@Bai96a and [@Zho11a]. For later use, note that the distribution of $(y,z)$ in – is characterized by $\vartheta$ and $\theta$, by $\mathcal L(\epsilon)$, by $\Sigma$ and $\mu$, by $f_{\tilde{z}}$, and by $R$.

Fix an integer $k\geq 1$ and positive (finite) constants $D$ and $E$. With this, write ${\mathcal F}_{d,k}(D,E)$ for the class of Lebesgue densities on ${{\mathbb R}}^d$ that are products of univariate marginal densities such that each such marginal density is bounded from above by $D$, and such that each univariate marginal density has absolute moments of order up to $k$ that are bounded by $E$.

The sub-model and the $F$-test {#theworkingmodel}
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Consider a sub-model where $y$ is regressed on $x$, with $x$ given by $$\label{x}
x \quad=\quad  M'z$$ for some full-rank $d\times p$ matrix $M$ with $p<d$. For example, $M$ can be a selection matrix that picks out $p$ components of the $d$-vector $z$. Submodels with regressors of the form $x=M'z$ also occur in principal component regression, partial least squares, and certain sufficient dimension reduction methods. We are particularly interested in situations where $d$ is *much* larger than $p$, i.e., $p \ll d$. Trivially, we can write $$\label{workingmodel}
    y = \alpha + \beta' x + e$$ with 