Statistical inference with F-statistics when fitting simple models to high-dimensional data
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We study linear subset regression in the context of the high-dimensional overall model y = ϑ+θ' z + ϵ with univariate response y and a d-vector of random regressors z, independent of ϵ. 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 sub-models where y is regressed on a set of p regressors given by x = M'z, for some d × p matrix M of full rank p < n. The corresponding simple model, i.e., y=α+β' x + e, can be justified by imposing appropriate restrictions on the unknown parameter θ in the overall model; otherwise, this simple model can be grossly misspecified. In this paper, we establish asymptotic validity of the standard F-test on the surrogate parameter β, in an appropriate sense, even when the simple model is misspecified.
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