Linear Hypothesis Testing in Dense High-Dimensional Linear Models

We propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, that is, model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensions is extremely challenging—especially without making restrictive or unverifiable assumptions on the number of nonzero elements. We propose to test the moment conditions related to the newly designed restructured regression, where the inputs are transformed and augmented features. These new features incorporate the structure of the null hypothesis directly. The test statistics are constructed in such a way that lack of sparsity in the original model parameter does not present a problem for the theoretical justification of our procedures. We establish asymptotically exact control on Type I error without imposing any sparsity assumptions on model parameter or the vector representing the linear hypothesis. Our method is also shown to achieve certain optimality in detecting deviations from the null hypothesis. We demonstrate the favorable finite-sample performance of the proposed methods, via a number of numerical and a real data example. Supplementary materials for this article are available online.