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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
ADMISSIBLE INVARIANT SIMILAR TESTS FOR
INSTRUMENTAL VARIABLES REGRESSION
Victor
Chemozhukov
Christian
Hansen
Michael Jansson
Working Paper 07-09
March 12, 2007
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=97061
Admissible Invariant Similar Tests for Instrumental Variables Regression*
Victor Chernozhukov
Department of Economics
Christian Hansen
Graduate School of Business
M.I.T.
University of Chicago
Michael Jansson
Department of Economics
UC Berkeley
March
12,
2007
Abstract.
This paper studies a model widely used in the weak instruand establishes admissibility of the weighted average power
likelihood ratio tests recently derived by Andrews, Moreira, and Stock (2004).
The class of tests covered by this admissibility result contains the Anderson
and Rubin (1949) test. Thus, there is no conventional statistical sense in which
the Anderson and Rubin (1949) test "wastes degrees of freedom". In addition,
it is shown that the test proposed by Moreira (2003) belongs to the closure of
(i.e., can be interpreted as a limiting case of) the class of tests covered by our
ments
literature
admissibility result.
1.
Introduction
Conducting valid (and preferably "optimal") inference on structural coefficients in
instrumental variables (IVs) regression models is known to be nontrivial when the
IVs are weak.^ Influential papers on this subject include Dufour (1997) and Staiger
and Stock (1997), both of which highhght the inadequacy of conventional asymptotic
approximations to the behavior of two-stage least squares and point out that valid inference can be based on the Anderson and Rubin (1949, henceforth AR) test. Several
methods intended to enjoy improved power properties relative to the AR test have
been proposed, prominent examples being the conditional likelihood ratio (CLR) and
Lagrange multiplier (LM) tests of Moreira (2003) and Kleibergen (2002), respectively.
All of the abovementioned methods and results have been or can be deduced within
an IV regression model with a single endogenous regressor, fixed (i.e., non-stochastic)
IVs, and i.i.d. homoskedastic Gaussian errors. Studying that model, Andrews, Moreira,
and Stock (2004, henceforth AMS) obtain a family
of tests, the so-called weighted
*The authors thank Jim Powell, Paul Ruud, a co-editor, and (in particular) two referees for very
valuable comments.
^Recent reviews include Stock, Wright, and Yogo (2002), Dufour (2003), Hahn and Hausman
(2003), and Andrews and Stock (2006).
Admissible Invariant Similar Tests
average power likelihood ratio
(WAP-LR)
tests,
each
member
2
of which enjoys
demon-
strable optimality properties within the class of tests satisfying a certain invariance
restriction.
The
invariance restriction in question, namely that inference
to transformations of the
minimal
corresponding to a rotation of the instruments,
LM
is
invariant
iS,T) (defined in (3) below)
satisfied by the AR, CLR, and
sufficient statistic
is
Furthermore, testing problems involving the structural coefficient are rotation invariant under the distributional assumptions employed by AMS. For these
tests.
reasons (and others), the rotation invariance restriction seems "natural", in which
AMS's numerical
case
CLR
finding that the
test is "nearly" efficient relative to the
class of rotation invariant tests provides strong evidence in favor of the
Nevertheless, because best invariant procedures can
they
exist,
it is
not entirely obvious whether
when developing
rotation invariant tests
model of AMS. In
it is
CLR
test.
to be admissible even
fail
if
"natural" to confine attention to
optimality theory for hypothesis tests in the
would appear to be an open question whether the
members of the WAP-LR family, upon which the construction of the two-sided power
envelope of AMS is based, are even admissible (in the Gaussian model with fixed IVs).
We
show that
all
particular,
members
it
of the
WAP-LR
family are indeed admissible, essentially
because the defining optimality property of these tests can be reformulated in such a
way that rotation invariance becomes a conclusion rather than an assumption. The
AR
test belongs to the
the
CLR
WAP-LR family and is therefore admissible. In contrast,
and LM tests do not seem to admit WAP-LR representations, though we
demonstrate here that these test statistics "nearly" admit WAP-LR interpretations
in the sense that they belong to the closure (appropriately defined) of the class of
WAP-LR
tests.
Section 2 introduces the model and defines
velopment of the formal results of the paper,
proven in Section
some terminology needed
all
of
which are stated
for the de-
in Section 3
and
-
4.
Preliminaries
Consider the model
y2
where
TT
G
yi, y2
M'' are
^It
is
S
IR"
Z
and
G
M"''''
Ztt
+
V2,
(1)
some A; > 2); ^ G
G M" are unobserved errors.^
are observed variables (for
unknown parameters; and
straightforward to
=
u, V2
accommodate exogenous
regrcssors in (1)
such regressors have been "partialed out". This assumption
loss of generality (for details, see Section 2 of
AMS).
is
made
.
