This note provides Monte Carlo evidence illustrating that feasible and true GLS estimators of the Baltagi and Griffin (1988) generalized error component model do not have the same sampling behavior. Indeed, while the true GLS estimator is consistent, a feasible GLS estimator need not be, an observation corroborated by the Monte Carlo results.

Robert Phillips

English

Research Papers in Economics

Some Monte Carlo results for a generalized error component
model with heteroskedastic disturbances 
Robert Phillips
George Washington University
Abstract
This note provides Monte Carlo evidence illustrating that feasible and true GLS estimators of
the Baltagi and Griffin (1988) generalized error component model do not have the same
sampling behavior. Indeed, while the true GLS estimator is consistent, a feasible GLS
estimator need not be, an observation corroborated by the Monte Carlo results.
Citation: Phillips, Robert, (2003) "Some Monte Carlo results for a generalized error component model with heteroskedastic
disturbances." Economics Bulletin, Vol. 3, No. 15 pp. 1−4
Submitted: June 13, 2003.  Accepted: July 16, 2003.
URL: http://www.economicsbulletin.com/2003/volume3/EB−03C20007A.pdf1. Introduction 
 
Baltagi and Griffin (1988) suggested a generalization of a standard panel data model given by 
 
   (1)  ,         ,       ( 1, , ,   1, , ) it it it it i it yx u u v iN t αβ µ ′ =+ + = + = = …… T
 
In this expression  and   are   vectors of regressors and slope coefficients, α  is an overall 
intercept term, and the regression error (u ) consists of a heterogeneity component ( ) and a 
remainder term ( ).  The error components   and   are typically assumed to satisfy 
it x
it v
β 1 K ×
it i µ
i µ it v
2 ~( 0 , ) i µ µσ
i µ
 all  , and   all i and t, with the  ’s and  ’s all mutually independent.  
Baltagi and Griffin (1988), however, recognizing that heteroskedasticity may be a common 
problem when working with panel data, generalized this model by allowing for the possibility that 
each   has a distinct variance 
i
2) v σ
2
i
~( 0 , vit i µ it v
µ σ  ( i ).  This model is also discussed by Baltagi (1995, 
pp. 77-80), Greene (2000, pp. 580-581; 2003, pp. 316-317), and Phillips (2003). 
1 , …, N =
Baltagi and Griffin (1988) proposed feasible generalized least-squares (GLS) estimators 
based on estimators of 
2
1,,N
2
µµ σ  and σ , feasible GLS estimators which they claimed have the 
same asymptotic distribution as the true GLS estimator, that is, the GLS estimator based on the true 
variance components 
σ …
22
1,,N
2
v
µµ σ  and σ . That claim, however, has been questioned by Greene 
(2003) and Phillips (2003). The usual asymptotic equivalence between feasible and true GLS 
breaks down because, regardless of how large   and T  are, we have only one draw from the 
distribution generating   ( i ), which makes consistent estimation of 
σ …
i µ 1 , = …
2
v
N
N
,
2
i µ σ  impossible. This 
observation led Greene to point out that the efficiency of GLS estimators based on estimated 
2
i µ σ ’s 
“seems unlikely” (Greene 2003, p. 617), and Phillips (2003) goes even further by providing an 
example in which a feasible GLS estimator is not even consistent while the true GLS estimator is 
consistent. This note provides Monte Carlo evidence confirming that, for the Baltagi-Griffin model, 
one can learn little about the sampling behavior of a feasible GLS estimator by studying the 
sampling behavior of the true GLS estimator. 
 
2. Monte Carlo Experiments 
 
The finite sample behavior of one of the feasible GLS estimators considered in Baltagi and Griffin 
(1988) was investigated with Monte Carlo experiments. The sampling design relied on the model in 
(1) with  ,  , and  . The explanatory variable   was generated according to 
, where ε  for all i and t, while the d ’s were generated as 
independent Bernoulli random variables with  . Moreover, for most of the 
experiments, the regression errors were generated as  , with  ,  , 
and  . For each combination of   and   considered, the parameter values 
1 K =
36 i ε +
(0 IIN
2
N
1 α=
t
1)
1 β=
~ IIN
it x
2
it i xd =− +
~ , i ω
2
1,,
(0,1) it i
~ it
Pr(
N
1)
i µ =
T
1/ i d ==
it it uv + (0,1) vI I N ii µ µσ ω = i
µµ σ … σ  were generated once from a uniform distribution over the interval from   to 5 and 
then held fixed over all samples drawn for that combination of   and T . The values of   and T  
considered were 5, 10, 50, 100, and  , for a total of   different  , T combinations. 
0
N N
25 200 N
  1 TABLE I:  Relative efficiency of slope estimators: the RMSE of within (W) and feasible GLS 
(FGLS) estimates over the RMSE of true GLS estimates. Each relative efficiency estimate is based 
on 2500 independent samples. 
 
