The system GMM estimator in dynamic panel data models combines moment conditions for hte differenced equation with moment conditions for the model in levels. An initial optimal weight matrix under homoscedasticity and non-serial correlation is not known for this estimation procedure. It is common practice to use the inverse of the moment matrix of the instruments as the initial weight matrix. This paper assesses the potential efficiency loss from the use of this weight matrix using the efficiency bounds as derived by Liu and Neudecker (1997).

Frank Windmeijer

English

Research Papers in Economics

E±ciency Comparisons for a System GMM
Estimator in Dynamic Panel Data Models
Frank Windmeijer
Institute for Fiscal Studies
7 Ridgmount Street
London WC1E 7AE
IFS Working Paper Series No. W98/1
Abstract
The system GMM estimator in dynamic panel data models combines mo-
ment conditions for the di®erenced equation with moment conditions for
the model in levels. An initial optimal weight matrix under homoscedastic-
ity and non-serial correlation is not known for this estimation procedure.
It is common practice to use the inverse of the moment matrix of the in-
struments as the initial weight matrix. This paper assesses the potential
e±ciency loss from the use of this weight matrix using the e±ciency bounds
as derived by Liu and Neudecker (1997).
1 Introduction
A standard practice to estimate the parameters in dynamic panel data models is
to take ¯rst di®erences to eliminate the correlated individual speci¯c e®ects, and
estimate the di®erenced model by Generalised Method of Moments (GMM) us-
ing appropriately lagged level variables as instruments. As the information of the
instruments for the di®erenced model decreases as the series become more per-
sistent, Arellano and Bover (1995) and Blundell and Bond (1997) have proposed
use of a system GMM estimator that combines the di®erenced equation with the
level equation. The instruments for the level equation are lagged di®erences of
1the variables, which are valid when these di®erences are uncorrelated with the
individual e®ects. Blundell and Bond (1997) show that the system estimator
has superior properties in terms of small sample bias and RMSE, especially for
persistent series.
The GMM estimator is a two-step estimator. In the ¯rst step, an initial
positive semide¯nite weight matrix is used to obtain consistent estimates of the
parameters. Given these consistent estimates, a weight matrix can be constructed
that is consistent for the e±cient weight matrix, and this weight matrix is used
for the asymptotically e±cient two-step estimates. It is well known, see e.g.
Arellano and Bond (1991), that the two-step estimated standard errors have a
small sample downward bias in this dynamic panel data setting, and one-step
estimates with robust standard errors are often preferred. Although an e±cient
weight matrix for the di®erenced model with errors that are homoscedastic and
that are not serially correlated is easily derived, this is not the case for the system
estimator, combining di®erences and levels information.
It is common practice to use the inverse of the moment matrix of the instru-
ments as an initial weight matrix. In this paper the potential e±ciency loss will
be considered in a model with homogeneous and non-serially correlated errors.
To do this, upper bounds for the e±ciency loss will be calculated as derived by
Liu and Neudecker (1997) based on the Kantorovich Inequality (KI). These upper
bounds indicate that the e±ciency loss could potentially be quite severe. When
the variance of the individual unobserved heterogeneity is small, e±ciency can
be gained by using a weight matrix that is optimal under the assumption that
the variance of the unobserved heterogeneity is equal to zero.
In section 2, an AR(1) dynamic panel data model is considered and a descrip-
tion of the system GMM estimator is given. In section 3 the upper bounds of
the e±ciency loss are calculated for 3 and 4 time periods respectively. Section 4
concludes.
