Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol101.Issue2.2001:
Journal of Econometrics 101 (2001) 357}381
The unbalanced nested error component
regression model
Badi H. Baltagi!,*, Seuck Heun Song", Byoung Cheol Jung"
!Department of Economics, Texas A&M University, College Station, TX 77843-4228, USA
"Department of Statistics, Korea University, Sungbuk-Ku, Seoul 136-701, South Korea
Received 1 December 1998; received in revised form 31 August 2000; accepted 2 October 2000
Abstract
This paper considers a nested error component model with unbalanced data and
proposes simple analysis of variance (ANOVA), maximum likelihood (MLE) and minimum norm quadratic unbiased estimators (MINQUE)-type estimators of the variance
components. These are natural extensions from the biometrics, statistics and econometrics literature. The performance of these estimators is investigated by means of Monte
Carlo experiments. While the MLE and MINQUE methods perform the best in estimating the variance components and the standard errors of the regression coe$cients, the
simple ANOVA methods perform just as well in estimating the regression coe$cients.
These estimation methods are also used to investigate the productivity of public capital
in private production. ( 2001 Published by Elsevier Science S.A.
JEL: C23
Keywords: Panel data; Nested error component; Unbalanced ANOVA; MINQUE;
MLE; Variance components
1. Introduction
The analysis of panel data in econometrics have relied on the error component regression model which has its origin in the statistics and biometrics
* Corresponding author. Tel.: #1-979-845-7380; fax: #1-979-847-8757.
E-mail address: [email protected] (B.H. Baltagi).
0304-4076/01/$ - see front matter ( 2001 Published by Elsevier Science S.A.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 8 9 - 0
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B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
literature, see Hsiao (1986), Baltagi (1995) and MaH tyaH s and Sevestre (1996).
A huge bulk of this econometrics literature focuses on the complete or balanced
panels, yet the empirical applications face missing observations or incomplete
panels. Exceptions are Baltagi (1985), Wansbeek and Kapteyn (1989) and
Baltagi and Chang (1994). This paper considers the incomplete panel data
regression model in which the economic data has a natural nested groupings.
For example, data on "rms may be grouped by industry, data on states by
region and data on individuals by profession. In this case, one can control for
unobserved industry and within industry "rm e!ects using a nested error
component model. See Montmarquette and Mahseredjian (1989) for an empirical application of the nested error component model to study whether schooling
matters in educational achievements in Montreal's Francophone public elementary schools. More recently, see Antweiler (1999) for an application of the
determinants of pollution concentration as measured by observation stations in
various countries over time.
This paper proposes natural extensions of the analysis of variance (ANOVA),
maximum likelihood (MLE) and minimum norm quadratic unbiased estimators
(MINQUE) and compares their performance by means of Monte Carlo experiments. Statisticians and biometricians are more interested in the estimates of the
variance components per se, see Harville (1969, 1977), Hocking (1985), LaMotte
(1973a, b), Rao (1971a, b), Searle (1971, 1987) and Swallow and Monahan (1984)
to mention a few. Econometricians, on the other hand, are more interested in the
regression coe$cients, see Hsiao (1986) and Baltagi (1995). Monte Carlo results
on the balanced error component regression model include Nerlove (1971),
Maddala and Mount (1973) and Baltagi (1981). For the unbalanced error
component regression model, see Wansbeek and Kapteyn (1989) and Baltagi
and Chang (1994). None of these studies deal with the nested and unbalanced
error component model. The only exception is Fuller and Battese (1973). This
paper generalizes several estimators in the literature to the nested unbalanced
setting and reports the results of Monte Carlo experiments comparing the
performance of these proposed estimators. The type of unbalancedness considered in this paper allows for unequal number of "rms in each industry as well
as di!erent number of time periods across industries. Section 2 describes the
model and the estimation methods to be compared. Section 3 gives the design of
the Monte Carlo experiment and summarizes the results, while Section 4 gives an
empirical illustration applying these estimation methods to the study of productivity of public capital in private production. Section 5 gives our conclusion.
2. The model
We consider the following unbalanced panel data regression model:
y "x@ b#u , i"1,2, M, j"1,2, N and t"1,2, ¹ ,
ijt
ijt
i
i
ijt
(1)
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
359
where y could denote the output of the jth "rm in the ith industry for the tth
ijt
time period. x denotes a vector of k nonstochastic inputs. The disturbance of
ijt
(1) is given by
u "k #l #e , i"1,2, M, j"1,2, N and t"1,2, ¹ , (2)
ijt
i
ij
ijt
i
i
where k denotes the ith unobservable industry speci"c e!ect which is assumed
i
to be i.i.d. (0, p2),l denotes the nested e!ect of the jth "rm within the ith
k ij
industry which is assumed to be i.i.d. (0, p2) and e denotes the remainder
l
ijt
disturbance which is also assumed to be i.i.d. (0, p2). The k 's, l 's and e 's are
e
i
ij
ijt
independent of each other and among themselves. This is a nested classi"cation
in that each successive component of the error term is imbedded or &nested'
within the preceding component, see Graybill (1961, p. 350). This model allows
for unequal number of "rms in each industry as well as di!erent number of
observed time periods across industries. Model (1) can be rewritten in matrix
notation as
y"Xb#u,
(3)
where y is a +M N ¹ ]1, X is a +M N ¹ ]k, b is a k]1 parameter vector,
i/1 i i
i/1 i i
and u is a +M N ¹ ]1 disturbance vector. Eq. (2) in vector form yields
i/1 i i
u"Z k#Z l#e,
(4)
k
l
where k@"(k ,2, k ), l@"(l ,2,l 1 ,2,l M ), e@"(e ,2, e 1 ,2,
MN
111
11T
1
M
11
1N
e M M ), Z "diag(ι i ?ι i ), Z "diag(I i ?ι i ), ι i and ι i are vectors of
MN T
k
N
T
l
N
T
N
T
ones of dimension N and ¹ , respectively. By diag(ι i ?ι i ) we mean
i
i
N
T
diag(ι 1 ?ι 1 ,2,ι M ?ι M ). I i is an identity matrix of dimension N , and
N
T
N
T
N
i
? denotes the Kronecker product. Note that the observations are stacked such
that the slowest running index is the industry index i, the next slowest running
index is the "rm index j and the fastest running index is time.
Under these assumptions, the disturbance covariance matrix E(uu@) can be
written as
X"p2 Z Z@ #p2Z Z@ #p2 diag(I i ?I i )
T
e
N
l l l
k k k
(5)
"diag[p2(J i ?J i )#p2(I i ?J i )#p2(I i ?I i )],
l N
e N
T
T
T
k N
where J i "ι i ι@ i and J i "ι i ι@ i are matrices of ones of dimension N and ¹ .
N
N N
T
T T
i
i
It is clear from Eq. (5) that X is a block diagonal matrix with the ith block given
by
K "p2 (J i ?J i )#p2(I i ?J i )#p2(I i ?I i ), i"1,2, M.
(6)
k N
i
l N
e N
T
T
T
Replacing J i by its idempotent counterpart ¹JM i where JM i "J i /¹ and
T
T
T
T i
J i by N JM i where JM i "J i /N , we get
N
i N
N
N i
(7)
K "N ¹ p2 (JM i ?JM i )#¹ p2(I i ?JM i )#p2(I i ?I i ).
e N
T
i l N
T
T
i
i i k N
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B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Replacing I i by E i #JM i and I i by E i #JM i , where E i "I i !JM i and
N
N
N
T
T
T
N
N
N
E i "I i !JM i and collecting terms with the same matrices, see Wansbeek and
T
T
T
Kapteyn (1982, 1983), one gets the spectral decomposition of K :
i
K "j Q #j Q #j Q ,
(8)
i
1i 1i
2i 2i
3i 3i
where j "p2, j "¹ p2#p2 and j "N ¹ p2#¹ p2#p2. Corree
i l
e
3i
i i k
e
2i
i l
1i
spondingly, Q "I i ?E i , Q "E i ?JM i and Q "JM i ?JM i . The
1i
N
T
2i
N
T
3i
N
T
j , p"1, 2, 3, are the distinct characteristic roots of K of multiplicity
pi
i
N (¹ !1), N !1 and 1, respectively. Note that each Q , for p"1, 2, 3 is
i i
i
pi
symmetric, idempotent with its rank equal to its trace. Moreover, the Q 's are
pi
pairwise orthogonal and sum to the identity matrix. The advantages of this
spectral decomposition are that
(9)
Kp"jp Q #jp Q #jp Q ,
3i 3i
2i 2i
1i 1i
i
where p is an arbitrary scalar, see Baltagi (1993). Therefore, we can easily obtain
X~1 as
X~1"diag[K~1]"diag[j~1Q #j~1Q #j~1Q ]
3i 3i
2i 2i
1i 1i
i
(10)
and
C
D
p
p
p
e Q # e Q # e Q
p X~1@2"diag
1i Jj
2i Jj
3i
e
Jj
2i
3i
1i
"diag[I i ?I i ]!diag[h (I i ?JM i )]!diag[h (JM i ?JM i )],
N
T
1i N
T
2i N
T
(11)
where h "1!p /Jj and h "p /Jj !p /Jj . This allows us to ob1i
e
2i
2i
e
2i
e
3i
tain GLS on (3) as an ordinary least squares (OLS) of yH"p X~1@2y on
e
XH"p X~1@2X. The typical element of yH is given by (y !h y6 !h y6 )
e
ijt
1i ij.
2i i..
where y6 "+Ti y /¹ and y6 "+Ni +Ti y /N ¹ . This is known in the
t/1 ijt i
i..
j/1 t/1 ijt i i
ij.
econometrics literature as the Fuller and Battese (1973) transformation.
Note that the OLS estimator is given by
bK
"(X@X)~1X@y.
(12)
OLS
This is the best linear unbiased estimator when the variance components p2 and
k
p2 are both equal to 0. Even when these variance components are positive, the
l
OLS estimator is still unbiased and consistent, but its standard errors are biased,
see Moulton (1986). The OLS residuals are denoted by u(
"y!XbK
.
OLS
OLS
The within estimator in this case can be obtained by transforming the model
in (3) by Q "diag(I i ?E i ) and then applying OLS. Note that
T
1
N
Q Z "Q Z "0 because E i ι i "0. Therefore, Q sweeps away the k 's and
1 k
1 l
T T
1
i
l 's whether they are "xed or random e!ects. This yields
ij
(13)
bI "(X@ Q X )~1X@ Q y,
4 1
4 1 4
4
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
361
where X denotes the exogenous regressors excluding the intercept and b de4
4
notes the corresponding (k!1) vector of slope coe$cients. b@"(a, b@ ) and the
4
estimate of the intercept can be retrieved as follows: a8"(y6 !XM bI ), where
...
4... 4
the dots indicate summation and the bar indicates averaging. Following
Amemiya (1971), the within residuals u8
for the unbalanced nested e!ect
WTN
model are given by
u8
"y!a8ι !X bI
WTN
m
4 4
where m"+M N ¹ .
i/1 i imethods of estimating the variance components.
Next, we consider
(14)
2.1. Analysis of variance methods
These are methods of moments-type estimators that equate quadratic sums of
squares to their expectations and solve the resulting equations for the unknown
variance components. These ANOVA estimators are best quadratic unbiased
(BQU) estimators of the variance components in the balanced error component
model case, see Graybill (1961). Under normality of the disturbances they are
even minimum variance unbiased. However, for the unbalanced model, BQU
estimators of the variance components are a function of the variance components themselves, see Searle (1987). Unbalanced ANOVA methods are available
but optimal properties beyond unbiasedness are lost. We consider four
ANOVA-type methods which are natural extensions of those proposed in the
balanced error component literature:
(1) A modi"ed Wallace and Hussain (WH) estimator: Consider the three
quadratic forms of the disturbances using the Q , Q and Q matrices obtained
1 2
3
from the spectral decomposition of X in (8):
q "u@Q u, q "u@Q u, q "u@Q u,
(15)
1
1
2
2
3
3
where Q "diag(Q ), Q "diag(Q ) and Q "diag(Q ). Substituting OLS
1
1i
2
2i
3
3i
residuals u(
for u in (15) we get q( , q( and q( , see Wallace and Hussain (1969)
OLS
1 2
3
and Baltagi and Chang (1994). Taking expected values, we obtain
)"d p2#d p2#d p2,
E(q( )"E(u( @ Q u(
13 l
12 k
OLS 1 OLS
11 e
1
)"d p2#d p2#d p2,
E(q( )"E(u( @ Q u(
23 l
22 k
OLS 2 OLS
21 e
2
)"d p2#d p2#d p2,
E(q( )"E(u( @ Q u(
33 l
32 k
OLS 3 OLS
31 e
3
where the d 's are given by
ij
d "m!n!tr(X@Q X(X@X)~1),
11
1
d "tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
1
12
k k
d "tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
13
l l
1
(16)
362
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
d "n!M!tr(X@Q X(X@X)~1),
21
2
d "tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
2
22
k k
d "m!t!2tr[(X@Z Z@ Q X)(X@X)~1]
23
l l 2
# tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
2
l l
d "M!tr(X@Q X(X@X)~1),
31
3
d "m!2tr[(X@Z Z@ X)(X@X)~1]
32
k k
# tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
3
k k
d "t!2tr[(X@Z Z@ Q X)(X@X)~1]
33
l l 3
(17)
# tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
3
l l
with m"+M N ¹ , n"+M N and t"+M ¹ . Equating the q( 's to their
i i
i
i
i/1 E(q
i/1 i of equations, one
expected values
( ) in (16)i/1
and solving the system
gets the
i
Wallace and Hussain (1969)-type estimators of the variance components.1 These
are denoted by WH.
