Symmetrically
Normalized
Instrumental-Variable
Estimation
Using
Panel
Data
César ALONSO-BORREGO
Universidad Carlos
111
de Madrid, 28903 Getafe, Madrid, Spain (alonso1@eco.uc3m.es)
Manuel ARELLANO
CEMFI, 28014 Madrid, Spain (arellano@cemfi.es)
We
discuss the estimation
of
linear panel-data models with sequential moment restrictions using
symrnetrically normalized generalized method
of
moments (GMM) estimators (SNM) and limíted
information maximum likelihood (LIML) analogues. These estimators are asymptotically equivalent
to standard GMM but are invariant to normalization and tend to have a smaller finite-sample bias,
especially when the instruments are poor.
We
study theír properties in relation to ordinary GMM
and mínimum distance estimators for AR(1) models with individual etfects by mean
of
simulations.
Finally, as empírical illustrations, we estimate by SNM and LIML employment and wage equations
using panels
of
U.K. and Spanish firms.
KEY WORDS: Autoregressive models; Dynamíc panel data; Employment equations; Generalized
method
of
moments; Monte Cario methods; Symmetric normalization.
This work is motivated by a concern with the finite-
sample bias in panel data instrumental-variable (IV) esti-
mators when the instruments are weak. A linear panel-data
model with predetermined variables [like vector autoregres-
sions (VAR's) or linear Euler equations] is typically esti-
mated by IV techniques in first-differences using
aH
the
available lags
of
the predetermined variables as instruments.
The specification
of
the equation error in first-differences
reflects the fact that the analysis is conditional on an un-
observable individual effect. Because the number
of
instru-
ments increases with the time series dimension
(T), the
model generates many overidentifying restrictions even for
moderate values
of
T,
although the quality
of
these instru-
ments is often poor.
The effect
of
weak instruments on the distributions
of
two-stage least squares (2SLS) and limited information
maximum likelihood (LIML) differs substantiaHy, despite
the fact that both estimators have the same asymptotic dis-
tribution. Although the distribution
of
LIML is centered at
the parameter value, 2SLS is biased toward ordinary least
squares (OLS), and in the completely
unidentified case con-
verges to a random variable with the OLS probability limit
as its central value. On the other hand, LIML has no
fi-
nite moments regardless
of
the sample size, and as a con-
sequence its distribution has thicker tails than that
of
2SLS
and a higher
probability
of
extreme values [see Phillips
(1983) for a good survey
of
the literature]. As a result
of
nu-
merical comparisons
of
the two distributions involving me-
dian bias, interquartile ranges, and rates
of
approach to nor-
mality, Anderson, Kunitomo, and Sawa (1982) concluded
that LIML was to be strongly preferred to 2SLS, particu-
larly
if
the number
of
instruments is large. Similar conclu-
sions emerge from the results
of
asymptotic approximations
based on an increasing number
of
instruments as the sam-
pIe size tends to infinity; under these sequences, LIML is
a consistent estimator but 2SLS is inconsistent (Kunitomo
1980; Morimune 1983; and, more recentIy, Bekker 1994).
36
(In our context, these approximations would amount to al-
lowing
T to increase to infinity at a chosen rate as opposed
to the standard fixed
T,
large N asymptotics.)
Despite this favorable evidence, LIML has not been used
as much in applications as IV estimators. In the past,
LIML was at a disadvantage relative to 2SLS on compu-
tational grounds. More fundamentaHy, applied econometri-
cians have often regarded 2SLS as a more "flexible" choice
than LIML from the point
of
view
of
the restrictions they
were willing to impose on their models. In effect, the IV
techniques used for a panel-data model with predetermined
instruments are not standard 2SLS estimators because the
model gives rise to a system
of
equations (one for each time
period) with a different number
of
instruments available
for each equation. Moreover, concern with heteroscedas-
ticity has led to considering alternative ("two-step") gener-
alized method
of
moments (GMM) estimators that use
as
weighting matrix more robust estimators
of
the variances
and covariances
of
the orthogonality conditions (foHowing
the work
of
Chamberlain 1982; Hansen 1982; White 1982).