22 Model and System GMM Estimator
Consider the AR(1) panel data speci¯cation
yit = ®0yit¡1 + ´i + "it (1)
for i = 1;:::;N, t = 2;:::;T, with N large, and T ¯xed. The error terms follow
the error components structure in which
E (´i) = 0 ; E ("it) = 0;
E
³
"
2
it
´
= ¾
2
" ; E
³
´
2
i
´
= ¾
2
´;
E (´i"it) = 0 ; E ("it"is) = 0; t 6= s:
The yit series are assumed stationary with an in¯nite time horizon and therefore
the series can alternatively be written as
yit =
´i
1 ¡ ®0
+
1 X
j=0
®
j
0"it¡j: (2)
The OLS and within groups estimators of ®0 in model (1) are biased and
inconsistent. A consistent estimator for ®0 is the system GMM estimator, as
proposed by Arellano and Bover (1995) and Blundell and Bond (1997), utilising
the following (T + 1)(T ¡ 2)=2 moment conditions1
E [(¢yit ¡ ®0¢yit¡1)(yit¡2;:::;yi1)] = 0 (3)
E [(yit ¡ ®0yit¡1)¢yit¡1] = 0; (4)
for t = 3;:::;T. Moment conditions (3) are for the model in ¯rst di®erences,
utilising appropriately lagged levels information as instruments, whereas condi-
tions (4) are for the model in levels, utilising lagged di®erences as instruments.
As Blundell and Bond (1997) show, the system estimator is considerably more
1Under homoscedasticity there are additional moment conditions available to improve e±-
ciency, see Ahn and Schmidt (1995). These extra moment conditions are not considered here.
3e±cient than the traditional GMM estimator utilising the moment conditions of
the di®erenced model only.
De¯ne
Zi =
2
6 6 6 6 6 6 6 6 6 6 6 6 6
4
yi1 0 0 ::: 0 ::: 0 0 0 ::: 0
0 yi1 yi2 ::: 0 ::: 0 0 0 ::: 0
0 0 0 ::: 0 ::: 0 0 0 ::: 0
0 0 0 ::: yi1 ::: yiT¡2 0 0 ::: 0
0 0 0 ::: 0 ::: 0 ¢yi2 0 ::: 0
0 0 0 ::: 0 ::: 0 0 ¢yi3 ::: 0
0 0 0 ::: 0 ::: 0 0 0 ::: 0
0 0 0 ::: 0 ::: 0 0 0 ::: ¢yT¡1
3
7 7 7 7 7 7 7 7 7 7 7 7 7
5
vi = vi (®0) =
2
6 6 6 6 6 6 6 6 6 6 6 6 6
4
¢yi3 ¡ ®0¢yi2
¢yi4 ¡ ®0¢yi3
:::
¢yiT ¡ ®0¢yiT¡1
yi3 ¡ ®0yi2
yi4 ¡ ®0yi3
:::
yiT ¡ ®0yiT¡1
3
7 7 7 7 7 7 7 7 7 7 7 7 7
5
;
and
fi(®0) = Z
0
ivi:
Moment conditions (3) and (4) imply that E (fi (®0)) = 0. The GMM estimator2
b ® for ®0 minimises
"
1
N
N X
i=1
fi (®)
#0
WN
"
1
N
N X
i=1
fi (®)
#
;
with respect to ®; where WN is a positive semide¯nite weight matrix which
satis¯es plimN!1 WN = W, with W a positive de¯nite matrix. Regularity con-
ditions are in place such that limN!1
1
N
PN
i=1 fi(®) = E (f (®)) and 1 p
Nfi(®0) !
N (0;ª): Let F (®) = E (@fi (®)=@®) and F0 ´ F (®0), then
p
N (b ® ¡ ®0) has a
limiting normal distribution,
p
N (b ® ¡ ®0) ! N (0;VW), where
VW = (F
0
0WF0)
¡1 F
0
0WªWF0(F
0
0WF0)
¡1 : (5)
2See Hansen (1982), Ogaki (1993). Here, the same notation as in Liu and Neudecker (1997)
is used.
