(
My apologies. I am often too busy to get back here very often and missed your followup posting. I had thought that I had answered the question and I just saw the updated thread today. Please feel free to followup on the discussion with the wider audience at cross validated.)
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That said, I thought that I was clear in my earlier responses on October 23 which said that EViews
does not use the "common way" (your term) OLS standard error calculation on the random effects transformed data but instead uses the GLS estimator of the standard errors. As I indicated, the differences in the standard error estimators were due to to this different choice of individual variance estimate and that by rescaling using the ratio of the standard error estimates you would obtain the desired matching results.
In practical terms, note that the fixed effects residual standard error estimate is about 0.1302487477370731 while the transformed data standard error estimate that you are using is 0.1352415529193776. If you take the ratio of the former to the latter you'll find that it is roughly equal to 0.9630823140186739, which as you note is exactly the constant scaling factor difference between the two estimates. This equivalence is not an accident.
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In full detail.
I just grabbed my old 1986 Hsiao off of the shelf. As I suspected (and speculated in my earlier responses), Hsiao pulls the individual error variance out of the GLS transform when solving the normal equations, which means that his moment matrix doesn't have the appropriate scale. As a result his estimator of the coefficient standard errors is not based on the standard GLS estimator but instead uses the pseudo-weighted least squares residual variance to obtain properly scaled standard errors.
Let's be more specific. Because he is focusing on the moment conditions, Hsiao employs a transformation that isn't the actual GLS transform, V^{-1/2}, but rather a pseudo-GLS transformation S^{-1/2} = sigma_v * V^{-1/2} that removes the individual scale. The Hsiao form of the RE estimator is (using the Z = S^{-1/2} X notation we used in our earlier discussion):
b_pgls = (Z'Z)^{-1} Z'y
= (X' S^{-1/2} S^{-1/2} X)^{-1} X' S^{-1/2} S^{-1/2} y
= (X' V^{-1/2} sigma_v^2 V^{-1/2} X)^{-1} X' V^{-1/2} sigma_v^2 V^{-1/2} Y
= (X' V^{-1/2} V^{-1/2} X)^{-1} X' V^{-1/2} V^{-1/2} Y
= (X' V^{-1} X)^{-1} X' V^{-1} Y
= b_gls
Note that for estimation of the coefficients, the use of the pseudo-GLS estimator yields estimates that are the same as the standard GLS estimator b_gps as we can simply include or exclude the common sigma_v from both terms since it factors out and cancels.
Now let's consider the b_pgls estimator of the variance. The "common way" OLS method you use is given by Hsiao as:
sigma_v^2 * ( Z'Z)^-1 = sigma_v^2 * (X' S^{-1/2} S^{-1/2} X)^-1
= sigma_v^2 * (X' V^{-1/2} sigma_v^2 V^{-1/2} X)^{-1}
which, if we assume that the two sigma_v's in the second line are identical, simplifies to the standard GLS estimator
= (X' V^{-1/2) V^{-1/2} X)^{-1}
= (X' V^{-1} X)^{-1}
Replacing sigma_v^2 in the first line with the variance from the pseudo-GLS transformed data (let's call this sigma_v(2)^2) yields your estimator. This is a perfectly reasonable estimator that follows from the pseudo-GLS transformation of the data.
I reiterate, however, that this is not the calculation that EViews is performing. For estimation, EViews uses the full GLS estimator:
b_gls = (X' V(1)^{-1/2} V(1)^{-1/2} X)^-1 X' V(1)^{-1/2} V(1)^{-1/2} Y.
where I am using V(1) to indicate that there is a first-stage estimator of the individual idiosyncratic variance embedded in this GLS transformation.Then the variance estimator that we use is the standard GLS estimator
(X' V(1)^{-1} X)^{-1} = (X' V(1)^{-1/2} V(1)^{-1/2} X)^{-1}
= (Z*'Z*)^{-1}
but where, as I wrote earlier, for our Z* transform "(t)he overall scaling is built into our GLS transform".
Substituting in the definition of the pseudo-GLS transform form, we have
= (X' S^{-1/2} 1/sigma_v(1)^2 S^{-1/2} X)^{-1}
= sigma_v(1)^2 (X' S^{-1/2} S^{-1/2} X)^{-1}
= sigma_v(1)^2 (Z'Z)^{-1}
where sigma_v(1)^2 is the variance estimate embedded in the GLS transformation.
The bottom line is that the difference between the two relevant estimators of the coefficient standard error scomes down to whether you wish to use the initial GLS estimator of the individual standard error, sigma_v(1), or the pseudo-GLS estimator of the individual standard error, sigma_v(2). The GLS estimate of the individual variance is obtained from the fixed effects model while the pseudo-GLS estimate comes from the weighted regression. EViews does the former, you are doing the latter. Both are consistent, but as you have seen can differ in finite sample.
One final note. EViews follows Baltagi's formulation for the random effects estimator, which is why we use the full GLS transformation, and which is why the EViews standard errors are computed as simple GLS standard errors. In our view (and this is not meant pejoratively), the Hsiao formulation relies on a calculational trick that applies only in a narrow range of cases where there is a variance scale that can be pulled out of the GLS transformation [Edit: though as noted below, this trick does have a nice property]. That approach is not wrong, but the EViews GLS calculation was consciously chosen so that conceptually, EViews is performing the same form of GLS variance calculation for RE as for estimation in panel models with various GLS cross-section or period weights.
Parenthetically, as noted in the discussion above, a third method that could be employed is
Another consistent estimator uses both a first and second stage estimator for Omega, as in
(X' Omega1^-1 X)^-1 (X' Omega^-1 Omega2 Omega^-1 X) (X' Omega1^-1 X)^-1
where Omega1 is the estimator obtained from the pre-GLS estimates, and the Omega2 are obtained using the GLS transform components and the GLS estimator residual scaling.
While I rather prefer this approach, it is not generally employed in GLS models.
[Edit: Though to be fair, in this setting this latter approach yields the Hsiao estimator.]