can someone derive the last formula for the variance ? I dont get the result and its in the script of my university:
"
(ii) Regression methodology: now assume the model $\mathrm{x}=\boldsymbol{\mu}+\mathrm{Lf}+\boldsymbol{\epsilon}$, such that \begin{equation*} \left[\begin{array}{l} \mathrm{x} \\ \mathrm{f} \end{array}\right] \sim \mathcal{N}_{p+k}\left(\left[\begin{array}{l} \boldsymbol{\mu} \\ 0 \end{array}\right],\left[\begin{array}{cc} \mathrm{LL}^{\prime}+\mathrm{V} & \mathrm{L} \\ \mathrm{L}^{\prime} & \mathrm{l} \end{array}\right]\right) . \end{equation*}
Then the conditional expectation $E[\mathrm{f} \mid \mathrm{x}]$ can be obtained as \begin{equation*} \hat{f}_R=L^{\prime}\left(L L^{\prime}+V\right)^{-1}(x-\mu) \end{equation*} using the result on the next slide. The variance of the estimator is given as \begin{equation*} \operatorname{Var}\left(\hat{\mathrm{f}}_{\mathrm{R}}\right)=E\left[\left(\hat{\mathrm{f}}_{\mathrm{R}}-\mathrm{f}\right)\left(\hat{\mathrm{f}}_{\mathrm{R}}-\mathrm{f}\right)^{\prime}\right]=\mathrm{I}-\mathrm{L}^{\prime}\left(\mathrm{LL}^{\prime}+\mathrm{V}\right)^{-1} \mathrm{~L} . \end{equation*} "
Thank you in advance