Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol101.Issue2.2001:

Journal of Econometrics 101 (2001) 295}313

Nested random e!ects estimation in
unbalanced panel data
Werner Antweiler*
Faculty of Commerce, The University of British Columbia, 2053 Main Mall, Vancouver,
BC Canada V6T 1Z2
Received 27 October 1997; received in revised form 26 July 2000; accepted 2 October 2000

Abstract
Panel data in many econometric applications exhibit a nested (hierarchical) structure.
For example, data on "rms may be grouped by industry, or data on air pollution may be
grouped by observation station within a city, city within a country, and by country. In
these cases, one can control for unobserved group and sub-group e!ects using a
nested-error component model. A double-nested unbalanced panel is examined and
a corresponding maximum likelihood estimator is derived. A generalization to even
higher-order nesting is feasible. A practical example and a Monte-Carlo simulation
compare the new estimator against the non-nested ML estimator. The style of presentation is intended to aid applied econometricians in implementing the new ML
estimator. ( 2001 Elsevier Science S.A. All rights reserved.
JEL classixcation: C13; C23
Keywords: Panel data; Nested e!ects; Error component model; Econometrics

1. Introduction and motivation
Data often exhibit a nested (or hierarchical) structure. For example, "rms may
be grouped by industry. This is a single-nested structure. Data on air pollution

* Corresponding author. Tel.: #1-604-822-8484; fax: #1-604-822-8477.
E-mail address: werner.antweiler@ubc.ca (W. Antweiler).

0304-4076/01/$ - see front matter ( 2001 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 8 6 - 5

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W. Antweiler / Journal of Econometrics 101 (2001) 295}313

may be grouped by observation station within a city and city within a country.
This is a double-nested structure. Although there can be any number of nestings,
most applications will be of the single-or double-nested type. This paper explores the double-nested error component model, but estimators for higherorder nestings can be obtained using the methodology presented here. This
paper is aimed at the econometric practitioner. Therefore, care has been taken to
use notation that will allow the practitioner to implement the methods described

in this paper into suitable econometric software. In what follows, Section
2 provides a brief overview of the literature on nested panels and motivates the
analysis by presenting the case of balanced panels. Section 3 addresses the more
complex case of unbalanced panels and introduces a suitable maximum likelihood (ML) estimator. A concrete example is provided in Section 4 where this
new ML estimator for double-nested panels is compared to the non-nested
panel ML estimator. Section 5 concludes with a small Monte-Carlo study of the
new ML estimator.
While the problem of hierarchical panels appears to be one of great practical
importance, not much work has been published on this topic. Early work on
balanced panels include Ghosh (1976), who developed an ANOVA-type estimator for single-nested panels, and Fuller and Battese (1973), who investigated
the error structure of double-nested data sets with cross-sectional data but no
time dimension. This was later improved upon by Baltagi (1987). Pakes (1983)
discusses regression analysis where populations are grouped. Searle (1987,
Chapter 3) discusses nested data structures in a variance analysis context. It
appears that Baltagi (1993) "rst proposed the term `nested e!ects,a adapting the
term `nested-error structurea by Fuller and Battese (1973) to the context of
panel data.
New and complementary work to this paper appears in Baltagi et al. (1999a, b)
and Davis (1999).
One econometric problem in data sets with a nested structure relates to the

possibility that individual e!ects may be associated with each level. For
example, in a model with cities and countries there can be city-speci"c e!ects as
well as country-speci"c e!ects. Using country-speci"c e!ects alone would exclude the possibility of city-speci"c e!ects, and vice versa.1
Another econometric problem is introduced by estimating a nested panel
with a non-nested error structure. This leads to a phenomenon similar to the

1 For single-nested panels one alternative is the estimation of &mixed e!ects' where a "xed-e!ects
approach is used for the top-level group (e.g. countries) and a random-e!ects approach is used for
the low-level group (e.g. cities). This method may be appropriate in cases where the top level covers
an entire population (such as all industries in an economy or all countries in the world); in other
cases, however, this method may be inappropriate.

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

297

well-known Moulton bias in standard errors.2 There are two issues. First, the
standard errors computed under the assumption that the error term is i.i.d. will
be biased (downward if the error terms are positively correlated). Second, the
assumption of independence is unlikely to be satis"ed when aggregate (macro)

data are merged with micro data. For example, it is common in labour economics to merge micro data on individual workers with macro data on industry,
occupation, or geographic location. More concretely, estimation of a wage
equation for individual workers could involve characteristics of the individual
worker as well as aggregate (e.g., state or country) unemployment rates.
Consider a regression equation
(1)
y "x@ b#u ,
ijkt
ijkt
ijkt
for i"1,2, ¸; j"1,2, M ; k"1,2, N and t"1,2, ¹ . For example, the
i
ij
ijk
dependent variable y could denote the air pollution measured at station k in
ijkt
city j of country i in time period t. This means that there are ¸ countries, and
each country i has M cities in which N observation stations are located. At
i
ij

such a station, air pollution is observed during ¹ periods. The x denotes
ijk
ijkt
a vector of K explanatory variables, and the disturbance is given by
u "j #k #l #e ,
(2)
ijkt
i
ij
ijk
ijkt
where j &IID(0, p2), k &IID(0, p2 ), l &IID(0, p2), and e &IID(0, p2),
e
l
ijkt
k ijk
j ij
i
are independent of each other and among themselves. When considering the ML
estimator, the assumption of normality will be added.

