What Is the Necessary and Sufficient Condition for a Given Linear Transformation to Be
Open Journal of Social Sciences, 2014, 2, 19-24
Published Online July 2014 in Due southciR es. http://www.scirp.org/periodical/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 Academy, Saint Louis, USA
Electronic mail: bednarhl@slu.edu, * qianh@slu.edu
Received April 2014
Abstract
In this newspaper, we consider the redundancy of an actress fix of moment conditions, given an initial set
of moment conditions, for the efficient interpretation of an arbitrary linear transformation of an
original parameter vector. The redundancy status 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 Weather condition, Redunda ncy, Partial Back-up
ane. Introduction
In their seminal paper, [one] derived, respectively, the necessary and sufficient status for the full and partial
redundancy of an extra set of moment conditions, given an original set up of moment weather, for the efficient
generalized method of moments (GMM) interpretation of parameters of interest. Still, they treated full and
partial redundancy of moment conditions as contained conditions and derived them separately.
In this newspaper, nosotros will extend their back-up concept to interpretation of an capricious linear transformation of an
original parameter vector and derive the necessary and sufficient condition for an actress ready of moment conditions
to be redundant, given an initial set of moment weather, for the efficient estimation of the linear tra ns fo r thoua-
tion of parameters. More specifically, the current newspaper makes 3 new contributions. Firstly, nosotros 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 total and partial redundancy condi-
tions of [1] . As a result, our redundancy status given in the theorem of Department two 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 chief results instead of using "brute- force" matrix algebra, as adopted by [ane] and [two] .
The residuum of the paper is organized equally follows. Section 2 presents the main results, while Section 3 briefly codue north-
cludes.
*
H. Bednarek, H. L. Qian
2.Redundancy of Moment Conditions for Linear Transformation of Parameters
Let
exist the optimal GMM estimator of
, based on moment conditions:
, t=1, 2, ..., T, (1)
where
is a vector of appreciable variables,
is a
vector of unkno wn parameters to exist estimated,
is an
vector of moment functions, and T is the sample size. For identification purpose, we
presume
. Nosotros also assume that the vector of moment functions
satisfies the usual regularity
atmospheric condition; for a listing of usual regularity weather condition, see for example [iii]. As adopted past [i] and for simplicity of
the derivation beneath, we brand 3 usual assumptions, as follows.
Assu mption s: (A.one) The sample
is independent and identically distributed.
(A.2)
is positive-defi nite.
(A.3)
has full column rank.
Under these assumptions,
is identified and the optimal GMM estimator
of
is consistent and
asymptotically normal, with its asymptotic variance matrix equal to
; run across for example
[4].
At present, suppose that nosotros have available an extra prepare of moment conditions:
, t = 1, two, ..., T, (2)
westhere
is an
vector of moment functions satisfying usual regularity conditions. And then, it is
well-known in the GMM literature that the optimal GMM estimator of
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 (i) alone. However, there are circumstances when adding an extra set of moment conditions (2) to the due eastx-
isting fix of moment weather (ane) does not improve the asymptotic efficiency of the resultant GMM estimator
of
. This is the so-called (full) redundancy of moment weather (2), given moment conditions (i), for the
efficient estimation of
; [ane] der ived the necessary and sufficient condition for the set of moment conditions
(2) to exist (fully) redundant given the set of moment conditions (1). At that place are also circumstances when calculation the
2nd set of moment conditions (2) onto the first ready of moment conditions (1) does not ameliorate the asymptotic
efficiency of the resultant GMM estimator of a sub- vector of
. This is the so-called partial redundancy of
moment atmospheric condition (2), given moment weather condition (1), for the efficient interpretation of a sub- vector of
; [one]
used "creature-force" matrix algebra to derive the necessary and sufficient condition for the partial redundancy of
an extra gear up of moment atmospheric condition, while [v] used moment projection approach to deriving the same partialre-
dundancy condition.
1 main weakness of [ane] is that information technology treats the full and partial back-up of moment conditions as independent
of each other and derives them separately. In what follows, nosotros volition 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 prepare of moment conditions to be redundant, given an initial gear up of mo-
ment weather, for the efficient estimation of the linear transformation of original parameters. The necessary
and sufficient status given in the theorem below unifies the total and fractional redundancy of moment condi-
tions.
