Role of Hat Matrix in Regression Analysis

Hat matrix is a $n\times n$ symmetric and idempotent matrix with many special properties play an important role in diagnostics of regression analysis by transforming the vector of observed responses Y into the vector of fitted responses $\hat{Y}$.

The model $Y=X\beta+\varepsilon$ with solution $b=(X’X)^{-1}X’Y$ provided that $(X’X)^{-1}$ is non-singular. The fitted values are ${\hat{Y}=Xb=X(X’X)^{-1} X’Y=HY}$.

Like fitted values ($\hat{Y}$), the residual can be expressed as linear combinations of the response variable Y_i.

\begin{align*}
e&=Y-\hat{Y}\\
&=Y-HY\\&=(I-H)Y
\end{align*}

• Hat matrix only involves the observation in the predictor variable X  as $H=X(X’X)^{-1}X’$. It plays an important role in diagnostics for regression analysis.
• The hat matrix plays an important role in determining the magnitude of a studentized deleted residual and therefore in identifying outlying Y observations.
• The hat matrix is also helpful in directly identifying outlying X observation.
• In particular the diagonal elements of the hat matrix are indicator of in a multi-variable setting of whether or not a case is outlying with respect to X values.
• The elements of hat matrix have their values between 0 and 1 always and their sum is p i.e. $0 \le h_{ii}\le 1$  and  $\sum _{i=1}^{n}h_{ii} =p $
  where p is number of regression parameter with intercept term.
• h_ii is a measure of the distance between the X values for the ith case and the means of the X values for all n cases.

Mathematical Properties of Hat Matrix

• HX=X
• (I-H)X=0
• HH=H²=H=H^p
• H(I-H)=0
• $Cov(\hat{e},\hat{Y})=Cov\left\{HY,(I-H)Y\right\}=\sigma ^{2} H(I-H)=0$
• 1-H is also symmetric and idempotent.
• H1=1 with intercept term. i.e. every row of H adds  upto 1. 1’=1H’=1’H  & 1’H1=n
• The elements of H are denoted by h_ii  i.e.
  \[H=\begin{pmatrix}{h_{11} } & {h_{12} } & {\cdots } & {h_{1n} } \\ {h_{21} } & {h_{22} } & {\cdots } & {h_{2n} } \\ {\vdots } & {\vdots } & {\ddots } & {\vdots } \\ {h_{n1} } & {h_{n2} } & {\vdots } & {h_{nn} }\end{pmatrix}\]
  The large value of h_ii indicates that the ith case is distant from the center for all n cases. The diagonal element h_ii in this context is called leverage of the ith case.h_ii is a function of only the X values, so h_ii measures the role of the X values in determining how important Y_i is affecting the fitted $\hat{Y}_{i} $ values.
  The larger the h_ii, the smaller the variance of the residuals e_i for h_ii =1, σ²(e_i)=0.
• Variance, Covariance of e
  \begin{align*}
  e-E(e)&=(I-H)Y(Y-X\beta )=(I-H)\varepsilon \\
  E(\varepsilon \varepsilon ‘)&=V(\varepsilon )=I\sigma ^{2} \,\,\text{and} \,\, E(\varepsilon )=0\\
  (I-H)’&=(I-H’)=(I-H)\\
  V(e) & =  E\left[e-E(e_{i} )\right]\left[e-E(e_{i} )\right]^{{‘} } \\
  & = (I-H)E(\varepsilon \varepsilon ‘)(I-H)’ \\
  & = (I-H)I\sigma ^{2} (I-H)’ \\
  & =(I-H)(I-H)I\sigma ^{2} =(I-H)\sigma ^{2}
  \end{align*}V(e_i) is given by the ith diagonal element 1-h_ii  and Cov(e_i ,e_j ) is given by the (i, j)th  element of −h_ij of the matrix (I-H)σ².\begin{align*}
  \rho _{ij} &=\frac{Cov(e_{i} ,e_{j} )}{\sqrt{V(e_{i} )V(e_{j} )} } \\
  &=\frac{-h_{ij} }{\sqrt{(1-h_{ii} )(1-h_{jj} )} }\\
  SS(b) & = SS({\rm all\; parameter)=}b’X’Y \\
  & = \hat{Y}’Y=Y’H’Y=Y’HY=Y’H^{2} Y=\hat{Y}’\hat{Y}
  \end{align*}The average $V(\hat{Y}_{i} )$ to all data points is
  \begin{align*}
  \sum _{i=1}^{n}\frac{V(\hat{Y}_{i} )}{n} &=\frac{trace(H\sigma ^{2} )}{n}=\frac{p\sigma ^{2} }{n} \\
  \hat{Y}_{i} &=h_{ii} Y_{i} +\sum _{j\ne 1}h_{ij} Y_{j}
  \end{align*}

Role of Hat Matrix in Regression Diagnostic

Internally Studentized Residuals

$V(e_i)=(1-h_{ii}\sigma^2$ where σ² is estimated by s²

i.e. $s^{2} =\frac{e’e}{n-p} =\frac{\Sigma e_{i}^{2} }{n-p} $  (RMS)

we can studentized the residual as $s_{i} =\frac{e_{i} }{s\sqrt{(1-h_{ii} )} } $

These studentized residuals are said to be internally studentized because s has within it e_i itself.

Extra Sum of Squares attributable to $e_i$

\begin{align*}
e&=(1-H)Y\\
e_{i} &=-h_{i1} Y_{1} -h_{i2} Y_{2} -\cdots +(1-h_{ii} )Y_{i} -h_{in} Y_{n} =c’Y\\
c’&=(-h_{i1} ,-h_{i2} ,\cdots ,(1-h_{ii} )\cdots -h_{in} )\\
c’c&=\sum _{i=1}^{n}h_{i1}^{2}  +(1-2h_{ii} )=(1-h_{ii} )\\
SS(e_{i})&=\frac{e_{i}^{2} }{(1-h_{ii} )}\\
S_{(i)}^{2}&=\frac{(n-p)s^{2} -\frac{e_{i}^{2}}{e_{i}^{2}  (1-h_{ii} )}}{n-p-1}
\end{align*}
provides an estimate of σ² after deletion of the contribution of e_i.

Externally Studentized Residuals

$t_{i} =\frac{e_{i} }{s(i)\sqrt{(1-h_{ii} )} }$ are externally studentized residuals. Here if e_i is large, it is thrown into emphases even more by the fact that s_i has excluded it. The t_i follows a t_n-p-1 distribution under the usual normality of errors assumptions.

Read more about Role of Hat Matrix in Regression Anbalysis https://en.wikipedia.org/wiki/Hat_matrix

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