In a recent article, Hillier (1990) showed that the al-
ternative normalization rules adopted by LIML and 2SLS
are at the root
of
their different sampling behavior. Hillier
also showed that a syrnmetrically normalized 2SLS esti-
mator has properties similar to those
of
LIML. This re-
sult motivates our focus on symmetricalIy normalized esti-
mation. Syrnmetrically normalized 2SLS, unlike LIML, is
a GMM estimator based on structural-form orthogonality
conditions, and it therefore can be readily extended to two-
step weighting matrices and the nonstandard IV situations
that are
of
interest in dynarnic panel-data models, while re-
lying on standard GMM asymptotic theory. In this article,
we discuss both nonrobust and robust LIML analogues and
1
Alonso-Borrego and Arellano: Symmetrically Normalized IV Estimation
symmetrically normalized
GMM
estimates in the panel-data
context.
The symmetrically normalized estimator can be de-
scribed in a simple example as follows. Consider a struc-
tural equation with a single endogenous explanatory vari-
able and a matrix
of
instruments Z,
y =
f3
0
x + u,
with associated reduced-form equations
y
=
Z7r
o
+
VI
X =
Z"fo+V2.
(1)
(2)
Both symmetrically normalized 2SLS and LIML are least
squares estimators
of
the reduced form (2) imposing the
overidentifying restrictions
7r =
"ff3.
Let
us define
(fiv,
i'v) =
arg
min ( y -
Zz"f
13
)'
{3,'Y
X - "f
X
(V-I
®
1)
( Y - Z"ff3 )
X-Z"f
=
argmin
(
nA
-
"ff3
)'
{3,'Y
"f -
"f
X
(V-I
®
Z'
Z)
(
nA
-
"ff3
) . (3)
"f-"f
Concentrating
"f
out
of
the least squares criterion, we obtain
13
- - .
(y-f3x),Z(Z'Z)-IZ'(y_f3x)
(4)
v -
argmJn
(1,
-f3')V(l,
-13')'
.
LIML
is
fiv
with V equal to the reduced-form resid-
ual covariance matrix, whereas symmetrically normalized
2SLS is
fiv
with V equal to an identity matrix (Malinvaud
1970; Goldberger and OIkin 1971; Keller 1975; Anderson
1976) so that both
LIML
and symmetrically normalized
2SLS solve minimum eigenvalue problems. Symmetrically
normalized 2SLS can also
be
described as a
GMM
estima-
tor
based
on
the unit-Iength orthogonality conditions
Note that the asymptotic distribution
of
fiv does not de-
pend on the choice
of
V because optimal minimum dis-
tance estimators (MDE)
of
13
based
on
(n -
"ff3,
i'
- "f) and
on
(n -
i'(3)
are asymptotically equivalent. Note also that
ordinary and symmetrically normalized 2SLS are given, re-
spectively, by the ordinary and the orthogonal regressions
of
y
on
x(Y =
Zn
and x = Zi'), and although the former
differs from indirect 2SLS (the inverse regression
of
x
on
y),
the latter is invariant to normalization.
This article is organized as follows. Section 1 develops
the relationship between symmetrically normalized
GMM
(SNM) and
LIML
in the context
of
a linear equation
for panel data with sequential moment restrictions. We
also present two-step
SNM
estimators and test statistics
of
overidentifying restrictions and compare them with ro-
bust
LIML
analogues.
The
latter are the "continuously up-
37
dated
GMM"
estimators considered by Hansen, Heaton, and
Yaron (1995). Section 2 compares the finite-sample proper-
ties
of
SNM
and
LIML
to those
of
ordinary
GMM
and
MDE
for first-order autoregressive [AR(1)] models with
individual effects. Section 3 reestimates the employment
equations for a sample
of
U.K. firms reported
by
Arellano
and Bond (1991) using SNM, LIML, and indirect
GMM
es-
timators. This section further illustrates the techniques by
presenting symmetrically normalized estimates and boot-
strap confidence intervals
of
employment and wage VAR's
from a larger panel
of
Spanish firms. Finally, Section 4 con-
eludes.
