simulation to investigate how soon these asymptotic approximations become accurate. We consider a rather small sample size of $n=50$ and look at different configurations of the model dimensions $d$ and $p$ with $p<d$, and also at different points in parameter space.

The theorem contains two asymptotic statements, one about the distribution of the $F$-statistic and one about the size of the set $\mathbb U$. For the distribution of the $F$-statistic, we compare the rejection probability of the $F$-test under the null hypothesis with the nominal significance level $\alpha=0.05$. The nominal significance level provides a natural benchmark. \[Clearly, one can also investigate the power of the $F$-test through simulation experiments, but, unlike the significance level, it is less obvious what the right benchmark for the power should be.\] In particular, we simulate 1000 independent realizations $F_{j,r}$, $j=1,\dots,1000$ of the $F$-statistic at sample size $n=50$ under the null for each point in parameter space (the index $r$ will be explained shortly), and compare the empirical significance level $\overline{p}_r = 1000^{-1} \sum_{j=1}^{1000} {\mathbf 1}\{ F_{j,r} > 
    F^{-1}_{p,n-p-1,0}(1-\alpha)\}$ with the nominal level $\alpha$.

Gauging the size of $\mathbb U$ is more difficult, because that set is not given explicitly. We proceed as follows: We 