in potentially misspecified settings; see, for example, @Che10b [@Eic67a; @Hub67a; @Whi80a; @Whi80b; @Zho11a]. For further results on hypothesis testing and marginal screening in misspecified models, see, for example, @Boo13a [@Cho11a; @Fom03a; @Jen91a; @Ram91a], and the references therein.

On a technical level, this paper relies on @Wan13a, the corresponding extensions and corrections in [@Ste16a], and also on @Ste18a [@Ste18b]; all but the first of these references are based on @Ste15.

The rest of the paper is structured as follows: In Section \[thetruemodel\], we describe the true data-generating model and the underlying parameter space. The (typically misspecified) working model and the corresponding $F$-statistic are described in Section \[theworkingmodel\]. Our main theoretical result is given in Section \[mainresult\], and a simulation study in Section \[numericalresults\] demonstrates that our asymptotic approximations can ‘kick-in’ reasonably fast.

The true model {#thetruemodel}
==============

Throughout, we consider the (true) linear model $$\label{y}
y \quad = \quad  \vartheta+\theta' z + \epsilon$$ with $\vartheta\in {{\mathbb R}}$ and $\theta \in {{\mathbb R}}^d$ for some $d \in {{\mathbb N}}$. We assume that the error $\epsilon$ is independent of $z$, with mean zero and finite variance $\sigma^2>0$; its distribution will be denoted by $\mathcal L(\epsilon)$. Moreover, we assume that the vector of regressors 