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ON RANK ESTIMATION IN SYMMETRIC MATRICES: THE CASE OF INDEFINITE MATRIX ESTIMATORS

Published online by Cambridge University Press:  06 September 2007

Stephen G. Donald ,
Natércia Fortuna  and
Vladas Pipiras
Stephen G. Donald
Affiliation:
University of Texas at Austin
Natércia Fortuna
Affiliation:
CEMPRE, Faculdade de Economia, Universidade do Porto
Vladas Pipiras
Affiliation:
University of North Carolina at Chapel Hill
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Abstract

In this paper we consider estimating the rank of 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 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 sum of eigenvalues of the matrix estimator. We discuss two applications where rank estimation in symmetric matrices is of interest, and we also provide a small simulation study.The first author acknowledges the support of an Alfred P. Sloan Foundation Research Fellowship and NSF Grant SES-0196372. We thank the co-editor and the two referees for useful comments and suggestions. CEMPRE—Centro de Estudos Macroeconómicos e Previsão—is supported by the Fundação para a Ciência e a Tecnologia, Portugal, through the Programa Operacional Ciência, Tecnologia e Inovação (POCTI) of the Quadro Comunitário de Apoio III, which is financed by FEDER and Portuguese funds.

Type
MISCELLANEA
Information
Econometric Theory , Volume 23 , Issue 6 , December 2007 , pp. 1217 - 1232
Copyright
© 2007 Cambridge University Press

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References

REFERENCES

Bunch, J.R. & B.N. Parlett (1971) Direct methods for solving symmetric indefinite systems of linear equations. SIAM Journal on Numerical Analysis 8, 639–655.Google Scholar
Cragg, J.G. & S.G. Donald (1993) Testing identifiability and specification in instrumental variable models. Econometric Theory 9, 222–240.Google Scholar
Cragg, J.G. & S.G. Donald (1996) On the asymptotic properties of LDU-based tests of the rank of a matrix. Journal of the American Statistical Association 91, 1301–1309.Google Scholar
Cragg, J.G. & S.G. Donald (1997) Inferring the rank of a matrix. Journal of Econometrics 76, 223–250.Google Scholar
Donald, S.G. (1997) Inference concerning the number of factors in a multivariate nonparametric relationship. Econometrica 65, 103–131.Google Scholar
Donald, S.G., N. Fortuna, & V. Pipiras (2005) On Rank Estimation in Symmetric Matrices: The Case of Indefinite Matrix Estimators. FEP Working paper 167, Faculdade de Economia do Porto, Porto, Portugal.
Donkers, B. & M. Schafgans (2003) A Derivative Based Estimator for Semiparametric Index Models. Econometric Institute Report 2003-08, Erasmus University Rotterdam, Rotterdam, the Netherlands.
Fortuna, N. (2004) Local Rank Tests in a Multivariate Nonparametric Relationship. FEP Working paper 137, Faculdade de Economia do Porto, Porto, Portugal.
Gill, L. & A. Lewbel (1992) Testing the rank and definiteness of estimated matrices with applications to factor, state-space and ARMA models. Journal of the American Statistical Association 87, 766–776.Google Scholar
Gouriéroux, C., A. Monfort, & A. Trognon (1985) Moindres carrés asymptotiques. Annales de l'I.N.S. E.E. 58, 91–122.Google Scholar
Kapetanios, G. & G. Camba-Mendez (1999) A Bootstrap Test of Cointegration Rank. NIESR Discussion paper 151, National Institute of Economic and Social Research, London, United Kingdom.
Kleibergen, F. & R. Paap (2006) Generalized reduced rank tests using the singular value decomposition. Journal of Econometrics 133, 97–126.Google Scholar
Kneip, A. (1994) Nonparametric estimation of common regressors for similar curve data. Annals of Statistics 22, 1386–1427.Google Scholar
Magnus, J.R. & H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley. Revised reprint of the 1988 original.
Ratsimalahelo, Z. (2002) Rank Test Based on Matrix Perturbation Theory. Preprint, U.F.R. Science Economique, University of Franche-Comté.
Ratsimalahelo, Z. (2003) Strongly Consistent Determination of the Rank of Matrix. Preprint, U.F.R. Science Economique, University of Franche-Comté.
Robin, J.-M. & R.J. Smith (1995) Tests of Rank. DAE Working paper 9521, Department of Applied Economics, University of Cambridge, Cambridge, United Kingdom.
Robin, J.-M. & R.J. Smith (2000) Tests of rank. Econometric Theory 16, 151–175.Google Scholar