(2) A modi"ed Wansbeek and Kapteyn (WK) estimator: Alternatively, one
can substitute within residuals in the quadratic forms given by (15) to get q8 ,
1
q8 and q8 , see Amemiya (1971) and Wansbeek and Kapteyn (1989). Taking
2
3
expected values of q8 , q8 and q8 we get
1 2
3
)"(m!n!k#1)p2,
E(q8 )"E(u8 @ Q u8
e
WTN 1 WTN
1
E(q8 )"E(u8 @ Q u8
)
2
WTN 2 WTN
"[n!M#trM(X@ Q X )~1X@ Q X N]p2#(m!t)p2,
l
e
4 2 4
4 1 4
)
E(q8 )"E(u8 @ Q u8
WTN 3 WTN
3
"[M!1#trM(X@ Q X )~1X@ Q X N!trM(X@ Q X )~1X@ JM X N]p2
e
4 m 4
4 1 4
4 3 4
4 1 4
(18)
# [t!+N ¹2/m]p2#[m!+N2¹2/m]p2 .
k
i i
l
i i
Equating q8 to its expected value E(q8 ) in (18) and solving the system of
i
i
equations, we get the following Wansbeek and Kapteyn-type estimator of the
variance components which we denote by WK:
/(m!n!k#1),
p82"u8 @ Q u8
WTN 1 WTN
e
u8 @ Q u8
![n!M#trM(X@ Q X )~1(X@ Q X )Np82]
4 1 4
4 2 4 e ,
p82" WTN 2 WTN
l
m!t
1 Most of the algebra involved is simple but tedious and all proofs are available upon request from
the authors.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
363
![M!1#trM(X@ Q X )~1X@ Q X N
p82"(u8 @ Q u8
4 3 4
4 1 4
WTN 3 WTN
k
! trM(X@ Q X )~1X@ JM X N]p82
4 m 4 e
4 1 4
(19)
![t!+N ¹2/m]p82)/[m!+N2¹2/m].
i i
l
i i
(3) A Modi"ed Swamy and Arora (SA) estimator: Following Swamy and
Arora (1972), we transform the regression model in (3) by premultiplying it by
Q , Q and Q and we obtain the transformed residuals u8 , u8 and u8 ,
1
2
3
1 2
3
respectively. Let q8 `"u8 @ Q u8 , q8 `"u8 @ Q u8 and q8 `"u8 @ Q u8 . Since q8 ` is
1
3 3 3
3
2 2 2
1 1 1 2
1
exactly the same as q8 the resulting expected value of q8 ` is the same as that
1
1
given in (18). The expected values of q8 ` and q8 ` are
3
2
E(q8 `)"E(u8 @ Q u8 )
2 2 2
2
"(n!M!k#1)p2#[m!t!trM(X@ Z Z@ Q X )
4 l l 2 4
e
(X@ Q X )~1N]p2,
l
4 2 4
E(q8 `)"E(u8 @ Q u8 )
3 3 3
3
"(M!k)p2#[t!trM(X@Z Z@ Q X)(X@Q X)~1N]p2
l
3
e
l l 3
(20)
# [m!trM(X@Z Z@ X)(X@Q X)~1N]p2 .
k
3
k k
Equating q8 ` to its expected value E(q8 `) and solving the system of equations, we
i
i
get the following Swamy and Arora-type estimators of the variance components
which we denote by SA:
/(m!n!k#1),
p82"u8 @ Q u8
WTN 1 WTN
e
u8 @ Q u8 !(n!M!k#1)p82
e
2 2 2
,
p82"
l m!t!trM(X@ Z Z@ Q X )(X@ Q X )~1N
4 l l 2 4 4 2 4
u8 @ Q u8 !(M!k)p82![t!trM(X@Z Z@ Q X)(X@Q X)~1N]p82
3
e
l l 3
l.
p82" 3 3 3
k
m!trM(X@Z Z@ X)(X@Q X)~1N
3
k k
(21)
(4) Henderson Method III: Fuller and Battese (1973) suggest an estimation of
the variance components using the "tting constants methods. This method uses
. Also, the residual
the within residual sums of squares given by q8 H"u8 @ u8
WTN WTN
1
sum of squares obtained by transforming the regression in (3) by (Q #Q ) (i.e.,
1
2
the regression of y !y6 on x l !x6 l , for l"1,2, k). This is denoted by
i..
ijt
i..
ijt
q8 H"u8 H@u8 H where u8 H is the residual vector of the (Q #Q ) transformed regres2
1
2
2 2
2
sion. Finally, this method uses the conventional OLS residual sum of squares
. If the x variables do not have constant values for
denoted by q8 H"u( @ u(
OLS OLS
3
measurement of group and nested subgroups, q8 H is exactly the same as that for
1
364
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
the WK method, the resulting expected value of q8 H is the same as that given in
1
(18). Also, the expected value of q8 H and q8 H are given by
3
2
E(q8 H)"p2[m!M!k#1]
e
2
# p2[m!t!trM(X@ Z Z@ Q X )(X@ (Q #Q )X )~1N],
2 4
4 l l 2 4 4 1
l
E(q8 H)"p2[m!k]#p2[m!trM(X@Z Z@ X)(X@X)~1N]
l
l l
e
3
(22)
# p2 [m!trM(X@Z Z@ X)(X@X)~1N].
k
k k
Equating q8 H, for i"1, 2, 3 to its expected value E(q8 H) in (22), we obtain the
i
i
Henderson Method III estimator of the variance components, see Fuller and
Battese (1973). These are denoted by HFB:
/(m!n!k#1),
p82"u8 @ Q u8
WTN 1 WTN
e
u8 H@u8 H!(m!M!k#1)p82
e
2 2
p82"
,
l m!t!trM(X@ Z Z@ Q X )(X@ (Q #Q )X )~1N
2 4
4 l l 2 4 4 1
!(m!k)p82![m!trM(X@diag(I i ?J i )X)(X@X)~1Np82]
u( @ u(
e
N
l .
T
p82" OLS OLS
k
m!trM(X@Z Z@ X)(X@X)~1N
k k
(23)
2.2. Maximum likelihood estimator
Since j , for p"1, 2, 3 are the distinct characteristic roots of K then
pi
i
DK D"(j )(jNi ~1)(jNi (Ti ~1)). Let o "p2/p2, o "p2/p2 and X"p2R, then the
e
l e
i
3i 2i
k e 2
1i
1
log-likelihood function can be written as
m
1 M
log ¸"C! log p2! + log(N ¹ o #¹ o #1)
i i 1
i 2
e 2
2
i/1
1
1 M
! + (N !1) log(¹ o #1)! u@R~1u/2p2.
(24)
i
i 2
e
2
2
i/1
The "rst-order conditions give closed form solutions for b and p2 conditional on
e
o and o :
1
2
bK "(X@RK ~1X)~1X@RK ~1y,
(25)
ML
(26)
p( 2"(y!Xb)@RK ~1(y!Xb)/m.
e
However, the "rst-order conditions based on o( and o( are nonlinear in o and
1
2
1
o even for known values of b and p2. Following Hemmerle and Hartley (1973),
e
2
we get
1
1
L log ¸
"! tr[Z@ R~1Z ]#
(y!Xb)@R~1Z Z@ R~1(y!Xb),
k
k
k k
2p2
2
Lo
e
1
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
365
L log ¸
1
1
"! tr[Z@ R~1Z ]#
(y!Xb)@R~1Z Z@ R~1(y!Xb). (27)
l
l
l l
Lo
2p2
2
2
e
Therefore, a numerical solution by means of iteration is needed. The Fisher
scoring procedure is used to estimate o and o . The partition of the informa1
2
tion matrix corresponding to o and o is given by
1
2
(N ¹ )2
1 M
L2 log ¸
i i
,
" +
E !
(1#o ¹ #o N ¹ )2
2
Lo2
2 i
1 i i
1
i/1
N ¹2
L2 log ¸
1 M
i i
E !
,
" +
(1#o ¹ #o N ¹ )2
Lo Lo
2
2
i
1
i
i
1 2
i/1
L2 log ¸
¹2
1 M
1 M (N !1)¹2
i # +
i
i
E !
, (28)
" +
Lo2
(1#o ¹ )2 2
(1#o ¹ #o N ¹ )2
2
2
2 i
2 i
1 i i
i/1
i/1
see Harville (1977). Starting with an initial value, the (r#1)th updated value of
o and o is given by
1
2
L2 log ¸ ~1 L log ¸
L2 log ¸
E !
E !
o(
o(
Lo Lo
Lo2
Lo
1
1 2
1
1
,
" 1 #
L
log ¸
L2 log ¸
L2 log ¸
o( r`1
o( r
2
2
E !
E !
Lo
Lo Lo
Lo2
2 r
1 2
2
r
(29)
C
C
C
C D
D
D
D
C D
C
C
C
D C
D C
DC D
D
D
where at each step, L log ¸/Lo and L log ¸/Lo are obtained from Eq. (27), bK and
1
2
p( 2 are obtained from (25) and (26), the information matrix is obtained from
e
Eq. (28). The subscript r means this is evaluated at the rth iteration. For a review
of the advantages and disadvantages of MLE, see Harville (1977).
2.3. Restricted maximum likelihood estimator
Patterson and Thompson (1971) suggested a restricted maximum likelihood
(REML) estimation method that takes into account the loss of degrees of
freedom due to the regression coe$cients in estimating the variance components. REML is based on a transformation that partitions the likelihood
function into two parts, one being free of the "xed regression coe$cients.
Maximizing this part yields REML. Patterson and Thompson (1971) suggest the
singular transformation y@[C F R~1X/p2], where C"I!X(X@X)~1X@. Cy is
e
distributed as N(0, CRC@/p2), and from the fact that CX"0, it is independent of
e
X@R~1y/p2 which is also distributed as N(X@R~1Xb/p2, X@R~1X/p2). It is clear
e
e
e
that Cy does not depend on b. Since C is an idempotent matrix of rank m!k,
there exists an (m!k)]m matrix A such that
A@A"C, AA@"I.
(30)
366
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Using the A@y transformation instead of C@y, we get
p2ARA@ 0
e
.
(31)
0
X@R~1X/p2
X@R~1y/p2
X@R~1Xb/p2
e
e
e
Following Corbeil and Searle (1976), the log-likelihood function of A@y and
X@R~1y/p2 are given by log ¸ and log ¸ , respectively:
e
1
2
1
m!k
m!k
log(2p)!
log(p2)! logDARA@D
log ¸ "!
e
1
2
2
2
C
A@y
D AC
&N
0
DC
,
DB
1
y@[A@(ARA@)~1A]y,
2p2
e
1
k
log ¸ "! log(2p)! log(p2)!logDX@R~1XD
e
2
2
2
!
1
(y!Xb)@R~1X(X@R~1X)~1X@R~1(y!Xb).
(32)
2p2
e
Using the results of Hocking (1985) and Corbeil and Searle (1976), we obtain
!
A@(ARA@)~1A"R~1[I!X(X@R~1X)~1X@R~1]"R~1(I!M),
(33)
where M"X(X@R~1X)~1X@R~1.
Using log ¸ which is free from b, the "rst-order derivatives of log ¸ with
1
1
respect to p2, o and o are given by
e 1
2
m!k
1
L log ¸
1 "!
#
y@A@(ARA@)~1Ay
2p2
2p4
Lp2
e
e
e
m!k
1
"!
#
y@R~1(I!M)y,
2p2
2p4
e
e
1
L log ¸
1 "! tr[Z@ A@(ARA@)~1AZ ]
k
k
2
Lo
1
1
#
y@[A@(ARA@)~1AZ Z@ A@(ARA@)~1A]y,
k k
2p2
e
1
L log ¸
1 "! tr[Z@ A@(ARA@)~1AZ ]
l
l
2
Lo
2
1
#
(34)
y@[A@(ARA@)~1AZ Z@ A@(ARA@)~1A]y.
l l
2p2
e
Equating the equations in (34) to 0's yield the REML estimates. For example,
solving L log ¸ /Lp2"0 conditional on o and o , we obtain
1
2
1 e
(35)
p( 2"y@A@(ARK A@)~1Ay/(m!k)"y@[RK ~1(I!M)]y/(m!k).
e
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
367
But there are no closed-form solutions on o and o . Thus a numerical solution
1
2
by means of iteration is needed. The Fisher scoring procedure is used to estimate
o and o . Using the results of Harville (1977) and Eq. (33), the information
1
2
matrix with respect to o and o is given by
1
2
1
L2 log ¸
1 " tr[Z Z@ R~1(I!M)Z Z@ R~1(I!M)],
E !
k k
k k
Lo2
2
1
1
L2 log ¸
1 " tr[Z Z@ R~1(I!M)Z Z@ R~1(I!M)],
E !
l l
k k
2
Lo Lo
1 2
1
L2 log ¸
1 " tr[Z Z@ R~1(I!M)Z Z@ R~1(I!M)].
(36)
E !
l l
l l
2
Lo2
2
The updated values of o and o can be obtained as in (29).
1
2
C
C
C
D
D
D
2.4. MINQUE and MIVQUE
Rao (1971a) proposed a general procedure for variance components estimation which requires no distributional assumptions other than the existence of the
"rst four moments. This procedure yields MINQUE of the variance components. Under normality of the disturbances, MINQUE and minimum variance
quadratic unbiased estimators (MIVQUE) are identical. Since we assume normality, we will focus on MIVQUE. Let
R"R~1[I!X(X@R~1X)~1X@R~1]/p2,
e
S"Ms N"Mtr(< R< R)N, i, j"1, 2, 3
ij
i
j
(37)
(38)
and
u"Mu N"My@R< RyN, i"1, 2, 3,
(39)
i
i
where < "I , < "Z Z@ and < "Z Z@ . Rao (1971b) shows that the vector
3
l l
1
m 2
k k
of MIVQUEs is given by
hK "S~1u,
(40)
where hK @"(p( 2, p( 2, p( 2). However, MIVQUE requires a priori values of
e k l
the variance components. Therefore, MIVQUE is only &locally minimum
variance', see LaMotte (1973a, b), and &locally best', see Harville (1969). Three
priors of the MIVQUE estimator are considered in our Monte Carlo study:
(1) the identity matrix, which we denote by MV1, and (2) all values of
the variance components equal to 1, see Swallow and Searle (1978) which
we denote by MV2, and (3) the ANOVA estimator of WK, which we denote
by MV3. Note that the MIVQUE estimator can produce negative estimates
of the variance components. In this case, we replace the negative variance
estimate by 0.
368
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
3. Monte Carlo results
3.1. Design of the Monte Carlo study
We consider the following simple regression equation:
y "a#x b#u , i"1,2, M, j"1,2, N , t"1,2, ¹ , (41)
ijt
ijt
ijt
i
i
with u "k #l #e . The exogenous variable x was generated by a similar
ijt
i
ij
ijt
ijt
method to that of Nerlove (1971). In fact, x "0.3t#0.8x
#w , where
ijt
ij,t~1
ijt
w is uniformly distributed on the interval [!0.5, 0.5]. The initial values
ijt
x were chosen as (100#250w ). Throughout the experiment a"5 and
ij0
ij0
b"2. For generating the u
disturbances, we let k &IIN(0, p2),
k
ijt
i
l &IIN(0, p2) and e &IIN(0, p2). We "x p2"p2 #p2#p2"20 and de"ne
e
l
k
e
l
ijt
ij
c "p2 /p2 and c "p2/p2. These are varied over the set (0, 0.2, 0.4, 0.6, 0.8)
l
k
2
1
such that (1!c !c ) is always positive. Extending a measure of unbalanced1
2
ness given by Ahrens and Pincus (1981) to the unbalanced nested model, we
de"ne
c "M/NM + (1/N ) where NM "+ N /M,
1
i
i
c "M/¹M + (1/¹ ) where ¹M "+¹ /M,
2
i
i
(42)
c "M/N¹ + (1/N ¹ ) where N¹"+ N ¹ /M,
i i
i i
3
where c , c and c denote the measures of subgroup unbalancedness, observed
1 2
3
time unbalancedness and group unbalancedness due to each group size. Note
that c , c and c take the value 1 when the data are balanced but take smaller
1 2
3
values than 1 as the data pattern gets more unbalanced. Table 1 gives the
(N , ¹ ) pattern used along with the corresponding unbalancedness measures for
i i
M"10. The "rst parentheses gives the N pattern, while the second parentheses
i
below it gives the corresponding ¹ pattern. For example, P observes the "rst
i
1
grouping of eight individuals over six time periods and the last grouping of 12
individuals over four time periods. The sample size is "xed at 500 for every
pattern. Two other values of M are used, M"6 and 15. For each experiment,
1000 replications are performed. For each replication, we calculate OLS, WTN,
WH, SA, WK, HFB, ML, REML, MV1, MV2, MV3 and true GLS. The last
estimator is obtained for comparison purposes.
