Fixed effects model

Summary

Setup

• Stationary balanced panel with one explanatory variable
• A linear regression model

$$y_{i,t}=β_1+β_2x_{i,t}+ε_{i,t} , i=1,…,n , t=1,…,T$$

• The one-way error component model with individual specific effects

$$ε_{i,t}=α_i+μ_{i,t}, i=1,…,n , t=1,…,T$$

• $α_i$ is a fixed individual specific effect ( $α_i$ may be correlated with $x_{i,t}$ )
• The explanatory variable is exogenous with respect to the error terms $μ_{i,t}$ , $E\left( μ_{i,t}|x_i \right)=0$
• The error terms $μ_{i,t}$ are homoscedastic and not autocorrelated.

The fixed effects model, OLS

• OLS is inconsistent and biased (the explanatory variable is endogenous)

The fixed effects estimator

• The fixed effects (FE) estimator is consistent.
• The standard errors of the FE estimator are consistent and it is possible to estimate the individual specific effects.
• The model can be extended to several explanatory variables, one-way error component model with time specific effects and a two-way error component model.

Individual specific variation

• We say that there is no individual specific variation over time in the explanatory variable if $x_{i,t}=c_i$ for all $i=1,…,n, t=1,…,T$
• The FE estimator is available only if there is some individual specific variation over time in all the explanatory variables.