We
tacitly
K
and
assume that any
and entails no
for simplicity
Admissible Invariant Similar Tests
Suppose P
in
is
the parameter of the interest. Specifically, suppose
we
are interested
a testing problem of the form
//o
:
/3
=
H,:p^ /?„.
vs.
/?o
Writing the model in reduced form, we have:
Hi
y2
=
=
+ Vi,
+ V2,
ZttP
Z7r
'•
(2)
.
where vi = u + V20- Following AMS (and many others), we treat Z as a fixed n x k
matrix with full column rank and we assume that {v[,V2)' ~ A^(0, fi ® /„) where Q
is a known, positive definite 2x2 matrix. Without loss generality we normalize a
variety of unimportant constants by assuming that /?q = 0, Z'Z = Ii^, and that fl is
of the form
,
{6
where 6 G
M
is
known.
1
+
'^
Under the stated assumptions, the model
is
fully parametric, the (multivariate
normal) distribution of (yi,y2) being completely specified up to the parameters
and
n.
A
minimal
-,
sufficient statistic for
(r)
•
(3
and
tt is
/3
given by
= (z'fe'\,,)-^("'>«--^-)-
»)
,
where
Because
(5,
T)
is
sufficient,
the totality of attainable power functions
•'These assumptions correspond to the model in which (Z,
where uj,j is the
the parameter S
6 = p/,/l-p'-.
{i,j)
is
element of
fi
and
0^22.1
=
"^22
—
j/i,j/2)
'^r/'^i2-
related to the correlation coefficient p
^^^
is
spanned
has been replaced by
'^e chosen parameterization,
computed from
Cl
through the formula
Admissible Invariant Similar Tests
by the
set of
4
power functions associated with (possibly randomized)
tests
based on
Any such test can be represented by means of a [0, l]-valued function </>()
such that Ho is rejected with probabiHty 4> {s, t) if {S, T) = (s, t) The power function
of this test is the function (with arguments P and vr) Ep^^^cf) (5, T) where the subscript
{S,T)
.
.
,
on
E
indicates the distribution with respect to which the expectation
For any a €
(0, 1)
,
a test with test function
is
of level
a
is
taken.
if
sup^Eo,^0(5,T) <a.
A
level
a
test
with test function
(4)
said to be a-admissible
is
if
E0^^<l>{S,T)<Ep,,^{S,T)
V(/3,7r)
(5)
=
V(/3,7r)
(6)
implies
Ef,,^<P{S,T)
Ep,^ip{S,T)
whenever the test associated with (p is of level a^
The main purpose of the present paper is to investigate the a-admissibihty properties of certain recently
developed rotation invariant, a-similar
a rotation invariant test
is
for every
orthogonal k x k matrix
.
The
tests of
one whose
-
O
and an
E^,,(i^{S,T)
4>
satisfies
Of-similar test
=a
Vtt.
is
.
</>
one
::
By definition,
OT) = (p {S, T)
tests.
[OS,
which
for
.:..
:•,
(7)
rotation invariant, a-similar tests under consideration here are the
AMS. By
construction, a size
the form
a
WAP-LR
test
maximizes a
WAP
criterion of
£;^X5,r)dH/(/3,7r)
(8),
rotation invariant, a-similar tests, where the weight function
mulative distribution function (cdf) on M!'^^
it is
not obvious whether a
W
is
some
cu-
.
In spite of the fact that the defining property of a
property,
WAP-LR
.
/
among
test function
WAP-LR
is
WAP-LR
test
is
an optimality
a-admissible, the reason being that
^In the model under study here, the present notion of Q-admissibihty agrees with that of Lchmann
and Romano (2005, Section 6.7) (in which (5) and (6) are required to hold for all /? 7^ and all tt)
because (i) all power functions arc continuous and (ii) the set {/3 /3 7^ 0} is a dense subset of R.
:
Admissible Invariant Similar Tests
optimum
admissibility of
invariant tests cannot be taken for granted
and Romano (2005, Section
imizes a
WAP
among
criterion
that optimahty property
is
We
6.7)].
show
in Section 3 that
the class of a-similar tests.
sufficiently strong to
any
[e.g.,
Lehmann
WAP-LR test
max-
In the present context,
imply that any
WAP-LR
test
is
a-
admissible.
Remark. The
Romano
fact that the alternative hypothesis
that any a-admissible test
esis implies
The
(2005).
is
6.7.2
whether these
Lehmann and
Lehmann and Romano
[e.g.,
but because these tests are not necessarily similar
(i))],
tests are also a-admissible (for
some
it is
(2005,
unclear
a).
