Admissible invariant similar tests for instrumental variables regression

MIT LIBRARIES

3

9080 02618 0908

Digitized

by

the

Internet

Archive

in

2011

with

funding

from

Boston

Library

Consortium

Member

Libraries

HB31

.M415

DEWEV

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

Thispaper can be

downloaded

withoutcharge fromthe

SocialScience

Research

Network

Paper

Collection at

http://ssrn.com/abstract=97061 6

Admissible

Invariant Similar Tests for

Instrumental

Variables Regression*

Victor

Chernozhukov

Christian

Hansen

Department

of

Economics

Graduate School

of Business

M.I.T.

University of

Chicago

Michael

Jansson

Department

of

Economics

UC

Berkeley

March

12,2007

Abstract. Thispaperstudiesamodelwidelyusedintheweak

instru-ments literature and establishes admissibility of the weighted average power

likelihoodratio testsrecentlyderivedbyAndrews,Moreira,andStock(2004).

The classoftests covered bythis admissibilityresult contains the Anderson

and Rubin(1949) test. Thus,thereisnoconventionalstatisticalsenseinwhich

theAndersonand Rubin(1949) test "wastes degrees of freedom". In addition,

itisshownthatthetestproposed by Moreira (2003) belongs to the closure of

(i.e.,canbeinterpreted asa limiting caseof) theclassoftestscoveredbyour admissibilityresult.

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 areweak.^ Influentialpapersonthissubject includeDufour (1997)and Staiger

andStock(1997),bothofwhichhighhght theinadequacyofconventionalasymptotic

approximationstothebehavioroftwo-stageleastsquaresandpointout thatvalid

in-ferencecanbe basedontheAnderson and

Rubin

(1949, henceforth

AR)

test. Several

methods intended to enjoyimproved powerproperties relative tothe

AR

test have

beenproposed,prominent examplesbeing the conditionallikelihood ratio

(CLR)

and

Lagrangemultiplier

(LM)

testsofMoreira(2003)andKleibergen(2002), respectively. Alloftheabovementionedmethodsandresultshavebeenorcanbededucedwithin

an

IV

regressionmodelwith asingleendogenousregressor, fixed(i.e.,non-stochastic)

IVs, andi.i.d. homoskedastic Gaussianerrors. Studyingthat model, Andrews,

Mor-eira,andStock(2004,henceforth

AMS)

obtain a familyoftests,theso-calledweighted

*Theauthors thankJimPowell,PaulRuud,aco-editor,and(inparticular)tworefereesforvery

valuablecomments.

^Recent reviewsinclude Stock, Wright, andYogo (2002), Dufour (2003), Hahnand Hausman

Admissible Invariant

Similar

Tests

2

averagepowerlikelihood ratio

(WAP-LR)

tests,each

member

ofwhichenjoys

demon-strableoptimality propertieswithin the classof tests satisfyingacertaininvariance

restriction.

The

invariancerestriction inquestion, namelythat inferenceisinvariant

to transformations ofthe minimal sufficient statistic iS,T) (defined in (3) below)

corresponding to a rotation ofthe instruments, is satisfied by the

AR,

CLR,

and

LM

tests. Furthermore, testingproblems involving thestructural coefficient are

ro-tation invariant underthe distributional assumptions employed by

AMS.

For these reasons (and others), the rotation invariance restriction seems "natural", in which

case

AMS's

numerical finding thatthe

CLR

test is "nearly" efficientrelativetothe

class ofrotationinvariant testsprovides strong evidenceinfavor ofthe

CLR

test.

Nevertheless, because best invariant procedures can failtobe admissible even if

they exist, it isnot entirelyobviouswhetherit is "natural" to confine attentionto

rotationinvariant tests

when

developing optimality theoryforhypothesistests inthe

modelof

AMS.

In particular, itwould appear to be an openquestion whether the

members

ofthe

WAP-LR

family,

upon

whichtheconstructionofthetwo-sidedpower

envelopeof

AMS

isbased,areeven admissible(intheGaussianmodelwithfixedIVs).

