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WORKING PAPER
DEPARTMENT
OF ECONOMICS
A NEW SPECIFICATION TEST FOR THE VALIDITY
OF INSTRUMENTAL VARIABLES
Hahn
Hausman
Jinyong
Jerry
No. 99-11
May, 1999
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
LIBF?ARIES
WP No. 99-11
Abstract
for:
A New Specification Test for the Validity of Instrumental Variables
By Hahn and Hausman
We develop a new specification test for tlie iV estimators adopting a particular second
order approximation of Bekker (1994). The new specification test compares the
difference of the forward (conventional) 2SLS estimator of the coefficient of the right
hand side endogenous variable with the reserve 2SLS estimator of the same unknown
parameter when the normalization is changed. Under the null hypothesis that
conventional first order asymptotics provides a reliable guide, the two estimates should
be very similar. Our test sees whether the resulting difference in the two estimates satisfies
the results of second order asymptotic theory. Essentially the same idea is applied to
develop another new specification test using second-order unbiased estimators of the
type first proposed by Nagar (1959). If the forward and reverse Nagar-type estimators
are not significantly different we recommend estimation by LIML, which we demonstrate
is the optimal linear combination of the Nagar-type estimators (to second order). We
also demonstrate the high degree of similarity for k-class estimators between the
approach of Bekker (1994) and the Edgeworth expansion approach of Rothenberg
(1983). Empirical example and Monte Carlo evidence are provided.
—
Preliminary
First draft
Please do not cite without authors' permission
A New Specification Test for the Validity of
Instrumental Variables*
Jinyong Hahn
University of Pennsylvania
And
Jerry
Hausman
MIT
May, 1999
Excellent research assistance has been provided by Derek
Ho Cheung and Karen Hull. Whitney Newey
provided some helpful suggestions. Please send email to jhausnian@,miLedu
.
1
Introduction
.
A significant understanding has emerged over the past few years that instrumental
variable (FV) estimation of the simultaneous equation model can lead to problems of
inference in the situation of "weak instruments," which can arise
when
the instruments do
not have a high degree of explanatory power for the jointly endogenous variable(s) or
when the number of instruments becomes
large.
The
situation
of limited information
estimation of a single equation has been studied extensively in the presence of "weak
instruments." These problems of inference in the
when
conventional
conventional
weak instrument
situation can arise
order) asymptotic inference techniques are used. In particular,
(first
order asymptotics can lead to a lack of indication of a problem even
first
though significant (large sample) bias
is
present because estimated standard errors are not
very accurate.
A number of papers have recommended possible diagnostics for the presence of
the problem, e.g. Shea (1997).
The usual form of the recommended
diagnostics
is
to
examine the R^ or the associated
F
included endogenous variable(s).
A more refined recommendation is to consider the
partial
R^
(or
partialled out
rank
its
associated
F
statistic
of the reduced form regression for the
statistic) after
the predetermined variables have been
of the equation being estimated. Another approach has been to consider the
statistic originally
put forward by Anderson and Rubin (1949). While both
'
approaches yield valuable information, the R^ approach lacks a distribution theory and
the rank condition
test, in
some
sense, does not
answer the question
conventional asymptotic theory does in forming
In this paper,
we take a new approach
and
and use higher order asymptotic
test
IV asymptotics
new
estimator since
it
is
by
far the
most commonly
specification test takes the general approach as the specification
approach of Hausman (1978) and estimates the same parameter(s)
ways. In particular,
are reliable
We recommend a new specification test for the IV estimators,
we concentrate initially on the 2SLS
used estimator. Our
of how well
statistics for inference.
distribution theory to determine if the conventional first order
in a particular situation.
at issue
we compare the
difference
in
two
different
of the forward (conventional) 2SLS
estimator of the coefficient of the right hand side endogenous variable with the reverse
2SLS
estimator of the
same unknown parameter when the normalization
is
changed.
Under the
null hypothesis that conventional first order asymptotics provides a reliable
guide, the
two estimates should be very
according to
first
Our test
difference in the
it
Indeed, they have unitary correlation
when second
order asymptotic distribution theory. However,
asymptotic distribution theory
bias terms.
similar.
is
two estimators
used, the
will differ
order
due to second order
subtracts off these bias terms and then sees whether the resulting
two estimates
satisfies the results
does and the second order bias term
is
small,
of second order asymptotic theory. If
we do
not reject the use of first order
asymptotic theory. Furthermore, the second order asymptotic theory
reliable basis for inference.
An added
attraction
of our approach
is
may
that
it
provide a more
permits the
econometrician to compare two estimates of a structural parameter, which will have a
straightforward economic interpretation in
many
situations.
can use economic knowledge to determine
if the
two estimates
Thus, the econometrician
are very different or are
close together in terms of the economic problem under study.
If the
new
specification test rejects
we then
second-order unbiased estimators of the type
first
consider estimation of the equation by
We again
proposed by Nagar (1959).
consider forward and reverse estimation by the Nagar-type estimators to determine if the
estimates are significantly different according to the
not significantly different
we recommend
new
estimation
specification test. If they are
by LIML, which we demonstrate
is
the optimal linear combination of the Nagar-type estimators (to second order). If the
second specification
test rejects or the
based on economic considerations,
LIML, may provide
two Nagar-type estimators
we conclude that
neither set of estimates,
2SLS
also provides
recommendations found
some
in the literature.
possible lessons about previous
To second
reduced form affects the asymptotic bias of the
2SLS
order, not only the
estimator, but also
R^ of the
two other terms
These terms are the covariance between stochastic disturbances
structural equation
and
in the
reduced form equation(s), and
instruments. These three factors interact in a nonlinear
in the
manner so
does not perform well.
in the
number of
it is
unlikely that any
single first order asymptotic theory based test statistic will suffice to indicate
errors in variables
or
reliable results for inference in the particular situation.
Our approach
are important.
differ substantially
when 2SLS
We point out similarities between this result and the well-knov^
model
in
econometrics to demonstrate
how this outcome might be
We also develop conditions under which the true parameter will be lie in the
expected.
of the forward and reverse estimates.
interval
Lastly,
IV
estimators.
may
we
investigate the performance of Nagar-type second order bias corrected
While these estimators and
also not perform well in the
LIML need
weak instrument
not be significantly better than
particular, inferences
based on
situation.
2SLS over
LIML may not do well
While Rothenberg (1983) uses
situation.
LIML can lead to
results
when
Thus,
demonstrate that
a range of possible situations. In
"weak instruments"
in the
LIML is
second order
reliable inference
We also demonstrate the high degree of similarity for
of LIML.
we
of Pfanzagl and Wefelmeyer (1978,
1979) to demonstrate that, under certain conditions,
specification test should help determine
improved performance, they
efficient,
our
can be based on the use
k -class estimators
between the approach of Bekker (1994) and the Edgeworth expansion approach of
Rothenberg (1983).
We analyze an empirical problem of a simultaneous equation specification of a
demand
who
equation. This type of model formed the original model consider by Haavelmo,
first
demand
demonstrated that least squares would lead to biased
for railroad
particular bulk
movements between
commodity. The
different origin
and destination pairs for a
original specification has quantity as the left
variable and price along with other variables on the right hand side.
have short run marginal cost
regression using price as the
endogenous
variable.
finds that the
asymptotic
variables.
We find that the 2SLS
two
left
elasticity increases, but the
new
right
hand side
specification test
estimates are close together enough so as not to reject the
results.
demand
We then reverse the
hand side variable and quantity as the
The estimated
hand side
As instruments we
estimate of the
about 2 times larger than the least squares estimate.
elasticity is
We consider the
results.
first
order
We then include many more instruments by interacting the cost
instruments with the indicator variables for each origin-destination
pair.
The estimated
price elasticity decreases significantly in magnitude, back toward the least squares
estimate.
When we run the reverse 2SLS
estimation,
we find that the
estimate
is
about 6
times higher than then forward estimate. Here our specification test easily rejects the use
of the
first
order asymptotics. Also,
LIML does not do well
in this latter situation.
The previous
literature
on the presence of weak instruments begins with Nelson
and Startz (1990 a and b) and Bound, Jaeger, and Baker (1995)
performance of IV estimators
when the weak
in the
instruments problem
weak instruments
may
who
situation.
exist are given
demonstrate the poor
Analysis of conditions
by Hall, Rudebusch, and Wilcox
(1996), Shea (1997), and Staiger and Stock (1997). Improved inferential techniques are
recommended by
Startz,
Startz, Nelson,
and Zivot (1998),
Wang
and Zivot (1999), and Zivot,
and Nelson (1999). All of these approaches are essentially
first
order asymptotic
approximation approaches in terms of recognizing the weak instruments problem and
offering alternative approaches to inference.
inference and to estimation that
by a number of researchers.
we use was
The second order asymptotic approach
initiated
to
by Nagar (1959) and has been used
We follow the particular second
order approximation of
Bekker(1994).
While many
different conclusions can
we tend to recommend
that the
IV
be drawn
estimates, or even the
used when the specification
test rejects (unless the
reason for this conclusion
that the
situations
when
is
in the
IV
weak instruments
situation,
"improved" IV estimates not be
two estimates
are close together).
The
estimators typically have significant bias in these
the specification test rejects
which recommends against
their use.
First
order asymptotics assumes that no bias exists, but the second order approach can find
significant bias
depending on the underlying primitive conditions.
present as demonstrated by the specification
may
test,
When this
bias
we believe that use of the IV
is
estimates
lead to misguided conclusions.
2.
Model
We begin with the simplest model specification with one right hand side (RHS)
jointly
endogenous variable so that the
single jointly
jointly
endogenous
endogenous
left
RHS variable.
variable,
which
is
by
hand side variable (LHS) depends only on the
In the class of models with only one
far the
RHS
most common specification used
econometrics, this model specification accounts for other
RHS
in
predetermined (or
exogenous) variables, which have been "partialled out" of the specification. Thus,
not lose any generality by not including predetermined variables in the
initial
we do
specification.
in the
We demonstrate below how RHS
predetermined variables
may be
included
formulae and computations.
We will assume that
(2.1)
Jl
=Py2^S\
+V,
=fiz7t^
(2.2)
where d^m{7i^) =
equation (2.2)
is
K
Thus, the matrix z
.
is
the matrix of all predetermined variables, and
the reduced form equation for y^ with coefficient vector
n:^
.
We also
assume homoscedastic normality:
r
r
V
L'
~A^(0,Q)~A^
(2.3)
«I2
^22
We use the following notation:
(yA
y=
,
o-^'^Varfei),
o-^^
=Cov(^„,V2,),
K^nJ
\yn.
