Greene W.H. Econometric Analysis
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CHAPTER 13
✦
Minimum Distance Estimation and GMM
489
instrumental variables. The assumptions thus far have implied a set of orthogonality
conditions,
E [z
i
ε
i
] = 0,
which may be sufficient to identify (if L= K) or even overidentify (if L> K) the pa-
rameters of the model. (See Section 8.3.4.)
For convenience, define
e(X,
ˆ
β) = y
i
− h(x
i
,
ˆ
β), i = 1,...,n,
and
Z = n × L matrix whose ith row is z
i
.
By a straightforward extension of our earlier results, we can produce a GMM estimator
of β. The sample moments will be
¯
m
n
(β) =
1
n
n
i=1
z
i
e(x
i
, β) =
1
n
Z
e(X, β).
The minimum distance estimator will be the
ˆ
β that minimizes
q =
¯
m
n
(
ˆ
β)
W
¯
m
n
(
ˆ
β) =
1
n
[e(X,
ˆ
β)
Z]
W
1
n
[Z
e(X,
ˆ
β)]
(13-27)
for some choice of W that we have yet to determine. The criterion given earlier produces
the nonlinear instrumental variable estimator.IfweuseW =(Z
Z)
−1
, then we have
exactly the estimation criterion we used in Section 8.6, where we defined the nonlinear
instrumental variables estimator. Apparently (13-27) is more general, because we are
not limited to this choice of W. For any given choice of W, as long as there are enough
orthogonality conditions to identify the parameters, estimation by minimizing q is, at
least in principle, a straightforward problem in nonlinear optimization. The optimal
choice of W for this estimator is
W
GMM
=
Asy. Var[
√
n
¯
m
n
(β)]
−1
=
+
Asy. Va r
1
√
n
n
i=1
z
i
ε
i
,
−1
=
Asy. Va r
1
√
n
Z
e(X, β)
−1
.
(13-28)
For our model, this is
W =
⎡
⎣
1
n
n
i=1
n
j=1
Cov[z
i
ε
i
, z
j
ε
j
]
⎤
⎦
−1
=
⎡
⎣
1
n
n
i=1
n
j=1
σ
ij
z
i
z
j
⎤
⎦
−1
=
Z
Z
n
−1
.
If we insert this result in (13-27), we obtain the criterion for the GMM estimator:
q =
1
n
e(X,
ˆ
β)
Z
Z
Z
n
−1
1
n
Z
e(X,
ˆ
β)
.
There is a possibly difficult detail to be considered. The GMM estimator involves
1
n
Z
Z =
1
n
n
i=1
n
j=1
z
i
z
j
Cov[ε
i
,ε
j
] =
1
n
n
i=1
n
j=1
z
i
z
j
Cov[(y
i
− h(x
i
, β)), (y
j
− h(x
j
, β))].
490
PART III
✦
Estimation Methodology
The conditions under which such a double sum might converge to a positive definite
matrix are sketched in Section 9.2.2. Assuming that they do hold, estimation appears to
require that an estimate of β be in hand already, even though it is the object of estimation.
It may be that a consistent but inefficient estimator of β is available. Suppose for the
present that one is. If observations are uncorrelated, then the cross-observation terms
may be omitted, and what is required is
1
n
Z
Z =
1
n
n
i=1
z
i
z
i
Var[(y
i
− h(x
i
, β))].
We can use a counterpart to the White (1980) estimator discussed in Section 9.4.4 for
this case:
S
0
=
1
n
n
i=1
z
i
z
i
(y
i
− h(x
i
,
ˆ
β))
2
. (13-29)
If the disturbances are autocorrelated but the process is stationary, then Newey and
West’s (1987a) estimator is available (assuming that the autocorrelations are sufficiently
small at a reasonable lag, p):
S =
S
0
+
1
n
p
=1
w()
n
i=+1
e
i
e
i−
(z
i
z
i−
+ z
i−
z
i
)
=
p
=0
w()S
, (13-30)
where
w() = 1 −
p + 1
.