We

show

thatall

members

ofthe

WAP-LR

familyareindeedadmissible, essentially

because thedefiningoptimality propertyofthesetestscan bereformulatedinsuch a

way

that rotation invariance becomesa conclusion ratherthan an assumption.

The

AR

test belongs to the

WAP-LR

family and is therefore admissible. In contrast,

the

CLR

and

LM

testsdonot seem to admit

WAP-LR

representations, though

we

demonstratehere that thesetest statistics "nearly" admit

WAP-LR

interpretations

in the sense that they belongto theclosure (appropriately defined) oftheclass of

WAP-LR

tests.

Section 2 introduces the model anddefines

some

terminology neededforthe

de-velopmentofthe formalresultsofthe paper,all ofwhicharestatedinSection3 and

proveninSection4.

-Consider the model

Preliminaries

y2

=

Ztt

+

V2, (1)

whereyi, y2 S IR"and

Z

GM"'''' areobservedvariables (for

some

A;

>

2);

^

G

K

and

TT GM'' are

unknown

parameters; andu, V2G

M"

areunobservederrors.^

^Itisstraightforward toaccommodateexogenousregrcssorsin (1).

We

tacitlyassumethatany

suchregressorshavebeen "partialed out". This assumptionismadeforsimplicityandentailsno

Admissible

Invariant

Similar

Tests

Suppose

_P

is

the

parameter

of

the

interest. Specifically,

suppose

we

are interested

in

a

testing

problem

of

the

form

//o : /3

=

_/?o vs.

H,:p^

/?„.

Writing

the

model

in

reduced

form,

we

have:

Hi

=

ZttP

+

Vi,

y2

=

Z7r

+

V2, '•

. (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

1

+

where

6

G

M

is

known.

'^

Under

the stated

assumptions, the

model

is fully

parametric, the

(multivariate

normal)

_{distribution of (yi,y2)}

being completely

specified

up

to

the

parameters

/3

and

n.

A

minimal

sufficient statistic for (3

and

tt is

given

by

-, •

(r)

=

(z'fe'\,,)-^("'>«--^-)-,

»)

where

Because

(5,

T)

is sufficient,

the

totality

of

attainable

power

functions

is

spanned

•'These

assumptions correspond

to the

model

in

which

(Z,j/i,j/2) has

been

replaced

by

where

uj,j is the {i,j)

element

of fi

and

0^22.1

=

"^22

—

_'^r/'^i2- ^^^ '^e

chosen

parameterization, the

parameter

S is related to the correlation coefficient

p

computed

from

Cl

through

the

formula

Admissible Invariant

Similar

Tests

4

bytheset ofpower functions associated with (possibly randomized) testsbasedon {S,T).

Any

suchtest can be represented by

means

ofa [0,l]-valued function </>()

such that

Ho

isrejectedwith probabiHty4>{s, t) if{S,T)

=

(s, t) .

The

powerfunction of thistestisthe function (witharguments

_P

andvr)Ep^^^cf)(5,T),wherethesubscript

on

E

indicatesthedistributionwith respecttowhichtheexpectationistaken.

Forany

a

€ (0, 1), atestwithtestfunction isoflevel

a

if

sup^Eo,^0(5,T)

<a.

(4)

A

level

a

test withtestfunction issaid tobea-admissible if

E0^^<l>{S,T)<Ep,,^{S,T) V(/3,7r) ₍₅₎

implies

Ef,,^<P{S,T)

=

Ep,^ip{S,T) V(/3,7r) ₍₆₎

wheneverthetest associatedwith(pisoflevel

a^

The main

purposeofthe presentpaperisto investigatethea-admissibihty

prop-ertiesofcertainrecentlydeveloped rotationinvariant, a-similartests.

By

definition,

a rotationinvariant testisonewhose testfunction 4>satisfies</>[OS,

OT)

=

(p{S,T)

foreveryorthogonal k x kmatrix

O

and anOf-similartestisoneforwhich

.

