WITHIN
ESTIMATOR (The xtreg,fe
command)
The individual-spesific-effects
model for the scalar dependent variable \({{y}_{it}}\) specifies that;
\({{y}_{it}}=\alpha +{{x}_{it}}\beta
+{{\varepsilon }_{t}}\) (1)
where \({{x}_{it}}\) are regressor, \({{\alpha }_{i}}\) are random
individual-spesific-effects, and \({{\varepsilon }_{it}}\) is and idiosyncratic error.
In the fixed-effect (FE) model, the \({{\alpha }_{i}}\) in the model
Eq(1) can be eliminated by subtraction of the corresponding model for
individual means;
\({{\bar{y}}_{i}}={{\alpha
}_{i}}+{{\bar{x}}_{i}}\beta +{{\bar{\varepsilon }}_{i}}\) (2)
where, for example \(\bar{x}=T_{i}^{-1}\sum\nolimits_{t=1}^{{{T}_{i}}}{{{x}_{it}}}\)
Subtracts Eq(1) to Eq (2)
\(\left(
{{y}_{it}}-{{{\bar{y}}}_{i}} \right)=\left( {{x}_{it}}-{{{\bar{x}}}_{it}}
\right)'\beta +\left( {{\varepsilon }_{it}}-{{{\bar{\varepsilon }}}_{i}}
\right)\) (3)
Because \({{\alpha }_{i}}\) has been eliminated, OLS leads to
consistent estimates of \(\beta\) even
if \({{\alpha }_{i}}\) is correlated
with \({{x}_{it}}\) as in case of the FE
model.
This results give great advantage of panel data.
The disadvantage is inability to estimate the coefficients or a
time-invariant regressor.
Also within estimator will be relatively imprecise for time-varying
regressors that vary little over time.
Stata actually fit the model;
\(\left(
{{y}_{it}}-{{{\bar{y}}}_{i}}+\bar{\bar{y}} \right)=\alpha +\left(
{{x}_{it}}-{{{\bar{x}}}_{it}}+\bar{\bar{x}} \right)'\beta +\left( {{\varepsilon
}_{it}}-{{{\bar{\varepsilon }}}_{i}}+\bar{\bar{\varepsilon }} \right)\) (4)
where, for example \(\bar{\bar{y}}=\left(
1/N \right){{\bar{y}}_{i}}\) is the
grand mean of \({{y}_{it}}\) .
This parameterization has the advantage of providing an intercept
estimate, the average of the individual effects \({{\alpha }_{i}}\), while
yielding the same slope estimate \(\beta \) as that from the within
model.
The within estimator is computed by using xtreg command
with the fe option.
The default standard error assume that after controlling for \({{\alpha
}_{i}}\) , the error \({{\varepsilon }_{it}}\) is i.i.d.
The vce (robust)
option relaxes this assumption and provides cluster-robust standard error,
provided that observations are independent over \(i\) and \(N\to \infty \) .
To estimate Eq(4) using same variables before when we discuss within
and between variation ,
xtreg
lwage exp exp2 wks ed,fe vce(cluster id)
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|
Coefficient of
edu is not identified because
the data on education is time-invariant.
WITHIN ESTIMATOR (LSDV
Regression- areg command)
Another name for the within estimator is the least-square
dummy-variable (LSDV) estimator.
This because it can be shown to equal the estimator obtained from
OLS estimation of \({{y}_{it}}\) on \({{x}_{it}}\)
and \(N\) individual-specific indicator variables \({{d}_{j,it,}}j=1,...,N\),
where \({{d}_{j,it,}}=1\) for the it-th observation if \(j=1\) , and \({{d}_{j,it,}}j=0\)
otherwise.
Thus, we fit the model;
\({{y}_{it}}=\left(
\sum\nolimits_{j=1}^{N}{{{\alpha }_{i}}{{d}_{j,it}}{{x}_{it}}\beta }
\right)+{{\varepsilon }_{it}}\) (5)
This equivalence of LSDV and within estimators does not carry over
to nonlinear models.
To estimate Eq(5) using same variable before,
areg
lwage exp exp2 wks ed, absorb(id) vce(cluster id)
The coefficient estimates are the same as those from xtreg,fe.
The cluster-robust standard error differ because of difference
small-sample correction.
Thus, xtreg,fe should be used.
This difference arises because inference for areg is
designed for case where \(N\) is fixed and \(T\to \infty \) , whereas we are considering short
panel case, where \(T\) is fixed and \(N\to \infty \).
BETWEEN ESTIMATOR (The xtreg,be command)
Uses only between or cross-section variation in the data and is the
OLS estimator from the regression of \({{\bar{y}}_{i}}\) on \({{\text{x}}_{it}}\).
Because only cross-section variation in the data is used, the
coefficient of any individual-invariant
regressor (such as time dummies) cannot be identified.
The between estimator is inconsistent in the FE model but consistent
in the RE model.
To explain this, average the individual-effects model Eq(1) to
obtain between model;
\({{\bar{y}}_{i}}={{\alpha
}_{i}}+{{\bar{x}}_{i}}\beta +\left( {{\alpha }_{i}}-\alpha +{{{\bar{\varepsilon
}}}_{i}} \right)\) (6)
The between estimator is the OLS estimator in this model.
Consistency requires that the error term \(\left( {{\alpha
}_{i}}-\alpha +{{{\bar{\varepsilon }}}_{i}} \right)\) be uncorrelated with \({{x}_{it}}\)
This is the case if \({{\alpha }_{i}}\) is a random effect but not
if \({{\alpha }_{i}}\) is a fixed effect.
To estimate Eq(6) using same variable before, in Command window;
xtreg
lwage exp exp2 ed,be
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The estimates and standard error are closer to those obtained from
pooled OLS than from within estimation.