 
      N
 
      T   5 =
  W    FGLS 
 
     T   10 =
  W    FGLS 
 
     T   50 =
  W    FGLS 
 
    T   100 =
  W    FGLS 
 
    T   200 =
  W    FGLS 
 
    5 
  10 
  50 
100 
200 
 
 
1.30 1.19 
1.47 1.30 
1.52 1.39 
1.55 1.44 
1.53    1.46 
 
1.10 1.17 
1.20 1.33 
1.32    1.48 
1.27 1.47 
1.41    1.50 
 
 
1.03 1.25 
1.08 1.48 
1.07 1.71 
1.16 1.85 
1.06    1.79 
 
1.03 1.24 
1.02 1.44 
1.03 1.79 
1.04 1.93 
1.06    2.01 
 
1.02 1.26 
1.02 1.52 
1.03 2.04 
1.02 2.11 
1.02    2.16 
 
 
 
 
For estimators of the slope coefficient I considered the within estimator, the true GLS 
estimator, and a feasible GLS estimator proposed by Baltagi and Griffin (1988). To construct the 
latter estimator, the variance components had to be estimated. To that end, I used one of the 
iterative methods recommended by Baltagi and Griffin (1988). The method requires first applying 
ordinary least-squares (OLS) to obtain OLS residuals (u ) and then estimating 
 with σ .  Then the sample residual variance from 
the within regression ( ) is subtracted from σ  to obtain σσ , an estimate of the 
variance component 
ˆit
22 ˆˆ iu =
22 var( ) ui it i v u µ σσ == +
s
2
i
2 22
1 ˆˆ /( 1)
T
ui it t uTK
= =− ∑
2 ˆui
σ −
2
v
2
v i s µ −
µ σ . The estimates   and 
2
v s
2
1 ˆ ,,
2 ˆ N µµ σ  are then used to construct a feasible 
GLS estimate of  . After the feasible GLS estimate of   is obtained, the procedure is 
repeated using the feasible GLS residuals to re-estimate the 
σ …
( , b = ) ′′ αβ b
2
i µ σ ’s, which are used in turn to 
construct a new feasible GLS estimate, and so on, until the iterative scheme converges. Negative 
estimated 
2
i µ σ ’s were set to zero, as recommended by Baltagi and Griffin (1988). 
The sampling behavior of the feasible GLS estimator is illustrated in Table I. Table I reports 
Monte Carlo relative efficiency estimates for estimators of the slope coefficient. For each 
combination of   and T  two numbers are given. One number is the ratio of the root mean squared 
error (RMSE) of the within estimates of the slope coefficient over the RMSE of the true GLS 
estimates. (Relative efficiency estimates for the within estimator appear in the columns headed by 
“W”.) The other figure is the RMSE of the feasible GLS estimates over the RMSE of the true GLS 
estimates.  For each combination of   and T , each pair of relative efficiency estimates is based on 
2500 independent samples. 
N
N
The data in Table I do not provide a compelling case for abandoning the conventional 
within estimator in favor of the feasible GLS estimator. Only for T  is the feasible GLS 
estimator of the slope coefficient more efficient than the within estimator. Moreover, we can only 
conjecture as to what the sampling distribution of the feasible GLS estimator might be. For all 
combinations of   and T , its RMSE is too large relative to the RMSE of the true GLS estimator 
to warrant the conclusion that its sampling behavior approximates that of the true GLS estimator. 
5 =
N
  2 Indeed, for the sample sizes considered, the feasible GLS estimator’s efficiency relative to the true 
GLS estimator tends to deteriorate as   and   increase, with its RMSE increasing to over twice 
that of the true GLS estimator for large   and T . On the other hand, for every  , the efficiency 
of the within estimator approaches that of the true GLS estimator as T  increases, a result that is 
consistent with the observation that the within estimator has the same asymptotic (as T ) 
efficiency as the true GLS estimator of the slope coefficient (see Baltagi and Griffin, 1988). 
N
N
i
T
N
→∞
=
2
i µ
2
i µ
2
v
2 µ …
,
2
=+
1
2
N
i= ∑
( j µ
αα
1 ) /
j ∑
=+
1
N
= /
() =
FGLS αα =
1/2)] xp(1
Moreover, the feasible GLS estimator can be biased and inconsistent in situations where the 
true GLS estimator is not. In particular, the symmetry of the distribution of   is important for the 
unbiasedness and consistency of the feasible estimator. To illustrate this numerically, I generated 
one additional set of   independent samples with   and T  while using the same 
experimental design and estimators used in the other Monte Carlo experiments except for one 
alteration. Instead of generating the ω ’s as standardized normal random variates, they were 
generated as standardized lognormal random variables with zero mean and unit variance.
i µ
2500
i µ ω
200 N = 200
1 Thus, the 
distribution of   conditional on  i µσ = i
2
i µ σ  had a mean of zero, but it was asymmetric about zero. 
For this set of Monte Carlo samples, the averages of the true GLS estimates for both the intercept 
and slope parameters were close to the true value of one. On the other hand, although the average 
of the slope coefficient feasible GLS estimates was close to the true value of one, the average of the 
intercept feasible GLS estimates was significantly less than one: it was 0.603 with a standard error 
of  .  0.009
To see why symmetry is important, consider a simple example, which draws on an example 
used in Phillips (2003). Phillips (2003) points out that, although σ  can be estimated consistently, 
at best we can expect an estimator of 
2
v
σ  to actually only estimate   consistently rather than 
2
i µ σ . 
In light of this, consider a “feasible” GLS estimator that is based on σ  and  . Also, for 
the sake of simplicity, let the model be   ( iN ,  ).  Then, from 
results provided in Phillips (2003), the “feasible” GLS estimator of α  is 
1 ,, N µ
, T … it i it yv αµ + 1 , …, = 1 t =
 
