5.6 Feasible group level Random Effects

  1. Estimate the following equation using OLS

\[ y_{gi} = \mathbf{x}_{gi}\beta + \epsilon_{gi} \] and obtain the residuals \(e_{gi} = y_{gi} - \mathbf{x}_{gi}\hat{\beta}\) 2. Estimate the variance using the usual $s^2 estimator

\[ s^2 = \frac{1}{n-k}\sum_{i=1}^{n}e_i^2 \] as an estimator for \(\sigma^2_c + \sigma^2_u\) and estimate the within group correlation,

\[ \hat{\sigma}^2_c = \frac{1}{G} \sum_{g=1}^{G} (\frac{1}{\sum_{i=1}^{n_g-1}i}\sum_{i\neq j}\sum_{j}^{n_g}e_{gi}e_{gj}) \]

and plug in the estimates to obtain \(\hat{\Omega}\)

  1. The feasible group level RE estimator is obtained as

\[ \hat{\beta}= \mathbf{(X'\hat{\Omega}^{-1}X)^{-1}X'\hat{\Omega}^{-1}y} \]

Properties of the Feasible group level Random Effects Estimator

  • The infeasible group RE estimator is a linear estimator and is unbiased under A1-A3 for the unweighted equation
    • A3 requires \(E(\epsilon_{gi}|\mathbf{x}_i) = E(c_{g}|\mathbf{x}_i)+ (u_{gi}|\mathbf{x}_i)=0\) so we generally assume \(E(c_{g}|\mathbf{x}_i)+ (u_{gi}|\mathbf{x}_i)=0\). The assumption \(E(c_{g}|\mathbf{x}_i)=0\) is generally called random effects assumption
  • The Feasible group level Random Effects is biased
  • The Feasible group level Random Effects is consistent under A1-A3a, and A5a for the unweighted equation.
    • A3a requires \(E(\mathbf{x}_i'\epsilon_{gi}) = E(\mathbf{x}_i'c_{g})+ (\mathbf{x}_i'u_{gi})=0\)
  • The Feasible group level Random Effects estimator is asymptotically more efficient than OLS if the errors follow the random effects specification
    • If the errors do follow the random effects specification than the usual standard errors are consistent
    • If there might be a more complex dependence structure or heteroskedasticity, then we need cluster robust standard errors.