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