its applicability is debatable because underlying assumptions may not be met. We study a situation of this kind: An $F$-test after fitting a (possibly misspecified) working model. We focus, in particular, on a scenario where the fitted model has $p$ explanatory variables while the true model has $d$ explanatory variables, with $p \ll d$, and where sample size is of the same order as $p$, i.e., $p = O(n)$. Scenarios like this occur, for example, in quality control studies like [@Sou91a], where a model with 18 explanatory variables (out of a total of about 8,000) is fit based on a sample of size 50; in time series forecasting with principal components as in [@Sto02a], who extract a handful of factors from 149 explanatory variables based on 480 monthly observations; or in genetic analyses like [@Vee02a], who select and fit a model with 70 genes (out of a total of about 25,000) based on a sample of size 78. In situations like these, the question whether the fitted model has any explanatory value is of particular interest. We show that, approximately, the usual $F$-statistic is $F$-distributed under a corresponding null-hypothesis, and that it is non-central $F$-distributed in a local neighborhood of 