<p>The result is not the <a href="https://en.wikipedia.org/wiki/Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem" rel="nofollow noreferrer">Frisch–Waugh–Lovell Theorem</a>. Rather, the result shows the extent of <a href="https://en.wikipedia.org/wiki/Omitted-variable_bias" rel="nofollow noreferrer">Omitted variable bias</a>.</p>

<p>The Frisch-Waugh-Lovell Theorem states that the <em>residuals</em> of the regression of <span class="math-container">$\boldsymbol y$</span> on <span class="math-container">$\boldsymbol Z$</span>, regressed on the <em>residuals</em> of the regression of <span class="math-container">$\boldsymbol X$</span> on <span class="math-container">$\boldsymbol Z$</span>, gives <span class="math-container">$\boldsymbol{{\hat\beta}}$</span>. To put it another way, let <span class="math-container">$\boldsymbol{M}$</span> be the residual maker. Then the regression of <span class="math-container">$\boldsymbol{M_Z y}$</span> on <span class="math-container">$\boldsymbol{M_Z X}$</span> gives <span class="math-container">$\boldsymbol{{\hat\beta}}$</span>. The equation is:
<span class="math-container">$$\boldsymbol{\hat\beta} = \left(\boldsymbol{X}'\boldsymbol{M_Z}'\boldsymbol{M_Z X}\right)^{-1}\boldsymbol X'\boldsymbol{M_Z}' \boldsymbol{M_Z y}$$</span>
Note that the residual maker matrix is symmetric and idempotent.</p>

<p>An analogous result holds for <span class="math-container">$\boldsymbol{\hat\gamma}$</span>.</p>

CC BY-SA 4.0
2019-10-05T16:49:46.107Z
2019-10-05T16:49:46.107Z
ahorn
<p><span class="math-container">$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}
$</span></p>

<p>where <span class="math-container">$\hat{\beta}_{short}$</span> is the OLS estimator of the coefficient on <span class="math-container">$X$</span> from the misspecified model that omits <span class="math-container">$Z$</span>, and <span class="math-container">$\hat{\delta}$</span> is from the regression <span class="math-container">$Z = X\hat{\delta} + \eta$</span>.</p>

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

<regression>
Is this a valid derivation of FWL?
user24465
# Is this a valid derivation of FWL?

(Asked by: username_0 on 2019-10-05)

$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?

**Tags:** regression


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## 1 answers

### Answer by username_1 on 2019-10-05. Score: 1

The result is not the [Frisch–Waugh–Lovell Theorem](https://en.wikipedia.org/wiki/Frisch%E2%80%93Waugh%E2%80%93Lovell_theorem). Rather, the result shows the extent of [Omitted variable bias](https://en.wikipedia.org/wiki/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 $\boldsymbol{\hat\gamma}$.


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