5.6 Feasible group level Random Effects

\[ y_{gi} = \mathbf{x}_{gi}\beta + \epsilon_{gi} \]

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 $\hat{\Omega}$

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

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.