The simultaneous equation problem, which causes
<T^
^
This situation
.
(1978) and others
3.
o"^^ =Cov(ff,,,v,,).
is
what
least squares to
be biased,
specification tests of the type proposed
arises
when
by Hausman
test.
Motivation
3
.
1
Errors in Variables
We first consider an analogy between the simultaneous equation model
specification and the errors in variable (EIV)
model
were replaced with a mismeasured exogenous
specification. If
variable,
jj
i" equation (2. 1)
under classical assumptions of
uncorrelated measurement error, the least squares estimate of J3 would be biased (in
magnitude) towards zero. Hausman,
Newey and Powell
(1995)
call this result
the "iron
law" of econometrics - the magnitude of the coefficient estimates are less than expected.
.
While
this result
does not always occur when
least squares is
since the direction of bias depends on the sign of 0,2
2SLS
used the coefficient estimate increases
is
under certain situations even when 2SLS
an estimate of n^ fi"om equation (2.2)
is
is
.
in
a
,
used on equation
common
finding
is
magnitude. However, in
used on equation
(2.1), bias
(2.1),
that
finite
when
samples
remains because
used, since the true parameters are unknown.
We now demonstrate how this result occurs.
Suppose
that
zn^
is
measured without
unbiased. Instead, z;T2must be estimated,
the
stage
first
., ,.
(31)
OLS
estimator.
i.e.,
error.
Then,
we have to
OLS
rely
of
y^
on zn^ would be
on 2SLS. Let
n.^
denote
We have
X1,(^..->^.'-(^2-^2))-^;^2
n
b^sLs-fi^
,
E,:,fer
Observe that
^lz,1,k -a;
-(^2
-^2))-^'a]= 1,1,4..
= ^u
Also note that
^J
^
i^'fit
regression to obtain ^^
(3.2)
Rf
Y = ^j
'
2^,"_
Therefore,
we
z,!,
yi^
>
-^.'(^'^r ^'^2]-/?
^'(^'^y
^i
where Rj
is
M2
the
•
R^
^
in the first stage
expect bias approximately equal to
x=y2i
We make some observations.
-
-4(^2 -^2)'(^'*2 -^2)
Other things being equal,
•
Bias
is
a monotonically increasing function of cr^
•
Bias
is
a monotonically increasing function of
K
•
Bias
is
a monotonically decreasing function of
Rj
.
Note that conventional asymptotics, which
influence of <T^
In terms
« -^
lets
oo
of the analogy with the EIV model
RHS variable,
covariance term
DGP
fixed, ignores the
,K, Rj.
specification, note that the bias in the
EIV model depends on the ratio of the variance of the
variance of the
keeping
a^
observation error divided by the
a result which occurs in equation (3.1) except that the
The
replaces the variance of the observation error.
model specification also depends inversely on the R^ of the EIV model
we demonstrate below,
we also
a result
bias in the
specification, as
find in equation (3.1). Thus, the finite sample
bias in the simultaneous equation problem has similarities with the bias in the
specification, with the
of instruments.
No
major difference that equation (3.1) has a term
similar variable arises in the
instruments are used
when OLS
EIV
estimation
is
EIV model
in
K
,
EIV model
the
number
bias formula because
no
undertaken.
Forward and Reverse Regressions
3.2
A well-known result in the EIV model specification is that the forward regression and the
reverse regression,
/?
,
when the
coefficient estimate
is
inverted,
where by reverse regression we mean interchanging the
bound the
RHS
true coefficient
variable with the
variable in the regression specification. Perhaps a less well-known resuU
is
LHS
that the
product of the forward and reverse estimates equal the R^ of the regression model (which
is
the
same
for the forward or reverse regression). Thus, a high
bounds for the true
coefficient
^
R^
are very tight, and vice versa.'
implies that the
We now explore a
similar result in the context of the simultaneous equation model. Let
(3.3)
b^^^
,
and
denote the forward and reverse
'
OLS
c^^=
'
^
estimates of the model (2.1). Note that
Thus, the use of the "permanent income" consumption model specification cannot explain the finding of
different estimates of the marginal
and average propensity to consume, when estimates are done on
aggregate time series data because the R^ of the regression specification
is
extremely high.
.
(3.4)
'^OLS
where E^^^
the
the
is
EIV model
parameter
~ ^y.x)
^OLS
R^
in the
»
OLS
regression of (2.
specification although here the
1),
a result that
is
the
same
as that of
two estimates need not bound the true
/?
Now,
(3.5)
'
let
h2SLS
2
'
and
'2SLS
Zyi
Z,yl.
denote forward and reverse
2SLS
estimates,
where
and
j)2,
orthogonal projections onto the subspace spanned by z
.
are the results of
j),,.
They
are based
on moment
restrictions
(3.6)
It
£k-Cy„-/?-jJ] = 0,
can easily be shown
(3.7)
V^.
that,
under conventional
z.-
(first
y2i--^yu
order) asymptotics.
=oM
^2SLS
'2SLS
which implies
1
and
J
that the forward
estimates are exactly the
same
and reverse estimates are perfectly correlated,
in a
given sample up to
first
i.e.
the
two
order asymptotics. Thus,
using equation (3.2) amounts to the implicit assumption that
(3.8)
^...P:)*^'
asymptotically to
reverse
2SLS
first
order. Empirically, the authors
have observed that the forward and
estimates can differ by large amounts numerically even with quite large
samples, which by equation (3.2) implies that in these situations conventional
asymptotics
question.
may
first
order
not provide a particularly good guide to the actual sample situation in
We use this observation and implication of equation (3.2) to provide an
.
approach that attempts to determine when conventional
when
relied on, or
alternative approaches need to
Bekker's (1994) Asymptotics:
4.
Since conventional
guide,
we need to use
Is It
first
order asymptotics can be
be employed.
Sensible?
order asymptotics do not necessarily provide a reliable
first
a different approach to the asymptotics.
of Bekker (1994) and see whether
main features of the bias
his
We explore the approach
approach to asymptotic expansion captures the
in the estimators that
concern
us.
We assume as in Bekker
(1994) that
K
1
>a
(4.1)
and
n
Below,
we examine whether his
4.
It
asymptotics captures our motivation.
Errors-in- Variables Motivation
1
can be shown that^
(4.2)
It
—Tr'^z'zn^^®n
plim&2si^
= fi-\-a—^
—
^
+ a(022
can also be shown that
1
(4.3)
plim
Using the
(4.4)
"
-Y,yli =
fact that
1
+ fi>22
and
plim
^
=
+ acD^^
a^ = co^^ - fico^ we may rewrite equation (4.2)
p\imb,,^=fi +
,
"
-
'
pUmn-Xy^^
^
"
-^yl
See Bekker (1994).
See Bekker (1994).
10
plim/?;
as
which coincides with equation
EIV model
(3.1),
and again points out the close similarity with the
specification result with the addition of the parameter
By a
similar argument,
a
.
we can show that
^12-^^11
g25i^=4-^" /?2r>/
(4.5)
+ ''/'W'
and therefore.
-^ = p^a^-^^^^o^{l).
(4.6)
Analogous to equation
(4.4),
we may also
1
Here,
i?^
is
from the
first
write
-y-'"
a
.
stage regression of equation (2.1). Thus,
equations (4.4) and (4.7) that if the (asymptotic) i?^
one, conventional
situations
first
when this
's
we
see from
of the reduced form equations were
order asymptotics yields the correct results. However, in actual
does not hold the asymptotic approximation to the bias
result
depends on the covariance term which creates the simultaneous equation problem
first place,
the size of the
B}
in the
reduced form equation and the parameter
in the
a which
approximates the dimension of the subspace spanned by the predetermined variables
relative to the
how well
sample
size.
conventional
first
Thus, under this approach sample size alone does not indicate
order asymptotics do, but instead the dimension of the
subspace spanned by z must also be considered.
4.2
IC-
Motivation
By using Lemma
5 in Appendix,
we can show that
11
,
which avoids the
and false R^ assumption of equation
implicit
Bekker asymptotic approach does
result
5.
we attempt to
capture and that affects the bias in the
•/?^,.yj
= b^sis
^Y
'^isls
Thus, the use of the
•
2SLS
estimates.
2SLS
definition, equation (4.8) implies that
< pWmb^sLs Plimcj^^ <
(5.1)
(3.3).
yield an implication consistent with the empirical
Biases of Forward and Reverse
Because
which
.
1
in turn implies that
< plim ^^^
(5.2)
plim
^^^ < 1
l/;9
;9
Inequality (5.2) suggests that the forward and reverse
parameter
J3 as in the
2SLS
estimates
may bound
EIV case.
In order to understand the inequality
from a
different perspective, rewrite
equations (4.4) and (4.8) as
P"""""" "
(3 3)
^^
p\imb2sr^-j3
1
o-..
P CT^^
V^rnR}
plimR) plim«-'^"^^j;;^,
where the second equality
is
Assume that a^ja^^ >
We would then have
.
^.^ yl
plimR^ p\imn-'Y,l,yu
1
1
plim;.-'
based on
12
the true
or
/3e{p\\mb^sLs,p\iml/c2SLs)
(5.4)
>9
e (plim
l/c^.,^ .plim^'^si^).
We can see that the ratio
determines the relative magnitude of the bias of the two estimators.
bias of c^sis
is
small relative to that of b^s^s if -^/
We can see that the
« ^r
A Specification Test based on Forward and Reverse 2SLS
6.
We now turn to the main contribution of the paper. We attempt to provide an
answer to the question:
When
can you trust the conventional
the well documented problems of the
As our
cases?
first
first
order asymptotics given
order asymptotic approximation in certain
derivations demonstrate above, the
2SLS
bias depends
on
3 factors: the
covariance of the stochastic terms in equations (2.1) and (2.2), the R^ of the reduced
form equations, and the parameter
single statistic, e.g. the
seems
likely to
is
doing
we turn to one
and see
and n
.
Thus, no simple
(or the associated
F
statistic),
in a particular situation.
of the basic ideas of the specification
estimate the same parameter,
difference between the estimates
conclusion.
K
be sufficient to answer the question of how well the conventional
Hausman (1978) and
of the model
which depends on both
R^ of the reduced form equation
asymptotic approximation
Instead
a
is
small,
how far apart they are.
is
,
one will not
specification. If the difference
Here a possible approach
yff
is large,
in
two
test
approach of
different ways. If the
reject the underlying
assumption
one will come to the opposite
to use the forward and reverse
2SLS
estimates
Thus, the specification test will be used in model
specifications with overidentification, but this situation holds in
most
instances.