The maximum lag length p must be determined in advance. We will require that observa-
tions that are far apart in time—that is, for which |i −|is large—must have increasingly
smaller covariances for us to establish the convergence results that justify OLS, GLS, and
now GMM estimation. The choice of p is a reflection of how far back in time one must
go to consider the autocorrelation negligible for purposes of estimating (1/n)Z
Z.
Current practice suggests using the smallest integer greater than or equal to n
1/4
.
Still left open is the question of where the initial consistent estimator should be
obtained. One possibility is to obtain an inefficient but consistent GMM estimator by
using W = I in (13-27). That is, use a nonlinear (or linear, if the equation is linear)
instrumental variables estimator. This first-step estimator can then be used to construct
W, which, in turn, can then be used in the GMM estimator. Another possibility is that
β may be consistently estimable by some straightforward procedure other than GMM.
Once the GMM estimator has been computed, its asymptotic covariance matrix
and asymptotic distribution can be estimated based on Theorem 13.2. Recall that
¯
m
n
(β) =
1
n
n
i=1
z
i
ε
i
,
which is a sum of L×1 vectors. The derivative, ∂
¯
m
n
(β)/∂β
, is a sum of L× K matrices,
so
¯
G(β) = ∂
¯
m(β)/∂β
=
1
n
n
i=1
G
i
(β) =
1
n
n
i=1
z
i
∂ε
i
∂β
. (13-31)
CHAPTER 13
✦
Minimum Distance Estimation and GMM
491
In the model we are considering here,
∂ε
i
∂β
=
−∂h(x
i
, β)
∂β
.
The derivatives are the pseudoregressors in the linearized regression model that we
examined in Section 7.2.3. Using the notation defined there,
∂ε
i
∂β
=−x
0
i
,
so
¯
G(β) =
1
n
n
i=1
G
i
(β) =
1
n
n
i=1
−z
i
x
0
i
=−
1
n
Z
X
0
. (13-32)
With this matrix in hand, the estimated asymptotic covariance matrix for the GMM
estimator is
Est. Asy. Var[
ˆ
β] =
¯
G(
ˆ
β)
1
n
Z
ˆ
Z
−1
¯
G(
ˆ
β)
−1
= [(X
0
Z)(Z
ˆ
Z)
−1
(Z
X
0
)]
−1
.
(13-33)
(The two minus signs, a 1/n
2
, and an n
2
, all fall out of the result.)
If the that appears in (13-33) were σ
2
I, then (13-33) would be precisely the asymp-
totic covariance matrix that appears in Theorem 8.1 for linear models and Theorem 8.2
for nonlinear models. But there is an interesting distinction between this estimator
and the IV estimators discussed earlier. In the earlier cases, when there were more in-
strumental variables than parameters, we resolved the overidentification by specifically
choosing a set of K instruments, the K projections of the columns of X or X
0
into the
column space of Z. Here, in contrast, we do not attempt to resolve the overidentifi-
cation; we simply use all the instruments and minimize the GMM criterion. Now, you
should be able to show that when = σ
2
I and we use this information, when all is said
and done, the same parameter estimates will be obtained. But, if we use a weighting
matrix that differs from W = (Z
Z/n)
−1
, then they are not.
13.6.3 SEEMINGLY UNRELATED REGRESSION MODELS
In Section 10.4, we considered FGLS estimation of the equation system
y
1
= h
1
(X, β) + ε
1
,
y
2
= h
2
(X, β) + ε
2
,
.
.
.
y
M
= h
M
(X, β) + ε
M
.
The development there extends backwards to the linear system as well. However, none
of the estimators considered are consistent if the pseudoregressors, x
0
tm
, or the actual
regressors, x
tm
for the linear model, are correlated with the disturbances, ε
tm
. Suppose
we allow for this correlation both within and across equations. (If it is, in fact, absent,
then the GMM estimator developed here will remain consistent.) For simplicity in this
section, we will denote observations with subscript t and equations with subscripts
i and j. Suppose, as well, that there are a set of instrumental variables, z
t
, such that
E[z
t
ε
tm
] = 0, t = 1,...,T and m = 1,...,M. (13-34)
492
PART III
✦
Estimation Methodology
(We could allow a separate set of instrumental variables for each equation, but it would
needlessly complicate the presentation.)
