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I am trying to understand the result of the Frisch-Waugh-Lovell Theorem that we can partial out a set out regressors. The model I am looking at is $y=X_1\beta_1 + X_2\beta_2 +u$

So the first step would be to regress $X_2$ on $X_1$: $$ \begin{align} X_2&=X_1\hat{\gamma}_1+\hat{w}\\ &=X_1\hat{\gamma}_1+M_{X_1}X_2 \end{align} $$ with $M_X$ being the orthogonal projection matrix ($M_X=I-P_X$). The second step is then to regress $y$ on $X_1$: $$ \begin{align} y&=X_1\hat{\gamma}_2+\hat{v}\\ &=X_1\hat{\gamma}_2+M_{X_1}y \end{align} $$ And the third step would be to regress the residuals of the previous regressions on each other: $$ \begin{align} \hat{w}&=\hat{v}\beta_2+u\\ M_{X_1}y&=M_{X_1}X_2\beta_2+u \end{align} $$ Can somebody tell me whether this would be a correct way of describing this result and in particular whether the residuals in the last equation $u$ are equal to the residuals in the model?

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To see the residuals are the same consider that by definition of the residuals $$y = X_1\hat \beta_1 + X_2\hat\beta_2 + \hat u$$

which is the first regression and its residuals $\hat u$. Now multiply with $M_{X_1}$ to get

$$M_{X_1}y = M_{X_1}X_2\hat\beta_2 + \hat u,$$

because $M_{X_1} X_1 = \mathbf 0$ and $M_{X_1}\hat u = \hat u - P_{X_1}\hat u = \hat u$ because $P_{X_1}\hat u = \mathbf 0$. But the second equation is the regression of the residuals $M_{X_1}y$ from regression $y$ on $X_1$, regressed on the residuals $M_{X_1}X_2$ being residuals from regression of $X_2$ on $X_1$. The residuals in the second equation are however also $\hat u$ so the residuals from the two regressions must be the same.

Heres R code to illustrate:

N <- 1000
z <- rnorm(N) # just a way to make x1 and x2 correlated
x1 <- rnorm(N) + z
x2 <- rnorm(N) + z
cov(x1,x2)
y <- 2 + 2*x1 - 2*x2 + rnorm(N) 

e_y_x1 <- lm(y~x1)$residuals
e_x2_x1 <- lm(x2~x1)$residuals

u1 <- lm(e_y_x1~e_x2_x1)$residuals
u2 <- lm(y ~ x1 + x2)$residuals

plot(u1,u2)
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