Below is a simulation of the consequence of controlling for fixed effects. In some cases this is desirable, in others not.
In the wage equation below, the wage depends on gender (b = 2), effort (b = 10), and ability (b = 3). There is no unobserved heterogeneity. A straightforward OLS estimation will return the correct b's.
However, as gender and ability are panel-invariant, the fixed effect regression does not yield any effect -- although the effect of effort is unbiased.
On the other hand, say that ability and effort are correlated, and ability is not observed, the estimated beta if effort in OLS would be biased, but not in a fixed effects estimation. The latter, however, would not allow estimating the beta for gender.
Good riddance.
clear
gene byte gender = .
gene float effort = .
gene float wage = .
gene float ability = .
gene long id = .
forvalues i = 1/1000 {
local a = runiform()
local g = runiform() > .5
set obs `=`i'*5'
replace ability = `a' if missing(id)
replace gender = `g' if missing(id)
replace id = `i' if missing(id)
}
replace effort = runiform()
replace wage = 2*gender + 10*effort + 3*ability + .5*runiform()
regress wage gender effort ability
areg wage gender effort ability, abs(id)
/*
. regress wage gender effort ability
Source | SS df MS Number of obs = 5,000
-------------+---------------------------------- F(3, 4996) > 99999.00
Model | 50282.7806 3 16760.9269 Prob > F = 0.0000
Residual | 104.339671 4,996 .020884642 R-squared = 0.9979
-------------+---------------------------------- Adj R-squared = 0.9979
Total | 50387.1202 4,999 10.0794399 Root MSE = .14452
------------------------------------------------------------------------------
wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
gender | 1.999089 .0040916 488.58 0.000 1.991067 2.00711
effort | 10.00101 .0070513 1418.33 0.000 9.987187 10.01483
ability | 3.004343 .0071066 422.75 0.000 2.990411 3.018275
_cons | .2498592 .0058902 42.42 0.000 .2383118 .2614065
------------------------------------------------------------------------------
.
end of do-file
. do "/var/folders/p2/2v2ckxtd2794655ypfbmfg9w0000gn/T//SD31177.000000"
. areg wage gender effort ability, abs(id)
note: gender omitted because of collinearity
note: ability omitted because of collinearity
Linear regression, absorbing indicators Number of obs = 5,000
F( 1, 3999) = 1602779.17
Prob > F = 0.0000
R-squared = 0.9983
Adj R-squared = 0.9979
Root MSE = 0.1450
------------------------------------------------------------------------------
wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
gender | 0 (omitted)
effort | 10.00054 .0078993 1266.01 0.000 9.985056 10.01603
ability | 0 (omitted)
_cons | 2.810143 .0044419 632.65 0.000 2.801435 2.818852
-------------+----------------------------------------------------------------
id | F(999, 3999) = 401.808 0.000 (1000 categories)
*/
