Hi,
I need to implement the Frisch Waugh-Lovell-theorem in Stata 15 MP (64-bit) in the context of a research project. To illustrate my problem, I would like to abstract from my actual problem and focus on the following MWE. In the example, I'd like to show that the coeffcient on headroom can be obtained in two ways, either through a standard OLS estimation with two regressors in total or through partialling out of the first regressor, trunk.

Code:
sysuse auto2, clear

* Multivariate regression
reg price trunk headroom  

* Partialling out
reg headroom trunk, vce(robust)
predict double resid_x2, res

reg price trunk
predict double resid_y, res

reg resid_y resid_x2
​​​​​​
My trivial question is: Why are the standard errors of the variable headroom from the multiple regression and the standard error of the partialled out coeffcient on headroom not exactly equal? The coeffcients themselves correspond (which I wanted to see), however, I seem to not understand this procedure properly, since actually also the standard errros should correspond, right? Where am I not seeing the mistake?

Thank you very much in advance.