3.2. A comparison of regression coezcient estimates
Table 2 gives the mean square error (MSE) of the estimate of bK relative to that
of true GLS for the case when M"10.2 From this table it is clear that OLS is
2 Similar MSE tables for the regression coe$cients and the variance components estimates are
generated for M"6 and 15, but they are not produced here to save space. These tables are
available upon request from the authors.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
369
Table 1
(N , ¹ ) patterns considered and their corresponding unbalancedness measures when M"10
i i
Pattern
(N , N ,2, N )!
1 2
10
(¹ , ¹ ,2, ¹ )
1 2
10
c
P
1
(8,8,8,10,10,10,10,12,12,12)
(6,6,6,5,5,5,5,5,4,4)
0.976
0.980
0.996
P
2
(6,6,6,10,10,10,10,12,12,12)
(9,9,9,9,8,3,3,3,3,3)
0.925
0.757
0.8238
P
3
(5,5,5,10,10,10,10,11,11,11)
(2,2,3,3,3,6,7,8,8,9)
0.893
0.734
0.504
P
4
(4,4,4,5,5,9,9,10,10,10)
(14,15,15,15,15,3,3,4,4,4)
0.854
0.619
0.881
P
5
(3,3,3,3,3,8,8,8,8,8)
(2,2,2,3,3,11,11,12,12,12)
0.793
0.550
0.258
P
6
(2,2,6,6,6,10,10,10,13,13)
(16,16,16,16,16,2,2,3,3,3)
0.656
0.465
0.718
P
7
(2,2,2,10,10,10,10,13,13,13)
(2,1,1,1,1,8,8,8,8,8)
0.552
0.424
0.133
P
8
(20,20,15,15,15,3,3,3,2,2)
(1,1,6,6,6,10,10,10,25,25)
0.444
0.347
0.732
P
9
(16,16,16,16,16,2,2,2,2,2)
(2,2,3,3,3,28,28,30,30,30)
0.395
0.290
0.949
P
10
(20,20,20,20,20,2,2,2,2,1)
(2,2,2,3,3,25,30,30,30,30)
0.282
0.272
0.945
P
11
(1,1,1,1,5,5,25,25,25,25)
(1,2,2,35,35,2,2,3,3,3)
0.192
0.280
0.091
P
12
(1,1,1,1,5,5,30,30,30,30)
(27,27,28,28,28,2,2,2,2,2)
0.165
0.252
0.626
1
c
2
c
3
!The "rst parentheses gives the N pattern, while the parentheses below it gives the corresponding
i
¹ pattern.
i
inferior to true GLS, ML-type (ML, REML) estimators and all feasible GLStype estimators except when c " c "0. For all experiments, the e!ect of an
1
2
increase in c on the MSE of OLS is much larger than that of an increase in c .
1
2
This is because c a!ects the primary group while c a!ects only the nested
1
2
subgroup. The WTN estimator performs poorly for small c and c values. The
1
2
performance of WTN is in some cases worse than OLS if either c or c is 0.
1
2
However, its performance improves as c and c increase and the unbalanced1
2
ness pattern gets more severe. The ANOVA-type (WH, WK, SA and HFB)
370
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Table 2
MSE of bK relative to that of true GLS when M"10
c
1
c
2
OLS
WTN
WH
WK
SA
HFB
MLE
REML MV1
MV2
MV3
P
1
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.069
1.396
2.073
4.341
1.649
1.633
2.274
4.358
3.537
2.748
4.407
6.235
5.673
13.606
4.864
2.732
1.985
1.406
1.212
3.979
2.333
1.549
1.219
4.149
2.049
1.276
4.077
1.555
4.473
0.998
1.014
1.008
1.000
1.000
1.003
1.002
1.010
1.011
1.001
1.004
1.000
1.003
1.000
1.007
1.006
1.016
1.011
1.001
1.000
1.001
0.998
1.011
1.010
0.998
1.008
0.998
1.004
0.998
1.011
0.996
1.011
1.009
1.001
1.001
1.003
1.002
1.014
1.017
1.004
1.006
1.001
1.007
1.003
1.004
0.999
1.014
1.009
1.000
1.000
1.002
1.001
1.011
1.010
0.999
1.006
1.000
1.005
0.998
1.006
0.998
1.012
1.008
0.999
1.000
1.003
1.001
1.011
1.011
1.001
1.006
0.999
1.005
0.998
1.006
0.999
1.015
1.009
0.999
1.000
1.004
1.001
1.010
1.010
1.000
1.007
0.999
1.005
0.998
1.006
0.998
1.017
1.010
1.000
1.004
1.002
0.999
1.014
1.015
1.004
1.012
1.003
1.036
1.013
1.135
0.999
1.013
1.009
1.001
0.999
1.002
1.000
1.010
1.010
0.999
1.007
0.999
1.004
0.998
1.006
0.999
1.014
1.009
1.000
0.999
1.004
1.001
1.010
1.010
1.000
1.007
0.999
1.005
0.998
1.006
P
3
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.236
1.653
2.590
5.815
1.970
2.106
3.005
6.389
3.932
3.467
6.324
8.675
7.699
19.474
3.578
2.233
1.735
1.296
1.133
3.400
1.816
1.508
1.097
3.022
1.605
1.180
3.474
1.433
3.528
1.003
1.025
1.009
0.999
1.015
1.001
1.016
1.015
1.018
1.003
1.004
1.004
1.011
1.017
1.028
1.013
1.024
1.009
1.000
1.012
1.007
1.010
1.016
1.011
1.004
1.003
1.003
1.010
1.015
1.026
1.002
1.022
1.005
1.002
1.014
1.004
1.025
1.022
1.023
1.001
1.003
1.009
1.012
1.017
1.015
1.004
1.025
1.010
0.998
1.015
1.001
1.014
1.014
1.014
1.001
1.003
1.004
1.007
1.016
1.016
0.998
1.021
1.009
1.000
1.011
1.006
1.016
1.011
1.010
1.000
1.005
1.001
1.008
1.017
1.011
1.002
1.025
1.011
0.999
1.012
1.005
1.016
1.011
1.008
0.999
1.004
1.001
1.008
1.017
1.011
1.001
1.020
1.008
1.003
1.026
1.002
1.024
1.035
1.040
1.012
1.016
1.038
1.028
1.048
1.054
1.016
1.031
1.011
0.999
1.014
1.006
1.014
1.011
1.009
1.002
1.004
1.001
1.006
1.017
1.023
1.004
1.025
1.010
0.999
1.012
1.004
1.016
1.011
1.008
0.999
1.004
1.001
1.008
1.017
1.012
P
5
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.473
2.237
4.354
9.091
1.979
2.374
3.961
8.756
3.404
4.385
9.748
6.053
9.692
17.824
2.541
1.560
1.328
1.127
1.056
2.066
1.212
1.162
1.047
1.922
1.174
1.094
1.918
1.093
1.893
1.020
1.020
1.007
1.005
1.004
1.018
1.028
1.019
1.008
1.011
1.030
1.005
1.009
1.018
1.020
1.029
1.019
1.009
1.004
1.002
1.021
1.027
1.018
1.006
1.014
1.022
1.004
1.010
1.014
1.020
1.013
1.021
1.006
1.004
1.003
1.021
1.033
1.022
1.008
1.014
1.036
1.005
1.009
1.021
1.018
1.023
1.019
1.009
1.005
1.003
1.017
1.029
1.018
1.007
1.011
1.025
1.004
1.007
1.016
1.017
1.014
1.019
1.009
1.004
1.003
1.015
1.026
1.015
1.005
1.012
1.021
1.006
1.006
1.013
1.008
1.020
1.021
1.011
1.005
1.003
1.012
1.022
1.015
1.004
1.011
1.018
1.005
1.006
1.012
1.008
1.017
1.020
1.005
1.005
1.005
1.028
1.039
1.024
1.013
1.021
1.040
1.015
1.014
1.035
1.041
1.046
1.031
1.014
1.006
1.002
1.014
1.017
1.014
1.004
1.015
1.017
1.004
1.020
1.011
1.030
1.021
1.020
1.011
1.005
1.003
1.014
1.024
1.016
1.004
1.011
1.019
1.004
1.006
1.012
1.010
P
7
0.0
0.0
0.0
0.0
0.2
0.4
1.000
1.293
2.008
2.894
1.806
1.470
1.010
1.008
1.015
1.016
1.010
1.013
1.008
1.007
1.016
1.011
1.010
1.014
1.009
1.005
1.015
1.012
1.006
1.014
1.011
1.006
1.017
1.020
1.012
1.014
1.012
1.006
1.014
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
371
Table 2 (Continued)
c
1
c
2
OLS
WTN
WH
WK
SA
HFB
MLE
REML MV1
MV2
MV3
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
3.153
6.276
1.916
1.910
3.302
7.084
3.178
3.232
7.325
6.502
7.435
16.032
1.298
1.156
2.684
1.644
1.300
1.140
3.043
1.535
1.225
2.779
1.296
2.662
1.001
1.012
1.012
1.014
1.022
1.012
1.037
1.010
1.005
1.018
1.010
1.034
1.001
1.012
1.015
1.017
1.020
1.011
1.045
1.014
1.001
1.026
1.007
1.035
1.002
1.011
1.019
1.015
1.023
1.017
1.045
1.013
1.005
1.020
1.008
1.028
1.001
1.012
1.012
1.014
1.021
1.011
1.037
1.012
1.002
1.018
1.009
1.028
1.001
1.010
1.016
1.010
1.015
1.006
1.028
1.012
1.003
1.008
1.011
1.019
1.002
1.010
1.015
1.008
1.014
1.006
1.027
1.012
1.002
1.009
1.010
1.019
1.001
1.016
1.021
1.026
1.035
1.019
1.065
1.014
1.012
1.024
1.030
1.104
1.002
1.011
1.014
1.003
1.008
1.006
1.034
1.013
1.003
1.032
1.009
1.039
1.001
1.010
1.015
1.010
1.015
1.006
1.029
1.012
1.003
1.009
1.010
1.019
P
9
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.254
1.939
2.871
5.502
3.053
3.198
4.132
8.514
8.563
6.631
10.603
14.027
14.553
37.097
5.028
3.273
2.228
1.670
1.340
4.067
2.904
2.137
1.348
4.782
2.555
1.568
4.090
1.899
4.310
1.033
1.014
1.035
1.014
1.013
1.009
1.012
1.044
1.044
1.004
1.007
1.032
1.014
1.022
1.018
1.045
1.025
1.033
1.016
1.011
1.010
1.024
1.038
1.035
1.007
1.006
1.027
1.008
1.019
1.009
1.027
1.013
1.034
1.014
1.012
1.014
1.013
1.049
1.048
1.012
1.014
1.040
1.016
1.030
1.007
1.037
1.017
1.035
1.015
1.013
1.006
1.012
1.037
1.038
1.000
1.006
1.030
1.007
1.021
1.007
1.022
1.008
1.016
1.010
1.007
1.008
1.018
1.018
1.011
1.001
1.001
1.015
1.009
1.010
1.007
1.032
1.011
1.019
1.012
1.007
1.006
1.014
1.019
1.007
1.000
1.002
1.014
1.008
1.011
1.007
1.028
1.014
1.069
1.075
1.120
1.024
1.037
1.134
1.251
1.083
1.174
1.355
1.187
1.267
1.396
1.048
1.021
1.027
1.014
1.007
1.012
1.012
1.019
1.015
1.004
1.000
1.014
1.007
1.011
1.012
1.031
1.015
1.024
1.012
1.009
1.007
1.011
1.018
1.007
1.000
1.003
1.015
1.008
1.011
1.008
P
11
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.489
1.821
2.856
4.494
4.970
5.078
5.933
8.409
13.184
10.965
14.110
28.659
21.438
73.642
8.900
4.513
3.154
2.322
1.455
6.283
3.682
2.392
1.643
7.260
3.193
1.818
6.450
2.374
5.963
1.061
1.014
1.007
1.032
1.036
1.011
1.016
1.033
1.007
1.001
1.005
1.009
1.017
1.024
1.009
1.091
1.009
1.012
1.033
1.036
1.019
1.013
1.025
1.008
1.009
1.006
1.007
1.007
1.021
1.000
1.054
1.012
1.005
1.030
1.032
1.032
1.042
1.037
1.003
1.012
1.009
1.015
1.012
1.031
1.009
1.063
1.014
1.010
1.033
1.039
1.010
1.011
1.033
1.008
1.003
1.006
1.007
1.007
1.019
1.001
1.035
1.008
1.002
1.016
1.014
1.010
1.010
1.017
1.003
1.002
1.006
1.004
1.004
1.008
0.998
1.048
1.007
1.005
1.017
1.016
1.007
1.006
1.013
1.003
1.001
1.006
1.003
1.004
1.008
0.998
1.047
1.034
1.050
1.159
1.366
1.023
1.052
1.238
1.253
1.050
1.079
1.227
1.268
1.218
1.142
1.092
1.015
1.007
1.017
1.021
1.027
1.015
1.013
1.008
1.002
1.006
1.002
1.006
1.007
1.000
1.054
1.007
1.005
1.017
1.016
1.007
1.006
1.011
1.002
1.001
1.006
1.004
1.005
1.008
0.999
estimators, the ML-type and MV2 and MV3 estimators all perform well relative
to true GLS. See also Fig. 1 which plots the MSE of bK relative to that of true
GLS for the 12 unbalanced patterns for M"10 and c " c "0.4. It is
1
2
important to point out that the simple ANOVA-type estimators compare well in
372
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Fig. 1. MSE of bK relative to that of true GLS when M"10, c "c "0.4.
1
2
MSE performance to the ML- and MIVQUE-type estimators which are relatively more di$cult to compute. They have at most 9.1% higher MSE than true
GLS, see the WK estimator for pattern P when c "c "0. MV1 performs
11
1
2
well for c (0.4 and c (0.4. This is to be expected since the prior values for
1
2
MV1 are c "0 and c "0. As c and c increase or the degree of unbalanced1
2
1
2
ness gets large, the performance of MV1 deteriorates relative to the other
estimators, see Fig. 1.
