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An Extension of Cointegration to Fractional Autoregressive Processes

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  • Søren Johansen

    (Department of Economics, University of Copenhagen)

Abstract

This paper contains an overview of some recent results on the statistical analysis of cofractional processes, see Johansen and Nielsen (2010b). We first give an brief summary of the analysis of cointegration in the vector autoregressive model and then show how this can be extended to fractional processes. The model allows the process X_{t} to be fractional of order d and cofractional of order d-b≥0; that is, there exist vectors β for which β′X_{t} is fractional of order d-b. We analyse the Gaussian likelihood function to derive estimators and test statistics. The asymptotic properties are derived without the Gaussian assumption, under suitable moment conditions. We assume that the initial values are bounded and show that they do not influence the asymptotic analysis. The estimator of β is asymptotically mixed Gaussian and estimators of the remaining parameters are asymptotically Gaussian. The asymptotic distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion.

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Bibliographic Info

Paper provided by University of Copenhagen. Department of Economics in its series Discussion Papers with number 10-28.

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Length: 15 pages
Date of creation: Oct 2010
Date of revision:
Handle: RePEc:kud:kuiedp:1028

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Related research

Keywords: cofractional processes; cointegration rank; fractional cointegration; likelihood inference; vector autoregressive model;

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References

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  1. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March.
  2. Søren Johansen & Morten Ørregaard Nielsen, 2007. "Likelihood inference for a nonstationary fractional autoregressive model," CREATES Research Papers 2007-33, School of Economics and Management, University of Aarhus.
  3. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
  4. Katarzyna Lasak, 2008. "Likelihood based testing for no fractional cointegration," CREATES Research Papers 2008-52, School of Economics and Management, University of Aarhus.
  5. Javier Hualde & Peter M Robinson, 2003. "Cointegration in Fractional Systems with Unkown Integration Orders," STICERD - Econometrics Paper Series /2003/449, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  6. Juselius, Katarina, 2006. "The Cointegrated VAR Model: Methodology and Applications," OUP Catalogue, Oxford University Press, number 9780199285679.
  7. John Y. Campbell & Robert J. Shiller, 1986. "Cointegration and Tests of Present Value Models," Cowles Foundation Discussion Papers 785, Cowles Foundation for Research in Economics, Yale University.
  8. Jeganathan, P., 1999. "On Asymptotic Inference In Cointegrated Time Series With Fractionally Integrated Errors," Econometric Theory, Cambridge University Press, vol. 15(04), pages 583-621, August.
  9. Johansen, SØren, 2008. "A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes," Econometric Theory, Cambridge University Press, vol. 24(03), pages 651-676, June.
  10. Granger, Clive W J, 1986. "Developments in the Study of Cointegrated Economic Variables," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 48(3), pages 213-28, August.
  11. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-80, November.
  12. Katarzyna Lasak, 2008. "Maximum likelihood estimation of fractionally cointegrated systems," CREATES Research Papers 2008-53, School of Economics and Management, University of Aarhus.
  13. Johansen, Soren, 1995. "The Role of Ancillarity in Inference for Non-stationary Variables," Economic Journal, Royal Economic Society, vol. 105(429), pages 302-20, March.
  14. Hualde, J. & Robinson, P.M., 2010. "Semiparametric inference in multivariate fractionally cointegrated systems," Journal of Econometrics, Elsevier, vol. 157(2), pages 492-511, August.
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Cited by:
  1. Claudio Morana, 2014. "Factor Vector Autoregressive Estimation of Heteroskedastic Persistent and Non Persistent Processes Subject to Structural Breaks," Working Papers 273, University of Milano-Bicocca, Department of Economics, revised May 2014.
  2. Luis A. Gil-Alana & Guglielmo Maria Caporale, 2012. "Fractional Integration and Cointegration in US Financial Time Series Data," Faculty Working Papers 12/12, School of Economics and Business Administration, University of Navarra.

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