Improved Inference on the Rank of a Matrix

This paper develops a general framework for conducting inference on the rank of an unknown matrix $\Pi_0$. A defining feature of our setup is the null hypothesis of the form $\mathrm H_0: \mathrm{rank}(\Pi_0)\le r$. The problem is of first order importance because the previous literature focuses on $\mathrm H_0': \mathrm{rank}(\Pi_0)= r$ by implicitly assuming away $\mathrm{rank}(\Pi_0)<r$, which may lead to invalid rank tests due to over-rejections. In particular, we show that limiting distributions of test statistics under $\mathrm H_0'$ may not stochastically dominate those under $\mathrm{rank}(\Pi_0)<r$. A multiple test on the nulls $\mathrm{rank}(\Pi_0)=0,\ldots,r$, though valid, may be substantially conservative. We employ a testing statistic whose limiting distributions under $\mathrm H_0$ are highly nonstandard due to the inherent irregular natures of the problem, and then construct bootstrap critical values that deliver size control and improved power. Since our procedure relies on a tuning parameter, a two-step procedure is designed to mitigate concerns on this nuisance. We additionally argue that our setup is also important for estimation. We illustrate the empirical relevance of our results through testing identification in linear IV models that allows for clustered data and inference on sorting dimensions in a two-sided matching model with transferrable utility.