To summarize, for the regression parameters, the computationally simple
ANOVA estimators compare well with the more complicated estimators such as
ML, REML, MV2 and MV3 in terms of MSE criteria. Also, MV1 is not
recommended when the primary group e!ects or nested subgroup e!ects are
suspected to be large or the unbalanced pattern is severe.
3.3. A comparison of variance components estimates
Tables 3}5 report the MSE of the variance components relative to that of
MLE for the 12 unbalancedness patterns for M"10 and c "c "0.4. Similar
1
2
tables for other values of M"6 and 15 and various values of c and c are
1
2
available upon request from the authors. These tables are plotted in Figs. 2}4 for
ease of comparison.
For the estimation of p2, see Table 3 or Fig. 2, MLE ranks "rst; REML, MV2
k
and MV3 rank second and the ANOVA methods rank third by the MSE
criteria. MV1 does well only when c and c are close to 0. For c "c "0.4,
1
2
1
2
MV1 performs the worst. The ANOVA methods yield almost twice the MSE of
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
373
Table 3
MSE of p( 2 relative to that of MLE when M"10, c "c "0.4
k
1
2
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
11
P
12
WH
WK
SA
HFB
REML
MV1
MV2
MV3
1.170
1.284
1.523
1.307
1.866
1.814
1.897
1.473
1.446
1.427
1.680
1.338
1.170
1.267
1.487
1.295
1.855
1.750
1.885
1.476
1.362
1.360
1.863
1.379
1.302
1.448
1.722
1.413
1.951
2.022
2.034
1.558
1.579
1.585
1.787
1.465
1.168
1.274
1.512
1.297
1.863
1.779
1.891
1.472
1.402
1.383
1.755
1.336
1.158
1.170
1.183
1.140
1.140
1.170
1.178
1.189
1.159
1.177
1.211
1.165
1.204
1.669
2.175
1.602
2.244
3.124
2.216
2.196
1.407
1.459
2.558
2.170
1.160
1.165
1.176
1.133
1.156
1.162
1.202
1.183
1.159
1.179
1.171
1.074
1.157
1.170
1.173
1.134
1.140
1.159
1.159
1.169
1.146
1.170
1.129
1.100
Table 4
MSE of p( 2 relative to that of MLE when M"10, c "c "0.4
l
1
2
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
11
P
12
WH
WK
SA
HFB
REML
MV1
MV2
MV3
1.010
1.205
1.116
1.458
1.086
1.584
1.105
1.475
2.615
2.717
2.694
2.863
1.010
1.181
1.115
1.428
1.066
1.566
1.103
1.453
2.561
2.672
2.583
2.697
1.012
1.194
1.117
1.430
1.096
1.567
1.110
1.466
2.581
2.681
2.583
2.692
1.004
1.179
1.104
1.412
1.065
1.549
1.092
1.442
2.551
2.650
2.559
2.667
0.999
0.999
1.000
0.999
0.995
0.999
0.999
0.998
0.993
1.000
1.001
1.000
1.312
2.514
1.450
3.224
1.181
3.563
1.333
16.188
13.130
21.473
20.234
23.894
0.997
1.012
1.001
1.002
0.989
1.015
1.024
1.039
0.998
1.021
1.001
1.000
0.998
0.998
1.002
0.990
1.015
0.993
1.007
0.985
0.977
0.992
1.004
0.989
MLE for patterns P , P and P . On the other hand, REML, MV2 and MV3
5 6
7
have MSE that are only 14}20% higher than that of MLE. MV1 has 2}3 times
the MSE of MLE for these patterns and is not included in Fig. 2.
For the estimation of p2, see Table 4 or Fig. 3, REML, MLE, MV2 and MV3
l
rank "rst by the MSE criteria. They are followed by the ANOVA methods with
MV1 performing the worst. For patterns P , P , P and P , these ANOVA
9 10 11
12
methods yield more than 2.5 times the MSE of MLE. In contrast, MV1 yields
13}23 times the MSE of MLE and is not included in Fig. 3.
For the estimation of p2, there is not much di!erence among WK, SA, HFB,
e
REML, MV2 and MV3. However, WH performs slightly worse than the other
374
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Table 5
MSE of p( 2 relative to that of MLE when M"10, c "c "0.4
e
1
2
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
11
P
12
WH
WK
SA
HFB
REML
MV1
MV2
MV3
1.038
1.053
1.050
1.202
1.201
1.173
1.078
1.126
1.121
1.128
1.193
1.148
1.001
1.005
1.000
1.004
1.006
1.005
1.000
1.004
1.000
1.003
1.004
1.001
1.001
1.005
1.000
1.004
1.006
1.005
1.000
1.004
1.000
1.003
1.004
1.001
1.001
1.005
1.000
1.004
1.006
1.005
1.000
1.004
1.000
1.003
1.004
1.001
1.001
1.002
1.000
1.002
1.006
1.002
1.000
1.002
1.002
1.001
1.002
0.997
5.190
34.983
27.465
54.860
8.669
58.511
6.469
124.852
124.467
154.966
144.898
119.133
1.004
0.999
1.002
1.002
1.012
1.009
1.012
1.018
1.010
1.006
1.006
0.994
1.002
1.003
1.000
1.003
1.010
1.003
1.001
1.005
1.006
1.005
1.004
1.000
Fig. 2. MSE of p( 2 relative to that of MLE when M"10, c "c "0.4.
k
1
2
ANOVA methods yielding 3.8}20% higher MSE than that of MLE. MV1
performs the worst yielding MSE that is 5}154 times that of MLE and is
therefore not included in Fig. 4.
This con"rms that if one is interested in the estimates of the variance
components per se one is better o! with MLE, REML or MV2 and MV3-type
estimators. The ANOVA methods suggested here are second best. MV1 is not
recommended unless one suspects c and c are close to 0. However, for the
1
2
estimation of the regression coe$cients, the ANOVA methods compare well
and are recommended.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
375
Fig. 3. MSE of p( 2 relative to that of MLE when M"10, c "c "0.4.
l
1
2
Fig. 4. MSE of p( 2 relative to that of MLE when M"10, c "c "0.4.
e
1
2
For ANOVA and MIVQUE-type estimators, negative estimates of p2 or
k
p2 occur in about 50% of the replications when c "0 or c "0. When
l
1
2
the negative estimates of variance components are replaced by 0, the corresponding estimator forfeits its unbiasedness property. But, replacing these
negative estimates by 0 did not lead to much loss in e$ciency using the MSE
criterion.
376
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Fig. 5. MSE of standard errors of bK relative to that of MLE when M"10, c "c "0.4.
1
2
Finally, better estimates of the variance components by the MSE criterion, do
not necessarily imply better estimates of the regression coe$cients. A similar
result was obtained by Baltagi and Chang (1994) for the unbalanced one way
model and by Taylor (1980) and Baltagi (1981) for the balanced error component model. However, MV1 has worse relative MSE performance than other
ANOVA, ML, REML and MIVQUE-type estimators of the variance components when c and c are large and the pattern is severely unbalanced and this
1
2
clearly translates into a corresponding worse relative MSE performance of the
regression coe$cients. Similar conclusions can be drawn for M"6 and 15 and
are not produced here to save space.
3.4. A comparison of standard errors of the regression coezcients
Fig. 5 plots the MSE of the standard error of bK relative to that of MLE for the
12 unbalanced patterns for M"10 and c "c "0.4. Besides the relative
1
2
e$ciency of the parameter estimates, one is also interested in proper inference
on the parameter values. This is where the computationally involved estimators
(like MV2, MV3 and REML) perform well producing a MSE for the standard
error of bK that is close to that of MLE. The computationally simple ANOVA
methods (WH, WK, SA, HFB) have MSE for the standard error of bK that are
2 times that of MLE for severely unbalanced patterns like P , P and P .
10 11
12
However, these ANOVA methods perform reasonably well in patterns P }P
1 8
giving MSEs of the standard error of bK that are no more than 30% higher than
that of MLE.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
377
4. Empirical example
Baltagi and Pinnoi (1995) estimated a Cobb}Douglas production function
investigating the productivity of public capital in each state's private output.
This is based on a panel of 48 states over the period 1970}1986. The data were
provided by Munnell (1990). These states can be grouped into nine regions with
the Middle Atlantic region for example containing three states: New York, New
Jersey and Pennsylvania and the Mountain region containing eight states:
Montana, Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah and
Nevada. The primary group would be the regions, the nested group would be
the states and these are observed over 17 years. The dependent variable y is the
gross state product and the regressors include the private capital stock (K)
computed by apportioning the Bureau of Economic Analysis (BEA) national
estimates. The public capital stock is measured by its components: highways and
streets (KH), water and sewer facilities (KW), and other public buildings and
structures (KO), all based on the BEA national series. Labor (¸) is measured by
the employment in nonagricultural payrolls. The state unemployment rate is
included to capture the business cycle in a given state. See Munnell (1990) for
details on the data series and their construction. All variables except the
unemployment rate are expressed in natural logarithm
y "a#b K #b KH #b KW #b KO
ijt
1 ijt
2
ijt
3
ijt
4
ijt
# b ¸ #b Unemp #u ,
(43)
5 ijt
6
ijt
ijt
where i"1, 2,2, 9 regions, j"1,2, N with N equaling 3 for the Middle
i
i
Atlantic region and 8 for the Mountain region and t"1, 2,2, 17. The data is
unbalanced only in the di!ering number of states in each region. The disturbances follow the nested error component speci"cation given by (2).
Table 6 gives the OLS, WTN, ANOVA, MLE, REML and MIVQUE-type
estimates using this unbalanced nested error component model. The OLS
estimates show that the highways and streets and water and sewer components
of public capital have a positive and signi"cant e!ect upon private output
whereas that of other public buildings and structures is not signi"cant. Because
OLS ignores the state and region e!ects, the corresponding standard errors and
t-statistics are biased, see Moulton (1986). The within estimator shows that the
e!ect of KH and KW are insigni"cant whereas that of KO is negative and
signi"cant. The primary region and nested state e!ects are signi"cant using
several LM tests developed in Baltagi et al. (1999). This justi"es the application
of the feasible GLS, MLE and MIVQUE methods. For the variance components estimates, there are no di!erences in the estimate of p2. But estimates of
e
p2 and p2 vary. p( 2 is as low as 0.0015 for SA and MLE and as high as 0.0029 for
k
l
k
HFB. Similarly, p( 2 is as low as 0.0043 for SA and as high as 0.0069 for WK. This
l
variation had little e!ect on estimates of the regression coe$cients or their
378
Variable
OLS
WTN
WH
WK
SA
HFB
MLE
REML
MV1
MV2
MV3
Intercept
1.926
(0.053)
0.312
(0.011)
0.550
(0.016)
0.059
(0.015)
0.119
(0.012)
0.009
(0.012)
!0.007
(0.001)
*
2.082
(0.152)
0.273
(0.021)
0.742
(0.026)
0.075
(0.023)
0.076
(0.014)
!0.095
(0.017)
!0.006
(0.001)
2.131
(0.160)
0.264
(0.022)
0.758
(0.027)
0.072
(0.024)
0.076
(0.014)
!0.102
(0.017)
!0.006
(0.001)
2.089
(0.144)
0.274
(0.020)
0.740
(0.025)
0.073
(0.022)
0.076
(0.014)
!0.094
(0.017)
!0.006
(0.001)
2.084
(0.150)
0.272
(0.021)
0.743
(0.026)
0.075
(0.022)
0.076
(0.014)
!0.096
(0.017)
!0.006
(0.001)
2.129
(0.154)
0.267
(0.021)
0.754
(0.026)
0.071
(0.023)
0.076
(0.014)
!0.100
(0.017)
!0.006
(0.001)
2.127
(0.157)
0.266
(0.022)
0.756
(0.026)
0.072
(0.023)
0.076
(0.014)
!0.101
(0.017)
!0.006
(0.001)
2.083
(0.152)
0.272
(0.021)
0.742
(0.026)
0.075
(0.023)
0.076
(0.014)
!0.095
(0.017)
!0.006
(0.001)
2.114
(0.154)
0.269
(0.021)
0.750
(0.026)
0.072
(0.023)
0.076
(0.014)
!0.098
(0.017)
!0.006
(0.001)
2.127
(0.156)
0.267
(0.021)
0.755
(0.026)
0.072
(0.023)
0.076
(0.014)
!0.100
(0.017)
!0.006
(0.001)
0.0014
0.0027
0.0045
0.0014
0.0022
0.0069
0.0014
0.0015
0.0043
0.0014
0.0029
0.0044
0.0013
0.0015
0.0063
0.0014
0.0019
0.0064
0.0014
0.0027
0.0046
0.0014
0.0017
0.0056
0.0014
0.0017
0.0063
K
¸
KH
KW
KO
Unemp
p2
e
p2
k
p2
l
0.235
(0.026)
0.801
(0.030)
0.077
(0.031)
0.079
(0.015)
!0.115
(0.018)
!0.005
(0.001)
0.0073
*
*
0.0013
*
*
!The dependent variable is log of gross state product. Standard errors are given in parentheses.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Table 6
Cobb}Douglas production function estimates with unbalanced nested error components 1970}1986, Nine regions, 48 states!
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
379
standard errors. For all estimators of the random e!ects model, the highways
and streets and water and sewer components of public capital had a positive and
signi"cant e!ect, while the other public buildings and structures had a negative
and signi"cant e!ect upon private output.
5. Conclusion
For the regression coe$cients of the nested unbalanced error component
model, the simple ANOVA methods proposed in this paper performed well in
Monte Carlo experiments as well as in the empirical example and are recommended. However, for the variance components estimates themselves, as well as
the standard errors of the regression coe$cients, the computationally more
demanding MLE, REML or MIVQUE (MV2 and MV3) estimators are recommended especially if the unbalanced pattern is severe. Further research should
extend the unbalanced nested error component model considered in this paper
to allow for endogeneity of the regressors, a dynamic speci"cation, ignorability
of the sample selection and serial correlation in the disturbances.
Acknowledgements
The authors would like to thank two anonymous referees and an Associate
Editor for their helpful comments and suggestions. A preliminary version of this
paper was presented at the European meetings of the Econometric Society held
in Santiago de Compostela, Spain, August, 1999. Also, at the University of
Chicago, University of Pennsylvania and the University of Rochester. Baltagi
would like to thank the Texas Advanced Research Program and the Bush
Program in economics of public policy for their "nancial support.
References
Ahrens, H., Pincus, R., 1981. On two measures of unbalancedness in a one-way model and their
relation to e$ciency. Biometric Journal 23, 227}235.
Amemiya, T., 1971. The estimation of variances in a variance components model. International
Economic Review 12, 1}13.
Antweiler, W., 1999. Nested random e!ects estimation in unbalanced panel data. Working Paper,
University of British Columbia.
Baltagi, B.H., 1981. Pooling: an experimental study of alternative testing and estimation procedures
in a two-way error components model. Journal of Econometrics 17, 21}49.
Baltagi, B.H., 1985. Pooling cross-sections with unequal time series lengths. Economics Letters
18, 133}136.
