Book Econometric Analysis of Cross Section and Panel Data By Wooldridge - Chapter 14 pptx

14 Generalized Method of Moments and Minimum Distance Estimation
In Chapter 8 we saw how the generalized method of momen ts (GMM) approach to
estimation can be applied to multiple-equation linear models, including systems of
equations, with exogenous or endogenous explanatory variables, and to panel data
models. In this chapter we extend GMM to nonlinear estimation problems. This
setup allows us to treat various e‰ciency issues that we have glossed over until now.
We also cover the related method of minimum distance estimation. Because the
asymptotic analysis has many features in common with Chapters 8 and 12, the anal-
ysis is not quite as detailed here as in previous chapters. A good reference for this
material, which fills in most of the gaps left here, is Newey and McFadden (1994).
14.1 Asymptotic Properties of GMM
Let fw
i
A R
M
: i ¼ 1; 2; g denote a set of independent, identically distributed ran-
dom vectors, where some feature of the distribution of w
i
is indexed by the P Â 1
parameter vector y. The assumption of identical distribution is mostly for notational
convenience; the following methods apply to independently pooled cross sections
without modification.
We assume that for some function gðw
i
; yÞ A R
L
, the parameter y
o
A Y H R
P
sat-

isfies the moment assumptions
E½gðw
i
; y
o
Þ ¼ 0 ð14:1Þ
As we saw in the linear case, where gðw
i
; yÞ was of the form Z
0
i
ðy
i
À X
i
yÞ, a minimal
requirement for these moment conditions to identify y
o
is L b P.IfL ¼ P, then
the analogy principle suggests estimating y
o
by setting the sample counterpart,
N
À1
P
N
i¼1
gðw
i
; yÞ, to zero. In the linear case, this step leads to the instrumental vari-

ables estimator [see equation (8.22)]. When L > P, we can choose
^
yy to make the
sample average close to zero in an appropriate metric. A generalized method of
moments (GMM) estimator,
^
yy, minimizes a quadratic form in
P
N
i¼1
gðw
i
; yÞ:
min
y AY
X
N
i¼1
gðw
i
; yÞ
"#
0
^
XX
X
N
i¼1
gðw
i

; yÞ
"#
ð14:2Þ
where
^
XX is an L Â L symmetric, positive semidefinite weighting matrix.
Consistency of the GMM estimator follows along the lines of consistency of the
M-estimator in Chapter 12. Under standard moment conditions, N
À1
P
N
i¼1
gðw
i
; yÞ
satisfies the uniform law of large numbers (see Theorem 12.1). If,
^
XX !
p
X
o
, where X
o
is an L ÂL positive definite matrix, then the random function
Q
N
ðyÞ1 N
À1
X
N

i¼1
gðw
i
; yÞ
"#
0
^
XX N
À1
X
N
i¼1
gðw
i
; yÞ
"#
ð14:3Þ
converges uniformly in probability to
fE½gðw
i
; yÞg
0
X
o
fE½gðw
i
; yÞg ð14:4Þ
Because X
o
is positive definite, y

o
uniquely minimizes expression (14.4). For com-
pleteness, we summarize with a theorem containing regu larity conditions:
theorem 14.1 (Consistency of GMM): Assume that (a) Y is compact; (b) for each
y A Y, gðÁ; yÞ is Borel measurable on W; (c) for each w A W, gðw; ÁÞ is continuous on
Y;(d)jg
j
ðw; yÞja bðwÞ for all y A Y and j ¼ 1; ; L, where bðÁÞ is a nonnegative
function on W such that E½bðwÞ < y; (e)
^
XX !
p
X
o
,anL ÂL positive definite matrix;
and (f ) y
o
is the unique solution to equation (14.1). Then a random vector
^
yy exists
that solves problem (14.2), and
^
yy !
p
y
o
.
If we assume only that X
o
is positive semi definite, then we must directly assume that

y
o
is the unique minimizer of expression (14.4). Occasionally this generality is useful,
but we will not need it.
Under the assumption that gðw; ÁÞ is continuously di¤erentiable on intðYÞ, y
o
A
intðYÞ, and other standard regularity conditions, we can easily derive the limiting
distribution of the GMM estimator. The first-order condition for
^
yy can be written as
X
N
i¼1
‘
y
gðw
i
;
^
yyÞ
"#
0
^
XX
X
N
i¼1
gðw
i

;
^
yyÞ
"#
1 0 ð14:5Þ
Define the L ÂP matrix
G
o
1 E½‘
y
gðw
i
; y
o
Þ ð14:6Þ
which we assume to have full rank P. This assumption essentially means that the
moment conditions (14.1) are nonredundant. Then, by the WLLN and CLT,
N
À1
X
N
i¼1
‘
y
gðw
i
; y
o
Þ!
p

G
o
and N
À1=2
X
N
i¼1
gðw
i
; y
o
Þ¼O
p
ð1Þð14:7Þ
respectively. Let g
i
ðyÞ1 gðw
i
; yÞ. A mean value expansion of
P
N
i¼1
gðw
i
;
^
yyÞ about y
o
,
appropriate standardizations by the sample size, and rep lacing random averages with

their plims gives
0 ¼ G
0
o
X
o
N
À1=2
X
N
i¼1
g
i
ðy
o
ÞþA
o
ffiffiffiffiffi
N
p
ð
^
yy Ày
o
Þþo
p
ð1Þð14:8Þ
Chapter 14422
where
A

o
1 G
0
o
X
o
G
o
ð14:9Þ
Since A
o
is positive definite under the given assumptions, we have
ffiffiffiffiffi
N
p
ð
^
yy Ày
o
Þ¼ÀA
À1
o
G
0
o
X
o
N
À1=2
X

N
i¼1
g
i
ðy
o
Þþo
p
ð1Þ!
d
Normalð0; A
À1
o
B
o
A
À1
o
Þ
ð14:10Þ
where
B
o
1 G
0
o
X
o
L
o

X
o
G
o
ð14:11Þ
and
L
o
1 E½g
i
ðy
o
Þg
i
ðy
o
Þ
0
¼Var½g
i
ðy
o
Þ ð14:12Þ
Expression (14.10) gives the influence function representation for the GMM estima-
tor, and it also gives the limiting distribution of the GMM estimator. We summarize
with a theorem, which is essentially given by Newey and McFadden (1994, Theorem
3.4):
theorem 14.2 (Asymptotic Normality of GMM): In addition to the assumptions in
Theorem 14.1, assume that (a) y
o

is in the interior of Y; (b) gðw; ÁÞ is continuously
di¤erentiable on the interior of Y for all w A W; (c) each element of gðw; y
o
Þ has finite
second moment; (d) each element of ‘
y
gðw; yÞ is bounded in absolute value by a
function bðwÞ, where E½bðwÞ < y; and (e) G
o
in expression (14.6) has rank P. Then
expression (14.10) holds, and so Avarð
^
yyÞ¼A
À1
o
B
o
A
À1
o
=N.
Estimating the asymptotic variance of the GMM estimator is easy once
^
yy has been
obtained. A consistent estimator of L
o
is given by
^
LL 1 N
À1

X
N
i¼1
g
i
ð
^
yyÞg
i
ð
^
yyÞ
0
ð14:13Þ
and Avarð
^
yyÞ is estimated as
^
AA
À1
^
BB
^
AA
À1
=N, where
^
AA 1
^
GG

0
^
XX
^
GG;
^
BB 1
^
GG
0
^
XX
^
LL
^
XX
^
GG ð14:14Þ
and
^
GG 1 N
À1
X
N
i¼1
‘
y
g
i
ð

^
yyÞð14:15Þ
Generalized Method of Moments and Minimum Distance Estimation 423
As in the linear case in Section 8.3.3, an optimal weighting matrix exists for the
given moment conditions:
^
XX should be a consistent estimator of L
À1
o
.WhenX
o
¼ L
À1
o
,
B
o
¼ A
o
and Avar
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ¼ðG
0
o

L
À1
o
G
o
Þ
À1
. Thus the di¤erence in asymptotic
variances between the general GMM estimato r and the estimator with plim
^
XX ¼ L
À1
o
is
ðG
0
o
X
o
G
o
Þ
À1
ðG
0
o
X
o
L
o

X
o
G
o
ÞðG
0
o
X
o
G
o
Þ
À1
ÀðG
0
o
L
À1
o
G
o
Þ
À1
ð14:16Þ
This expression can be shown to be positive semidefinite using the same argument as
in Chapter 8 (see Problem 8.5).
In order to obtain an asymptotically e‰cient GMM estimator we nee d a prelimi-
nary estimator of y
o
in order to obtain

^
LL. Let
^
^
yy
^
yy be such an estimator, and define
^
LL as
in expression (14.13) but with
^
^
yy
^
yy in place of
^
yy. Then, an e‰cient GMM estimator
[given the funct ion gðw; yÞ] solves
min
y AY
X
N
i¼1
gðw
i
; yÞ
"#
0
^
LL