2. Nested balanced panels
The "rst reference of a nested error structure can be found in Ghosh (1976).
This study considers a balanced panel with a single-nested (e.g., country and
region) structure as well as individual time e!ects. Baltagi (1987), using methodology developed by Wansbeek and Kapteyn (1982), provides a more rigorous
algebraic derivation of this type of estimator based on the spectral decomposition of the variance}covariance matrix X.
Xiong (1995) elegantly derives a nested-e!ects estimator for single-nested
balanced panels (without separate time e!ects). I generalize this estimator to the
double-nested case to provide a reference point for the discussion of the doublenested unbalanced panel in Section 3. To begin with, let I be an identity matrix
s
of dimension s, and let J be a square matrix of dimension s with all elements 1,
s
and for convenience of algebraic manipulation, introduce P ,J /s and
s
s

2 See Moulton (1986, 1987, 1990).

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W. Antweiler / Journal of Econometrics 101 (2001) 295}313

Q ,I !P . Also note that I "I ?I and P "P ?P for any positive
s
s
s
rs
r
s
rs
r
s
integers r and s. Expressing (2) in matrix form and computing X"E(uu@), it
follows that
.
(3)
?J )#p2I
)#p2(I ?J )#p2(I
X"p2 (I ?J
e LMNT

l LMN
T
k LM
NT
j L
MNT
First transforming the J matrices into P matrices, then recursively expanding
the multi-group I matrices into the sum of P and Q matrices, collapsing
contiguous P matrices, and "nally collecting terms, yields
?P )#p2I
)#N¹p2(I ?P )#¹p2(I
X"MN¹p2(I ?P
e LMNT
l LMN
T
k LM
NT
j L
MNT
)

"(MN¹p2 #N¹p2 #¹p2#p2)(I ?P
e L
MNT
l
k
j
# (N¹p2#¹p2#p2)(I ?Q ?P )
e L
M
NT
l
k
?Q )
# (¹p2#p2)(I ?Q ?P )#p2(I
e LMN
T
e LM
N
T
l

?Q )#p2 (I ?Q ?P )#p2 (I ?Q ?P )
"p2(I
2 L
M
NT
1 LM
N
T
e LMN
T
),
(4)
# p2 (I ?P
3 L
MNT
where in the last line p2, p2 , p2 , and p2 were introduced as the characteristic
3
e 1 2
roots of the spectral decomposition of X. They are de"ned as follows:
(5)

p2 ,¹p2#p2,
e
l
1
(6)
p2 ,N¹p2#¹p2#p2,
e
l
k
2
(7)
p2 ,MN¹p2#N¹p2#¹p2#p2.
e
l
k
j
3
As the terms with the Kronecker products are all orthogonal to each other and
sum to I
, it follows that3
LMNT
?Q )#p~1(I ?Q ?P )
X~1@2"p~1(I
1 LM
N
T
e
LMN
T
).
(8)
# p~1(I ?Q ?P )#p~1(I ?P
3 L
MNT
2 L
M
NT
Now expand all the Q matrices as the di!erence of I and P, multiply both sides
of the equation by p , and collect terms
e
p
?P )
p X~1@2"I
! 1! e (I
LMN
T
e
LMNT
p
1
p
p
p
p
! e ! e (I ?P )! e ! e (I ?P
).
(9)
LM
NT
L
MNT
p
p
p
p
2
3
1
2

B

A

A

B

A

B

3 For scalars f and an arbitrary scalar r, it holds that Xr"+ frZ when the matrices Z are
i
i i i
i
pairwise orthogonal to each other and sum to the identity matrix, and each Z is symmetric and
i
idempotent with its rank equal to its trace. See Nerlove (1971b).

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

299

To perform generalized least squares, the following transformed system of y (and
likewise x) is estimated via OLS:

A

B

A

B

A

B

p
p
p
p
p
yH "y ! 1! e y6 ! e ! e y6 ! e ! e y6
ijkt
ijkt
ijk
ij
i
p
p
p
p
p
1
1
2
2
3
v

vv

vvv

,
(10)

indicate group averages. The pattern exhibited in Eq.
where y6 , y6 , and y6
i
ijk ij
(10) is suggestive of solutions for higher-order nested panels. For feasible
generalized least squares, estimates of the variances can be obtained from the
three group-wise between estimators and the within estimator for the innermost
group.
v

vv

vvv

3. An ML estimator for unbalanced panels
Unbalanced panels cannot be handled easily in the framework developed in
the previous section. The Kronecker product can only be used in the case of
balanced panels. Thus, unbalanced panels introduce quite a bit of notational
inconvenience into the algebra.4 At the same time, it will become apparent that
unbalanced panels cannot be easily moulded into a feasible generalized least
squares (FGLS) transformation for OLS estimation. Provided that introducing
the normality assumption on the error structure is unproblematic in the given
application, maximum likelihood estimation provides a suitable alternative.5
Two other points make ML estimation appealing: the variances are estimated
directly, and ML generally performs well for unbalanced panels.6
In deriving a practical estimator for nested panels, the key challenge is "nding
an expression for X~1 that is computationally feasible. The expression I derive
below turns out to be just that, and by exploiting the recursive nature of the
panel's hierarchic structure it suggests an immediate generalization to arbitrary
orders of nesting.
I begin by introducing some notation and de"ning certain types of matrices
that are used in the derivation of an inverse and determinant of X. An unbalanced panel is made up of ¸ top level groups, each containing M second-level
i
groups. The second-level groups contain the innermost N subgroups, which in
ij
turn contain ¹ observations. The number of observations in the higher-level
ijk
groups are thus ¹ ,+Nij ¹ and ¹ ,+Mi ¹ , and the total number of
k/1 ijk
i
j/1 ij
ij

4 For a discussion, see Baltagi (1995, Chapter 9), and in particular, Wansbeek and Kapteyn (1989).
5 Computational cost is hardly a consideration anymore.
6 For the latter, see Maddala and Mount (1973). Also see Section 5 below.

