Let $mathbf{x}_{j}$ be the $j$ th row of $mathbf{X}$, and $mathbf{X}_{-j}$ be the $mathbf{X}$ matrix with the
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Let $\mathbf{x}_{j}$ be the $j$ th row of $\mathbf{X}$, and $\mathbf{X}_{-j}$ be the $\mathbf{X}$ matrix with the $j$ th row removed. Show that
\[\operatorname{Var}\left[\hat{\beta}_{j}\right]=\sigma^{2}\left[\mathbf{x}_{j}^{\prime} \mathbf{x}_{j}-\mathbf{x}_{j}^{\prime} \mathbf{X}_{-j}\left(\mathbf{X}_{-j}^{\prime} \mathbf{X}_{-j}\right)^{-1} \mathbf{X}_{-j}^{\prime} \mathbf{x}_{j}\right]\]
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Related Book For 
Introduction To Linear Regression Analysis
ISBN: 9781119578727
6th Edition
Authors: Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining
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