À1
X
N
i¼1
gðw
i
; yÞ
"#
ð14:17Þ
and its asymptotic variance is estimated as
Av
^
aarð
^
yyÞ¼ð
^
GG
0
^
LL
À1
^
GGÞ
À1
=N ð14:18Þ
As in the linear case, an optimal GMM estimator is called the minimum chi-square
estimator because
N
À1=2
X

N
i¼1
g
i
ð
^
yyÞ
"#
0
^
LL
À1
X
N
i¼1
N
À1=2
g
i
ð
^
yyÞ
"#
ð14:19Þ
has a limiting chi-square distribution with L À P degrees of freedom under the con-
ditions of Theorem 14.2. Therefore, the value of the objective function (properly
standardized by the sample size) can be used as a test of any overidentifying restric-
tions in equation (14.1) when L > P. If statistic (14.19) exceeds the relevant critical
value in a w
2

LÀP
distribution, then equation (14.1) must be rejected: at least some of
the moment conditions are not supported by the data. For the linear model, this is
the same statistic given in equation (8.49).
As always, we can test hypotheses of the form H
0
: cðy
o
Þ¼0, where cðy Þ is a Q Â 1
vector, Q a P, by using the Wald approach and the appropriate variance matrix
estimator. A statistic based on the di¤erence in objective functions is also available if
the minimum chi-square estimator is used so that B
o
¼ A
o
. Let
~
yy denote the solution
to problem (14.17) subject to the restrictions cðyÞ¼0, and let
^
yy denote the unrestricted
estimator solving problem (14.17); importantly, these both use the same weighting
Chapter 14424
matrix
^
LL
À1
. Typically,
^
LL is obtained from a first-stage, unrestricted estimator.

Assuming that the constraints can be written in implicit form and satisfy the condi-
tions discussed in Section 12.6.2, the GMM distance statistic (or GMM criterion
function statistic) has a limiting w
2
Q
distribution:
X
N
i¼1
g
i
ð
~
yyÞ
"#
0
^
LL
À1
X
N
i¼1
g
i
ð
~
yyÞ
"#
À
X

N
i¼1
g
i
ð
^
yyÞ
"#
0
^
LL
À1
X
N
i¼1
g
i
ð
^
yyÞ
"#()
=N !
d
w
2
Q
ð14:20Þ
When applied to linear GMM problems, we obtain the statistic in equation (8.45).
One nice feature of expression (14.20) is that it is invariant to reparameterization of
the null hypothesis, just as the quasi-LR statistic is invariant for M-estimation.

Therefore, we might prefer statistic (14.20) over the Wald statistic (8.48) for testing
nonlinear restrictions in linear models. Of course, the computation of expression
(14.20) is more di‰cult because we would actually need to carry out estimation sub-
ject to nonlinear restrictions.
A nice application of the GMM methods discussed in this section is two-step esti-
mation procedures, which arose in Chapters 6, 12, and 13. Suppose that the estimator
^
yy—it could be an M-estimator or a GMM estimator—depends on a first-stage esti-
mator,
^
gg. A unified approach to obtaining the asymptotic variance of
^
yy is to stack the
first-order conditions for
^
yy and
^
gg into the same function gðÁÞ. This is always possible
for the estimators encountered in this book. For example, if
^
gg is an M-estimator
solving
P
N
i¼1
sðw
i
;
^
ggÞ¼0, and

^
yy is a two-step M-estimator solving
X
N
i¼1
hðw
i
;
^
yy;
^
ggÞ ¼ 0 ð14:21Þ
then we can obtain the asymptotic variance of
^
yy by defining
gðw; y; gÞ¼
hðw; y; gÞ
sðw; gÞ

and applying the GMM formulas. The first-order condition for the full GMM prob-
lem reproduces the first-order conditions for each estimator separately.
In general, either
^
gg,
^
yy, or both might themselves be GMM estimators. Then,
stacking the orthogonality conditions into one vector can simplify the derivation of
the asymptotic variance of the second-step estimator
^
yy while also ensuring e‰cient

estimation when the optimal weighting matrix is used.
Finally, sometimes we want to know whether adding additional moment con-
ditions does not improve the e‰ciency of the minimum chi-square e stimator. (Adding
Generalized Method of Moments and Minimum Distance Estimation 425
additional moment conditions can never reduce asymptotic e‰ciency, provided an
e‰cient weighting matrix is used.) In other words, if we start with equation (14.1) but
add new moments of the form E½hðw; y
o
Þ ¼ 0, when does using the extra moment
conditions yield the same asymptotic variance as the original moment conditions?
Breusch, Qian, Schmidt, and Wyhowski (1999) prove som e general redundancy
results for the minimum chi-square estimator. Qian and Schmidt (1999) study the
problem of adding moment conditions that do not depend on unknown parameters,
and they characterize when such moment conditions improve e‰ciency.
14.2 Estimation under Orthogonality Conditions
In Chapter 8 we saw how linear systems of equations can be estimated by GMM
under certain orthogonality conditions. In general applications, the moment con-
ditions (14.1) almost always arise from assumptions that disturbances are uncorre-
lated with exogenous variables. For a G Â 1 vector rðw
i
; yÞ and a G Â L matrix Z
i
,
assume that y
o
satisfies
E½Z
0
i
rðw

i
; y
o
Þ ¼ 0 ð14:22Þ
The vector function rðw
i
; yÞ can be thought of as a generalized residual function. The
matrix Z
i
is usually called the matrix of instruments. Equation (14.22) is a special case
of equation (14.1) with gðw
i
; yÞ1 Z
0
i
rðw
i
; yÞ. In what follows, write r
i
ðyÞ1 rðw
i
; yÞ.
Identification requires that y
o
be the only y A Y such that equation (14.22) holds.
Condition e of the asy mptotic normality result Theorem 14.2 requires that rank
E½Z
0
i
‘

y
r
i
ðy
o
Þ ¼ P (necessary is L b P). Thus, while Z
i
must be orthogonal to r
i
ðy
o
Þ,
Z
i
must be su‰ciently correlated with the G ÂP Jacobian, ‘
y
r
i
ðy
o
Þ. In the linear case
where rðw
i
; yÞ¼y
i
À X
i
y, this requirement reduces to EðZ
0
i

X
i
Þ having full column
rank, which is simply Assumption SIV.2 in Chapter 8.
Given the instruments Z
i
, the e‰cient estimator can be obtained as in Section 14.1.
A preliminary estimator
^
^
yy
^
yy is usually obtained with
^
XX 1 N
À1
X
N
i¼1
Z
0
i
Z
i
!
À1
ð14:23Þ
so that
^
^

yy
^
yy solves
min
y A Y
X
N
i¼1
Z
0
i
r
i
ðyÞ
"#
0
N
À1
X
N
i¼1
Z
0
i
Z
i
"#
À1
X
N

i¼1
Z
0
i
r
i
ðyÞ
"#
ð14:24Þ
Chapter 14426
The solution to proble m (14.24) is called the nonlinear system 2SLS estimator;itisan
example of a nonlinear instrumental variables estimator.
From Section 14.1, we know that the nonlinear system 2SLS estimator is guaran-
teed to be the e‰cient GMM estimator if for some s
2
o
> 0,
E½Z
0
i
r
i
ðy
o
Þr
i
ðy
o
Þ
0

Z
i
¼s
2
o
EðZ
0
i
Z
i
Þ
Generally, this is a strong assumption. Instead, we can obtain the minimum chi-square
estimator by obtaining
^
LL ¼ N
À1
X
N
i¼1
Z
0
i
r
i
ð
^
^
yy
^
yyÞr

i
ð
^
^
yy
^
yyÞ
0
Z
i
ð14:25Þ
and using this in expression (14.17).
In some cases more structure is available that leads to a three-stage least squares
estimator. In particular, suppose that
E½Z
0
i
r
i
ðy
o
Þr
i
ðy
o
Þ
0
Z
i
¼EðZ

0
i
W
o
Z
i
Þð14:26Þ
where W
o
is the G Â G matrix
W
o
¼ E½r
i
ðy
o
Þr
i
ðy
o
Þ
0
ð14:27Þ
When E½r
i
ðy
o
Þ ¼ 0, as is almost always the case under assumption (14.22), W
o
is the

variance matrix of r
i
ðy
o
Þ. As in Chapter 8, assumption (14.26) is a kind of system
homoskedasticity assumption.
By iterated expectations, a su‰cient condition for assumption (14.26) is
E½r
i
ðy
o
Þr
i
ðy
o
Þ
0
jZ
i
¼W
o
ð14:28Þ
However, assumption (14.26) can hold in cases where assumption (14.28) does not.
If assumption (14.26) holds, then L
o
can be estimated as
^
LL ¼ N
À1
X