Results
3.
Admissible a-similar
3.1.
dense in the maintained hypoth-
(rotation invariant) posterior odds ratio tests of Chamberlain
(2006) are d-admissible almost by construction
Theorem
is
d-admissible in the sense of
Two
tests.
basic facts about exponential families
greatly simplify the construction of o-admissible, a-similar tests. First, because the
power function
of the test with test function
E0,^cP{S,T)=
where fs
T,
{-{P, tt)
and fr
a continuous function of
is
dominated by a
level
with test function
with
(f)
[e.g.,
denote the densities (indexed by
(/3, tt)
test
is
a-admissible
is
not a-similar.
if
tt is
unrestricted,
T
is
and n) of S and
implication, an a-similar test
whenever the
test associated
a complete, sufficient statistic for
Moreira (2001)]. As a consequence, a
and only
if it is
test
with test function
(f)
is
it
under
a-similar
conditionally a-similar in the sense that, almost surely,
Vvr.
(10)
and the Neyman-Pearson lemma that
(8) is maximized among a-similar tests
follows from the preceding considerations
It
W
By
(5) implies (6)
Eo,.[</)(5,T)|T]=a
if
/?
2.7.1) that Ep^T,(j){S,T)
Therefore, an o-similar similar test cannot be
.
a
which
Theorem
(2005,
(9)
a-similar.
if is
Second, because
Hq
if
(-1/3, vr)
Lehmann and Romano
follows from
it
can be represented as
(j){s,t)fs{s\P,n)fT{t\P,n)dsdt,
[
[
(p
is
by the
a cdf on R''"+\ then the
test
lW
where
1 []
WAP criterion
with test function given by
is
(s, t;
a)
=
the indicator function,
1
[LR^
(5, t)
> k^^
{t;
a)]
-
(11)
,
'
Admissible Invariant Similar Tests
^'''^"
^^
and
KYii{t;(y)
is
the
^'^^
fsis\0,0)frmO)
- a
I
quantile of the distribution of
LR^ {Z^J.)
,
where Z^
distributed J\f{0,lk)-^ Because the maximizer cp^R is essentially unique (in the
measure theoretic sense), the test with test function (/"^(sQ;) is a-admissible.'' To
is
demonstrate a-admissibihty of a test, it therefore suffices to show its test function
can be represented as
(•;«) for some W. Section 3.2 uses that approach to show
^^
that the
AMS
tests of
are
all
a-admissible.
Admissibility of WAP-LR tests.
X R+.' Accordingly, let u; be a cdf on
3.2.
on
WAP-LR
R
C-{s,t)=
where qFi
is
exp(-^)
I
The
M
7/^ is
size
a
indexed by cdfs
dw{p,X),
(13)
the regularized confluent hypergeometric function,
defined as in Section
WAP-LR
test associated
By
2.
with
=
'/':^Ms(5,t;«)
where k'^ms
tests are
define
„A
c(M)=(;;:
and
WAP-LR
R+ and
x
(^5 '^) i^
the
I
—a
Corollary
w
(.4)
;;;).
1
of
AMS,
the test function of the
is
l[£-(s,t)>«:^,,s(t;a)],
quantile of the distribution of £"' {Zk,
(15)
t)
.^'^
upon request.
^As pointed out by a referee, this fact is a special case of the more general decision theoretic
result that (essentially) unique Bayes rules are admissible [e.g., Ferguson (1967, Theorem 2.3.1)].
^Because the power of an invariant test depends on n only through the scalar tt'tt, the correonly through the cdf on R x 1R+ given by
sponding WAP criterion (8) depends on
^Details are provided in an Appendix, available from the authors
W
W(/?,A)
*£'" {s,t)
is
proportional to
tional to 2~*''"^"''''*/(fc-2)/2 (\A)
of order i/.
^The
'^AMS
(*;
size
a
WAP-LR
^) depends on
t
test
ip^,
.
/
iqi,qT) in
where
is
=
Jj, (•)
dW{b,TT).
Lemma 1 of AMS because qFi (; fc/2; z/4) is propordenotes the modified Bessel function of the first kind
a-similar by construction and
only through
t't.
is
rotation invariant because
.
Admissible Invariant Similar Tests
For our purposes,
of
(pjj^fs
has cdf
of
B
w
and Uk
convenient to characterize the defining optimaUty property
it is
denote the cdf (on R^+i) of (b, y/KU'S
W^^s
as follows. Let
7
,
where [B, A)'
uniformly distributed on the unit sphere in R*" (independently
and A). In terms of W^j^^g, Theorem 3 of AMS asserts that the test with test
is
function (f>^MS (s ^) maximizes
E0,,cPiS,T)dW:^^siP^^)
among
rotation invariant, a-similar tests.
of rotation invariance
is
rAMs{s,t;a)
The
=
it
turns out that
<p^t"is,t;a).
