In linear panel analysis, it can be desirable to estimate the magnitude of the fixed effects, as they provide measures of the unobserved components.
For instance, in wage equation regressions, fixed effects capture unobservables that are constant over time, such as motivation.
[1] Rather than differencing out the unobserved effect ci, Chamberlain proposed to replace it with the linear projection of it onto the explanatory variables in all time periods.
Specifically, this leads to the following equation where the conditional distribution of ci given xit is unspecified, as is standard in fixed effects models.
For instance, Mundlak follows a very similar approach, but rather projects the unobserved effect ci onto the average of all xit across all T time periods, more specifically [4] It can be shown that the Chamberlain method is a generalization of Mundlak's model.