th variable randomly permuted. Note that <span class="arithmatex">\(X_j\)</span> is random. We have </p>
<div class="arithmatex">\[\begin{eqnarray}
    \text{RSS} &amp;=&amp; (\bb{Y}-\bb{X}\hat\beta)^T(\bb{Y}-\bb{X}\hat\beta)\non\\
    \text{RSS}_j^\ast &amp;=&amp; (\bb{Y}-\bb{X}_j\hat\beta)^T(\bb{Y}-\bb{X}_j\hat\beta).\non
\end{eqnarray}\]</div>
<p>Therefore, </p>
<div class="arithmatex">\[\begin{eqnarray}
    \text{RSS}_j^\ast - \text{RSS} = 2\bb{Y}^T(\bb{X}-\bb{X}_j)\hat\beta + \hat\beta^T(\bb{X}_j^T\bb{