The Hausman test for random effects

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

\[ε_{i,t}=α_i+μ_{i,t}, i=1,…,n , t=1,…,T\]

• 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.
• There is some individual specific variation over time in the explanatory variable

The Hausman test

• If the individual specific effects are fixed, then the RE estimator and the FE estimator of \(β_2\) will converge to different values .
• If the individual specific effects are random, then the RE estimator and the FE estimator of \(β_2\) will converge to the same value . The RE estimator is more efficient .
• The null-hypothesis is \(H_0\) : the individual specific effects are random. If the difference between the RE-estimator and the FE estimator is significantly different from zero then \(H_0\) is rejected.