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Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models

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Abstract

This paper derives some exact finite sample distributions and characterizes the tail behavior of maximum likelihood estimators of the cointegrating coefficients in error correction models. It is shown that the reduced rank regression estimator has a distribution with Cauchy-like tails and no finite moments of integer order. The maximum likelihood estimator of the coefficients in a particular triangular system representation is studied and shown to have matrix t-distribution tails with finite integer moments to order T - n + r where T is the sample size, n is the total number of variables in the system and r is the dimension of the cointegration space. These results help to explain some recent simulation studies where extreme outliers are found to occur more frequently for the reduced rank regression estimator than for alternative asymptotically efficient procedures that are based on the triangular representation. In a simple triangular system, the Wald statistic for testing linear hypotheses about the columns of the cointegrating matrix is shown to have an F distribution, analogous to Hotelling's T^{2} distribution in multivariate linear regression.

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  • Peter C.B. Phillips, 1992. "Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models," Cowles Foundation Discussion Papers 1039, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1039
    Note: CFP 864.
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    References listed on IDEAS

    as
    1. Park, Joon Y, 1992. "Canonical Cointegrating Regressions," Econometrica, Econometric Society, vol. 60(1), pages 119-143, January.
    2. Phillips, Peter C B, 1985. "The Exact Distribution of LIML: II," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 21-36, February.
    3. repec:cup:etheor:v:7:y:1991:i:1:p:1-21 is not listed on IDEAS
    4. Phillips, P. C. B., 1989. "Spherical matrix distributions and cauchy quotients," Statistics & Probability Letters, Elsevier, vol. 8(1), pages 51-53, May.
    5. Peter C.B. Phillips, 1985. "Fractional Matrix Calculus and the Distribution of Multivariate Tests," Cowles Foundation Discussion Papers 767, Cowles Foundation for Research in Economics, Yale University.
    6. Peter C. B. Phillips & Mico Loretan, 1991. "Estimating Long-run Economic Equilibria," Review of Economic Studies, Oxford University Press, vol. 58(3), pages 407-436.
    7. Phillips, P C B, 1986. "The Distribution of FIML in the Leading Case," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 27(1), pages 239-243, February.
    8. Gregory, Allan W, 1994. "Testing for Cointegration in Linear Quadratic Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(3), pages 347-360, July.
    9. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers 872, Cowles Foundation for Research in Economics, Yale University.
    10. Phillips, P C B, 1988. "Reflections on Econometric Methodology," The Economic Record, The Economic Society of Australia, vol. 64(187), pages 344-359, December.
    11. Park, J.Y. & Ogaki, M., 1991. "Inference in Cointegrated Models Using VAR Prewhitening to Estimate Shortrun Dynamics," RCER Working Papers 281, University of Rochester - Center for Economic Research (RCER).
    12. Phillips, P C B, 1986. "The Exact Distribution of the Wald Statistic," Econometrica, Econometric Society, vol. 54(4), pages 881-895, July.
    13. Stock, James H & Watson, Mark W, 1993. "A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems," Econometrica, Econometric Society, vol. 61(4), pages 783-820, July.
    14. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    15. Constantine, A. G. & Muirhead, R. J., 1976. "Asymptotic expansions for distributions of latent roots in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 6(3), pages 369-391, September.
    16. Saikkonen, Pentti, 1991. "Asymptotically Efficient Estimation of Cointegration Regressions," Econometric Theory, Cambridge University Press, vol. 7(01), pages 1-21, March.
    17. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-1580, November.
    18. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
    19. Hiro Y. Toda & Peter C.B. Phillips, 1991. "Vector Autoregression and Causality: A Theoretical Overview and Simulation Study," Cowles Foundation Discussion Papers 1001, Cowles Foundation for Research in Economics, Yale University.
    20. Peter C.B. Phillips & Bruce E. Hansen, 1988. "Statistical Inference in Instrumental Variables," Cowles Foundation Discussion Papers 869R, Cowles Foundation for Research in Economics, Yale University, revised Apr 1989.
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