# Is this a valid derivation of FWL?

$$Y = X\hat{\beta} + Z\hat{\gamma} + e\\ (X'X)^{-1}X'Y = (X'X)^{-1}X'X\hat{\beta} + (X'X)^{-1}X'Z\hat{\gamma} + (X'X)^{-1}X'e\\ \hat{\beta}_{short} = \hat{\beta} + \hat{\delta}\hat{\gamma}$$

where $$\hat{\beta}_{short}$$ is the OLS estimator of the coefficient on $$X$$ from the misspecified model that omits $$Z$$, and $$\hat{\delta}$$ is from the regression $$Z = X\hat{\delta} + \eta$$.

This seems infinitely more simple than FWL proofs I've seen in econometrics textbooks. So what is this missing or getting wrong?

The Frisch-Waugh-Lovell Theorem states that the residuals of the regression of $$\boldsymbol y$$ on $$\boldsymbol Z$$, regressed on the residuals of the regression of $$\boldsymbol X$$ on $$\boldsymbol Z$$, gives $$\boldsymbol{{\hat\beta}}$$. To put it another way, let $$\boldsymbol{M}$$ be the residual maker. Then the regression of $$\boldsymbol{M_Z y}$$ on $$\boldsymbol{M_Z X}$$ gives $$\boldsymbol{{\hat\beta}}$$. The equation is: $$\boldsymbol{\hat\beta} = \left(\boldsymbol{X}'\boldsymbol{M_Z}'\boldsymbol{M_Z X}\right)^{-1}\boldsymbol X'\boldsymbol{M_Z}' \boldsymbol{M_Z y}$$ Note that the residual maker matrix is symmetric and idempotent.
An analogous result holds for $$\boldsymbol{\hat\gamma}$$.