Baltagi, B.H., 1993. Useful matrix transformations for panel data analysis: a survey. Statistical
Papers 34, 281}301.
380
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Baltagi, B.H., 1995. Econometric Analysis of Panel Data. Wiley, New York.
Baltagi, B.H., Chang, Y.J., 1994. Incomplete panels: a comparative study of alternative estimators for
the unbalanced one-way error component regression model. Journal of Econometrics 62, 67}89.
Baltagi, B.
The unbalanced nested error component
regression model
Badi H. Baltagi!,*, Seuck Heun Song", Byoung Cheol Jung"
!Department of Economics, Texas A&M University, College Station, TX 77843-4228, USA
"Department of Statistics, Korea University, Sungbuk-Ku, Seoul 136-701, South Korea
Received 1 December 1998; received in revised form 31 August 2000; accepted 2 October 2000
Abstract
This paper considers a nested error component model with unbalanced data and
proposes simple analysis of variance (ANOVA), maximum likelihood (MLE) and minimum norm quadratic unbiased estimators (MINQUE)-type estimators of the variance
components. These are natural extensions from the biometrics, statistics and econometrics literature. The performance of these estimators is investigated by means of Monte
Carlo experiments. While the MLE and MINQUE methods perform the best in estimating the variance components and the standard errors of the regression coe$cients, the
simple ANOVA methods perform just as well in estimating the regression coe$cients.
These estimation methods are also used to investigate the productivity of public capital
in private production. ( 2001 Published by Elsevier Science S.A.
JEL: C23
Keywords: Panel data; Nested error component; Unbalanced ANOVA; MINQUE;
MLE; Variance components
1. Introduction
The analysis of panel data in econometrics have relied on the error component regression model which has its origin in the statistics and biometrics
* Corresponding author. Tel.: #1-979-845-7380; fax: #1-979-847-8757.
E-mail address: [email protected] (B.H. Baltagi).
0304-4076/01/$ - see front matter ( 2001 Published by Elsevier Science S.A.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 8 9 - 0
358
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
literature, see Hsiao (1986), Baltagi (1995) and MaH tyaH s and Sevestre (1996).
A huge bulk of this econometrics literature focuses on the complete or balanced
panels, yet the empirical applications face missing observations or incomplete
panels. Exceptions are Baltagi (1985), Wansbeek and Kapteyn (1989) and
Baltagi and Chang (1994). This paper considers the incomplete panel data
regression model in which the economic data has a natural nested groupings.
For example, data on "rms may be grouped by industry, data on states by
region and data on individuals by profession. In this case, one can control for
unobserved industry and within industry "rm e!ects using a nested error
component model. See Montmarquette and Mahseredjian (1989) for an empirical application of the nested error component model to study whether schooling
matters in educational achievements in Montreal's Francophone public elementary schools. More recently, see Antweiler (1999) for an application of the
determinants of pollution concentration as measured by observation stations in
various countries over time.
This paper proposes natural extensions of the analysis of variance (ANOVA),
maximum likelihood (MLE) and minimum norm quadratic unbiased estimators
(MINQUE) and compares their performance by means of Monte Carlo experiments. Statisticians and biometricians are more interested in the estimates of the
variance components per se, see Harville (1969, 1977), Hocking (1985), LaMotte
(1973a, b), Rao (1971a, b), Searle (1971, 1987) and Swallow and Monahan (1984)
to mention a few. Econometricians, on the other hand, are more interested in the
regression coe$cients, see Hsiao (1986) and Baltagi (1995). Monte Carlo results
on the balanced error component regression model include Nerlove (1971),
Maddala and Mount (1973) and Baltagi (1981). For the unbalanced error
component regression model, see Wansbeek and Kapteyn (1989) and Baltagi
and Chang (1994). None of these studies deal with the nested and unbalanced
error component model. The only exception is Fuller and Battese (1973). This
paper generalizes several estimators in the literature to the nested unbalanced
setting and reports the results of Monte Carlo experiments comparing the
performance of these proposed estimators. The type of unbalancedness considered in this paper allows for unequal number of "rms in each industry as well
as di!erent number of time periods across industries. Section 2 describes the
model and the estimation methods to be compared. Section 3 gives the design of
the Monte Carlo experiment and summarizes the results, while Section 4 gives an
empirical illustration applying these estimation methods to the study of productivity of public capital in private production. Section 5 gives our conclusion.
2. The model
We consider the following unbalanced panel data regression model:
y "x@ b#u , i"1,2, M, j"1,2, N and t"1,2, ¹ ,
ijt
ijt
i
i
ijt
(1)
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
359
where y could denote the output of the jth "rm in the ith industry for the tth
ijt
time period. x denotes a vector of k nonstochastic inputs. The disturbance of
ijt
(1) is given by
u "k #l #e , i"1,2, M, j"1,2, N and t"1,2, ¹ , (2)
ijt
i
ij
ijt
i
i
where k denotes the ith unobservable industry speci"c e!ect which is assumed
i
to be i.i.d. (0, p2),l denotes the nested e!ect of the jth "rm within the ith
k ij
industry which is assumed to be i.i.d. (0, p2) and e denotes the remainder
l
ijt
disturbance which is also assumed to be i.i.d. (0, p2). The k 's, l 's and e 's are
e
i
ij
ijt
independent of each other and among themselves. This is a nested classi"cation
in that each successive component of the error term is imbedded or &nested'
within the preceding component, see Graybill (1961, p. 350). This model allows
for unequal number of "rms in each industry as well as di!erent number of
observed time periods across industries. Model (1) can be rewritten in matrix
notation as
y"Xb#u,
(3)
where y is a +M N ¹ ]1, X is a +M N ¹ ]k, b is a k]1 parameter vector,
i/1 i i
i/1 i i
and u is a +M N ¹ ]1 disturbance vector. Eq. (2) in vector form yields
i/1 i i
u"Z k#Z l#e,
(4)
k
l
where k@"(k ,2, k ), l@"(l ,2,l 1 ,2,l M ), e@"(e ,2, e 1 ,2,
MN
111
11T
1
M
11
1N
e M M ), Z "diag(ι i ?ι i ), Z "diag(I i ?ι i ), ι i and ι i are vectors of
MN T
k
N
T
l
N
T
N
T
ones of dimension N and ¹ , respectively. By diag(ι i ?ι i ) we mean
i
i
N
T
diag(ι 1 ?ι 1 ,2,ι M ?ι M ). I i is an identity matrix of dimension N , and
N
T
N
T
N
i
? denotes the Kronecker product. Note that the observations are stacked such
that the slowest running index is the industry index i, the next slowest running
index is the "rm index j and the fastest running index is time.
Under these assumptions, the disturbance covariance matrix E(uu@) can be
written as
X"p2 Z Z@ #p2Z Z@ #p2 diag(I i ?I i )
T
e
N
l l l
k k k
(5)
"diag[p2(J i ?J i )#p2(I i ?J i )#p2(I i ?I i )],
l N
e N
T
T
T
k N
where J i "ι i ι@ i and J i "ι i ι@ i are matrices of ones of dimension N and ¹ .
N
N N
T
T T
i
i
It is clear from Eq. (5) that X is a block diagonal matrix with the ith block given
by
K "p2 (J i ?J i )#p2(I i ?J i )#p2(I i ?I i ), i"1,2, M.
(6)
k N
i
l N
e N
T
T
T
Replacing J i by its idempotent counterpart ¹JM i where JM i "J i /¹ and
T
T
T
T i
J i by N JM i where JM i "J i /N , we get
N
i N
N
N i
(7)
K "N ¹ p2 (JM i ?JM i )#¹ p2(I i ?JM i )#p2(I i ?I i ).
e N
T
i l N
T
T
i
i i k N
360
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Replacing I i by E i #JM i and I i by E i #JM i , where E i "I i !JM i and
N
N
N
T
T
T
N
N
N
E i "I i !JM i and collecting terms with the same matrices, see Wansbeek and
T
T
T
Kapteyn (1982, 1983), one gets the spectral decomposition of K :
i
K "j Q #j Q #j Q ,
(8)
i
1i 1i
2i 2i
3i 3i
where j "p2, j "¹ p2#p2 and j "N ¹ p2#¹ p2#p2. Corree
i l
e
3i
i i k
e
2i
i l
1i
spondingly, Q "I i ?E i , Q "E i ?JM i and Q "JM i ?JM i . The
1i
N
T
2i
N
T
3i
N
T
j , p"1, 2, 3, are the distinct characteristic roots of K of multiplicity
pi
i
N (¹ !1), N !1 and 1, respectively. Note that each Q , for p"1, 2, 3 is
i i
i
pi
symmetric, idempotent with its rank equal to its trace. Moreover, the Q 's are
pi
pairwise orthogonal and sum to the identity matrix. The advantages of this
spectral decomposition are that
(9)
Kp"jp Q #jp Q #jp Q ,
3i 3i
2i 2i
1i 1i
i
where p is an arbitrary scalar, see Baltagi (1993). Therefore, we can easily obtain
X~1 as
X~1"diag[K~1]"diag[j~1Q #j~1Q #j~1Q ]
3i 3i
2i 2i
1i 1i
i
(10)
and
C
D
p
p
p
e Q # e Q # e Q
p X~1@2"diag
1i Jj
2i Jj
3i
e
Jj
2i
3i
1i
"diag[I i ?I i ]!diag[h (I i ?JM i )]!diag[h (JM i ?JM i )],
N
T
1i N
T
2i N
T
(11)
where h "1!p /Jj and h "p /Jj !p /Jj . This allows us to ob1i
e
2i
2i
e
2i
e
3i
tain GLS on (3) as an ordinary least squares (OLS) of yH"p X~1@2y on
e
XH"p X~1@2X. The typical element of yH is given by (y !h y6 !h y6 )
e
ijt
1i ij.
2i i..
where y6 "+Ti y /¹ and y6 "+Ni +Ti y /N ¹ . This is known in the
t/1 ijt i
i..
j/1 t/1 ijt i i
ij.
econometrics literature as the Fuller and Battese (1973) transformation.
Note that the OLS estimator is given by
bK
"(X@X)~1X@y.
(12)
OLS
This is the best linear unbiased estimator when the variance components p2 and
k
p2 are both equal to 0. Even when these variance components are positive, the
l
OLS estimator is still unbiased and consistent, but its standard errors are biased,
see Moulton (1986). The OLS residuals are denoted by u(
"y!XbK
.
OLS
OLS
The within estimator in this case can be obtained by transforming the model
in (3) by Q "diag(I i ?E i ) and then applying OLS. Note that
T
1
N
Q Z "Q Z "0 because E i ι i "0. Therefore, Q sweeps away the k 's and
1 k
1 l
T T
1
i
l 's whether they are "xed or random e!ects. This yields
ij
(13)
bI "(X@ Q X )~1X@ Q y,
4 1
4 1 4
4
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
361
where X denotes the exogenous regressors excluding the intercept and b de4
4
notes the corresponding (k!1) vector of slope coe$cients. b@"(a, b@ ) and the
4
estimate of the intercept can be retrieved as follows: a8"(y6 !XM bI ), where
...
4... 4
the dots indicate summation and the bar indicates averaging. Following
Amemiya (1971), the within residuals u8
for the unbalanced nested e!ect
WTN
model are given by
u8
"y!a8ι !X bI
WTN
m
4 4
where m"+M N ¹ .
i/1 i imethods of estimating the variance components.
Next, we consider
(14)
2.1. Analysis of variance methods
These are methods of moments-type estimators that equate quadratic sums of
squares to their expectations and solve the resulting equations for the unknown
variance components. These ANOVA estimators are best quadratic unbiased
(BQU) estimators of the variance components in the balanced error component
model case, see Graybill (1961). Under normality of the disturbances they are
even minimum variance unbiased. However, for the unbalanced model, BQU
estimators of the variance components are a function of the variance components themselves, see Searle (1987). Unbalanced ANOVA methods are available
but optimal properties beyond unbiasedness are lost. We consider four
ANOVA-type methods which are natural extensions of those proposed in the
balanced error component literature:
(1) A modi"ed Wallace and Hussain (WH) estimator: Consider the three
quadratic forms of the disturbances using the Q , Q and Q matrices obtained
1 2
3
from the spectral decomposition of X in (8):
q "u@Q u, q "u@Q u, q "u@Q u,
(15)
1
1
2
2
3
3
where Q "diag(Q ), Q "diag(Q ) and Q "diag(Q ). Substituting OLS
1
1i
2
2i
3
3i
residuals u(
for u in (15) we get q( , q( and q( , see Wallace and Hussain (1969)
OLS
1 2
3
and Baltagi and Chang (1994). Taking expected values, we obtain
)"d p2#d p2#d p2,
E(q( )"E(u( @ Q u(
13 l
12 k
OLS 1 OLS
11 e
1
)"d p2#d p2#d p2,
E(q( )"E(u( @ Q u(
23 l
22 k
OLS 2 OLS
21 e
2
)"d p2#d p2#d p2,
E(q( )"E(u( @ Q u(
33 l
32 k
OLS 3 OLS
31 e
3
where the d 's are given by
ij
d "m!n!tr(X@Q X(X@X)~1),
11
1
d "tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
1
12
k k
d "tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
13
l l
1
(16)
362
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
d "n!M!tr(X@Q X(X@X)~1),
21
2
d "tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
2
22
k k
d "m!t!2tr[(X@Z Z@ Q X)(X@X)~1]
23
l l 2
# tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
2
l l
d "M!tr(X@Q X(X@X)~1),
31
3
d "m!2tr[(X@Z Z@ X)(X@X)~1]
32
k k
# tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
3
k k
d "t!2tr[(X@Z Z@ Q X)(X@X)~1]
33
l l 3
(17)
# tr[(X@Z Z@ X)(X@X)~1(X@Q X)(X@X)~1],
3
l l
with m"+M N ¹ , n"+M N and t"+M ¹ . Equating the q( 's to their
i i
i
i
i/1 E(q
i/1 i of equations, one
expected values
( ) in (16)i/1
and solving the system
gets the
i
Wallace and Hussain (1969)-type estimators of the variance components.1 These
are denoted by WH.