"economic sense" of the difference of the two estimates can be gained because
cases the econometrician will
know how big
iinportant, since the parameter will
a change in the true coefficient >9
have a marginal
13
interpretation.
An
in
is
many
To do a
two
statistical test,
Here
estimates.
first
we need to
determine the variance of the difference of the
order asymptotics will not suffice, since because the forward
and reverse coefficient estimates have unit correlation, the variance of the difference of
the
two estimates
we turn to
will
be zero when a
first
order asymptotic approximation
is
used. Thus,
second order asymptotic approximations, which were pioneered by Nagar
(1959) and have been used since by Kadane (1973), Sargan (1976), Rothenberg (1983),
and numerous other authors.
Note that the
p
estimators of
ci.oi.^2
Bekker (1994) shows
'^^v^isLs
general,
~y^2SLs
we would
two possible
equal to
9<^.'-^°Mn)
(0 + aojjz \p^ +
B = -a
(6.1)
is
probability limit of the difference between the
that
~^)
's
2SLS
is
)
asymptotically normal. Therefore,
also asymptotically normal.
like to deal
we do
Because
not
know
B
in
with an asymptotic result of the form
.^
"2SLS
^2SLS
V
for our specification test
B
^
1
V«
(6.2)
will
-> A^(0,F)
J
4
be a consistent estimator of the difference of the
biases.
Let P^ and
denote the projection matrices onto the column space spanned by z and
complement.
(6.3)
It
can be shown that
Q-k-aa}^^-^\mi—y\P^y^,
and
;99
n
Further,
(6.4)
it
can be shown by using
plim «
+ ao,^ = plim — j^2^^;^r
n
Lemma 5
in
= 0(T^ + adet(Q),
14
Appendix
that
its
M^
orthogonal
where
i-an
\n
1
a
.,
j\-an
1
a 1
y'2^.y2
J X-d n
,
,
l-an
\r
1
+a
1
^l-d n
and
d
is
\
1
..
y'M.y.
J\-an
\
+ -y'2P.y2-T^-y2M,y2
\n
\-an
yn
1
l-d n
..
y'2^.y2
any consistent estimator for
a\
--y\M,y,
l-d n
a We may therefore use
.
B = -d
(6.6)
-y2Pzy2--y'2P2y\
n
By the
delta
Theorem
1.
n
method based on
Lemma 5
in
\
(
4n
-B
^2SLS
'2SLS
Thus,
we compare the
Appendix,
it
can be shown that
kN'
la
'l-a (0 + a6)22)'(/?9 + a«,J'
J
difference of the forward
2SLS
estimator and the reverse
2SLS
estimator after subtracting off the bias term which arises to the second order of
approximation. Note that the order of the variance in Theorem
0(n~'
j
1 is
0\rf^) rather than
because of the second order approximation. Thus, the specification
form of an asymptotic
/
test takes the
statistic:
m=
(6.7)
w°'
where
d
Theorem
''^
is
1
.
the
LHS
of Theorem
1,
and
w
is
a consistent estimate of the variance in
We discuss later how to estimate this variance term.
A similar result was given in Hausman (1978) for the bias of the least squares coefficient when tlie RHS
15
7.
A Specification Test based on Nagar-Type Estimators
We also explore an alternative approach, which is closely related to comparing
the forward and reverse
2SLS
estimators.
Nagar (1959) calculated the second-order
of the 2SLS (and other k class) estimators.
estimators to second order. Thus,
we
bias
He demonstrated how to bias-adjust these
can estimate Nagar-type bias corrected IV
estimators and then again compare forward and reverse bias-corrected estimators.
The
estimates should be very similar if the asymptotic approximations are sufficient for the
particular simultaneous equation
as in the last section, but here
model
we
specification. Thus,
we use bias-corrected
follow a similar strategy
forward and reverse regression
estimators.
We use the B2SLS
estimator of Donald and
forward and reverse regressions. Note that
member of the Nagar
class
Newey
(1998) to estimate the
this estimator is a
class estimator
/:
of estimators. The forward IV estimator of
/3
and
is
is:
K-l
where
(7.1)
;i
=
-
n
K-l
n
We can also estimate p
by the reverse IV
specification:
(7.2)
y\P.y^-^y'^.y^
c
By the
delta
Theorem
method based on
Lemma 5
in
Appendix,
we
can show that
2.
_
-I
V" \^%-(p.d -^N
1-
Q'
\-a
1-a
J3Q'
^2^2
2
,
^2
£V,
\
variable
was
\-a
L
correlated with the stochastic term in the equation.
16
p^Q
2/:^2
a
Proof: See Appendix B.
As
in
Theorem
1,
the terms of the variance matrix are of order 0\n'^) due to the second
order nature of the asymptotic approximation.
We will use Theorem 3
below
to
compare
the forward and reverse bias adjusted estimators of fi to form a test of the model
specification.
We may want to consider linear combinations of b
and
—
for
improved
c
inference.
It
can easily be shown that the asymptotic variance, to second order, for the
optimal linear combination
is
ra.,(«_) =
(7.3)
given by
^.-^M2,
l-a
which coincides with the asymptotic variance of LIML
Therefore,
forward
we may interpret LIML
2SLS and
reverse 2SLS.
LIML
is
also
known to be median unbiased
of the stochastic disturbance of equation
(1977), and,
more
(2. 1),
(2.1), as
for normal
shown by Anderson
generally, for symmetric distributions of the stochastic disturbance of
by Rothenberg (1983). Thus, the optimality
Wefelmeyer (1978, 1979) are applicable
admissible, while other
definition
by Bekker (1994).
as an optimal linear combination of bias corrected
distributions
equation
as derived
k
results
to claim that the resulting
class estimators are inadmissible unless
of Pfanzagl and
LIML
A
estimator
is
in the estimator
above has a coefficient of unity.
We now calculate our second specification test by comparing the forward and
reverse
and
B2SLS
estimators.
Note that no
in the first specification test since
The variance of the
2,
as in
Theorem
Mb--\-^N
_
1
our estimators here have no bias to second order.
we obtain
f
3:
made
difference of the estimators thus has a very simple form.
consequence of Theorem
Theorem
bias correction need be
(
n\t\
2a (a]f
0,
17
As a
Proof: Follows from
hO-f
V
<T^C7.'
r,
+
0'
\-a
frr^
"l
+0"?.
"l
2r\2
p^Q
\-a
-2
\-a
J
/50'
\-a p'Q'
1
a
\-a P'Q'
2a iplf
l-a/?'0''
Note that the denominator of the variance
order approximation
we use. Our second
is
again of order 0\n~^) because of the second
specification test has the
form of an asymptotic
/ statistic:
(7.4)
m,=/^
0.5
^2
where the numerator is the difference of the two estimators multiplied by
denominator
the square root of the variance term in
is
Theorem
3.
discuss
how to
8.
Similarity of Bekker's (1994) Asymptotics to the
and the
n^^^
We subsequently
consistently estimate the variance term.
Edgeworth
Expansion for /:-Class Estimators
In this section,
we
approximation
is
demonstrate that the relevance of Bekker's (1994) asymptotic
not necessarily confined to the case where
Bekker's alternative limiting distribution
is
a - Kfn
is large.
driven by the assumption that the
Given
number of
instruments grows to infinity as a function of the sample size, his approximation
seem of limited
applicability
when the number of instruments
is 'small.'
that Bekker's approximation is in fact quite similar to the second order
expansion with symmetrically distributed
errors.
18
that
may
We demonstrate
Edgeworth
Unlike the Edgeworth expansion based
approximation, Bekker's approximation produces limiting normal distributions, which
causes the resulting tests to be quite convenient. Normal approximations turn out to be
quite reasonable approximations as supported by our
Monte Carlo
simulation discussed
in Section 13.
Rothenberg (1983) computes higher order moments of A:-class estimators. For
symmetrically distributed errors,
Jn{p2sis
can be shown by Rothenberg (1983, Theorem 2) that
- P) has an (approximate) mean
^"^
^
(8.1)
'
%4^
mean of h^^^
which predicts
that the
(8.2)
P + ^^-
Observe
it
that equation (8.2)
is
is
approximately equal to
similar to the probability limit (4.2) of 2SLS under
Bekker's asymptotics except that equation (4.2) uses
bias.
As
for
Q + aco^.^
LIML, using an Edgeworth expansion we
(approximate)
as the denominator of the
find that 4n{bjjj^
- 0) has an
mean
ctI
--^ =
(8-3)
o{y),
0V«
and (approximate) variance
^^£<^;<;<^„(i),£i^„detp
(8.4)^
which
„
is
02
similar to the Bekker-based result
the approximating factor a/(l
- a)
02
'^^
we derived
for
in equation (7. 1) has
19
LIML in equation (7. 1),
changed to
a
except
in equation (8.4).
As
for the (forward) A:-class estimator b considered
an Edgeworth expansion
it
by Donald and Newey (1998), using
can be shown that 4n{b - fi) has (approximate) mean
0,
and
(approximate) variance
£l+£^>il<,„(,).i,.->-.*<
''0
(8.5)
0/7
0'
0'
Notice that equation (8.5) again agrees with a Bekker-based asymptotic variance of the
Donald-Newey estimator in Theorem 2 except
a
correction terms are of order
a/(l-a). These
,
Rothenberg's Edgeworth
whereas Bekker's correction terms are of order
results suggest that
interpreted as a convenient
that, again,
Bekker's asymptotic approximation can be
method of Edgeworth expansion with wider
might be thought considering Bekker-type asymptotics
applicability than
in isolation.
Bekker-type asymptotics or Edgeworth expansions do not always provide
reasonable approximation to
finite
sample distribution of IV estimators.
should be noted that variance predicted by the Edgeworth expansion
guaranteed to be positive.
4n{p2SLs
^
~ P)
It
Observe that equation (8.6)
the variance of 2SLS
not
tell
is
is
not always
is
equal to
02
smaller than that of a Nagar-type estimator or LIML.'
We could
whether Bekker's asymptotics predicts the same pattern of variances. There
(8.6)
may be
in certain situations: It is not difficult to
such that equation (8.6)
is
negative, especially
The bias of 2SLS
is larger,
come up with a parameter combination
when
which leads to the optlmality
20
is
overly optimistic about the variance of
the
first
stage i?^
,
and hence
extremely small which can correspond to the "weak instrument" situation. Because
'
it
smaller than equations (8.4) or (8.5), which suggests that
good reason to believe that equation
2SLS
^^0
02
w
'
of all,
can be shown that the (approximate) variance of
by Rothenberg (1983, Theorem 2)
calculated
is
First
results of Rotlienberg (1983).
