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Co-integration rank tests under conditional heteroskedasticity

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  • Giuseppe Cavaliere
  • Anders Rahbek
  • A. M. Robert Taylor

Abstract

In this paper we analyse the properties of the conventional Gaussian-based co-integrating rank tests of Johansen (1996) in the case where the vector of series under test is driven by possibly non-stationary, conditionally heteroskedastic (martingale difference) innovations. We first demonstrate that the limiting null distributions of the rank statistics coincide with those derived by previous authors who assume either i.i.d. or stationary martingale difference innovations. We then propose wild bootstrap implementations of the co-integrating rank tests and demonstrate that the associated bootstrap rank statistics replicate the first- order asymptotic null distributions of the rank statistics. We show that the same is also true of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the re-sampled data the pattern of heteroskedasticity present in the original shocks. Consistent with this, numerical evidence suggests that, relative to tests based on the asymptotic critical values or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small samples under a variety of conditionally heteroskedastic innovation processes. An empirical application to the term structure of interest rates is also given.

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Bibliographic Info

Paper provided by University of Nottingham, Granger Centre for Time Series Econometrics in its series Discussion Papers with number 09/02.

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Date of creation: Mar 2009
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Handle: RePEc:not:notgts:09/02

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Web page: http://www.nottingham.ac.uk/economics/grangercentre/
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Keywords: Co-integration; trace and maximum eigenvalue rank tests; conditional heteroskedasticity; IID bootstrap; wild bootstrap;

References

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  1. Park, Joon, 2002. "Bootstrap Unit Root Tests," Working Papers 2003-04, Rice University, Department of Economics.
  2. Davidson, Russell & Flachaire, Emmanuel, 2008. "The wild bootstrap, tamed at last," Journal of Econometrics, Elsevier, Elsevier, vol. 146(1), pages 162-169, September.
  3. Kristensen, Dennis & Rahbek, Anders, 2005. "ASYMPTOTICS OF THE QMLE FOR A CLASS OF ARCH(q) MODELS," Econometric Theory, Cambridge University Press, vol. 21(05), pages 946-961, October.
  4. Gonçalves, Sílvia & Kilian, Lutz, 2002. "Bootstrapping autoregressions with conditional heteroskedasticity of unknown form," Working Paper Series, European Central Bank 0196, European Central Bank.
  5. Sílvia Gonçalves & Lutz Kilian, 2003. "Asymptotic and Bootstrap Inference for AR( Infinite ) Processes with Conditional Heteroskedasticity," CIRANO Working Papers 2003s-28, CIRANO.
  6. Bruce E. Hansen, 2000. "Sample Splitting and Threshold Estimation," Econometrica, Econometric Society, Econometric Society, vol. 68(3), pages 575-604, May.
  7. Trenkler, Carsten, 2009. "Bootstrapping Systems Cointegration Tests With A Prior Adjustment For Deterministic Terms," Econometric Theory, Cambridge University Press, vol. 25(01), pages 243-269, February.
  8. Hansen, Bruce E., 2000. "Testing for structural change in conditional models," Journal of Econometrics, Elsevier, Elsevier, vol. 97(1), pages 93-115, July.
  9. Hall, Anthony D & Anderson, Heather M & Granger, Clive W J, 1992. "A Cointegration Analysis of Treasury Bill Yields," The Review of Economics and Statistics, MIT Press, vol. 74(1), pages 116-26, February.
  10. Silvia Goncalves & Lutz Kilian, 2007. "Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity," Econometric Reviews, Taylor & Francis Journals, Taylor & Francis Journals, vol. 26(6), pages 609-641.
  11. Giese, Julia V., 2008. "Level, Slope, Curvature: Characterising the Yield Curve in a Cointegrated VAR Model," Economics Discussion Papers 2008-13, Kiel Institute for the World Economy.
  12. Burridge, P. & Taylor, A.M.R., 1999. "On Regression-Based Tests for Seasonal Unit Roots in the Presence of Periodic Heteroscedasticity," Discussion Papers, Department of Economics, University of Birmingham 99-10, Department of Economics, University of Birmingham.
  13. Giuseppe Cavaliere & Anders Rahbek & A.M.Robert Taylor, 2008. "Testing for Co-integration in Vector Autoregressions with Non-Stationary Volatility," CREATES Research Papers 2008-50, School of Economics and Management, University of Aarhus.
  14. James G. MacKinnon & Alfred A. Haug & Leo Michelis, 1996. "Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration," Working Papers, York University, Department of Economics 1996_07, York University, Department of Economics.
  15. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, Econometric Society, vol. 64(2), pages 413-30, March.
  16. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 347-70, March.
  17. Hansen, Bruce E., 1992. "Convergence to Stochastic Integrals for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 8(04), pages 489-500, December.
  18. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2008. "Bootstrap Unit Root Tests For Time Series With Nonstationary Volatility," Econometric Theory, Cambridge University Press, vol. 24(01), pages 43-71, February.
  19. Nielsen, Bent & Rahbek, Anders, 2000. " Similarity Issues in Cointegration Analysis," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 62(1), pages 5-22, February.
  20. Efstathios Paparoditis & Dimitris N. Politis, 2003. "Residual-Based Block Bootstrap for Unit Root Testing," Econometrica, Econometric Society, Econometric Society, vol. 71(3), pages 813-855, 05.
  21. Atsushi Inoue & Lutz Kilian, 2002. "Bootstrapping Autoregressive Processes with Possible Unit Roots," Econometrica, Econometric Society, Econometric Society, vol. 70(1), pages 377-391, January.
  22. Harris, R. I. D. & Judge, G., 1998. "Small sample testing for cointegration using the bootstrap approach," Economics Letters, Elsevier, vol. 58(1), pages 31-37, January.
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