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International Statistical Review (2004), 72, 3, 397–408, Printed in Wales by Cambrian Printers
c
International Statistical Institute
A Comparative Analysis of Different IV and
GMM Estimators of Dynamic Panel Data
Models
Mark N. Harris &L
´
aszl
´
oM
´
aty
´
as
Department of Econometrics and Business Statistics, Monash University, Australia
Central European University, Budapest, Hungary
Erudite, Universite Paris XII
Summary
It is well known that the usual procedures for estimating panel data models are inconsistent in the
dynamic setting. A large number of consistent estimators however, have been proposed in the literature.
This paper provides a survey of the majority of mainstream estimators, which tend to consist of IV and
GMM ones. It also considers a newly proposed extension to the promising Wansbeek–Bekker estimator
(Harris & M
´
aty
´
as, 2000). To provide guidance to the applied researcher working on micro-datasets, the
small sample performance of these estimators is evaluated using a set of Monte Carlo experiments.
Key words: Panel data; Dynamic models; Monte Carlo; IV and GMM estimators.
1 Introduction
As a consequence of the increase in both the availability of panel data sets, and the stock of tools
the applied researcher has to analyse them, the area of panel data econometrics has become very
popular over the last decade. Given the vast array of economic theories espousing some form of
partial adjustment of economic variables to an equilibrium level, it has become increasingly obvious
that special attention must be paid to the estimation of dynamic panel models. This topic has been the
focus of many recent theoretical and simulation papers (see, for example, Arellano & Bond, 1991;
Arellano & Bover, 1995; Ahn & Schmidt, 1995; Kiviet, 1995; Ahn & Schmidt, 1997; Blundell &
Bond, 1998; Crepon, Kramarz & Trognon, 1998; Harris & M
´
aty
´
as, 2000).
Estimation of dynamic panel models is unfortunatelyproblematic. Essentially, the problemarises
from the fact that the equation’s disturbance terms and the lagged dependent variable are correlated
(see Nickell, 1981; Sevestre & Trognon, 1985). This causes the traditional panel estimators to be
biased and inconsistent in the usual large N and fixed T case.
The most favoured approaches to consistent estimation are instrumental variables (IV) and Gen-
eralised Method of Moments (GMM). Indeed, GMM estimation has spawned much interest in
attempting to identify the maximum (and optimal) number of orthogonality conditions (see, for
example,Ahn & Schmidt,1995,1997;Blundell & Bond, 1998; Crepon, Kramarz & Trognon,1998).
Some techniques infrequently used in empirical studies, such as Chamberlain’s (1982) minimum
distance approach, maximum likelihoodestimation (MLE) and Kiviet’s (1995) bias adjusted estima-
tor (amongst others), are not consideredin this paper (althoughMLE might be a viable alternative as
routines are now available in good computer programs, such as S
). A large number of estimators
are considered however,including a general minimum distance one.
398 M.N. HARRIS &L.M
´
ATY
´
AS
In this paper, we survey the majority of the existing mainstream GMM (IV) estimators and com-
pare their small sample performance both in terms of bias and coverage probabilities, using a set of
Monte Carlo experiments.The results suggest that no one estimator clearly dominates. However, the
(nonlinear) GMM estimator and a simple IV estimator based on lagged exogenous variables, tend to
perform well across most settings.
2 The Basic Dynamic Panel Data Model
Our interest lies in the estimation of the population regression function
E
y y y φ (2.1)
where
can contain time-variant and invariant observed characteristics ( and , respectively). As
the typical panel setting is characterised by “small” T and “large” N (an assumption maintained
throughout this paper), it is convenient to treat the cross-sectional observations as independent,
identicallydistributeddrawsfromthe population(Wooldridge,2002).Theimplied sampleregression
function is then
y
it
y
i t
it
φ u
it
u
it i it
i N t T (2.2)
where
i
are the individual (or unobserved) effects and
it
are idiosyncratic error terms. The
usual method of estimating equation (2.2) when there is no lagged dependent variable, consists of
transforming the variables in the model into differences from time means for each individual and
applying OLS—the so-calledfixed-effects (Within) estimator, which is appropriate if the unobserved
effects are (arbitrarily) correlated with the explanatory variables. Alternatively, one could adopt
a random effects approach and estimate the model by feasible generalised least squares (FGLS).
However, when
both of these techniques yield biased and inconsistent estimates (see Nickell,
1981; Sevestre & Trognon, 1985).
3 Orthogonality Conditions and Consistent Estimators
3.1 Strictly Exogenous Explanatory Variables
Arellano&Bover(1995),Arellano&Honor
´
e(2001)andWooldridge(2002)allpresentconvenient
frameworks for consistent estimators of the dynamic panel data model. The following summary is
based on these representations.
The first assumption we make is
E
it i iT
y
i t
y
i i
t T (3.1)
which together with equation (2.2) imply that given
it
, y
i t
y
i
and
i
,
is
has no partial
effect on y
it
for t s; the
it
are strictly exogenous conditional on the unobserved effect, such that
E
y
it i iT
y
i t
y
i i
E y
it it
y
i t i
(3.2)
y
i t
it
φ
i
Note that equation (3.2) is an example of Wooldridge’s (2002) dynamic completeness conditional
on
i
; once one has conditioned on
it
y
i t i
, just one lag of y
it
is sufficient to completely
capture all of the dynamics. Such dynamic completeness also implicitly assumes
E
it i i t
(3.3)
although this is not required for all of the consistent estimators.
