the null. Approximation errors go to zero as $n\to \infty$ if $n^2 / \log d \to 0$ and if, at the same time, $p$ is of the same, or of slower, order as $n$; cf. Theorem \[t1\] and Remark \[rateofp\], respectively. Our results are uniform over a large region of the parameter space that we consider. In particular, our results also cover situations where the fitted model is misspecified. The setting of our analysis is non-standard in that we require a particular constellation of $d$, $p$ and $n$. This is a challenging setting of practical relevance, for which few theoretical results are available so far. Our findings, which are given for independent observations, also prompt the question whether similar results can be obtained under serial correlation.

The $F$-statistic is exactly $F$-distributed in a correctly specified linear model with Gaussian errors; and it is asymptotically $F$-distributed under the strong Gau[ß]{}-Markov condition on the errors if $n\to\infty$ while the model dimension stays fixed; cf. @And58a. $F$-tests in correctly specified models in settings where $p$ is allowed to increase with $n$ are studied, among others, by [@Akr00a; @Bat05a; @Boo95a; @Har08a; @Por84a; @Por85a; @Wan13a]. In addition, there are several viable alternatives to the $F$-test 