Under these assumptions, the nonlinear FGLS and ML estimators given earlier will
be inconsistent. But a relatively minor extension of the instrumental variables technique
developed for the single-equation case in Section 8.4 can be used instead. The sample
analog to (13-34) is
1
T
T
t=1
z
t
[y
ti
− h
i
(x
t
, β)] = 0, i = 1,...,M.
If we use this result for each equation in the system, one at a time, then we obtain exactly
the GMM estimator discussed in Section 13.6.2. But, in addition to the efficiency loss
that results from not imposing the cross-equation constraints in β, we would also neglect
the correlation between the disturbances. Let
1
T
Z
ij
Z = E
Z
ε
i
ε
j
Z
T
. (13-35)
The GMM criterion for estimation in this setting is
q =
M
i=1
M
j=1
[(y
i
− h
i
(X, β))
Z/T][Z
ij
Z/T]
ij
[Z
(y
j
− h
j
(X, β))/T]
(13-36)
=
M
i=1
M
j=1
[ε
i
(β)
Z/T][Z
ij
Z/T]
ij
[Z
ε
j
(β)/T],
where [Z
ij
Z/T]
ij
denotes the ijth block of the inverse of the matrix with the ijth
block equal to Z
ij
Z/T. (This matrix is laid out in full in Section 13.6.4.)
GMM estimation would proceed in several passes. To compute any of the variance
parameters, we will require an initial consistent estimator of β. This step can be done with
equation-by-equation nonlinear instrumental variables—see Section 8.6—although if
equations have parameters in common, then a choice must be made as to which to use.
At the next step, the familiar White or Newey–West technique is used to compute, block
by block, the matrix in (13-35). Because it is based on a consistent estimator of β (we
assume), this matrix need not be recomputed. Now, with this result in hand, an iterative
solution to the maximization problem in (13-36) can be sought, for example, using the
methods of Appendix E. The first-order conditions are
∂q
∂β
=−2
M
i=1
M
j=1
X
0
i
(β)
Z/T
[Z
W
ij
Z/T]
ij
[Z
ε
j
(β)/T] = 0. (13-37)
Note again that the blocks of the inverse matrix in the center are extracted from the
larger constructed matrix after inversion. [This brief discussion might understate the
complexity of the optimization problem in (13-36), but that is inherent in the pro-
cedure.] At completion, the asymptotic covariance matrix for the GMM estimator is
estimated with
V
GMM
=
1
T
M
i=1
M
j=1
X
0
i
(β)
Z/T
[Z
W
ij
Z/T]
ij
Z
X
0
j
(β)/T
−1
.
CHAPTER 13
✦
Minimum Distance Estimation and GMM
493
13.6.4 SIMULTANEOUS EQUATIONS MODELS WITH
HETEROSCEDASTICITY
The GMM estimator in Section 13.6.1 is, with a minor change of notation, precisely
the set of procedures we used in Section 10.6.4 and 10.6.5 to estimate the equations in
a simultaneous equations model. Using a GMM estimator, however, will allow us to
generalize the covariance structure for the disturbances. We assume that
y
tj
= z
tj
δ
j
+ ε
tj
, t = 1,...,T,
where z
tj
= [Y
tj
, x
tj
]. (We use the capital Y
tj
to denote the L
j
included endogenous
variables. Note, as well, that to maintain consistency with Chapter 10, the roles of the
symbols x and z are reversed here; x is now the vector of exogenous variables.) We have
assumed that ε
tj
in the jth equation is neither heteroscedastic nor autocorrelated. There
is no need to impose those assumptions at this point. Autocorrelation in the context of a
simultaneous equations model is a substantial complication, however. For the present,
we will consider the heteroscedastic case only.
The assumptions of the model provide the orthogonality conditions,
E [x
t
ε
tj
] = E [x
t
(y
tj
− z
tj
δ
j
)] = 0.
