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Strongly Consistent Determination of the Rank of Matrix

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  • Zaka Ratsimalahelo
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    Abstract

    In this paper, we develop methods of the determination of the rank of random matrix. Using the matrix perturbation theory to construct or find a suitable bases of the kernel (null space) of the matrix and to determine the limiting distribution of the estimator of the smallest singular values. We propose a new rank test for an unobserved matrix for which a root-N-consistent estimator is available and construct a Wald- type test statistic (generalized Wald test). The test, based on matrix perturbation theory, enable to determine how many singular values of the estimated matrix are insignificantly different from zero and we fully characterise the asymptotic distribution of the generalized Wald statistic under the most general conditions. We show that it is chi- square distribution under the null. In particular case, when the asymptotic covariance matrix has a Kronecker product form, the test statistic is equivalent to likelihood ratio test statistic and to Multiplier Lagrange test statistic. Two approaches to be considered are sequential testing strategy and information theoretic criterion. We establish a strongly consistent of the determination of the rank of matrix using the two approaches.

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    Bibliographic Info

    Paper provided by Economics and Econometrics Research Institute (EERI), Brussels in its series EERI Research Paper Series with number EERI_RP_2003_04.

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    Length: 40 pages
    Date of creation: Jun 2003
    Date of revision:
    Handle: RePEc:eei:rpaper:eeri_rp_2003_04

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    Related research

    Keywords: Rank Testing; Matrix Perturbation Theory; Rank Estimation; Subspace Methods; Singular Value Decomposition; Weighting Matrices; Sequential Testing Strategy; Information Theoretic Criterion.;

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    References

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    1. Hall, Alastair R & Rudebusch, Glenn D & Wilcox, David W, 1996. "Judging Instrument Relevance in Instrumental Variables Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 37(2), pages 283-98, May.
    2. Lutkepohl, Helmut & Burda, Maike M., 1997. "Modified Wald tests under nonregular conditions," Journal of Econometrics, Elsevier, vol. 78(2), pages 315-332, June.
    3. Cragg, John G. & Donald, Stephen G., 1997. "Inferring the rank of a matrix," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 223-250.
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    6. Anderson, T. W. & Kunitomo, Naoto, 1994. "Asymptotic robustness of tests of overidentification and predeterminedness," Journal of Econometrics, Elsevier, vol. 62(2), pages 383-414, June.
    7. Robinson, P. M., 1973. "Generalized canonical analysis for time series," Journal of Multivariate Analysis, Elsevier, vol. 3(2), pages 141-160, June.
    8. Andrews, Donald W. K., 1987. "Asymptotic Results for Generalized Wald Tests," Econometric Theory, Cambridge University Press, vol. 3(03), pages 348-358, June.
    9. Robin, J.M. & Smith, R.J., 1995. "Tests of Rank," Cambridge Working Papers in Economics 9521, Faculty of Economics, University of Cambridge.
    10. Andrews, Donald W K, 1988. "Chi-Square Diagnostic Tests for Econometric Models: Theory," Econometrica, Econometric Society, vol. 56(6), pages 1419-53, November.
    11. Rao, C. Radhakrishna, 1979. "Separation theorems for singular values of matrices and their applications in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 9(3), pages 362-377, September.
    12. Lwebel Arthur & Perraudin William, 1995. "A Theorem on Portfolio Separation with General Preferences," Journal of Economic Theory, Elsevier, vol. 65(2), pages 624-626, April.
    13. Sin, Chor-Yiu & White, Halbert, 1996. "Information criteria for selecting possibly misspecified parametric models," Journal of Econometrics, Elsevier, vol. 71(1-2), pages 207-225.
    14. Lewbel, Arthur, 1991. "The Rank of Demand Systems: Theory and Nonparametric Estimation," Econometrica, Econometric Society, vol. 59(3), pages 711-30, May.
    15. Vuong, Quang H., 1987. "Generalized inverses and asymptotic properties of Wald tests," Economics Letters, Elsevier, vol. 24(4), pages 343-347.
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    Cited by:
    1. Donald, Stephen G. & Fortuna, Nat rcia & Pipiras, Vladas, 2007. "On Rank Estimation In Symmetric Matrices: The Case Of Indefinite Matrix Estimators," Econometric Theory, Cambridge University Press, vol. 23(06), pages 1217-1232, December.

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