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.