43 E±ciency Comparisons
As is clear from the expression of the asymptotic variance matrix VW, (5), the
e±ciency of the GMM estimator is a®ected by the choice of the weight matrix
WN. An optimal choice is a weight matrix for which W = ª¡1. The asymptotic
variance matrix is then given by (F 0
0ª¡1F0)
¡1. For any other W the GMM
estimator is less e±cient as
³
F
0
0ª
¡1F0
´¡1
· (F
0
0WF0)
¡1 F
0
0WªWF0(F
0
0WF0)
¡1 :
In panel data models the e±cient estimator is obtained in a two-step pro-
cedure. The one-step GMM estimator e ® is obtained using an arbitrary weight
matrix WN1. Let e vi = vi (e ®). The e±cient two-step estimator is then based on
the weight matrix
WN2 =
Ã
1
N
N X
i=1
Z
0
ie vie viZi
!¡1
; plimWN2 = ª
¡1.
Although the e±cient estimator is easily obtained, there is a serious problem
associated with it as the estimated standard errors of the two-step estimator
can be biased downwards quite severely for moderate sample sizes N, as has
been documented by Arellano and Bond (1991) and Blundell and Bond (1997),
who performed Monte Carlo simulations with sample sizes N = 200. Therefore,
inference based on the two-step estimator can be very unreliable. In contrast,
the one-step estimated standard errors based on the asymptotic variance matrix
(5), using WN2 as an estimate for ª and substituting e ® for ®0, are found to be
much less biased, and inference, like Wald tests, much more reliable. In practice
therefore, one can often only rely on inference based on the less e±cient one-step
estimator.
For the GMM estimator that only utilises the moment conditions (3) for the
di®erenced model, an optimal weight matrix is
³
1
N
PN
i=1 D
0
iHDi
´¡1
, where Di is
the left upper block of Zi and H is a (T ¡ 2) square matrix which has 2's on the
5main diagonal, -1's on the ¯rst subdiagonals and zeros elsewhere. Setting WN1 =
³
1
N
PN
i=1 D
0
iHDi
´¡1
results therefore in an e±cient one-step estimator. For the
system GMM estimator such an e±cient one-step weight matrix is not known,
and in practice one uses as an initial weight matrix WN1 =
³
1
N
PN
i=1 Z0
iZi
´¡1
. To
assess the potential loss in e±ciency from using this initial weight matrix, the
following expression for the upper bound of the e±ciency loss has been derived
by Liu and Neudecker (1997, p.350) on the basis of the Kantorovich Inequality
(KI):
(F
0
0WF0)
¡1 F
0
0WªWF0(F
0
0WF0)
¡1 ·
(¸1 + ¸p)
2
4¸1¸p
³
F
0
0ª
¡1F0
´¡1
(6)
where ¸1 ¸ ::: ¸ ¸p are the eigenvalues of the matrix ªW.
For T = 3, there is one overidentifying moment condition, as the system
estimator utilises the following two moment conditions
E [(¢yi3 ¡ ®¢yi2)yi1] = 0
E [(yi3 ¡ ®yi2)¢yi2] = 0;
and ª is given by
ª = plim
1
N
N X
i=1
"
y2
1 (¢"i3)
2 yi1¢yi2(´i + "i3)¢"i3
yi1¢yi2(´i + "i3)¢"i3 (¢yi2)
2(´i + "i3)
2
#
= ¾
2
"
"
2¾2
y ¡(1 ¡ ®)¾2
y
¡(1 ¡ ®)¾2
y 2
¾2
´+¾2
"
1+®
#
where
¾
2
y =
¾2
´
(1 ¡ ®)
2 +
¾2
"
1 ¡ ®2:
Further,
W1 =
Ã
plim
1
N
N X
i=1
Z
0
iZi
!¡1
=
Ã
plim
1
N
N X
i=1
"
y2
i1 0
0 (¢yi2)
2
#!¡1
=
"
¾2
y 0
0 2
¾2
"
1+®
#¡1
6and the matrix G = ªW1 is given by
G =
"
2¾2
" ¡(1 ¡ ®2)¾2
y=2
¡(1 ¡ ®)¾2
" ¾2
" + ¾2
´
#
:
Figure 1 presents the plot of the function bKI = (¸G1 + ¸Gp)
2=4¸G1¸Gp, where
the ¸G's are the eigenvalues of G, for various values of ®0 and ¾2
´=¾2
". When
¾2
´ = ¾2
", bKI is constant for di®erent values of ® and equal to 4/3, indicating
that the asymptotic variance of the one-step estimator could potentially be 33%
larger than the e±cient estimator. When ¾2
´=¾2
" < 1, bKI is declining with ®0,
whereas it is increasing with ®0 when ¾2
´=¾2
" > 1. The value of bKI increases with
¾2
´=¾2
" when ¾2
´=¾2
" > 1.
