*Corresponding author. Tel.:#1-979-845-7380; fax:#1-979-847-8757. E-mail address:[email protected] (B.H. Baltagi).
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 min-imum norm quadratic unbiased estimators (MINQUE)-type estimators of the variance components. These are natural extensions from the biometrics, statistics and econo-metrics literature. The performance of these estimators is investigated by means of Monte Carlo experiments. While the MLE and MINQUE methods perform the best in estima-ting 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 compon-ent regression model which has its origin in the statistics and biometrics
literature, see Hsiao (1986), Baltagi (1995) and MaHtyaHs 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 empiri-cal application of the nested error component model to study whether schooling
matters in educational achievements in Montreal's Francophone public
elemen-tary 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 experi-ments. 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
con-sidered 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 produc-tivity of public capital in private production. Section 5 gives our conclusion.
2. The model
We consider the following unbalanced panel data regression model:
y
wherey
ijtcould denote the output of thejth"rm in theith industry for thetth
time period.x
ijt denotes a vector ofknonstochastic inputs. The disturbance of
(1) is given by
u
ijt"ki#lij#eijt, i"1,2,M, j"1,2,Ni and t"1,2,¹i, (2)
whereki denotes theith unobservable industry speci"c e!ect which is assumed
to be i.i.d. (0, p2k),lij denotes the nested e!ect of the jth "rm within the ith
industry which is assumed to be i.i.d. (0,p2l) and eijt denotes the remainder
disturbance which is also assumed to be i.i.d. (0,p2e). Theki's,lij's andeijt's are
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
?denotes the Kronecker product. Note that the observations are stacked such
that the slowest running index is the industry indexi, the next slowest running
index is the"rm indexjand the fastest running index is time.
Under these assumptions, the disturbance covariance matrix E(uu@) can be
written as
It is clear from Eq. (5) thatXis a block diagonal matrix with theith block given
ReplacingI
Ni byENi#JMNi and ITi by ETi#JMTi, whereENi"INi!JMNi and
E
Ti"ITi!JMTi and collecting terms with the same matrices, see Wansbeek and
Kapteyn (1982, 1983), one gets the spectral decomposition ofK
i:
pi,p"1, 2, 3, are the distinct characteristic roots of Ki of multiplicity
N
i(¹i!1), Ni!1 and 1, respectively. Note that each Qpi, for p"1, 2, 3 is
symmetric, idempotent with its rank equal to its trace. Moreover, theQ
pi's are
pairwise orthogonal and sum to the identity matrix. The advantages of this spectral decomposition are that
Kpi"jp
1iQ1i#jp2iQ2i#jp3iQ3i, (9)
wherepis an arbitrary scalar, see Baltagi (1993). Therefore, we can easily obtain
X~1as
econometrics literature as the Fuller and Battese (1973) transformation. Note that the OLS estimator is given by
bKOLS"(X@X)~1X@y. (12)
This is the best linear unbiased estimator when the variance componentsp2kand
p2l are both equal to 0. Even when these variance components are positive, the
OLS estimator is still unbiased and consistent, but its standard errors are biased,
see Moulton (1986). The OLS residuals are denoted byu(
OLS"y!XbKOLS.
The within estimator in this case can be obtained by transforming the model
in (3) by Q
1"diag(INi?ETi) and then applying OLS. Note that
Q
1Zk"Q1Zl"0 becauseETiιTi"0. Therefore,Q1 sweeps away theki's and l
ij's whether they are"xed or random e!ects. This yields
bI4"(X@4Q
whereX
4 denotes the exogenous regressors excluding the intercept andb4
de-notes the corresponding (k!1) vector of slope coe$cients.b@"(a, b@4) and the
estimate of the intercept can be retrieved as follows:a8"(y6...!XM 4...bI4), where
the dots indicate summation and the bar indicates averaging. Following
Amemiya (1971), the within residuals u8WTN for the unbalanced nested e!ect
model are given by
u8WTN"y!a8ι
m!X4bI4 (14)
wherem"+M
i/1Ni¹i.
Next, we consider methods of estimating the variance components.
