Pooled or Population-Average Estimators with Stata (Panel)

Pooled estimator simply regress ${{y}_{it}}$ on an intercept and ${{x}_{it}}$ using both between (cross-section) and within (time-series) variation in the data.

Pooled or population-average estimator are consistent if the RE model is appropriate and are inconsistent if the FE model is appropriate.

From discussion in within and between variation,  the individual-spesific-effects model for the scalar dependent variable ${{y}_{it}}$ specifies that;

${{y}_{it}}={{\alpha }_{i}}+{{\text{{x}'}}_{it}}\beta +{{\varepsilon }_{it}}$                                                            (1)

where ${{\text{{x}'}}_{it}}$ are regressor, ${{\alpha }_{i}}$ are random individual-spesific-effects, and ${{\varepsilon }_{it}}$ is and idiosyncratic error

Pooled OLS estimator can be motivated from the individual-effects model by rewriting Eq (1) as the pooled model.

$lwage=\alpha +{{\beta }_{1}}ex{{p}_{it}}+{{\beta }_{2}}exp{{2}_{it}}+{{\beta }_{3}}wk{{s}_{it}}+{{\beta }_{4}}e{{d}_{it}}+\left( {{\alpha }_{i}}-\alpha +{{u}_{it}} \right)$                                (2)

Consistency of OLS requires that the error term $\left( {{\alpha }_{i}}-\alpha +{{u}_{it}} \right)$ be uncorrelated with independent variables.

So pooled OLS is consistent in the in the RE model but inconsistent in the FE model because then ${{\alpha }_{i}}$  is correlated with independent variables.

The pooled estimator, or PA estimator is obtained by using the xtreg command with the pa option.

The option is corr( ) for place different restriction on the error correlation, and vce(robust) to obtain cluster-robust standard errors.

We use the data Paneldata01.

xtset id t

Assumed that there are AR(2) error process in model Eq(2), to estimate PA,

xtreg lwage exp exp2 wks ed, pa corr(2) vce(robust)