Why does Frisch-Waugh produce inconsistent estimates with Weighted Least Squares?

Question

Frisch-Waugh's theorem states that in the setup $Y = X^T \beta $, where $\beta = [\beta_1,\beta_2]^T$, $X = [X_1, X_2]$, $\hat{\beta}_2$ obtained from the multiple regression is the same as that obtained from a two-stage specification, i.e. $\hat{\beta}_2 = (X_2^T X_2)^{-1} X_2^T(Y-X_1\hat{\beta}_1)$

This procedure should also hold for WLS:

$\hat{\beta}_2 = (X_2^T \Omega^{-1}X_2)^{-1}X_2^T \Omega^{-1}(Y-X_1\hat{\beta}_1)$

However, I find different coefficient estimates for a single multiple regression than for two separate regression.

From the equation I derived, weights should be used in both parts of the two-part regressions, but just in case, I tried adding weights to only the first or second regression as well.

A simple reproducible example in R:

# load dataset from MASS package
data(cabbages)
# Here we see exactly what the coef on Cult should be:
target <- lm(VitC ~ Date + Cult, weight=HeadWt, data=cabbages)
# We see that the coef of Cultc52 is 12.69

# Weights on both equations: 
one_f <- lm(VitC ~ Date,weight = HeadWt,data=cabbages)
(one_s <- lm(resid(one_f) ~ Cult,weight=HeadWt,data=cabbages))
# 12.42 

# Weights on the first equation only
two_f <- lm(VitC ~ Date, weight=HeadWt,data=cabbages)
(two_s <- lm(resid(two_f) ~ Cult, data=cabbages))
# 12.90... hmmm... 

# Weights on second equation only
three_f <- lm(VitC ~ Date,data=cabbages)
(three_s <- lm(resid(three_f) ~ Cult,weight=HeadWt,data=cabbages))
# 12.79

Answer 1

Your statement of Frisch-Waugh is actually incorrect; essentially you can partial $X_1$ out of $Y$, but you have to partial it out of $X_2$. you want the following:

$$ \hat{\beta}_2=(X_2M_{x1}X_2)^{-1}X_2M_{x1}Y $$

Where

$$ M_{x2}=(I-X_2(X_2X_2)^{-1}X_2X_1) $$

That does work with weights in the first and second stage.