Is this a valid derivation of FWL?

Question

$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?

Answer 1

The result is not the Frisch–Waugh–Lovell Theorem. Rather, the result shows the extent of Omitted variable bias.

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 $\boldsymbol{\hat\gamma}$.