Consider we have the following regression model:
where we have data on $N$
individuals for $T$
time periods. Now, if we estimate $beta$
by fixed effects, we only use within individual variation. As a result, the effect of time invariant regressors are not identified.
Now, if, lets say, $x$
contains the individual's state of residence. If we include state dummies, what do coefficients on those dummies identify? Where does the variation come from? In the extreme case, if all the variation in state residence is across individuals (i.e no individual moves states), then will this coefficient be identified?
If you have individual fixed effects, your estimate of the state dummy will be based upon within individual variation (i.e. it will be based upon the people that move across state lines). If no one switches state, then the state dummy will not be identified.