N
i¼1
Z
0
i
^
WWZ
i
ð14:29Þ
where
^
WW ¼ N
À1
X
N
i¼1
r
i
ð
^
^
yy
^
yyÞr
i
ð
^
^
yy
^

yyÞð14:30Þ
and
^
^
yy
^
yy is a preliminary estimator. The resulting GMM estimator is usually called the
nonlinear 3SLS (N3SLS) estimator. The name is a holdover from the traditional
Generalized Method of Moments and Minimum Distance Estimation 427
3SLS estimator in linear systems of equations; there are not really three estimation
steps. We should remember that nonlinear 3SLS is generally ine‰cient when as-
sumption (14.26) fails.
The Wald statistic and the QLR statistic can be computed as in Section 14.1. In
addition, a score statistic is sometimes useful. Let
~
~
yy
~
yy be a preliminary ine‰cient esti-
mator with Q restrictions imposed. The estimator
~
~
yy
~
yy would usually come from prob-
lem (14.24) subject to the restrictions cðyÞ¼0. Let
~
LL be the estimated weighting
matrix from equation (14.25) or (14.29), based on
~

~
yy
~
yy. Let
~
yy be the minimum chi-
square estimator using weighting matrix
~
LL
À1
. Then the score statistic is based on the
limiting distribution of the score of the unrestricted objective function evaluated at
the restricted estimates, properly standardized:
N
À1
X
N
i¼1
Z
0
i
‘
y
r
i
ð
~
yyÞ
"#
0

~
LL
À1
N
À1=2
X
N
i¼1
Z
0
i
r
i
ð
~
yyÞ
"#
ð14:31Þ
Let
~
ss
i
1
~
GG
0
~
LL
À1
Z

0
i
~
rr
i
, where
~
GG is the first matrix in expression (14.31), and let s
o
i
1
G
0
o
L
À1
o
Z
0
i
r
o
i
. Then, following the proof in Section 12.6.2, it can be shown that equa-
tion (12.67) holds with A
o
1 G
0
o
L

À1
o
G
o
. Further, since B
o
¼ A
o
for the minimum chi-
square estimator, we obtain
LM ¼
X
N
i¼1
~
ss
i
!
0
~
AA
À1
X
N
i¼1
~
ss
i
!
=N ð14:32Þ

where
~
AA ¼
~
GG
0
~
LL
À1
~
GG. Under H
0
and the usual regularity conditions, LM has a limit-
ing w
2
Q
distribution.
14.3 Systems of Nonlinear Equations
A leading application of the results in Section 14.2 is to estimation of the parameters
in an implicit set of nonlinear equations, such as a nonlinear simultaneous equations
model. Partition w
i
as y
i
A R
J
, x
i
A R
K

and, for h ¼ 1; ; G, suppose we have
q
1
ðy
i
; x
i
; y
o1
Þ¼u
i1
.
.
.
q
G
ðy
i
; x
i
; y
oG
Þ¼u
iG
ð14:33Þ
where y
oh
is a P
h
 1 vector of parameters. As an example, write a two-equation

SEM in the population as
Chapter 14428
y
1
¼ x
1
d
1
þ g
1
y
g
2
2
þ u
1
ð14:34Þ
y
2
¼ x
2
d
2
þ g
3
y
1
þ u
2
ð14:35Þ

(where we drop ‘‘o’’ to index the parameters). This model, unlike those covered in
Section 9.5, is nonlinear in the parameters as well as the endogenous variables. Nev-
ertheless, assuming that Eðu
g
jxÞ¼0, g ¼ 1; 2, the parameters in the system can be
estimated by GMM by defining q
1
ðy; x; y
1
Þ¼y
1
À x
1
d
1
À g
1
y
g
2
2
and q
2
ðy; x; y
2
Þ¼
y
2
À x
2

d
2
À g
3
y
1
.
Generally, the equations (14.33) need not actually determine y
i
given the exoge-
nous variables and disturbances; in fact, nothing requires J ¼ G. Sometimes equations
(14.33) represent a system of orthogonality conditions of the form E½q
g
ðy; x; y
og
Þjx¼
0, g ¼ 1; ; G. We will see an example later.
Denote the P Â1 vector of all parameters by y
o
, and the parameter space by Y H
R
P
. To identify the parameters we need the errors u
ih
to satisfy some orthogonality
conditions. A general assumption is, for some subvector x
ih
of x
i
,

Eðu
ih
jx
ih
Þ¼0; h ¼ 1; 2; ; G ð14:36Þ
This allows elements of x
i
to be correlated with some errors, a situation that some-
times arises in practice (see, for example, Chapter 9 and Wooldridge, 1996). Under
assumption (14.36), let z
ih
1 f
h
ðx
ih
Þ be a 1 ÂL
h
vector of possibly nonlinear func-
tions of x
i
. If there are no restrictions on the y
oh
across equations we should have
L
h
b P
h
so that each y
o h
is identified. By iterated expectations, for all h ¼ 1; ; G,

Eðz
0
ih
u
ih
Þ¼0 ð14:37Þ
provided appropriate moments exist. Therefore, we obtain a set of orthogonality
conditions by defining the G Â L matrix Z
i
as the block diagonal matrix with z
ig
in
the gth block:
Z
i
1
z
i1
00ÁÁÁ 0
0z
i2
0 ÁÁÁ 0
.
.
.
.
.
.
000ÁÁÁ z
iG

2
6
6
6
4
3
7
7
7
5
ð14:38Þ
where L 1 L
1
þ L
2
þÁÁÁþL
G
. Letting rðw
i
; yÞ1 qðy
i
; x
i
; yÞ1 ½q
i1
ðy
1
Þ; ; q
iG
ðy

G
Þ
0
,
equation (14.22) holds under assumption (14.36).
When there are no restrictions on the y
g
across equations and Z
i
is chosen as in
matrix (14.38), the system 2SLS estimator reduces to the nonlinear 2SLS (N2SLS)
estimator (Amemiya, 1974) equation by equation. That is, for each h, the N2SLS
estimator solves
Generalized Method of Moments and Minimum Distance Estimation 429
min
y
h
X
N
i¼1
z
0
ih
q
ih
ðy
h
Þ
"#
0

N
À1
X
N
i¼1
z
0
ih
z
ih
!
À1
X
N
i¼1
z
0
ih
q
ih
ðy
h
Þ
"#
ð14:39Þ
Given only the orthogonality conditions (14.37), the N2SLS estimator is the e‰cient
estimator of y
oh
if
Eðu

2
ih
z
0
ih
z
ih
Þ¼s
2
oh
Eðz
0
ih
z
ih
Þð14:40Þ
where s
2
oh
1 Eðu
2
ih
Þ; su‰cient for condition (14.40) is Eðu
2
ih
jx
ih
Þ¼s
2
oh

. Let
^
^
yy
^
yy
h
denote
the N2SLS estim ator. Then a consistent estimator of s
2
oh
is
^
ss
2
h
1 N
À1
X
N
i¼1
^
^
uu
^
uu
2
ih
ð14:41Þ
where

^
^
uu
^
uu
ih
1 q
h
ðy
i
; x
i
;
^
^
yy
^
yy
h
Þ are the N2SLS residuals. Under assumptions (14.37) and
(14.40), the asymptotic variance of
^
^
yy
^
yy
h
is estimated as
^
ss

2
h
X
N
i¼1
z
0
ih
‘
y
h
q
ih
ð
^
^
yy
^
yy
h
Þ
"#
0
X
N
i¼1
z
0
ih
z

ih
!
À1
X
N
i¼1
z
0
ih
‘
y
h
q
ih
ð
^
^
yy
^
yy
h
Þ
"#
8
<
:
9
=
;
À1

ð14:42Þ
where ‘
y
h
q
ih
ð
^
^
yy
^
yy
h
Þ is the 1 ÂP
h
gradient.
If a ssumption (14.37) holds but assumption (14.40) does not, the N2SLS estimator
is still
ffiffiffiffiffi
N
p
-consistent, but it is not the e‰cient estimator that uses the orthogonality
condition (14.37) whenever L
h
> P
h
[and expression (14.42) is no longer valid]. A
more e‰cient estimator is obtained by solving
min
y

h
X
N
i¼1
z
0
ih
q
ih
ðy
h
Þ
"#
0
N
À1
X
N
i¼1
^
^
uu
^
uu
2
ih
z
0
ih
z

ih
!
À1
X
N
i¼1
z
0
ih
q
ih
ðy
h
Þ
"#
with asymptotic variance estimated as
X
N
i¼1
z
0
ih
‘
y
h
q
ih
ð
^
^

yy
^
yy
h
Þ
"#
0
X
N
i¼1
^
^
uu
^
uu
2
ih
z
0
ih
z
ih
!
À1
X
N
i¼1
z
0
ih

‘
y
h
q
ih
ð
^
^
yy
^
yy
h
Þ
"#
8
<
:
9
=
;
À1
This estimator is asymptotically equivalent to the N2SLS estimator if assumption
(14.40) happens to hold.
Rather than focus on one equation at a time, we can increase e‰ci ency if we esti-
mate the equations simultaneously. One reason for doing so is to impose cross
equation restrictions on the y
oh
. The system 2SLS estimator can be used for these
Chapter 14430
purposes, where Z

i
generally has the form (14.38). But this estimator does not exploit
correlation in the errors u
ig
and u
ih
in di¤erent equations.
The e‰cient estimator that uses all orthogonality conditions in equation (14.37) is
just the GMM estimato r with
^
LL given by equation (14.25), where r
i
ð
^
^
yy
^
yyÞ is the G Â1
vector of system 2SLS residuals,
^
^
uu
^
uu
i
. In other words, the e‰cient GMM estimator
solves
min
y A Y
X