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W. Antweiler / Journal of Econometrics 101 (2001) 295}313

observations is H,+L ¹ . The number of top-level groups is ¸, the number of
i/1 i
second-level groups is F,+L M , and the number of bottom-level groups is
i/1 i
G,+L +Mi N .
i/1 j/1 ij
For balanced panels it was possible to neatly stack the J matrices of all &1's
using I?J products. For unbalanced panels I rede"ne the J matrices to be
blockdiagonal of size H]H, corresponding in structure to the groups or
subgroups they represent. They can be constructed explicitly by using `group
membershipa matrices consisting of ones and zeros that uniquely assign each of
the H observations to one of the G (or F or ¸) groups. Let R be such an H]G
l
matrix corresponding to the innermost group level. Then the blockdiagonal
H]H matrix J can be expressed as the outer product of its membership
l
matrices: J "R R@ . Further note that the inner product R@ R produces a diagl
l l
l l
onal matrix ¹I of size G]G that contains the number of observations for each
l
group. Similarly, ¹I "R@ R . In what follows the tilde accent will always denote
k k
k
such an `interiora matrix corresponding to the `exteriora H]H matrix of the
same name. De"ne ¹ as an H]H matrix of observations with elements
l
¹
"¹ , and then it is apparent that R ¹I "¹ R . De"ne ¹ analogously
l,ijkt
ijk
l l
l l
k
as the H]H matrix corresponding to the F]F matrix ¹I . The properties of the
k
¹ and J matrices can be exploited advantageously. Note that
J J "R (R@ R )R@ "R ¹I R@ "¹ J . Similarly, J J "¹ J .
l l
k k
k k
l l l
l l
l l l l
The mapping from groups to subgroups can also be captured through
membership matrices. Let Rl ,R@ R ¹I ~1 be an F]G matrix mapping
k l l
k
F groups to G subgroups. De"ne Rk correspondingly as an ¸]F matrix. Also
j
note the recursive nature of the mapping process: R "R (Rl )@.
k
l k
Similar to the P matrices that were de"ned for balanced panels, projection
matrix P ,R ¹I ~1R@ "¹~1J is idempotent: P P "R ¹I ~1R@ R ¹I ~1R@ "
l l
l l
l l l l l l
l
l l l
R (¹I ~1¹I )¹I ~1R@ "P . The second property of this projection matrix establ
l l l l l
lishes that such a matrix can be pre- or post-multiplied with a J or P matrix of
equal nesting without changing it, for example, P J "R ¹I ~1R@ R R@ "
l l
l l l l l
R ¹I ~1¹I R@ "R R@ "J . Using this second property I establish the third
l l l l
l l
l
property which allows a projection matrix to be applied to a J or P of
higher-order
nesting
without
changing
it:
P J "P P J "
l k
l k k
(R ¹I ~1R@ )(R ¹I ~1R@ )J "¹~1(R R@ )(R R@ )¹~1J "(¹~1¹ )(¹~1¹ )J "J .
k
l l k k k
l
l l k k k k
l l l k k k k
Now it is apparent that J J "¹ J , and analogously, J J "J ¹ . To see
l k
l k
k l
k l
this, observe that by construction J "¹ P , and that P J "J by property
l
l l
l k
k
3 of the projection matrix.
In the next step I introduce a further type of matrix. It will turn out that the
diagonal S matrices introduced here are made up of eigenvalues of yet another
type of matrix (the Z matrices) introduced later. Let
S ,I#o ¹ ,
l
l l
S ,I#o U ,
k
k k
S ,I#o U ,
j
j j

(11)
(12)
(13)

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

301

where U and U are diagonal matrices with elements U
"/ and
k
j
k,ijkt
ij
U
"/ . The elements of the S and U matrices are de"ned recursively
j,ijkt
i
Nij ¹
Mi /
h ,1#o ¹ , / , + ijk , h ,1#o / , / , + ij ,
ijk
l ijk
ij
ij
k
ij
i
h
h
k/1 ijk
j/1 ij
h ,1#o /
i
j i
where o ,p2/p2, o ,p2 /p2, and o ,p2/p2 are variance ratios.
j e
j
j e
l e k
l
For the following proofs it is helpful to consider the G]G, F]F, and ¸]¸
`interiora counterparts of the S and U matrices; they are indicated again by a
tilde accent on their top.7 Corresponding to (11)}(13) are SI "I #o ¹I ;
l
G
l l
SI "I #o UI ; and SI "I #o UI . The UI matrices can then be derived as
k
F
k k
j
L
j j
follows:
(14)
UI "Rl SI ~1¹I Rl@"R@ R SI ~1¹I ~1R@ R "R@ S~1R ,
k l k
k l l l l k
k l l k
k
UI "Rk SI ~1UI Rk@"R@ R ¹~1(R@ S~1R )SI ~1¹~1R@ R
k l k k k k j
j k k
j k k j
j
"R@ S~1S~1R .
(15)
j k l j
In the above simpli"cations it should be noted that R SI "S R , R SI "S R ,
l l
l l k k
k k
and R SI "S R . Also remember that R ¹I ~1R@ is an idempotent matrix and
j j
j j
k k k
that P R "R . S matrices and J matrices commute in one special case: if S can
k k
k
be represented by an `interiora matrix S that corresponds to the nesting level of
J. This means that the commutativity holds only when for each block in the
J matrix the corresponding elements in the S matrix are all the same. Thus it is
true that S J "J S , but note that S J OJ S . Also, S J "J S . To see
k k
k k
l k
k l
k l
l k
this, consider the case where S J "S R R@ "R SI R@ "R R@ S "J S . As
k k
k k k
k k k
k k
k k k
long as S of size H]H is reducible to SI of size F]F corresponding to R,
commutativity is guaranteed.
Having dealt with these preliminaries, the variance}covariance matrix for an
unbalanced panel can be written as
(16)
X"p2[I#o J #o J #o J ].
e
l l
k k
j j
The log-likelihood function L corresponding to the error structure given in
(2) is8
L"!1MH ln(2n)#lnDXD#u@X~1uN.
2

(17)

7 By comparison, the matrices with the tilde accent contain one unique value for each group, while
the matrices without a tilde accent replicate the value from each group for all observations in that
group.
8 See also Baltagi (1995, Chapters 2.4 and 3.4) and Hsiao (1986, Chapter 3.3.3).