1.
SYMMETRICALLY NORMALlZED
IV
ESTIMATION
Consider a model with individual effects for panel data
given by
Yit =
x~t80
+ Uit,
Uit
=
r¡i
+
Vito
t =
1,
...
,Tj
i =
1,
...
,N,
(6)
The model specifies sequential moment conditions
of
the
form
E(vitlzf)
=
O,
where
zf
=
(zh
...
Z~t)'
is a vector
of
instruments, which may inelude current
or
lagged values
of
Yit and Xit. Thus, this setting is sufficiently general to cover
models with strictly exogenous, predetermined, and endoge-
nous explanatory variables. Observations across individuals
are assumed to be independent and identically distributed.
Estimation will be based
on
a sequence
of
orthogonality
conditions
of
the form
t =
1,
...
, T
-1,
where starred variables denote forward orthogonal devia-
tions
of
the original variables (Arellano and Bover 1995).
It is convenient to rewrite the transformed model as
(8)
where Yi = (Yil
...
Y:(T-I))"
and so forth.
The
k x 1 parameter vector 8
0
is usually estimated by
GMM
leading to estimators
of
the form (Ho1tz-Eakin,
Newey, and Rosen 1988; Arellano and Bond 1991; Cham-
berlain 1992; Arellano and Bover 1995; Ahn and Schmidt
1995)
where
y*
= (yi'
..
.
Yl$)',
X* =
(Xi'
..
. X;¡)', and Z =
(Zi
...
Z~)'.
Zi
is a
(T
-1)
x q block diagonal matrix whose
tth block is
zf
and an optimal choice
of
AN
is such that
it is a consistent estimate
of
the inverse
of
E(Z;uiui'
Zi).
Under "classical" errors [i.e., when
E(v~tlzf)
=
0'2
and
E(VitVi(t+j)lzf)
= O for j > O and all t],
E(Z~uiui'Zi)
=
0'2
E(Z;Zi),
and hence the "one-step" nonrobust choice
AN
=
(a-
2
Z'
Z)-I
is optimal
(a-
2
,
which denotes the residual
variance, is irrelevant for estimation, but it is kept here for
notational convenience). Altematively, the standard "two-
step" robust choice is
AN
=
(¿i
Z;üiüi'
Zi)-l,
where
üi
is a vector
of
residuals evaluated using sorne prelimi-
nary consistent estimate
of
8
0
•
Given identification,
8GMM
is consistent and asymptotically normal as N -+
00
for
2
38
fixed T. In addition, for either choice
of
A
N
,
provided the
conditions under which they are optimal choice s are satis-
fied, a consistent estimator
of
the asymptotic variance
of
8
GMM
is
Vai'(8
GMM
) =
(X*'ZA
N
Z'X*)-I.
Moreover, let-
ting
u* =
y*
- X*8
GMM
, the Sargan or GMM statistic
of
overidentifying restrictions is given by
S
-
'*'ZA
Z"*
~
X
2
- U N U
----r
q-k'
(10)
Now, partition X*
=
(Xi,
X2')
and 8
0
=
(8~1'
8~2)'
to dis-
tinguish between nonexogenous and exogenous variables,
such that the
k
2
columns
of
X2'
are linear combinations
of
those
of
Z but the k
1
columns
of
Xi
are not. SNM is the
GMM estimator
of
8
0
based on the orthogonality conditions
Journal of Business & Economic Statistics, January 1999
is a minimized optimal GMM criterion it can be used as an
alternative test statistic
of
overidentifying restrictions.
We
have that
" , - d 2
(1
+
81SNM81SNM).x
--+
Xq-k,
(17)
which is asymptotical1y equivalent to the Sargan test.
Let us now turn to consider LIML analogues or "con-
tinuously updated GMM" estimators in the terminology
of
Hansen et al. (1996). The nonrobust LIML analogue 8
LI
MLl
minimizes a criterion
of
the form
l(8) = (y* -
X*8)'ZA
N
(8)Z'(y* -
X*8)
(18)
with
AN(8) =
(Z'
Z)-1
/(y*
-X*8)'(y*
-X*8).