2.1. Analysis ofvariance 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 compo-nents 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 theQ
1,Q2andQ3matrices obtained
from the spectral decomposition ofXin (8):
q
and Baltagi and Chang (1994). Taking expected values, we obtain
1Most of the algebra involved is simple but tedious and all proofs are available upon request from
i) in (16) and solving the system of equations, one gets the
Wallace and Hussain (1969)-type estimators of the variance components.1These
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 getq81,
q82 and q83, see Amemiya (1971) and Wansbeek and Kapteyn (1989). Taking
expected values ofq81,q82 andq83 we get
equations, we get the following Wansbeek and Kapteyn-type estimator of the variance components which we denote by WK:
p82e"u8 @WTNQ
1u8WTN/(m!n!k#1),
p82l"u8 @WTNQ2u8WTN![n!M#trM(X@4Q1X4)~1(X@4Q2X4)Np82e]
p82k"(u8 @WTNQ
(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
1, Q2 and Q3 and we obtain the transformed residuals u81, u82 and u83,
respectively. Let q8`1"u8 @
1Q1u81, q82`"u8 @2Q2u82 and q8`3"u8 @3Q3u83. Since q8`1 is
exactly the same as q81 the resulting expected value of q8`1 is the same as that
given in (18). The expected values ofq8`2 andq8`3 are
get the following Swamy and Arora-type estimators of the variance components which we denote by SA:
p82e"u8 @WTNQ
(4) Henderson Method III: Fuller and Battese (1973) suggest an estimation of
the variance components using the"tting constants methods. This method uses
the within residual sums of squares given byq8H1"u8 @
WTNu8WTN. Also, the residual
sum of squares obtained by transforming the regression in (3) by (Q
1#Q2) (i.e.,
sion. Finally, this method uses the conventional OLS residual sum of squares
denoted by q8H3"u(@OLSu(
OLS. If the x variables do not have constant values for
the WK method, the resulting expected value ofq8H1is the same as that given in
Henderson Method III estimator of the variance components, see Fuller and Battese (1973). These are denoted by HFB:
p82e"u8 @
Since jpi, for p"1, 2, 3 are the distinct characteristic roots of K
i then
DKiD"(j3i)(jNi~1
2i )(jN1ii(Ti~1)). Leto1"p2k/p2e,o2"p2l/p2e andX"p2eR, then the
log-likelihood function can be written as
log¸"C!m
The"rst-order conditions give closed form solutions forbandp2e conditional on
o1 ando2:
bKML"(X@RK~1X)~1X@RK~1y, (25)
p(2e"(y!Xb)@RK~1(y!Xb)/m. (26)
However, the"rst-order conditions based ono(1ando(2are nonlinear ino1and
o2even for known values ofbandp2e. Following Hemmerle and Hartley (1973),
Llog¸
Therefore, a numerical solution by means of iteration is needed. The Fisher
scoring procedure is used to estimateo1ando2. The partition of the
informa-tion matrix corresponding too1 ando2 is given by
E
C
!L2log¸see Harville (1977). Starting with an initial value, the (r#1)th updated value of
o1 ando2 is given by
p(2e are obtained from (25) and (26), the information matrix is obtained from
Eq. (28). The subscriptrmeans this is evaluated at therth 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$cients in estimating the variance
compo-nents. REML is based on a transformation that partitions the likelihood
function into two parts, one being free of the "xed regression coe$cients.
Maximizing this part yields REML. Patterson and Thompson (1971) suggest the
Using theA@ytransformation instead ofC@y, we get
Following Corbeil and Searle (1976), the log-likelihood function of A@y and
X@R~1y/p2e are given by log¸
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)
whereM"X(X@R~1X)~1X@R~1.
Using log¸
1 which is free from b, the "rst-order derivatives of log¸1 with
respect top2e, o1 ando2 are given by
Equating the equations in (34) to 0's yield the REML estimates. For example,
solvingLlog¸
1/Lp2e"0 conditional on o1ando2, we obtain
But there are no closed-form solutions ono1ando2. Thus a numerical solution by means of iteration is needed. The Fisher scoring procedure is used to estimate
o1 ando2. Using the results of Harville (1977) and Eq. (33), the information
matrix with respect too1ando2is given by
Rao (1971a) proposed a general procedure for variance components estima-tion which requires no distribuestima-tional assumpestima-tions other than the existence of the
"rst four moments. This procedure yields MINQUE of the variance compo-nents. Under normality of the disturbances, MINQUE and minimum variance quadratic unbiased estimators (MIVQUE) are identical. Since we assume nor-mality, we will focus on MIVQUE. Let
R"R~1[I!X(X@R~1X)~1X@R~1]/p2e, (37)
of MIVQUEs is given by
hK"S~1u, (40)
where hK@"(p(2e, p(2k, p(2l). However, MIVQUE requires a priori values of
the variance components. Therefore, MIVQUE is only &locally minimum
variance', see LaMotte (1973a, b), and&locally best', see Harville (1969). Three
2Similar 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.