Equation (3.2) does not preclude correlations between
i
and
it
, although assumption (3.1)
A Comparative Analysis of Different IV and GMM Estimators 399
implies
E
is
it
s t (3.4)
and
E
i it
t (3.5)
Any correlation between the unobservedeffects and any of the included explanatory variables (be
it from the presence of y
i t
, which is correlated with
i
by construction, or from any elements of
it
) renderstraditionalestimators ofthe modelinconsistent(see, Nickell,1981;Sevestre& Trognon,
1985). Using the partition
it
it it
, where
it
are a subset of
it
that are independent of
i
, suggests that one can base GMM estimation on the orthogonality conditions
E
it
i
(3.6)
where
it
are based on
it
. Note that in the subsequent Monte Carlo experiments (Section 4),
it
is assumed to be an empty set. The optimal (linear) GMM estimator has the weighting matrix
u
(3.7)
where use is made of the traditional error components structure from assumption (3.5) and that of
homoscedasticity, such that E
u
i
u
i
i
E u
i
u
i
u
, that is Var u
i
does not depend on i,
and
is the matrix stacked version of
it
.
Using the same partitions as for
it
, the HT (Hausman & Taylor, 1981) estimator has
i
i
i
; the AM (Amemiya & MaCurdy, 1986) one with
i
i i iT
and the BN
(Balestra & Nerlove, 1966) one with
i
i i t
.
If the correlation between
it
and
i
is constant over time, one can use the further orthogonality
conditions that deviations from time means are also valid instruments (Breusch, Mizon & Schmidt,
1989). However, due to the difficulties of identifying such restrictions in practice, this estimator is
not considered in this paper.
Sevestre & Trognon (1996), ST, also suggest an estimator based on assumption (3.4), STA, which
implies
E
is
it
s t (3.8)
as
i
,
i t
are valid instruments for y
i t
in the differenced equation.
From equation (3.2)
it
is uncorrelated with y
i t
y
i
, but by construction subsequent
values of y
it
cannot be. Wansbeek & Bekker (1996) show, however, that estimation can be based on
orthogonality conditions implied by linear transformations of y
it
defined by the matrix
i
, provided
i
conforms to particular restrictions. Explicitly, consistency requires
E
i
υ E
i
υ tr
i
E υy (3.9)
with
y
i
y
iT
. Harris & M
´
aty
´
as (2000) suggest “operationalising” the estimatorfrom the
simpleautoregressivemodeltoonewith additionally
it
,bynotingthatastheusual transformation
removestheunobservedeffects,
it
cannow be consideredstrictly exogenousbecauseofassumption
(3.4). This defines
i i i
and estimation can again be based on conditions of the form
(3.9). However,
i
is unspecified apart from the noted restrictions. The variance-covariance matrix
of the resulting estimator is a function of
i
and the optimal choice of
i
is taken to be that
which (numerically) minimises the trace of this variance-covariance matrix, whilst simultaneously
ensuring that the necessary restrictions hold. Once
i
is known, WBA has the usual (linear) GMM
form. Following from equation (3.4) an “expanded” estimator can also be considered, WBB, which
has
i i i i
with similarly defined.
400 M.N. HARRIS &L.M
´
ATY
´
AS
3.2 Sequentially Exogenous Explanatory Variables
A weaker assumption than that of (3.1), is that the
it
are Wooldridge’s (2002) sequentially
exogenous conditional on the unobserved effect, such that
E
it it i t i
y
i t
y
i i
t T (3.10)
which implies
E
y
it it i t i
y
i t
y
i i
E y
it it
y
i t i
(3.11)
y
i t
it
i
This appears to be a more realistic assumption as opposed to conditioning on all (past, current and
future) values of
it
. In such a situation, it is convenient to apply a transformation—typically first
differencing—andto find an appropriate set of instruments.
Following from assumption (3.10) one has
E
is
it
s t (3.12)
where
it
y
i t it
, which implies the orthogonality conditions
E
is
it
s t (3.13)
Defining
it
i i it
, then
i t
is an appropriate instrument for
it
in the
differencedequation,as from equation (3.13) it is independentof
it
and obviouslycorrelated with
it
. Moreover, one could also use lagged differences (which are simply linear combinations of the
validinstruments in levels) of
i t
as instruments for
it
. Several estimators are based on these
kind of orthogonality conditions, with the focus being on the y
i t h
subset of
it
. Thus the estimator
ofAnderson&Hsiao(1982),AHA, uses only
y
i t
asaninstrument,whilst thatofArellano (1988),
ARA, only y
i t
. The estimator of Arellano & Bond (1991), ABA, uses the over-identifying set of
sequenceof instruments y
i
y
i
y
i
y
i
y
i T
for successive time periods, in a (linear)GMM
framework, with the weighting matrix taking into account the nature of the transformed disturbance
term
u
it
.
