Open Journal of Social Sciences, 2014, 2, 19-24
Published Online July 2014 in SciRes. http://www.scirp.org/journal/jss
http://dx.doi.org/10.4236/jss.2014.27004
How to cite this paper: Bednarek, H. and Qian, H.L. (2014) Redundancy of Moment Conditions for Linear Transformation of
Parameters. Open Journal of Social Sciences, 2, 19-24. http://dx.doi.org/10.4236/jss.2014.27004
Redundancy of Moment Conditions for
Linear Transformation of Parameters
Heather Bednarek, Hailong Qian*
Department of Economics, Saint Louis University, Saint Louis, USA
Received April 2014
Abstract
In this paper, we consider the redundancy of an extra set of moment conditions, given an initial set
of moment conditions, for the efficient estimation of an arbitrary linear transformation of an
original parameter vector. The redundancy condition derived in the current paper unifies the full
and partial redundancy of moment conditions of Breusch, Qian, Schmidt and Wyhowski (Journal of
Econometrics, 1999).
Keywords
GMM Estimation, Moment Conditions, Redunda ncy, Partial Redundancy
1. Introduction
In their seminal paper, [1] derived, respectively, the necessary and sufficient condition for the full and partial
redundancy of an extra set of moment conditions, given an original set of moment conditions, for the efficient
generalized method of moments (GMM) estimation of parameters of interest. However, they treated full and
partial redundancy of moment conditions as independent conditions and derived them separately.
In this paper, we will extend their redundancy concept to estimation of an arbitrary linear transformation of an
original parameter vector and derive the necessary and sufficient condition for an extra set of moment conditions
to be redundant, given an initial set of moment conditions, for the efficient estimation of the linear tra ns fo r ma-
tion of parameters. More specifically, the current paper makes three new contributions. Firstly, we extend the
moment redundancy concept of [1] to estimation of an arbitrary linear transformation of original parameters.
Secondly, the redundancy condition derived in the current paper unifies the full and partial redundancy condi-
tions of [1]. As a result, our redundancy condition given in the theorem of Section 2 includes as two special cas-
es the full and partial redundancy conditions of [1]. Lastly, we use a much simpler approach based on matrix
ranks to deriving our main results instead of using “brute-force” matrix algebra, as adopted by [1] and [2].
The rest of the paper is organized as follows. Section 2 presents the main results, while Section 3 briefly con-
cludes.
*
Corresponding author.

