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# More on the Hat Matrix
### Dr. D’Agostino McGowan

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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

`$$\begin{align}E[\mathbf{e}^T\mathbf{e}]=&E[\mathbf{y}^T(\mathbf{I-H})\mathbf{y}]\\
=&E[\mathbf{y}]^T(\mathbf{I-H})E[\mathbf{y}]+\textrm{tr}((\mathbf{I-H})\textrm{var}(\mathbf{y}))\\
=&\beta^T\mathbf{X}^T(\mathbf{I-H})\mathbf{X}\beta+\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{H}\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\underbrace{(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{X}}_{\mathbf{I}}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\beta^T\mathbf{X}^T(\mathbf{X}\beta-\mathbf{X}\beta)+\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\beta^T\mathbf{X}^T(0)+\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\sigma^2\textrm{tr}(\mathbf{I-H})\\
=&\sigma^2(\textrm{tr}(\mathbf{I}_{n\times n})-\textrm{tr}(\mathbf{H}))\\
=&\sigma^2(n-\textrm{tr}(\mathbf{H}))\\
=&\sigma^2(n-(p+1))
\end{align}$$`

Notes for current slide

Notes for next slide

More on the Hat MatrixDr. D’Agostino McGowan1 / 32

Dr. Lucy D'Agostino McGowan

 Matrix factIf a matrix is symmetric, then A=AT
2 / 32

Dr. Lucy D'Agostino McGowan

 Matrix factIf a matrix is symmetric, then A=AT
If a matrix is idempotent, then ATA=A
2 / 32

Dr. Lucy D'Agostino McGowan

 Matrix factIf a matrix is symmetric, then A=AT
If a matrix is idempotent, then ATA=A
The hat matrix, H, is both symmetric and idempotent
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The hat matrix is symmetric

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The hat matrix is symmetric

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The hat matrix is symmetric

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The hat matrix is idempotent

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The hat matrix is idempotent

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The hat matrix is idempotent

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The hat matrix is idempotent

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The hat matrix is idempotent

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The hat matrix is idempotent

is also idempotent

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The trace of the hat matrix is p+1

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The trace of the hat matrix is p+1

`Matrix fact`

The trace of a square matrix, written as , is the sum of the diagonal elements

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The trace of the hat matrix is p+1

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The trace of the hat matrix is p+1

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The trace of the hat matrix is p+1

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The trace of the hat matrix is p+1

What are the dimensions of this ?

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The trace of the hat matrix is p+1

What are the dimensions of this ?

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The trace of the hat matrix is p+1

What are the dimensions of this ?

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The trace of the hat matrix is p+1

What are the dimensions of this ?

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Dr. Lucy D'Agostino McGowan

 Matrix factIf a matrix is symmetric, then A=AT
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