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The asymptotic variance of the estimated roots in a cointegrated vector autoregressive model


  • Søren Johansen


We show that the asymptotic distribution of the estimated stationary roots in a vector autoregressive model is Gaussian. A simple expression for the asymptotic variance in terms of the roots and the eigenvectors of the companion matrix is derived. The results are extended to the cointegrated vector autoregressive model and we discuss the implementation of the results for complex roots. Copyright 2003 Blackwell Publishing Ltd.

Suggested Citation

  • Søren Johansen, 2003. "The asymptotic variance of the estimated roots in a cointegrated vector autoregressive model," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(6), pages 663-678, November.
  • Handle: RePEc:bla:jtsera:v:24:y:2003:i:6:p:663-678

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    References listed on IDEAS

    1. Hylleberg, S. & Engle, R. F. & Granger, C. W. J. & Yoo, B. S., 1990. "Seasonal integration and cointegration," Journal of Econometrics, Elsevier, vol. 44(1-2), pages 215-238.
    2. Ghysels, Eric & Lee, Hahn S. & Noh, Jaesum, 1994. "Testing for unit roots in seasonal time series : Some theoretical extensions and a Monte Carlo investigation," Journal of Econometrics, Elsevier, vol. 62(2), pages 415-442, June.
    3. Osborn, Denise R., 1993. "Seasonal cointegration," Journal of Econometrics, Elsevier, vol. 55(1-2), pages 299-303.
    4. Castro, Tomas del Barrio & Osborn, Denise R., 2008. "Testing For Seasonal Unit Roots In Periodic Integrated Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 24(04), pages 1093-1129, August.
    5. Hylleberg, Svend & Jorgensen, Clara & Sorensen, Nils Karl, 1993. "Seasonality in Macroeconomic Time Series," Empirical Economics, Springer, vol. 18(2), pages 321-335.
    6. Ghysels,Eric & Osborn,Denise R., 2001. "The Econometric Analysis of Seasonal Time Series," Cambridge Books, Cambridge University Press, number 9780521565882, March.
    7. Rodrigues, Paulo M.M. & Taylor, A.M. Robert, 2004. "Asymptotic Distributions For Regression-Based Seasonal Unit Root Test Statistics In A Near-Integrated Model," Econometric Theory, Cambridge University Press, vol. 20(04), pages 645-670, August.
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    9. Denise Osborn & Paulo Rodrigues, 2002. "Asymptotic Distributions Of Seasonal Unit Root Tests: A Unifying Approach," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 221-241.
    10. Smith, Richard J. & Taylor, A.M. Robert & del Barrio Castro, Tomas, 2009. "Regression-Based Seasonal Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 25(02), pages 527-560, April.
    11. Smith, Richard J. & Taylor, A. M. Robert, 1998. "Additional critical values and asymptotic representations for seasonal unit root tests," Journal of Econometrics, Elsevier, vol. 85(2), pages 269-288, August.
    12. Phillips, Peter C B & Ouliaris, S, 1990. "Asymptotic Properties of Residual Based Tests for Cointegration," Econometrica, Econometric Society, vol. 58(1), pages 165-193, January.
    13. Rodrigues, Paulo M. M. & Taylor, A. M. Robert, 2004. "Alternative estimators and unit root tests for seasonal autoregressive processes," Journal of Econometrics, Elsevier, vol. 120(1), pages 35-73, May.
    14. Taylor, A.M. Robert, 2003. "On The Asymptotic Properties Of Some Seasonal Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 19(02), pages 311-321, April.
    15. Burridge, Peter & Taylor, A M Robert, 2001. "On the Properties of Regression-Based Tests for Seasonal Unit Roots in the Presence of Higher-Order Serial Correlation," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(3), pages 374-379, July.
    16. Hansen, Bruce E., 1992. "Efficient estimation and testing of cointegrating vectors in the presence of deterministic trends," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 87-121.
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    Cited by:

    1. Onatski, Alexei & Uhlig, Harald, 2012. "Unit Roots In White Noise," Econometric Theory, Cambridge University Press, vol. 28(03), pages 485-508, June.
    2. Bent Nielsen & Heino Bohn Nielsen, 2008. "Properties of etimated characteristic roots," Economics Papers 2008-W07, Economics Group, Nuffield College, University of Oxford.
    3. Alain Hecq & Franz Palm & Jean-Pierre Urbain, 2002. "Separation, Weak Exogeneity, And P-T Decomposition In Cointegrated Var Systems With Common Features," Econometric Reviews, Taylor & Francis Journals, vol. 21(3), pages 273-307.
    4. Mauricio, Jose Alberto, 2006. "Exact maximum likelihood estimation of partially nonstationary vector ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 50(12), pages 3644-3662, August.

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