H. Bednarek, H. L. Qian
20
2.Redundancy of Moment Conditions for Linear Transformation of Parameters
Let
θ
ˆ
be the optimal GMM estimator of
0
θ
, based on moment conditions:
0)],w(g[E
0t1
=θ
, t=1, 2, ..., T, (1)
where
t
w
is a vector of observable variables,
0
θ
is a
1p×
vector of unkno wn parameters to be estimated,
),w(g
t1
θ
is an
1m×
vector of moment functions, and T is the sample size. For identification purpose, we
assume
pm ≥
. We also assume that the vector of moment functions
),w(g
t 1
θ
satisfies the usual regularity
conditions; for a list of usual regularity conditions, see for example [3]. As adopted by [1] and for simplicity of
the derivation below, we make three usual assumptions, as follows.
Assu mption s: (A.1) The sample
}T ,..., 2, 1 t: w{
t
=
is independent and identically distributed.
(A.2)
)],w(gvar[
0t111
θ≡Ω
is positive-defi nite.
(A.3)
]'/),w(g[ED
0t11
θ∂θ∂≡
has full column rank.
Under these assumptions,
0
θ
is identified and the optimal GMM estimator
θ
ˆ
of
0
θ
is consistent and
asymptotically normal, with its asymptotic variance matrix equal to
1
1
1
111
)D'D()
ˆ
(AV
−−
Ω=θ
; see for example
[4].
Now, suppose that we have available an extra set of moment conditions:
0)],w(g[E 0t2 =θ
, t = 1, 2, ..., T, (2)
where
),w(g
0t2
θ
is an
1n×
vector of moment functions satisfying usual regularity conditions. Then, it is
well-known in the GMM literature that the optimal GMM estimator of
0
θ
usin g both sets of moment condi-
tions (1) and (2) is usuall y a sympto tically more efficient than the optimal GMM estimator using moment condi-
tions (1) alone. However, there are circumstances when adding an extra set of moment conditions (2) to the ex-
isting set of moment conditions (1) does not improve the asymptotic efficiency of the resultant GMM estimator
of
0
θ
. This is the so-called (full) redundancy of moment conditions (2), given moment conditions (1), for the
efficient estimation of
0
θ
; [1] der ived the necessary and sufficient condition for the set of moment conditions
(2) to be (fully) redundant given the set of moment conditions (1). There are also circumstances when adding the
second set of moment conditions (2) onto the first set of moment conditions (1) does not improve the asymptotic
efficiency of the resultant GMM estimator of a sub-vector of
0
θ
. This is the so-called partial redundancy of
moment conditions (2), given moment conditions (1), for the efficient estimation of a sub-vector of
0
θ
; [1]
used “brute-force” matrix algebra to derive the necessary and sufficient condition for the partial redundancy of
an extra set of moment conditions, while [5] used moment projection approach to deriving the same partialre-
dundancy condition.
One main weakness of [1] is that it treats the full and partial redundancy of moment conditions as independent
of each other and derives them separately. In what follows, we will ext end the moment redundancy concept of
[5] to the estimation of an arbitrary linear transformation of an original parameter vector and derive the neces-
sary and sufficient condition for an extra set of moment conditions to be redundant, given an initial set of mo-
ment conditions, for the efficient estimation of the linear transformation of original parameters. The necessary
and sufficient condition given in the theorem below unifies the full and partial redundancy of moment condi-
tions.
More specifically, suppose that we are interested in the estimation of a linear transformation of the original
parameter vector
0
θ
:
00
Aθ=δ
, (3)
where A is a
pq×
matrix of known constants, with full row rank. Then, since
θ
ˆ
is the optimal GMM esti-
mator of
0
θ
based on moment conditions (1), it is easy to verify that
ˆˆ
δ= Aθ
is a consistent and efficient es-
timator of
0
δ
, with its asymptotic variance equal to:
'A)D' D(A)
ˆ
(AV
1
1
1
111
−−
Ω= δ
. (4)
Now, let
θ
~
be the optimal GMM estimator of
0
θ
, based on the joint moment conditions (1) and (2). [1]
show that the optimal GMM estimator
θ
~
based on moment conditions (1) and (2) is equivalent to the optimal

H. Bednarek, H. L. Qian
21
GMM estimator based on the following set of orthogonalized moment conditions:
0)
),w(r
),w(g
(E
0t2
0t1
=






θ
θ
, (5)
where
) ,w (g ) ,w (g),w(r t
1
1
11 21 t2t2 θ
ΩΩ − θ≡θ −
, with
] )', w (g ), w (g [E 0
t 10 t2 21 θ
θ≡ Ω
. As a result,
we can express the asymptotic variance of
θ
~
as:
11
AV()(G 'G)
−−
θ= Σ

, (6)
whe r e






ΩΩ−
=






θ∂θ∂
θ∂θ∂
≡






=−
1
1
11212
1
0t2
0t1
2
1
DD
D
)
'/),w(r
'/),w(g
(E
G
D
G
,
(7 A)






Σ
Ω
=






θ
θ
≡Σ
22
11
0t2
0t1
0
0
)
),w(r
),w(g
var(
, (7B )
and where
]'/),w(g[ED
0t22
θ∂θ∂≡
and
12
1
11212222 ΩΩΩ−Ω≡Σ−
.
Now, given
θ
~
, we can also estimate
0
δ
by
θ=δ~ A
~
. Then, it is easy to verify that
δ
~
is a consistent es-
timator of
0
δ
and its asymptotic variance is equal to:
'A)G'G(A)
~
(AV 11 −− Σ=δ
. (8)
Since
θ

is asymptotically at least as efficient as
ˆ
θ
, it is easy to verify that
δ

is also asymptotically no
less efficient than
ˆ
δ
. Then, an interesting question is when
ˆ
δ
is asymptotically as efficient as
δ