In the simulation study two further variants of the AR and AH estimators are considered(ARB and
AHB) which additionally include
i t
as valid instruments to ensure that the estimator has finite
moments (Kinal, 1980). Also, various two-step variants are also considered (ARC, AHC, ABB, STB,
WBC and WBD, respectively) where the weighting matrix E
i
i
i
i
E
i
u i uZ
(under standard assumptions), is estimated as
uZ
N
N
i
i
i
i
i
(3.14)
Estimators using past values of y
i t
and transformations of this, unlike the earlier ones, specifi-
cally requirethe assumption(3.3),that is that the errorsin first differences,at most,exhibitfirst-order
autocorrelation.
3.3 Additional Moment Restrictions, Non-Linear GMM and Initial Conditions
As shown, many orthogonality conditions are implied by standard assumptions, although the
proposed estimators are (generally) only based upon subsets of these. The ABov estimator (Arellano
& Bover, 1995), for example, uses the additional moment restrictions implied by the assumptionthat
thepredeterminedvariablehas constantcorrelationwiththe unobservedeffects. Ingeneral,important
recent advances have been made with regard to nonlinear GMM estimation of dynamic panel data
models (see, inter alia, Ahn & Schmidt, 1995, 1997; Blundell & Bond, 1998; Crepon, Kramarz
& Trognon, 1998; Ahn & Schmidt, 1999; Breitung & Lechner, 1999) with regard to exploiting
A Comparative Analysis of Different IV and GMM Estimators 401
furtherorthogonalityconditions.Forexample,Ahn&Schmidt(1995)exploitquadraticorthogonality
conditions implied by the lack of serial correlation and theassumption of homoscedasticity, whereas
Crepon, Kramarz & Trognon (1998)focus on the additional linear conditions implied by assumption
(3.1).
In the subsequent analysis, the orthogonality conditions used were those reported in Crepon,
Kramarz & Trognon (1998). No conditions relating to the disturbances of the initial conditionswere
used. From an empirical perspective, one would like to be unconcerned that the initial conditions
have been specified “correctly”. Whilst using the true initial conditions equation within a GMM-
framework would increase efficiency (Blundell & Bond, 1998), due to the risk of an incorrect
specification in practice, such conditions are not considered here. Two nonlinear GMM estimators
are analysed, GMMW which uses the asymptotically efficient weighting matrix (Hansen, 1982) and
GMMI, the identity matrix.
4 The Monte Carlo Experiments
To analyse the small sample performanceof these various estimators, a limited set of Monte Carlo
experiments was conducted. The basic data generating process was of the form
y
it i
y
i t
x
it
x
it
it
(4.1)
where
i
N and
it
N . Following Kiviet (1995) is given by
, with taking the values 1 and 5. The reason for this, is that if for example, the
impact of the disturbances and the individual effects on y
it
are of an equal magnitude. Conversely,
if
, the effect of
i
is double that of
it
, irrespective of any changes in . If this were not the
case, changes in
would also affect the impact ratio of the two error components, thus confounding
the separate effects of
and (Kiviet, 1995). As in Kiviet (1995) and Arellano & Bond (1991) N is
set here to 100, and two values of T are considered, 4 and 6.
Following Kiviet (1995), Arellano & Bond (1991) and others, initial conditions were specified as
y
i i
x
i
i
(4.2)
where the coefficients of the exogenous variables approach their long-run values. The exogenous
variables were generated as
x
k
it
k
x
x
k
i t
uniform (4.3)
with
x
and
x
, in an attempt to coverconfigurationslikely to be encountered in practice.
Thus the former attempts to proxy macro-economic type variables, and the latter the more typical
small T panel datasets, in which it is well-known that stationarity is not an issue. Different distribu-
tional assumptions were made regardingthe stochastic elementsof equation (4.3).This did not affect
the rankings of estimators and the uniform distribution results gave the greatest variance of results.
Giventhe variousparameterisations,sample sizes, and so on, this yielded twelve experimentsin total
(I to XII).
Frequently in practice, explanatory variables are correlated with the unobserved effects. To ascer-
tain the performance of the estimators in such a situation, additional experiments (XIII to XVIII)
were undertaken whereby the
i
were now generated as
i
iid N x
i
(4.4)
That is, the unobserved effects were independent random drawings from a normal distribution with
expected value equal to the time mean of the first exogenous variable, for each individual, and
variance as before. This represents a movementaway from the basic data generating process, in that
the assumed process has been deliberately misspecified. The initial conditions were generated as
402 M.N. HARRIS &L.M
´
ATY
´
AS
before, such that the unobserved effects are correlated with not only the explanatory variables, but
also with the initial conditions.
Thefullrangeof(varying)parametervaluesforall oftheMonteCarloexperimentsaresummarised
in Table 4.1, with M
(the number of Monte Carlo repetitions). (An anonymousreferee has
pointed out that as the signal to noise ratios, defined as
j
x
j
u
, are unaltered in all experiments,
an interesting line of future inquiry would be to relax this, especially as relative increases in
u
may
adversely affect one of the better performing estimators, STA—see Section 5).
Table 4.1
Monte Carlo design.