,
is
Bekker's asymptotic variance of -Jnib^^j^ -
/?) is
based on the delta method,
guaranteed to be nonnegative. Therefore, Bekker's asymptotics
way to
fix
may be
such undesirable predictions of Edgeworth expansions
However, a
further caution should be recognized
or Edgeworth expansions for
LIML
when using
in
it
is
interpreted as a
extreme
situations.^
either Bekker's asymptotics
or Nagar-type estimators. Neither
LIML nor Nagar-
type estimators possess finite sample second moments.^ Thus, the performance of the
asymptotic approximations
instruments" situation
is
may
present.
vary depending on sample size and whether a "weak
We explore this possibility in
we
Section 13 where
perform Monte Carlo experiments.
Estimation of Asymptotic Variance Terms
9.
9.1
2SLS
For the
first
specification test,
.,^^
(9.1)
we
need to estimate the asymptotic variance
fc£®^
For this purpose we note that a] may be consistently estimated by
a/ =
(9.2)
for
some
for
,
The
,=1
consistent estimator fl for >9
.
We also note that LEML is consistent for
;5
.
As
we note that
^ -y[P,y,
n
(9.3)
*
n- 1
--^-y[M^y,
\-a n
=
+ o^(l)
Edgeworth expansion predicts a smaller variance for 2SLS suggests tliat if tlie bias of 2SLS
2SLS may dominate both Nagar-type estimators and LIML under reasonable loss functions. In
Section 13, we investigate such a potential outcome by Monte Carlo simulation
'
See Mariano and Sawa (1972). As for the forward Nagar estimator, it does not even possess first moments
as established by Sawa (1972).
is
fact that
negligible
21
by
Lemma 5. As
for
(9.4)
«-^w-1
Finally,
we note that
« we follow Bekker and
,
+ aa>22 = p'im — >'2-''r^2
(9. 5)
To summarize, our consistent
.
^^^
use
fi^ + ^^12 -
Pl''" ~>'2-^7^i
estimator for asymptotic variance
is
given by
^2
J
fciO^.
-/?LZML>'2,)7f >'2^.>'2
-^^7Y^^^'^2
Bias Corrected 2SLS
9.2
For the second
specification test based
on the bias corrected 2SLS estimators,
need to estimate the asymptotic variance
,„ ^,
(9.7)
w, =
'
2a cj]
r^.
l-a>9'0'
By the same calculation
as in the previous section, a consistent estimator
is
given by
In either of the variance estimates of equations (9.6) and (9.8), a different consistent
estimator other than
estimated test
LIML can be used, with no change in the distribution of the
statistic.
22
we
.
.
Included Exogenous and Predetermined Variables
10.
We have so far assumed that a single jointly endogenous RHS variable exhausts the list
of explanatory variables. The
results
we have
derived are fully general with respect to the
inclusion of predetermined variables in equation (2.1). In this section,
that our procedure
would need
Suppose
that the full
''
Z,, is
a
k^
M^
if
equations
"
(2. 1)
K
all
in
equation
other predetermined variables.
out of equation (10.
Z,,
1),
and
let
and (2.2) be understood to be the resultant expression: Let Yj denote a
Yj^
.
Define Z,, Z^, E, and V^ similarly. With
y,=M,Y„ y,=M,Y„ z=M,Z„
we obtain
partialled out.
'
dimensional vector containing
column vector consisting of
(10.2)
and (2.2) are understood
is
denote the projection operator partialling
equations
(2. 1)
demonstrate
dimensional vector of included predetermined variables
(10.1) and Zj, isa
Let
model
'
1)
where
be modified
where included exogenous variables are
to be equations
(10
to
we
£ = M,E,
v,=M,V„
equations (2.1) and (2.2) premultiplying equation (10.1) by
We ask if there is any simple way to compute
projection of Z^
J2^^>'2' y-J'^z}'!
M^
»
.
^^c avoiding the
on Z^ Simple projection algebra based arguments show that they could
be characterized quite
.
The following then provides a convenient computational
easily.
procedure:
ff
Obtain residuals, and label them W^ and W^
•
Regress
7,
and Y^ on Z,
•
Regress
7,
and Y^ on Z, and Z^ Obtain
•
Let
j),
.
.
residuals,
=W, -W, and y^^W^ -W^.
23
and label them W^ and W^
Compute
.
y'^p^y^
y'M.yi = yly
As
•
for
.,-1
As
•
V
for
y'jP.y^
,
We can replace
,
y\Pzy^
= y'xy y'M.yi = Kyi
>
plug into equations
^
,
1,
= i"^,
yM^yi = yiJi ^nd
>
al we can estimate
regression:
/
= y'jh
by the average squared
it
2^J^,(y,
- PumyiJ
(6. 1), (9. 1),
and
LIML residuals
'" equations (9.1)
.
(9.2).
on the
full
and (9.2) by
~ HUML^2i ~ ^mYuML )
K
in equations (6.6), (9.6),
and
(9.8),
we may
conservatively use
K = dim(Z,.)+ dim(Z2,), although K = dim(zj= dim(Z2,)
may
also be a reasonable
choice.
Note
also that
=n-k^
n*
one may want to adjust the sample
to take account
of the
loss
size in the
above equations to
of degrees of freedom from partialling out the
Z,,.
variables.
11.
Additional
To this
jointly
RHS Jointly Endogenous
point in the paper,
endogenous
we have
which
variable,
is
empirical application of IV estimators
specifications to allow for additional
second specification
test for
generalize our results to
r,
2
>2
by
only considered the situation of one
far the
{e.g.
Variables
most
2SLS).
common
RHS
situation encountered in
We now extend the model
RHS jointly endogenous variables. We derive the
RHS jointly endogenous,
which demonstrates how to
RHS jointly endogenous variables. We
leave the
derivation of the first specification test in this situation to further research.
We extend our original simultaneous model specification of equations (2.1) and
(2.2) to the situation of 2
(11.1)
y^=P2y2+P^y^+e^
^^
(11.2)
RHS jointly endogenous variables:
'
'
J3=z;r3+V3
24
)
where we use the same matrix and vector notation
P^ and
yffj
in equation (11.1)
by use of the Donald and Newey (1998) B2SLS estimator.
We will refer to the estimator as
estimate
- P^lPi)
(l/y^j
or
(6, ,c, )
.
Changing the normalization
- P^j P^. Thus, we would have
(l/yffj
estimators for O^z' A)- '^he question
potential estimators to achieve the
would
naturally arise
in
Technical Appendices
cannot stack the estimates to derive a more powerful
variance matrix of the three tests
will
have the same operating
singular.
is
characteristics,
Thus,
1/Z»2
is
test
C
we will
of a given
and D,
it
size.
turns out that
asymptotic
based on a single difference
and a more powerful
using additional differences (contrasts). Thus,
could also
three potential
test since the
all tests
we
of how to combine these
most powerful specification
However, as we demonstrate
we
We consider estimation of
as before.
test
cannot be derived
use the estimator
A,
-l/^j
>
where
the estimator derived from application of B2SLS (or another Nagar-type
estimator) to the equation:
(11.3)
''~-h^ A
In Appendix
D we derive a consistent estimate of the asymptotic variance of the scaled
difference of the
two estimators d^ =
^-1
(11.4)
2
y^+s^.
j
(z,ii
n^'^ip^
-yb^)
to
^u - P2.um.y2i - pyumy^i y
n-K
P2.UML y2P.y2
—
\y'2P.yz
,
,
be
J.
A-l
i^
77y2^.y2 - ^
n-K-
n-K-j^y'i^zyi
•^'^'•^'"w^-^'^'^'
As before,
formula.
other Nagar-type estimators
The
may
replace
specification test will take the form:
(11.5)
25
LIML estimators in the above
where w^
is
the estimated variance in the above equation.
Inclusion of exogenous and predetermined variables in the specification as in
equation (10.
Section 10 raises no
1) in
methodology we used
more)
in Section
9
is
new
by regressing
Y^
on Zj
The
partialling-out
directly applicable to the current situation with 2 (or
RHS jointly endogenous variables.
partialled out
complications.
.
The new jointly endogenous
variable, Y^
All other formulae follow as before, and the
,
is
above
variance formula can be used on the partialled out variables to form the second form of
our specification
12.
test.
An Empirical Example
We analyze an empirical example of a simultaneous equation specification of a
demand
function. This type of specification
Haavelmo, who demonstrated
variable of the
first
origin-destination
fi-eight
in log
(OD)
movement. As
movement which
The
left
by
hand side
movements of a homogenous bulk chemical
of ton miles. Data were collected on approximately 50
pairs over a 33
right
the original type of problem studied
that least squares lead to bias results.
specification represents
commodity measured
is
month
period.
hand side variables,
we
Each data point
is
an individual
include the log of the price of the
a jointly endogenous variable, a measure of economic activity, and
is
OD indicator variables which change each year to allow for fixed effects for OD pairs.
We also used a trucking price index variable, which was assumed to be predetermined.
Altogether,
As
we have
132 right hand side variables, one of which
instruments for the jointly endogenous variable
cost variable for the appropriate
index for diesel
*
of a short run marginal
variable that
is
available
we use is the monthly price
fuel.
In Table
estimate of the
we use the log
jointly endogenous.
movements of the bulk commodity, which
The other instrumental
for each shipment.*
is
A in the first column we give the estimated price elasticity (and an
first
order asymptotic standard error) along with the estimated standard
While these data are accounting data
that
unUkely to be tnie measures of marginal
cost, potential errors in
variables in instniments do not create a problem in instnrniental variable estimation under the usual
assumptions. See
Hausman (1977).
26
error and the
R^
estimator.
which
is
Note that the
The R^
quite precisely.
2SLS
.
is
price elasticity estimate
also quite high at .962. In
The estimated
we find
bias of least squares. Again,
we
consider possible diagnostics,
an
F
statistic
RHS
-1.36 (.147) and
statistic,
F
estimated
column 2 we use the conventional
direction of the simultaneous equation
a relatively small estimated standard error.
we find
that the
R^ of the reduced form
of 154.5. The R^ of the reduced form model
after all
variables of the structural equation have been partialled out
associated
is
price elasticity increases in magnitude to -2.03 (.465)
outcome given the expected
the expected
is
is
When
0.941 with
is
of the predetermined
.093 with an
of 74.6. While the partialled out model has a lower R^ and
statistic
as expected, they
do not indicate a problem according
F
to rules of thumb
previously put forward in the literature.
