Random 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

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

• 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.

Random and fixed individual specific effect

• If the explanatory variable is exogenous with respect to the individual specific effects \(α_i\) ,

\[E\left( x_i \right)=0, i=1,…,n \]

• then we call it a random individual specific effect . Otherwise we call it a fixed individual specific effect .
• If \(α_i\) is correlated with any \(x_{i,t}\) : fixed
• If \(α_i\) is independent of all \(x_{i,t}\) : random

Random effects model, OLS

• A linear regression model with random specific is called a random effects model
• OLS will still be consistent and unbiased.
• OLS is no longer efficient and the standard errors will be inconsistent as the error terms will no longer satisfy the GM conditions.

The random effects estimator

• The random effects (RE) estimator of this model is consistent and asymptotically efficient.
• The standard errors of the RE estimator are consistent and it is possible to estimate the individual specific effects from the RE estimator.
• 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.