T E
i
I 4 0.1 1 0
II 4 0.5 1 0
III 4 0.9 1 0
IV 6 0.1 1 0
V 6 0.5 1 0
VI 6 0.9 1 0
VII 4 0.1 5 0
VIII 4 0.5 5 0
IX 4 0.9 5 0
X 6 0.1 5 0
XI 6 0.5 5 0
XII 6 0.9 5 0
XIII 4 0.1 1
x
i
XIV 4 0.5 1 x
i
XV 4 0.9 1 x
i
XVI 6 0.1 1 x
i
XVII 6 0.5 1 x
i
XVIII 6 0.9 1 x
i
5 Monte Carlo Results
For reasons of space, not all of the results are presented here. In particular, attention is only paid
to the estimation of
and also some estimators are disregarded. Estimators are not reported if they:
suffered from computational problems; exhibited excessive small sample bias; or had very similar
performance to other estimators.
Those estimators suffering unduly from excessive small sample bias, were the expanded versions
oftheABandABov ones.This is an importantfinding, as both ofthese are frequentlyused inpractice.
The bias was presumably a function of the excessive number of instruments (see, for example, Arel-
lano & Bover, 1995; Altonji & Segal, 1996). Note, that although the results from the AM estimator
are reported, this could easily be placed in the same bracket (heavy small sample bias and an excess
of instruments). Those that tended to exhibit computational problems were the simple AHA and
ARA estimators, due to the lack of finite moments. Finally, those estimators exhibiting very similar
performance, were the one- and two-step variants of particular estimators (though not for the WB
estimators).
For comparison purposes, several inconsistent traditional panel estimators: simple OLS (OLS1);
Within; FGLS; and OLS on the differenced model (DOLS), are also included (a further inconsis-
tent estimator could also have been considered—generalised least squares on the first differenced
model—which,presumably,wouldhavebetterperformancethan thatofDOLS).In offeringpreferred
estimator(s), focus is primarily on bias (and later on, on coverage probabilities) although some re-
A Comparative Analysis of Different IV and GMM Estimators 403
searchers may prefer to base their choice simply on rootmean squared error (RMSE), as it combines
both bias and variance properties. The Monte Carlo results contain
Mean
M
M
i
i
and RMSE
M
M
i
i
5.1 Experiments I to XII
The results for Experiments I to XII can be found in Table 5.1.
Table 5.1
Monte Carlo results: mean parameter estimates (RMSE in parentheses).
Exp I Exp II Exp III Exp IV Exp V Exp VI
OLS1 0.530 (0.43) 0.691 (0.20) 0.905 (0.02) 0.530 (0.43) 0.694 (0.20) 0.905 (0.01)
Within -0.254 (0.36) 0.010 (0.50) 0.289 (0.62) -0.113 (0.22) 0.199 (0.31) 0.528 (0.37)
FGLS 0.560 (0.47) 0.704 (0.21) 0.905 (0.02) 0.490 (0.40) 0.696 (0.20) 0.904 (0.01)
HT 0.129 (0.28) 0.488 (0.16) 0.899 (0.03) 0.146 (0.23) 0.506 (0.13) 0.900 (0.02)
AM 0.450 (0.46) 0.624 (0.21) 0.904 (0.03) 0.666 (0.61) 0.720 (0.25) 0.906 (0.02)
BN 0.088 (0.34) 0.496 (0.15) 0.900 (0.03) 0.094 (0.27) 0.497 (0.13) 0.900 (0.03)
STA 0.101 (0.37) 0.502 (0.32) 0.902 (0.29) 0.110 (0.23) 0.506 (0.19) 0.903 (0.15)
WBA 0.142 (0.21) 0.661 (0.18) 0.907 (0.02) 0.075 (0.08) 0.449 (0.10) 0.901 (0.02)
WBB 0.115 (0.20) 0.652 (0.18) 0.907 (0.02) 0.075 (0.08) 0.450 (0.10) 0.900 (0.02)
WBC 0.085 (0.13) 0.481 (0.16) 0.880 (0.10) 0.090 (0.08) 0.469 (0.10) 0.861 (0.09)
WBD 0.069 (0.14) 0.463 (0.15) 0.865 (0.11) 0.080 (0.09) 0.458 (0.10) 0.859 (0.08)
DOLS -0.425 (0.53) -0.202 (0.70) 0.046 (0.86) -0.429 (0.53) -0.211 (0.71) 0.047 (0.85)
AHB 0.072 (0.25) 0.345 (0.52) 0.489 (0.93) 0.092 (0.15) 0.456 (0.28) 0.763 (0.55)
ARB 0.090 (0.13) 0.482 (0.16) 0.874 (0.22) 0.098 (0.08) 0.495 (0.10) 0.892 (0.12)
AB 0.087 (0.13) 0.478 (0.17) 0.866 (0.25) 0.087 (0.08) 0.472 (0.10) 0.852 (0.12)
GMMI 0.111 (0.13) 0.493 (0.15) 0.882 (0.08) 0.114 (0.09) 0.495 (0.09) 0.881 (0.05)
GMMW 0.120 (0.12) 0.504 (0.12) 0.893 (0.03) 0.112 (0.08) 0.498 (0.09) 0.886 (0.04)
Exp VII Exp VIII Exp IX Exp X Exp XI Exp XII
OLS1 0.962 (0.86) 0.970 (0.47) 0.962 (0.06) 0.962 (0.86) 0.970 (0.47) 0.961 (0.06)