302

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

There are thus two challenges: computing the determinant DXD and the inverse
X~1. The following technique can be used to tackle these two challenges.
Introduce
Z ,I#o J ,
(18)
l
l l
(19)
Z ,I#Z~1o J "I#S~1o J ,
l k k
l k k
k
(20)
Z ,I#Z~1Z~1o J "I#S~1S~1o J ,
k l j j
k l j j
j
so that instead of writing (16) as a sum it can be written as the product
X"Z Z Z , which is veri"ed by substituting (18)}(20) into this expression and
l k j
expanding. Notice the introduction of the S matrices in (19)}(20). They are direct
counterparts to the Z matrices, but unlike these, the S matrices are diagonal and
made up of eigenvalues of the corresponding Z matrices. Proving the equality
parts of (19)}(20) is the primary challenge. It is now apparent that
DXD"(p2)HDZ DDZ DDZ D,
e
l k j
X~1"Z~1Z~1Z~1/p2,
e
j k l
and it needs to be shown that9

(21)
(22)

(23)
Z~1"I!Z~1o J "I!S~1o J ,
l l l
l l l
l
(24)
Z~1"I!Z~1Z~1o J "I!S~1S~1o J ,
k l k k
k l k k
k
(25)
Z~1"I!Z~1Z~1Z~1o J "I!S~1S~1S~1o J .
j k l j j
j k l j j
j
It is easily veri"ed that the left and middle expressions in (23)}(25) are identical.
By construction Z~1Z "I, Z~1Z "I, and Z~1Z "I. In the "rst case,
j j
k k
l l
rearranging expressions yields Z~1(I#o J )"I, where the part in parenthesis
l
l l
is of course Z . The other two cases work analogously. The remaining task is to
l
show the equality of the middle and right parts of (23)}(25). I will only show the
most di$cult case (25); the two simpler cases are embedded within. Proving (25)
is equivalent to demonstrating that
J "Z Z Z S~1S~1S~1J ,
j
l k j j k l j
"[I#o J #o J #o J ]S~1S~1S~1J ,
l l
k k
j j j k l j
"S~1[S~1(S~1(J #o J J )#o J S~1J )#o J S~1S~1J ],
j j k l j
l
j
l l j
k k l j
k
j
"S~1[S~1(S~1(J #o ¹ J )#o U J )#o U J ],
l
j
l l j
k k j
j j j
k
j

(26)
(27)

9 Note that the following is a highly specialized version of rule 19.18 in Sydsvter et al. (1999,
p. 125).

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

303

"S~1[S~1(S~1(I#o ¹ )#o U )#o U ]J ,
l
l l
k k
j j j
k
j
"S~1[S~1(I#o U )#o U ]J ,
k
k k
j j j
j
"S~1[I#o U ]J ,
j
j j j
"J .
j
Arriving at step (26) simply makes use of the commutativity of the S and
J matrices. The crucial step from (26) to (27) makes use of three equalities. First,
it was shown earlier that J J "¹ J . Second, using the simpli"cation from
l l
l l
(14), it is true that J S~1J "R (R@ S~1R )¹I ~1R@ J "R UI ¹I ~1R@ J "
k k k k j
k k l k k k j
k l j
U P J "U J . And third, J S~1S~1J "U J is trivially true from (15).
k k j
k j
j k l j
j j
The above proof is also helpful in the following step. Using the same technique it can be shown that J Z~1Z~1"J S~1S~1 and J Z~1"J S~1. I use
j k l
j k l
k l
k l
these equalities below in proceeding from steps (28) to (29) when I expand and
simplify p2X~1:
e
Z~1Z~1Z~1"Z~1Z~1!S~1S~1S~1J Z~1Z~1,
j k l j k l
k l
j k l
"Z~1!S~1S~1o J Z~1!S~1S~1S~1J Z~1Z~1,
j k l j k l
k l k k l
l
"I!S~1o J !S~1S~1o J Z~1!S~1S~1S~1J Z~1Z~1,
j k l j k l
k l k k l
l l l
(28)
"I!S~1o J !S~1S~1o J S~1!S~1S~1S~1J S~1S~1.
j k l j k l
k l k k l
l l l
(29)
In passing I note the di$culty of "nding a simple projection matrix C satisfying
C@C"X~1 that would produce an observation-by-observation GLS-to-OLS
transformation. The J and J matrices are bordered on both sides by some
k
j
non-commutable S matrices. Thus, an observation-by-observation transformation cannot be obtained from (29) except in the non-nested case where
o "o "0.
k
j
In the next step I derive the determinant of X. The following lemma applies to
the sum of the identity matrix and the product of an arbitrary diagonal matrix
D and a block-diagonal J matrix stacked with k"1,2, N blocks of 1's with size
¹ ]¹ .
k
k
N
Tk
(30)
DI#oDJD" < 1#o + d .
kt
t/1
k/1
This lemma can be applied immediately to (18). In the case of (19) and (20), it is
true that Z~1J "S~1J and Z~1Z~1J "S~1S~1J . Then DXD can be written
k l j
k l j
l k
l k
as follows:

A

B

Nij
L
Mi
DXD"(p2)H < h < h < h .
ij
ijk
i
e
i/1 j/1 k/1

(31)

304

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

In the case of a balanced panel, DXD simpli"es to
DXD"(p2)H(1#o ¹)LM(N~1)(1#o ¹#o N¹)L(M~1)
e
l
l
k
(1#o ¹#o N¹#o MN¹)L.
l
k
j

(32)

Having dealt with these lengthy preliminaries, the log-likelihood function (17)
can now be fully expanded by taking the log of (31), using (29) in u@X~1u, and
rearranging. In order to simplify the resulting expression, I abbreviate the
residual sum of squares as < ,+Tijk u2 , and further de"ne recursively
t/1 ijkt
ijk
Nij ;
Mi ;
Tijk
; , + u , ; , + ijk , ; , + ij .
ijkt
ij
i
ijk
h
h
t/1
k/1 ijk
j/1 ij
Thus,

C

G
H

G
HD

G

L
Mi
1
Nij
<
L"! H ln(2np2)# + ln h # + ln h # + ln h # ijk
e
i
ij
ijk
p2
2
e
i/1
j/1
k/1

H

o ;2
o ;2
o ;2
! l ijk ! k ij ! j i
h p2
h p2
p2
h
ij e
i e
ijk e

.