3. Monte Carlo results
3.1. Design of the Monte Carlo study
We consider the following simple regression equation:
y
ijt"a#xijtb#uijt, i"1,2,M, j"1,2,Ni, t"1,2,¹i, (41)
withu
ijt"ki#lij#eijt. The exogenous variablexijtwas generated by a similar
method to that of Nerlove (1971). In fact,x
ijt"0.3t#0.8xij,t~1#wijt, where
w
ijt is uniformly distributed on the interval [!0.5, 0.5]. The initial values
x
ij0 were chosen as (100#250wij0). Throughout the experiment a"5 and
b"2. For generating the u
ijt disturbances, we let ki&IIN(0,p2k),
1!c2) is always positive. Extending a measure of
unbalanced-ness given by Ahrens and Pincus (1981) to the unbalanced nested model, we de"ne
1,c2andc3denote the measures of subgroup unbalancedness, observed
time unbalancedness and group unbalancedness due to each group size. Note
thatc
1, c2andc3take the value 1 when the data are balanced but take smaller
values than 1 as the data pattern gets more unbalanced. Table 1 gives the
(N
i,¹i) pattern used along with the corresponding unbalancedness measures for
M"10. The"rst parentheses gives theN
ipattern, while the second parentheses
below it gives the corresponding¹
i pattern. For example,P1observes the"rst
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 ofMare 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 ofbK relative to that
Table 1 (N
i,¹i) patterns considered and their corresponding unbalancedness measures whenM"10
Pattern (N
1,N2,2,N10)! c1 c2 c3 (¹
1,¹2,2,¹10)
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) 0.925 0.757 0.8238
(9,9,9,9,8,3,3,3,3,3)
P
3 (5,5,5,10,10,10,10,11,11,11) 0.893 0.734 0.504
(2,2,3,3,3,6,7,8,8,9)
P
4 (4,4,4,5,5,9,9,10,10,10) 0.854 0.619 0.881
(14,15,15,15,15,3,3,4,4,4)
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) 0.395 0.290 0.949
(2,2,3,3,3,28,28,30,30,30)
P
10 (20,20,20,20,20,2,2,2,2,1) 0.282 0.272 0.945
(2,2,2,3,3,25,30,30,30,30)
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
!The"rst parentheses gives theN
ipattern, while the parentheses below it gives the corresponding ¹
ipattern.
inferior to true GLS, ML-type (ML, REML) estimators and all feasible
GLS-type estimators except whenc1"c
2"0. For all experiments, the e!ect of an
increase inc1on the MSE of OLS is much larger than that of an increase inc2.
This is because c1 a!ects the primary group whilec2 a!ects only the nested
subgroup. The WTN estimator performs poorly for smallc1andc2values. The
performance of WTN is in some cases worse than OLS if eitherc1 orc2 is 0.
