I can't seem to figure this out algebraically, or intuitively.

Same with a result like $M_X M_X = M_X$ which I know is the idempotent property for a residual maker.

If I think of $M_X X = 0$, this makes sense to me, because it's saying the vector of least-squares residuals of regressing X on itself is 0 - because X perfectly predicts itself.

But what's the interpretation when we have two residual makers? What exactly is the regression?

  • $\begingroup$ I have edited the title pls. see if this is what you meant. Also not that you have to specify the relationship between $X_2$ and $X$. Is it for example the case that $X = [X_1 X_2]$ hence $X_2$ being part of $X$ or what? $\endgroup$ Commented Mar 16, 2020 at 7:55
  • $\begingroup$ Sorry for the ambiguity. The title is correct, and yes, X_2 is a column vector of X $\endgroup$ Commented Mar 16, 2020 at 8:21

1 Answer 1


If you take some vector $y$ and regress on $X$ you get residuals $\hat \epsilon = M_Xy$ which is why $M_X$ is sometimes referred to as the "residual maker matrix". Let assume you decide to take these residuals and regress them on $X_2$ where $X = [X_1 \ X_2]$ then the identity that $M_{X_2}M_X=M_X$ for $X = [X_1 \ X_2]$ tells you that the residuals $\hat u$ from that regression will be $\hat u=M_{X_2}\hat \epsilon =M_{X_2} M_Xy = M_Xy = \hat \epsilon$ the same.

You could say that the first regression takes all the variation in $y$ that systematically varies with $X$ and puts the unsystematic variation in $\hat \epsilon$. Since $X_2$ is included in $X$ there is no variation left in these residuals that systematically varies with $X_2$.


To show that $M_{X_2}M_X=M_X$ for $X = [X_1 \ X_2]$.

It should makes sense to you that $$P_X X = X,$$ which can be read as for any column in $X$ the predicted values from a regression of the column unto $X$ is the column itself.

And by partioning this implies that $P_X [X_1 \ X_2] = [X_1 \ X_2]$ hence $$P_XX_2 = X_2,$$ stating simply that since it holds for any column of $X$ it must hold for some selection of the columns in $X$ the selection being $X_2$.

Using this it is easy to show that $P_{X_2}P_X=P_{X_2}$. Consider that

$$P_{X_2}P_X=X_2(X_2^\top X_2)^{-1}X_2^{\top}P_X$$

and notice that the final part $X_2^{\top}P_X$ is simply $(P_XX_2)^\top$ due to symmetry of $P_X$. But you know from above that $P_XX_2=X_2$ so $(P_XX_2)^\top = X_2^\top$. It follows then that

$$P_{X_2}P_X=X_2(X_2^\top X_2)^{-1}X_2^{\top}P_X = X_2(X_2^\top X_2)^{-1}X_2^{\top} =P_{X_2}$$ Finally consider that

$$M_{X_2}M_X = (I-P_{X_{2}})(I-P_X) = I -P_{X} - P_{X_2}+ P_{X_2}P_X = I -P_{X} = M_X$$ using from above that $P_{X_2}P_X=P_{X_2}$.

  • $\begingroup$ Thank you so much for your answer. Can you also write an econometrics textbook? I've never seen someone give such a clear explanation of what is going on. Instead I have to cross-reference three textbooks to check the behind-the-scene lin alg proofs and intuitive meaning behind the concepts being taught. $\endgroup$ Commented Mar 16, 2020 at 10:05
  • $\begingroup$ I can't get over how good your answer is ffs im gonna cry. Where have you been my whole life $\endgroup$ Commented Mar 16, 2020 at 10:12
  • $\begingroup$ The beauty derives from the underlying mathematical theory of projections and vector spaces. So in general the point is to think about what the matrices $P_X$ and $M_X$ does to a vector. There is a rich geometric interpretation Gilbert Strang lectures on youtube are a good source to understand vectorspaces and Davidson and McKinnon an advanced econometrics book has a chapter on regression where they give a geometric interpretation using these projection matrices $P_X$ and $M_X$. I am happy you liked the answer. $\endgroup$ Commented Mar 16, 2020 at 10:27
  • $\begingroup$ I'll be sure to check them out. It's been a while since I did linear algebra. Unfortunately, I never really did grasp the geometric intuition, but Greene's appendix has helped a lot with that. $\endgroup$ Commented Mar 16, 2020 at 10:33

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