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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 sub-models 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, i.e., $y=\alpha+\beta'' x + e$, can be justified by imposing appropriate restrictions on the unknown parameter $\theta$ 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 $\beta$, in an appropriate sense, even when the simple model is misspecified.'
author:
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    Hannes Leeb (University of Vienna and DataScience@UniVienna)\
    Lukas Steinberger (University of Freiburg)
bibliography:
- 'lit.bib'
title: 'Statistical inference with $F$-statistics when fitting simple models to high-dimensional data'
---

Introduction
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The $F$-test is a staple tool of applied statistical analyses. It is widely used, sometimes also in situations where 