the size of the $F$-test from the nominal level, as the dimension $p$ of the sub-model increases, which was also suggested by our theory. However, if we fix the ratio $p/d=1/2$, i.e., if we move along the staircase pattern in each of the panels, except for the heavy tailed distributions $t(3)$ and $t(2)$, we still see the effect of misspecification decrease as $d$ increases. This suggests that convergence of $n^2 /\log(d)\sim p^2/\log(d)$ to zero, as required in Theorem \[t1\], may not be necessary, at least in the scenarios considered here.

[c|cccc | Hcccc]{} $d\backslash p$ & 1 & 2 & 5 & 25 & $d\backslash p$ & 1 & 2 & 5 & 25\
& && & & &&\
2 & 0.077 & & & & 2 & 0.141 & & &\
4 & 0.056 & 0.076 & & & 4 & 0.093 & 0.140 & &\
10 & 0.032 & 0.047 & 0.066 & & 10 & 0.052 & 0.071 & 0.109 &\
50 & 0.009 & 0.013 & 0.017 & 0.019 & 50 & 0.014 & 0.015 & 0.020 & 0.033\
100 & 0.007 & 0.008 & 0.009 & 0.010 & 100 & 0.009 & 0.009 & 0.012 