[Figure 1 about here]
When T = 4, there are 4 overidentifying moment conditions, and the matrices
ª and W1 are given by
ª = ¾
2
"
2
6 6 6 6 6 6 6 6
4
2¾2
y ¡¾2
y ¡± ¡(1 ¡ ®)¾2
y
2¡®
1¡®¾2
´
¡¾2
y 2¾2
y 2±
¾2
"
1+® ¡(1 ¡ ®)±
¡± 2± 2¾2
y ¡
¾2
"
1+® ¡(1 ¡ ®)¾2
y
¡(1 ¡ ®)¾2
y
¾2
"
1+® ¡
¾2
"
1+® 2
¾2
´+¾2
"
1+® ¡1¡®
1+®¾2
´
2¡®
1¡®¾2
´ ¡(1 ¡ ®)± ¡(1 ¡ ®)¾2
y ¡1¡®
1+®¾2
´ 2
¾2
´+¾2
"
1+®
3
7 7 7 7 7 7 7 7
5
W1 =
2
6 6 6 6 6 6 6
4
¾2
y 0 0 0 0
0 ¾2
y ± 0 0
0 ± ¾2
y 0 0
0 0 0 2
¾2
"
1+® 0
0 0 0 0 2
¾2
"
1+®
3
7 7 7 7 7 7 7
5
;
where
± = ¾
2
y ¡
¾2
"
1 + ®
:
Figure 2 presents the e±ciency bounds for the one-step system estimator
when T = 4. The values for bKI are larger than for the T = 3 case. When
¾2
´=¾2
" = 1, bKI is no longer constant for di®erent values of ®0, and takes values
around 3, indicating that the asymptotic variance of the one-step estimator could
7be 3 times the variance of the e±cient estimator. Again, bKI increases with ¾2
´=¾2
"
when ¾2
´=¾2
" > 1, and bKI reaches the value 6 when ¾2
´=¾2
" = 2:5.3
[Figure 2 about here]
3.1 An Optimal Weight Matrix when ¾2
´=0
An optimal weight matrix for the system GMM estimator when ¾2
´ = 0 is given
by
WN;¾2
´=0 =
Ã
1
N
N X
i=1
Z
0
iAZi
!¡1
;
where
A =
"
H C
C0 IT¡2
#
with H as de¯ned above, IT¡2 is the identity matrix of order (T ¡ 2), and
C =
2
6 6 6 6 6 6
4
1 0 0 0
¡1 1 0 0
0 ¡1 1 0
::: ::: ::: ::: :
0 0 0 ::: 1
3
7 7 7 7 7 7
5
:
Using this weight matrix instead of WN1 =
³
1
N
PN
i=1 Z0
iZi
´¡1
may improve on
e±ciency when ¾2
´ is small. Figures 3 and 4 display the values for bKI when
WN;¾2
´=0 is used in the one-step estimator, for T = 3 and T = 4 respectively.
Indeed, for small values of ¾2
´ the potential loss in e±ciency is seen to be smaller
than when WN1 is used. However, when ¾2
´=¾2
" is large, the potential e±ciency
loss gets larger for WN;¾2
´=0, which is what one would expect.