More than specifically, suppose that we are interested in the estimation of a linear transformation of the original
parameter vector
:
, (3)
where A is a
matrix of known constants, with full row rank. And then, since
is the optimal GMM esti-
mator of
based on moment conditions (ane), information technology is like shooting fish in a barrel to verify that
is a consequent and efficient es-
timator of
, with its asymptotic variance equal to:
. (4)
Now, let
exist the optimal GMM estimator of
, based on the articulation moment conditions (1) and (2). [i]
show that the optimal GMM computer
based on moment weather (i) and (ii) is equivalent to the optimal
H. Bednarek, H. L. Qian
GMM estimator based on the follofly ready of orthogonalized moment conditions:
0)
),w(r
),w(yard
(E
0t2
0t1
=
θ
θ
, (5)
where
)
,w
(one thousand
)
,w
(g),due west(r t
1
i
11
21
t2t2 θ
ΩΩ
−
θ≡θ −
, with
]
)',
w
(g
),
west
(g
[E 0
t
x
t2
21 θ
θ≡
Ω
. As a result,
we tin can express the asymptotic variance of
every bit:
, (6)
whe r east
ΩΩ−
=
θ∂θ∂
θ∂θ∂
≡
=−
one
i
11212
one
0t2
0t1
ii
1
DD
D
)
'/),w(r
'/),w(g
(Due east
G
D
G
,
(7 A)
Σ
Ω
=
θ
θ
≡Σ
22
11
0t2
0t1
0
0
)
),w(r
),w(g
var(
, (7B )
and where
and
.
Now, given
, we tin can besides approximate
by
. Then, it is easy to verify that
is a consequent es-
timator of
and its asymptotic variance is equal to:
. (8)
Since
is asymptotically at least as efficient as
, information technology is easy to verify that
is also asymptotically no
less efficient than
. Then, an interesting question is when
is asymptotically as efficient as
, or equiva-
lently, under what circumstances will adding the extra gear up of moment conditions (2) to moment weather condition (i)
not improve the asymptotic efficiency of the GMM estimator of the transformed parameter vector,
?
When
, we will say that the extra set of moment conditions (2) is redundant, given the original
prepare of moment conditions (1), for the (efficient) estimation of the transformed parameter vector
.
Nosotros at present go on to find theast necessary and sufficient condition for
.To this end, nosotros beginning stat e
a well-known rank formula for partitioned-matrices; see for instance Masarglia and [half dozen]).
Lemma. Let A exist a nonsingular matrix of lodge
, B, C and D be
,
, and
matrices,
respectively. And then,
)BCAD(rk)A
(rk
)
DC
BA
(
rk i−
−+=
, (9A)
or equivalently,
)A(
rk)
DC
B
A
(rk
)BCA
D(rk 1 −
=− −
. (9B)
Then, using (9B), nosotros take:
(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 i
1
ane
ane
111
one−
−−
−
Σ−
Ω
Σ
=
H. Bednarek, H. L. Qian
p)
'A)D'D(AA
'AG'GD'D
(rk 1
ane
1
111
two
ane
2221
1
111 −
Ω
Σ+Ω
=−−
−−
, (10B )
Using
two
1
2221
ane
111
1Chiliad'GD'DG'G−−− Σ+Ω=Σ
from (7A)-(7B) and
. Note that,
Ω
Σ+Ω
−−
−−
'A)D'D(AA
'AG'GD'D
1
1
i
111
ii
ane
2221
1
111
Ω
Ω
Σ
Ω
=
−−
−
−
−−
'A)D'D(I
0G
D'D0
0
)D'D(A0
I'G
ane
1
one
111p
2
1
1
111
1
22
1
1
1
111
p2
.
Substituting it into (10B), nosotros have:
(11A)
p)
'A)D'D(I
0G
D'D0
0
)D'D(A0
I'Chiliad
(rk
1
i
1
111p
2
1
1
111
1
22
1
1
1
111
p2
−
Ω
Ω
Σ
Ω
=
−−
−
−
−−
p)
'A)D'D(I
0G
(rk
i
i
1
111p
ii
−
Ω
=
−−
(using
, with B positive definite)
p)
0G
'A)D'D
(I
(rk
ii
i
1
1
111p
−
Ω
=
−−
(exchanging rows)
p)'A)D'D(IG0(rk)I(rk
1
1
1
111
i
p2p
−Ω−+=
−−−
(usi ng (9A))
. (11B)
Now, given the equality of expressions (11A) and (11B), we are set up to establish the principal result of the pa-
per.