If x
t
is taken to be the full set of exogenous variables in the model, then we obtain the
criterion for the GMM estimator for the j th equation,
q =
e(z
t
, δ
j
)
X
T
W
−1
jj
X
e(z
t
, δ
j
)
T
=
¯
m(δ
j
)
W
−1
jj
¯
m(δ
j
),
where
¯
m(δ
j
) =
1
T
T
t=1
x
t
(y
tj
− z
tj
δ
j
) and W
−1
jj
= the GMM weighting matrix.
Once again, this is precisely the estimator defined in Section 13.6.1. If the disturbances
are assumed to be homoscedastic and nonautocorrelated, then the optimal weighting
matrix will be an estimator of the inverse of
W
jj
= Asy. Var[
√
T
¯
m(δ
j
)]
= plim
1
T
T
t=1
x
t
x
t
(y
tj
− z
tj
δ
j
)
2
= plim
1
T
T
t=1
σ
jj
x
t
x
t
= plim σ
jj
X
X
T
.
The constant σ
jj
is irrelevant to the solution. If we use (X
X)
−1
as the weighting matrix,
then the GMM estimator that minimizes q is the 2SLS estimator.
The extension that we can obtain here is to allow for heteroscedasticity of un-
known form. There is no need to rederive the earlier result. If the disturbances are
494
PART III
✦
Estimation Methodology
heteroscedastic, then
W
jj
= plim
1
T
T
t=1
ω
jj,t
x
t
x
t
= plim
X
jj
X
T
.
The weighting matrix can be estimated with White’s heteroscedasticity consistent
estimator—see (13-24)—if a consistent estimator of δ
j
is in hand with which to com-
pute the residuals. One is, because 2SLS ignoring the heteroscedasticity is consistent,
albeit inefficient. The conclusion then is that under these assumptions, there is a way to
improve on 2SLS by adding another step. The name 3SLS is reserved for the systems
estimator of this sort. When choosing between 2.5-stage least squares and Davidson
and MacKinnon’s suggested “heteroscedastic 2SLS,” or H2SLS, we chose to opt for the
latter. The estimator is based on the initial two-stage least squares procedure. Thus,
ˆ
δ
j,H2SLS
= [Z
j
X(S
0, jj
)
−1
X
Z
j
]
−1
[Z
j
X(S
0, jj
)
−1
X
y
j
],
where
S
0, jj
=
T
t=1
x
t
x
t
(y
tj
− z
tj
ˆ
δ
j,2SLS
)
2
.
The asymptotic covariance matrix is estimated with
Est. Asy. Var[
ˆ
δ
j,H2SLS
] = [Z
j
X(S
0, jj
)
−1
X
Z
j
]
−1
.
Extensions of this estimator were suggested by Cragg (1983) and Cumby, Huizinga, and
Obstfeld (1983).
The GMM estimator for a system of equations is described in Section 13.6.3. As
in the single-equation case, a minor change in notation produces the estimators for a
simultaneous equations model. As before, we will consider the case of unknown het-
eroscedasticity only. The extension to autocorrelation is quite complicated. [See Cumby,
Huizinga, and Obstfeld (1983).] The orthogonality conditions defined in (13-34) are
E [x
t
ε
tj
] = E [x
t
(y
tj
− z
tj
δ
j
)] = 0.
If we consider all the equations jointly, then we obtain the criterion for estimation of
all the model’s parameters,
q =
M
j=1
M
l=1
e(z
t
, δ
j
)
X
T
[W]
jl
X
e(z
t
, δ
l
)
T
=
M
j=1
M
l=1
¯
m(δ
j
)
[W]
jl
¯
m(δ
l
),
where
¯
m(δ
j
) =
1
T
T
t=1
x
t
(y
tj
− z
tj
δ
j
),
and
[W]
jl
= block jl of the weighting matrix, W
−1
.
As before, we consider the optimal weighting matrix obtained as the asymptotic covari-
ance matrix of the empirical moments,
¯
m(δ
j
). These moments are stacked in a single
CHAPTER 13
✦
Minimum Distance Estimation and GMM
495
vector
¯
m(δ). Then, the jlth block of Asy. Var[
√
T
¯
m(δ)]is
jl
= plim
+
1
T
T
t=1
[x
t
x
t
(y
tj
− z
tj
δ
j
)(y
tl
− z
tl
δ
l
)]
,
= plim
1
T
T
t=1
ω
jl,t
x
t
x
t
.
