Advertisement

Journal of the Italian Statistical Society

, Volume 7, Issue 2, pp 111–127 | Cite as

An alternative proof of Granger’s Representation Theorem forI(1) systems through Jordan matrices

  • Fragiskos Archontakis
Article

Summary

The Jordan form of the VAR’s companion matrix is used for proving the equivalence between the statement that there are no Jordan blocks of order two or higher and the condition of Granger’s Representation Theorem for anI(1) series. Furthermore, a diagonal polynomial matrix containing the unit roots associated to the VAR system is derived and related to the Granger’s Representation Theorem.

Keywords

VAR Systems Integrated Series Granger’s Representation Jordan matrices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, T. W. (1971),The Statistical Analysis of Time Series. New York: John Wiley & Sons.MATHGoogle Scholar
  2. Banerjee, A., Dolado, J., Galbraith, J. W. andHendry, D. F. (1993),Co-integration, Error-Correction and the Econometric Analysis of Non-Stationary Data. Oxford: Oxford University Press.MATHCrossRefGoogle Scholar
  3. D’AUTUME, A. (1990), Cointegration of Higher Orders: AClarification.DELTA, Working Paper 90-22.Google Scholar
  4. Engle, R. F. andGranger, C. W. J. (1987), Cointegration and Error Correction: Representation, Estimation and Testing.Econometrica, 55, 251–271.MATHCrossRefMathSciNetGoogle Scholar
  5. Engle, R. F. andYoo, B. S. (1991), Cointegrated Economic Time Series: An overview with New Results. In Engle R. F. and Granger C. W. J. (eds.),Long-run economic relationships, readings in cointegration, 237–266. Oxford: Oxford University Press.Google Scholar
  6. Gantmacher, F. R. (1959),The Theory of Matrices, vols. I and II. New York: Chelsea.MATHGoogle Scholar
  7. Gohberg, I., Lancaster, P. andRodman, L. (1982),Matrix Polynomials. New York: Academic Press.MATHGoogle Scholar
  8. GRANGER, C. W. J. and LEE, T.-H. (1990), Multicointegration. In Rhodes G. F. and Fomby T. B. (eds.),Advances in econometrics: Cointegration, spurious regressions and unit roots, 71–84. JAI Press Inc.Google Scholar
  9. Gregoir, S. andLaroque, G. (1993), Multivariate Time Series: APolynomial Error Correction Theory,Econometric Theory, 9, 329–342.CrossRefMathSciNetGoogle Scholar
  10. Haldrup, N. andSalmon, M. (1998), Representations ofI(2) Cointegrated Systems using the Smith-McMillian Form.Journal of Econometrics, 84, 303–325.MATHCrossRefMathSciNetGoogle Scholar
  11. Hansen, P. R. andJohansen, S. (1998),Workbook on Cointegration. Oxford: Oxford University Press.Google Scholar
  12. Hendry, D. F. (1995),Dynamic Econometrics. Oxford: Oxford University Press.MATHCrossRefGoogle Scholar
  13. Hylleberg, S. andMizon, G. (1989), Cointegration and Error Correction Mechanisms.The Economic Journal, 99, 113–125.CrossRefGoogle Scholar
  14. Johansen, S. (1991), Estimation and Hypothesis Testing ofCointegration Vectors in Gaussian Vector Autoregressive Models.Econometrica, 59, 1551–1580.MATHCrossRefMathSciNetGoogle Scholar
  15. Johansen, S. (1992), A Representation of Vector Autoregressive Processes Integrated of Order 2.Econometric Theory, 8, 188–202.MathSciNetGoogle Scholar
  16. Johansen, S. (1996),Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, 2nd edition. Oxford: Oxford University Press.Google Scholar
  17. Kailath, T. (1980),Linear Systems. New York: Prentice-Hall, Englewood Cliffs.MATHGoogle Scholar

Copyright information

© Società Italiana di Statistica 1998

Authors and Affiliations

  1. 1.Department of EconomicsEuropean University InstituteFirenzeItaly

Personalised recommendations