scaling of these distributions is inconsequential, because of the scale-invariance of the $F$-statistic $\hat{F}(X,Y)$ in both arguments and the fact that we do not include an error term in the full model, i.e., scaling of $\tilde{z}_i$ is equivalent to scaling of both $y_i = \theta'z_i$ and $x_i = B'z_i$. Similarly, also the scaling of $\theta$ and $\Sigma$ has no impact on the value of the $F$-statistic.\] For $\Sigma$, we chose a spiked covariance matrix $\Sigma = U\text{diag}(\lambda_1,\dots,\lambda_n) U'$ with eigenvalues $\lambda_1 = \lambda_2 = 400$ and $\lambda_3=\dots=\lambda_d=1$ and an orthogonal matrix of eigenvectors $U$ chosen randomly from the uniform distribution on the orthogonal group.[^2] The intercept terms $\vartheta$ and $\mu$ are set to zero, for convenience. For the matrix $M$, which describes the working model, we take $M$ equal to the $d\times p$ matrix whose $k$-th column is the $k$-th standard basis vector in $\mathbb R^d$, $1\leq k \leq p$. In other words, we consider a sub-model that includes only the first $p$ regressors (out of $d$). For the parameter $\theta\in \mathbb R^d$, we need to ensure that the null hypothesis is satisfied, i.e., that $\beta = (M'\Sigma M)^{-1}M'\Sigma\theta = 0$. By construction of $\Sigma$, $M'\Sigma M$ is regular, and 