Sluisu4

## Answered question

2022-09-30

Statistics (Regression Analysis): Show that the residuals from a linear regression model can be expressed as $\mathbf{e}=\left(\mathbf{I}-\mathbf{H}\right)ϵ$
The bold represents vectors or matrices.
I know that $\mathbf{e}=\mathbf{y}-\mathbf{H}\mathbf{y}$
So I tried expanding this to,
$\mathbf{e}=\mathbf{X}\beta +ϵ-\mathbf{H}\mathbf{X}\beta \mathbf{-}\mathbf{H}ϵ$
At this point I can see how to derive the more traditional,
$\mathbf{e}\mathbf{=}\mathbf{\left(}\mathbf{I}\mathbf{-}\mathbf{H}\mathbf{\right)}\mathbf{y}$
how to solve the original problem?

### Answer & Explanation

recepiamsb

Beginner2022-10-01Added 9 answers

Recall that $H$ is an orthogonal projection onto the space spanned by the columns of $X$, hence $HX\beta =X\beta$, thus
$\begin{array}{rl}e& =Y-\stackrel{^}{Y}\\ & =X\beta +ϵ-HY\\ & =X\beta +ϵ-H\left(X\beta +ϵ\right)\\ & =X\beta -X\beta +ϵ-Hϵ\\ & =\left(I-H\right)ϵ.\end{array}$

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