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A General Test for the Cointegrating Rank in Vector Autoregressive Models


  • Ahlgren, Niklas

    () (Swedish School of Economics and Business Administration)

  • Nyblom, Jukka

    () (University of Joensuu)


The article proposes a general test for the cointegrating rank in vector autoregressive models. The test is based on the eigenvalues of the companion matrix, more precisely on the sum of the real parts of those closest to one. The roots of the companion matrix are often inspected as a diagnostic tool. Here this practice is elevated to the level of a formal test. The asymptotic distribution of the test statistic is derived and tabulated by simulation. The new test is compared with the likelihood ratio test. Neither one of the tests is dominating the other over the entire parameter space. The new test safeguards the researcher against making spurious inferences on the cointegrating rank in the presence of explosive roots and of integration of higher order than one. A numerical illustration is given. It is shown that the limiting distribution of the eigenvalues closest to one of the companion matrix is free of nuisance parameters, a useful result in its own right. When applied to a univariate autoregressive model, the augmented Dickey—Fuller test is obtained. The new test can be regarded as the multivariate version of the Dickey—Fuller coefficent test of a unit root. When there is a single common trend, the suggested test statistic has the same asymptotic distribution as the least squares estimate of the autoregressive coeficent in a univariate first order autoregressive model with a unit root. In the numerical application the test is applied to Swedish money demand data, and it is shown how to determine the cointegrating rank using the new test. The results from a small sample simulation experiment mimicking the actual data used in the application are also discussed.

Suggested Citation

  • Ahlgren, Niklas & Nyblom, Jukka, 2003. "A General Test for the Cointegrating Rank in Vector Autoregressive Models," Working Papers 499, Hanken School of Economics.
  • Handle: RePEc:hhb:hanken:0499 Note: This paper is published as: Ahlgren, Niklas and Nyblom, Jukka (2008), 'Tests against Stationary and Explosive Alternatives in Vector Autoregressive Models', Journal of Time Series Analysis, 29, 421-443.

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    References listed on IDEAS

    1. Saikkonen, Pentti & L tkepohl, Helmut, 2000. "Testing For The Cointegrating Rank Of A Var Process With An Intercept," Econometric Theory, Cambridge University Press, vol. 16(03), pages 373-406, June.
    2. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    3. repec:cup:etheor:v:13:y:1997:i:4:p:529-57 is not listed on IDEAS
    4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    5. Kirstin Hubrich & Helmut Lutkepohl & Pentti Saikkonen, 2001. "A Review Of Systems Cointegration Tests," Econometric Reviews, Taylor & Francis Journals, vol. 20(3), pages 247-318.
    6. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-144, January.
    7. Harris, David, 1997. "Principal Components Analysis of Cointegrated Time Series," Econometric Theory, Cambridge University Press, vol. 13(04), pages 529-557, August.
    8. Gonzalo, Jesus & Lee, Tae-Hwy, 1998. "Pitfalls in testing for long run relationships," Journal of Econometrics, Elsevier, vol. 86(1), pages 129-154, June.
    9. repec:cup:etheor:v:11:y:1995:i:5:p:1015-32 is not listed on IDEAS
    10. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    11. Phillips, P. C. B. & Ouliaris, S., 1988. "Testing for cointegration using principal components methods," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 205-230.
    12. Toda, Hiro Y., 1995. "Finite Sample Performance of Likelihood Ratio Tests for Cointegrating Ranks in Vector Autoregressions," Econometric Theory, Cambridge University Press, vol. 11(05), pages 1015-1032, October.
    13. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
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    More about this item


    Common trends; Companion matrix; Eigenvalue test; Integration of order 2; Rank determination; Unit roots;

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