N
i¼1
Z
0
i
q
i
ðyÞ
"#
0
N
À1
X
N
i¼1
Z
0
i
^
^
uu
^
uu
i
^
^
uu
^
uu
0

i
Z
i
!
À1
X
N
i¼1
Z
0
i
q
i
ðyÞ
"#
ð14:43Þ
The asymptotic variance of
^
yy is estimated as
X
N
i¼1
Z
0
i
‘
y
q
i
ð

^
yyÞ
"#
0
X
N
i¼1
Z
0
i
^
^
uu
^
uu
i
^
^
uu
^
uu
0
i
Z
i
!
À1
X
N
i¼1

Z
0
i
‘
y
q
i
ð
^
yyÞ
"#
8
<
:
9
=
;
À1
Because this is the e‰cient GMM estimator, the QLR statistic can be used to test
hypotheses about y
o
. The Wald statistic can also be applied.
Under the homoskedasticity assumption (14.26) with r
i
ðy
o
Þ¼u
i
, the nonlinear
3SLS estimator, which solves

min
y A Y
X
N
i¼1
Z
0
i
q
i
ðyÞ
"#
0
N
À1
X
N
i¼1
Z
0
i
^
WWZ
i
!
À1
X
N
i¼1
Z

0
i
q
i
ðyÞ
"#
is e‰cient, and its asymptotic variance is estimated as
X
N
i¼1
Z
0
i
‘
y
r
i
ð
^
yyÞ
"#
0
X
N
i¼1
Z
0
i
^
WWZ

i
!
À1
X
N
i¼1
Z
0
i
‘
y
r
i
ð
^
yyÞ
"#
8
<
:
9
=
;
À1
The N3SLS estimator is used widely for systems of the form (14.33), but, as we dis-
cussed in Section 9.6, there are many cases where assumption (14.26) must fail when
di¤erent instruments are needed for di¤erent equations.
As an example, we show how a hedonic price system fits into this framework.
Consider a linear demand and supply system for G attributes of a good or service (see
Epple, 1987; Kahn and Lang, 1988; and Wooldridge, 1996). The demand and supply

system is written as
demand
g
¼ h
1g
þ wa
1g
þ x
1
b
1g
þ u
1g
; g ¼ 1; ; G
supply
g
¼ h
2g
þ wa
2g
þ x
2
b
2g
þ u
2g
; g ¼ 1; ; G
Generalized Method of Moments and Minimum Distance Estimation 431
where w ¼ðw
1

; ; w
G
Þ is the 1 Â G vector of attribute prices. The demand equations
usually represent an individual or household; the supply equations can represent an
individual, firm, or employer.
There are several tricky issues in estimating either the demand or supply function
for a particular g. First, the attribute prices w
g
are not directly observed. What is
usually observed are the equilibrium quantities for each attribute and each cross
section unit i; call these q
ig
, g ¼ 1; ; G. (In the hedonic systems literature these are
often denoted z
ig
, but we use q
ig
here because they are endogenous variables, and we
have been using z
i
to denote exogenous variables.) For example, the q
ig
can be fea-
tures of a house, such as size, number of bathrooms, and so on. Along with these
features we observe the equilibrium price of the good, p
i
, which we assume follows a
quadratic hedonic price function:
p
i

¼ g þ q
i
c þ q
i
Pq
0
i
=2 þ x
i3
d þ x
i3
Gq
0
i
þ u
i3
ð14:44Þ
where x
i3
is a vector of variabl es that a¤ect p
i
, P is a G Â G symmetric matrix, and G
is a G ÂG matrix.
A key point for identifying the demand and supply functions is that w
i
¼ qp
i
=qq
i
,

which, under equation (14.44), becomes w
i
¼ q
i
P þ x
i3
G,orw
ig
¼ q
i
p
g
þ x
i3
g
g
for
each g. By substitution, the equilibrium estimating equations can be written as
equation (14.44) plus
q
ig
¼ h
1g
þðq
i
P þ x
i3
GÞa
1g
þ x

i1
b
1g
þ u
i1g
; g ¼ 1; ; G ð14:45Þ
q
ig
¼ h
2g
þðq
i
P þ x
i3
GÞa
2g
þ x
i2
b
2g
þ u
i2g
; g ¼ 1; ; G ð14:46Þ
These two equations are linear in q
i
; x
i1
; x
i2
, and x

i3
but nonlinear in the parameters.
Let u
i1
be the G Â1 vector of attribut e demand disturbances and u
i2
the G Â1
vector of attribute supply disturbances. What are reasonable assumptions about
u
i1
; u
i2
,andu
i3
? It is almost always assumed that equation (14.44) represents a con-
ditional expectation with no important unobserved factors; this assumption means
Eðu
i3
jq
i
; x
i
Þ¼0, where x
i
contains all elements in x
i1
; x
i2
, and x
i3

. The properties of
u
i1
and u
i2
are more subtle. It is clear that these cannot be uncorrelated with q
i
, and
so equations (14.45) and (14.46) contain endogenous explanatory variables if P 0 0.
But there is another problem, pointed out by Bartik (1987), Epple (1987), and Kahn
and Lang (1988): because of matching that happens between individual buyers and
sellers, x
i2
is correlated with u
i1
, and x
i1
is correlated with u
i2
. Consequently, what
would seem to be the obvious IVs for the demand equations (14.45)—the factors
shifting the supply curve—are endogenous to equation (14.45). Fortunately, all is not
lost: if x
i3
contains exogenous factors that a¤ect p
i
but do not appear in the struc-
Chapter 14432
tural demand and supply functions, we can use these as instruments in both the de-
mand and supply equations. Specifically, we assume

Eðu
i1
jx
i1
; x
i3
Þ¼0; Eðu
i2
jx
i2
; x
i3
Þ¼0; Eðu
i3
jq
i
; x
i
Þ¼0 ð14:47Þ
Common choices for x
i3
are geographical or industry dummy indicators (for exam-
ple, Montgomery, Shaw, and Benedict, 1992; Hagy, 1998), where the assumption is
that the demand and supply functions do not change across region or industry but the
type of matching does, and therefore p
i
can di¤er systematically across region or in-
dustry. Bartik (1987) discusses how a randomized experiment can be used to create
the elements of x
i3

.
For concreteness, let us focus on estimating the set of demand functions. If P ¼ 0,
so that the quadratic in q
i
does not appear in equation (14.44), a simple two-step
procedure is available: (1) estimate equation (14.44) by OLS, and obtain
^
ww
ig
¼
^
cc
g
þ x
i3
^
gg
g
for each i and g; (2) run the regression q
ig
on 1,
^
ww
i
; x
i1
; i ¼ 1; ; N. Under
assumptions (14.47) and identification assumptions, this method produces
ffiffiffiffiffi
N

p
-
consistent, asymptotically normal estimators of the parameters in demand equation
g. Because the second regression involves generated regressors, the standard errors
and test statistics should be adjusted.
It is clear that, without restrictions on a
1g
, the order condition necessary for iden-
tifying the demand parameters is that the dimension of x
i3
, say K
3
, must exceed G.If
K
3
< G then E½ðw
i
; x
i1
Þ
0
ðw
i
; x
i1
Þ has less than full rank, and the OLS rank condition
fails. If we make exclusion restrictions on a
1g
, fewer elements are needed in x
i3

. In the
case that only w
ig
appears in the demand equation for attri bute g, x
i3
can be a scalar,
provided its interaction with q
ig
in the hedonic price system is significant ðg
gg
0 0Þ.
Checking the analogue of the rank condition in general is somewhat complicated; see
Epple (1987) for discussion.
When w
i
¼ q
i
P þ x
i3
G, w
i
is correlated with u
i1g
, so we must modify the two-step
procedure. In the second step, we can use instruments for
^
ww
i
and perform 2SLS rather
than OLS. Assuming that x

i3
has enough elements, the demand equations are still
identified. If only w
ig
appears in demand
ig
, su‰cient for identification is that an ele-
ment of x
i3
appears in the linear projection of w
ig
on x
i1
, x
i3
. This assumption can
hold even if x
i3
has only a single element. For the matching reasons we discussed
previously, x
i2
cannot be used as instruments fo r
^
ww
i
in the demand equation.
Whether P ¼ 0 or not, more e‰cient estimators are obtained from the fu ll demand
system and the hedonic price function. Write
q
0

i
¼ h
1
þðq
i
P þ x
i3
GÞA
1
þ x
i1
B
1
þ u
i1
Generalized Method of Moments and Minimum Distance Estimation 433
along with equation (14.44). Then ðx
i1
; x
i3
Þ (and functions of these) can be used as
instruments in any of the G demand equations, and ðq
i
; x
i
Þ act as IVs in equation
(14.44). (It may be that the supply function is not even specified, in which case x
i
contains only x
i1