(33)

The solution for a single-nested unbalanced panel is obtained by setting ¸"1
and o "0. A gradient of this log-likelihood function can be obtained analytij
cally (see the appendix), but it can also be obtained through numeric approximation.10 Using a set of suitable starting values (for example, OLS or non-nestedpanel GLS estimates combined with setting the variance ratios equal to values
so that their sum is less than one), convergence can be achieved reasonably fast.
A technical concern in carrying out the maximization is that the variances must
not become negative. It is thus necessary to constrain the optimization such that
the variance p2 remains positive and that the variance ratios o , o , and
e
j k
o remain non-negative.
l

10 While an analytic gradient is not strictly necessary for computational purposes (as many
software packages provide numeric approximations), they signi"cantly speed up computations.
Even with an analytic gradient, however, the Hessian matrix W is typically obtained through
numeric approximation methods. W is used to obtain standard errors s for the estimates of the
b
B regressors. The diagonal elements of the inverse of the Hessian, corrected for the degrees of
freedom, are approximately following a t-distribution with H!G!B degrees of freedom. More
formally, this can be expressed as s "Jabs([W~1H/(H!G!B)] )&t(H!G!B).
b
bb

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

305

4. Empirical example
To illustrate the practical value of the method introduced in Section 3, Table
1 compares the estimates from a conventional non-nested random e!ects ML
estimation with those from a nested-e!ects ML estimation.
The example is a constant-elasticity closed-economy version of the regression
model in Antweiler et al. (1998). The original model passed a number of
standard robustness and speci"cation checks. The dependent variable in this
model is the log of atmospheric sulfuric dioxide concentration at observation
stations around the world. A total of 2621 observations are obtained from 293
observation stations located in 44 countries, spanning a time period (not
necessarily continuous) from 1971 to 1996. In this highly unbalanced panel,
about a third of the observations are from stations in the United States. The
second column in Table 1 indicates the dimensions of the independent variables.
The indicators &C', &S', and &T' symbolize the dimensions country, observation
station and time; a dot indicates that the variable does not vary over this
dimension. The three key regressors in the model are the log of economic
intensity (measured as GDP per square kilometer) at a particular location, the

Table 1
Example regression: Simple vs. Nested!
Dep. Var. log(SO )
2
Intercept
log(GDP/km2)
log(K/¸)
log(Income)
Suburban location
Rural location
Communist country
log(Oilprice)
Average temperature
Time trend
p2
e
o
l
o
k
log(L)

Dim.

Simple

C
C
C
C
C
C
.
C
.

!10.787***
0.445***
0.255*
!0.714***
!0.627***
!0.834*
0.471*
!0.083*
!0.045***
!0.043***

S
.
.
S
S
.
.
S
.

T
T
T
T
T
.
T
T
T

Nested
(12.037)
(7.921)
(1.999)
(5.004)
(3.685)
(2.181)
(2.241)
(2.267)
(4.299)
(11.666)

0.330*** (31.992)
1.807*** (9.389)

!2645.4

!7.103***
0.202*
0.371*
!0.477**
!0.720***
!1.061***
0.613
!0.089*
!0.044***
!0.046***

(5.613)
(2.531)
(2.345)
(2.620)
(4.531)
(3.439)
(1.443)
(2.410)
(3.719)
(10.927)

0.329*** (31.986)
1.017*** (8.243)
1.347** (3.229)
!2608.0

!¹ ratios are given in parenthesis. The second column indicates over which dimensions the variable
varies: &C' indicates country, &S' indicates observation station, and &T' indicates time.
*Signi"cance at 95% con"dence level.
**Signi"cance at 99% con"dence level.
***Signi"cance at 99.9% con"dence level.

306

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

log of a country's capital abundance, and the log of its lagged 3-year
moving income per capita. These regressors correspond (in that order)
to the scale, composition, and technique e!ect that determine pollution
concentration. Additional regressors are station-speci"c factors such as
average temperature and the geographic proximity to urban centres (suburban
and rural location dummies). A communist-country dummy, a time series
for the real price of crude oil, and a time trend conclude the set of regressors for
this model.
The results of this comparison shed some light on the problems with nested
and highly unbalanced panels. First, the magnitudes of the estimates of the key
regressors are noticeably di!erent across the two models. The scale elasticity is
only one-half the value from the non-nested panel estimation, while the importance of the composition e!ect has increased, and the income e!ect has decreased. While these results still (and strongly) support the conclusions in
Antweiler et al. (1998), they cast doubt on the reliability of the point estimates.
Second, the decomposition of the error term permits a clearer view on the
aggregation level where the disturbances are introduced. In the simple model,
the station and country variations are lumped together. When they are disaggregated it appears that more variation is at the country level than at the station
level. Third, insofar as the simple model can be viewed as being nested within the
more general nested-panel model, a likelihood ratio test between the two models
with a test statistic of 74.8 is highly signi"cant and favours the nested-panel
model.

5. Monte-Carlo simulation
Comparing variance components estimators for unbalanced panel data requires comparison under a variety of panel patterns and true values of the
variance components. An early Monte-Carlo study by Maddala and Mount
(1973) evaluated the performance of di!erent estimators for panel data. A recent
and very thorough study by Baltagi and Chang (1994) comes to the conclusion
that the ML estimator performs better than ANOVA-type estimators in severely
unbalanced panels and when variance component ratios are large. The Baltagi
and Chang study also introduces important concepts for the design of a MonteCarlo study for such unbalanced panels which I will follow in this section. Here
I investigate the performance of the new ML estimator for a double-nested panel
and compare its performance with the conventional non-nested random-e!ects
ML estimator.
The degree of `unbalancednessa of a panel can be captured through a generalization of the Ahrens and Pincus (1981) statistic 0(u)1 which captures the
ratio of harmonic to arithmetic mean of the number of observations (or subgroups) in a group, and which is equal to one if the panel is completely balanced.

