Random effects versus fixed 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 error terms \(μ_{i,t}\) are homoscedastic and not autocorrelated.
• There is some individual specific variation over time in the explanatory variable

The RE and the FE estimator

• The RE estimator of \(β_2\) is an estimator of

\[ \frac{dE\left( x \right)}{dx}\]

• The FE estimator of \(β_2\) is an estimator of

\[ \frac{dE\left( x,α \right)}{dx}\]

With random specific effects:

\[ \frac{dE\left( x,α \right)}{dx}= \frac{dE\left( x \right)}{dx}\]

• \(b_2\) from pooled OLS, RE and FE are all consistent estimators of \(dE\left( x,α \right)/dx\)
• \(SE\left( b_2 \right)\) from pooled OLS will be inconsistent, \(SE\left( b_2 \right)\) from RE and FE is consistent
• The RE estimator is more efficient (asymptotically) than pooled OLS and FE.

With fixed specific effects:

• \(b_2\) from pooled OLS and RE are inconsistent estimators of \(dE\left( x,α \right)/dx\)
• The fixed effects estimator is consistent