$e = y - \alpha-\beta'x$, where $\alpha$ and $\beta$ minimize ${{\mathbb E}}[(y-\alpha-\beta' x)^2]$. The ‘error’ $e$ has mean zero (because both and include an intercept), and we denote its variance by $s^2 = {{\mathbb E}}[e^2]$. Note that $\alpha= \vartheta+\mu'\theta - \mu' M (M' \Sigma M)^{-1} M' \Sigma \theta$ and, for later use, that $$\begin{aligned}
 \label{betas2}
\begin{split}
\beta \quad & = \quad (M'\Sigma M)^{-1} M'\Sigma \theta\quad\text{and}\\
s^2 \quad   & = \quad \theta'\Sigma\theta +
        \theta'\Sigma M (M'\Sigma M)^{-1} M'\Sigma\theta+\sigma^2.
\end{split}\end{aligned}$$ Irrespective of whether the working model is correctly specified, the ‘surrogate’ parameters $\alpha$, $\beta$ and $s^2$ are always well-defined. Here, $\beta$ is our main object of interest, instead of the underlying true parameter $\theta$. Such surrogate parameters are well-known in the statistics literature, certainly since @Hub67a, and have recently gained new popularity, as witnessed by, e.g., @Aba14a [@Bra14a; @Bac15a; @Buj14a]. In particular, such surrogate parameters can be consistently estimated, in a standard $M$-estimation setting, by the OLS estimator or by robust alternatives, provided that $p$ is not too large relative to $n$ [see @Por84a; @Por85a; @Whi80a; @Whi80b]; cf. also Lemma A.3 in @Ste15 and Lemma A.4 in @Ste18a for analyses tailored to our present setting.

The working model is correct (in the 