However, its performance improves asc1andc2increase and the
Table 2
MSE ofbK relative to that of true GLS whenM"10
c1 c2 OLS WTN WH WK SA HFB MLE REML MV1 MV2 MV3
P
1 0.0 0.0 1.000 4.864 0.998 1.006 0.996 0.999 0.998 0.999 0.998 0.999 0.999
0.0 0.2 1.069 2.732 1.014 1.016 1.011 1.014 1.012 1.015 1.017 1.013 1.014
0.0 0.4 1.396 1.985 1.008 1.011 1.009 1.009 1.008 1.009 1.010 1.009 1.009
0.0 0.6 2.073 1.406 1.000 1.001 1.001 1.000 0.999 0.999 1.000 1.001 1.000
0.0 0.8 4.341 1.212 1.000 1.000 1.001 1.000 1.000 1.000 1.004 0.999 0.999
0.2 0.0 1.649 3.979 1.003 1.001 1.003 1.002 1.003 1.004 1.002 1.002 1.004
0.2 0.2 1.633 2.333 1.002 0.998 1.002 1.001 1.001 1.001 0.999 1.000 1.001
0.2 0.4 2.274 1.549 1.010 1.011 1.014 1.011 1.011 1.010 1.014 1.010 1.010
0.2 0.6 4.358 1.219 1.011 1.010 1.017 1.010 1.011 1.010 1.015 1.010 1.010
0.4 0.0 3.537 4.149 1.001 0.998 1.004 0.999 1.001 1.000 1.004 0.999 1.000
0.4 0.2 2.748 2.049 1.004 1.008 1.006 1.006 1.006 1.007 1.012 1.007 1.007
0.4 0.4 4.407 1.276 1.000 0.998 1.001 1.000 0.999 0.999 1.003 0.999 0.999
0.6 0.0 6.235 4.077 1.003 1.004 1.007 1.005 1.005 1.005 1.036 1.004 1.005
0.6 0.2 5.673 1.555 1.000 0.998 1.003 0.998 0.998 0.998 1.013 0.998 0.998
0.8 0.0 13.606 4.473 1.007 1.011 1.004 1.006 1.006 1.006 1.135 1.006 1.006
P
3 0.0 0.0 1.000 3.578 1.003 1.013 1.002 1.004 0.998 1.002 1.001 1.016 1.004
0.0 0.2 1.236 2.233 1.025 1.024 1.022 1.025 1.021 1.025 1.020 1.031 1.025
0.0 0.4 1.653 1.735 1.009 1.009 1.005 1.010 1.009 1.011 1.008 1.011 1.010
0.0 0.6 2.590 1.296 0.999 1.000 1.002 0.998 1.000 0.999 1.003 0.999 0.999
0.0 0.8 5.815 1.133 1.015 1.012 1.014 1.015 1.011 1.012 1.026 1.014 1.012
0.2 0.0 1.970 3.400 1.001 1.007 1.004 1.001 1.006 1.005 1.002 1.006 1.004
0.2 0.2 2.106 1.816 1.016 1.010 1.025 1.014 1.016 1.016 1.024 1.014 1.016
0.2 0.4 3.005 1.508 1.015 1.016 1.022 1.014 1.011 1.011 1.035 1.011 1.011
0.2 0.6 6.389 1.097 1.018 1.011 1.023 1.014 1.010 1.008 1.040 1.009 1.008
0.4 0.0 3.932 3.022 1.003 1.004 1.001 1.001 1.000 0.999 1.012 1.002 0.999
0.4 0.2 3.467 1.605 1.004 1.003 1.003 1.003 1.005 1.004 1.016 1.004 1.004
0.4 0.4 6.324 1.180 1.004 1.003 1.009 1.004 1.001 1.001 1.038 1.001 1.001
0.6 0.0 8.675 3.474 1.011 1.010 1.012 1.007 1.008 1.008 1.028 1.006 1.008
0.6 0.2 7.699 1.433 1.017 1.015 1.017 1.016 1.017 1.017 1.048 1.017 1.017
0.8 0.0 19.474 3.528 1.028 1.026 1.015 1.016 1.011 1.011 1.054 1.023 1.012
P5 0.0 0.0 1.000 2.541 1.020 1.029 1.013 1.023 1.014 1.020 1.017 1.046 1.021
0.0 0.2 1.473 1.560 1.020 1.019 1.021 1.019 1.019 1.021 1.020 1.031 1.020
0.0 0.4 2.237 1.328 1.007 1.009 1.006 1.009 1.009 1.011 1.005 1.014 1.011
