class: center, middle, inverse, title-slide # More on the Hat Matrix ### Dr. D’Agostino McGowan --- layout: true <div class="my-footer"> <span> Dr. Lucy D'Agostino McGowan </span> </div> --- ## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 448 512"><path d="M144 479H48c-26.5 0-48-21.5-48-48V79c0-26.5 21.5-48 48-48h96c26.5 0 48 21.5 48 48v352c0 26.5-21.5 48-48 48zm304-48V79c0-26.5-21.5-48-48-48h-96c-26.5 0-48 21.5-48 48v352c0 26.5 21.5 48 48 48h96c26.5 0 48-21.5 48-48z"/></svg> `Matrix fact` * If a matrix is **symmetric**, then `\(\mathbf{A}=\mathbf{A}^T\)` -- * If a matrix is **idempotent**, then `\(\mathbf{A}^T\mathbf{A}=\mathbf{A}\)` -- * The **hat matrix**, `\(\mathbf{H}\)`, is both symmetric **and** idempotent --- ## The hat matrix is symmetric `$$\begin{align}\mathbf{H}& = \mathbf{H}^T\\ \end{align}$$` --- ## The hat matrix is symmetric `$$\begin{align}\mathbf{H}& = \mathbf{H}^T\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=(\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T)^T\\ \end{align}$$` --- ## The hat matrix is symmetric `$$\begin{align}\mathbf{H}& = \mathbf{H}^T\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=(\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T)^T\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T \end{align}$$` --- ## The hat matrix is idempotent `$$\begin{align}\mathbf{H}^T\mathbf{H}&=\mathbf{H}\\ \end{align}$$` --- ## The hat matrix is idempotent `$$\begin{align}\mathbf{H}^T\mathbf{H}&=\mathbf{H}\\ \mathbf{H}\mathbf{H}&=\mathbf{H}\\ \end{align}$$` --- ## The hat matrix is idempotent `$$\begin{align}\mathbf{H}^T\mathbf{H}&=\mathbf{H}\\ \mathbf{H}\mathbf{H}&=\mathbf{H}\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\\ \end{align}$$` --- ## The hat matrix is idempotent `$$\begin{align}\mathbf{H}^T\mathbf{H}&=\mathbf{H}\\ \mathbf{H}\mathbf{H}&=\mathbf{H}\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\underbrace{\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}}_{\mathbf{I}}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\\ \end{align}$$` --- ## The hat matrix is idempotent `$$\begin{align}\mathbf{H}^T\mathbf{H}&=\mathbf{H}\\ \mathbf{H}\mathbf{H}&=\mathbf{H}\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\underbrace{\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}}_{\mathbf{I}}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\\ \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T&=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\\ \end{align}$$` -- * `\(\mathbf{I-H}\)` is also **idempotent** --- ## The trace of the hat matrix is p+1 `$$\textrm{tr}(\mathbf{H})$$` -- ## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 448 512"><path d="M144 479H48c-26.5 0-48-21.5-48-48V79c0-26.5 21.5-48 48-48h96c26.5 0 48 21.5 48 48v352c0 26.5-21.5 48-48 48zm304-48V79c0-26.5-21.5-48-48-48h-96c-26.5 0-48 21.5-48 48v352c0 26.5 21.5 48 48 48h96c26.5 0 48-21.5 48-48z"/></svg> `Matrix fact` * The **trace** of a square matrix, written as `\(\textrm{tr}(\mathbf{A})\)`, is the sum of the diagonal elements * `\(\textrm{tr}(\mathbf{A}+\mathbf{B}) = \textrm{tr}(\mathbf{A})+\textrm{tr}(\mathbf{B})\)` * `\(\textrm{tr}(c\mathbf{A}) = c\textrm{tr}(\mathbf{A})\)` * `\(\textrm{tr}(\mathbf{ABC})=\textrm{tr}(\mathbf{BCA})=\textrm{tr}(\mathbf{CAB})\)` --- ## The trace of the hat matrix is p+1 `$$\begin{align}\textrm{tr}(\mathbf{H})=&\textrm{tr}(\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T)\end{align}$$` --- ## The trace of the hat matrix is p+1 `$$\begin{align}\textrm{tr}(\mathbf{H})=&\textrm{tr}(\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T)\\ =&\textrm{tr}(\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1})\end{align}$$` --- ## The trace of the hat matrix is p+1 `$$\begin{align}\textrm{tr}(\mathbf{H})=&\textrm{tr}(\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T)\\ =&\textrm{tr}(\mathbf{X}^T\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1})\\ =&\textrm{tr}(\mathbf{I})\end{align}$$` -- .question[ What are the dimensions of this `\(\mathbf{I}\)`? ] -- * `\(\mathbf{X}_{n\times (p+1)}\)` -- * `\(\mathbf{I}_{(p+1)\times(p+1)}\)` -- * `\(\textrm{tr}(\mathbf{I}_{(p+1)\times(p+1)})=p+1\)` --- ## RSS `$$\begin{align}\mathbf{e}^T\mathbf{e}=&(\mathbf{y}-\hat{\mathbf{y}})^T(\mathbf{y}-\hat{\mathbf{y}})\\ \end{align}$$` --- ## RSS `$$\begin{align}\mathbf{e}^T\mathbf{e}=&(\mathbf{y}-\hat{\mathbf{y}})^T(\mathbf{y}-\hat{\mathbf{y}})\\ =&(\mathbf{y} - \mathbf{H}\mathbf{y})^T(\mathbf{y}-\mathbf{H}\mathbf{y})\\ \end{align}$$` --- ## RSS `$$\begin{align}\mathbf{e}^T\mathbf{e}=&(\mathbf{y}-\hat{\mathbf{y}})^T(\mathbf{y}-\hat{\mathbf{y}})\\ =&(\mathbf{y} - \mathbf{H}\mathbf{y})^T(\mathbf{y}-\mathbf{H}\mathbf{y})\\ =&((\mathbf{I}-\mathbf{H})\mathbf{y})^T(\mathbf{I}-\mathbf{H})\mathbf{y}\\ \end{align}$$` --- ## RSS `$$\begin{align}\mathbf{e}^T\mathbf{e}=&(\mathbf{y}-\hat{\mathbf{y}})^T(\mathbf{y}-\hat{\mathbf{y}})\\ =&(\mathbf{y} - \mathbf{H}\mathbf{y})^T(\mathbf{y}-\mathbf{H}\mathbf{y})\\ =&((\mathbf{I}-\mathbf{H})\mathbf{y})^T(\mathbf{I}-\mathbf{H})\mathbf{y}\\ =&\mathbf{y}^T(\mathbf{I-H})^T(\mathbf{I}-\mathbf{H})\mathbf{y} \end{align}$$` --- ## RSS `$$\begin{align}\mathbf{e}^T\mathbf{e}=&(\mathbf{y}-\hat{\mathbf{y}})^T(\mathbf{y}-\hat{\mathbf{y}})\\ =&(\mathbf{y} - \mathbf{H}\mathbf{y})^T(\mathbf{y}-\mathbf{H}\mathbf{y})\\ =&((\mathbf{I}-\mathbf{H})\mathbf{y})^T(\mathbf{I}-\mathbf{H})\mathbf{y}\\ =&\mathbf{y}^T(\mathbf{I-H})^T(\mathbf{I}-\mathbf{H})\mathbf{y}\\ =&\mathbf{y}^T(\mathbf{I-H})\mathbf{y} \end{align}$$` --- ## RSS `$$E[\mathbf{e}^T\mathbf{e}]=E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]$$` -- ## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 448 512"><path d="M144 479H48c-26.5 0-48-21.5-48-48V79c0-26.5 21.5-48 48-48h96c26.5 0 48 21.5 48 48v352c0 26.5-21.5 48-48 48zm304-48V79c0-26.5-21.5-48-48-48h-96c-26.5 0-48 21.5-48 48v352c0 26.5 21.5 48 48 48h96c26.5 0 48-21.5 48-48z"/></svg> `Matrix fact` `$$E[\mathbf{X}^T\mathbf{CX}]=E[\mathbf{X}]^T\mathbf{C}E[\mathbf{X}] +\textrm{tr}(\mathbf{C}\textrm{var}(\mathbf{X}))$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(0)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(0)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2\textrm{tr}(\mathbf{I-H})\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(0)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2(\textrm{tr}(\mathbf{I}_{n\times n})-\textrm{tr}(\mathbf{H}))\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(0)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2(\textrm{tr}(\mathbf{I}_{n\times n})-\textrm{tr}(\mathbf{H}))\\ =&\sigma^2(n-\textrm{tr}(\mathbf{H}))\\ \end{align}$$` --- ## RSS `$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\ =&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\ =&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\beta^T\mathbf{X}^T(0)+\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2\textrm{tr}(\mathbf{I-H})\\ =&\sigma^2(\textrm{tr}(\mathbf{I}_{n\times n})-\textrm{tr}(\mathbf{H}))\\ =&\sigma^2(n-\textrm{tr}(\mathbf{H}))\\ =&\sigma^2(n-(p+1)) \end{align}$$`