and x
i3
.) A first-stage estimator is the nonlinear system 2SLS esti-
mator. Then the system can be estimated by the mini mum chi-square estimator that
solves problem (14.43). When restricting attention to demand equations plus the
hedonic price equation, or supply equations plus the hedonic price equation, nonlinear
3SLS is e‰cient under certain assumptions. If the demand and supply equations are
estimated together, the key assumption (14.26) that makes nonlinear 3SLS asymp-
totically e‰cient cannot be expected to hold; see Wooldridge (1996) for discussion.
If one of the demand functions is of primary interest, it may make sense to estimate
it along with equation (14.44), by GMM or nonlinear 3SLS. If the demand functions
are written in inverse form, the resulting system is linear in the parameters, as shown
in Wooldridge (1996).
14.4 Panel Data Applica tions
As we saw in Chapter 11, system IV methods are needed in certain panel data con-
texts. In the current case, our interest is in nonlinear panel data models that cannot
be estimated using linear methods. We hold o¤ on discussing nonlinear panel data
models explicitly containing unobserved e¤ects until Part IV.
One increasingly popular use of panel data is to test rationality in economic models
of individual, family, or firm behavior (see, for example, Shapiro, 1984; Zeldes, 1989;
Keane and Runkle, 1992; Shea, 1995). For a random draw from the population we
assume that T time periods are available. Suppose that an economic theory implies
that
E½r
t
ðw
t
; y
o
Þjw

tÀ1
; ; w
1
Þ¼0; t ¼ 1; ; T ð14:48Þ
where, for simplicity, r
t
is a scalar. These conditional moment restrictions are often
implied by rational expectations, under the assumption that the decision horizon is
the same length as the sampling period. For example, consi der a standard life-cycle
model of consumption. Let c
it
denote consumption of family i at time t,leth
it
denote
taste shifters, let d
o
denote the common rate of time preference, and let a
j
it
denote the
return for family i from holding asset j from period t À 1tot. Under the assumption
that utility is given by
uðc
it
; y
it
Þ¼expðh
it
b
o

Þc
1Àl
o
it
=ð1 À l
o
Þð14:49Þ
the Euler equation is
Chapter 14434
E½ð1 þ a
j
it
Þðc
it
=c
i; tÀ1
Þ
Àl
o
jI
i; tÀ1
¼ð1 þ d
o
Þ
À1
expðx
it
b
o
Þð14:50Þ

where I
it
is family i’s information set at time t and x
it
1 h
i; tÀ1
À h
it
; equation (14.50)
assumes that h
it
À h
i; tÀ1
A I
i; tÀ1
, an assumption which is often reasonable. Given
equation (14.50), we can define a residual function for each t:
r
it
ðyÞ¼ð1 þ a
j
it
Þðc
it
=c
i; tÀ1
Þ
Àl
À expðx
it

bÞð14:51Þ
where ð1 þ dÞ
À1
is absorbed in an intercept in x
it
. Let w
it
contain c
it
, c
i; tÀ1
, a
it
, and
x
it
. Then condition (14.48) holds, and l
o
and b
o
can be estimated by GMM.
Returning to condition (14.48), valid instruments at time t are functions of infor-
mation known at time t À 1:
z
t
¼ f
t
ðw
tÀ1
; ; w

1
Þð14:52Þ
The T Â1 residual vector is rðw; yÞ¼½r
1
ðw
1
; yÞ; ; r
T
ðw
T
; yÞ
0
, and the matrix of
instruments has the same form as matrix (14.38) for each i (with G ¼ T). Then, the
minimum chi-square estimator can be obtained after using the system 2SLS estima-
tor, although the choice of instruments is a nontrivial matter. A common choice is
linear and quadratic functions of variables lagged one or two time periods.
Estimation of the optimal weighting matrix is somewhat simplified under the con-
ditional moment restrictions (14.48). Recall from Section 14.2 that the optimal esti-
mator uses the inverse of a consistent estimator of L
o
¼ E½Z
0
i
r
i
ðy
o
Þr
i

ðy
o
Þ
0
Z
i
. Under
condition (14.48), this matrix is blo ck diagonal. Dropping the i subscript, the ðs ; tÞ
block is E½r
s
ðy
o
Þr
t
ðy
o
Þz
0
s
z
t
. For concreteness, assume that s < t. Then z
t
; z
s
, and r
s
ðy
o
Þ

are all functions of w
tÀ1
; w
tÀ2
; ; w
1
. By iterated expectations it follows that
E½r
s
ðy
o
Þr
t
ðy
o
Þz
0
s
z
t
¼Efr
s
ðy
o
Þz
0
s
z
t
E½r

t
ðy
o
Þjw
tÀ1
; ; w
1
g ¼ 0
and so we only need to estimate the diagonal blocks of E½Z
0
i
r
i
ðy
o
Þr
i
ðy
o
Þ
0
Z
i
:
N
À1
X
N
i¼1
^

^
rr
^
rr
2
it
z
0
it
z
it
ð14:53Þ
is a consistent estimator of the tth block, where the
^
^
rr
^
rr
it
are obtained from an ine‰cient
GMM estimator.
In cases where the data frequency does not match the horizon relevant for decision
making, the optimal matrix does not have the block diagonal form: some o¤-diagonal
blocks will be nonzero. See Hansen (1982) for the pure time series case.
Ahn and Schmidt (1995) apply nonlinear GMM methods to estimate the linear,
unobserved e¤ects AR(1) model. Some of the orthogonality restrictions they use are
nonlinear in the parameters of interest. In Part IV we will cover nonlinear panel data
Generalized Method of Moments and Minimum Distance Estimation 435
models with unobserved e¤ects. For the consumption example, we would like to allow
for a family-specific rate of time preference, as well as unobserved family tastes.

Orthogonality conditions can often be obtained in such cases, but they are not as
straightforward to obtain as in the previous example.
14.5 E‰cient Estimation
In Chapter 8 we obtained the e‰cient weighting matrix for GMM estimation of
linear models, and we extended that to nonlinear models in Section 14.1. In Chapter
13 we asserted that maximum likelihood estimation has some important e‰ciency
properties. We are now in a position to study a framework that allows us to show the
e‰ciency of an estimator within a particular class of estimators, and also to find
e‰cient estimators with in a stated class. Our approach is essentially that in Newey
and McFadden (1994, Section 5.3), although we will not use the weakest possible
assumptions. Bates and White (1993) proposed a very similar framework and also
considered time series problems.
14.5.1 A General E‰ciency Framework
Most estimators in econometrics—and all of the ones we have studied—are
ffiffiffiffiffi
N
p
-
asymptotically normal, with variance matrices of the form
V ¼ A
À1
E½sðwÞsðwÞ
0
ðA
0
Þ
À1
ð14:54Þ
where, in most cases, sðwÞ is the score of an objective function (evaluated at y
o

) and
A is the expected value of the Jacobian of the score, again evaluated at y
o
.(We
suppress an ‘‘o’’ subscript here, as the value of the true paramete r is irrelevant.) All
M-estimators with twice continuously di¤erentiable objective functions (and even
some without) have variance matrices of this form, as do GMM estimators. The fol-
lowing lemma is a useful su‰cient condition for showing that one estimator is more
e‰cient than another.
lemma 14.1 (Relative E‰ciency): Let
^
yy
1
and
^
yy
2
be two
ffiffiffiffiffi
N
p
-asymptotically normal
estimators of the P Â 1 parameter vector y
o
, with asymptotic variances of the form
(14.54) (with appropriate subscripts on A, s, and V). If for some r > 0,
E½s
1
ðwÞs
1

ðwÞ
0
¼rA
1
ð14:55Þ
E½s
2
ðwÞs
1
ðwÞ
0
¼rA
2
ð14:56Þ
then V
2
À V
1
is positive semi definite.
Chapter 14436
The proof of Lemma 14.1 is given in the chapter appendix.
Condition (14.55) is essentially the generalized information matrix equality (GIME)
we introduced in Section 12.5.1 for the estimator
^
yy
1
. Notice that A
1
is necessarily
symmetric and positive definite under condition (14.55). Condition (14.56) is new. In

most cases, it says that the expected outer product of the scores s
2
and s
1
equals the
expected Jacobian of s
2
(evaluated at y
o
). In Section 12.5.1 we claimed that the
GIME plays a role in e‰ciency, and Lemma 14.1 shows that it does so.
Verifying the conditions of Lemma 14.1 is also very convenient for constructing
simple forms of the Hausman (1978) statistic in a variety of contexts. Provided that
the two estimators are jointly asymptotically normally distributed—something that is
almost always true when each is
ffiffiffiffiffi
N
p
-asymptotically normal, and that can be verified
by stacking the first-order representations of the estimators—assumptions (14.55) and
(14.56) imply that the asymptotic covariance between
ffiffiffiffiffi
N
p
ð
^
yy
2
À y
o

Þ and
ffiffiffiffiffi
N
p
ð
^
yy
1
À y
o
Þ
is A
À1
2
Eðs
2
s
0
1
ÞA
À1
1
¼ A
À1
2
ðrA
2
ÞA
À1
1

