STATISTICAL INFERENCE WITH F-STATISTICS WHEN FITTING SIMPLE MODELS TO HIGH-DIMENSIONAL DATA

We study linear subset regression in the context of the high-dimensional overall model y = upsilon + 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 x 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.