Within -0.254 (0.36) 0.010 (0.50) 0.289 (0.62) -0.113 (0.22) 0.199 (0.31) 0.528 (0.37)
FGLS 0.983 (0.88) 0.979 (0.48) 0.961 (0.06) 0.987 (0.89) 0.983 (0.48) 0.959 (0.06)
HT 0.432 (0.61) 0.611 (0.32) 0.896 (0.03) 0.383 (0.51) 0.612 (0.25) 0.898 (0.03)
AM 0.930 (0.84) 0.922 (0.44) 0.922 (0.04) 0.982 (0.88) 0.967 (0.47) 0.928 (0.04)
BN 0.635 (0.98) 0.649 (0.68) 0.900 (0.03) 0.536 (1.57) 0.604 (0.47) 0.899 (0.03)
STA 0.101 (0.37) 0.502 (0.32) 0.902 (0.29) 0.110 (0.23) 0.506 (0.19) 0.903 (0.15)
WBA 0.956 (0.86) 0.969 (0.47) 0.963 (0.06) 0.752 (0.71) 0.944 (0.45) 0.959 (0.06)
WBB 0.958 (0.86) 0.971 (0.47) 0.964 (0.06) 0.728 (0.70) 0.945 (0.45) 0.959 (0.06)
WBC 0.328 (0.39) 0.758 (0.36) 0.953 (0.07) 0.215 (0.17) 0.544 (0.14) 0.866 (0.10)
WBD 0.242 (0.35) 0.536 (0.29) 0.930 (0.09) 0.212 (0.19) 0.551 (0.14) 0.861 (0.09)
DOLS -0.425 (0.53) -0.202 (0.70) 0.046 (0.86) -0.429 (0.53) -0.211 (0.71) 0.047 (0.85)
AHB 0.072 (0.25) 0.345 (0.52) 0.489 (0.93) 0.092 (0.15) 0.456 (0.28) 0.763 (0.55)
ARB 0.065 (0.27) 0.455 (0.31) 0.860 (0.27) 0.087 (0.17) 0.482 (0.21) 0.888 (0.16)
AB 0.083 (0.22) 0.413 (0.41) 0.843 (0.37) 0.080 (0.10) 0.424 (0.17) 0.820 (0.16)
GMMI 0.387 (0.47) 0.667 (0.30) 0.891 (0.09) 0.513 (0.50) 0.751 (0.30) 0.889 (0.06)
GMMW 0.415 (0.46) 0.716 (0.30) 0.903 (0.04) 0.496 (0.49) 0.757 (0.30) 0.895 (0.05)
As , simple OLS is consistent (Sevestre & Trognon, 1985) and in such situations, either
OLS1orindeed,FGLS, couldwell be preferredin terms of simplicity,especiallywhen
is small
(Experiments III and VI, although,of course, this could be simply due to the particular experimental
design). Increasing the ratio of these variances to
(Experiments IX and XII), does marginally
adversely effect the performance of both OLS1 and FGLS. However, in all of these scenarios, there
is effectively very little to choose between any of the consistent estimators, which all have extremely
satisfactory performance both in terms of bias and RMSE, with the notable exception of AHB (and
404 M.N. HARRIS &L.M
´
ATY
´
AS
possibly also, AB). Considering these results, the cases where appear to be of more interest.
With small
the inconsistent estimators can clearly be disregarded (Experiments I, IV, VII
and X). When
and T many estimators can be disregarded in terms of excessive
bias, and/or dispersion. However, both nonlinear GMM estimators perform very well, with very
little bias and variability. Indeed, these would be the preferred estimators for this scenario, although
both the ARB and AB estimators have reasonable (and similar) performance, as do all variants of
the WB estimator. Raising T to
improves most of the estimators’ performance, at least in terms of
variability, but generally also in terms of bias (most noticeably all of the WB variants).The nonlinear
GMM-type estimators again perform well, although clearly thebest performingestimator is the ARB
one, with effectively zero bias
and small RMSE.
Holding
at but letting the standard deviationof the unobservedeffects being times that of
the idiosyncratic disturbance terms, clearly has strong adverse consequences on all of the estimators’
performance (most noticeable for those operating in levels i.e., the BN, HT, AM, WBA, WBB, WBC,
WBD, GMMW and GMMI ones—Experiments VII and X). There is little to choose from between
the first differenced estimators of: AHB; ARB; STA; and AB, of which STA has the smallest bias,
but the largest variability. Although increasing T in this scenario does improve performance across
the board (Experiment X compared to VII), the choice remains one between the first differenced
estimators AHB, ARB, STA and AB.
So, as
it appears to make little difference what estimator is chosen (even OLS), above
caveats noted. On the other hand, as
, the first differenced estimators of AHB, ARB, STA and
AB, tend to stand out. What then, of the intermediate case, when
(Experiments II, V, VIII
and XI)?