, or equiva-
lently, under what circumstances will adding the extra set of moment conditions (2) to moment conditions (1)
not improve the asymptotic efficiency of the GMM estimator of the transformed parameter vector,
00
Aθ=δ
?
When
ˆ
AV( )AV( )δ= δ

, we will say that the extra set of moment conditions (2) is redundant, given the original
set of moment conditions (1), for the (efficient) estimation of the transformed parameter vector
0
δ
.
We now proceed to find the necessary and sufficient condition for
ˆ
AV( )AV( )δ= δ

.To this end, we first stat e
a well-known rank formula for partitioned-matrices; see for example Masarglia and [6]).
Lemma. Let A be a nonsingular matrix of order
1
r
, B, C and D be
21
c r×
,
12 r
r×
, and
22
cr ×
matrices,
respectively. Then,
)BCAD(rk)A
(rk
)
DC
BA
(
rk 1−
−+=






, (9A)
or equivalently,
)A(
rk)
DC
B A
(rk
)BCA
D(rk 1−






=− −
. (9B)
Then, using (9B), we have:
)]
~
(AV)
ˆ
(AV[ rk δ−δ
(10A)
]'A)G'G(A'A)D'D(A[rk111
1
1
111
−−−− Σ−Ω=
)G'G(rk)
'A)D'D(AA
'AG'G
(rk 1
1
1
1
111
1−
−−
−
Σ−






Ω
Σ
=

H. Bednarek, H. L. Qian
22
p)
'A)D'D(AA
'AG'GD'D
(rk 1
1
1
111
2
1
2221
1
111 −






Ω
Σ+Ω
=−−
−−
, (10B )
Using
2
1
2221
1
111
1G'GD'DG'G−−− Σ+Ω=Σ
from (7A)-(7B) and
p)G'G(rk 1=Σ−
. Note that,






Ω
Σ+Ω
−−
−−
'A)D'D(AA
'AG'GD'D
1
1
1
111
2
1
2221
1
111






Ω






Ω
Σ






Ω
=
−−
−
−
−−
'A)D'D(I
0G
D'D0
0
)D'D(A0
I'G
1
1
1
111p
2
1
1
111
1
22
1
1
1
111
p2
.
Substituting it into (10B), we have:
)]
~
(AV)
ˆ
(AV[rk δ−δ
(11A)
p)
'A)D'D(I
0G
D'D0
0
)D'D(A0
I'G
(rk
1
1
1
111p
2
1
1
111
1
22
1
1
1
111
p2
−






Ω






Ω
Σ






Ω
=
−−
−
−
−−
p)
'A)D'D(I
0G
(rk
1
1
1
111p
2
−






Ω
=
−−
(using
)C(rk)BC'C(rk =
, with B positive definite)
p)
0G
'A)D'D (I
(rk
2
1
1
1
111p
−





Ω
=
−−
(exchanging rows)
p)'A)D'D(IG0(rk)I(rk
1
1
1
111
1
p2p
−Ω−+=
−−−
(usi ng (9A))
)'A)D'D(G(rk1
1
1
1112
−−
Ω=
. (11B)
Now, given the equality of expressions (11A) and (11B), we are ready to establish the main result of the pa-
per.
THEOR EM . The extra set of moment conditions (2) is redundant, given the set of moment conditions (1), for
the efficient estimation of the linear transformation of
00 Aθ=δ
, if and only if
0' A)D'D(G
1
1
1
111
2
=Ω
−−
.
Pr oo f: By the definition, moment conditions (2) is redundant, given moment conditions (1), for the efficient
estimation of
0 0Aθ
= δ
, if and only if
ˆ
AV()AV() δ= δ

