## 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.