We now interchange the jointly endogenous variable and put price on the LHS
and quantity on the RHS. The results are given in Column 3 of Table A. We use the
same instruments and
find our estimate
of
—
to
be -0.433 (.094) so the reverse estimate
c
of the price
elasticity is -2.3
1
(.500) so that the difference between the forward and
reverse estimates of the price elasticity
is
which should be exactly the same under
The question
0.275.
first
whether these estimates,
is
order asymptotics, are different enough to
reject the conventional first order asymptotic approach.
Using the second order approximation of equation
difference in the bias of the
two estimators to be
-.0012,
(6.6),
which
we
is
estimate the
quite small.
use equation (9.1) to calculate the variance and estimate our specification
We then
test statistic to
be:
(12.1)
;;,
=
d = __
10.03
__
= 1.85,
~
w"-'
5.43
Thus, up to a second order asymptotic approximation
we do
not reject the
first
order
asymptotic approach or the associated estimators.
We now use the Nagar-type estimator of Donald and Newey in columns four and
five.
Here we
find the
same estimates
degree of overidentification
is 1.
in the
forward and reverse direction because the
Using Theorem 2 to form the specification
27
test,
we find
it
to
be 1.82, which
is
very similar to our previous estimate. Lastly,
estimate to be -2.05 (.469), which does
expected, but note that
it is
lie
we
find the
LIML
between the forward and reverse estimates, as
2SLS
quite close to the forward
estimate.
We now increase the number of instruments by 13 by interacting the cost
instrumental variable with the corresponding OD indicator variables. This new variable
allows for unobserved cost differences across the different OD pairs. The resuhs are
1
given in Table B. The
first
column has the forward 2SLS estimate of-1 .24
(.194)
which
has decreased significantly in magnitude back towards the least squares estimate from
Table
B
A situation of weak instruments may well be present.
of -1.36.
reduced form
is
The R^ of the reduced form
0.972.
with an associated
F
statistic
of 2.79, which gives
The R^ of the
for the partialled out
little
indication of a
model
is
.219
weak instruments
problem.
In the second
(.789),
which
is
column of Table
B we present the reverse 2SLS
approximately 6.5 times higher than the forward
difference between the
two estimates of -6. 77 would
likely
estimate of -8. 01
2SLS
estimate.
be considered
significant,
economic terms, by most researchers. Here the R^ of the reduced form of the
out model
is
.038 with an associated
"weak instruments" problem
literature.
The
F
on
partialled
of .280, which could indicate that a
statistic
exists according to rules
difference in second order bias terms
of thumb put forward
is
in the past
estimated to be -.041,
smaller than the actual difference in the forward and reverse estimates.
is
The
The
much
test statistic.
estimated to be
(12.2)
;;,
=
d
247.3
__
__
=
= 14.21.
~
w'-'
17.4
Thus, the specification term rejects the conventional
first
order asymptotic approach, and
we would recommend that the estimates not be used.
In columns 3 and 4 of Table
B we present the Nagar-type bias corrected forward
and reverse IV estimates recommended by Donald and Newey. The forward estimator
now -1.21
(.194), while the reverse estimator is
has been made, the two estimates
still
differ
-4.64 (.293). While some improvement
by a
large amount.
estimated to be 5.78, which again rejects. Lastly, the
28
is
The
specification test
LIML estimate is -1
.
1
8 (.2 1
1),
is
which, again,
is
quite close to the forward regression. Thus,
use of LIML in the weak instrument situation
estimators differ significantly because
it
when
we
do not recommend the
the forward and reverse Nagar-type
often has a significant asymptotic bias, as
indicated in this example and in other empirical examples
we have
investigated.
We conclude that in a real world example that the IV estimators can perform
poorly in the
weak instrument
situation.
Using the forward and reverse estimate seems to
give a convenient metric to analyze the performance of the estimators. The specification
tests
we have proposed
work
also
as
we would
Carlo results to explore further the performance of the
13.
We now turn to some Monte-
expect.
tests.
Monte Carlo Experiments
We generated data from the model specification
J^2i
=-2>2+^2,
i
= \,...,n
such that
^.-HoJkI
12
©,12
Here,
Rj
^2^k^'k2
-J2 _
Ri^
0),
n=
K(/>'
1
denotes the theoretical
R^
in the first stage regression.
We use following
parameter combinations:
«=
100,
250,
o-„^=-.9,
-.5,
.5,
.9
.01,
.1,
.3
-R^=.001,
K = 5,
10,
1000,
10000
30
We examined performance of our tests by 5000 Monte Carlo replications.
1-4 report results using a range of instruments from 5-30,
29
Tables
sample sizes of 100-10,000,
and a range of covariances (correlations) where
regression.'
Columns
(a)
and
(b) report the actual size
2SLS with 10% and 5% nominal
reverse
we vary the Rj of the
sizes.
The
of the
test
actual sizes
reduced form
based on forward and
of the
test are generally
quite close to the nominal sizes, with only a small falling off above the nominal size
when the number of instruments becomes
Columns
column
(c)
large and the
quite
low
(.001).
and (d) report actual biases of forward and reverse 2SLS estimators, and
(e) reports the
expected value of
B
.
The estimates of the
order bias terms are typically quite accurate, although
biases
R^ becomes
becomes
quite large, the estimates can vary
situations, the test statistic should
still
work
difference of second
when the expected
by quite a
lot.
difference of
However,
in these
well because the presence of a large expected
bias (even if not measured totally accurately) will alert the econometrician to the dangers
of using 2SLS, or other IV estimators,
expected value of
in this situation. Importantly, the estimates
B appear to do a good job
instruments," e.g. column (e) in Table 3 with
of the
of indicating the presence of "weak
"weak instruments" compared
to
column
(e)
of Table 4 where the instruments are better because the R^ of the reduced form equation
is
much
higher. Thus, the second order asymptotic approach seems to provide a useful
tool to indicate
when the "weak
Columns
(f)
not perform well
test
in
previous literature.
1
of
sizes equal to
10%, and 5%.'°
a large variety of situations, as has been noted numerous times in the
As shown
the correct size. This result
«
traditional test
of overidentification, based on the forward 2SLS estimates, does
thei?^ of the reduced form
Tables
present.
on forward 2SLS) with nominal
has actual size of above 0.3,
' In
is
and (g) report the actual size of the
overidentification (based
The conventional
instrument" problem
in
Table
1
the conventional test of overidentification often
when the nominal
becomes
is
size is smaller than 0.1.
Note that when
high, the test of overidentification has approximately
expected since the test of overidentification assumes
- 4, we set tlie values of yffsuch tliat Var(e)=l.
We use
i? of the regression of the forward 2SLS residuals on instruments as the test statistic.
Because forward and reverse 2SLS should be perfectly correlated under conventional asymptotics, tests of
overidentification based on forward and reverse 2SLS should have the same operating characteristics if
conventional asymptotics provides reasonable approximations to sampling distributions of various IV
'"
•
estimators.
30
implicitly that this
coefficients are
of the
test
R^
is unity, in
known with
the sense that
assumes that the reduced form
Also, for very large samples, e.g. 10,000, the size
certainty.
becomes approximately
it
correct.
When the number of instruments begins to
When the
increase, the size performance of the test of overidentifi cation falls off again.
number of instruments becomes
quite large (30) in Tables
1
-
4, the actual size
of the
conventional test of overidentification becomes abysmally large, sometimes exceeding
0.5 in the
low Rj
situation.
Thus,
we
conclude that the second order asymptotic
approximations work considerably better than the conventional
approximations
when
Columns
(h)
applied to the
and
bias corrected estimator.
(i)
2SLS
first
order asymptotic
estimator.
report results for cases
We find that the actual
on Donald and Newey's estimator with 10% and
where we consider the Nagar-type
size
5%
of the new specification
traditional test
Donald and Newey's forward estimator) with nominal
sizes equal
0.2,
the reduced form
when
the nominal size
becomes
is
Columns
tolO% and 5%.^'
traditional test
overidentification, the conventional test of overidentification
of above
test.
of overidentification (based on
While the use of the Nagar-type estimator improves the
size
based
nominal sizes again approximates
the nominal size quite well with no tendency to be too large a size for the
(m) and (n) report the actual size of the
test
of
sometimes has an actual
smaller than 0.1. Note that
when the Rj of
high, the test of overidentification has approximately the
correct size once again.
Also, note that in columns 0)-0) where
type and
and
we report the means biases of the Nagar-
LIML estimators, the mean bias of the Donald-Newey (Nagar-type)
LIML estimators occasionally
the non-existence of finite sample
discussed in Section
8.
These
estimators
are found to be very large. This finding results fi"om
moments of Nagar-type and LIML
results should
LIML estimates even with the second
estimators that
be a caution about using Nagar-type or
order asymptotic approximations without
fijrther
investigation or specificafion tests in a given empirical problem.
"
We
use
n-R
of the regression of the residuals from Donald and Newey's forward estimator on
instnmients as the test
we
statistic.
31
Table 5 report Monte Carlo results
instruments
the
new
sizes,
is
large, AT
=
theRJ of the reduced form
30, and
specification test in
columns
although in a few cases the
large size.
However, these
overidentification based
(a)-(b)
in
but they
still
(h)-(i) are
low.
The
actual sizes
of
again close to the nominal
be compared to the
results should
on 2SLS
and
is
number of
the
based on the Nagar-type estimator does have too
test
columns
exceed 0.85, even though the nominal size
overidentification based
some "extreme cases" where
in
(f)
is 0.
10!
Similarly, the traditional tests of
in
exceed the nominal size by factors of 2 to
(e),
of
and (g) where the actual sizes always
on the Nagar-type estimators
second order bias estimates of column
traditional test
columns (m) and
5.
These
results,
which are again successful
(n)
do
better,
along with the
in indicating the
presence of "weak instruments," demonstrate that tests based on the second order
asymptotic approximations do considerably better than
first
based on the conventional
order asymptotic approximations in these extreme situations.
As we
2SLS
than for
discussed in Section
LIML.
negligible andi?^
loss.
tests
This result
is
In Table 6,
small. In
is
all
8,
Edgeworth expansions predict smaller variances
we compare 2SLS
cases,
LIML does not possess second
However, the dispersion of LIML around
asymptotics
much
may be
LIML when the bias
2SLS dominates LIML under mean
not surprising because
interdecile range is
and
y9 measured in
larger than that of 2SLS.'^
a poor approximation
when Rj
for
of 2SLS
square error
moments.
the interquartile range or
We conclude that Bekker's
is
extremely small, which leads to
the suggestion of using the specification test to help determine the usefulness of second
order asymptotics in a given situation.
14.