The result-
ing estimator is
[
Z;(Yi -
X;8
0
)]
E'l/Ji(8
0
)
= E
(1
+
8~1801)1/2
=
O.
(11) 8
LIML1
=
[X*
Z(Z'
Z)-1
Z'
X* -
ix*'
X*t
1
Because
E['l/Ji(8
0
)'l/J;(8
0
))
=
E(Z;uiui'Zi)/(l
+
8~1801),
a
consistent estimate
of
the inverse
of
E(Z;uiui'
Zi)
remains
an optimal weighting matrix for the SNM estimator. There-
fore,
, . (y* -
X*8)'
M(y*
-
X*8)
8
SNM
=
argmm
(8'8
)
,(12)
6 1 + 1 1
where M =
ZANZ'.
Minimizing the criterion with respect
to 8
2
we obtain a concentrated criterion that only depends
on 8
1
. This gives us
8
1
s
N
M
=
argmind~Wi'(M
-
M2)Widt!d~dl
6
1
=
[Xt(M
- M
2
)Xi
-
,Ü)-IXt(M
- M2)Y*
(13)
and
8
2SNM
=
(X2"MX;)-IX:;'M(y*
-Xi8
1SNM
), (14)
where
Wi
= (y*,
Xi),
dI =
(1,
-8D',
M
2
= MX2'
(X2"MX2)-
I
X2"M,
and X =
mineigen[Wi'(M
-M
2
)Wi).
Notice also that
X = min(y* -
X*8)'M(y*
-
X*8)/(1
+
8~81)'
(15)
EquivalentIy,
8
SNM
=
(X*'MX*
-
Xtl.)-IX*'My*
(16)
with
tl. =
(I~1
~)
[if no columns
of
X* are perfectIy predictable from Z, or
if
the entire vector
of
coefficients is normalized to unity,
then
tl. = 1 and X = min eigen(W*'
MW*),
with
W*
=
(y*,
X*)).
In the just-identified case, X =
O,
with the result
that GMM and SNM coincide.
Because
8GMM
and
8S
NM
are asymptotically equivalent,
Vai'(8
GMM
) is also a consistent estimate
of
the asymp-
totic variance
of
8S
N
M.
An alternative natural estimator
of
var(8
SNM
),
however, suggested by the previous expression,
is
Vai'(8
SNM
) =
(X*'MX*
-
Xtl.)-I.
Moreover, because X
X
[X*'
Z(Z'
Z)-1
Z'y*
-
ix*'y*)'
(19)
where, letting d =
(1,
-8')',
i =
mind'W*'Z(Z'Z)-IZ'W*d/(d'W*'W*d)
= mineigen[W*'
Z(Z'
Z)-1
Z'W*(W*'W*)-I).
(20)
On the other hand, the robust LIML analogue
8LIML2
minimizes a criterion
of
the same form as (18) with
AN(ój
~
(t,Z;U;(ójUi(ój'Z;)
-, (21)
where
ui(8)
=
yi
-X;8.
Therefore, LIML2, unlike LIML1
or the SNM estimators, does not solve a simple minimum
eigenvalue problem and requires the use
of
numerical opti-
mization methods.
Both the SNM and the LIML analogues are invariant to
normalization, but the ordinary GMM estimator is
noto
That
is,
if
the equation is solved for an endogenous variable other
than
Yi,
contrary to the case with ordinary GMM, the in-
direct estimates obtained from SNM or LIML analogues
coincide with the direct SNM or LIML estimates, respec-
tively. [Notice that empirical likelihood estimators
of
the
type considered by Qin and Lawless (1994) and Imbens
(1997) will also be invariant to normalization due to the
invariance property
of
maximum likelihood estimators.)
