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The formulation is just a special case of FWL. Say we have a partitioned regression, $Y=X_1\beta_1+X_2\beta_2+\epsilon$ but with $X_2$ be $n\times 1$ and $\beta_2$ a constant. Let $b_1,b_2$ be two OLS estimators of $\beta_1,\beta_2$ by regressing $Y$ on $X_1,X_2$. $\tilde \beta_1$ be ols estimator when regress $Y$ on $X_1$ only, and $\delta$ be estimator when regress $X_2$ on $X_1$. We want to show $\tilde\beta_1=b_1+\delta b_2$.

I think this should be a corollary of FWL, and I can write out the values of these coefficients using matrix multiplications of $X_1,X_2,Y$, but I don't know what to do next. Should I just try to manipulate matrices? But I can't get to the result.

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I figured it out, the idea is just matrix multiplication.

Obtain $b_1,b_2$ by regressing $Y$ on $X_1,X_2$. We then have $$Y=X_1b_1+X_2b_2+MY.\tag{1}\label{1}$$

Obtain $\tilde\beta_1$ by regressing $Y$ on $X_1$:$$Y=X_1\tilde\beta_1+M_1Y.\tag{2}\label{2}$$

Finally regress $X_2$ on $X_1$ to obtain $\delta$, $$X_2=X_1\delta+M_1X_2.$$

Substitute $X_2$ into the first equation \eqref{1}, $$Y=X_1b_1+(X_1\delta+M_1X_2)b_2+MY,$$ so we have $$Y=X_1(b_1+\delta b_2)+M_1X_2b_2+MY.$$

Premultiply the first equation by $M_1$ to get $$M_1Y=M_1X_1b_1+M_1X_2b_2+M_1MY=M_1X_2b_2+MY,$$ where we use that the residuals of a regression on $X_1$ of the residuals of a regression on $X_1$ and $X_2$ are still $MY$, i.e. $M_1MY=MY$.

So the second equation \eqref{2} becomes $$Y=X_1\tilde\beta_1+M_1X_2b_2+MY.$$

Combining them gives the desired result.

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  • $\begingroup$ +1 The argument also works for general $X_2$ with, then $\delta$ a matrix and $b_2$ a vector of regression coefficients. $\endgroup$ Commented Apr 30 at 9:04

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