  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, i.e. 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 non-zero 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.