-

E^,,(i^{S,T)

=

a

Vtt. . :: .:.. :•, (7)

The

rotationinvariant,a-similar testsunderconsiderationhere arethe

WAP-LR

testsof

AMS.

By

construction,asize

a

WAP-LR

testmaximizes a

WAP

criterionof

theform .

/

£;^X5,r)dH/(/3,7r)

(8),

among

rotation invariant, a-similartests, wherethe weight function

W

is

some

cu-mulativedistributionfunction (cdf) onM!'^^.

In spite ofthefactthatthedefiningpropertyofa

WAP-LR

test isan optimality

property,itisnot obviouswhethera

WAP-LR

isa-admissible,the reason being that

^Inthemodelunder studyhere,the presentnotionofQ-admissibihtyagreeswiththat ofLchmann

andRomano(2005, Section6.7) (inwhich (5) and(6)are required toholdforall/?7^ andall tt)

Admissible Invariant Similar Tests

admissibility of

optimum

invariant testscannot betakenforgranted [e.g.,

Lehmann

and

Romano

(2005,Section6.7)].

We

show

inSection3 thatany

WAP-LR

test

max-imizes a

WAP

criterion

among

the classofa-similar tests. In the presentcontext,

that optimahtypropertyissufficientlystrongtoimply that any

WAP-LR

test is

a-admissible.

Remark.

The

factthatthealternativehypothesisisdenseinthemaintained

hypoth-esis implies thatany a-admissible test isd-admissible inthe senseof

Lehmann

and

Romano

(2005).

The

(rotation invariant) posteriorodds ratio testsofChamberlain

(2006) are d-admissible almost byconstruction [e.g.,

Lehmann

and

Romano

(2005,

Theorem

6.7.2 (i))], but becausethesetestsare notnecessarily similar it isunclear

whetherthesetestsare alsoa-admissible (for

some

a).

3.

Results

3.1.

Admissible

a-similar tests.

Two

basic facts about exponential families greatly simplifytheconstruction ofo-admissible, a-similartests. First,because the

powerfunctionofthetestwithtest function(p canbe representedas

E0,^cP{S,T)=

[

[

(j){s,t)fs{s\P,n)fT{t\P,n)dsdt, (9)

wherefs{-{P,tt) and _fr(-1/3,vr) denote thedensities (indexedby/?and n) of

S

and

T, it follows from

Lehmann

and

Romano

(2005,

Theorem

2.7.1) that Ep^T,(j){S,T)

is a continuous function of (/3, tt). Therefore, an o-similar similar test cannot be

dominatedbyalevel

a

test whichisnota-similar.

By

implication,ana-similar test

withtest function (f) isa-admissible if (5) implies (6) whenever the test associated

withifisa-similar.

Second,becausettisunrestricted,

T

isa complete, sufficientstatistic forit under

Hq

[e.g., Moreira(2001)].

As

a consequence, atestwithtest function(f)isa-similar

ifand onlyifitisconditionallya-similarin the sensethat, almostsurely,

Eo,.[</)(5,T)|T]=a Vvr. (10)

Itfollowsfromthepreceding considerationsandtheNeyman-Pearson

lemma

that

if

W

isa cdfonR''"+\then the

WAP

criterion(8)ismaximized

among

a-similartests

bythetest withtest functiongivenby

lW

(s,t;a)

=

1

[LR^

(5, t)

>

k^^{t;a)] , (11)

Admissible Invariant

Similar

Tests

^^

^'''^"

fsis\0,0)frmO)

^'^^

and KYii{t;(y) is the I

-

a

quantile of the distribution of

LR^

{Z^J.), where Z^

is distributed J\f{0,lk)-^ Because the maximizer cp^R is essentially unique (in the

measure theoretic sense), thetestwith test function _(/"^(sQ;) isa-admissible.''

To

demonstrate a-admissibihtyofatest, it thereforesuffices to

show

its test function

canberepresented as

_^^

(•;«)for

some

W. Section3.2 usesthat approachto

show

thatthe

WAP-LR

tests of

AMS

arealla-admissible.

