\hat{s}^2)$ if the numerator is well-defined and the denominator is positive and $\hat{F}_n=0$ otherwise. Here, $P_{\dots}$ denotes the orthogonal projection on the space spanned by the column-vectors indicated in the subscript and $\iota$ denotes the $n$-vector $\iota=(1,\dots,1)'$. Note that $\hat{F}_n>0$ with probability one by our assumptions.

$H_0$ may be re-phrased as the hypothesis that the best linear predictor of $y$ given $x$ is constant. An alternative to $H_0$ is the hypothesis that the Bayes-estimator of $y$ given $x$ is constant, i.e., $$\tilde{H}_0: {{\mathbb E}}[y\|x] \text{ is constant.}$$ Testing this non-parametric hypothesis is more difficult. In the asymptotic setting that we consider in the next section, however, we find that $H_0$ and $\tilde{H}_0$ are close to each other in the sense that the Bayes predictor and the best linear predictor (of $y$ given $x$) are close in terms of mean-squared prediction error; see Remark \[pinsker\] for details.

Main result {#mainresult}
===========

Our main result is concerned with the asymptotic distribution of the $F$-statistic in a local neighborhood of the null-hypothesis. Here, the local neighborhood is defined through the requirement that $$\Delta \quad=\quad 
    \text{Var}(  \beta' x) / \text{Var}( e)
    \quad=\quad
    \beta' M' \Sigma M \beta /s^2$$ is small. This quantity can be 