(2) A modi"ed Wansbeek and Kapteyn (WK) estimator: Alternatively, one
can substitute within residuals in the quadratic forms given by (15) to get q8 ,
1
q8 and q8 , see Amemiya (1971) and Wansbeek and Kapteyn (1989). Taking
2
3
expected values of q8 , q8 and q8 we get
1 2
3
)"(m!n!k#1)p2,
E(q8 )"E(u8 @ Q u8
e
WTN 1 WTN
1
E(q8 )"E(u8 @ Q u8
)
2
WTN 2 WTN
"[n!M#trM(X@ Q X )~1X@ Q X N]p2#(m!t)p2,
l
e
4 2 4
4 1 4
)
E(q8 )"E(u8 @ Q u8
WTN 3 WTN
3
"[M!1#trM(X@ Q X )~1X@ Q X N!trM(X@ Q X )~1X@ JM X N]p2
e
4 m 4
4 1 4
4 3 4
4 1 4
(18)
# [t!+N ¹2/m]p2#[m!+N2¹2/m]p2 .
k
i i
l
i i
Equating q8 to its expected value E(q8 ) in (18) and solving the system of
i
i
equations, we get the following Wansbeek and Kapteyn-type estimator of the
variance components which we denote by WK:
/(m!n!k#1),
p82"u8 @ Q u8
WTN 1 WTN
e
u8 @ Q u8
![n!M#trM(X@ Q X )~1(X@ Q X )Np82]
4 1 4
4 2 4 e ,
p82" WTN 2 WTN
l
m!t
1 Most of the algebra involved is simple but tedious and all proofs are available upon request from
the authors.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
363
![M!1#trM(X@ Q X )~1X@ Q X N
p82"(u8 @ Q u8
4 3 4
4 1 4
WTN 3 WTN
k
! trM(X@ Q X )~1X@ JM X N]p82
4 m 4 e
4 1 4
(19)
![t!+N ¹2/m]p82)/[m!+N2¹2/m].
i i
l
i i
(3) A Modi"ed Swamy and Arora (SA) estimator: Following Swamy and
Arora (1972), we transform the regression model in (3) by premultiplying it by
Q , Q and Q and we obtain the transformed residuals u8 , u8 and u8 ,
1
2
3
1 2
3
respectively. Let q8 `"u8 @ Q u8 , q8 `"u8 @ Q u8 and q8 `"u8 @ Q u8 . Since q8 ` is
1
3 3 3
3
2 2 2
1 1 1 2
1
exactly the same as q8 the resulting expected value of q8 ` is the same as that
1
1
given in (18). The expected values of q8 ` and q8 ` are
3
2
E(q8 `)"E(u8 @ Q u8 )
2 2 2
2
"(n!M!k#1)p2#[m!t!trM(X@ Z Z@ Q X )
4 l l 2 4
e
(X@ Q X )~1N]p2,
l
4 2 4
E(q8 `)"E(u8 @ Q u8 )
3 3 3
3
"(M!k)p2#[t!trM(X@Z Z@ Q X)(X@Q X)~1N]p2
l
3
e
l l 3
(20)
# [m!trM(X@Z Z@ X)(X@Q X)~1N]p2 .
k
3
k k
Equating q8 ` to its expected value E(q8 `) and solving the system of equations, we
i
i
get the following Swamy and Arora-type estimators of the variance components
which we denote by SA:
/(m!n!k#1),
p82"u8 @ Q u8
WTN 1 WTN
e
u8 @ Q u8 !(n!M!k#1)p82
e
2 2 2
,
p82"
l m!t!trM(X@ Z Z@ Q X )(X@ Q X )~1N
4 l l 2 4 4 2 4
u8 @ Q u8 !(M!k)p82![t!trM(X@Z Z@ Q X)(X@Q X)~1N]p82
3
e
l l 3
l.
p82" 3 3 3
k
m!trM(X@Z Z@ X)(X@Q X)~1N
3
k k
(21)
(4) Henderson Method III: Fuller and Battese (1973) suggest an estimation of
the variance components using the "tting constants methods. This method uses
. Also, the residual
the within residual sums of squares given by q8 H"u8 @ u8
WTN WTN
1
sum of squares obtained by transforming the regression in (3) by (Q #Q ) (i.e.,
1
2
the regression of y !y6 on x l !x6 l , for l"1,2, k). This is denoted by
i..
ijt
i..
ijt
q8 H"u8 H@u8 H where u8 H is the residual vector of the (Q #Q ) transformed regres2
1
2
2 2
2
sion. Finally, this method uses the conventional OLS residual sum of squares
. If the x variables do not have constant values for
denoted by q8 H"u( @ u(
OLS OLS
3
measurement of group and nested subgroups, q8 H is exactly the same as that for
1
364
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
the WK method, the resulting expected value of q8 H is the same as that given in
1
(18). Also, the expected value of q8 H and q8 H are given by
3
2
E(q8 H)"p2[m!M!k#1]
e
2
# p2[m!t!trM(X@ Z Z@ Q X )(X@ (Q #Q )X )~1N],
2 4
4 l l 2 4 4 1
l
E(q8 H)"p2[m!k]#p2[m!trM(X@Z Z@ X)(X@X)~1N]
l
l l
e
3
(22)
# p2 [m!trM(X@Z Z@ X)(X@X)~1N].
k
k k
Equating q8 H, for i"1, 2, 3 to its expected value E(q8 H) in (22), we obtain the
i
i
Henderson Method III estimator of the variance components, see Fuller and
Battese (1973). These are denoted by HFB:
/(m!n!k#1),
p82"u8 @ Q u8
WTN 1 WTN
e
u8 H@u8 H!(m!M!k#1)p82
e
2 2
p82"
,
l m!t!trM(X@ Z Z@ Q X )(X@ (Q #Q )X )~1N
2 4
4 l l 2 4 4 1
!(m!k)p82![m!trM(X@diag(I i ?J i )X)(X@X)~1Np82]
u( @ u(
e
N
l .
T
p82" OLS OLS
k
m!trM(X@Z Z@ X)(X@X)~1N
k k
(23)
2.2. Maximum likelihood estimator
Since j , for p"1, 2, 3 are the distinct characteristic roots of K then
pi
i
DK D"(j )(jNi ~1)(jNi (Ti ~1)). Let o "p2/p2, o "p2/p2 and X"p2R, then the
e
l e
i
3i 2i
k e 2
1i
1
log-likelihood function can be written as
m
1 M
log ¸"C! log p2! + log(N ¹ o #¹ o #1)
i i 1
i 2
e 2
2
i/1
1
1 M
! + (N !1) log(¹ o #1)! u@R~1u/2p2.
(24)
i
i 2
e
2
2
i/1
The "rst-order conditions give closed form solutions for b and p2 conditional on
e
o and o :
1
2
bK "(X@RK ~1X)~1X@RK ~1y,
(25)
ML
(26)
p( 2"(y!Xb)@RK ~1(y!Xb)/m.
e
However, the "rst-order conditions based on o( and o( are nonlinear in o and
1
2
1
o even for known values of b and p2. Following Hemmerle and Hartley (1973),
e
2
we get
1
1
L log ¸
"! tr[Z@ R~1Z ]#
(y!Xb)@R~1Z Z@ R~1(y!Xb),
k
k
k k
2p2
2
Lo
e
1
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
365
L log ¸
1
1
"! tr[Z@ R~1Z ]#
(y!Xb)@R~1Z Z@ R~1(y!Xb). (27)
l
l
l l
Lo
2p2
2
2
e
Therefore, a numerical solution by means of iteration is needed. The Fisher
scoring procedure is used to estimate o and o . The partition of the informa1
2
tion matrix corresponding to o and o is given by
1
2
(N ¹ )2
1 M
L2 log ¸
i i
,
" +
E !
(1#o ¹ #o N ¹ )2
2
Lo2
2 i
1 i i
1
i/1
N ¹2
L2 log ¸
1 M
i i
E !
,
" +
(1#o ¹ #o N ¹ )2
Lo Lo
2
2
i
1
i
i
1 2
i/1
L2 log ¸
¹2
1 M
1 M (N !1)¹2
i # +
i
i
E !
, (28)
" +
Lo2
(1#o ¹ )2 2
(1#o ¹ #o N ¹ )2
2
2
2 i
2 i
1 i i
i/1
i/1
see Harville (1977). Starting with an initial value, the (r#1)th updated value of
o and o is given by
1
2
L2 log ¸ ~1 L log ¸
L2 log ¸
E !
E !
o(
o(
Lo Lo
Lo2
Lo
1
1 2
1
1
,
" 1 #
L
log ¸
L2 log ¸
L2 log ¸
o( r`1
o( r
2
2
E !
E !
Lo
Lo Lo
Lo2
2 r
1 2
2
r
(29)
C
C
C
C D
D
D
D
C D
C
C
C
D C
D C
DC D
D
D
where at each step, L log ¸/Lo and L log ¸/Lo are obtained from Eq. (27), bK and
1
2
p( 2 are obtained from (25) and (26), the information matrix is obtained from
e
Eq. (28). The subscript r means this is evaluated at the rth iteration. For a review
of the advantages and disadvantages of MLE, see Harville (1977).
2.3. Restricted maximum likelihood estimator
Patterson and Thompson (1971) suggested a restricted maximum likelihood
(REML) estimation method that takes into account the loss of degrees of
freedom due to the regression coe$cients in estimating the variance components. REML is based on a transformation that partitions the likelihood
function into two parts, one being free of the "xed regression coe$cients.
Maximizing this part yields REML. Patterson and Thompson (1971) suggest the
singular transformation y@[C F R~1X/p2], where C"I!X(X@X)~1X@. Cy is
e
distributed as N(0, CRC@/p2), and from the fact that CX"0, it is independent of
e
X@R~1y/p2 which is also distributed as N(X@R~1Xb/p2, X@R~1X/p2). It is clear
e
e
e
that Cy does not depend on b. Since C is an idempotent matrix of rank m!k,
there exists an (m!k)]m matrix A such that
A@A"C, AA@"I.
(30)
366
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Using the A@y transformation instead of C@y, we get
p2ARA@ 0
e
.
(31)
0
X@R~1X/p2
X@R~1y/p2
X@R~1Xb/p2
e
e
e
Following Corbeil and Searle (1976), the log-likelihood function of A@y and
X@R~1y/p2 are given by log ¸ and log ¸ , respectively:
e
1
2
1
m!k
m!k
log(2p)!
log(p2)! logDARA@D
log ¸ "!
e
1
2
2
2
C
A@y
D AC
&N
0
DC
,
DB
1
y@[A@(ARA@)~1A]y,
2p2
e
1
k
log ¸ "! log(2p)! log(p2)!logDX@R~1XD
e
2
2
2
!
1
(y!Xb)@R~1X(X@R~1X)~1X@R~1(y!Xb).
(32)
2p2
e
Using the results of Hocking (1985) and Corbeil and Searle (1976), we obtain
!
A@(ARA@)~1A"R~1[I!X(X@R~1X)~1X@R~1]"R~1(I!M),
(33)
where M"X(X@R~1X)~1X@R~1.
Using log ¸ which is free from b, the "rst-order derivatives of log ¸ with
1
1
respect to p2, o and o are given by
e 1
2
m!k
1
L log ¸
1 "!
#
y@A@(ARA@)~1Ay
2p2
2p4
Lp2
e
e
e
m!k
1
"!
#
y@R~1(I!M)y,
2p2
2p4
e
e
1
L log ¸
1 "! tr[Z@ A@(ARA@)~1AZ ]
k
k
2
Lo
1
1
#
y@[A@(ARA@)~1AZ Z@ A@(ARA@)~1A]y,
k k
2p2
e
1
L log ¸
1 "! tr[Z@ A@(ARA@)~1AZ ]
l
l
2
Lo
2
1
#
(34)
y@[A@(ARA@)~1AZ Z@ A@(ARA@)~1A]y.
l l
2p2
e
Equating the equations in (34) to 0's yield the REML estimates. For example,
solving L log ¸ /Lp2"0 conditional on o and o , we obtain
1
2
1 e
(35)
p( 2"y@A@(ARK A@)~1Ay/(m!k)"y@[RK ~1(I!M)]y/(m!k).
e
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
367
But there are no closed-form solutions on o and o . Thus a numerical solution
1
2
by means of iteration is needed. The Fisher scoring procedure is used to estimate
o and o . Using the results of Harville (1977) and Eq. (33), the information
1
2
matrix with respect to o and o is given by
1
2
1
L2 log ¸
1 " tr[Z Z@ R~1(I!M)Z Z@ R~1(I!M)],
E !
k k
k k
Lo2
2
1
1
L2 log ¸
1 " tr[Z Z@ R~1(I!M)Z Z@ R~1(I!M)],
E !
l l
k k
2
Lo Lo
1 2
1
L2 log ¸
1 " tr[Z Z@ R~1(I!M)Z Z@ R~1(I!M)].
(36)
E !
l l
l l
2
Lo2
2
The updated values of o and o can be obtained as in (29).
1
2
C
C
C
D
D
D
2.4. MINQUE and MIVQUE
Rao (1971a) proposed a general procedure for variance components estimation which requires no distributional assumptions other than the existence of the
"rst four moments. This procedure yields MINQUE of the variance components. Under normality of the disturbances, MINQUE and minimum variance
quadratic unbiased estimators (MIVQUE) are identical. Since we assume normality, we will focus on MIVQUE. Let
R"R~1[I!X(X@R~1X)~1X@R~1]/p2,
e
S"Ms N"Mtr(< R< R)N, i, j"1, 2, 3
ij
i
j
(37)
(38)
and
u"Mu N"My@R< RyN, i"1, 2, 3,
(39)
i
i
where < "I , < "Z Z@ and < "Z Z@ . Rao (1971b) shows that the vector
3
l l
1
m 2
k k
of MIVQUEs is given by
hK "S~1u,
(40)
where hK @"(p( 2, p( 2, p( 2). However, MIVQUE requires a priori values of
e k l
the variance components. Therefore, MIVQUE is only &locally minimum
variance', see LaMotte (1973a, b), and &locally best', see Harville (1969). Three
priors of the MIVQUE estimator are considered in our Monte Carlo study:
(1) the identity matrix, which we denote by MV1, and (2) all values of
the variance components equal to 1, see Swallow and Searle (1978) which
we denote by MV2, and (3) the ANOVA estimator of WK, which we denote
by MV3. Note that the MIVQUE estimator can produce negative estimates
of the variance components. In this case, we replace the negative variance
estimate by 0.
368
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
3. Monte Carlo results
3.1. Design of the Monte Carlo study
We consider the following simple regression equation:
y "a#x b#u , i"1,2, M, j"1,2, N , t"1,2, ¹ , (41)
ijt
ijt
ijt
i
i
with u "k #l #e . The exogenous variable x was generated by a similar
ijt
i
ij
ijt
ijt
method to that of Nerlove (1971). In fact, x "0.3t#0.8x
#w , where
ijt
ij,t~1
ijt
w is uniformly distributed on the interval [!0.5, 0.5]. The initial values
ijt
x were chosen as (100#250w ). Throughout the experiment a"5 and
ij0
ij0
b"2. For generating the u
disturbances, we let k &IIN(0, p2),
k
ijt
i
l &IIN(0, p2) and e &IIN(0, p2). We "x p2"p2 #p2#p2"20 and de"ne
e
l
k
e
l
ijt
ij
c "p2 /p2 and c "p2/p2. These are varied over the set (0, 0.2, 0.4, 0.6, 0.8)
l
k
2
1
such that (1!c !c ) is always positive. Extending a measure of unbalanced1
2
ness given by Ahrens and Pincus (1981) to the unbalanced nested model, we
de"ne
c "M/NM + (1/N ) where NM "+ N /M,
1
i
i
c "M/¹M + (1/¹ ) where ¹M "+¹ /M,
2
i
i
(42)
c "M/N¹ + (1/N ¹ ) where N¹"+ N ¹ /M,
i i
i i
3
where c , c and c denote the measures of subgroup unbalancedness, observed
1 2
3
time unbalancedness and group unbalancedness due to each group size. Note
that c , c and c take the value 1 when the data are balanced but take smaller
1 2
3
values than 1 as the data pattern gets more unbalanced. Table 1 gives the
(N , ¹ ) pattern used along with the corresponding unbalancedness measures for
i i
M"10. The "rst parentheses gives the N pattern, while the second parentheses
i
below it gives the corresponding ¹ pattern. For example, P observes the "rst
i
1
grouping of eight individuals over six time periods and the last grouping of 12
individuals over four time periods. The sample size is "xed at 500 for every
pattern. Two other values of M are used, M"6 and 15. For each experiment,
1000 replications are performed. For each replication, we calculate OLS, WTN,
WH, SA, WK, HFB, ML, REML, MV1, MV2, MV3 and true GLS. The last
estimator is obtained for comparison purposes.