¼ rA
À1
1
¼ Avar½
ffiffiffiffiffi
N
p
ð
^
yy
1
À y
o
Þ. In other words, the
asymptotic covariance between the (
ffiffiffiffiffi
N
p
-scaled) estimators is equal to the asymptotic
variance of the e‰cient estimator. This equality implies that Avar½
ffiffiffiffiffi
N
p
ð
^
yy
2
À
^
yy

1
Þ ¼
V
2
þ V
1
À C À C
0
¼ V
2
þ V
1
À 2V
1
¼ V
2
À V
1
, where C is the asymptotic covariance.
If V
2
À V
1
is actually positive definite (rather than just positive semidefinite), then
½
ffiffiffiffiffi
N
p
ð
^

yy
2
À
^
yy
1
Þ
0
ð
^
VV
2
À
^
VV
1
Þ
À1
½
ffiffiffiffiffi
N
p
ð
^
yy
2
À
^
yy
1

Þ@
a
w
2
P
under the assumptions of Lemma
14.1, where
^
VV
g
is a consistent estimator of V
g
, g ¼ 1; 2. Statistically significant di¤er-
ences between
^
yy
2
and
^
yy
1
signal some sort of model misspecification. (See Section
6.2.1, where we discussed this form of the Hausman test for comparing 2SLS and
OLS to test whether the explanatory variables are exogenous.) If assumptions (14.55)
and (14.56) do not hold, this standard form of the Hausman statistic is invalid.
Given Lemma 14.1, we can state a condition that implies e‰ciency of an estimator
in an entire class of estimators. It is useful to be somewhat formal in defining the
relevant class of estimators. We do so by introducing an index, t. For each t in an
index set, say, T, the estimator
^

yy
t
has an associated s
t
and A
t
such that the asymp-
totic variance of
ffiffiffiffiffi
N
p
ð
^
yy
t
À y
o
Þ has the form (14.54). The index can be very abstract; it
simply serves to distinguish di¤erent
ffiffiffiffiffi
N
p
-asymptotically normal estimators of y
o
. For
example, in the class of M-estimators, the set T consists of objective functions qðÁ; ÁÞ
such that y
o
uniquely minimizes E½qðw; yÞ over Y, and q satisfies the twice con-
tinuously di¤erentiable and bounded moment assumptions imposed for asymptotic

normality. For GMM with given moment conditions, T is the set of all L Â L posi-
tive definite matrices. We will see another example in Section 14.5.3.
Lemma 14.1 immediately implies the following theorem.
theorem 14.3 (E‰ciency in a Class of Estimators): Let f
^
yy
t
: t A Tg be a class of
ffiffiffiffiffi
N
p
-asymptotically normal estimators with variance matrices of the form (14.54). If
Generalized Method of Moments and Minimum Distance Estimation 437
for some t
Ã
A T and r > 0
E½s
t
ðwÞs
t
Ã
ðwÞ
0
¼rA
t
; all t A T ð14:57Þ
then
^
yy
t

Ã
is asymptotically relatively e‰cient in the class f
^
yy
t
: t A Tg.
This theorem has many applications. If we specify a class of estimators by defining
the index set T, then the estimator
^
yy
t
Ã
is more e‰cient than all other estimators in the
class if we can show condition (14.57). [A partial converse to Theorem 14.3 also
holds; see Newey and McFadden (1994, Section 5.3).] This is not to say that
^
yy
t
Ã
is
necessarily more e‰cient than all possible
ffiffiffiffiffi
N
p
-asymptotically normal estimators. If
there is an estimator that falls outside of the specified class, then Theorem 14.3 does
not help us to compare it with
^
yy
t

Ã
. In this sense, Theorem 14.3 is a more general (and
asymptotic) version of the Gauss-Markov theorem from linear regression analysis:
while the Gauss-Markov theorem states that OLS has the smallest variance in the
class of linear, unbiased estimators, it does not allow us to compare OLS to unbiased
estimators that are not linear in the vector of observations on the dependent variable.
14.5.2 E‰ciency of MLE
Students of econometrics are often told that the maximu m likelihood estimator is
‘‘e‰cient.’’ Unfortunately, in the context of conditional MLE from Chapter 13, the
statement of e‰ciency is usually ambiguous; Manski (1988, Chapter 8) is a notable
exception. Theorem 14.3 allows us to state precisely the class of estimators in which
the conditional MLE is relatively e‰cient. As in Chap ter 13, we let E
y
ðÁjxÞ denote
the expectation with respect to the conditional density f ðy jx; yÞ.
Consider the class of estimators solving the first-order condition
N
À1
X
N
i¼1
gðw
i
;
^
yyÞ1 0 ð14:58Þ
where the P Â 1 function gðw; yÞ such that
E
y
½gðw; yÞjx¼0; all x A X; all y A Y ð14:59Þ

In other words, the class of estimators is indexed by functions g satisfying a zero
conditional moment restriction. We assume the standard regularity conditions from
Chapter 12; in particular, gðw; ÁÞ is continuously di¤erentiably on the interior of Y.
As we showed in Section 13.7, functions g satisfying condition (14.59) generally
have the property
ÀE½‘
y
gðw; y
o
Þjx¼E½gðw; y
o
Þsðw; y
o
Þ
0
jx
Chapter 14438
where sðw; yÞ is the score of log f ðy jx; yÞ (as always, we must impose certain regu-
larity conditons on g and log f ). If we take the expectation of both sides with respect
to x, we obtain condition (14.57) with r ¼ 1, A
t
¼ E½‘
y
gðw; y
o
Þ, and s
t
Ã
ðwÞ¼
Àsðw; y

o
Þ. It follows from Theorem 14.3 that the conditional MLE is e‰cient in
the class of estimators solving equation (14.58), where gðÁÞ satisfies condition (14.59)
and appropriate regularity conditions. Recall from Section 13.5.1 that the asymp-
totic variance of the (centered and standardized) CMLE is fE½sðw; y
o
Þsðw; y
o
Þ
0
g
À1
.
This is an example of an e‰ciency bound because no estimator of the form
(14.58) under condition (14.59) can have an asymptotic variance smaller than
fE½sðw; y
o
Þsðw; y
o
Þ
0
g
À1
(in the matrix sense). When an estimator from this class has
the same asymptotic variance as the CMLE, we way it achieves the e‰ciency bound.
It is important to see that the e‰ciency of the conditional MLE in the class of
estimators solving equation (14.58) under condition (14.59) does not require x to be
ancillary for y
o
: except for regularity conditions, the distribution of x is essentially

unrestricted, and could depend on y
o
. Conditional MLE simply ignore s information
on y
o
that might be contained in the distributio n of x, but so do all other estimators
that are based on condition (14.59).
By choosing x to be empty, we conclude that the unconditional MLE is e‰cient in
the class of estimators based on equation (14.58) with E
y
½gðw; yÞ ¼ 0,ally A Y. This
is a very broad class of estimators, including all of the estimators requiring condition
(14.59): if a function g satisfies condition (14.59), it has zero unconditional mean, too.
Consequently, the unconditional MLE is generally more e‰cient than the condi-
tional MLE. This e‰ciency comes at the price of having to model the joint density of
ðy; xÞ, rather than just the conditional density of y given x. And, if our model for the
density of x is incorrect, the unconditional MLE generally would be inconsistent.
When is CMLE as e‰cient as unconditional MLE for estimating y
o
? Assume th at
the model for the joint density of ðx; yÞ can be expressed as f ðy jx; yÞhðx; dÞ, where y
is the parameter vector of interest, and hðx; d
o
Þ is the marginal density of x for some
vector d
o
. Then, if d does not depend on y in the sense that ‘
y
hðx; dÞ¼0 for all x and
d, x is ancillary for y

o
. In fact, the CMLE is identical to the unconditional MLE. If d
depends on y, the term ‘
y
log½hðx; dÞ generally contains information for estimating
y
o
, and unconditional MLE will be more e‰cient than CMLE.
14.5.3 E‰cient Choice of Instruments under Conditional Moment Restrictions
We can also apply Theorem 14.3 to find the optimal set of instrumental variables
under general conditional moment restrictions. For a G Â 1 vector rðw
i
; yÞ, where
w
i
A R
M
, y
o
is said to satisfy conditional moment restrictions if
E½rðw
i
; y
o
Þjx
i
¼0 ð14:60Þ
Generalized Method of Moments and Minimum Distance Estimation 439
where x
i