3.2. Admissibility of

WAP-LR

tests.

The

WAP-LR

testsareindexedbycdfs

on

R

XR+.' Accordingly,letu; bea cdfon

M

x

R+

anddefine

C-{s,t)=

I

exp(-^)

„A

dw{p,X),

(13)

whereqFi isthe regularized confluent hypergeometric function,

c(M)=(;;:

;;;). (.4)

and 7/^ is definedas in Section 2.

By

Corollary 1 of

AMS,

the test function ofthe

size

a

WAP-LR

test associatedwith

w

is

'/':^Ms(5,t;«)

=

l[£-(s,t)>«:^,,s(t;a)], (15)

where_k'^ms(^5'^) i^theI

—

a

quantileofthedistribution of £"' {Zk,t).^'^

^Details areprovidedinan Appendix,availablefromtheauthorsuponrequest.

^Aspointed out by areferee,this fact is aspecial case ofthemoregeneral decision theoretic

resultthat(essentially)unique Bayesrulesareadmissible[e.g.,Ferguson(1967,Theorem2.3.1)].

^Becausethe powerofaninvarianttest dependsonnonly throughthe scalartt'tt, the

corre-sponding

WAP

criterion(8)depends on

W

onlythroughthe cdfon

R

x1R+givenby W(/?,A)

=

/ dW{b,TT).

*£'"{s,t) isproportional toip^,_iqi,qT) in

Lemma

1 of

AMS

because qFi(;fc/2;z/4) is

propor-tionalto2~*''"^"''''*/(fc-2)/2(\A).where Jj,(•)denotesthemodifiedBesselfunction of thefirstkind

oforderi/.

^The size a

WAP-LR

test is a-similar by construction and is rotation invariant because '^AMS(*;^) depends ontonly throught't.

Admissible Invariant

Similar

Tests

7

For our purposes,itisconvenientto characterizethedefiningoptimaUtyproperty

of(pjj^fsas follows. Let

W^^s

denote the cdf (on R^+i)of(b,

y/KU'S

,where[B, A)'

has cdf

w

and Uk is uniformly distributedon theunit sphere inR*" (independently

of

B

and A). In termsof W^j^^g,

Theorem

3 of

AMS

asserts that thetest withtest

function(f>^MS (s^) maximizes

E0,,cPiS,T)dW:^^siP^^)

among

rotation invariant, a-similartests. In thisoptimality result, the assumption

ofrotation invarianceisunnecessary becauseitturns out that

rAMs{s,t;a)

=

<p^t"is,t;a). (16)

The

displayed equality fohows fromacalculationperformedintheproofofthe

fol-lowing strengtheningof

Theorem

3 of

AMS.

Theorem

1. Let

w

bea cdfon

M

x R+. If(/>satisfies (7), then

f

E0,^<f^ (5,T)

dW^^s

iP^

^)< [

Ep^^rAMS(5,T-a)

dW^^s

(/?,^)-.

wlieve the inequahty is strict unless PiCff^.^[0 [S,T)

=

4>^ms{S,T;a)]

=

1 for

some

(andhenceforall) (/?, tt). In particular, thesize

a

WAP-LR

testassociated with

w

isa-admissible.

The

testingfunctionofthesize

a

AR

test (for

known

Q) isgivenby

4>^j,{s,t;a)

=

l[s's>xl{k)],

(17)

where xi(^)isthe1

—

a

quantileofthe x^distribution-withk degreesoffreedom.

As

remarked by

AMS,

£"'_{(s, t) isanincreasingfunctionofs'swhenevertheweight} func-tion

w

assigns unitmass totheset _{(^5,A) G

R

x

R+

:/?

=

6^^j.

As

aconsequence,

the followingisanimmediate consequenceof

Theorem

1

.

Corollary2.

The

size

a

AR

testisa-admissible.