With
, the nonlinear GMM-type estimators again perform very well, with negligible
bias and small RMSE. Now those estimators using lagged x’s and transformations of such as
instruments, also appear to perform quite well, most notably the BN, HT and STA estimators.
Performance of the two-step WB estimators (WBC and WBD) is quite good, although the one-step
variants (WBA and WBB) tend to over-estimate the true parameter value by about
. Increasing
T to 6 again generally benefits all of the estimators, primarily in the form of lower RMSE’s. Several
estimators have effectivelyzero averagebias and verysmall RMSE’s (BN, HT, STA, ARB, AB, GMMI
and GMMW). It is hard though, to go past the GMMW estimator, with an average bias of
and
RMSE of
.
Holding
at , but increasing (to ) once more adversely affects the performance of
the estimators, in some cases quite dramatically, especially in the small T sample. The least affected
are those estimators operating in first differences, although only two estimators that offer anywhere
near reasonable performance,are the STA and WBD ones. Finally, increasing T to 6, again generally
improvestheestimators’performance.Intermsofbias,theSTAestimatorclearlydominates,although
both of the WBC and WBD estimators have reasonable bias properties, but with lower RMSE’s than
that of the former.
Overall,then, all estimatorsare adverselyaffected by increasing
relative to , but benefit from
an increase in the temporal sample size. At the extreme of the likely values of
, the first differenced
estimators of AHB, ARB, STA and AB appear appropriate. For an intermediate value of
, many
estimators have reasonable performance (for example,GMMW), although notably STA, ARB and AB
again all have desirable finite sample properties. Indeed, any of these three estimators appear to be
an appropriate choice for the applied researcher in a variety of settings, and are, moreover,relatively
simple to execute.
An interesting question is how do these results compare to previous ones (see, inter alia, Arellano
& Bond, 1991;Arellano & Bover, 1995; Kiviet, 1995)? In all these studies, the simple AH estimator
faredpoorly. When it was considered, the AR onehad much better performance,as did the AB one(s),
although often their relative performance was quite close. That is, the AB estimator(s) generally
A Comparative Analysis of Different IV and GMM Estimators 405
outperformed the AR one(s), which in turn outperformed the AH one(s), although often the latter
two exhibited very close performance.The results of this papergenerally concur with these findings.
Here many more estimators are considered such that the AR and AB ones are often comfortably
surpassed by ST, nonlinear GMM and WB estimators, for example.
5.2 Experiments XIII to XVIII
The results for Experiments XIII to XVIII, where the unobserved effects are correlated with an
explanatory variable, can be found in Table 5.2. Note that estimators were constructed as if this
correlation were nil (c.f. the HT, AM and nonlinear GMM estimators).
Table 5.2
Monte Carlo results: mean parameter estimates (RMSE in parentheses).
Exp XIII Exp XIV Exp XV Exp XVI EXP XVII Exp XVIII
OLS1 0.569 (0.47) 0.726 (0.23) 0.933(0.03) 0.582 (0.48) 0.743 (0.25) 0.938(0.04)
Within -0.235 (0.34) 0.048 (0.46) 0.359(0.55) -0.103 (0.21) 0.224 (0.28) 0.590(0.31)
FGLS 0.610 (0.51) 0.742 (0.25) 0.933(0.03) 0.578 (0.48) 0.761 (0.26) 0.938(0.04)
HT 0.513 (0.44) 0.739 (0.25) 0.944 (0.05) 0.606 (0.52) 0.789 (0.29) 0.952 (0.05)
AM 0.586 (0.50) 0.762 (0.27) 0.950 (0.05) 0.706 (0.61) 0.806 (0.31) 0.952 (0.05)
BN 0.417 (0.34) 0.679 (0.19) 0.936 (0.04) 0.414 (0.33) 0.683 (0.19) 0.937 (0.04)
STA 0.095 (0.24) 0.496 (0.21) 0.897 (0.19) 0.099 (0.16) 0.499 (0.13) 0.898 (0.11)
WBA 0.552 (0.45) 0.720 (0.22) 0.934 (0.04) 0.129 (0.09) 0.658 (0.17) 0.938 (0.01)
WBB 0.547 (0.45) 0.720 (0.22) 0.935 (0.04) 0.150 (0.09) 0.617 (0.14) 0.958 (0.06)
WBC 0.153 (0.15) 0.549 (0.17) 0.935 (0.06) 0.143 (0.10) 0.598 (0.13) 0.940 (0.05)
WBD 0.174 (0.15) 0.581 (0.15) 0.953 (0.06) 0.153 (0.10) 0.603 (0.13) 0.960 (0.06)
DOLS -0.425 (0.53) -0.162 (0.67) 0.119 (0.78) -0.411 (0.51) -0.179 (0.68) 0.123 (0.78)
AHB 0.074 (0.23) 0.369 (0.50) 0.751 (0.70) 0.097 (0.15) 0.472 (0.27) 0.845 (0.35)
ARB 0.085 (0.14) 0.478 (0.18) 0.876 (0.21) 0.095 (0.09) 0.490 (0.13) 0.890 (0.14)
AB 0.082 (0.15) 0.462 (0.21) 0.858 (0.28) 0.081 (0.09) 0.448 (0.13) 0.824 (0.15)
GMMI 0.223 (0.19) 0.604 (0.17) 0.936(0.08) 0.269 (0.20) 0.649 (0.17) 0.940 (0.06)
GMMW 0.256 (0.21) 0.659 (0.18) 0.936(0.04) 0.255 (0.18) 0.662 (0.18) 0.941 (0.05)
In all of these experiments, increasing T again generally benefits all of the estimators, primarily
so in terms of reduced variability. A large
effectively cancels out the bias instigated by this
correlation between the unobserved effects and x. With small T, for computational ease, one could
opt again for OLS1 (or FGLS). Of the consistent estimators, any of the BN, HT, AM and WB ones,
might be preferred over the ARB, STA, AB, GMMI and GMMW ones, in terms of tighter sampling
distributions. The same first difference estimators continue to perform well when T is increased to
6, but are dominated by most of the levels estimators in terms of variability. For example, STA has
the smallest average bias (effectively zero), but a RMSE which is more than double that of most of
the levels estimators.
