12 c =
G G
− 1 N
− 1 N
− K .
In general c
G G − 1 , though Section IIIB addresses an important exception
when fi xed effects are directly estimated. Some other packages such as SAS use c = G G – 1, a simpler correction that is also used by Stata for extensions to nonlinear
models. Either choice of c
usually lessens, but does not fully eliminate, the usual downward bias in the CRVE. Other fi nite- cluster corrections are discussed in Section
VI but there is no clear best correction.
D. Feasible GLS
If errors are correlated within cluster, then in general OLS is ineffi cient and feasible GLS may be more effi cient.
Suppose we specify a model for Ω
g
= E[u
g
′
u
g
| X
g
] in Equation 9, such as within-
cluster equicorrelation. Then the GLS estimator is ′
X Ω
−1
X
−1
′ X
Ω
−1
y
, where Ω = Diag[⍀
g
] . Given a consistent estimate
ˆ Ω
of Ω
, the feasible GLS estimator of 
is 13
ˆ 
FGLS
=
g =1 G
∑
′ X
g
ˆ Ω
g −1
X
g
⎛ ⎝
⎜ ⎞
⎠ ⎟
−1 g =1
G
∑
′ X
g
ˆ Ω
g −1
y
g
. The FGLS estimator is second- moment effi cient, with variance matrix
14 ˆ
V
def
[ ˆ 
FGLS
] = ′ X ˆ
Ω
−1
X
−1
, under the strong assumption that the error variance
Ω is correctly specifi ed.
Remarkably, the cluster- robust method of the previous section can be extended to FGLS. Essentially, OLS is the special case where
Ω
g
=
2
I
Ng
. The cluster- robust es- timate of the asymptotic variance matrix of the FGLS estimator is
15 ˆ
V
clu
[ ˆ 
FGLS
] = ′ X ˆ
Ω
−1
X
−1 g =1
G
∑
′ X
g
ˆ Ω
g −1
ˆ
u
g
ˆ ′
u
g
ˆ Ω
g −1
X
g
⎛ ⎝
⎜ ⎞
⎠ ⎟ ′
X ˆ Ω
−1
X
−1
, where
ˆ u
g
= y
g
− X
g
ˆ 
FGLS
. This estimator requires that u
g
and u
h
are uncorrelated when
g ≠ h , and that
G → ∞ , but permits
E[u
g
′
u
g
| X
g
] ≠ Ω
g
. The approach of speci- fying a model for the error variances and then doing inference that guards against
misspecifi cation of this model is especially popular in the biostatistics literature that calls
Ω
g
a “working” variance matrix see, for example, Liang and Zeger 1986. There are many possible candidate models for
Ω
g
, depending on the type of data being analyzed.
For individual- level data clustered by region, the example in Section IIB1, a com- mon starting point model is the random effects RE model. The error in Model 4 is
specifi ed to have two components: 16
u
ig
= ␣
g
+
ig
, where
␣
g
is a cluster- specifi c error or common shock that is assumed to be independent and identically distributed iid
0,
␣ 2
, and
ig
is an idiosyncratic error that is assumed to be iid
0,
2
. Then V[u
ig
] =
␣ 2
+
2
and Cov[u
ig
, u
jg
] =
␣ 2
for i ≠ j
. It follows that the intraclass correlation of the error
u
= Cor[u
ig
, u
jg
] =
␣ 2
␣ 2
+
2
so this model
implies equicorrelated errors within cluster. Richer models that introduce heteroskedas- ticity include random coeffi cients models and hierarchical linear models.
For panel data, the example in Section IIB2, a range of time series models for u
it
may be used, including autoregressive and moving average error models. Analysis of within- cluster residual correlation patterns after OLS estimation can be helpful in se-
lecting a model for Ω
g
. Note that in all cases if cluster- specifi c fi xed effects are included as regressors and
N
g
is small then bias- corrected FGLS should be used; see Section IIIC. The effi ciency gains of FGLS need not be great. As an extreme example, with
equicorrelated errors, balanced clusters, and all regressors invariant within cluster x
ig
= x
g
the FGLS estimator equals the OLS estimator—and so there is no effi ciency gain
to FGLS. With equicorrelated errors and general X, Scott and Holt 1982 provides an upper bound to the maximum proportionate effi ciency loss of OLS, compared to the
variance of the FGLS estimator, of 1 [1 + 41 −
u
[1 + N
max
− 1
u
N
max
×
u 2
] ,
N
max
= max{N
1
, ..., N
G
} . This upper bound is increasing in the error correlation
u
and the maximum cluster size N
max
. For low
u
, the maximal effi ciency gain can be low. For example, Scott and Holt 1982 notes that for
u
= 0.05 and N
max
= 20 there is at most a 12 percent effi ciency loss of OLS compared to FGLS. With
u
= 0.2 and N
max
= 100, the effi ciency loss could be as much as 86 percent, though this depends on the nature of X.
There is no clear guide to when FGLS may lead to considerable improvement in effi ciency, and the effi ciency gains can be modest. However, especially in models
without cluster- specifi c fi xed effects, implementation of FGLS and use of Equation 15 to guard against misspecifi cation of
Ω
g
is straightforward. And even modest effi ciency gains can be benefi cial. It is remarkable that current econometric practice with clus-
tered errors ignores the potential effi ciency gains of FGLS.
E. Implementation for OLS and FGLS