[Figures 3 and 4 about here]
One way to detect whether use of WN;¾2
´=0 could be bene¯cial, without actually
calculating the variances of the components, is to calculate the e±ciency bounds
3The value of bKI increases with T as the number of moment restrictions increases. For
example when T = 6 and ¾2
´=¾2
" = 1, bKI is approximately 14.
8bKI for the e±ciency di®erence between the one-step and two-step estimators,
i.e. calculate bKI from eigenvalues of the matrices WN2WN;¾2
´=0, and WN2WN1.
If the former are closer to 1 than the latter, this is an indication that there could
be an e±ciency gain from using WN;¾2
´=0 instead of WN1.
4 Discussion
Upper bounds for the e±ciency loss of the one-step system GMM estimator for a
dynamic AR(1) panel data model as compared to the e±cient two-step estimator,
show that the e±ciency loss could be quite severe when the weight matrix WN1 =
³
1
N
PN
i=1 Z0
iZi
´¡1
is used, especially when T gets large. When the variance of the
unobserved individual e®ects, ¾2
´, is small, an e±ciency gain can be made by
using a weight matrix that is optimal under the assumption that ¾2
´ = 0.
The upper bounds were shown to be quite large, for example when T = 4,
WN1 =
³
1
N
PN
i=1 Z0
iZi
´¡1
and ¾2
´=¾2
" = 2:5, the ratio of the asymptotic variance
of the ine±cient estimator to that of the e±cient estimator can be as large as
6 for high values of ®0. In Monte Carlo studies however, such large di®erences
of the variances are not found, using normal and non-normal data generating
processes. This could mean that the KI upper bounds, bKI, are too large to be
informative for these cases. When the bKI are close to one, there is evidence of
an e±cient one-step estimator. The opposite statement for large bKI may not
be true. Further research is needed to assess whether the bKI are informative to
rank di®erent one-step estimators on the basis of their relative KI -values.
In empirical settings, one can easily compute bKI from the eigenvalues of the
matrix WN2WN1, where WN2 is the two-step e±cient weight matrix.
Acknowledgments
I would like to thank Steve Bond for helpful comments and for drawing my
attention to the use of WN;¾2
´=0. The usual disclaimer applies.
9References
[1] Ahn, S.C. and P. Schmidt (1995), E±cient Estimation of Models for Dynamic
Panel Data, Journal of Econometrics, 68, 5-28.
[2] Arellano, M. and S. Bond (1991), Some Tests of Speci¯cation for Panel Data:
Monte Carlo Evidence and an Application to Employment Equations, Review
of Economic Studies, 58, 277-98.
[3] Arellano, M. and O. Bover (1995), Another Look at the Instrumental-Variable
Estimation of Error-Components Models, Journal of Econometrics, 68, 29-51.
[4] Blundell, R. and S. Bond (1997), Initial Conditions and Moment Restrictions
in Dynamic Panel Data Models, Journal of Econometrics, in press.
[5] Hansen, L.P. (1982) `Large Sample Properties of Generalized Method of Mo-
ments Estimators', Econometrica, 50, 1029-1054.
[6] Liu, S. and H. Neudecker (1997), Kantorovich Inequalities and E±ciency
Comparisons for Several Classes of Estimators in Linear Models, Statistica
Neerlandica, 51, 345-355.
[7] Ogaki, M. (1993) `Generalized Method of Moments: Econometric Applica-
tions', in: Maddala, G.S., C.R. Rao, and H.D. Vinod (eds.), `Econometrics',
Handbook of Statistics, 11, North-Holland, Amsterdam.
10Figure 1: KI E±ciency Bounds, T = 3; WN1 =
³
1
N
P
i Z0
iZi
´¡1
Figure 2: KI E±ciency Bounds, T = 4; WN1 =
³
1
N
P
i Z0
iZi
´¡1
11Figure 3: KI E±ciency Bounds, T = 3; WN1 = WN;¾2
´=0
Figure 4: KI E±ciency Bounds, T = 4; WN1 = WN;¾2
´=0
12

Efficiency comparisons for a system GMM estimator in dynamic panel data models

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