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On rank estimation in symmetric matrices: the case of indefinite matrix estimators

Author

Listed:
  • Stephen G. Donald

    () (University of Texas at Austin)

  • Natércia Fortuna

    () (CEMPRE, Faculdade de Economia do Porto)

  • Vladas Pipiras

    () (University of North Carolina at Chapel Hill)

Abstract

We focus on the problem of rank estimation in an unknown symmetric matrix based on a symmetric, asymptotically normal estimator of the matrix. The related positive definite limit covariance matrix is assumed to be estimated consistently, and to have either a Kronecker product or an arbitrary structure. These assumptions are standard although they also exclude the case when the matrix estimator is positive or negative semidefinite. We adapt and reexamine here some available rank tests, and introduce a new rank test based on the eigenvalues of the matrix estimator. We discuss several applications where rank estimation in symmetric matrices is of interest, and also provide a small simulation study and an application.

Suggested Citation

  • Stephen G. Donald & Natércia Fortuna & Vladas Pipiras, 2005. "On rank estimation in symmetric matrices: the case of indefinite matrix estimators," FEP Working Papers 167, Universidade do Porto, Faculdade de Economia do Porto.
  • Handle: RePEc:por:fepwps:167
    as

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    File URL: http://www.fep.up.pt/investigacao/workingpapers/05.02.14_WP167_natercia.pdf
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    References listed on IDEAS

    as
    1. Kleibergen, Frank & Paap, Richard, 2006. "Generalized reduced rank tests using the singular value decomposition," Journal of Econometrics, Elsevier, vol. 133(1), pages 97-126, July.
    2. Fortuna, Natercia, 2008. "Local rank tests in a multivariate nonparametric relationship," Journal of Econometrics, Elsevier, vol. 142(1), pages 162-182, January.
    3. Stephen G. Donald, 1997. "Inference Concerning the Number of Factors in a Multivariate Nonparametric Relationship," Econometrica, Econometric Society, vol. 65(1), pages 103-132, January.
    4. Efstathia Bura & R. Dennis Cook, 2001. "Estimating the structural dimension of regressions via parametric inverse regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 393-410.
    5. 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.
    6. Robin, Jean-Marc & Smith, Richard J., 2000. "Tests Of Rank," Econometric Theory, Cambridge University Press, vol. 16(02), pages 151-175, April.
    7. Donkers, A.C.D. & Schafgans, M., 2003. "A derivative based estimator for semiparametric index models," Econometric Institute Research Papers EI 2003-08, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    8. Tatiana Kirsanova, 2001. "A Comparison of Personal Sector Saving Rates in the UK, US and Italy," National Institute of Economic and Social Research (NIESR) Discussion Papers 192, National Institute of Economic and Social Research.
    9. Cragg, John G. & Donald, Stephen G., 1997. "Inferring the rank of a matrix," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 223-250.
    10. 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.
    11. repec:cup:etheor:v:9:y:1993:i:2:p:222-40 is not listed on IDEAS
    12. Bura, Efstathia & Cook, R. Dennis, 2003. "Rank estimation in reduced-rank regression," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 159-176, October.
    13. Lewbel, Arthur, 1991. "The Rank of Demand Systems: Theory and Nonparametric Estimation," Econometrica, Econometric Society, vol. 59(3), pages 711-730, May.
    14. Zaka Ratsimalahelo, 2003. "Strongly Consistent Determination of the Rank of Matrix," Econometrics 0307007, EconWPA.
    15. Camba-Mendez, Gonzalo, et al, 2003. "Tests of Rank in Reduced Rank Regression Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(1), pages 145-155, January.
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    Cited by:

    1. Ahn, Seung C. & Perez, M. Fabricio, 2010. "GMM estimation of the number of latent factors: With application to international stock markets," Journal of Empirical Finance, Elsevier, vol. 17(4), pages 783-802, September.
    2. repec:eee:econom:v:199:y:2017:i:1:p:49-62 is not listed on IDEAS
    3. Majid M. Al-Sadoon, 2014. "A general theory of rank testing," Economics Working Papers 1411, Department of Economics and Business, Universitat Pompeu Fabra, revised Feb 2015.
    4. Martins Luis Filipe & Gabriel Vasco J., 2013. "Time-varying cointegration, identification, and cointegration spaces," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(2), pages 199-209, April.
    5. Jin, Fei & Lee, Lung-fei, 2013. "Cox-type tests for competing spatial autoregressive models with spatial autoregressive disturbances," Regional Science and Urban Economics, Elsevier, vol. 43(4), pages 590-616.

    More about this item

    Keywords

    rank; symmetric matrix; eigenvalues; matrix decompositions; estimation; asymptotic normality; consistency;

    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

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