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

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Author Info
Peter C.B. Phillips () (Cowles Foundation, Yale University)

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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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Publisher Info
Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 1039.

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Length: 28 pages
Date of creation: Nov 1992
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Publication status: Published in Econometrica, 62(1), 1994
Handle: RePEc:cwl:cwldpp:1039

Note: CFP 864.
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Hypergeometric function invariant measure matrix Cauchy maximum likelihood orthogonal group reduced rank regression spherical distribution Stiefel manifold triangular system

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Park, Joon Y, 1992. "Canonical Cointegrating Regressions," Econometrica, Econometric Society, vol. 60(1), pages 119-43, January. [Downloadable!] (restricted)
  2. 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 239-43, February. [Downloadable!] (restricted)
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  3. 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).
  4. Peter C.B. Phillips & Mico Loretan, 1989. "Estimating Long Run Economic Equilibria," Cowles Foundation Discussion Papers 928, Cowles Foundation, Yale University. [Downloadable!]
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  5. Phillips, P C B, 1986. "The Exact Distribution of the Wald Statistic," Econometrica, Econometric Society, vol. 54(4), pages 881-95, July. [Downloadable!] (restricted)
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  6. Phillips, Peter C B, 1984. "The Exact Distribution of LIML: I," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(1), pages 249-61, February. [Downloadable!] (restricted)
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  7. repec:cup:etheor:v:7:y:1991:i:1:p:1-21 is not listed on IDEAS
  8. James H. Stock & Mark W. Watson, 1991. "A simple estimator of cointegrating vectors in higher order integrated systems," Working Paper Series, Macroeconomic Issues 91-3, Federal Reserve Bank of Chicago.
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  9. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December. [Downloadable!] (restricted)
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  10. Allan W. Gregory, 1991. "Testing for Cointegration in Linear Quadratic Models," Working Papers 811, Queen's University, Department of Economics.
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  11. Phillips, P C B, 1988. "Reflections on Econometric Methodology," The Economic Record, The Economic Society of Australia, vol. 64(187), pages 344-59, December.
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  12. Peter C.B. Phillips, 1988. "Spectral Regression for Cointegrated Time Series," Cowles Foundation Discussion Papers 872, Cowles Foundation, Yale University. [Downloadable!]
  13. 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, Yale University. [Downloadable!]
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  14. Johansen, Soren, 1991. "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models," Econometrica, Econometric Society, vol. 59(6), pages 1551-80, November. [Downloadable!] (restricted)
  15. Peter C.B. Phillips, 1987. "Spherical Matrix Distributions and Cauchy Quotients," Cowles Foundation Discussion Papers 823, Cowles Foundation, Yale University. [Downloadable!]
  16. Peter C.B. Phillips & Bruce E. Hansen, 1988. "Statistical Inference in Instrumental Variables," Cowles Foundation Discussion Papers 869R, Cowles Foundation, Yale University, revised Apr 1989. [Downloadable!]
  17. Peter C.B. Phillips, 1985. "Fractional Matrix Calculus and the Distribution of Multivariate Tests," Cowles Foundation Discussion Papers 767, Cowles Foundation, Yale University. [Downloadable!]
  18. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Bernd Hayo, 2004. "Review of PcGive 10," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 19(4), pages 525-531.
  2. Cubadda, Gianluca & Omtzigt, Pieter, 2003. "Small Sample Improvements in the Statistical Analysis of Seasonally Cointegrated Systems," Economics & Statistics Discussion Papers esdp03012, University of Molise, Dept. SEGeS. [Downloadable!]
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  3. Bernd Hayo, 1999. "The Demand For Money In Austria," Macroeconomics 9902012, EconWPA. [Downloadable!]
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  4. D. Nautz, . "Die empirische Relevanz des Monetären Modells für die Erklärung des DM/Dollar Wechselkurses," Sonderforschungsbereich 373 1999-63, Humboldt Universitaet Berlin.
  5. Breitung, J. & Pesaran, M.H., 2005. "Unit Roots and Cointegration in Panels," Cambridge Working Papers in Economics 0535, Faculty of Economics, University of Cambridge. [Downloadable!]
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  6. Tilak Abeysinghe & Keen Meng Choy, 2005. "Modelling Small Economy Exports: The Case of Singapore," SCAPE Policy Research Working Paper Series 0501, National University of Singapore, Department of Economics, SCAPE. [Downloadable!]
  7. Anders Warne, 2006. "Bayesian inference in cointegrated VAR models - with applications to the demand for euro area M3," Working Paper Series 692, European Central Bank. [Downloadable!]
  8. Kim, In-Moo & Park, Joon Y., 2005. "Iterative Maximum Likelihood Estimation of Cointegrating Vectors," Working Papers 2005-02, Rice University, Department of Economics. [Downloadable!]
  9. John C. Chao & Peter C.B. Phillips, 1998. "Jeffreys Prior Analysis of the Simultaneous Equations Model in the Case with n+1 Endogenous Variables," Cowles Foundation Discussion Papers 1198, Cowles Foundation, Yale University. [Downloadable!]
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  10. Kleibergen, F., 1996. "Reduced rank regression using generalized method of moments estimators," Discussion Paper 20, Tilburg University, Center for Economic Research. [Downloadable!]
  11. Gary Koop & Rodney Strachan & Herman van Dijk & Mattias Villani, 2004. "Bayesian Approaches to Cointegration," Discussion Papers in Economics 04/27, Department of Economics, University of Leicester. [Downloadable!]
  12. James D. Hamilton & Daniel F. Waggoner & Tao Zha, 2004. "Normalization in econometrics," Working Paper 2004-13, Federal Reserve Bank of Atlanta. [Downloadable!]
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