Rank Test Based Matrix Perturbation Theory

Establishing the rank of a matrix is an important problem in a wide variety of econometric and statistical contexts that includes the classical identification problem in linear simultaneous equation models, determining the rank of demand systems, the order of lag polynomials in ARMA models, and the composition of factor models. In an important paper, Gill and Lewbel (1992) introduce a rank test based on the Lower-Diagonal-Upper triangular (LDU) decomposition. Unfortunately, the asymptotic distribution given for this test statistic is incorrect except in a limited special case. Cragg and Donald (1996) provides an appropriate modification of the LDU decomposition.This paper proposes another means for determining the rank of a matrix. We consider a rank test for a matrix for which a root-n consistent estimator is available whose limiting (normal) distribution's variance matrix may be of either unknown or less than full rank. The test statistic is based on certain estimated singular values. Using Matrix Perturbation Theory, we determine that the asymptotic distribution of the test statistic is chi-squared. The test has certain advantages over standard tests in being easier to compute and not requiring knowledge of the rank of the limiting variance matrix of the root-n consistent estimator. Simulation evidence is given for this test and its performance is compared with that of the LDU test modified.