usual sense) if ${{\mathbb E}}[y\|z] = {{\mathbb E}}[y\|x]$, i.e., if $\vartheta+\theta'z = \alpha + \beta'x$ or, equivalently, if $\epsilon=e$. This is the case if $\theta$ lies in the column space of $M$; if $M$ is a selection matrix, this means that $M'\theta$ selects all the non-zero components of $\theta$. Here, we do not assume that the working model is correct. In particular, we stress that $e$ may differ from $\epsilon$, and that $e$ may depend on $x$.

When working with the simple sub-model , a natural question is whether $x$ has any explanatory value for the response variable $y$. Given a sample of $n > p+1$ independent and identically distributed (i.i.d.) observations of $y$ and $x$ from , a classical approach to this question is to use the $F$-test of the hypotheses $$\label{hypotheses}
H_0: \beta = 0 \quad\text{versus}\quad H_1: \beta\neq 0.$$ Let $Y$ and $X$ denote the $n\times 1$ vector of responses and the $n\times p$ matrix of explanatory variables, respectively. Write $\hat{\beta}$ for the OLS-estimator for $\beta$ when $Y$ is regressed on $X$ and a constant, set $\hat{s}^2 = \|(I_n - P_{\iota,X})Y\|^2/(n-p-1)$, and write $\hat{F}_n=\hat{F}_n(X,Y)$ for the usual $F$-statistics for testing $H_0$, i.e., $\hat{F}_n= \|(I_n - P_\iota)X \hat{\beta}\|^2/ (p 