Symmetrical1y normalized estimators are potential1y at-
tractive alternatives to ordinary GMM on at least two
grounds (aside from the desirability
of
invariance to nor-
malization in its own right). First, they tend to have a
smaller finite-sample bias than the GMM estimators. Hillier
(1990) showed that, for the normal case in a standard linear
structural equation with two endogenous variables, sym-
metrically normalized 2SLS and LIML are
"spherically
unbiased" in finite samples [meaning that the density
of
ii
=
dt!(did
1
)1/2
defined on the unit circle is symmetric
about the true points
±a
=
±dt!(didt)I/2
having modes at
±a). However, 2SLS does not have this property.
Second, the concentration
of
the densities
of
the symmet-
rically normalized estimators depends on the quality
of
the
instruments. In the completely unidentified case, as shown
by Hillier, these estimators have a uniform distribution on
3
Alonso-Borrego and Arellano: Symmetrically Normalized IV Estimation
the unit circle. This is in contrast with 2SLS, which con-
verges to the same limit
as
OLS and whose distribution is
determined exclusively by the normalization adopted. When
the instruments are poor,
as
well
as
when the number
of
in-
struments is large relative to the sample size, 2SLS tends
to provide results that are biased in the direction
of
OLS
and also large discrepancies between "direct" and "indirect"
2SLS when using different normalizations. This situation
has been stressed in several recent works (Bekker 1994;
Bound, Jaeger, and Baker 1995; Angrist and Krueger 1995;
Staiger and Stock 1997, among others). In contrast, with
poor instruments the distributions
of
LIML and symmetri-
cally normalized 2SLS accurately reproduce the fact that
the information on the structural parameters is very small.
Although the LIML analogues and the SNM estimators
are asymptotical1y equivalent (and in the Hillier setting ex-
hibit similar finite-sample properties
as
well), SNM has
sorne disadvantages relative to the other estimators. The
main one is that in general the results are not independent
of
the units in which the variables are measured, so that a
sensible choice
of
units may be important. In contrast, or-
dinary GMM is invariant to units but not to normalization,
and LIML is invariant to units and normalization. This prob-
lem does not arise in the autoregressive panel-data models
Table
1.
Model
1:
Nonrobust Estimates
ex
=.5
ex
=.8
GMM1 SNM1 LlML1 GMM1
SNM1 LlML1
T=4
~
= o
Median .49 .50
.50 .76 .80 .80
% bias 2.5
.3
.6 5.6
.1
.1
iqr
.18 .19 .19 .28
.29 .29
iq80 .35 .36 .36
.56
.61 .61
MAE .09 .09
.09 .15 .15 .15
~
=.2
Median .47 .49 .49
.66
.77
.77
% bias 6.9 1.7 1.7
17.8 3.7
4.1
iqr .23 .25
.24 .45 .57 .58
iq80
.44
.47 .47 .93 1.26 1.29
MAE .12 .12
.12 .25 .28 .29
~
= 1
Median
.43 .48 .48 .44
.65
.61
% bias
14.8 3.8
3.1
44.7
19.0 23.8
iqr
.33 .36 .36
.67 .95 1.02
iq80 .68
.77
.77
1.39
2.81
2.89
MAE .18
.18 .18 .44 .50 .53
T=7
~
= O
Median
.47 .50 .49
.75 .80 .79
% bias 5.0
.7 2.0 6.0
.3
1.1
iqr .09 .09
.09
.11
.12 .12
iq80
.16 .17 .17 .22
.23 .24
MAE .05 .04
.04 .07 .06 .06
~
=.2
Median
.47 .50 .49 .70
.81
.78
% bias 6.7
.8
1.8 13.0 1.2 2.7
iqr
.11
.11
.11
.18 .18
.21
iq80 .20
.21
.21
.34 .39 .45
MAE .06
.06 .06 .12 .09
.11
~
= 1
Median
.45 .50 .48
.61
.82
.74
% bias 10.4
1.0 3.3 24.0 3.0
8.1
iqr
.13 .14 .14
.23 .26
.38
iq80
.24 .26 .27 .45 .54
.86
MAE .07
.07 .07 .20 .13
.19
NOTE:
1,000 replications.
N = 1 00,
CT~
=
1.
% bias gives the percentage median bias Ior all
the estimates; iqr is the 75th-25th interquartile range; iq80 is the 9Oth-l0th interquantile range;
MAE denotes the median absoluta error.