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

307

A generalization of this concept to nested panels is as follows:
G/+L +Mi +Nij (1/¹ )
F/+L +Mi (1/N )
i/1 j/1 k/1
ijk , u ,
ij
i/1 j/1
u ,
l
k
H/G
G/F
¸/+L (1/M )
i/1
i .
u ,
j
F/¸
I consider two degrees of unbalancedness: strong unbalancedness (;) corresponds to a pattern of seven groups with M16, 11, 10, 7, 3, 1, 1N subgroups
(implying u"0.366), while weak unbalancedness (u) corresponds to a pattern of
seven groups with a uniform M2,2,12N distribution of subgroups (implying an
average of u"0.827). In all cases, groups have an average of seven subgroups,
which implies a sample size of H"74"2,401 observations. In varying the
degree of unbalancedness over the three hierarchical levels in the panel, I consider the six patterns ;uu, u;u, uu;, ;;u, ;u;, and u;;.
The variance decomposition will be measured by the familiar o , o , and
l k
o variance ratios, "xing p2 at 5. I consider 10 permutations of the variance
e
j
ratios 0.1, 0.4, and 0.7 for each of the three o's that satisfy 1!o !o !o '0.
j
k
l
With H and p2 "xed, the parameter space (u , u , u , o , o , o ) encompasses
e
l k j l k j
60 di!erent models. The Monte-Carlo simulation was conducted with 500
replications of each model, making use of the analytic gradient for the loglikelihood function documented in the appendix. The true model is assumed to
be
y "b #b x
#b x #b x #j #k #l #e ,
(34)
ijkt
0
1 1,ijkt
2 2,ijt
3 3,it
i
ij
ijk
ijt
with x varying only at the top (e.g., country) level, x varying at the mid (e.g.,
3
2
province) level, and x available for all observations at the most detailed (e.g.,
1
city) level.11
Results of the simulation are shown in Table 2. The "rst group (six columns)
in the table identi"es the parameter space (suppressing the "xed H and p2). The
e
second group } consisting of the p( 2 and o( columns } shows results for the
e
u
simple non-nested model, while the third group } consisting of the p( 2, o( , o(
e l k
and o( columns } contains the results from the double-nested model. All results
j
in these six columns are ratios of the estimated variance relative to the true

11 The data generation process (DGP) is an adaptation from Nerlove (1971a); see also Baltagi
et al. (1999b). Here, x
"0.2t#0.9x
w , x "0.3t#0.8x
#w , and x "0.3t
1,ijkt
1,ijk,t~1 ijt 2,ijt
2,ij,t~1
ij
3,it
#0.8x
#w , where the w , w and w are uniformly distributed on the interval [!z,#z]
3,ij,t~1
i
ijk ij
i
with z equal to 0.5, 5, and 1, respectively. The initial observations are de"ned as
x
"4#2w , x
"40#20w , and x "8#5w . The true b is [10,3,0.2,!1.5]. Start1,ijk0
ij0 2,ij0
ij0
3,i0
i0
ing values for the ML routine are OLS estimates for the non-nested ML, and estimates from the
non-nested ML were used in turn as starting values for the nested ML.