(16)
displayed equality fohows from a calculation performed in the proof of the
Theorem
f
1.
Let
w
E0,^<f^ (5,
be a cdf on
T)
wlieve the inequahty
(and hence for
dW^^s
is strict
all) (/?, tt)
.
M
iP^
x R+.
If
satisfies (7)
(/>
^)< [
unless
fol-
AMS.
lowing strengthening of Theorem 3 of
is
In this optimality result, the assumption
unnecessary because
Ep^^rAMS
PiCff^.^
[0 [S,
In particular, the size
T)
=
,
then
(5, T-
a)
dW^^s
4>^ms {S, T; a)]
a WAP-LR
=
(/?,
^)
1
for
test associated
-.
some
with
w
a-admissible.
The
testing function of the size
a
AR test
4>^j,{s,t;a)
where xi (^) is the
remarked by AMS,
tion
w
1
—a
(for
Corollary
2.
is
given by
(17)
quantile of the x^ distribution -with k degrees of freedom. As
t) is an increasing function of s's whenever the weight func-
£"' (s,
mass to the set {(^5, A) G R x R+
an immediate consequence of Theorem
The
is
= l[s's>xl{k)],
assigns unit
the following
known Q)
:
size
a
AR
/?
=
6^^j
.
As a consequence,
1
test is a-admissible.
In spite of the fact that the
AR
test
is
a
A:
degrees of freedom test appUed to
a testing problem with a single restriction, a fact which suggests that
properties should be poor [e.g., Kleibergen (2002, p. 1781), Moreira (2003,
its
p.
power
1031)],
.
.
.
Admissible Invariant Similar Tests
Corollary 2 implies that there
test "wastes degrees of
In addition to the
ant similar tests of
is
no conventional
statistical sense in
which the
AR
freedom"
AR
AMS,
Theorem
test,
also covers the two-point optimal invari-
1
the power functions of which trace out a two-sided power
envelope for rotation invariant, a-similar
LM
8
On
tests.
closure (appropriately defined) of the class
CLR
the other hand, the
do not seem to belong to the class of WAP-LR
appear to be an open question whether one or both of these
tests
WAP-LR
tests.
tests.
tests
Indeed,
it
and
would
even belong to the
Section 3.3 provides an
affirmative answer to that question.
Remark. As
pointed out by a referee, two distinct generalizations of the results
of this section
seem
to a distribution
ti;
First, the conclusion that a
feasible.
on
R
x
R_,_
can be
distribution on R'^+^ (by introducing a Uk which
sphere in
R'^
and independent
of
(/3,
is
Bayes rule corresponding
a Bayes rule corresponding to a
"lifted" to
uniformly distributed on the unit
A)) appHes to
more general decision problems
than the one considered. Second, using Muirhead (1982, Theorems 2.1.14 and 7.4.1)
it should be possible to generalize the results to a model with multiple endogenous
To conserve
regressors.
3.3.
The
CLR
test.
we do not pursue
space,
The
these extensions here.
test function of the size
a
CLR
test is
= l[LR{s,t)>KcLR{t;a)],
(l>CLR{s,t]a)
(18)
where
LR {s, t) =
and KcLR
[t] ct)
is
the
^ (s's
1
—a
-
ft
+
^{s's
+ t'tf -
4 [{s's) [ft)
quantile of the distribution of
CLR
-
{s'tf] ]
(19)
LR {Z^, t)
been found to perform remarkably
well in terms of power. For instance, AMS find that the power of the size a CLR
test is "essentially the same" as the two-sided power envelope for rotation invariant,
Q-similar tests. In hght of this numerical finding, it would appear to be of interest to
In numerical investigations, the
analytically characterize the relation
WAP-LR
(if
test has
any) between the
CLR
test
and the
class of
tests.
CLR
Theorem 3 shows that
(s, t) can be represented as the hmit as
suitably normalized version of C"clr,n (^^ {^ where {wclr,n TV G N}
^
chosen collection of cdfs on
To motivate
R
x
:
A'' -^-
oo of a
is
a carefully
it is
convenient
R+
this representation
to express the test function of the
and the functional form
CLR
test as
of wclr,n,
.