0.0 0.6 4.354 1.127 1.005 1.004 1.004 1.005 1.004 1.005 1.005 1.006 1.005
0.0 0.8 9.091 1.056 1.004 1.002 1.003 1.003 1.003 1.003 1.005 1.002 1.003
0.2 0.0 1.979 2.066 1.018 1.021 1.021 1.017 1.015 1.012 1.028 1.014 1.014
0.2 0.2 2.374 1.212 1.028 1.027 1.033 1.029 1.026 1.022 1.039 1.017 1.024
0.2 0.4 3.961 1.162 1.019 1.018 1.022 1.018 1.015 1.015 1.024 1.014 1.016
0.2 0.6 8.756 1.047 1.008 1.006 1.008 1.007 1.005 1.004 1.013 1.004 1.004
0.4 0.0 3.404 1.922 1.011 1.014 1.014 1.011 1.012 1.011 1.021 1.015 1.011
0.4 0.2 4.385 1.174 1.030 1.022 1.036 1.025 1.021 1.018 1.040 1.017 1.019
0.4 0.4 9.748 1.094 1.005 1.004 1.005 1.004 1.006 1.005 1.015 1.004 1.004
0.6 0.0 6.053 1.918 1.009 1.010 1.009 1.007 1.006 1.006 1.014 1.020 1.006
0.6 0.2 9.692 1.093 1.018 1.014 1.021 1.016 1.013 1.012 1.035 1.011 1.012
0.8 0.0 17.824 1.893 1.020 1.020 1.018 1.017 1.008 1.008 1.041 1.030 1.010
P7 0.0 0.0 1.000 2.894 1.010 1.016 1.008 1.011 1.009 1.012 1.011 1.020 1.012
0.0 0.2 1.293 1.806 1.008 1.010 1.007 1.010 1.005 1.006 1.006 1.012 1.006
Table 2 (Continued)
c1 c2 OLS WTN WH WK SA HFB MLE REML MV1 MV2 MV3
0.0 0.6 3.153 1.298 1.001 1.001 1.002 1.001 1.001 1.002 1.001 1.002 1.001
0.0 0.8 6.276 1.156 1.012 1.012 1.011 1.012 1.010 1.010 1.016 1.011 1.010
0.2 0.0 1.916 2.684 1.012 1.015 1.019 1.012 1.016 1.015 1.021 1.014 1.015
0.2 0.2 1.910 1.644 1.014 1.017 1.015 1.014 1.010 1.008 1.026 1.003 1.010
0.2 0.4 3.302 1.300 1.022 1.020 1.023 1.021 1.015 1.014 1.035 1.008 1.015
0.2 0.6 7.084 1.140 1.012 1.011 1.017 1.011 1.006 1.006 1.019 1.006 1.006
0.4 0.0 3.178 3.043 1.037 1.045 1.045 1.037 1.028 1.027 1.065 1.034 1.029
0.4 0.2 3.232 1.535 1.010 1.014 1.013 1.012 1.012 1.012 1.014 1.013 1.012
0.4 0.4 7.325 1.225 1.005 1.001 1.005 1.002 1.003 1.002 1.012 1.003 1.003
0.6 0.0 6.502 2.779 1.018 1.026 1.020 1.018 1.008 1.009 1.024 1.032 1.009
0.6 0.2 7.435 1.296 1.010 1.007 1.008 1.009 1.011 1.010 1.030 1.009 1.010
0.8 0.0 16.032 2.662 1.034 1.035 1.028 1.028 1.019 1.019 1.104 1.039 1.019
P
9 0.0 0.0 1.000 5.028 1.033 1.045 1.027 1.037 1.022 1.032 1.028 1.048 1.031
0.0 0.2 1.254 3.273 1.014 1.025 1.013 1.017 1.008 1.011 1.014 1.021 1.015
0.0 0.4 1.939 2.228 1.035 1.033 1.034 1.035 1.016 1.019 1.069 1.027 1.024
0.0 0.6 2.871 1.670 1.014 1.016 1.014 1.015 1.010 1.012 1.075 1.014 1.012
0.0 0.8 5.502 1.340 1.013 1.011 1.012 1.013 1.007 1.007 1.120 1.007 1.009
0.2 0.0 3.053 4.067 1.009 1.010 1.014 1.006 1.008 1.006 1.024 1.012 1.007
0.2 0.2 3.198 2.904 1.012 1.024 1.013 1.012 1.018 1.014 1.037 1.012 1.011
0.2 0.4 4.132 2.137 1.044 1.038 1.049 1.037 1.018 1.019 1.134 1.019 1.018
