Improved inference on the rank of a matrix

Abstract

This paper develops a general framework for conducting inference on the rank of an unknown matrix Π0. A defining feature of our setup is the null hypothesis of the form normalH0:normalrankfalse(Π0false)≤r. The problem is of first‐order importance because the previous literature focuses on normalH0′:normalrankfalse(Π0false)=r by implicitly assuming away normalrankfalse(Π0false)<r, which may lead to invalid rank tests due to overrejections. In particular, we show that limiting distributions of test statistics under normalH0′ may not stochastically dominate those under normalrankfalse(Π0false)<r. A multiple test on the nulls normalrankfalse(Π0false)=0,…,r, though valid, may be substantially conservative. We employ a testing statistic whose limiting distributions under normalH0 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.

A Projection Framework for Testing Shape Restrictions That Form Convex Cones

“…Analogous rate conditions on γ n have appeared in parametric settings-see Chen and Fang (2019) and references therein. In nonparametric settings, the practice of obtaining data-driven tuning parameters through quantile estimation has appeared in Chernozhukov, Lee, and Rosen (2013), Chernozhukov, Newey, and Santos (2015), and Fang and Santos (2019), though a formal theory seems to be lacking.…”

“…However, discretization entails an extra tuning parameter whose choice may be a delicate matter. Moreover, Dette, Hoderlein, and Neumeyer (2016)'s test, as the authors noted, is in general conservative, while validity of Hoderlein (2011)'s test has not been formally proven-see Chen and Fang (2019) for the challenges involved in a related but different problem. EXAMPLE B.4-Supermodularity: Let d ≥ 2 and ⊂ H be the set of supermodular functions, that is, f ∈ if and only if, for any y z…”

“…Alternatively, one may employ a numerical estimator following Hong and Li (2018), but there are no data-driven procedures to date for selecting the step size (needed to carry out the numerical differentiation). This raises substantive concerns because the resulting bootstrap may be sensitive to the choice of the step size, as documented in Masten and Poirier (2021) and Chen and Fang (2019).…”

“…One may also appeal to the m out of n bootstrap or subsampling, but the choice of the sub-sample size may be difficult, among other issues-see Remark 3.1 in Chen and Fang (2019). While our development is undertaken outside the scope of the Delta method, there is an intriguing connection to the general theory of Fang and Santos (2019), as we now flesh out.…”

Implementing an Improved Test of Matrix Rank in Stata

“…In this paper, we present an introduction to the improved test of matrix rank developed by Chen and Fang (2019), and then describe the Stata command bootranktest for implementing the test. The central message in Chen and Fang (2019) is that existing rank tests may fail to control Type I error, even in simple examples-see p.1791 of the paper. This in particular includes the influential test of Kleibergen and Paap (2006) and conceivably other analogous tests such as Robin and Smith (2000).…”