A unifying theory of tests of rank

The general principles underlying tests of matrix rank are investigated. It is demonstrated that statistics for such tests can be seen as implicit functions of null space estimators. In turn, the asymptotic behaviour of the null space estimators is shown to determine the asymptotic behaviour of the statistics through a plug-in principle. The theory simplifies the asymptotics under a variety of alternatives of empirical relevance as well as misspecification, clarifies the relationships between the various existing tests, makes use of important results in the numerical analysis literature, and motivates numerous new tests. A brief Monte Carlo study illustrates the results.