0.2 0.6 8.514 1.348 1.044 1.035 1.048 1.038 1.011 1.007 1.251 1.015 1.007
0.4 0.0 8.563 4.782 1.004 1.007 1.012 1.000 1.001 1.000 1.083 1.004 1.000
0.4 0.2 6.631 2.555 1.007 1.006 1.014 1.006 1.001 1.002 1.174 1.000 1.003
0.4 0.4 10.603 1.568 1.032 1.027 1.040 1.030 1.015 1.014 1.355 1.014 1.015
0.6 0.0 14.027 4.090 1.014 1.008 1.016 1.007 1.009 1.008 1.187 1.007 1.008
0.6 0.2 14.553 1.899 1.022 1.019 1.030 1.021 1.010 1.011 1.267 1.011 1.011
0.8 0.0 37.097 4.310 1.018 1.009 1.007 1.007 1.007 1.007 1.396 1.012 1.008
P
11 0.0 0.0 1.000 8.900 1.061 1.091 1.054 1.063 1.035 1.048 1.047 1.092 1.054
0.0 0.2 1.489 4.513 1.014 1.009 1.012 1.014 1.008 1.007 1.034 1.015 1.007
0.0 0.4 1.821 3.154 1.007 1.012 1.005 1.010 1.002 1.005 1.050 1.007 1.005
0.0 0.6 2.856 2.322 1.032 1.033 1.030 1.033 1.016 1.017 1.159 1.017 1.017
0.0 0.8 4.494 1.455 1.036 1.036 1.032 1.039 1.014 1.016 1.366 1.021 1.016
0.2 0.0 4.970 6.283 1.011 1.019 1.032 1.010 1.010 1.007 1.023 1.027 1.007
0.2 0.2 5.078 3.682 1.016 1.013 1.042 1.011 1.010 1.006 1.052 1.015 1.006
0.2 0.4 5.933 2.392 1.033 1.025 1.037 1.033 1.017 1.013 1.238 1.013 1.011
0.2 0.6 8.409 1.643 1.007 1.008 1.003 1.008 1.003 1.003 1.253 1.008 1.002
0.4 0.0 13.184 7.260 1.001 1.009 1.012 1.003 1.002 1.001 1.050 1.002 1.001
0.4 0.2 10.965 3.193 1.005 1.006 1.009 1.006 1.006 1.006 1.079 1.006 1.006
0.4 0.4 14.110 1.818 1.009 1.007 1.015 1.007 1.004 1.003 1.227 1.002 1.004
0.6 0.0 28.659 6.450 1.017 1.007 1.012 1.007 1.004 1.004 1.268 1.006 1.005
0.6 0.2 21.438 2.374 1.024 1.021 1.031 1.019 1.008 1.008 1.218 1.007 1.008
0.8 0.0 73.642 5.963 1.009 1.000 1.009 1.001 0.998 0.998 1.142 1.000 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 ofbK relative to that of true
GLS for the 12 unbalanced patterns for M"10 and c1" c
2"0.4. It is
Fig. 1. MSE ofbK relative to that of true GLS whenM"10,c1"c 2"0.4.
MSE performance to the ML- and MIVQUE-type estimators which are
rela-tively more di$cult to compute. They have at most 9.1% higher MSE than true
GLS, see the WK estimator for patternP
11whenc1"c2"0. MV1 performs
well forc1(0.4 andc
2(0.4. This is to be expected since the prior values for
MV1 arec1"0 andc
2"0. Asc1andc2increase or the degree of
unbalanced-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 ofvariance components estimates
Tables 3}5 report the MSE of the variance components relative to that of
MLE for the 12 unbalancedness patterns forM"10 andc
1"c2"0.4. Similar
tables for other values of M"6 and 15 and various values of c
1 andc2 are
available upon request from the authors. These tables are plotted in Figs. 2}4 for
ease of comparison.
