Both the coefs and the standard errors are incorrect. For instance, consider this example:
On the other hand, approximations like the one you mentioned have been used and published in the past because they were thought as "good enough". However, they are not really "good enough" as e.g. Gormley & Matsa (2014) discuss.
Code:
* Create dataset
sysuse auto, clear
global vars price weight length
rename turn id
sort id, stable
by id: gen time = _n
xtset id time
* Wrong
foreach var of varlist $vars {
by id: egen `var'_m = mean(`var')
gen `var'_dm = `var' - `var'_m
}
areg *_dm, absorb(time)
* Correct
areg price weight length i.time, a(id)
* Correct
reghdfe price weight length, a(id time) keepsingleton

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