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For a linear regression model $Y = β_0 + β_1X$, consider the matrix $X$ and $X_c$ with centering the mean.

How do you use algebra to show directly that centering does not change the leverage?

I have tried to expand the $X'_c(X'_cX_c)^{-1}X_c$ matrix but that does not seem to work out.

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It is easier to see it by geometry than by algebra: The leverages $h_i$ are diagonal elements of hat matrix $H$, which is an orthogonal projector. Being an orthogonal projector is a geometric property which is independent of coordinate systems. Centering the variables gives just a reparametrization of the same model. Reparametrization of a linear model is just a change of basis.

Note that this argument depends on the model having an intercept. Without an intercept, centering will change the model.

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