For the estimation ofp2k, see Table 3 or Fig. 2, MLE ranks"rst; REML, MV2
and MV3 rank second and the ANOVA methods rank third by the MSE
criteria. MV1 does well only whenc1 andc2are close to 0. Forc1"c
2"0.4,
Table 3
MSE ofp(2krelative to that of MLE whenM"10,c1"c 2"0.4
WH WK SA HFB REML MV1 MV2 MV3
P
1 1.170 1.170 1.302 1.168 1.158 1.204 1.160 1.157
P
2 1.284 1.267 1.448 1.274 1.170 1.669 1.165 1.170
P
3 1.523 1.487 1.722 1.512 1.183 2.175 1.176 1.173
P
4 1.307 1.295 1.413 1.297 1.140 1.602 1.133 1.134
P
5 1.866 1.855 1.951 1.863 1.140 2.244 1.156 1.140
P
6 1.814 1.750 2.022 1.779 1.170 3.124 1.162 1.159
P
7 1.897 1.885 2.034 1.891 1.178 2.216 1.202 1.159
P
8 1.473 1.476 1.558 1.472 1.189 2.196 1.183 1.169
P
9 1.446 1.362 1.579 1.402 1.159 1.407 1.159 1.146
P
10 1.427 1.360 1.585 1.383 1.177 1.459 1.179 1.170
P
11 1.680 1.863 1.787 1.755 1.211 2.558 1.171 1.129
P
12 1.338 1.379 1.465 1.336 1.165 2.170 1.074 1.100
Table 4
MSE ofp(2lrelative to that of MLE whenM"10,c1"c 2"0.4
WH WK SA HFB REML MV1 MV2 MV3
P
1 1.010 1.010 1.012 1.004 0.999 1.312 0.997 0.998
P
2 1.205 1.181 1.194 1.179 0.999 2.514 1.012 0.998
P
3 1.116 1.115 1.117 1.104 1.000 1.450 1.001 1.002
P
4 1.458 1.428 1.430 1.412 0.999 3.224 1.002 0.990
P
5 1.086 1.066 1.096 1.065 0.995 1.181 0.989 1.015
P
6 1.584 1.566 1.567 1.549 0.999 3.563 1.015 0.993
P
7 1.105 1.103 1.110 1.092 0.999 1.333 1.024 1.007
P
8 1.475 1.453 1.466 1.442 0.998 16.188 1.039 0.985
P
9 2.615 2.561 2.581 2.551 0.993 13.130 0.998 0.977
P
10 2.717 2.672 2.681 2.650 1.000 21.473 1.021 0.992
P
11 2.694 2.583 2.583 2.559 1.001 20.234 1.001 1.004
P
12 2.863 2.697 2.692 2.667 1.000 23.894 1.000 0.989
MLE for patternsP
5, P6andP7. On the other hand, REML, MV2 and MV3
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 ofp2l, see Table 4 or Fig. 3, REML, MLE, MV2 and MV3
rank"rst by the MSE criteria. They are followed by the ANOVA methods with
MV1 performing the worst. For patternsP9,P10,P11 andP12, these ANOVA
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 ofp2e, there is not much di!erence among WK, SA, HFB,
Table 5
MSE ofp(2e relative to that of MLE whenM"10,c1"c 2"0.4
WH WK SA HFB REML MV1 MV2 MV3
P
1 1.038 1.001 1.001 1.001 1.001 5.190 1.004 1.002
P
2 1.053 1.005 1.005 1.005 1.002 34.983 0.999 1.003
P
3 1.050 1.000 1.000 1.000 1.000 27.465 1.002 1.000
P
4 1.202 1.004 1.004 1.004 1.002 54.860 1.002 1.003
P
5 1.201 1.006 1.006 1.006 1.006 8.669 1.012 1.010
P
6 1.173 1.005 1.005 1.005 1.002 58.511 1.009 1.003
P
7 1.078 1.000 1.000 1.000 1.000 6.469 1.012 1.001
P
8 1.126 1.004 1.004 1.004 1.002 124.852 1.018 1.005
P
9 1.121 1.000 1.000 1.000 1.002 124.467 1.010 1.006
P
10 1.128 1.003 1.003 1.003 1.001 154.966 1.006 1.005
P
11 1.193 1.004 1.004 1.004 1.002 144.898 1.006 1.004
P
12 1.148 1.001 1.001 1.001 0.997 119.133 0.994 1.000
Fig. 2. MSE ofp(2krelative to that of MLE whenM"10,c1"c 2"0.4.
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 suspectsc1 and c2 are close to 0. However, for the
estimation of the regression coe$cients, the ANOVA methods compare well
Fig. 3. MSE ofp(2lrelative to that of MLE whenM"10,c1"c 2"0.4.
Fig. 4. MSE ofp(2e relative to that of MLE whenM"10,c
1"c2"0.4.