If the disturbances are homoscedastic, then
jl
=σ
jl
[plim(X
X/T)] is produced. Other-
wise, we obtain a matrix of the form
jl
=plim[X
jl
X/T]. Collecting terms, then, the
criterion function for GMM estimation is
q =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
[X
(y
1
− Z
1
δ
1
)]/T
[X
(y
2
− Z
2
δ
2
)]/T
.
.
.
[X
(y
M
− Z
M
δ
M
)]/T
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
11
12
···
1M
21
22
···
2M
.
.
.
.
.
. ···
.
.
.
M1
M2
···
MM
⎤
⎥
⎥
⎥
⎥
⎥
⎦
−1
⎡
⎢
⎢
⎢
⎢
⎢
⎣
[X
(y
1
− Z
1
δ
1
)]/T
[X
(y
2
− Z
2
δ
2
)]/T
.
.
.
[X
(y
M
− Z
M
δ
M
)]/T
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
For implementation,
jl
can be estimated with
ˆ
jl
=
1
T
T
t=1
x
t
x
t
(y
tj
− z
tj
d
j
)(y
tl
− z
tl
d
l
),
where d
j
is a consistent estimator of δ
j
. The two-stage least squares estimator is a
natural choice. For the diagonal blocks, this choice is the White estimator as usual. For
the off-diagonal blocks, it is a simple extension. With this result in hand, the first-order
conditions for GMM estimation are
∂ ˆq
∂δ
j
=−2
M
l=1
Z
j
X
T
ˆ
jl
X
(y
l
− Z
l
δ
l
)
T
,
where
ˆ
jl
is the jlth block in the inverse of the estimate of the center matrix in q.
The solution is
⎡
⎢
⎢
⎢
⎣
ˆ
δ
1,GMM
ˆ
δ
2,GMM
.
.
.
ˆ
δ
M,GMM
⎤
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
Z
1
X
ˆ
11
X
Z
1
Z
1
X
ˆ
12
X
Z
2
··· Z
1
X
ˆ
1M
X
Z
M
Z
2
X
ˆ
21
X
Z
1
Z
2
X
ˆ
22
X
Z
2
··· Z
2
X
ˆ
2M
X
Z
M
.
.
.
.
.
. ···
.
.
.
Z
M
X
ˆ
M1
X
Z
1
Z
M
X
ˆ
M2
X
Z
2
··· Z
M
X
ˆ
MM
X
Z
M
⎤
⎥
⎥
⎥
⎥
⎦
−1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
M
j=1
Z
1
X
ˆ
1 j
y
j
M
j=1
Z
2
X
ˆ
2 j
y
j
.
.
.
M
j=1
Z
M
X
ˆ
Mj
y
j
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
The asymptotic covariance matrix for the estimator would be estimated with T times
the large inverse matrix in brackets.
Several of the estimators we have already considered are special cases:
•
If
ˆ
jj
= ˆσ
jj
(X
X/T) and
ˆ
jl
= 0 for j = l, then
ˆ
δ
j
is 2SLS.
•
If
ˆ
jl
= 0 for j = l, then
ˆ
δ
j
is H2SLS, the single-equation GMM estimator.
•
If
ˆ
jl
= ˆσ
jl
(X
X/T), then
ˆ
δ
j
is 3SLS.
As before, the GMM estimator brings efficiency gains in the presence of heteroscedas-
ticity. If the disturbances are homoscedastic, then it is asymptotically the same as 3SLS,
496
PART III
✦
Estimation Methodology
[although in a finite sample, it will differ numerically because S
jl
will not be identical to
ˆσ
jl
(X
X)].