308

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

Table 2
Monte-Carlo simulation results
u

l

u

k

u

j

o
l

o
k

o
j

p( 2
e

o(
u

p( 2
e

o(
l

o(
k

o(
j

s1
b

s2
b

s3
b

0.819
0.819
0.366
0.819
0.366
0.366

0.819
0.366
0.821
0.366
0.822
0.366

0.366
0.827
0.823
0.366
0.366
0.826

0.1
0.1
0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1
0.1
0.1

1.01
1.01
1.01
1.01
1.01
1.01

0.28
0.29
0.28
0.25
0.28
0.30

1.00
1.00
1.00
1.00
1.00
1.00

0.88
0.90
0.89
0.95
0.83
0.86

0.89
0.85
0.88
0.86
0.89
0.90

0.69
0.73
0.70
0.62
0.69
0.70

1.10
1.10
1.10
1.11
1.09
1.10

1.25
1.27
1.22
1.30
1.22
1.25

1.11
1.16
1.10
1.11
1.09
1.10

0.820
0.819
0.366
0.819
0.366
0.366

0.819
0.366
0.820
0.366
0.821
0.366

0.366
0.822
0.826
0.366
0.366
0.825

0.4
0.4
0.4
0.4
0.4
0.4

0.1
0.1
0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1
0.1
0.1

1.08
1.08
1.08
1.08
1.08
1.08

0.15
0.16
0.16
0.15
0.17
0.17

1.00
1.00
1.00
1.00
1.00
1.00

0.99
1.00
0.99
0.99
1.00
0.99

0.78
0.74
0.83
0.79
0.76
0.79

0.60
0.69
0.64
0.56
0.63
0.62

1.13
1.12
1.10
1.13
1.10
1.12

1.36
1.40
1.30
1.39
1.29
1.31

1.08
1.09
1.07
1.12
1.07
1.09

0.820
0.819
0.366
0.819
0.366
0.366

0.822
0.366
0.820
0.366
0.820
0.366

0.366
0.824
0.827
0.366
0.366
0.825

0.7
0.7
0.7
0.7
0.7
0.7

0.1
0.1
0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1
0.1
0.1

1.22
1.22
1.22
1.22
1.22
1.22

0.12
0.12
0.12
0.12
0.14
0.13

1.00
1.00
1.00
1.00
1.00
1.00

0.99
0.99
0.99
1.00
1.00
1.00

0.79
0.74
0.83
0.75
0.88
0.77

0.55
0.58
0.57
0.58
0.50
0.62

1.06
1.07
1.03
1.07
1.04
1.04

1.43
1.43
1.29
1.44
1.29
1.29

1.04
1.03
1.00
1.05
1.02
1.01

0.819
0.819
0.366
0.819
0.366
0.366

0.819
0.366
0.818
0.366
0.819
0.366

0.366
0.832
0.827
0.366
0.366
0.827

0.1
0.1
0.1
0.1
0.1
0.1

0.4
0.4
0.4
0.4
0.4
0.4

0.1
0.1
0.1
0.1
0.1
0.1

1.07
1.06
1.07
1.07
1.07
1.06

0.33
0.30
0.27
0.27
0.33
0.32

1.00
1.00
1.00
1.00
1.00
1.00

0.91
0.91
0.87
0.91
0.85
0.89

0.97
0.97
0.98
0.97
0.98
0.98

0.51
0.54
0.55
0.50
0.60
0.63

1.09
1.09
1.07
1.08
1.07
1.07

1.78
1.72
1.50
1.74
1.51
1.48

1.11
1.07
1.06
1.08
1.05
1.05

0.818
0.818
0.366
0.820
0.366
0.366

0.817
0.366
0.819
0.366
0.821
0.366

0.366
0.825
0.822
0.366
0.366
0.824

0.4
0.4
0.4
0.4
0.4
0.4

0.4
0.4
0.4
0.4
0.4
0.4

0.1
0.1
0.1
0.1
0.1
0.1

1.13
1.13
1.13
1.13
1.13
1.13

0.21
0.20
0.18
0.19
0.22
0.21

1.00
1.00
1.00
1.00
1.00
1.00

0.99
1.00
0.99
0.99
1.00
1.00

0.97
0.96
0.97
0.96
0.97
0.95

0.52
0.65
0.61
0.57
0.59
0.59

1.10
1.10
1.08
1.09
1.08
1.10

1.66
1.65
1.47
1.72
1.47
1.47

1.06
1.03
1.05
1.05
1.05
1.06

0.819
0.819
0.366
0.819
0.366
0.366

0.819
0.366
0.819
0.366
0.818
0.366

0.366
0.825
0.833
0.366
0.366
0.823

0.1
0.1
0.1
0.1
0.1
0.1

0.7
0.7
0.7
0.7
0.7
0.7

0.1
0.1
0.1
0.1
0.1
0.1

1.19
1.17
1.19
1.19
1.19
1.17

0.36
0.30
0.26
0.29
0.34
0.31

1.00
1.00
1.00
1.00
1.00
1.00

0.87
0.87
0.89
0.94
0.86
0.92

0.97
0.97
0.97
0.98
0.98
0.97

0.71
0.69
0.63
0.61
0.65
0.62

0.98
1.00
0.97
0.97
0.96
0.98

1.69
1.67
1.43
1.69
1.43
1.41

0.94
0.97
0.95
0.95
0.95
0.96

0.819
0.819
0.366
0.819
0.366
0.366

0.821
0.366
0.818
0.366
0.819
0.366

0.366
0.828
0.827
0.366
0.366
0.824

0.1
0.1
0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1
0.1
0.1

0.4
0.4
0.4
0.4
0.4
0.4

1.01
1.01
1.01
1.01
1.01
1.01

0.58
0.57
0.60
0.57
0.60
0.57

1.00
1.00
1.00
1.00
1.00
1.00

0.89
0.82
0.92
0.91
0.87
0.88

0.88
0.88
0.87
0.85
0.86
0.90

0.87
0.86
0.89
0.85
0.89
0.85

1.10
1.10
1.10
1.10
1.09
1.10

1.25
1.28
1.22
1.28
1.22
1.24

1.09
1.08
1.09
1.08
1.09
1.09

0.819
0.819
0.366

0.821
0.366
0.819

0.366
0.827
0.823

0.4
0.4
0.4

0.1
0.1
0.1

0.4
0.4
0.4

1.08
1.08
1.08

0.37
0.37
0.38

1.00
1.00
1.00

1.00
1.00
0.99

0.81
0.82
0.83

0.85
0.88
0.89

1.12
1.12
1.10

1.38
1.39
1.30

1.06
1.05
1.08

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

309

Table 2 (continued)
u

l

u

k

u

j

o
l

o
k

o
j

p( 2
e

o(
u

p( 2
e

o(
l

o(
k

o(
j

s1
b

s2
b

s3
b

0.819
0.366
0.366

0.366
0.822
0.366

0.366
0.366
0.830

0.4
0.4
0.4

0.1
0.1
0.1

0.4
0.4
0.4

1.08
1.08
1.08

0.37
0.37
0.39

1.00
1.00
1.00

0.99
0.99
1.00

0.86
0.79
0.84

0.83
0.84
0.90

1.12
1.11
1.10

1.41
1.30
1.31

1.05
1.08
1.08

0.818
0.818
0.366
0.819
0.366
0.366

0.822
0.366
0.818
0.366
0.819
0.366

0.366
0.832
0.827
0.366
0.366
0.825

0.1
0.1
0.1
0.1
0.1
0.1

0.4
0.4
0.4
0.4
0.4
0.4

0.4
0.4
0.4
0.4
0.4
0.4

1.07
1.06
1.07
1.07
1.07
1.06

0.42
0.41
0.40
0.41
0.42
0.42

1.00
1.00
1.00
1.00
1.00
1.00

0.93
0.87
0.92
0.87
0.86
0.90

0.99
0.99
0.99
1.00
0.99
0.99

0.80
0.82
0.83
0.80
0.82
0.82

1.07
1.08
1.06
1.08
1.06
1.07

1.72
1.71
1.51
1.75
1.51
1.49

1.04
1.05
1.05
1.05
1.05
1.05

0.819
0.819
0.366
0.819
0.366
0.366

0.820
0.366
0.822
0.366
0.821
0.366

0.366
0.824
0.827
0.366
0.366
0.834

0.1
0.1
0.1
0.1
0.1
0.1

0.1
0.1
0.1
0.1
0.1
0.1

0.7
0.7
0.7
0.7
0.7
0.7

1.01
1.01
1.01
1.01
1.01
1.01

0.66
0.69
0.69
0.67
0.68
0.70

1.00
1.00
1.00
1.00
1.00
1.00

0.87
0.86
0.83
0.88
0.85
0.85

0.89
0.91
0.91
0.89
0.86
0.90

0.85
0.89
0.90
0.86
0.87
0.90

1.10
1.10
1.09
1.10
1.09
1.10

1.26
1.29
1.22
1.31
1.22
1.24

1.08
1.08
1.09
1.08
1.09
1.09

variance; in the case of o( the ratio is relative to o #o #o . The last group of
u
j
k
l
three columns shows the standard errors obtained for the parameter estimates
b , b , and b , where each is expressed as the ratio of the standard error from
1 2
3
the nested-e!ects estimation relative to the standard error from the non-nestede!ects estimation. For brevity, information on bias of the regressor estimates is
not included as they were found to be negligible.
The Monte-Carlo simulation provides several interesting results. First, the
use of the non-nested panel estimation for a nested panel does not deliver a clear
picture of the variance decomposition. When estimating a nested model with the
conventional non-nested random-e!ects ML estimator, p( 2 is biased slightly
e
upwards and o "o #o #o is biased downwards. The bias seems to be
u
l
k
j
more pronounced when the variance ratio o is relatively large compared to
l
o and o . The degree of unbalancedness only has a modest impact. Second,
k
j
using the nested-e!ects estimator for a nested panel delivers much superior
estimates of the variance terms.12 Third, the standard errors of parameter
estimates obtained from the non-nested estimation appear to be biased. For the
estimate of the standard error of b (corresponding to the variable x which
2
2
varies only over mid-groups), the non-nested panel estimator introduces
a downward bias with a sizeable magnitude, indicating a problem resembling