A R
K
is a subvector of w
i
. Under assumption (14.60), the matrix Z
i
appearing in equation (14.22) can be any function of x
i
. For a given matrix Z
i
,we
obtain the e‰cient GMM estimator by using the e‰cient weighting matrix. However,
unless Z
i
is the optimal set of instruments, we can generally obtain a more e‰cient
estimator by adding any nonlinear function of x
i
to Z
i
. Because the list of potential
IVs is endless, it is useful to characterize the optimal choice of Z
i
.
The solution to this problem is now pretty well known, and it can be obtained by
applying Theorem 14.3. Let
W
o
ðx
i
Þ1 Var½rðw

i
; y
o
Þjx
i
ð14:61Þ
be the G Â G conditional variance of r
i
ðy
o
Þ given x
i
, and define
R
o
ðx
i
Þ1 E½‘
y
rðw
i
; y
o
Þjx
i
ð14:62Þ
Problem 14.3 asks you to verify that the optimal choice of instruments is
Z
Ã
ðx

i
Þ1 W
o
ðx
i
Þ
À1
R
o
ðx
i
Þð14:63Þ
The optimal instrument matrix is always G ÂP, and so the e‰cient method of
moments estimator solves
X
N
i¼1
Z
Ã
ðx
i
Þ
0
r
i
ð
^
yyÞ¼0
There is no need to use a weighting matrix. Incidentally, by taking gðw; yÞ1
Z

Ã
ðxÞ
0
rðw; yÞ, we obtain a function g satisfying condition (14.59). From our discus-
sion in Section 14.5.2, it follows immediately that the conditional MLE is no less
e‰cient than the optimal IV estimator.
In practice, Z
Ã
ðx
i
Þ is never a known function of x
i
. In some cases the function
R
o
ðx
i
Þ is a known function of x
i
and y
o
and can be easily estimated; this statement is
true of linear SEMs under conditional mean assumptions (see Chapters 8 and 9) and
of multivariate nonlinear regression, which we cover later in this subsection. Rarely
do moment conditions imply a parametric form for W
o
ðx
i
Þ, but sometimes homo-
skedasticity is assumed:

E½r
i
ðy
o
Þr
i
ðy
o
Þjx
i
¼W
o
ð14:64Þ
and W
o
is easily estimated as in equation (14.30) given a preliminary estimate of y
o
.
Since both W
o
ðx
i
Þ and R
o
ðx
i
Þ must be estimated, we must know the asymptotic
properties of GMM with generated instruments. Under conditional moment restric-
tions, generated instruments have no e¤ect on the asymptotic variance of the GMM
estimator. Thus, if the matrix of instruments is Zðx

i
; g
o
Þ for some unknown parame-
Chapter 14440
ter vector g
o
, and
^
gg is an estimator suc h that
ffiffiffiffiffi
N
p
ð
^
gg À g
o
Þ¼O
p
ð1Þ, then the GMM
estimator using the generated instruments
^
ZZ
i
1 Zðx
i
;
^
ggÞ has the same limiting dis-
tribution as the GMM estimator using instruments Zðx

i
; g
o
Þ (using any weighting
matrix). This result follows from a mean value expansion, using the fact that the de-
rivative of each element of Zðx
i
; gÞ with respect to g is orthogonal to r
i
ðy
o
Þ under
condition (14.60):
N
À1=2
X
N
i¼1
^
ZZ
0
i
r
i
ð
^
yyÞ¼N
À1=2
X
N

i¼1
Z
i
ðg
o
Þ
0
r
i
ðy
o
Þ
þ E½Z
i
ðg
o
Þ
0
R
o
ðx
i
Þ
ffiffiffiffiffi
N
p
ð
^
yy À y
o

Þþo
p
ð1Þð14:65Þ
The right-hand side of equation (14.65) is identical to the expansion with
^
ZZ
i
replaced
with Z
i
ðg
o
Þ.
Assuming now that Z
i
ðg
o
Þ is the matrix of e‰cient instruments, the asymptotic
variance of the e‰cient estimator is
Avar
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ¼fE½R
o
ðx

i
Þ
0
W
o
ðx
i
Þ
À1
R
o
ðx
i
Þg
À1
ð14:66Þ
as can be seen from Section 14.1 by noting th at G
o
¼ E½R
o
ðx
i
Þ
0
W
o
ðx
i
Þ
À1

R
o
ðx
i
Þ and
L
o
¼ G
À1
o
when the instruments are given by equation (14.63).
Equation (14.66) is another example of an e‰ciency bound, this time under the
conditional moment restrictions (14.54). What we have shown is that any GMM es-
timator has variance matrix that di¤ers from equation (14.66) by a positive semi-
definite matrix. Chamberlain (1987) has shown more: any estimator that uses only
condition (14.60) and satisfies regularity conditions has variance matrix no smaller
than equation (14.66).
Estimation of R
o
ðx
i
Þ generally requires nonpa rametric methods. Newey (1990)
describes one approach. Essentially, regress the elements of ‘
y
r
i
ð
^
^
yy

^
yyÞ on polynomial
functions of x
i
(or other functions with good approximating properties), where
^
^
yy
^
yy is
an initial estimate of y
o
. The fitted values from these regressions can be used as the
elements of
^
RR
i
. Other nonparametric approaches are available. See Newey (1990,
1993) for details. Unfortunately, we need a fairly large sample size in order to apply
such methods e¤ectively.
As an example of finding the optimal instruments, consider the problem of esti-
mating a cond itional mean for a vector y
i
:
Eðy
i
jx
i
Þ¼mðx
i

; y
o
Þð14:67Þ
Then the residual function is rðw
i
; yÞ1 y
i
À mðx
i
; yÞ and W
o
ðx
i
Þ¼Varðy
i
jx
i
Þ;
therefore, the optimal instruments are Z
o
ðx
i
Þ1 W
o
ðx
i
Þ
À1
‘
y

mðx
i
; y
o
Þ. This is an im-
Generalized Method of Moments and Minimum Distance Estimation 441
portant example where R
o
ðx
i
Þ¼À‘
y
mðx
i
; y
o
Þ is a known function of x
i
and y
o
.If
the homoskedasticity assumption
Varðy
i
jx
i
Þ¼W
o
ð14:68Þ
holds, then the e‰cient estimator is easy to obtain. First, let

^
^
yy
^
yy be the multivariate
nonlinear least squares (MNLS) estimator, which solves min
y A Y
P
N
i¼1
½y
i
Àmð x
i
; yÞ
0
Á
½y
i
À mðx
i
; yÞ. As discussed in Problem 12.11, the MNLS estimator is generally
consistent and
ffiffiffiffiffi
N
p
-asymptotic normal. Define the residuals
^
^
uu

^
uu
i
1 y
i
À mðx
i
;
^
^
yy
^
yyÞ, and
define a consistent estimator of W
o
by
^
WW ¼ N
À1
P
N
i¼1
^
^
uu
^
uu
i
^
^

uu
^
uu
0
i
. An e‰cient estimator,
^
yy,
solves
X
N
i¼1
‘
y
mðx
i
;
^
^
yy
^
yyÞ
0
^
WW
À1
½y
i
À mðx
i

;
^
yyÞ ¼ 0
and the asymptotic variance of
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ is fE½‘
y
m
i
ðy
o
Þ
0
W
À1
o
‘
y
m
i
ðy
o
Þg
À1

.An
asymptotically equivalent estimator is the nonlinear SUR estimator described in
Problem 12.7. In either case, the estimator of Avarð
^
yyÞ under assumption (14.68) is
Av
^
aarð
^
yyÞ¼
X
N
i¼1
‘
y
m
i
ð
^
yyÞ
0
^
WW
À1
‘
y
m
i
ð
^

yyÞ
"#
À1
Because the nonlinear SUR estimator is a two-step M-estimator and B
o
¼ A
o
(in the
notation of Chapter 12), the simplest forms of tests statistics are valid. If assumption
(14.68) fails, the nonlinear SUR estimator is consistent, but robu st inference should
be used because A
o
0 B
o
. And, the estimator is no longer e‰cient.
14.6 Classical Minimum Distance Estimation
We end this chapter with a brief treatment of classical minimum distance (CMD)
estimation. This method has features in common with GMM, and often it is a con-
venient substitute for GMM.
Suppose that the P Â 1 parameter vector of interest, y
o
, which often consists of
parameters from a structural model, is known to be related to an S Â1 vector of
reduced form parameters, p
o
, where S > P. In particular, p
o
¼ hðy
o
Þ for a known,

continuously di¤erentiable function h: R
P
! R
S
, so that h maps the structural
parameters into the reduced form parameters.
CMD estimation of y
o
entails first estimating p
o
by
^
pp, and then choosing an esti-
mator
^
yy of y
o
by making the distance between
^
pp and hð
^
yyÞ as small as possible. As
with GMM estimation, we use a weighted Euclidean measure of distance. While a
Chapter 14442
CMD estimator can be defined for any positive semidefinite weighting matrix, we
consider only the e‰cient CMD estimator given our choice of
^
pp. As with e‰cient
GMM, the CMD estimator that uses the e‰cient weighting matrix is also called the
minimum chi-square estimator.