13.6.5 GMM ESTIMATION OF DYNAMIC PANEL DATA MODELS
Panel data are well suited for examining dynamic effects, as in the first-order model,
y
it
= x
it
β + δy
i,t−1
+ c
i
+ ε
it
= w
it
θ + α
i
+ ε
it
,
where the set of right-hand-side variables, w
it
, now includes the lagged dependent vari-
able, y
i,t−1
. Adding dynamics to a model in this fashion creates a major change in the
interpretation of the equation. Without the lagged variable, the “independent vari-
ables” represent the full set of information that produce observed outcome y
it
. With the
lagged variable, we now have in the equation the entire history of the right-hand-side
variables, so that any measured influence is conditioned on this history; in this case,
any impact of x
it
represents the effect of new information. Substantial complications
arise in estimation of such a model. In both the fixed and random effects settings, the
difficulty is that the lagged dependent variable is correlated with the disturbance, even
if it is assumed that ε
it
is not itself autocorrelated. For the moment, consider the fixed
effects model as an ordinary regression with a lagged dependent variable that is depen-
dent across observations. In that dynamic regression model, the estimator based on T
observations is biased in finite samples, but it is consistent in T. The finite sample bias
is of order 1/T. The same result applies here, but the difference is that whereas before
we obtained our large sample results by allowing T to grow large, in this setting, T is
assumed to be small and fixed, and large-sample results are obtained with respect to
n growing large, not T. The fixed effects estimator of θ = [β, δ] can be viewed as an
average of n such estimators. Assume for now that T ≥ K + 1 where K is the number
of variables in x
it
. Then, from (11-13),
ˆ
θ =
n
i=1
W
i
M
0
W
i
−1
n
i=1
W
i
M
0
y
i
=
n
i=1
W
i
M
0
W
i
−1
n
i=1
W
i
M
0
W
i
d
i
=
n
i=1
F
i
d
i
,
where the rows of the T ×(K +1) matrix W
i
are w
it
and M
0
is the T × T matrix that
creates deviations from group means [see (11-14)]. Each group-specific estimator, d
i
,
is inconsistent, as it is biased in finite samples and its variance does not go to zero
as n increases. This matrix weighted average of n inconsistent estimators will also be
inconsistent. (This analysis is only heuristic. If T < K +1, then the individual coefficient
vectors cannot be computed.
13
)
13
Further discussion is given by Nickell (1981), Ridder and Wansbeek (1990), and Kiviet (1995).
CHAPTER 13
✦
Minimum Distance Estimation and GMM
497
The problem is more transparent in the random effects model. In the model
y
it
= x
it
β + δ y
i,t−1
+ u
i
+ ε
it
,
the lagged dependent variable is correlated with the compound disturbance in the
model, since the same u
i
enters the equation for every observation in group i.
Neither of these results renders the model inestimable, but they do make neces-
sary some technique other than our familiar LSDV or FGLS estimators. The general
approach, which has been developed in several stages in the literature,
14
relies on in-
strumental variables estimators and, most recently [by Arellano and Bond (1991) and
Arellano and Bover (1995)] on a GMM estimator. For example, in either the fixed or
random effects cases, the heterogeneity can be swept from the model by taking first
differences, which produces
y
it
− y
i,t−1
= (x
it
− x
i,t−1
)
β + δ(y
i,t−1
− y
i,t−2
) + (ε
it
− ε
i,t−1
).
This model is still complicated by correlation between the lagged dependent variable
and the disturbance (and by its first-order moving average disturbance). But without the
group effects, there is a simple instrumental variables estimator available. Assuming that
the time series is long enough, one could use the lagged differences, (y
i,t−2
−y
i,t−3
), or the
lagged levels, y
i,t−2
and y
i,t−3
, as one or two instrumental variables for (y
i,t−1
− y
i,t−2
).
(The other variables can serve as their own instruments.) This is the Anderson and Hsiao
estimator developed for this model in Section 11.8.2. By this construction, then, the
treatment of this model is a standard application of the instrumental variables technique
that we developed in Section 11.8
15
This illustrates the flavor of an instrumental variable
approach to estimation. But, as Arellano et al. and Ahn and Schmidt (1995) have shown,
there is still more information in the sample that can be brought to bear on estimation,
in the context of a GMM estimator, which we now consider.