, or equivalently, if and only if
0)]
~
(AV)
ˆ
(AV[ rk =θ−θ
(using the fact that the rank of a matrix is zero if and only of the matrix itself is a
zero matrix).Then, by using the equality of expressions (11A) and (11B), the result of this theorem follows im-
mediatel y.■
Given this theorem, we can now easily show that the full and partial redundancy conditions obtained by
Breusch et al. (1999) are just two special cases of it.
COROLLARY 1.When the transformation matrix A is nonsingular, the extra set of moment conditions (2) is
redundant, given the set of moment conditions (1), for the efficient estimation of
00 Aθ=δ
, if and only if
0G
2
=
, or equivalently,
1
1
11212 DD −
ΩΩ=
.
Pr oo f: When A is nonsingular, the redundancy condition
0'A)D'D(G
1
1
1
1112
=Ω
−−
of the theorem above
is equivalent to
0G
2
=
.Also, by the definition of
1
1
112122
DDG
−
ΩΩ−=
in (7A),
0 G
2
=
is the same
as
1
1
11212 DD −
ΩΩ=
. ■
A special case of Corollary 1 is when the transformation matrix A is an identity matrix; that is,
00
θ=δ
.
Then, the condition of
1
1
11212
DD
−
ΩΩ=
is just the full redundancy condition of Theorem 1 of Breusch et al.
(1999, p. 94).Thus, the full redundancy condition of [1] is just a special case of the necessary and sufficient con-

H. Bednarek, H. L. Qian
23
dition of the theorem above.
We now turn to using the theorem above to derive the partial redundancy of moment conditions (2), given
moment conditions (1), for the estimation of a sub-vector of
0
θ
. For this purpose, we partition the parameter
vector
0
θ
into
)'','( 20100 θθ=θ
, where
10
θ
is
1p1×
and
20
θ
is
1p
2
×
, with
ppp 21 =+
. We
also partition the expected derivative matrices accordingly:
)D,D(D
12111
=
, with
]' / ), w( g [E D
j0t1j 1
θ ∂θ ∂=
for j = 1, 2 (12 A)
)G,G(G
22212
=
, with
]'/),w(r[EG
j0t2j2
θ∂θ∂=
, for j = 1, 2. (12B)
Without loss of generality, suppose that we are now mainly interested in estimating the first subset of para-
meters,
10
θ
. That is, we wish to efficiently estimate
1000
Aθ=θ=δ
, with the transformation matrix A de-
fined as
]0,I[A
211
pp p×
=
. Now, substituting (12A)-(12B) and
]0 , I[ A
2 11
ppp ×
=
into the redundancy condi-
tion of the theorem above, we have:












ΩΩ
ΩΩ
=Ω
−
−−
−−
−−
0
I
D'DD'D
D'DD'D
)G,G('A)D'D(G
1
p
1
12
1
111211
1
1112
12
1
111111
1
1111
2221
1
1
1
1112






ΩΩ−
=−−−−
−
1
11
1
1112
1
12
1
1112
1
2221 ED'D)D'D(
E
)G,G(
1
11
1
1112
1
12
1
11122221
E] D'D) D 'D( GG[
−− −−
ΩΩ− =
,
where we used the formula for partitioned-matrix inverse in the second equation, with
11
1
1112
1
12
1
111212
1
111111
1
1111 D'D)D'D(D'DD'DE −−−−− ΩΩΩ−Ω≡
. That is, we establish:
=Ω
−−
'A)D'D( G
1
1
1
1112
(13)
Now, combining this expression with the theorem above, we obtain Corollary 2.
COROLLARY 2. The extra set of moment conditions (2) is partially redundant, given the set of moment
conditions (1), for the efficient estimation of
10
θ
, if and only if:
(A)
0'A)D'D(G
1
1
1
1112
=Ω
−−
, with
]0,I[A
211
ppp ×
=
, or equivalently,
(B)
11
1
1112
1
12
1
11122221
D'D)D'D(GG
−−−
ΩΩ=
.
Pr oo f: Condition (A) is just the redundancy condition of the theorem above, with the special transformation-
matr i x
] 0,I [A
21 1
ppp ×
=
. The equivalence of conditions (A) and (B) follows directly from the equality in
(13).■
Here we note that condition (B) of Corollary 2 is identical to the partial redundancy condition of [1] (Theorem
7) and [2], Theorem 2). Both of these papers used very tedious and brute-force matrix algebra to derive it, while
we straightforwardly derived it as a special case of the general redundancy condition for linear transformation of
parameters.
Before we conclude this paper, we want to briefly compare the redundancy conditions for transformed para-
meters of the current paper with the redundancy conditions in restricted GMM estimation of [5]. More specifi-
cally, [5] considers efficient GMM estimation of
0
θ
based on moment conditions (1), subject to a possibly
nonlinear set of q restrictions,
0)(r
0
=θ
. [5] Theorem 1) shows that the extra set of moment conditions (2) is
redundant, given moment conditions (1), for the efficient estimation of
0
θ
subjected to r (
0
θ
) = 0, if and only
if
2 [F']
GM 0=
, (14)
where
'/)(rF 0θ∂θ∂≡
. Comparing (14) with
0'A) D'D(G
1
1
1
1112
=Ω
−−
(the redundancy condition for the
linear transformation of parameters of the current paper), we can easily verify that they are the same as
0G2=
(the full redundancy condition of [1], when A is nonsingular and F is defined as the
pq×
zero ma-
trix for no restrictions. However, we want to emphasize that the redundancy condition for restricted GMM esti-