When
is small , many of the estimators severely overestimate the true parameter value,
especially when T is also small (BN, HT, AM, WBA, WBB, GMMI and GMMW). This excessive bias
is not surprising as most of these estimators specifically rely on the (assumed) independence of the
explanatory variables and the unobservedeffects. Not surprisingly, the favoured ones here would be
thosethat operateinfirst differences,therebyeliminating the
i
. The performanceof these estimators
is very similar to that observed in those previous experiments where
. In particular, all of the
AHB, ARB, STA and AB estimators have very small (average) bias and with little variation.
The performance of most of the estimators appears to improve as
moves away from . Again
the ones operating in levels (including the GMM ones), tend to exhibit positive bias, but not as
noticeable as for small
. However, their poor performance appears to be relatively independent of
the temporal sample size, which is to be expected as the correlation between the unobserved effects
406 M.N. HARRIS &L.M
´
ATY
´
AS
and x, is not decreasing in T. The STA estimator is the star performer here, with effectively zero
(average) bias and a tight sampling distribution.
Summarising these additional results, the choice of estimator appears to be heavily dependent on
the magnitude of the true value of
, but essentially independent of the temporal sample size. For
large values of
, any of the consistent estimators (or even OLS1 or FGLS) could quite confidently
be used. In the case of smaller values, applied researchers could do well to look to those estimators
which remove the unobserved effects, namely first difference estimators (or, in certain situations,
variants of the WB estimator).
5.3 Coverage Probabilities
Of great importance to the applied researcher is inference based upon estimated parametervalues.
Given a choice between two equally small-sample biased estimators, preference would be given to
that which yielded better inference in the sense of t-tests of
. To answer this question,empirical
coverageprobabilities (of confidence intervals at 95% nominal levels) were constructed. The results
are presented in Table 5.3 (although not presented here, some researchers may also be interested in
confidence intervals around the long-run multiplier effects).
The results suggest, in accordance with some previous results, that the standard errors of the
two-step variants of estimators appear to be unreliable (Arellano & Bond, 1991) and also that the
severelybiased estimators also tend to havepoor coverageprobabilities. Of the consistentestimators,
notwithstanding the earlier comments regarding the two-step variants, for
(Experiments I to
VI) only the AM, STB, WBC, WBD,GMMI and GMMW estimators havepoor coverageprobabilities,
whilst increasing
to 5 (VII to XII) adversely affects most of the estimators’ performance. In
summary, those estimators that have reasonable empirical size over most scenarios are: HT; AHB;
ARB; STA; STB; and ABA, but most notably the STA estimator (with a range of just 93% to 95%).
Those estimators with noticeably poor coverage probabilities are all variants of WB and GMM
estimators.
Table 5.3
Empirical coverage probabilities of confidence intervals at 95pc nominal levels.