Conclusions
Using the forward and reverse 2SLS estimates to
form a
specification test
seems to be a helpful approach.
asymptotic approximation to form a test
asymptotic approach
'^
is
statistic
test for
weak
We use a second order
accurate enough to provide reliable inferences.
The fact that 2SLS does better than
endogeneity of regressors.
instruments to
to see if the conventional
first
The
order
first
order
LIML suggests that 2SLS should be used for Hausman tests of
32
is
asymptotics implies that the two estimates should be the same, while the second order
asymptotic approach allows for different biases in the two estimators. The
econometrician can also consider the estimates and see whether the difference in the
estimates
is
is
large in
economic
terns relative to
what would be expected. The
test statistic
straightforward to compute using existing econometric software to calculate the
LIML as well
estimators, Nagar-type estimators, and
While giving guidance to inference
econometrician's beliefs,
specification test
2SLS
we
of equation
as the partialled out models.
suggest the following approach.
(6.7) based
small, the conventional first order asymptotics
all right.
We suggest that the new
on forward and reverse 2SLS be done.
estimates are close and the estimate of the bias term
should be
on the
often subjective based
is
2SLS
may be
B
from equation
used, and the
If the test rejects or the estimated bias term
is
2SLS
large,
If the
(6.6) is
estimates
we then
suggest
using Nagar-type estimates to perform the second specification test based on equation
(7.4).
If the forward and reverse estimates are close and the specification test does not
reject,
we
suggest using
If the test rejects,
we do
LIML, which
the optimal combination of the two estimators.^^
is
not suggest using these estimates as either a failure of the
orthogonality conditions or an extreme situation of "weak instruments"
is
likely to
be
present. If the Nagar-type forward and reverse estimates are not close but the
specification test does not reject, a decision cannot typically be
made based on the new
specification test.
Our approach can be generalized when more than one jointly endogenous
is
on the
right
hand side of the model
specification.
Two
variable
variables can be interchanged
as before to provide forward and reverse estimates. Second order asymptotic theory
is
again used to form the associated distributions of the second order distributions for the
bias terms and for the specification tests.
We derive the rather unexpected result that
only one set of differences provides the optimal specification
limited the extension to 2
test
So
far,
we have
RHS jointly endogenous variables for the second
based on the Nagar-type estimator.
RHS jointly
test.
specification
We expect to extend our results to 3
endogenous variables and to the
first
33
specification test,
which
is
or
more
based on the
2SLS
estimator.
" If the LIML estimate differs markedly from the forward and reverse Nagar-type estimates, the LIML
estimate should not be used because the problem of the absence of finite sample
present.
34
moments may
well be
References
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(1977): "Asymptotic Expansions of the Distributions of Estimates in
Simultaneous Equations for Alternative Parameter Sequences," Econometrica, 45,
509-518.
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Equation
in a
(1949): "Estimation of the Parameters of a Single
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Bekker, P.A. (1994): "Alternative Approximations
to the Distributions of Instrumental
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-
681.
and Baker, R. M. (1995): "Problems with Instrumental
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Bound,
J,
Jaeger, D. A.,
Endogenous Explanatory Variable
Association, 90,
Donald,
S.G.,
is
Weak," Journal of the American
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443 - 450.
ANDNewey, W.K.
(1998): "Choosing the
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mimeo.
Hall, A.
R.,
Rudebusch, G.
Relevance
i?ev/ew, 37,
Hausman,
and Wilcox,
D.
W.
(1996): "Judging Instrument
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D.,
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1271.
HAUSK4AN,
J.A. (1983): "Specification
Hausman,
J. A.,
and Estimation of Simultaneous Equation
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Vol. 1, Amsterdam: North Holland.
Newey, W.K., and Powell,
J.
(1995): "Nonlinear Errors in Variables:
Estimation of Some Engel Curves," Journal of Econometrics, 65, 205-233.
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J.
(1973): "Testing a Subset of the Overidentifying Restrictions,"
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Marlwo,
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ANDT. Sawa
"The Exact Finite-Sample Distribution of the
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35
"The Bias and Moment Matrix of the General k-CIass Estimators of
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AND
(1990b), "The Distribution of the Instrumental Variables
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Its t-ratio
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Sawa,
T.J. (1983):
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Inference
39,1119-1246.
36
Table A: Estimates with 1452 Observations, 134 Predetermined Variables and 136
Instruments,
1.
a - .002
Price elasticity
Least
2SLS
Forward
Reverse
Nagar
Forward
Nagar
Squares
-1.36
-2.03
-2.31
-2.03
-2.31
(.147)
(.465)
(.500)
(.465)
(.502)
.303
.133
.303
.133
—
—
—
2.
Standard Error
.301
3.
R^
.962
2SLS
Rev.
Reduced Form Regressions
4.
Standard Error
.053
.308
5.
r2
.941
.960
6.
F
154.5
234.1
statistic
Partialled
Out Reduced
Fom
1
Regression
7.
Standard Error
.016
.037
8.
R^
.093
.012
9.
F
74.6
8.54
'
statistic
37
Table B: Estimates with 1452 Observations, 134 Predetermined Variables and
266 Instruments a -.092
1.
Price elasticity
Least
2SLS
2SLS
Forward
Rev.
Nagar
Forward
Nagar
Squares
-1.36
-1.24
-8.01
-1.21
-4.64
(.147)
(.194)
(.789)
(.194)
(.293)
.301
.060
.317
.080
2.
Standard Error
.301
3.
R^
.962
Rev.
Reduced Form Regressions
4.
Standard Error
.038
.295
5.
r2
.972
.967
6.
F
154.2
130.5
statistic
Partialled
Out Reduced Form Regression
7.
Standard Error
.016
.038
8.
R^
.219
.027
9.
F
'
statistic
2.79
.280
38
Appendix
A
Technical
Lemmas
Let
U=
Note that the rows of
V
2/1
are
y2
i.i.d.
M = [P-z7r2,Z7T2]^Z7r2i0,l),
,
normal with zero mean and variance Q. Also
S = U'P,U
Lemma
1
5 and
V = U-M.
let
= U'M,U-
S-^
S-^ are independent of each other.
Proof. Follows easily from normality of U.
We need to
Lemma
establish asymptotic distributions of
2 Let
ei
=
(1,0)
and
e^
=
(0, 1)
.
S-^.
We
first
show that
Then,
J_ //
[/'P,[/ei
v^
U'P,Ue2
^
S and
\_^l
U'P^ei
\ U'PMe2
)
and
1
//
^\\
U'M^Uei \
( U'M.Uei
U'M,Ue2
\
)
VMM
€2
converge in distribution to normal distributions.
Proof. The proof will only be given for (([7'P,C/ei)' {U'P^ei)')'. The proof for (([/'M^f/ei)' (t/'M^C/ez)')'
,
is
omitted. Observe that the conclusion would follow
for arbitrary a,
that
U'PzUa
A.
is
By
if
{{JJ'PzUa)
,
,
(U'PzUA)
the Cramer- Wald device, the conclusion follows
asymptotically normal under appropriate conditions.
if
)
U'PzUa
is
is.
E [U'PzUa] = M'PzMa + K9.a
and
- a'Q.aM'PzM + a'M'PzMaQ + 9.aa'M'PzM
+ M'PzMaa'^ + Ka'TlaTt + KD.aa'?l.
39
Bekker (1994) shows
Also note that Pj
idempotent with rank equal to K. Therefore,
Var [U'PzUa]
asymptotically normal
is
symmetric
If
i Var [U'PMa]
is
asymptotically normal with
converges, say to
-M'P.M =
W,
mean
-1^1
n
then Bekker (1994,
zero and variance
n'2z'P,Z7T2 (/?,!)
=
Lemma 2)
W.
-^ {U'P^Ua - E [U'P^Ua])
notes that
In our case, because
f-TT^z' zn2]-
^
\
I
(/3,l)^e-A,
and
K
>
n
a,
we have
- Var [U'PzUa]
-> a'ila
•
n
+
6 A+6
•
Aaa'fi
•
+a
a'Aa -Q
a'Cla
+ G- Qua A
-Q + a-
flaa'n
convergent, where
P
A=
The
(/?,!)
conclusion follows.
Lemma
3 Let
A
denote a symmetric
3x3
matrix such that
+ 2au)l-^
Ai,i
^AuJiiQ0^
Ai,2
=
Ai,3
= A(3Quj\2 + 2qwi2
A2,2
= uJuQ + 0^Qlj22 + 29tJi2y9 + aa;iia;22 + a'^12
A2,3
= 2w220/3 + 20^12 + 2au;22'i^i2
A3,3
= 4^22© + 2aa;|2
2wii0/3
+ 2/3^9a;i2 + 2aa;iiWi2
Then
/
^11
1
-E
5i2
\
Proof.
We
will
i,
522
\
•AA(0,A).
5l2
i,
)
522
J /
denote the asymptotic variance and covariance as Vara and CoVa.Note that
U'P.Uer
=
5ei
'11
=
I
_
|
,
U'P^Ue2
S\2
- Se2
S12
S22
40
(1)
.
Therefore, (5ii, 5i2, 522)
consists of elements of
U'P,Ue2
and the asymptotic normahty follows
asymptotic variance. Using
(1),
easily
from
Lemma
2.
Therefore,
it
remains to characterize the
we obtain
Su
Vara
Ai,2
Ai,i
= Vara [U'P.Uei]
S12
A2,2
and
S12
Vara
= Vara [U'P,Ue2] =
A2,2
A2,3
S22
So
far,
we have
Therefore,
A3,3
characterized Vara (^11)1 VaXa (^12)1 Vara (5^22)1 Cova (5'ii,5i2), and Cova (S\2^S22)
remains to characterize CoVa (S 11,822)- For this purpose,
it
CoVa {u'P.Uei, {U'P,Ue2)')
+ CoVa
it is
useful to note that
(u'P,Ue2, {U'P.Uei)'^
CoVa (511,512)
CoVa (5ii,522)
\
/ CoVa(5ii,5i2)
Cov„
CoV„ (5i2,5i2)
CoVa (5i2,522) y
^ CoVa (5n ^22)
CoVa (5i2,522)
,
(5i2, ^iz)
= y(ei+ei)-17(ei)-F(e2),
where
V (a)
is
the asymptotic variance of l/'P^Ua as discussed in
Cova (5ii,522)
It
+ Cova
(812,812)
=
\V
(e,
+ 62)
(1).
