Consider the normal linear model where the $n$-vector of responses $Y$ satisfies $Y=X \beta+\varepsilon$ with $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$. Here $X$ is an $n \times p$ matrix of predictors with full column rank where $n \geqslant p+3$, and $\beta \in \mathbb{R}^{p}$ is an unknown vector of regression coefficients. Let $X_{0}$ be the matrix formed from the first $p_{0}<p$ columns of $X$, and partition $\beta$ as $\beta=\left(\beta_{0}^{T}, \beta_{1}^{T}\right)^{T}$ where $\beta_{0} \in \mathbb{R}^{p_{0}}$ and $\beta_{1} \in \mathbb{R}^{p-p_{0}}$. Denote the orthogonal projections onto the column spaces of $X$ and $X_{0}$ by $P$ and $P_{0}$ respectively.

It is desired to test the null hypothesis $H_{0}: \beta_{1}=0$ against the alternative hypothesis $H_{1}: \beta_{1} \neq 0$. Recall that the $F$-test for testing $H_{0}$ against $H_{1}$ rejects $H_{0}$ for large values of

$$F=\frac{\left\|\left(P-P_{0}\right) Y\right\|^{2} /\left(p-p_{0}\right)}{\|(I-P) Y\|^{2} /(n-p)} .$$

Show that $(I-P)\left(P-P_{0}\right)=0$, and hence prove that the numerator and denominator of $F$ are independent under either hypothesis.

Show that

$$\mathbb{E}_{\beta, \sigma^{2}}(F)=\frac{(n-p)\left(\tau^{2}+1\right)}{n-p-2}$$

where $\tau^{2}=\frac{\left\|\left(P-P_{0}\right) X \beta\right\|^{2}}{\left(p-p_{0}\right) \sigma^{2}}$.

[In this question you may use the following facts without proof: $P-P_{0}$ is an orthogonal projection with rank $p-p_{0}$; any $n \times n$ orthogonal projection matrix $\Pi$ satisfies $\|\Pi \varepsilon\|^{2} \sim \sigma^{2} \chi_{\nu}^{2}$, where $\nu=\operatorname{rank}(\Pi) ;$ and if $Z \sim \chi_{\nu}^{2}$ then $\mathbb{E}\left(Z^{-1}\right)=(\nu-2)^{-1}$ when $\left.\nu>2 .\right]$