,
Admissible Invariant Similar Tests
(J)CLR (s.
a)
^;
=
1
[CLR*
> KcLR-
(s, t)
{t;
a)
where
=
CLR*{s,t)
\s's
+ t't+J{s's +
^2[LR{s,t)
and KcLR*
The
{t;
a)
=
(defined in (14))
^/2[KcLR{t;a)
CLR*
statistic
(s, i)
is
t'tY-A[{s's){t't)-{s't)
+ t't]
(20)
+t't].
the square root of the largest eigenvahie of
Q
(s, t)
and therefore admits the following characterization:
CLR*
{s, t)
=
A
/max^eR2.^-^=i -q'Q
(s, t)
(21)
r).
Moreover, the integrand in (13) can be written as
^
exp
where
that
is
Tjp
its
=
V/s/ x/v'pVp i^
support
is
0^1
^ vector of unit length proportional to
the set of
all
pairs
(/3,
A) for
which
Ary^r/^
maximized (over the support of wclr,n) by setting
associated with the largest eigenvalue of
the
tail
behavior of oi^i
A^ behavior of C^clr.n
[;
k/2; •/4]
(^^ f)
is
Q (s, t)
.
7/^
=
wclr.n
is
such
then the integrand
equal to the eigenvector
This observation, and the fact that
similar to that of exp (i/^)
"should" depend on
7/^. If
A'',
Q (s, t)
,
suggests that the large
only through
CLR*
{s, t)
wclr,n denote the cdf of [B, N/^/rf^)' where ^ ~ A^ (0, 1) .^^
By construction, wclr,n is such that its support is the set of all pairs (/?, A) for which
= ^- Using that property, the relation (21) and basic facts about o^^i (; k/'2; )
^v'ffVp
For any A^
>
0, let
,
,
we obtain
the following result.
B ~ TV (0, 1) is made for concreteness. An inspection of tiie proof
shows that (22) is valid for any distribution (of B) whose support is R. Furthermore,
as pointed out by a referee it is possible to obtain analogous results without making the distribution
^"^The distributional assumption
of
Theorem
of
^VoVp degenerate.
3
—
.
Admissible Invariant Similar Tests
Theorem
3.
For any
CLR*
(s',t')'
(s, t)
=
10
G
lim7v-.oo
^p=
log
Z:""^^«''^ (s, t)
+
(22)
In particular,
CLR{s,t)
Define -C^^^^yv («>*)
=
lim^
=
[log £"'^^«"^ (s,i)
logC"^'^"-^ {s,t)
2N
+
A^/2] /\/iV.
+
N
(23)
t't.
The proof
of
Theorem
3
shows that as A'' — oo, C*qiii^n (') converges to CLR* (•) in the topology of uniform
convergence on compacta. Using this result, it fohows that
>•
Hmyv^oo E0,, I^Z-S'"
-
a)
(5, T-
In particular, the power function of the
<Pclr (S, T; a)
=
|
V
(/?,
n)
WAP-LR test associated with wclr,n conCLR test (as N -^ oo).
seems plausible that the CLR test enjoys an
verges (pointwise) to the power function of the
In hght of the previous paragraph,
it
property reminiscent of Andrews (1996, Theorem 1(c)). Verifying this conjecture is not entirely trivial, however, because the unboundedness (in
"admissibility at
cx)"
of K.CLR* [t] ci) makes it difficult (if not impossible) to adapt the proofs of Andrews
and Ploberger (1995) and Andrews (1996) to the present situation.
Theorem 3 also suggests a method of constructing tests which share the nice numerical properties of the CLR test and furthermore enjoy demonstrable optimality
properties, namely tests based on C^clr.n (^s,t) for some "large" (possibly sampledependent) value of N. Because the power improvements (if any) attainable in this
t)
way
are likely to be slight, the properties of tests constructed in this
way
are not
investigated in this paper.
Remark. The
test function of the size
[s,t;a)
=
a
LM
test
is
'{s'tf
>x;(i)
1
t't
where Xo (1) is the 1 — a quantile of the x^ distribution with 1 degree of freedom.
AMS show that a one-sided version of the LM test can be interpreted as a limit of
WAP-LR tests (and is locally most powerful invariant). On the other hand, we are
not aware of any such results which cover the (two-sided) LM test.
,
Admissible Invariant Similar Tests
A
slight variation
obtain a
WAP-LR
cdf of [Bn-,
on the argument used
N/ ^^v'bn^Bn]' where
^
in the previous subsection
LM
interpretation of the
11
test.
Indeed, letting
the distribution of
N^/^Bn
can be used to
Wlm,n denote the
uniform on [-1,
is
1]
can be shown that"
it
\s't\
l\fFt
=
hmA,_.oo
log
-TTJ
£^"-A^
(s, t)-\-
iVl/6
— - VnVFi
(24)
implying in particular that
n2
(s'ty
_
N
1
r— r—
(25)
4.