Therefore,
-V (e,) -V (62)]^^,,^
can be shown that
^ (ei + 62)
-l/(ei)-
40tJi2^^
60Puji2
1^(62)
=
+ 49/3a;ii + 4au;i2Wii
66^a;i2
+ QuJu + ©'^22/3^ + 3qwi2 + 0^22^11
+ Own + Qu220^ + Sawjj + au;22'i'ii
40a;i2
+ 40/3^22 + 4Q:a;i2u;22
from which we obtain
CoVa (5ii,522)
=
[F(ei
+ 62) - y (ei) -V{e2)],^ ^. - Gov a
- 4^0^12 +
The conclusion
2awi2.
follows.
41
(^12,
Su)
Lemma
4 Let
A""-
3x3
matrix such that
AJ-2
=
2(1
-
A^2
=
(1
denote a symmetric
~
a)wiia;i2
ci)
A^3 = 2 (1 -
UJ11UJ22
+ (1 -
a)
a;i2
q) W22'^12
= 2(l-a)a;^
^22
A^,3
T/ien,
/
^" )^
'-'11
1
-E
<?--
012
v \
Proof. Similar to the proof of
Leiruna 5 Assume
-^2^
N{0,A^)
'-'12
'^22
\
/
Lemma
f
/J
and omitted.
3,
that
K
i^„+«(„-/.).
and
that -k^z' z-Kil'n.
is
fixed at Q.
We
then have
( ( n-'-S,,
n-i5i2
•
+Q
/?
9+a
n ^^22
-
u;i2
A
a;22
•
AT
\/n
-^22
W^t
\ ""^-5^2
Proof. Using Bekker (1994,
/
)
\
Lemma
2),
E [[/'P;,[7a] =
Yj^'MJJa\
Combining
B
this observation
with
(1
-
a)
•
wii
(1
-
a)
•
U12
(1
-
a)
•
a;22
0,
A-L
))
we obtain
(;3,
1)' jr^Z'ZTTa (/9, 1)
a
+ KQ.a,
= {n-K)^a.
Lemmas
1, 3,
and
4,
we obtain the
Proof of Theorem 2
We note that
^
=
T^-("-'0
42
desired conclusion.
Therefore, without loss of generahty,
b=
and
utilize
we may assume
- T^^n-^S^
n-1522 - T^n-152^2
—:=
n-^Si2
the delta method.
It
and
-
""
^
asymptotically normal with
=n-^S,2 - j^n'^S^^
can be easily seen that
%^f(b,^y
is
=
mean
-(/?,/?)']
and variance
zero
A
r=A
A',
A-L
where
"
A=
e
1
e
n
"
_1_
"e
After some tedious algebra,
111
Tl2
"^
=
0^bJ22
—
2/90^12
2/3a;i2
H
Oi
"^
+ t^ii
e
UJ\\bJ22
Q
(2,
2)-elements are
+ W?2 " 4/?W22Wi2 + 2/3^^12
02
—UJUOJ22P
+
2a;iiwi2
l-a
+
and
+ 2/9^W22'^i2 —
Scjjj/J
'
pQ^
ti'iit<.'22/3'^
—
4u;iiWi2/?
l-a
+
2a;ii
+ 1^120^
^202
the desired conclusion from the observation that
C Two
It
(1, 1), (1,2),
l-a
+ ''^ii
Var
We
n:
e{-l+a)
a
"
e
—
QL
/30(-l+a)
can be shown that the
+ Wii
= /?^W22 - 20LO12
7^
e
/3^W22
f 22 =
We obtain
it
—a e(-l+a)
e(-l+a)
(e)
=
0^uj22
- W^i2 + wn,
Cov(£,i;i)
= wii
-/?a;i2.
Gov
=
—
(e,
V2)
W12
/Sw22.
Endogenous Regressors
will write
can be seen that
{(32-,
P2,)
yii
- P2y2i + PsVsi + £ii
y2i
=
ZiTV2
+ V2i
ySi
=
z[-K3
+ V3i
can be estimated by Donald and Newey's (1998) B2SLS applied to
yii
= /322/2i + /932/3i + £ii-
43
We
will call such estimator (61, ci).
(/^i~|^) and
Similarly,
f^,— |^jcan
be estimated by B2SLS
applied to
2/2'
=
j^yi'
+
ysi
=
-Q-yii
+
(-f
)y3.
+ e2.,
y2i
+ £31-
and
We will call such estimators
(62, C2),
and
Note that we have three estimators
]
(63,03).
for
(/32, Ps)'-
Technical Lemmcis
C.l
As
--Q-
I
before, let [/
=
covariance matrix
ilf
f2.
+
K, where
We
il/ is
examine the
a
fixed,
first
and n rows of
U
and
M.
{j
—
1,2,3)
We
would
are
like to characterize
z.
y'lPys
y'2Py2
J/2^2/3
6 By Bekker (1994,
Lemma
1),
y'sPys
J
we have
E [y'iPyj] = m[Pmj + ruiij
Therefore,
^ y[Pyi ^
'iPmi
+ run
y'iPy2
'iPm2
+ riJi2
y'iPy3
'iPms
+ rujis
\
Let
A
y'lPyi
7712
Pm2 + rW22
y'2Pyz
7712
Pma + '"'^23
y'zPyz
)
y^
denote the variance matrix of S.
44
is
rrij
an arbitrary projection
denote the
the expectation and variance of
yi-Pj/2
Lemma
P
Let yj and
^ y'lPyi ^
V
normal with zero mean and
i.i.d.
two moments of U'PU, where
matrix of rank r onto the subspace spanned by the columns of
of
V
7773 P7n3
+ rw33
J
j'th
columns
.
Lemma
7 By Bekker (1994,
Var
Therefore,
Lemma
(y'iPyj)
Lemma
1),
we have
= uium'^Pmj + ujjjm[Pmi + 2LJijm[Pmj + r {ujucojj + u)}^)
we have
All
=
A22
= wiim2Pm2 + bj'z^'m'-^Pmi + 2uJi2'm\Pm2 + r
(^11^22
+ ^12)
A33
= wiim'^Pm^ + wasTTi'iPmi + 2u)i2m\Pmz + r
(^11^33
+ w^g)
A44
= 4a;22"i2-P"^2 + 2rco'|2
A55
=
u;22"T'3-P?Ti3
(W22W33
+ a;23)
Aee
=
'iuiz^m'^Pmz
4a;iimiPmi
+ 2ru)li
+ W33m2Pm2 + 2a;23"^'2-P^3 + r
+ 2rujl^
8
= Uum'^Prnk + uJ^km^Pmi + uiijm[Pm.k + Uium^Pm^ + r {uJuOJjk + ^ij^ik)
Gov {y[Pyj,y[Pyk)
Therefore,
Proof.
we have
A12
= a;ii7niPTn2 + 0Ji2m[Pmi + uJiim[Pm.2 + oJurri'iPmi + r (wiia;i2
+W11W12)
Ai3
= wiim'iPms + LJi3m[Pm.i + uJiiTn[Pm3 + uJi3Tn[PTni + r (wiiWi3
+W11W13)
A23
= uwm^PTn:^ + u;237niPTni + LJi2'm\PTn^ + uJiT,Tn\PTn2 + r (a'iia;23 + ^12^13)
A24
= u;22"T'2-P^i + a;2im2Pm2 + U22m2Pmi + u)2\m!2Pm2 + r (^22^21
A25
= (^22^1 Pms + wi3Tn2Pm2 + W2im2Pm3 + W23?n2Pmi + r (a;22'^13 + '^2l'^23)
A35
= W33m'iPm2 + uJi2m'^Pm2 + u>3iTn'^Pm2 + u}22'm'2,Pmi + r (W33W12 + W3iu;32)
A36
= wasmaPmi + wsimgPms + wssmaPmi + W3im3Pm3 + r (w33a;3i + W33u;3i)
A45
= W22'Tl2Pm3
A56
= W33m3Pm2 + W32 rngPrns + a;33m3Pm2 + a;32m3Pm3 + r (W33W32 + W33a;32)
It follows
4- u;23"l'2-P"i2
+W22W21)
+ u;22m2Pm3 + W23"^2-f "^2 + ^ (tc'22'*^23 + W22W23)
from
2 Gov {y'iPyj,y'iPyk)
=
Vax
{y'^P [yj
+ y,-)) - Var (y^Py,) - Var (y'^Pyk)
= iJii {rrij + mk)' P (mj + rrik) + {oJjj + bJkk + 2wjfc) m'^Prrii
+ 2 (uij + Uik) m\P {mj + mk) + r
(lju {uJjj
+ Ukk + 2ujjk) +
- (uum'jPmj + Ujjm^Pmi + 2u:ijm[Pmj + r
{ojuojjj
- [uum'kPmk + uJkkmiPm.i + 2u}ikm[Pmk + r
=
2u)iim'jPmk
{i^ij
+ i^ikfj
+ Jfj))
{umiJkk
+ ^ik))
+ 2ujjkm'iPmi + 2u)ijm\Pmk + 2u}ikm'iPmj + 2r {uJucjjk + '^ij(^ik)
45
,
Lemma
9 Suppose that j
^
We
i.
Gov
Therefore,
,
.
then have
{y'.Pyu y'jPy,)
= AiO,,m.[Pm, + 2rujfj
we have
Ai4
= AuJi2m[Pm2 + 2rwf2
Ai6
— 4a;i37n'i Pma + 2ra;i3
A46
= Au)2zm'2Pmz + 2ruj23
Proof. Observe that
Gov
(y'iPyi.y'jPyj)
- Gov {y[Pyi + y'iPyjMPyj + y'jPVi) "
Gov {y[Pyuy[Pyj)
- Gov {y'jPyj, y'iPvj) - Var (xj\Pyj)
Also observe that
Gov
{y\Pyi
+ y'iPyj,y'iPyj + y'jPyj)
= Gov {{yi + j/j)' Pyu {yi + yj)' Pyj)
=
{(jJii
+ ijjj + 2iOij) m[Pmj + Uij (mj + nij)' P {mi + rrij)
+ (wii + u!ij) {rrii + TTij)' Prrij +
+ r {{ujii + uijj + 2wij) ujij +
where the
last line follows
Gov {y'iPyu y'iPyj)
Gov
{y'jPyj,
ylPyj)
from
Lemma
8.
