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Generalized Reduced Rank Tests using the Singular Value Decomposition

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  • Kleibergen, F.R.
  • Paap, R.

Abstract

We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: necessity of a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson (1951), sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald (1996) and Gill and Lewbel (1992), a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith (2000) and usage of numerical optimization for the objective function statistic of Cragg and Donald (1997). The new rank statistic consists of a quadratic form of a (orthogonal) transformation of the smallest singular values of a unrestricted estimate of the matrix of interest. The quadratic form is taken with respect to the inverse of a unrestricted covariance matrix that can be estimated using a heteroscedasticity autocorrelation consistent estimator. The rank statistic has a standard chi squared limiting distribution. In case of a Kronecker covariance matrix, the rank statistic simplifies to the canonical correlation rank statistic. In the non-stationary cointegration case, the limiting distribution of the rank statistic is identical to that of the Johansen trace statistic. We apply the rank statistic to test for the rank of a matrix that governs the identification of the parameters in the stochastic discount factor model of Jagannathan and Wang (1996). The rank statistic shows that non-identification of the parameters can not be rejected. We further use the stochastic discount factor model to illustrate the validity of the limiting distribution and to conduct a power comparison.

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Bibliographic Info

Paper provided by Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute in its series Econometric Institute Research Papers with number EI 2003-01.

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Date of creation: 17 Feb 2003
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Handle: RePEc:ems:eureir:1681

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Keywords: GMM; cointegration; stochastic discount factor model;

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  1. West, Kenneth D., 1997. "Another heteroskedasticity- and autocorrelation-consistent covariance matrix estimator," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 171-191.
  2. Pentti Saikkonen, 1999. "Testing normalization and overidentification of cointegrating vectors in vector autoregressive processes," Econometric Reviews, Taylor & Francis Journals, vol. 18(3), pages 235-257.
  3. Jagannathan, Ravi & Wang, Zhenyu, 1996. " The Conditional CAPM and the Cross-Section of Expected Returns," Journal of Finance, American Finance Association, vol. 51(1), pages 3-53, March.
  4. Robin, Jean-Marc & Smith, Richard J., 2000. "Tests Of Rank," Econometric Theory, Cambridge University Press, vol. 16(02), pages 151-175, April.
  5. Cragg, John G. & Donald, Stephen G., 1993. "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory, Cambridge University Press, vol. 9(02), pages 222-240, April.
  6. Andrews, Donald W. K., 1987. "Asymptotic Results for Generalized Wald Tests," Econometric Theory, Cambridge University Press, vol. 3(03), pages 348-358, June.
  7. Cragg, John G. & Donald, Stephen G., 1997. "Inferring the rank of a matrix," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 223-250.
  8. Kleibergen, F.R. & van Dijk, H.K., 1997. "Bayesian Simultaneous Equations Analysis using Reduced Rank Structures," Econometric Institute Research Papers EI 9714/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  9. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-54, July.
  10. Johansen, Soren, 1995. "Likelihood-Based Inference in Cointegrated Vector Autoregressive Models," OUP Catalogue, Oxford University Press, number 9780198774501, September.
  11. Whitney K. Newey & Kenneth D. West, 1986. "A Simple, Positive Semi-Definite, Heteroskedasticity and AutocorrelationConsistent Covariance Matrix," NBER Technical Working Papers 0055, National Bureau of Economic Research, Inc.
  12. Lewbel, Arthur, 1991. "The Rank of Demand Systems: Theory and Nonparametric Estimation," Econometrica, Econometric Society, vol. 59(3), pages 711-30, May.
  13. 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.
  14. Wright, Jonathan H., 2003. "Detecting Lack Of Identification In Gmm," Econometric Theory, Cambridge University Press, vol. 19(02), pages 322-330, April.
  15. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-58, May.
  16. Jagannathan, Ravi & Skoulakis, Georgios & Wang, Zhenyu, 2002. "Generalized Method of Moments: Applications in Finance," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(4), pages 470-81, October.
  17. Engle, Robert F & Granger, Clive W J, 1987. "Co-integration and Error Correction: Representation, Estimation, and Testing," Econometrica, Econometric Society, vol. 55(2), pages 251-76, March.
  18. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
  19. Kleibergen, Frank & Paap, Richard, 2002. "Priors, posteriors and bayes factors for a Bayesian analysis of cointegration," Journal of Econometrics, Elsevier, vol. 111(2), pages 223-249, December.
  20. Kleibergen, Frank & van Dijk, Herman K., 1994. "Direct cointegration testing in error correction models," Journal of Econometrics, Elsevier, vol. 63(1), pages 61-103, July.
  21. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245 Elsevier.
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