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)
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