Generalized Reduced Rank Tests Using the Singular Value Decomposition

Tinbergen Institute Discussion Paper No. 2003-003/4

25 Pages Posted: 5 Feb 2003  

Frank R. Kleibergen

Brown University - Department of Economics

Richard Paap

Erasmus University Rotterdam (EUR) - Department of Econometrics; Tinbergen Institute; Erasmus Research Institute of Management (ERIM)

Date Written: January 2003

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.

Keywords: stochastic discount factor model, cointegration, GMM

JEL Classification: C12, C13, C30

Suggested Citation

Kleibergen, Frank R. and Paap, Richard, Generalized Reduced Rank Tests Using the Singular Value Decomposition (January 2003). Tinbergen Institute Discussion Paper No. 2003-003/4. Available at SSRN: https://ssrn.com/abstract=373420 or http://dx.doi.org/10.2139/ssrn.373420

Frank R. Kleibergen (Contact Author)

Brown University - Department of Economics ( email )

64 Waterman Street
Providence, RI 02912
United States

Richard Paap

Erasmus University Rotterdam (EUR) - Department of Econometrics ( email )

P.O. Box 1738
3000 DR Rotterdam
Netherlands

Tinbergen Institute ( email )

P.O. Box 1738
3000 DR Rotterdam
Netherlands

Erasmus Research Institute of Management (ERIM) ( email )

P.O. Box 1738
3000 DR Rotterdam
Netherlands

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