Fixed Effects Model - Formal Description

Formal Description

Consider the linear unobserved effects model for observations and time periods:

for and

where is the dependent variable observed for individual at time is the time-variant regressor matrix, is the unobserved time-invariant individual effect and is the error term. Unlike, cannot be observed by the econometrician. Common examples for time-invariant effects are innate ability for individuals or historical and institutional factors for countries.

Unlike the Random effects (RE) model where the unobserved is independent of for all, the FE model allows to be correlated with the regressor matrix . Strict exogeneity, however, is still required.

Since is not observable, it cannot be directly controlled for. The FE model eliminates by demeaning the variables using the within transformation:

where and . Since is constant, and hence the effect is eliminated. The FE estimator is then obtained by an OLS regression of on .

Another alternative to the within transformation is to add a dummy variable for each individual . This is numerically, but not computationally, equivalent to the fixed effect model and only works if the number of time observations per individual, is much larger than the number of individuals in the panel.