H. Bednarek, H. L. Qian
24
mation, given in Theorem 1 of [5],is substantially different from the redundancy condition for transformed pa-
rameters, given in the theorem of the current paper. To see the difference between them, let’s consider a special
case, as follows. Let
)'','( 20100 θθ=θ
, as defined before. Now, suppose that we are only interested in esti-
mating
10
θ
(that is, the transformation matrix is
]0 , I[ A
2 11
ppp ×
=
, as we considered in Corollary 2 above)
and that the set of restrictions in the restricted GMM estimation is given by
0)(r
200
=θ=θ
(that is,
20
θ
is known). For this special case, we can easily verify that the redundancy
condition (14) for the restricted GMM estimation of
10
θ
becomes
0G
21
=
, which is actually the full redun-
dancy condition of moment conditions (2), given moment conditions (1), for the efficient estimation of
10
θ
,
because
20
θ
is known. On the other hand, the redundancy condition for the transformed parameter vector
10
00
Aθ=θ=δ
becomes the partial redundancy condition,
11
1
1112
1
12
1
11122221
D'D)D'D(GG
−−−
ΩΩ=
, as
shown in Corollary 2. Thus, we can interpret the redundancy condition of the current paper,
0'A)D'D(G
1
1
1
1112
=Ω
−−
, as the partial redundancy of moment conditions (2), given moment conditions (1),
for the efficient estimation of parameters of interest,
00
Aθ=δ
, while
2 [F']
GM 0=
(the condition for the
redundancy of moment conditions (2), given moment conditions (1), in the restricted GMM estimation consi-
dered by [5]), can be interpreted as the full redundancy condition of moment conditions (2), given moment con-
ditions (1), for the efficient estimation of the free parameters (after substituting out the restrictions).
3. Conclusions
In this paper, using the idea of redundancy of moment conditions for estimating a linear transformation of para-
meters, we successfully unify full and partial redundancy of moment conditions. As a consequence, the neces-
sary and sufficient condition for an extra set of moment conditions to be redundant for the estimation of a linear
transformation of original parameters encompasses the full and partial redundancy conditions of [1].
Two possible applications of the results of the current paper are the efficient estimation of parameters of in-
terest in panel data models and systems of equations. [7] (Section 8.4.2), for example, compares the relative ef-
ficiency of GMM, generalized instrumental variables (GIV) and the traditional 3SLS estimators of the whole
coefficient vector of a system of linear regressions. Using the results of the current paper and appropriately de-
fined sets of moment conditions, we could find useful conditions for GIV and the traditional 3SLS estimators of
a sub-vector of regression coefficients (e.g. the coefficient vector of the first equation in a system) to be as effi-
cient as the optimal GMM estimator applied to the entire system. This is a topic for future research.
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