Proofs
Proof of Theorem 1. It suffices to establish (16) To do
LI^AMs (5, t) is proportional to D" (s, i) Now,
.
so, it suffices to
show that
.
exp (-\^
/s(5|0,0)/T(i|0,0)
exp
[lis
- /?^f -
||s||-
+
\\t
^J!Mll\ exp [y\,\(is +
-
(1
-
-
(1
6/3) tj'yr)
where ||-|| signifies the Eucfidean norm, A^r = tt'tt, and tt = A^^^^tt.
Because tt has unit length, it follows from Muirhead (1982, Theorem
Lir'^Ms
{^s,t)
-
bH) vrf
jjtf ]")
,
7.4.1) that
fs{s\P,7r)fT{t\P,7r)
=
dW^,jsif3,7r)
/s(5|0,0)/r(t|0,0)
exp
as
Fi
was to be shown. (We are grateful to a
(1982,
Theorem
referee for suggesting the use of
3.
The
result
is
obvious
if
Q
(s, t)
=
Details arc provided in an Appendix, available from the authors
the proof given there shows that the distributional assumption on
insofar as the preceding representations arc valid
support
is
[—1,1].
dw{P,X),
Muirhead
7.4.1).)
Proof of Theorem
^^
-,-V'f3Q{S:t)V0
0,
so suppose
upon
request.
N^/^Bn
whenever N^/'^Bn has a
is
made
Q
An
(5, t) 7^ 0.
inspection of
for concrcteness
(fixed) distribution
whose
,
Admissible Invariant Similar Tests
The proof
2)]
that
<
C=
will
make use
C<C<
oo,
of the fact
Andrews and Ploberger
(1995,
Lemma
where
A[;k/2;z/i]
C=
inf.
exp (y^) max(2,
By
[e.g.,
12
^"
1)
,Fr[;k/2-z/4]
sup^
exp(v^)
''
construction,
C"
exp
{s,t)
=
exp
;^—
I
)
0^1
7;',-7V'i3Qis,t)ri0
T
2
dwr
(/?,A)
4
N
I
-—
d$(/?)
where
(•) is the standard normal
and monotonicity of oi^i /c/2; ]
<J>
[;
cdf.
Using
this representation, the relation (21)
,
TV
£"'c^^«'~(s,t)exp
d$(/3)
^F^
<
oFx
'24
^
'
< Cexp\VNCLR*is,t)\
from which
it
,
follows immediately that
1
Iq^jTwclr.n (5^ij
limr
+
N
< CLR*[s,t)
(26)
^A^
On
the other hand, for any
£"'"«.'^(s,t)exp(
y
)
=
<
e
< CLR*
3^1
{s,t)
-; jr?aQ(s,f)r?^
d«i>(/?)
d$(/3)
)i^i
(^,t)
> exp[\/A(CL/?*(s,f)-e:
xCmaxlN CLR*
-(fc-l)/4
{s,ty ,1]
rf$(/3)
'.(s,i)
.
Admissible Invariant Similar Tests
13
where
B,
the
first
(•)
{/3
:
has
full
o-^i
[;
k/2\ ]
.
>
^Jf]],Q{s,t)fi0
inequahty uses positivity of qFi
monotonicity of
$
=
{s,
The
[;
k/2\
•]
integral j^
.
,
CLW
and the
^,
d^
(/?)
(.s,
t)
- e],
last inequality uses (21)
is
strictly positive
and
because
support, so
log/:"'"«''^(s,i)
lim/v^c
+
A^
> CLR*{s,t)-£.
nv
Letting e tend to zero in the displayed inequality,
we obtain an
inequality which can
be combined with (26) to yield (22)
Indeed, because
sup(y _(,)'g/^ CLi?* (s,t)
for
any compact
set A'
C
hm N-^oo^^^V(s>,t')'eK
<
oo
and
inf^^/
^/j'^/,^
/
(i$
(/?)
>
M?^, the result (22) can be strengthened as follows:
]C"-'^^'''^ {s,t)
7n
+
N'
CLR*{s,t)
0.
Admissible Invariant Similar Tests
14
References
Anderson, T. W., and H. Rubin
Equation
Complete Set
in a
(1949): "Estimation of the Parameters of a Single
Annals of Mathematical
of Stochastic Equations,"
Statistics, 20, 46-63.
Andrews, D. W. K.
Parameter Space
Andrews,
D.
is
W.
(1996): "Admissibihty of the Likelihood Ratio Test
When
the
Restricted under the Alternative," Econometrica, 64, 705-718.
K.,
M.
Moreira, and
J.
J.
H.
Stock
ant Similar Tests for Instrumental Variables Regression,"
(2004): "Optimal Invari-
NBER Technical Working
Paper No. 299.