{uJii
(uJjj
+ Wij) {rm + rrij)
+ uiij) {tjjj + ojij))
Prrii
,
Because
= uJam^Pnij + Wijm'j^Pmi + Wiim\Pmj + u!ijm[Pmi + r {tJuujij + uJaUij)
= Ujjm'jPmi + Uijm'jPrrij + ujjjm'^Pmj + Uijm'jPrrij + r {oJjjUiij + i^jj^ij)
and
Var
we
{y'iPyj)
= uum'jPmj + UjjTn[Pmi + 2tLJijm\Pmj + r {ujuojjj + ufj)
,
obtain
Cov {y[PyuyrPyj)
—
{ijjij
+ Ujj + Wij — uJij — u!ij — u)jj) m'iPmi
+ {wii + Ljjj + 2ijJij + 2ijJij + Wii + cjij + ijjj + Wij — LJii - Wii — UJjj - Ujj — 2uJij) m'^Pnij
+ {uJij + Wii + LJij - Uij — Uij — UJii) rn'jPrrij
+ r {{u>ii + uijj + 2u;ij) ujtj +
{ua
+ Uij)
{ujjj
+ uJij) — uJaUJij - uJaUij - uJjjUJij - UjjUJij — uiaUjj - ufj)
= Auiijm'iPmj + 2rw?-
46
Lemma
10 Suppose
that j
Gov
Therefore,
^
i,
and k
^
{y'iPiJi.y'jPyk)
We
i.
then have
= 2uJikm.[Pm.j + 2uJijm[PTnk + 2roJijUik.
we have
Proof. By normality, we
Ai5
=
A26
= 2w32m3Fmi + 2w3im3Pm2 + 2rw3ia;32,
A34
=
may
+
2u>iz'm\Pm2
2u!i2'm\Pmz
+ 2rtJi2Wi3,
+ 2w2im2Pm3 + 2ra;2iW23.
2a;23'"2-P"^i
write
=
yj
—
+ Cj,
yk
=
—
yt
—rrii,
E
[efc]
=
mfe
i^ij
,
yi
Wife
+ e^,
where yiLej^yiLek, and
E [gj] =
We
TTij
—mi.
therefore have
Gov {y[Py,,y'^Pyk)
'
'
=
^^
urn
+
Gov {y[Py,,y[Py,)
^
Gov
+
{y',Py,,e'jPy,)
J
^
u>ii
ujii
^ Gov
/
(y,'Pyi, e'^Pyr)
+ Gov
(j/,'Pt/i,
e^Pet)
LJi;
Observe that the
where the
last
last
term on
RHS
is
zero by independence. Also observe that
Gov {y'iPyuy'iPyi)
=
Auji^m'^Pmi
Gov {y'iPyu y'iPek)
=
2uJiim'iP
imk
^'"i
Cov {y[Pyi,y[Pej) = 2u>iimiP
Iruj
-rui
+ 2rwfi,
two equalities can be deduced by an argument similar to
Gov
[y'iPyi, y'jPyk)
=
4
'^
'
m'iPmi
)
j
-
,
Lemma
8.
We
therefore have
+ 2rbJijUJik
UJii
+ 2uJikTniP
TUj
f
\
=
2u}ikm'iPmj
'-^nii ]
UJii
+ 2a;i,m'P mk
(
J
\
—mt
UJii
]
J
+ 2u)ijm'iPmk + 2ru)ijiJik.
Let
022
023
023
033
= plim -
n
_
m'2
Pz
[7712,7713]
m'3
47
= plim —
n
772 2
Pz[zTr2,ZTT3].
TTgZ
Utilizing Bekker (1994,
Lemma
(
(
and the previous
2),
y'lPzVl
\
I
(/3|G22
results,
we can conclude that
+ 2/32/^3023 + -9|033) + auy^
y'lPzVi
{P2Q22
+ P3Q23) + auii2
y'lPzVz
{P2Q23
+ P3Q33) + QW13
^ ^
y/n
\
y'2Pzy2
022
+
y'lPzVs
©23
+ aui23
033
+ aUJ33
y'^Pzyz
)
QW22
and
(
^
/
^
y'lM^vi
1
—
a)a;ii
y'iM^y2
1
—
a)
y{M^y3
1
-
a)u>i3
y'lMzVi
1
-
a) U22
y'lMzys
I- a) LJ23
,
^
1
0J12
\fn
{ y'zM.y^
\
\
J
^I
-a) W33
are independent of each other, and asymptotically normal with zero
A-"-,
J
mean and
variances equal to
A and
where
All
A12
Ai3
=
4u;ii (/3|022
= Wii (/32022 + ^3023) + Wi2
+ 2/32/33023 + ^|033) + ^n
(/5|022
+ 2/92/33023 + PI^Ss) + « (^^11^12 + ^11^12)
= Wll
(<52023
+ P3Q33) + Wl3
+ t^lS
(/32022
+ 2/32,93023 + /33033) + Oc (wll^is + Wn^ia)
Ai5
=
= ^^11022 + W22
= U;ii023 + W23
{P2Q22
(y9|022
(/3|022
+ 2P2P3Q23 + yS|033) + ^n
(/32022
+ P3Q23)
(^2023
+ /?3033)
= 4wi2 (/32022 + P3Q23) + 2aw?2
2wi3 {P2Q22
A16
A23
(/3|022
+ ^^12
Ai4
A22
+ 2/32^3023 + PIO33) + 2aujl^
+ P3Q23) + 2a;i2 {P2Q23 + P3Q33) + 2awi2a'i3
= 4wi3 (y92023 + P3Q33) + 2awj3
+ 2P2P3Q23 + PIQ33) + 2W12
+ 2/32,03023 + P3Q33) + ^U
+ a (0^11^23 + W12W13)
48
{P2Q22
{P2Q23
+ P3Q23) + a {uuiJ22 + Ji^)
+ P3Q33) + ^13
(,02022
+ /33023)
A24
=
^22 (/32©22
+ P3Q23) + W12G22 +
A25
=
^22 (/32023
+ P2Q33) + ^^13022 + ^^12023 + ^^23 {P2Q22 + /33023) + a (a;22Wi3 + Wl2'^23)
A26
A33
= '^11033 + W33
=
(/32022
A34
=
2u;23 (/32023
t<^22
(^92022
+ /S3023) + W12622 + Oc {u!220Jl2 + ^22^12)
+ /33033) + 2a;i3023 +
+ 2^2,53023 + /3|033) + 2^13
2a;23 (/32022
2Qtc;i3W23
(/32023
+ /33033) + a
(^11^33
+ wf^)
+ /33023) + 2^12023 + 2aWi2W23
A35
=
A36
= t^33 (/92023 + /93033) + ^^13033 + ^^33 (/92023 + ^^3033) + Wi3033 + a (W33W13 + W33W13)
W33 (,02022
+ /33023) + t^l2033 + ^13023 + W23
A44
A45
4^22022
+ /33033) + Q ('^33'^12 + W13W23)
+ 2aW22
= a;22023 + 1^23022 + ^22023 + ^23022 + Oi (a;22<^23 + '^22^23)
A46
A55
A56
=
(/32023
=
4^23023
+ 2aw^3
= a;22033 + ^33022 + 2^23023 + Oi (^22^33 + wfg)
= W33023 + a;23033 + ''-'33023 + ^23033 + a {uJz3U!23 + 1^330)23)
Aee
= 4^33033 + 2aw|3
A^i=2(l-a)a;fi
A5'2
=
(1
-
Ai3
=
(1
- a) ('^11^1^13 + wiiwis)
") ('^11^12
+ Wll'^12)
A^4=2(l-a)a;?2
Aj's
=
2 (1
-
a) a;i2Wi3
49
A^6
=
A^2
(1
=
2(l-a)a;23
-
a) (wilW22
+ i^u)
^23
=
(1
-
Q)
('^11'^23
+ '^12^13)
-'^24
=
(1
~
<^) ('^22'^12
+ ^^22^^12)
-^^25
=
(1
-
Q^) ('^22'^13
+ W12W23)
-^2(5
=
2
= (!-")
^^33
'^Si
'^SS
(1
=
2 (1
= (!-")
-
a) Wi3a;23
(^ll'^33
-
+ t^^s)
a) Wi2C^23
('^33'^12
+ '^13^23)
He = (!-") (wss'^is + W33W13)
Af4
Afs
=
(1
-
=2(1-Q)a;^2
a) (0^22^23
+ 'i^22'^23)
Ai-6=2(l-a)a;|3
A^5
^^56
=
(1
- a)
(a;22a;33
+ wfa)
= (!-») (W33W23 + ^330723)
A^6=2(l-Q)a;|3
50
,
D
Application: Nagar-Type Estimator
Applying the delta method, we can find that
v/n
Oi
-
V
is
— ,ci
— ,61
—
02
02
03
mean and
asymptotically normal with zero
Cl
variance equal to
2QVar(eii)^
a
times
623Q33
"/32^(e22833-e232)^
/32^(e22e33-e23'^)'
823633
(822833 -823'^)'
822833
/32/33(822833-823'^)'
822803
Q7-<Q33
^
/32^(e22e33-e23'^)'
/32^(e22e33-e23 = )''
8,3'
02^3(822833-823^)'
/32/33(e22e33-e23^)''
8?9033
022 Q 2.
/32/33(e22e33-e232)^
>2/33(822e33-823 = )'
With some
/Sife
/32/33(e22e33-e232)''
it
822823
e'.3"
/33'(e22833-e232)'
/33 =
822823
^3^(822833 -823 = )'
'H
tedious algebra,
/32/33(822'e33-e232)^
(e22833-e23'^)'
,832(822833 -823^)'
can be shown that the above asymptotic variance matrix
is
singular:
Postmultiplying the asymptotic variance by
G 23
PiO 22
,1,0,0
y
,633'
'
we
obtain zero.
Therefore,
'
V
'
/32e23
0,1,0
,0,0,1
y
/^3©33'
'
\p3Q33'
'
we cannot stack the estimates
to derive a
more
efficient test since all tests
based on a single difference will have the same operating characteristics. This implies that the test can
be applied only to one component of
bi
V
say
bi
— y.
It is
-
— ,ci
62
62
63
'
63
'
to be noted that the asymptotic variance of such a test
2aVar(£ii)^
1-a
is
given by
1
/52^K-f^)
Observe that Var (eu) and P2 can be estimated consistently
utilizing the consistency of
LIML. Also note
that
622
©23
plim
©23
for
1
—— ^—
an —
1
2/2
plim
Pz[y2,y3]-z
1
©33
any consistent estimator a of a.
We may therefore
y2
2
y'3
P2LIML
{y'2^
[ J/2^^y2
=
©22
©23
©23
©33
estimate the asymptotic variance consistently by
{Ya=1 (yn - PlLIMLVii - P3LIMLy3i)
K -1
n-K
M^[y2,y3]
„_K2/2-M^y2
51
j
y'^P^y,-^^y'^M.y,
J
'
in
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DEC
Date Due
Lib-26-67
MIT LIBRARIES
3 9080 01917 6343