# Special case of Frisch-Waugh-Lowell theorem

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.

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.

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