3.2. A comparison of regression coezcient estimates
Table 2 gives the mean square error (MSE) of the estimate of bK relative to that
of true GLS for the case when M"10.2 From this table it is clear that OLS is
2 Similar MSE tables for the regression coe$cients and the variance components estimates are
generated for M"6 and 15, but they are not produced here to save space. These tables are
available upon request from the authors.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
369
Table 1
(N , ¹ ) patterns considered and their corresponding unbalancedness measures when M"10
i i
Pattern
(N , N ,2, N )!
1 2
10
(¹ , ¹ ,2, ¹ )
1 2
10
c
P
1
(8,8,8,10,10,10,10,12,12,12)
(6,6,6,5,5,5,5,5,4,4)
0.976
0.980
0.996
P
2
(6,6,6,10,10,10,10,12,12,12)
(9,9,9,9,8,3,3,3,3,3)
0.925
0.757
0.8238
P
3
(5,5,5,10,10,10,10,11,11,11)
(2,2,3,3,3,6,7,8,8,9)
0.893
0.734
0.504
P
4
(4,4,4,5,5,9,9,10,10,10)
(14,15,15,15,15,3,3,4,4,4)
0.854
0.619
0.881
P
5
(3,3,3,3,3,8,8,8,8,8)
(2,2,2,3,3,11,11,12,12,12)
0.793
0.550
0.258
P
6
(2,2,6,6,6,10,10,10,13,13)
(16,16,16,16,16,2,2,3,3,3)
0.656
0.465
0.718
P
7
(2,2,2,10,10,10,10,13,13,13)
(2,1,1,1,1,8,8,8,8,8)
0.552
0.424
0.133
P
8
(20,20,15,15,15,3,3,3,2,2)
(1,1,6,6,6,10,10,10,25,25)
0.444
0.347
0.732
P
9
(16,16,16,16,16,2,2,2,2,2)
(2,2,3,3,3,28,28,30,30,30)
0.395
0.290
0.949
P
10
(20,20,20,20,20,2,2,2,2,1)
(2,2,2,3,3,25,30,30,30,30)
0.282
0.272
0.945
P
11
(1,1,1,1,5,5,25,25,25,25)
(1,2,2,35,35,2,2,3,3,3)
0.192
0.280
0.091
P
12
(1,1,1,1,5,5,30,30,30,30)
(27,27,28,28,28,2,2,2,2,2)
0.165
0.252
0.626
1
c
2
c
3
!The "rst parentheses gives the N pattern, while the parentheses below it gives the corresponding
i
¹ pattern.
i
inferior to true GLS, ML-type (ML, REML) estimators and all feasible GLStype estimators except when c " c "0. For all experiments, the e!ect of an
1
2
increase in c on the MSE of OLS is much larger than that of an increase in c .
1
2
This is because c a!ects the primary group while c a!ects only the nested
1
2
subgroup. The WTN estimator performs poorly for small c and c values. The
1
2
performance of WTN is in some cases worse than OLS if either c or c is 0.
1
2
However, its performance improves as c and c increase and the unbalanced1
2
ness pattern gets more severe. The ANOVA-type (WH, WK, SA and HFB)
370
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Table 2
MSE of bK relative to that of true GLS when M"10
c
1
c
2
OLS
WTN
WH
WK
SA
HFB
MLE
REML MV1
MV2
MV3
P
1
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.069
1.396
2.073
4.341
1.649
1.633
2.274
4.358
3.537
2.748
4.407
6.235
5.673
13.606
4.864
2.732
1.985
1.406
1.212
3.979
2.333
1.549
1.219
4.149
2.049
1.276
4.077
1.555
4.473
0.998
1.014
1.008
1.000
1.000
1.003
1.002
1.010
1.011
1.001
1.004
1.000
1.003
1.000
1.007
1.006
1.016
1.011
1.001
1.000
1.001
0.998
1.011
1.010
0.998
1.008
0.998
1.004
0.998
1.011
0.996
1.011
1.009
1.001
1.001
1.003
1.002
1.014
1.017
1.004
1.006
1.001
1.007
1.003
1.004
0.999
1.014
1.009
1.000
1.000
1.002
1.001
1.011
1.010
0.999
1.006
1.000
1.005
0.998
1.006
0.998
1.012
1.008
0.999
1.000
1.003
1.001
1.011
1.011
1.001
1.006
0.999
1.005
0.998
1.006
0.999
1.015
1.009
0.999
1.000
1.004
1.001
1.010
1.010
1.000
1.007
0.999
1.005
0.998
1.006
0.998
1.017
1.010
1.000
1.004
1.002
0.999
1.014
1.015
1.004
1.012
1.003
1.036
1.013
1.135
0.999
1.013
1.009
1.001
0.999
1.002
1.000
1.010
1.010
0.999
1.007
0.999
1.004
0.998
1.006
0.999
1.014
1.009
1.000
0.999
1.004
1.001
1.010
1.010
1.000
1.007
0.999
1.005
0.998
1.006
P
3
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.236
1.653
2.590
5.815
1.970
2.106
3.005
6.389
3.932
3.467
6.324
8.675
7.699
19.474
3.578
2.233
1.735
1.296
1.133
3.400
1.816
1.508
1.097
3.022
1.605
1.180
3.474
1.433
3.528
1.003
1.025
1.009
0.999
1.015
1.001
1.016
1.015
1.018
1.003
1.004
1.004
1.011
1.017
1.028
1.013
1.024
1.009
1.000
1.012
1.007
1.010
1.016
1.011
1.004
1.003
1.003
1.010
1.015
1.026
1.002
1.022
1.005
1.002
1.014
1.004
1.025
1.022
1.023
1.001
1.003
1.009
1.012
1.017
1.015
1.004
1.025
1.010
0.998
1.015
1.001
1.014
1.014
1.014
1.001
1.003
1.004
1.007
1.016
1.016
0.998
1.021
1.009
1.000
1.011
1.006
1.016
1.011
1.010
1.000
1.005
1.001
1.008
1.017
1.011
1.002
1.025
1.011
0.999
1.012
1.005
1.016
1.011
1.008
0.999
1.004
1.001
1.008
1.017
1.011
1.001
1.020
1.008
1.003
1.026
1.002
1.024
1.035
1.040
1.012
1.016
1.038
1.028
1.048
1.054
1.016
1.031
1.011
0.999
1.014
1.006
1.014
1.011
1.009
1.002
1.004
1.001
1.006
1.017
1.023
1.004
1.025
1.010
0.999
1.012
1.004
1.016
1.011
1.008
0.999
1.004
1.001
1.008
1.017
1.012
P
5
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.473
2.237
4.354
9.091
1.979
2.374
3.961
8.756
3.404
4.385
9.748
6.053
9.692
17.824
2.541
1.560
1.328
1.127
1.056
2.066
1.212
1.162
1.047
1.922
1.174
1.094
1.918
1.093
1.893
1.020
1.020
1.007
1.005
1.004
1.018
1.028
1.019
1.008
1.011
1.030
1.005
1.009
1.018
1.020
1.029
1.019
1.009
1.004
1.002
1.021
1.027
1.018
1.006
1.014
1.022
1.004
1.010
1.014
1.020
1.013
1.021
1.006
1.004
1.003
1.021
1.033
1.022
1.008
1.014
1.036
1.005
1.009
1.021
1.018
1.023
1.019
1.009
1.005
1.003
1.017
1.029
1.018
1.007
1.011
1.025
1.004
1.007
1.016
1.017
1.014
1.019
1.009
1.004
1.003
1.015
1.026
1.015
1.005
1.012
1.021
1.006
1.006
1.013
1.008
1.020
1.021
1.011
1.005
1.003
1.012
1.022
1.015
1.004
1.011
1.018
1.005
1.006
1.012
1.008
1.017
1.020
1.005
1.005
1.005
1.028
1.039
1.024
1.013
1.021
1.040
1.015
1.014
1.035
1.041
1.046
1.031
1.014
1.006
1.002
1.014
1.017
1.014
1.004
1.015
1.017
1.004
1.020
1.011
1.030
1.021
1.020
1.011
1.005
1.003
1.014
1.024
1.016
1.004
1.011
1.019
1.004
1.006
1.012
1.010
P
7
0.0
0.0
0.0
0.0
0.2
0.4
1.000
1.293
2.008
2.894
1.806
1.470
1.010
1.008
1.015
1.016
1.010
1.013
1.008
1.007
1.016
1.011
1.010
1.014
1.009
1.005
1.015
1.012
1.006
1.014
1.011
1.006
1.017
1.020
1.012
1.014
1.012
1.006
1.014
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
371
Table 2 (Continued)
c
1
c
2
OLS
WTN
WH
WK
SA
HFB
MLE
REML MV1
MV2
MV3
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
3.153
6.276
1.916
1.910
3.302
7.084
3.178
3.232
7.325
6.502
7.435
16.032
1.298
1.156
2.684
1.644
1.300
1.140
3.043
1.535
1.225
2.779
1.296
2.662
1.001
1.012
1.012
1.014
1.022
1.012
1.037
1.010
1.005
1.018
1.010
1.034
1.001
1.012
1.015
1.017
1.020
1.011
1.045
1.014
1.001
1.026
1.007
1.035
1.002
1.011
1.019
1.015
1.023
1.017
1.045
1.013
1.005
1.020
1.008
1.028
1.001
1.012
1.012
1.014
1.021
1.011
1.037
1.012
1.002
1.018
1.009
1.028
1.001
1.010
1.016
1.010
1.015
1.006
1.028
1.012
1.003
1.008
1.011
1.019
1.002
1.010
1.015
1.008
1.014
1.006
1.027
1.012
1.002
1.009
1.010
1.019
1.001
1.016
1.021
1.026
1.035
1.019
1.065
1.014
1.012
1.024
1.030
1.104
1.002
1.011
1.014
1.003
1.008
1.006
1.034
1.013
1.003
1.032
1.009
1.039
1.001
1.010
1.015
1.010
1.015
1.006
1.029
1.012
1.003
1.009
1.010
1.019
P
9
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.254
1.939
2.871
5.502
3.053
3.198
4.132
8.514
8.563
6.631
10.603
14.027
14.553
37.097
5.028
3.273
2.228
1.670
1.340
4.067
2.904
2.137
1.348
4.782
2.555
1.568
4.090
1.899
4.310
1.033
1.014
1.035
1.014
1.013
1.009
1.012
1.044
1.044
1.004
1.007
1.032
1.014
1.022
1.018
1.045
1.025
1.033
1.016
1.011
1.010
1.024
1.038
1.035
1.007
1.006
1.027
1.008
1.019
1.009
1.027
1.013
1.034
1.014
1.012
1.014
1.013
1.049
1.048
1.012
1.014
1.040
1.016
1.030
1.007
1.037
1.017
1.035
1.015
1.013
1.006
1.012
1.037
1.038
1.000
1.006
1.030
1.007
1.021
1.007
1.022
1.008
1.016
1.010
1.007
1.008
1.018
1.018
1.011
1.001
1.001
1.015
1.009
1.010
1.007
1.032
1.011
1.019
1.012
1.007
1.006
1.014
1.019
1.007
1.000
1.002
1.014
1.008
1.011
1.007
1.028
1.014
1.069
1.075
1.120
1.024
1.037
1.134
1.251
1.083
1.174
1.355
1.187
1.267
1.396
1.048
1.021
1.027
1.014
1.007
1.012
1.012
1.019
1.015
1.004
1.000
1.014
1.007
1.011
1.012
1.031
1.015
1.024
1.012
1.009
1.007
1.011
1.018
1.007
1.000
1.003
1.015
1.008
1.011
1.008
P
11
0.0
0.0
0.0
0.0
0.0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.8
0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.0
1.000
1.489
1.821
2.856
4.494
4.970
5.078
5.933
8.409
13.184
10.965
14.110
28.659
21.438
73.642
8.900
4.513
3.154
2.322
1.455
6.283
3.682
2.392
1.643
7.260
3.193
1.818
6.450
2.374
5.963
1.061
1.014
1.007
1.032
1.036
1.011
1.016
1.033
1.007
1.001
1.005
1.009
1.017
1.024
1.009
1.091
1.009
1.012
1.033
1.036
1.019
1.013
1.025
1.008
1.009
1.006
1.007
1.007
1.021
1.000
1.054
1.012
1.005
1.030
1.032
1.032
1.042
1.037
1.003
1.012
1.009
1.015
1.012
1.031
1.009
1.063
1.014
1.010
1.033
1.039
1.010
1.011
1.033
1.008
1.003
1.006
1.007
1.007
1.019
1.001
1.035
1.008
1.002
1.016
1.014
1.010
1.010
1.017
1.003
1.002
1.006
1.004
1.004
1.008
0.998
1.048
1.007
1.005
1.017
1.016
1.007
1.006
1.013
1.003
1.001
1.006
1.003
1.004
1.008
0.998
1.047
1.034
1.050
1.159
1.366
1.023
1.052
1.238
1.253
1.050
1.079
1.227
1.268
1.218
1.142
1.092
1.015
1.007
1.017
1.021
1.027
1.015
1.013
1.008
1.002
1.006
1.002
1.006
1.007
1.000
1.054
1.007
1.005
1.017
1.016
1.007
1.006
1.011
1.002
1.001
1.006
1.004
1.005
1.008
0.999
estimators, the ML-type and MV2 and MV3 estimators all perform well relative
to true GLS. See also Fig. 1 which plots the MSE of bK relative to that of true
GLS for the 12 unbalanced patterns for M"10 and c " c "0.4. It is
1
2
important to point out that the simple ANOVA-type estimators compare well in
372
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Fig. 1. MSE of bK relative to that of true GLS when M"10, c "c "0.4.