Assuming that for an S Â S positive definite matrix X
o
ffiffiffiffiffi
N
p
ð
^
pp Àp
o
Þ@
a
Normalð0; X
o
Þð14:69Þ
it turns out that an e‰cient CMD estimator solves
min
y A Y
f
^
pp Àhðy Þg
0
^
XX
À1
f
^
pp Àhðy Þg ð14:70Þ
where plim
N!y
^

XX ¼ X
o
. In other words, an e‰cient weighting matrix is the inverse
of any consistent estimator of Avar
ffiffiffiffiffi
N
p
ð
^
pp À p
o
Þ.
We can easily der ive the asymptotic variance of
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ. The first-order con-
dition for
^
yy is
Hð
^
yyÞ
0
^
XX

À1
f
^
pp Àhð
^
yyÞg1 0 ð14:71Þ
where HðyÞ1 ‘
y
hðyÞ is the S ÂP Jacobian of hðyÞ. Since hðy
o
Þ¼p
o
and
ffiffiffiffiffi
N
p
fhð
^
yyÞÀhðy
o
Þg ¼ Hðy
o
Þ
ffiffiffiffiffi
N
p
ð
^
yy À y
o

Þþo
p
ð1Þ
by a standard mean value expansion about y
o
, we have
0 ¼ Hð
^
yyÞ
0
^
XX
À1
f
ffiffiffiffiffi
N
p
ð
^
pp À p
o
ÞÀHðy
o
Þ
ffiffiffiffiffi
N
p
ð
^
yy À y

o
Þg þ o
p
ð1Þð14:72Þ
Because HðÁÞ is continuous and
^
yy !
p
y
o
, Hð
^
yyÞ¼Hðy
o
Þþo
p
ð1Þ; by assumption
^
XX ¼
X
o
þ o
p
ð1Þ. Therefore,
Hðy
o
Þ
0
X
À1

o
Hðy
o
Þ
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ¼Hðy
o
Þ
0
X
À1
o
ffiffiffiffiffi
N
p
ð
^
pp Àp
o
Þþo
p
ð1Þ
By assumption (14.69) and the asymptotic equivalence lemma,
Hðy

o
Þ
0
X
À1
o
Hðy
o
Þ
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ@
a
Normal½0; Hðy
o
Þ
0
X
À1
o
Hðy
o
Þ
and so
ffiffiffiffiffi

N
p
ð
^
yy À y
o
Þ@
a
Normal½0; ðH
0
o
X
À1
o
H
o
Þ
À1
ð14:73Þ
provided that H
o
1 Hðy
o
Þ has full-column rank P, as will generally be the case when
y
o
is identified and hðÁÞ contains no redundancies. The appropriate estimator of
Av
^
aarð

^
yyÞ is
Av
^
aarð
^
yyÞ1 ð
^
HH
0
^
XX
À1
^
HHÞ
À1
=N ¼ð
^
HH
0
½Av
^
aarð
^
ppÞ
À1
^
HHÞ
À1
ð14:74Þ

Generalized Method of Moments and Minimum Distance Estimation 443
The proof that
^
XX
À1
is the optimal weighting matrix in expression (14.70) is very
similar to the derivation of the optimal weighting matrix for GMM. (It can also
be shown by applying Theorem 14.3.) We will simply call the e‰cient estimator
the CMD estimator, where it is understood that we are using the e‰cient weighting
matrix.
There is another e‰ciency issue that arises when more than one
ffiffiffiffiffi
N
p
-asymptotically
normal estimator for p
o
is available: Which estimator of p
o
should be used? Let
^
yy be
the estimator based on
^
pp, and let
~
yy be the estimator based on another estimator,
~
pp.
You are asked to show in Problem 14.6 that Avar

ffiffiffiffiffi
N
p
ð
~
yy À y
o
ÞÀAvar
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þ is
p.s.d. whenever Avar
ffiffiffiffiffi
N
p
ð
~
pp Àp
o
ÞÀAvar
ffiffiffiffiffi
N
p
ð
^

pp Àp
o
Þ is p.s.d. In other words, we
should use the most e‰cient estimator of p
o
to obtain the most e‰cient estimator of y
o
.
A test of overidentifying restrictions is immediately available after estimation, be-
cause, under the null hypothesis p
o
¼ hðy
o
Þ,
N½
^
pp À hð
^
yyÞ
0
^
XX
À1
½
^
pp Àhð
^
yyÞ@
a
w

2
SÀP
ð14:75Þ
To show this result, we use
ffiffiffiffiffi
N
p
½
^
pp Àhð
^
yyÞ ¼
ffiffiffiffiffi
N
p
ð
^
pp Àp
o
ÞÀH
o
ffiffiffiffiffi
N
p
ð
^
yy À y
o
Þþo
p

ð1Þ
¼
ffiffiffiffiffi
N
p
ð
^
pp Àp
o
ÞÀH
o
ðH
0
o
X
À1
o
H
o
Þ
À1
H
0
o
X
À1
o
ffiffiffiffiffi
N
p

ð
^
pp Àp
o
Þþo
p
ð1Þ
¼½I
S
À H
o
ðH
0
o
X
À1
o
H
o
Þ
À1
H
0
o
X
À1
o

ffiffiffiffiffi
N

p
ð
^
pp Àp
o
Þþo
p
ð1Þ
Therefore, up to o
p
ð1Þ,
X
À1=2
o
ffiffiffiffiffi
N
p
f
^
pp À hð
^
yyÞg ¼ ½I
S
À X
À1=2
o
H
o
ðH
0

o
X
À1
o
H
o
Þ
À1
H
0
o
X
À1=2
o
Z 1 M
o
Z
where Z 1 X
À1=2
o
ffiffiffiffiffi
N
p
ð
^
pp Àp
o
Þ!
d
Normalð0; I

S
Þ. But M
o
is a symmetric idempotent
matrix with rank S À P,sof
ffiffiffiffiffi
N
p
½
^
pp Àhð
^
yyÞg
0
X
À1
o
f
ffiffiffiffiffi
N
p
½
^
pp Àhð
^
yyÞg@
a
w
2
SÀP

. Because
^
XX
is consistent for X
o
, expression (14.75) follows from the asymptotic equivalence
lemma. The statistic can also be expressed as
f
^
pp Àhð
^
yyÞg
0
½Av
^
aarð
^
ppÞ
À1
f
^
pp Àhð
^
yyÞg ð14:76Þ
Testing restrictions on y
o
is also straightforward, assuming that we can express the
restrictions as y
o
¼ dða

o
Þ for an R Â1 vector a
o
, R < P. Under these restrictions,
p
o
¼ h½dða
o
Þ1 gð a
o
Þ. Thus, a
o
can be estimated by minimum distance by solving
problem (14.70) with a in place of y and gða Þ in place of hðyÞ. The same estimator
^
XX
should be used in both minimization problems. Then it can be shown (under interi-
ority and di¤erentiability) that
Chapter 14444
N½
^
pp Àgð
^
aaÞ
0
^
XX
À1
½
^

pp Àgð
^
aaÞÀ N½
^
pp À hð
^
yyÞ
0
^
XX
À1
½
^
pp Àhð
^
yyÞ@
a
w
2
PÀR
ð14:77Þ
when the restri ctions on y
o
are true.
To illustrate the application of CMD estimation, we reconsider Chamberlain’s
(1982, 1984) approach to linear, unobserved e¤ects panel data models. (See Section
11.3.2 for the GMM approach.) The key equations are
y
it
¼ c þ x

i1
l
1
þÁÁÁþx
it
ðb þ l
t
ÞþÁÁÁþx
iT
l
T
þ v
it
ð14:78Þ
where
Eðv
it
Þ¼0; Eðx
0
i
v
it
Þ¼0; t ¼ 1; 2; ; T ð14:79Þ
(For notational simplicity we do not index the true parameters by ‘‘o’’.) Equation
(14.78) embodies the restrictions on the ‘‘structural’’ parameters y 1 ðc; l
0
1
; ;
l
0

T
; b
0
Þ
0
,að1 þ TK þ KÞÂ1 vector. To apply CMD, write
y
it
¼ p
t0
þ x
i
p
t
þ v
it
; t ¼ 1; ; T
so that the vector p is Tð1 þ TKÞÂ1. When we impose the restrictions,
p
t0
¼ c, p
t
¼½l
0
1
; l
0
2
; ; ðb þl
t

Þ
0
; ; l
0
T

0
; t ¼ 1; ; T
Therefore, we can write p ¼ Hy for a ðT þ T
2
KÞÂð1 þ TK þ KÞ matrix H. When
T ¼ 2, p can be written with restrictions imposed as p ¼ðc; b
0
þ l
0
1
; l
0
2
; c; l
0
1
; b
0
þ
l
0
2
Þ
0

, and so
H ¼
1 000
0I
K
0I
K
00 I
K
0
1 000
0I
K
00
00 I
K
I
K
2
6
6
6
6
6
6
6
6
4
3
7

7
7
7
7
7
7
7
5
The CMD estimator can be obtained in closed form, once we have
^
pp; see Prob lem
14.7 for the general case.
How should we obtain
^
pp, the vector of estimates without the restrictions imposed?
There is really only one way, and that is OLS for each time period. Condition (14.79)
ensures that OLS is consistent and
ffiffiffiffiffi
N
p
-asymptotically normal. Why not use a system
method, in particular, SUR? For one thing, we cannot generally assume that v
i
sat-
isfies the requisite homoskedasticity assumption that ensures that SUR is more e‰-
cient than OLS equation by equation; see Section 11.3.2. Anyway, because the same
Generalized Method of Moments and Minimum Distance Estimation 445