THEOR EM . The actress prepare of moment conditions (2) is redundant, given the set up of moment weather (1), for
the efficient interpretation of the linear transformation of
, if and but if
.
Pr oo f: Past the definition, moment conditions (2) is redundant, given moment conditions (ane), for the efficient
estimation of
, if and only if
, or equivalently, if and only if
(using the fact that the rank of a matrix is zero if and merely of the matrix itself is a
zero matrix).So, by using the equality of expressions (11A) and (11B), the result of this theorem follows im-
mediatel y.■
Given this theorem, nosotros can at present easily show that the full and partial redundancy conditions obtained past
Breusch et al. (1999) are just two special cases of it.
COROLLARY i.When the transformation matrix A is nonsingular, the extra gear up of moment conditions (2) is
redundant, given the fix of moment conditions (1), for the efficient estimation of
, if and but if
, or equivalently,
.
Pr oo f: When A is nonsingular, the redundancy status
of the theorem above
is equivalent to
.Also, by the definition of
in (7A),
is the same
every bit
. ■
A special instance of Corollary 1 is when the transformation matrix A is an identity matrix; that is,
.
And so, the condition of
is just the full back-up condition of Theorem ane of Breusch et al.
(1999, p. 94).Thus, the total redundancy condition of [1] is just a special case of the necessary and sufficient con-
H. Bednarek, H. 50. Qian
dition of the theorem above.
We now plow to using the theorem above to derive the partial redundancy of moment conditions (2), given
moment atmospheric condition (1), for the estimation of a sub-vector of
. For this purpose, nosotros division the parameter
vector
into
, where
is
and
is
, with
. We
also partition the expected derivative matrices accordingly:
, with
for j = 1, 2 (12 A)
, with
, for j = 1, 2. (12B)
Without loss of generality, suppose that we are now mainly interested in estimating the first subset of para-
meters,
. That is, we wish to efficiently gauge
, with the transformation matrix A de-
fined equally
. Now, substituting (12A)- (12B) and
into the ceriseundancy condi-
tion of the theorem above, we have:
ΩΩ
ΩΩ
=Ω
−
−−
−−
−−
0
I
D'DD'D
D'DD'D
)G,One thousand('A)D'D(M
1
p
ane
12
1
111211
ane
1112
12
1
111111
one
1111
2221
i
1
ane
1112
ΩΩ−
=−−−−
−
1
11
i
1112
1
12
1
1112
i
2221 ED'D)D'D(
E
)Thousand,K(
1
xi
1
1112
ane
12
one
11122221
E]
D'D)
D
'D(
GG[
−−
−−
ΩΩ−
=
,
where we used the formula for partitioned-matrix changed in the 2d equation, with
11
i
1112
1
12
1
111212
1
111111
ane
1111 D'D)D'D(D'DD'DE −−−−− ΩΩΩ−Ω≡
. That is, we constitute:
(thirteen)
Now, combining this expression with the theorem above, we obtain Corollary ii.
COROLLARY 2. The extra gear up of moment conditions (ii) is partially redundant, given the set of moment
conditions (1), for the efficient estimation of
, if and only if:
(A)
, with
, or equivalently,
(B)
11
1
1112
1
12
i
11122221
D'D)D'D(GG
−−−
ΩΩ=
.
