we choose $\theta = (I_d-P_{\Sigma M}) V/\|(I_d-P_{\Sigma M}) V\|$, for one realization of $V\thicksim N(0,I_d)$, to guarantee that $M'\Sigma\theta = 0$.

The results of the simulations are summarized in Table \[tab:NullMod\] and Figures \[fig:Boxplots1\] and \[fig:Boxplots2\]. From Table \[tab:NullMod\], the overall picture we get is consistent with what was predicted by our theory. For all distributions except the Gaussian, the average absolute difference between the true (simulated) rejection probabilities and the nominal level decreases as $d$ increases. This phenomenon is most pronounced for the exponential distribution, which has a finite moment generating function around the origin, and is weakest for the $t(2)$-distribution, which does not even have finite variance. For uniformly distributed design, which is bounded, the effect of misspecification on the size of the $F$-test is relatively mild already for small dimensions. In the Gaussian case, all sub-models of the form are correct in the sense that the error $e$ is Gaussian with mean zero and independent of $x$, so that theoretically the corresponding panel in Table \[tab:NullMod\] should contain only zeroes. The numbers therefore represent only the simulation error and serve as a benchmark for the other panels. We also see a monotonic increase, in the deviation of 