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.