Pr oo f: Status (A) is just the redundancy condition of the theorem in a higher place, with the special transformation-
matr i x
. The equivalence of conditions (A) and (B) follows direct from the equality in
(13).■
Here we note that condition (B) of Corollary ii is identical to the partial redundancy condition of [i] (Theorem
7) and [2], Theorem 2). Both of these papers used very irksome and brute- force matrix algebra to derive information technology, while
we straightforwardly derived it as a special case of the full 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 newspaper with the back-up conditions in restricted GMM interpretation of [5]. More specifi-
cally, [v] considers efficient GMM estimation of
based on moment weather condition (ane), field of study to a possibly
nonlinear set of q restrictions,
. [5] Theorem 1) shows that the actress ready of moment conditions (2) is
redundant, given moment conditions (1), for the efficient estimation of
subjected to r (
) = 0, if and only
if
, (14)
where
. Comparison (14) with
(the redundancy status for the
linear transformation of parameters of the electric current paper), we can easily verify that they are the aforementioned as
(the full redundancy condition of [ane] , when A is nonsingular and F is defined as the
zero ga-
trix for no restrictions. Withal, we want to emphasize that the redundancy status for restricted GMM esti-
H. Bednarek, H. L. Qian
mation, given in Theorem one of [five],is substantially unlike from the redundancy status for transformed pa-
rameters, given in the theorem of the electric current paper. To encounter the difference betwixt them, let's consider a special
case, as follows. Let
, as defined before. At present, suppose that we are only interested in esti-
mating
(that is, the transformation matrix is
, as we considered in Corollary 2 above)
and that the set of restrictions in the restricted GMM estimation is given past
(that is,
is known). For this special case, we can easily verify that the back-up
condition (14) for the restricted GMM estimation of
becomes
, which is actually the full redun-
dancy condition of moment conditions (2), given moment conditions (i), for the efficient estimation of
,
considering
is known. On the other manus, the redundancy condition for the transformed parameter vector
becomes the partial redundancy condition,
11
one
1112
1
12
1
11122221
D'D)D'D(GG
−−−
ΩΩ=
, as
shown in Corollary 2. Thus, we can translate the redundancy condition of the current paper,
, as the partial redundancy of moment atmospheric condition (2), given moment conditions (ane),
for the efficient estimation of parameters of interest,
, while
(the status for the
redundancy of moment conditions (2), given moment atmospheric condition (ane), in the restricted GMM estimation consi-
dered by [five]), can be interpreted every bit the full redundancy condition of moment conditions (two), given moment con-
ditions (1), for the efficient estimation of the free parameters (after substituting out the restrictions).
3. Conclusions
In this newspaper, using the idea of redundancy of moment atmospheric condition for estimating a linear transformation of para-
meters, we successfully unify full and partial redundancy of moment atmospheric condition. Equally a consequence, the nece south-
sary and sufficient status for an extra ready of moment weather to be redundant for the interpretation of a linear
transformation of original parameters encompasses the full and partial back-up conditions of [1] .
Two possible applications of the results of the current paper are the efficient estimation of parameters of inorthward-
terest in panel data models and systems of equations. [7] (Section 8.iv.ii), 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 organization of linear regressions. Using the results of the current paper and appropriately de-
fined sets of moment conditions, we could find useful weather condition for GIV and the traditional 3SLS estimators of
a sub- vector of regression coefficients (east.thou. the coefficient vector of the first equation in a arrangement) to exist as effi-
cient as the optimal GMM computer practical to the unabridged arrangement. This is a topic for future research.
References
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metrics, 91, 89 - 111. http://dx.doi.org/10.1016/S0304-4076( 98) 0005 0-5
[2] Qian , H. (2002) Partial Back-up of Moment Conditions. Econometric Theory , eighteen , 531 - 539.
http://dx.doi.org/ten.1017/S0266466602182132
[3] Newey, W. and McFadden, D. (1994) Big Sample Interpretation and Hypothesis Testing. In: Engle, R. and McFadden,
D., Eds., Handbook of Econometrics, Elsevier, Berlin, 21 11-22 45 .
[4] Hans en, Fifty.P. (1982) Large Sample Properties of Generalized Method of Moments Estimators. Econo metrica , 50 , 10 29 -
1054. http://dx.doi.org/10.2307/1912775
[5] Qian , H. (2013) Redu ndancy of Moment Conditions in Restricted GMM Estimation. Invited for Revision by Eco no -
metrics Theory; Existence Revised.
[6] Mars aglia, G. and Styan, G. (1974) Equalities and Inequalities for Ranks of Matrices. Linear and Multilinear Algebra,
2, 269- 292.
[seven] Woodridge, J. (2010) Econometric Analysis of Cross Section and Panel Information, MIT Press, Cambridge.
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