Andrews, D. W.
Ratio Test
Annals of
a Nuisance Parameter
K.,
(1995): "Admissibihty of the
is
Likehhood
Present Only under the Altenative,"
1609-1629.
Statistics, 23,
Andrews, D. W.
W. Ploberger
K., and
When
and
H.
J.
Stock
(2006): "Inference with
Weak Instruments,"
Economics and Econometrics, Theory and Applications: Ninth World
Congress of the Econometric Society, forthcoming.
Advances
in
Chamberlain, G.
"Decision Theory Applied to an Instrumental Variables
(2006):
Model," Econometrica, forthcoming.
DuFOUR, J.-M.
"Some Impossibihty Theorems in Econometrics with AppliDynamic Models," Econometrica, 65, 1365-1387.
(1997):
cations to Structural and
"Identification,
(2003):
Weak
Instruments, and Statistical Infei'ence in
Econometrics," Canadian Journal of Economics, 36, 767-808.
Ferguson, T.
New
Hahn,
S. (1967):
Mathematical
Statistics:
A
Decision Theoretic Approach.
York: Academic Press.
J.,
and
J.
Hausman
(2003):
"Weak Instruments: Diagnosis and Cures
in
Empirical Econometrics," American Economic Review, 93, 118-125.
Kleibergen,
F. (2002):
"Pivotal Statistics for Testing Structural Parameters in
Instrumental Variables Regression," Econometrica, 70, 1781-1803.
Lehmann,
Edition.
E. L., and
New
Moreira, M.
trarily
J.
P.
Romano
(2005):
Testing Statistical Hypotheses. Third
York: Springer.
J. (2001):
Weak,"
CLE
"Tests with Correct Size Wlien Instruments
Working Paper No.
37,
UC
Berkeley.
Can Be
Arbi-
Admissible Invariant Similar Tests
(2003):
metnca,
"A Conditional Likelihood Ratio Test
for Structural
15
Models," Econo-
71, 1027-1048.
MuiRHEAD, R.
J.
(1982):
Aspects of Multivariate Statistical Theory.
New
York:
Wiley.
Staiger, D., and
Weak
J.
H.
Stock
(1997):
"Instrumental Variables Estimation with
Instruments," Econometrica, 65, 557-586.
J. H., J. H. Wright, and M. Yogo (2002): "A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments," Journal of
Business and Economic Statistics, 20, 518-529.
Stock,
.
Admissible Invariant Similar Tests
16
Appendix: Omitted Proofs
5.
This Appendix provides a proof of (24) and a derivation of (11)
5.1.
Proof of
To motivate
(24).
support consists of
pairs
all
notice that
tlie result,
A) for which
(/3,
large A^ behavior of the integrand in (13)
is
\0\
dFi
(12)
wlm,n
if
< N'^^^ and
such that
is
Xripij^
=
its
N, then the
reminiscent of the behavior of
kN
TV
exp
—
2'T
{t't
+
2ps't)
Because
maX|^l<;v->/3 \/t't
+
= ^Vt +
2(3s't
2N-y^\s't\
Theorem 3 therefore suggests that |s'^| /Vt't can
oo of a suitably normalized version of 0"^m,n (^^ i^
reasoning similar to that leading to
be represented as the limit as A^
As
in Section 3.3, let
distribution of
N^^^Bn
is
—
>
Wlm,n denote the cdf
A:
exp
C"^'''" {sj)
—
exp
[B^
of
N/ ^Jv'bn ^Ibn
,
uniform on [—1,1]. Then
)
N\
,
finite
\Rn
iP;
A)
d/3.
2' 4
VpVis
as
N
--{t't +
constant
R
2l3s't
+ RN[^\s,t))
(depending only on
s
and
t),
i'0Q{s,t)v0
SUP|/3|<w-i/3
(/?,
7V1/3
)Fi
k
where, for some
where the
d'WLM,N
'
0-^1
'
1
oi^i
|-_Af-i/3_Ar-i/3]
The integrand can be written
]
.
S,t)\= SUP|^|<^-i/3
{t't
Nv'pVp
+
2(3s't)
< N-^I^R.
Admissible Invariant Similar Tests
As a consequence, by monotonicity of
Theorem 3),
qFi
[;
/c/2;
•]
17
and the defining property of
C
(appearing in the proof of
£Wi^M,N
k
<
-
4f
/
^
J\-\/Vn,i/Vn]
SUP|^|<yv-i/3
=
<
from which
it
A^
(^^ ^) gj.p
„A
N
2'T
[t't
+
2(5s't
7V-2/^i?)
dp
o-f^i
24V
0^1
+
/
Cexp ^N\t't + 2N'^l^\s't\+N-^I^R
,
follows that
7V-V6
<
A^^i/*^
logC
+
N^/\/t't
+
27V-1/3
+
|5'f
7V-2/3i?