In spite of the fact that the

AR

test is a A: degrees offreedom test appUed to

a testing problem with a single restriction, a fact which suggests that its power

Admissible Invariant Similar Tests

8

Corollary 2 implies that thereis noconventional statisticalsensein which the

AR

test "wastes degreesoffreedom"

.

In additiontothe

AR

test,

Theorem

1 also covers the two-pointoptimal

invari-antsimilar testsof

AMS,

thepowerfunctions ofwhichtrace out a two-sided power

envelope for rotationinvariant, a-similar tests.

On

the other hand, the

CLR

and

LM

tests do not seem to belong to the class of

WAP-LR

tests. Indeed, it would

appeartobeanopenquestion whether oneorbothofthesetestseven belongtothe

closure (appropriatelydefined) of theclass

WAP-LR

tests. Section 3.3 provides an

affirmativeanswertothat question.

Remark.

As

pointed out by a referee, two distinct generalizations of the results of this sectionseem feasible. First, theconclusion that a Bayes rule corresponding

toa distribution ti; on

R

xR_,_ can be "lifted" to a Bayes rule corresponding to a

distributionon R'^+^ (by introducingaUk which isuniformlydistributedon theunit

spherein R'^ and independent of (/3,A)) appHes to

more

generaldecision problems

thantheoneconsidered. Second, usingMuirhead (1982,

Theorems

2.1.14 and7.4.1)

itshould be possible togeneralizethe resultstoamodel with multiple endogenous

regressors.

To

conservespace,

we

donot pursue these extensionshere.

3.3.

The

CLR

test.

The

test function ofthesize

a

CLR

test is

(l>CLR{s,t]a)

=

l[LR{s,t)>KcLR{t;a)],

(18)

where

LR

{s, t)

=

^ (s's

-

ft

+

^{s's

+

t'tf

-

4[{s's)[ft)

-

{s'tf]] (19)

and

KcLR

[t]ct) isthe 1

—

a

quantileofthedistribution of

LR

{Z^,t)

.

Innumerical investigations,the

CLR

testhas been foundtoperform remarkably

well in terms ofpower. Forinstance,

AMS

find that the power ofthe size

a

CLR

testis "essentially thesame" asthe two-sidedpowerenvelopeforrotation invariant,

Q-similartests. Inhghtof thisnumericalfinding,itwould appeartobeof interest to

analyticallycharacterizetherelation (ifany) between the

CLR

testandtheclassof

WAP-LR

tests.

Theorem

3showsthat

CLR

(s, t)can berepresentedasthehmitas A''-^-oo ofa

suitably normalizedversionofC"clr,n(^^ {^^where {wclr,n : TVG

N}

isacarefully

chosencollectionof cdfson

R

x

R+

.

To

motivatethisrepresentationandthe functionalformofwclr,n,itisconvenient

Admissible Invariant Similar Tests

(J)CLR(s.^;a)

=

1

[CLR*

(s, t)

>

_KcLR-{t;a) where

CLR*{s,t)

=

\s's

+

t't+J{s's

+

t'tY-A[{s's){t't)-{s't)

^2[LR{s,t)

+

t't] (20) and KcLR*{t;a)

=

^/2[KcLR{t;a)+t't].

The

statistic

CLR*

(s, i) is the square root ofthe largest eigenvahie of

_Q

(s, t)

(defined in (14)) and thereforeadmitsthefollowing characterization:

CLR*

{s, t)

=

A/max^eR2.^-^=i-q'Q(s, t)r). (21)

Moreover, the integrand in (13) can be writtenas

exp

^

0^1

whereTjp

=

V/s/x/v'pVpi^^ vectorofunit lengthproportionalto7/^. If_wclr.nissuch thatitssupportistheset ofallpairs(/3,A) forwhichAry^r/^

=

A'',then the integrand

is maximized (over the support ofwclr,n) by setting7/^ equal to the eigenvector

associatedwith thelargesteigenvalueof

_Q

(s, t). This observation,andthefactthat

thetailbehaviorofoi^i [;k/2;•/4]issimilartothatofexp(i/^),suggests thatthelarge A^behaviorofC^clr.n(^^ f) "should" depend on

_Q

(s, t) onlythrough

CLR*

{s, t)

.