39
discussed later because in that case the SNM estimator is
invariant to units and to normalization (just because in the
autoregressive case a change in the units
of
the right-side
variable leads trivially to a similar change in the units
of
the
left-side variable). Another disadvantage
of
SNM is that the
distinction between exogenous and nonexogenous variables
is relevant for the specification
of
the estimator. This is so
because in the case
of
SNM only the length
of
the coeffi-
cient vector for the nonexogenous variables is normalized
to unity, and, contrary to LIML, this differs from normal-
izing to unity the entire coefficient vector. SNM, however,
does have a computational advantage over LIML when we
consider two-step or robust estimators. Indeed, LIML2, or
continuously updated GMM, no longer sol ves a minimum
eigenvalue problem, whereas two-step
SNM only involves
simple calculations that are similar to those performed for
two-step ordinary GMM.
Of
course, SNM is limited to lin-
ear models, but in such context it is
of
interest to see
if
SNM, which is considerably faster and simpler than LIML2,
can provide the benefits
of
the more complicated estimators
and perhaps avoid problems
of
nonconvergence in the case
ofLIML2.
2.
EXPERIMENTAL COMPARISONS
The purpose
of
this section is to study the finite-sample
properties
of
the syrnmetrically normalized estimators con-
sidered previously in relation to ordinary GMM for an
AR(l} model with individual effects. The IV restrictions im-
plied by various versions
of
the model can be represented
as
simple structures on the covariance matrix
of
the data,
so we can also make comparisons with the MDE
of
these
covariance structures. The emphasis is not on assessing the
value
of
enforcing particular restrictions in the model,
as
done, for example, by Ahn and Schmidt (1995), Arellano
and Bover (1995), and Blundell and Bond (1998). Rather,
we wish to evaluate the effects in small samples
of
using
alternative estimating criteria that produce asymptotically
equivalent estimators for fixed
T and large
N.
We
concen-
trate on a random-effects
AR(1}
model because
of
its sim-
plicity and the fact that it is a case that has received much
attention in the literature.
2.1
Models and Estimators
Let us consider a random sample
of
individual time series
of
size T, yT =
(Yil,""
YiT
)'(i =
1,
...
,
N)
with second-
order moment matrix
E(yTy'[,) = n = {Wts}'
We
as
sume
that the joint distribution
of
YT
and the unobservable time-
invariant effect
TU
satisfies Assumption A:
Yit
= aYi(t-l) +
"li
+
Vit,
t =
2,
...
,T,
(22)
E(Vitlyf-l) =
0,
(23)
where E("li) =
')',
E(v~t)
=
a'f,
and var("li) =
a~.
Notice that the dependence between
"li
and
Vit
is not re-
stricted by Assumption
A,
nor is the possibility
of
con-
ditional heteroscedasticity ruled out, because
E(v~tIYf-l)
need not coincide with
a'f.
Following Arellano and Bond (1991), Assumption A im-
plies
(T
-
2)(T
- 1)/2 linear moment restrictions
of
the
4
40
form
(24)
These restrictions can also be represented as constraints
on the elements
of
o.
Multiplying (22) by
Yis
for s < t
and taking expectations gives
Wts
= aW(t-l)s +
es
(t =
2,
...
T; s =
1,
...
,t
- 1), where es = E(YisT/i). This means
that, given Assumption A, the
T(T+1)/2
different elements
of
O can be written
as
functions
of
the
2T
x 1 parameter
vector
()
= (a,
el,··.,
eT-l,W11,··.,
WTT
)/.
We call this moment structure Model
1.
Because
it
is a
special case
of
the model in Section
1,
all the estimators
discussed in Section 1 can be particularized to the present
case. Here, however, we express the IV restrictions using er-
rors in first -differences
as
opposed to orthogonal deviations
to simplify the mapping with covariance structures. Notice
that with
T = 3 the parameters (a, el ,
e2)
are just -identified
as functions
of
the elements
of
O.