For ANOVA and MIVQUE-type estimators, negative estimates of p2k or
p2l occur in about 50% of the replications when c1"0 or c
2"0. When
the negative estimates of variance components are replaced by 0, the corre-sponding estimator forfeits its unbiasedness property. But, replacing these
negative estimates by 0 did not lead to much loss in e$ciency using the MSE
Fig. 5. MSE of standard errors ofbK relative to that of MLE whenM"10,c1"c 2"0.4.
Finally, better estimates of the variance components by the MSE criterion, do
not necessarily imply better estimates of the regression coe$cients. A similar
result was obtained by Baltagi and Chang (1994) for the unbalanced one way model and by Taylor (1980) and Baltagi (1981) for the balanced error compon-ent model. However, MV1 has worse relative MSE performance than other ANOVA, ML, REML and MIVQUE-type estimators of the variance
compo-nents whenc1andc2are large and the pattern is severely unbalanced and this
clearly translates into a corresponding worse relative MSE performance of the
regression coe$cients. Similar conclusions can be drawn forM"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 ofbK relative to that of MLE for the
12 unbalanced patterns for M"10 and c1"c
2"0.4. Besides the relative
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 ofbK that are
2 times that of MLE for severely unbalanced patterns likeP10, P11 andP12.
However, these ANOVA methods perform reasonably well in patternsP
1}P8
giving MSEs of the standard error ofbK that are no more than 30% higher than
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 variableyis 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
ijt"a#b1Kijt#b2KHijt#b3KWijt#b4KOijt
#b
5¸ijt#b6Unempijt#uijt, (43)
where i"1, 2,2, 9 regions, j"1,2,N
i with Ni equaling 3 for the Middle
Atlantic region and 8 for the Mountain region andt"1, 2,2, 17. The data is
unbalanced only in the di!ering number of states in each region. The
distur-bances 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
compo-nents estimates, there are no di!erences in the estimate ofp2e. But estimates of
p2kandp2lvary.p(2kis as low as 0.0015 for SA and MLE and as high as 0.0029 for
HFB. Similarly,p(2lis as low as 0.0043 for SA and as high as 0.0069 for WK. This
Table 6
Cobb}Douglas production function estimates with unbalanced nested error components 1970}1986, Nine regions, 48 states!
Variable OLS WTN WH WK SA HFB MLE REML MV1 MV2 MV3
Intercept 1.926 * 2.082 2.131 2.089 2.084 2.129 2.127 2.083 2.114 2.127 (0.053) (0.152) (0.160) (0.144) (0.150) (0.154) (0.157) (0.152) (0.154) (0.156)
K 0.312 0.235 0.273 0.264 0.274 0.272 0.267 0.266 0.272 0.269 0.267
(0.011) (0.026) (0.021) (0.022) (0.020) (0.021) (0.021) (0.022) (0.021) (0.021) (0.021)
¸ 0.550 0.801 0.742 0.758 0.740 0.743 0.754 0.756 0.742 0.750 0.755
(0.016) (0.030) (0.026) (0.027) (0.025) (0.026) (0.026) (0.026) (0.026) (0.026) (0.026) KH 0.059 0.077 0.075 0.072 0.073 0.075 0.071 0.072 0.075 0.072 0.072
(0.015) (0.031) (0.023) (0.024) (0.022) (0.022) (0.023) (0.023) (0.023) (0.023) (0.023) KW 0.119 0.079 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076
(0.012) (0.015) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) (0.014) KO 0.009 !0.115 !0.095 !0.102 !0.094 !0.096 !0.100 !0.101 !0.095 !0.098 !0.100
(0.012) (0.018) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) (0.017) Unemp !0.007 !0.005 !0.006 !0.006 !0.006 !0.006 !0.006 !0.006 !0.006 !0.006 !0.006
(0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001)
p2e 0.0073 0.0013 0.0014 0.0014 0.0014 0.0014 0.0013 0.0014 0.0014 0.0014 0.0014
p2k * * 0.0027 0.0022 0.0015 0.0029 0.0015 0.0019 0.0027 0.0017 0.0017
p2l * * 0.0045 0.0069 0.0043 0.0044 0.0063 0.0064 0.0046 0.0056 0.0063
!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
}
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 recom-mended. 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 recom-mended 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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