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Strongly Consistent Determination of the Rank of Matrix

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  • Zaka Ratsimalahelo

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.

Suggested Citation

  • Zaka Ratsimalahelo, 2003. "Strongly Consistent Determination of the Rank of Matrix," EERI Research Paper Series EERI_RP_2003_04, Economics and Econometrics Research Institute (EERI), Brussels.
    • Handle: RePEc:eei:rpaper:eeri_rp_2003_04
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    Cited by:

    1. Marcos A. Rangel & Duncan Thomas, 2019. "Decision-Making in Complex Households," Working Papers 2019-070, Human Capital and Economic Opportunity Working Group.
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    3. Thomas, Duncan & Rangel, Marcos, 2020. "Decision-Making in Complex Households," CEPR Discussion Papers 14278, C.E.P.R. Discussion Papers.
    4. Donald, Stephen G. & Fortuna, Natércia & Pipiras, Vladas, 2007. "On Rank Estimation In Symmetric Matrices: The Case Of Indefinite Matrix Estimators," Econometric Theory, Cambridge University Press, vol. 23(6), pages 1217-1232, December.

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    More about this item

    Keywords

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

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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