12 There appears to be a downward bias in the estimates of o , and more pronounced, o . While
k
j
this appears puzzling at "rst sight, closer inspection of the simulation results reveals that the
estimates of o and o are in some cases approaching zero, while in other cases, there are right on
k
j
target. The average shown in the table re#ects the mixing of these cases. This peculiarity may be
attributed to the high degree of non-linearity of the log-likelihood function (and its partial
derivatives) which in turn may lead to numerical instability in the maximization routine.

310

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

the Moulton bias. This downward bias is persistent over the entire parameter
space and is relevant for all three types of variables. The e!ect is noticeably
pronounced in the case of b when either o or o are large, or when a large
2
k
l
o coincides with the corresponding panel level being highly unbalanced.
k
In conclusion, the Monte-Carlo results suggest that inferences drawn from a
non-nested estimation of a nested panel may be inaccurate because the standard
errors are downward biased.

6. Conclusions
Hierarchically structured panel data are a common feature in many econometric studies. This paper has argued that a nested structure for the error term is
a suitable way to deal with related pertinent econometric problems. In particular, nested-e!ects models provide a method for addressing the problem of
merging micro and macro data. This paper has focused on unbalanced panels,
acknowledging their pervasiveness in applied work. The maximum-likelihood
estimator introduced in this paper provides a convenient estimation method.
The double-nested error structure discussed here also provides the basis for
generalizations to higher-order nested error structures.
I have compared the performance of the double-nested ML estimator with the
performance of the conventional non-nested ML estimator. The results from the
corresponding Monte-Carlo simulation reveal a downward bias in the standard
errors of the regressors when the conventional (non-nested) random-e!ects
estimator is applied to a nested panel. This key "nding, along with illustrating
the comparative performance of the estimator in a practical application, make
a solid case for using the nested-e!ects ML estimator.
A caveat applies to the exclusion of separate time e!ects in this model. One
way to deal with possible time e!ects is to introduce "xed time e!ects or a time
trend variable. It appears that extending the present model to a `two-waya
model with random time e!ects is very challenging because the time e!ects and
hierarchical individual e!ects overlap in a non-trivial manner.
The discussion of the nested-e!ects ML estimator in this paper is directed at
applied econometricians. Researchers interested in using this estimator can obtain
the software developed for this paper by sending an e-mail request to the author.

Acknowledgements
The idea for this paper originated in a joint project with Scott Taylor and
Brian Copeland on international trade and the environment where we were
dealing with hierarchically-structured environmental data as well as the problem of merging micro and macro data. I would like to thank Badi Baltagi, John

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

311

DiNardo and David Green for helpful comments. I am particularly grateful for
the suggestions and comments from one very helpful anonymous referee. This
project was supported, directly or indirectly, by grants from the Social Sciences
and Humanities Research Council of Canada, UBC's Centre for International
Business Studies, and UBC's Entrepreneurship and Venture Capital Research
Centre.

Appendix. Derivation of +L
In what follows I derive the analytic gradient +L of the log-likelihood
function. The derivatives are highly non-linear. For notational convenience let

A B
A B

A B
A B

2
Nij ¹
Mi / 2
ijk , W , +
ij ,
W ,+
ij
i
h
h
ijk
k/1
j/1 ij
2
Mi ; 2
Nij ;
ijk , < , +
ij ,
< ,+
i
ij
h
h
k/1 ijk
j/1 ij
Nij ; ¹
Mi ; /
N , + ijk ijk , N , + ij ij ,
ij
i
h2
h2
ijk
ij
k/1
j/1
Mi N
o
W
Mi W
ij ! k ; ij ,
WI , + ij , NI , +
i
ij h
i
h2
h
h
ij
ij
ij
j/1 ij
j/1
Nij X
Tijk
f,ijk ,
, X ,+
X
,+ x
f,ijkt
f,ij
f,ijk
h
t/1
k/1 ijk
Mi X
Tijk
f,ij , =
X ,+
,+ u
x
.
f,i
f,ijk
f,ijkt f,ijkt
h
j/1 ij
t/1

G

H

Then,

G G G
H
G G G
C
HD
H
G

H

H

LL
1 L
o
o
Mi
Nij
" +
+
+ =
; ! kX ;
! l X
f,ijk ijk
f,ij ij
f,ijk h
Lb
h
p2
f
ij
ijk
e i/1 j/1 k/1
o
! jX ; ,
h f,i i
i
LL
o
1
o
Mi
Nij
1 L
"!
+
+ < ! l ;2 ! k ;2
H! +
ijk
ij
ijk h
Lp2
h
2p2
p2
e
ij
ijk
e
e i/1 j/1 k/1
o
! j ;2 ,
h i
i
LL 1 L 1 ;2
i !/ ,
" +
i
h h p2
2
Lo
j
i/1 i i e

H

H

312

W. Antweiler / Journal of Econometrics 101 (2001) 295}313

C A

G

B DH

LL 1 L <
o ; o
i j ; W !2N #W
i !/ # j
" +
i
i
i
h p2 h i i
Lo
p2
2
i
i e
e
k
i/1
LL 1 L
" +
Lo
2
l
i/1

G G

C A

C A

B DH

,

B

o ; o
Mi <
ij k ; W !2N #W
ij !/ # k
+
ij
ij ij
ij
ij
h p2 h
p2
ij
ij e
e
j/1

DH

o ; o
i j ; WI !2NI #WI
.
# j
i
i
h p2 h i i
i
i e
where b is the estimate corresponding to the f th regressor in the model.
f
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