We can extend the Hausman and Taylor (HT) formulation of the random effects
model in Section 11.8.1 to include the lagged dependent variable;
y
it
= δ y
i,t−1
+ x
1it
β
1
+ x
2it
β
2
+ z
1i
α
1
+ z
2i
α
2
+ ε
it
+ u
i
= θ
w
it
+ ε
it
+ u
i
= θ
w
it
+ η
it
,
where
w
it
= [y
i,t−1
, x
1it
, x
2it
, z
1i
, z
2i
]
is now a (1+K
1
+K
2
+L
1
+L
2
)×1 vector. The terms in the equation are the same as in the
Hausman and Taylor model. Instrumental variables estimation of the model without the
lagged dependent variable is discussed in Section 11.8.1 on the HT estimator. Moreover,
by just including y
i,t−1
in x
2it
, we see that the HT approach extends to this setting as
well, essentially without modification. Arellano et al. suggest a GMM estimator and
show that efficiency gains are available by using a larger set of moment conditions.
14
The model was first proposed in this form by Balestra and Nerlove (1966). See, for example, Anderson and
Hsiao (1981, 1982), Bhargava and Sargan (1983), Arellano (1989), Arellano and Bond (1991), Arellano and
Bover (1995), Ahn and Schmidt (1995), and Nerlove (2003).
15
There is a question as to whether one should use differences or levels as instruments. Arellano (1989) and
Kiviet (1995) give evidence that the latter is preferable.
498
PART III
✦
Estimation Methodology
In the previous treatment, we used a GMM estimator constructed as follows: The set
of moment conditions we used to formulate the instrumental variables were
E
⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
x
1it
x
2it
z
1i
¯
x
1i.
⎞
⎟
⎟
⎠
(η
it
− ¯η
i
)
⎤
⎥
⎥
⎦
= E
⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
x
1it
x
2it
z
1i
¯
x
1i.
⎞
⎟
⎟
⎠
(ε
it
− ¯ε
i
)
⎤
⎥
⎥
⎦
= 0.
This moment condition is used to produce the instrumental variable estimator. We could
ignore the nonscalar variance of η
it
and use simple instrumental variables at this point.
However, by accounting for the random effects formulation and using the counterpart
to feasible GLS, we obtain the more efficient estimator in Section 11.8. As usual, this
can be done in two steps. The inefficient estimator is computed to obtain the residuals
needed to estimate the variance components. This is Hausman and Taylor’s steps 1 and
2. Steps 3 and 4 are the GMM estimator based on these estimated variance components.
Arellano et al. suggest that the preceding does not exploit all the information in
the sample. In simple terms, within the T observations in group i, we have not used the
fact that
E
⎡
⎢
⎢
⎣
⎛
⎜
⎜
⎝
x
1it
x
2it
z
1i
¯
x
1i.
⎞
⎟
⎟
⎠
(η
is
− ¯η
i
)
⎤
⎥
⎥
⎦
= 0 for some s = t.
Thus, for example, not only are disturbances at time t uncorrelated with these variables
at time t, arguably, they are uncorrelated with the same variables at time t −1, t −2,
possibly t +1, and so on. In principle, the number of valid instruments is potentially
enormous. Suppose, for example, that the set of instruments listed above is strictly
exogenous with respect to η
it
in every period including current, lagged, and future. Then,
there are a total of [T(K
1
+ K
2
) + L
1
+ K
1
)] moment conditions for every observation.
Consider, for example, a panel with two periods. We would have for the two periods,
E
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x
1i1
x
2i1
x
1i2
x
2i2
z
1i
¯
x
1i.
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(η
i1
− ¯η
i
)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= 0 and E
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x
1i1
x
2i1
x
1i2
x
2i2
z
1i
¯
x
1i.
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(η
i2
− ¯η
i
)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
= 0. (13-38)
How much useful information is brought to bear on estimation of the parameters is
uncertain, as it depends on the correlation of the instruments with the included exoge-
nous variables in the equation. The farther apart in time these sets of variables become
the less information is likely to be present. (The literature on this subject contains ref-
erence to “strong” versus “weak” instrumental variables.
16
) To proceed, as noted, we
can include the lagged dependent variable in x
2i
. This set of instrumental variables can
16
See West (2001).