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.
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References
Anderson, T. W. (1971),The Statistical Analysis of Time Series. New York: John Wiley & Sons.
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.
D’AUTUME, A. (1990), Cointegration of Higher Orders: AClarification.DELTA, Working Paper 90-22.
Engle, R. F. andGranger, C. W. J. (1987), Cointegration and Error Correction: Representation, Estimation and Testing.Econometrica, 55, 251–271.
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.
Gantmacher, F. R. (1959),The Theory of Matrices, vols. I and II. New York: Chelsea.
Gohberg, I., Lancaster, P. andRodman, L. (1982),Matrix Polynomials. New York: Academic Press.
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.
Gregoir, S. andLaroque, G. (1993), Multivariate Time Series: APolynomial Error Correction Theory,Econometric Theory, 9, 329–342.
Haldrup, N. andSalmon, M. (1998), Representations ofI(2) Cointegrated Systems using the Smith-McMillian Form.Journal of Econometrics, 84, 303–325.
Hansen, P. R. andJohansen, S. (1998),Workbook on Cointegration. Oxford: Oxford University Press.
Hendry, D. F. (1995),Dynamic Econometrics. Oxford: Oxford University Press.
Hylleberg, S. andMizon, G. (1989), Cointegration and Error Correction Mechanisms.The Economic Journal, 99, 113–125.
Johansen, S. (1991), Estimation and Hypothesis Testing ofCointegration Vectors in Gaussian Vector Autoregressive Models.Econometrica, 59, 1551–1580.
Johansen, S. (1992), A Representation of Vector Autoregressive Processes Integrated of Order 2.Econometric Theory, 8, 188–202.
Johansen, S. (1996),Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, 2nd edition. Oxford: Oxford University Press.
Kailath, T. (1980),Linear Systems. New York: Prentice-Hall, Englewood Cliffs.
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This work is part of my Ph.D. thesis under the guidance of Søren Johansen. His comments along with those of the two anonymous referees have greatly improved the content and the presentation of the paper.
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Archontakis, F. An alternative proof of Granger’s Representation Theorem forI(1) systems through Jordan matrices. J. Ital. Statist. Soc. 7, 111–127 (1998). https://doi.org/10.1007/BF03178924
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DOI: https://doi.org/10.1007/BF03178924