econometricsI’m studying economics and came across this matrix $M = (I-X(X’X)^{-1}X’)$ in my econometrics module. However, I don’t find much info on it online, could you explain the significance of this matrix in economics?
I tend to agree that this isn’t really an economics question… it is a regression question without any economic context. In any case…
We know the OLS formula for $\hat{\beta}$ in matrix notation is $\hat{\beta}=(X’X)^{-1}(X’Y)$.
Then $\hat{Y}=X\hat{\beta} =X(X’X)^{-1}(X’Y)$
So the residuals are $e=Y-\hat{Y}=Y-X(X’X)^{-1}(X’Y)$. Pulling out $Y$, We can write this as $e=(I-X(X’X)^{-1}X’)Y$.
Define $M\equiv I-X(X’X)^{-1}X’$ (the “residual maker”). Then, notice that for any vector, $Y$, $MY$ will yield the OLS residuals from a regression of $Y$ on$X$.
This is a commonly encountered type of “orthogonal projection matrix” (along with $P\equiv X(X’X)^{-1}X’$, so $M=I-P$). They are convenient largely because they are both idempotent (e.g. $MM=M$) and they are mutually orthogonal (e.g. $MP=PM=0$)–see Greene’s text for a number of proofs and derivations that rely on these properties.
could you explain the significance of this matrix in economics?
To my knowledge, the matrix has no particular economic interpretation or importance beyond its use in statistics/econometrics.
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