  (
2 1 ˆ () FGLS N i i i g
N
µ µ +  (2)  ) v
1 )
 
where   and 
22 2 2 ( ) (/ / Ni i v v gN T T µµ σ σ
−− =+ +
1
T
ii t vv
= =∑ .
2 Clearly 
2 [( Ni Eg µ ) ] 0 i v = j µ  if   and  i v  are independent for all   and  j 0 i Ev
i µ
i µ
T
. Moreover, 
 if the  ’s are independent and the distribution of   is symmetric about zero. 
Thus, under these conditions  . On the other hand, if   is not symmetrically 
distributed about zero, there is no guarantee that   and thus there is no guarantee 
that  . Moreover, this observation holds for all   and  , and since an estimator of 
2 [() ] Ni i Eg µµ=
ˆ () FGLS E α =
0
α
                                                          
j µ
ˆ () E
2 [() ] Ni i Eg µµ=
N
0
t T
 
1 Specifically, I set ω= , where   is a standard normal 
variate. 
1/2 [exp( ) exp( /{e )[exp(1) 1]} ii z − − i z
2 Eq. (2) follows from Eq. (3) in Phillips (2003) upon replacing 
2
i µ σ  in the latter expression with 
. 
2
i µ
  3 2
i µ σ  only consistently estimates  , at best, we see from the foregoing that we cannot expect a 
feasible GLS estimator, based on estimated 
2
i µ
2
i µ σ ’s, to consistently estimate α  when the  ’s have 
asymmetric distributions. 
i µ
 
 
References 
 
Baltagi, B. H. (1995) Econometric Analysis of Panel Data, Wiley: New York. 
 
Baltagi, B. H., and J. M. Griffin (1988) “A Generalized Error Component Model with 
Heteroscedastic Disturbances,” International Economic Review 29, 745-753. 
 
Greene, W. H. (2000) Econometric Analysis, 4th ed., Prentice Hall: Upper Saddle River. 
 
Greene, W. H. (2003) Econometric Analysis, 5th ed., Prentice Hall: Upper Saddle River. 
 
Phillips, R. F. (2003) “Estimation of a Stratified Error-Components Model,” International 
Economic Review 44, 501-521. 
 
  4 

Econometric Analysis of Panel Data,

Econometric Analysis, 4th ed., Prentice Hall: Upper Saddle River.

Econometric Analysis, 5th ed., Prentice Hall: Upper Saddle River.

Estimation of a Stratified Error-Components

Some Monte Carlo results for a generalized error component model with heteroskedastic disturbances

http://www.accessecon.com/pubs/EB/2003/Volume3/EB-03C20007A.pdf

Some Monte Carlo results for a generalized error component model with heteroskedastic disturbances

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