fix all the parameters in – except for the orthogonal matrix $R$ in . We then simulate 100 independent realizations $R_r$ of $R$, compute $\overline{p}_r$ as outlined above, $r=1,\dots, 100$, and finally compute $\overline{D}=100^{-1} \sum_{r=1}^{100} |\overline{p}_r - \alpha|$. If $R_r \in \mathbb U$, then $\overline{p}_r$ should be close to $\alpha$, in view of the last display in Theorem \[t1\]. We use $\overline{D}$ and the empirical distribution of the $\overline{p}_r$, $r=1,\dots,100$, as indicators for the size of $\mathbb U$.

The remaining parameters in – and the submodel matrix $M$ are chosen as follows for any fixed values of $d$ and $p$: We do not include an error term in the true model, i.e., we set $\sigma^2 = 0$, because the effect of misspecification becomes more pronounced when the error variance $\sigma^2$ is small.[^1] \[Note that the case where $\sigma^2=0$ is not covered by Theorem \[t1\], but inspection of the proof shows that our results also apply in this case; cf. Remark \[zerovariance\].\] For $\tilde{z}$, we consider product distributions with zero mean and i.i.d. components from the student-$t$ distribution with $2$, $3$ and $5$ degrees of freedom, as well as from the centered exponential, uniform, Bernoulli$\{-1,1\}$ and Gaussian distributions. \[Note that the 