  3. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March. [Downloadable!] (restricted)
    Other versions:
  4. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March. [Downloadable!] (restricted)
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  5. Rothenberg, Thomas J. & Stock, James H., 1997. "Inference in a nearly integrated autoregressive model with nonnormal innovations," Journal of Econometrics, Elsevier, vol. 80(2), pages 269-286, October. [Downloadable!] (restricted)
  6. Michael Jansson, 2008. "Semiparametric Power Envelopes for Tests of the Unit Root Hypothesis," Econometrica, Econometric Society, vol. 76(5), pages 1103-1142, 09. [Downloadable!] (restricted)
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  7. 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. [Downloadable!] (restricted)
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  8. Breitung, Jorg & Gourieroux, Christian, 1997. "Rank tests for unit roots," Journal of Econometrics, Elsevier, vol. 81(1), pages 7-27, November. [Downloadable!] (restricted)
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  9. Michael Jansson & Marcelo J. Moreira, 2006. "Optimal Inference in Regression Models with Nearly Integrated Regressors," Econometrica, Econometric Society, vol. 74(3), pages 681-714, 05. [Downloadable!] (restricted)
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  10. Luger, Richard, 2003. "Exact non-parametric tests for a random walk with unknown drift under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 115(2), pages 259-276, August. [Downloadable!] (restricted)
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  11. Jeganathan, P., 1995. "Some Aspects of Asymptotic Theory with Applications to Time Series Models," Econometric Theory, Cambridge University Press, vol. 11(05), pages 818-887, October. [Downloadable!]
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  16. José Angel Roldán Casas & Rafaela Dios-Palomares, 2004. "A Strategy for Testing the Unit Root in AR(1) Model with Intercept. A Monte Carlo Experiment," Economic Working Papers at Centro de Estudios Andaluces E2004/37, Centro de Estudios Andaluces. [Downloadable!]
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  18. Campbell, Bryan & Dufour, Jean-Marie, 1997. "Exact Nonparametric Tests of Orthogonality and Random Walk in the Presence of a Drift Parameter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(1), pages 151-73, February.
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  19. Bhargava, Alok, 1986. "On the Theory of Testing for Unit Roots in Observed Time Series," Review of Economic Studies, Blackwell Publishing, vol. 53(3), pages 369-84, July. [Downloadable!] (restricted)
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