due to random simulation error. Clearly, the results in the benchmark panel could have been equivalently obtained by repeating the previous simulation for the $F$-test with Gaussian design at significance levels $\alpha= 0.05,0.1, 0.15, 0.2$.

Acknowledgments {#acknowledgments .unnumbered}
===============

The first author’s research was partially supported by FWF projects P 26354-N26 and P 28233-N32.

Proofs
======

We begin with some preliminary considerations that connect this paper with the results of @Ste18b. In particular, we use Theorem 2.1, parts (ii) and (iii), in that reference with $Z=\tilde{z}$ and $\tau=1/2$: If $f_{\tilde{z}}\in\mathcal F_{d,20}(D,E)$, then the assumptions of that result are satisfied in view of Example 3.1 in @Ste18b. The theorem guarantees existence of a Borel subset $\mathbb G = \mathbb G(f_{\tilde{z}})\subseteq \mathcal V_{d,p}$ of the Stiefel manifold $\mathcal V_{d,p}$ of order $d\times p$, that depends on the density $f_{\tilde{z}}$, such that for all $t>0$ both $$\sup_{B\in\mathbb G} {{\mathbb P}}\left(\big\| {{\mathbb E}}[\tilde{z}\| 
    B'\tilde{z}] - BB'\tilde{z}\big\| > t \right)$$ and $$\sup_{B\in\mathbb G} {{\mathbb P}}\left(
    \big\|{{\mathbb E}}[\tilde{z}\tilde{z}'\| B'\tilde{z}] -
        (I_d-B B' + B B' \tilde{z}\tilde{z}' B B')\big\|
    > t \right)$$ are bounded from above by $$\begin{aligned}
\label{approxMeanVar}
    \frac{1}{t} d^{-1/20} + 4\gamma\frac{p}{\log{d}},\\\end{aligned}$$ such that $$\begin{aligned}
\label{eq:G1size}
\nu_{d,p}(\mathbb G^c) \;\le\; \kappa d^{-(1-20\gamma\frac{p}{\log{d}})/20},\end{aligned}$$ where $\nu_{d,p}$ denotes the uniform 