Tests of rank

This paper considers tests for the rank of a matrix for which a root-T cons istent estimator is available. However, in contrast to tests associated wit h the minimum chi-square and asymptotic least squares principles, the estim ator's asymptotic variance matrix is not required to be either full or of k nown rank. Test statistics based on certain estimated characteristic roots are proposed whose limiting distributions are a weighted sum of independent chi-squared variables. These weights may be simply estimated, yielding con venient estimators for the limiting distributions of the proposed statistic s. A sequential testing procedure is presented that yields a consistent est imator for the rank of a matrix. A simulation experiment is conducted compa ring the characteristic root statistics advocated in this paper with statis tics based on the Wald and asymptotic least squares principles.