Abstract
We study linear subset regression in the context of the high-dimensional overall model
$y = \vartheta +\theta ' z + \epsilon $
with univariate response y and a d-vector of random regressors z, independent of
$\epsilon $
. Here, “high-dimensional” means that the number d of available explanatory variables is much larger than the number n of observations. We consider simple linear submodels where y is regressed on a set of p regressors given by
$x = M'z$
, for some
$d \times p$
matrix M of full rank
$p < n$
. The corresponding simple model, that is,
$y=\alpha +\beta ' x + e$
, is usually justified by imposing appropriate restrictions on the unknown parameter
$\theta $
in the overall model; otherwise, this simple model can be grossly misspecified in the sense that relevant variables may have been omitted. In this paper, we establish asymptotic validity of the standard F-test on the surrogate parameter
$\beta $
, in an appropriate sense, even when the simple model is misspecified, that is, without any restrictions on
$\theta $
whatsoever and without assuming Gaussian data.