1
2
MSE performance to the ML- and MIVQUE-type estimators which are relatively more di$cult to compute. They have at most 9.1% higher MSE than true
GLS, see the WK estimator for pattern P when c "c "0. MV1 performs
11
1
2
well for c (0.4 and c (0.4. This is to be expected since the prior values for
1
2
MV1 are c "0 and c "0. As c and c increase or the degree of unbalanced1
2
1
2
ness gets large, the performance of MV1 deteriorates relative to the other
estimators, see Fig. 1.
To summarize, for the regression parameters, the computationally simple
ANOVA estimators compare well with the more complicated estimators such as
ML, REML, MV2 and MV3 in terms of MSE criteria. Also, MV1 is not
recommended when the primary group e!ects or nested subgroup e!ects are
suspected to be large or the unbalanced pattern is severe.
3.3. A comparison of variance components estimates
Tables 3}5 report the MSE of the variance components relative to that of
MLE for the 12 unbalancedness patterns for M"10 and c "c "0.4. Similar
1
2
tables for other values of M"6 and 15 and various values of c and c are
1
2
available upon request from the authors. These tables are plotted in Figs. 2}4 for
ease of comparison.
For the estimation of p2, see Table 3 or Fig. 2, MLE ranks "rst; REML, MV2
k
and MV3 rank second and the ANOVA methods rank third by the MSE
criteria. MV1 does well only when c and c are close to 0. For c "c "0.4,
1
2
1
2
MV1 performs the worst. The ANOVA methods yield almost twice the MSE of
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
373
Table 3
MSE of p( 2 relative to that of MLE when M"10, c "c "0.4
k
1
2
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
11
P
12
WH
WK
SA
HFB
REML
MV1
MV2
MV3
1.170
1.284
1.523
1.307
1.866
1.814
1.897
1.473
1.446
1.427
1.680
1.338
1.170
1.267
1.487
1.295
1.855
1.750
1.885
1.476
1.362
1.360
1.863
1.379
1.302
1.448
1.722
1.413
1.951
2.022
2.034
1.558
1.579
1.585
1.787
1.465
1.168
1.274
1.512
1.297
1.863
1.779
1.891
1.472
1.402
1.383
1.755
1.336
1.158
1.170
1.183
1.140
1.140
1.170
1.178
1.189
1.159
1.177
1.211
1.165
1.204
1.669
2.175
1.602
2.244
3.124
2.216
2.196
1.407
1.459
2.558
2.170
1.160
1.165
1.176
1.133
1.156
1.162
1.202
1.183
1.159
1.179
1.171
1.074
1.157
1.170
1.173
1.134
1.140
1.159
1.159
1.169
1.146
1.170
1.129
1.100
Table 4
MSE of p( 2 relative to that of MLE when M"10, c "c "0.4
l
1
2
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
11
P
12
WH
WK
SA
HFB
REML
MV1
MV2
MV3
1.010
1.205
1.116
1.458
1.086
1.584
1.105
1.475
2.615
2.717
2.694
2.863
1.010
1.181
1.115
1.428
1.066
1.566
1.103
1.453
2.561
2.672
2.583
2.697
1.012
1.194
1.117
1.430
1.096
1.567
1.110
1.466
2.581
2.681
2.583
2.692
1.004
1.179
1.104
1.412
1.065
1.549
1.092
1.442
2.551
2.650
2.559
2.667
0.999
0.999
1.000
0.999
0.995
0.999
0.999
0.998
0.993
1.000
1.001
1.000
1.312
2.514
1.450
3.224
1.181
3.563
1.333
16.188
13.130
21.473
20.234
23.894
0.997
1.012
1.001
1.002
0.989
1.015
1.024
1.039
0.998
1.021
1.001
1.000
0.998
0.998
1.002
0.990
1.015
0.993
1.007
0.985
0.977
0.992
1.004
0.989
MLE for patterns P , P and P . On the other hand, REML, MV2 and MV3
5 6
7
have MSE that are only 14}20% higher than that of MLE. MV1 has 2}3 times
the MSE of MLE for these patterns and is not included in Fig. 2.
For the estimation of p2, see Table 4 or Fig. 3, REML, MLE, MV2 and MV3
l
rank "rst by the MSE criteria. They are followed by the ANOVA methods with
MV1 performing the worst. For patterns P , P , P and P , these ANOVA
9 10 11
12
methods yield more than 2.5 times the MSE of MLE. In contrast, MV1 yields
13}23 times the MSE of MLE and is not included in Fig. 3.
For the estimation of p2, there is not much di!erence among WK, SA, HFB,
e
REML, MV2 and MV3. However, WH performs slightly worse than the other
374
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Table 5
MSE of p( 2 relative to that of MLE when M"10, c "c "0.4
e
1
2
P
1
P
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
11
P
12
WH
WK
SA
HFB
REML
MV1
MV2
MV3
1.038
1.053
1.050
1.202
1.201
1.173
1.078
1.126
1.121
1.128
1.193
1.148
1.001
1.005
1.000
1.004
1.006
1.005
1.000
1.004
1.000
1.003
1.004
1.001
1.001
1.005
1.000
1.004
1.006
1.005
1.000
1.004
1.000
1.003
1.004
1.001
1.001
1.005
1.000
1.004
1.006
1.005
1.000
1.004
1.000
1.003
1.004
1.001
1.001
1.002
1.000
1.002
1.006
1.002
1.000
1.002
1.002
1.001
1.002
0.997
5.190
34.983
27.465
54.860
8.669
58.511
6.469
124.852
124.467
154.966
144.898
119.133
1.004
0.999
1.002
1.002
1.012
1.009
1.012
1.018
1.010
1.006
1.006
0.994
1.002
1.003
1.000
1.003
1.010
1.003
1.001
1.005
1.006
1.005
1.004
1.000
Fig. 2. MSE of p( 2 relative to that of MLE when M"10, c "c "0.4.
k
1
2
ANOVA methods yielding 3.8}20% higher MSE than that of MLE. MV1
performs the worst yielding MSE that is 5}154 times that of MLE and is
therefore not included in Fig. 4.
This con"rms that if one is interested in the estimates of the variance
components per se one is better o! with MLE, REML or MV2 and MV3-type
estimators. The ANOVA methods suggested here are second best. MV1 is not
recommended unless one suspects c and c are close to 0. However, for the
1
2
estimation of the regression coe$cients, the ANOVA methods compare well
and are recommended.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
375
Fig. 3. MSE of p( 2 relative to that of MLE when M"10, c "c "0.4.
l
1
2
Fig. 4. MSE of p( 2 relative to that of MLE when M"10, c "c "0.4.
e
1
2
For ANOVA and MIVQUE-type estimators, negative estimates of p2 or
k
p2 occur in about 50% of the replications when c "0 or c "0. When
l
1
2
the negative estimates of variance components are replaced by 0, the corresponding estimator forfeits its unbiasedness property. But, replacing these
negative estimates by 0 did not lead to much loss in e$ciency using the MSE
criterion.
376
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Fig. 5. MSE of standard errors of bK relative to that of MLE when M"10, c "c "0.4.
1
2
Finally, better estimates of the variance components by the MSE criterion, do
not necessarily imply better estimates of the regression coe$cients. A similar
result was obtained by Baltagi and Chang (1994) for the unbalanced one way
model and by Taylor (1980) and Baltagi (1981) for the balanced error component model. However, MV1 has worse relative MSE performance than other
ANOVA, ML, REML and MIVQUE-type estimators of the variance components when c and c are large and the pattern is severely unbalanced and this
1
2
clearly translates into a corresponding worse relative MSE performance of the
regression coe$cients. Similar conclusions can be drawn for M"6 and 15 and
are not produced here to save space.
3.4. A comparison of standard errors of the regression coezcients
Fig. 5 plots the MSE of the standard error of bK relative to that of MLE for the
12 unbalanced patterns for M"10 and c "c "0.4. Besides the relative
1
2
e$ciency of the parameter estimates, one is also interested in proper inference
on the parameter values. This is where the computationally involved estimators
(like MV2, MV3 and REML) perform well producing a MSE for the standard
error of bK that is close to that of MLE. The computationally simple ANOVA
methods (WH, WK, SA, HFB) have MSE for the standard error of bK that are
2 times that of MLE for severely unbalanced patterns like P , P and P .
10 11
12
However, these ANOVA methods perform reasonably well in patterns P }P
1 8
giving MSEs of the standard error of bK that are no more than 30% higher than
that of MLE.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
377
4. Empirical example
Baltagi and Pinnoi (1995) estimated a Cobb}Douglas production function
investigating the productivity of public capital in each state's private output.
This is based on a panel of 48 states over the period 1970}1986. The data were
provided by Munnell (1990). These states can be grouped into nine regions with
the Middle Atlantic region for example containing three states: New York, New
Jersey and Pennsylvania and the Mountain region containing eight states:
Montana, Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah and
Nevada. The primary group would be the regions, the nested group would be
the states and these are observed over 17 years. The dependent variable y is the
gross state product and the regressors include the private capital stock (K)
computed by apportioning the Bureau of Economic Analysis (BEA) national
estimates. The public capital stock is measured by its components: highways and
streets (KH), water and sewer facilities (KW), and other public buildings and
structures (KO), all based on the BEA national series. Labor (¸) is measured by
the employment in nonagricultural payrolls. The state unemployment rate is
included to capture the business cycle in a given state. See Munnell (1990) for
details on the data series and their construction. All variables except the
unemployment rate are expressed in natural logarithm
y "a#b K #b KH #b KW #b KO
ijt
1 ijt
2
ijt
3
ijt
4
ijt
# b ¸ #b Unemp #u ,
(43)
5 ijt
6
ijt
ijt
where i"1, 2,2, 9 regions, j"1,2, N with N equaling 3 for the Middle
i
i
Atlantic region and 8 for the Mountain region and t"1, 2,2, 17. The data is
unbalanced only in the di!ering number of states in each region. The disturbances follow the nested error component speci"cation given by (2).
Table 6 gives the OLS, WTN, ANOVA, MLE, REML and MIVQUE-type
estimates using this unbalanced nested error component model. The OLS
estimates show that the highways and streets and water and sewer components
of public capital have a positive and signi"cant e!ect upon private output
whereas that of other public buildings and structures is not signi"cant. Because
OLS ignores the state and region e!ects, the corresponding standard errors and
t-statistics are biased, see Moulton (1986). The within estimator shows that the
e!ect of KH and KW are insigni"cant whereas that of KO is negative and
signi"cant. The primary region and nested state e!ects are signi"cant using
several LM tests developed in Baltagi et al. (1999). This justi"es the application
of the feasible GLS, MLE and MIVQUE methods. For the variance components estimates, there are no di!erences in the estimate of p2. But estimates of
e
p2 and p2 vary. p( 2 is as low as 0.0015 for SA and MLE and as high as 0.0029 for
k
l
k
HFB. Similarly, p( 2 is as low as 0.0043 for SA and as high as 0.0069 for WK. This
l
variation had little e!ect on estimates of the regression coe$cients or their
378
Variable
OLS
WTN
WH
WK
SA
HFB
MLE
REML
MV1
MV2
MV3
Intercept
1.926
(0.053)
0.312
(0.011)
0.550
(0.016)
0.059
(0.015)
0.119
(0.012)
0.009
(0.012)
!0.007
(0.001)
*
2.082
(0.152)
0.273
(0.021)
0.742
(0.026)
0.075
(0.023)
0.076
(0.014)
!0.095
(0.017)
!0.006
(0.001)
2.131
(0.160)
0.264
(0.022)
0.758
(0.027)
0.072
(0.024)
0.076
(0.014)
!0.102
(0.017)
!0.006
(0.001)
2.089
(0.144)
0.274
(0.020)
0.740
(0.025)
0.073
(0.022)
0.076
(0.014)
!0.094
(0.017)
!0.006
(0.001)
2.084
(0.150)
0.272
(0.021)
0.743
(0.026)
0.075
(0.022)
0.076
(0.014)
!0.096
(0.017)
!0.006
(0.001)
2.129
(0.154)
0.267
(0.021)
0.754
(0.026)
0.071
(0.023)
0.076
(0.014)
!0.100
(0.017)
!0.006
(0.001)
2.127
(0.157)
0.266
(0.022)
0.756
(0.026)
0.072
(0.023)
0.076
(0.014)
!0.101
(0.017)
!0.006
(0.001)
2.083
(0.152)
0.272
(0.021)
0.742
(0.026)
0.075
(0.023)
0.076
(0.014)
!0.095
(0.017)
!0.006
(0.001)
2.114
(0.154)
0.269
(0.021)
0.750
(0.026)
0.072
(0.023)
0.076
(0.014)
!0.098
(0.017)
!0.006
(0.001)
2.127
(0.156)
0.267
(0.021)
0.755
(0.026)
0.072
(0.023)
0.076
(0.014)
!0.100
(0.017)
!0.006
(0.001)
0.0014
0.0027
0.0045
0.0014
0.0022
0.0069
0.0014
0.0015
0.0043
0.0014
0.0029
0.0044
0.0013
0.0015
0.0063
0.0014
0.0019
0.0064
0.0014
0.0027
0.0046
0.0014
0.0017
0.0056
0.0014
0.0017
0.0063
K
¸
KH
KW
KO
Unemp
p2
e
p2
k
p2
l
0.235
(0.026)
0.801
(0.030)
0.077
(0.031)
0.079
(0.015)
!0.115
(0.018)
!0.005
(0.001)
0.0073
*
*
0.0013
*
*
!The dependent variable is log of gross state product. Standard errors are given in parentheses.
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
Table 6
Cobb}Douglas production function estimates with unbalanced nested error components 1970}1986, Nine regions, 48 states!
B.H. Baltagi et al. / Journal of Econometrics 101 (2001) 357}381
379
standard errors. For all estimators of the random e!ects model, the highways
and streets and water and sewer components of public capital had a positive and
signi"cant e!ect, while the other public buildings and structures had a negative
and signi"cant e!ect upon private output.
5. Conclusion
For the regression coe$cients of the nested unbalanced error component
model, the simple ANOVA methods proposed in this paper performed well in
Monte Carlo experiments as well as in the empirical example and are recommended. However, for the variance components estimates themselves, as well as
the standard errors of the regression coe$cients, the computationally more
demanding MLE, REML or MIVQUE (MV2 and MV3) estimators are recommended especially if the unbalanced pattern is severe. Further research should
extend the unbalanced nested error component model considered in this paper
to allow for endogeneity of the regressors, a dynamic speci"cation, ignorability
of the sample selection and serial correlation in the disturbances.
Acknowledgements
The authors would like to thank two anonymous referees and an Associate
Editor for their helpful comments and suggestions. A preliminary version of this
paper was presented at the European meetings of the Econometric Society held
in Santiago de Compostela, Spain, August, 1999. Also, at the University of
Chicago, University of Pennsylvania and the University of Rochester. Baltagi
would like to thank the Texas Advanced Research Program and the Bush
Program in economics of public policy for their "nancial support.
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Baltagi, B.