Statistical Tests and Estimators of the Rank of a Matrix and Their Applications in Econometric Modelling

Testing and estimating the rank of a matrix of estimated parameters is key in a large variety of econometric modelling scenarios. This article describes general methods to test for and estimate the rank of a matrix, and provides details on a variety of modelling scenarios in the econometrics literature where such methods are required. Four different methods to test for the true rank of a general matrix are described, as well as one method that can handle the case of a matrix subject to parameter constraints associated with defineteness structures. The technical requirements for the implementation of the tests of rank of a general matrix differ and hence there are merits to all of them that justify their use in applied work. Nonetheless, we review available evidence of their small sample properties in the context of different modelling scenarios where all, or some, are applicable.