- N^/^VVt
I
R
2yM
where the
last inequality uses
To obtain an
where t = and
concavity of y^.
inequality in the opposite direction,
t 7^ 0.
If t
=
0,
then
we
distinguish between the cases
Admissible Invariant Similar Tests
A^i/6
k
)Fi
> N-^/^
which completes the proof of
oFi
[;
the other hand,
k/2]
•]
¥
oFi
O {N-"^)
(24) in the case
if i 7^ 0,
then
Nri'f^Q{s,t)r]p
it
dp
2' 4
[_/V-l/3,iV-V3]
On
I— I—
A^
7V-V6
18
VpVf3
,
where t — Q.
from monotonicity and positivity of
follows
that
N
£t.^LM,N(5^^)exp
7V1/3
TV
fc
iF,
[t't
2'T
l-iV-i/3,Ar-V3]
+
2(5s't
- N-^I^R)
d/3
Ari/3
.L!l^t't
5F1
+ 2p\s't\-N-~/m) dp
[o,yv-i/3]
for
<
every
£
<
1
A
iV
=
inf {
and any
N
> K,
>
jru>LM,M
A
t't
:
- 2N~^'^
>
.
Therefore, for any
N
0-^1
k_N
2'T
o-f^l
^j{t't + 2P\s't\-N-'/m)
[(l~£)W-V3,iV-l/3]
9 "^f/3e[(l-e)A'-i/3,/v-i/3]
3F1
0}
(5^ ^) gj-p
iVi/3
^
- N'^^^R >
\s't\
;
-;
-
(i'i
+
'/_/vi/37^
-C\Nt't-N'i^R
2 (1
-
e)
A-i/^
-('-''/'
exp
+
't't
\s't\
N\
-
t't
2(3\s't\-N-''/'^R)
dp
iV^/^^R)
+
2 (1
-
e)
A-i/3
is'tl
-
A^-2/3i?
Admissible Invariant Similar Tests
where the
first
inequahty uses positivity of
o^^i
[;
k/'2\
the defining property of C_ (appearing in the proof of
19
and the
•]
Theorem
last inequality uses
3)
and monotonicity
„f[.]-(fc-l)/4_
By
implication,
N
Ar-i/6
>
iV"i/6
log {Cel2)
+N^'\lt't
>
The proof
- ^^-^ log
-
+
2 (1
(1
-£)\s't\
+ max
'ft
[2 (1
r- I—
-
e)
e)
yV^V3
-
(TVt'i
\s't\
-
-
N^I^'R)
N-ym - N^^^VFt
N~^/'^R/2
A^-V3
\s't\
+ 0{N-^/^\ogN)
- N-^m, O]
{l-e)\s't\/^t + 0(N-^^HogN).
of (24) can
now be completed by
letting e tend to zero in the preceding
display.
5.2.
Derivation of
(11)
—
(
The
12).
test
based on
(p
{S,
T)
is
a-similar
if
and only
if
/
Because
T
is
I
(p{s,t)fs{s\0,n)fT{t\0,n)dsdt
complete under Hq
/
:
(3
=
(f){s,t)
0,
=a
Vtt.
the preceding condition holds
fs {s\0,7r)ds
if
and only
if
=a
./«'
for
almost every
t
based on 0(5, T)
G
is
M.'^.
Furthermore, fs (s|0,7r) does not depend on
if and only if
a-similar
/ ((>{s,t)fsis\0,0)ds
for
almost every
t
€
=a
tt,
so the test
Admissible Invariant Similar Tests
Using (the Fubini theorem and) the representation
(9)
,
20
we can
fs{s\P,1T)fTit\(3,7T)dW{f3,Tv)
{s,t)
write (8) as
dsdt.
To maximize this expression subject to the condition that the test based on (5, T) is
a-similar, we can proceed on a "t by f basis and consider the problem of maximizing
/5(5|/?,^)/T(t|/?,7r)diy(/?,7r)
ds
subject to the condition
By
the Neyman-Pearson lemma, this maximization problem
[s,t-a)
= l\LR
is
solved by
{s,t)>KYi^{t;a:
where
~w
and k^u
/r
{t;
a)
(t|0, 0) is
is
the
positive
1
-a
,,,
J^,,,fs{s\P,n)fT{t\P,n)dW(A7r)
—w
quantile of the distribution of
we obtain the
equality
^LRis,ta) =
(l)^R{s,t]a)
LR
(2^., t)
.
Finally, because
Date Due