The orthogonality conditions (24) are the only restric-
tions implied by Assumption A on the second-order mo-
ments
of
the data. They are not the only restrictions avail-
able, however, because
(23) also implies that nonlinear func-
tions
of
y;-2
are uncorrelated with t!.Vit. The semiparamet-
ric efficiency bound for this model can be obtained from
the results
of
Chamberlain (1992). One reason estimators
Journal of Business & Economic Statistics, January 1999
based on (24) may not be fully efficient asymptotically is
that the dependence between
T/i
and
yf
may be nonlinear.
Another reason would be unaccounted conditional hetero-
scedasticity.
Model 1 is attractive because it is based on minimal as-
sumptions.
We
may be willing to impose additional struc-
ture, however,
if
this conforms to a priori beliefs. One pos-
sibility is to assume that the error s
Vit
are mean independent
of
the individual effect
'T}i
given
y;-l.
This situation gives
rise to Assumption
A':
(25)
Note that Assumption A' is more restrictive than As-
sumption A. When
T
2':
4,
Assumption A' implies the fol-
lowing additional
T - 3 moment restrictions:
E[(Yit -
aYi(t-l))(t!.Yi(t-l)
- at!.Yi(t-2))] =
O,
t = 4,
...
, T.
(26)
In effect, we can write E[(Yit -
aYi(t-l)
- 'T}i)(t!.Yi(t-l) -
at!.Yi(t-2))]
= ° and, because E('T}it!.Vi(t-l)) =
O,
the result
follows.
GMM estimators
of
a that exploit these restrictions in ad-
dition to those in
(24) were considered by Ahn and Schmidt
(1995), but because the additional restrictions are nonlinear
Table
2.
Model
1:
Robust Estimates
a
=.5
a
=.8
GMM2
SNM2 LlML2 MDE
GMM2 SNM2 LlML2 MDE
T=4
~
= O
Median .49 .50
.51 .51
.76
.80
.81
.80
% bias
2.1
.2 1.6
2.1
4.9
.3
1.7
.0
iqr
.19
.19 .19 .12 .29
.30
.31
.10
iq80
.36 .38 .38 .23
.58
.62
.63
.21
MAE
.09 .09 .09 .06 .15 .15 .16
.05
~
=.2
Median .47
.49
.50
.51
.65
.76 .84
.71
% bias 6.5
1.8
.3 1.3 19.0 4.6
5.1
11.3
iqr .24 .25 .25 .20 .47
.55
.56
.28
iq80 .47
.50
.51
.39 .97 1.33 1.23
.58
MAE
.12
.13
.13 .10
.27
.28 .28
.11
~
= 1
Median
.44 .47 .50 .49 .45 .64 .82
.65
% bias 12.8
5.4
.5 2.2 43.6 19.5 2.9 19.1
iqr .35 .38 .38 .82 .70 1.03 .94
.48
iq80 .75
.80
.80 .56 1.53 2.82
2.22 .94
MAE
.18 .19 .19 .16
.46
.54
.47 .18
T=7
~
=0
Median .48
.50
.50
.51
.75 .79 .80
.81
% bias 4.3 .4 .6 2.0 5.7
.8
.1
1.4
iqr
.10
.10
.10 .09 .13 .13 .14
.10
iq80 .18 .19
.21
.17 .24
.25 .28 .17
MAE
.05
.05
.05
.04 .07
.07 .07 .05
~
=.2
Median .47 .50 .50 .50
.69 .79
.81
.74
% bias 6.2 .5 .4
.1
13.7 1.7 .9 7.8
iqr .12 .12 .13 .12 .20 .20 .24
.17
iq80 .23 .23
.26
.23
.39
.41
.51
.34
MAE
.06 .06 .06 .06 .13 .10 .12
.09
~ = 1
Median .45 .49
.50
.50
.59 .77 .80
.71
% bias
9.8
1.5
.0
.2
26.0 3.9
.1
11.1
iqr
.14 .15 .16 .15 .27 .28 .36 .22
iq80 .28 .30 .33 .29 .53 .59 .80 .46
MAE
.08 .07 .08 .08 .22 .15 .18
.11
NOTE:
See
nole lo Table
1.
5