Experiment Number
I II III IV V VI VII VIII IX X XI XII
OLS1 0.00 0.01 0.94 0.00 0.00 0.92 0.00 0.00 0.00 0.00 0.00 0.00
Within 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
FGLS 0.00 0.01 0.81 0.00 0.00 0.78 0.00 0.00 0.00 0.00 0.00 0.00
HT 0.93 0.95 0.95 0.95 0.95 0.95 0.73 0.85 0.95 0.76 0.87 0.95
AM 0.67 0.84 0.94 0.29 0.54 0.94 0.05 0.13 0.88 0.00 0.01 0.77
BN 0.96 0.96 0.96 0.96 0.96 0.96 0.68 0.83 0.96 0.72 0.85 0.96
STA 0.94 0.93 0.93 0.95 0.95 0.95 0.94 0.93 0.93 0.95 0.95 0.95
STB 0.82 0.82 0.81 0.81 0.83 0.82 0.82 0.82 0.81 0.81 0.83 0.82
WBA 0.37 0.02 0.92 0.99 0.96 0.94 0.00 0.00 0.00 0.01 0.00 0.00
WBB 0.48 0.02 0.92 1.00 0.97 0.93 0.00 0.00 0.00 0.01 0.00 0.00
WBC 0.16 0.15 0.14 0.13 0.12 0.14 0.17 0.08 0.07 0.09 0.12 0.15
WBD 0.15 0.17 0.13 0.14 0.14 0.12 0.18 0.19 0.09 0.08 0.13 0.14
DOLS 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
AHB 0.93 0.89 0.86 0.95 0.92 0.91 0.93 0.89 0.86 0.95 0.92 0.91
AHC 0.14 0.12 0.12 0.12 0.15 0.14 0.14 0.12 0.12 0.12 0.15 0.14
ARB 0.95 0.93 0.94 0.95 0.95 0.94 0.93 0.93 0.93 0.95 0.95 0.94
ARC 0.15 0.16 0.17 0.15 0.16 0.15 0.16 0.16 0.15 0.16 0.16 0.17
ABA 0.94 0.93 0.94 0.95 0.93 0.92 0.94 0.91 0.94 0.94 0.90 0.92
ABB 0.14 0.15 0.15 0.12 0.13 0.14 0.15 0.14 0.14 0.12 0.11 0.14
GMMI 0.90 0.93 0.67 0.87 0.91 0.70 0.26 0.45 0.60 0.07 0.18 0.60
GMMW 0.74 0.74 0.70 0.53 0.48 0.34 0.31 0.30 0.66 0.05 0.08 0.36
A Comparative Analysis of Different IV and GMM Estimators 407
It is important however, not to overstress the importance of these coverage probability results due
to the potential bias/variance trade-off.A biased estimator with a large standard error, may appearto
exhibit favourable coverage probabilities, and vice versa (which appears to be the case for the GMM
and WB estimators). Arguably, an applied researcher would favour the latter.
6 Conclusions
This paper has been a summary of the most common linear (IV) and nonlinear GMM estimators
of the dynamic panel model. A set of Monte Carlo experiments was conducted to evaluate the small
sample performance, in the empirically relevant cases, of these estimators. Although, obviously, it
is impossible to cover all relevant situations by simulations, the key findings of the experiments can
be summarised as follows.
Those estimators that use very large instrument matrices (AM, expanded AB and ABov) tend to
suffer unduly from the resulting small sample bias. Also, if possible, one should use estimators
that have an excess number of instruments over endogenous variables to ensure that the resulting
estimator has finite moments (c.f. the simple and expanded AH and AR estimators). If there are
several variants of a particular estimator, for example one- and two-step ones, their performance is
likely to be very similar, such that it may be sufficient to consider just one of them. Moreover, the
standard errors of the two-step variants tend to be unreliable.
In general, all estimators’ performance increases with the sample size in T. This is especially so
for the AH estimator, such that it probably should not be considered for very short panels (say four
or five periods). Again, in general, all estimators’ performance decreases as the ratio of the variance
of the unobserved effects to that of the disturbance term increases, but appears to increase as
.
It is difficult to proffera preferredestimator,as performance across the experimentswas generally
quitevolatile.However,the nonlinearGMM-typeestimatorsgenerallyhadgoodperformance(except
when the variance of the unobserved effect was relatively large). The performance of the WB
estimators also appears quite favourable and robust across specifications. Both of these estimators
may prove computationally burdensome though, therefore a more attractive option might be any of
the ARB, ABA or STA estimators.
In terms of coverage probabilities, many of the empirical confidence intervals were very close to
the theoretical values, although those estimators with good bias properties often had poor empirical
coverageprobabilities (WB and GMM). That is, in choosinganappropriateestimator, one should first
focusonthosewithbetterbiaspropertiesand fromthese select one (or more),that also appeartohave
accurate empirical confidence intervals (coverage probabilities, by themselves, can be misleading).
An estimator which appears to perform well across most scenarios in terms of average bias (and the
variability of this) and coverage probabilities is the STA one, which looks an appropriate choice for
the applied researcher in many instances. Moreover, it also performed well when the unobserved
effects and explanatory variables were correlated, a scenario likely to occur frequently in practice.
Acknowledgements
We would like to thank one anonymous referee and Elja Arjas, Patrick Sevestre, Chris Cornwell,
John Earle, Max Gillman and, in particular, Jeffrey Wooldridge, for their helpful comments and
suggestions. We would also like to thank Attila Pataki for help in transcribing Gauss code into C and
Preety Ramful for research assistance. The usual caveats apply.
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R
´
esum
´
e
Il est maintenant bien connu que les estimateurs usuels de mod
´
eles de panel ne sont pas convergents lorsque le mod
´
ele est
de type autor
´
egressif. Toutefois, unnombre important d’estimateurs convergents ont
´
et
´
e propos
´
es dans la litt
´
erature. Cet article
propose un survey de la plupart de ces estimateurs qui, pour l’essentiel, reposent sur la m
´
ethode des variables instrumentales
ou celle des moments g
´
en
´
eralis
´
es. On consid
´
ere
´
egalement une extension de l’estimateur propos
´
e par Wansbeek et Bekker.
Afin d’aider les
´
econom
´
etres appliqu
´
es travaillant sur des panels de donn
´
ees micro-
´
economiques
´
e faire le meilleur choix, les
propri
´
et
´
es
´
e distance finie de ces estimateurs sont
´
etudi
´
ees
´
e partir de simulations de Monte-Carlo.
[Received June 2003, accepted December 2003]
