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In my econometrics notes (not a proper textbook) I find that given a partitioning of the sample matrix X into $$ X =(X_1 1),$$ where 1 is the (nx1) vector of all unity elements, then $$M_{[1]}=I - 1(1'1)^{-1}1'$$ transforms all variables in deviations from their sample means, where M is the "annihilator matrix" or also called the "residual maker".

Can someone please give me an intuition or even a full proof of this? Or maybe just a reference of where to look up the relationship between the unity vector being in the sample matrix and variables being in deviations from their sample means. I feel like I am stuck on this matrix as I am at a loss of what does this relationship look like. Thank you.

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  • $\begingroup$ This is just a matrix formula for recentering: subtracting the column mean from each of its entries. Its generalization is called the hat matrix, about which you can read many threads here on CV. $\endgroup$
    – whuber
    Commented Sep 27, 2022 at 14:44

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$\DeclareMathOperator{\1}{\mathbf 1}$

Few elementary observations:

  • \begin{align}\1\frac{1}n\1^\mathsf T\mathbf x &= \frac 1n\1\1^\mathsf T\mathbf x\\ &=\begin{bmatrix}\bar x\\\bar x\\\vdots \\ \bar x\end{bmatrix}.\tag 1\end{align}

  • \begin{align}\begin{bmatrix}x_1-\bar x\\ x_2-\bar x\\ \vdots \\x_n-\bar x\end{bmatrix}&=\begin{bmatrix}\mathbf x -\1\bar x\end{bmatrix}\\&= \begin{bmatrix}\mathbf x-\frac 1n\1\1^\mathsf T\mathbf x\end{bmatrix}\\&= \begin{bmatrix}\mathbf I\mathbf x-\frac 1n\1\1^\mathsf T\mathbf x\end{bmatrix}\\&= \underbrace{\begin{bmatrix}\mathbf I-\frac 1n\1\1^\mathsf T\end{bmatrix}}_{:=\mathbf M_{\1}}\mathbf x .\tag 2\end{align}

  • Since $\left(\1^\mathsf T\1\right)^{-1}=\frac 1n, ~\mathbf M_{\1}= \mathbf I-\1 \left(\1^\mathsf T\1\right)^{-1} \1^\mathsf T. $


Further Reading:

$[\rm I]$ Econometric Analysis, William H. Greene, Pearson Education, $2018, $ appendix $\rm A. 2.8,$ pp. $1059-1061.$

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