ForanyA^

>

0,letwclr,ndenote the cdfof [B,

N/^/rf^)'

,where

^

~

A^(0, 1).^^

By

construction,

wclr,n

issuch thatitssupportisthesetofallpairs(/?,A) forwhich

^v'ffVp

=

^- Usingthatproperty,therelation (21),andbasic factsabouto^^i(;k/'2;)

,

we

obtain the followingresult.

^"^The distributionalassumption

B ~

TV(0,1)ismadeforconcreteness.

An

inspection oftiieproof

ofTheorem3showsthat (22)isvalidforanydistribution(ofB)whosesupportisR.Furthermore,

aspointed outby arefereeitispossibletoobtainanalogousresultswithoutmakingthe distribution of^VoVpdegenerate.

Admissible Invariant

Similar

Tests

10

Theorem

3. Forany(s',t')' G

In particular,

CLR*

(s, t)

=

lim7v-.oo

^p=

logZ:""^^«''^ (s, t)

+

—

(22)

CLR{s,t)

=

lim^

2N

logC"^'^"-^{s,t)

+

N

t't. (23)

Define -C^^^^yv(«>*)

=

[log£"'^^«"^(s,i)

+

A^/2]/\/iV.

The

proofof

Theorem

3

shows thatasA''

—

>•oo, C*qiii^n(') converges to

CLR*

(•) in thetopologyofuniform

convergenceon compacta. Usingthisresult, it fohowsthat

Hmyv^ooE0,,_I^Z-S'"(5,T-a)

-

<Pclr(S,T; a)

|

=

V

(/?,

n) .

In particular, the powerfunction ofthe

WAP-LR

test associated withwclr,n

con-verges (pointwise) tothepowerfunctionofthe

CLR

test (as

N

-^ oo).

Inhghtofthe previousparagraph,itseemsplausiblethatthe

CLR

testenjoysan

"admissibility at cx)" property reminiscent of

Andrews

(1996,

Theorem

1(c)).

Veri-fyingthis conjectureisnot entirelytrivial,however, because theunboundedness (in

t)of K.CLR* [t]ci)

makes

itdifficult (ifnot impossible)toadapt the proofsof

Andrews

andPloberger (1995) and

Andrews

(1996) tothe presentsituation.

Theorem

3 alsosuggests a

method

ofconstructingtestswhich share thenice

nu-merical propertiesofthe

CLR

test andfurthermore enjoy demonstrable optimality

properties, namely tests based on C^clr.n(^s,t) for

some

"large" (possibly

sample-dependent) valueofN. Because thepower improvements (ifany) attainable inthis

way

are likelyto be slight, the properties of tests constructed in this

way

are not

investigatedin thispaper.

Remark.

The

testfunction ofthesize

a

LM

test is

'{s'tf [s,t;a)

=

1

t't

>x;(i)

where Xo(1) isthe 1

—

a

quantile of_{the x^} distribution with 1 degreeoffreedom.

AMS

show

that a one-sided version ofthe

LM

test can be interpreted asalimit of

WAP-LR

tests (andislocally mostpowerful invariant).

On

theother hand,

we

are

Admissible Invariant Similar Tests

11

A

slightvariationontheargumentusedinthe previous subsectioncan beusedto

obtain a

WAP-LR

interpretation ofthe

LM

test. Indeed, letting

Wlm,n

denote the

cdfof [Bn-,

N/

_^^v'bn^Bn]'^ wherethedistribution of

N^/^Bn

is uniformon [-1,

1]

,

itcan be

shown

that"

\s't\l\fFt

=

hmA,_.oo-TTJ

iVl/6 log

£^"-A^(s,t)-\-

—

-

VnVFi

implyinginparticular that

n2

(s'ty

_{_}

1

N

r—

r—

(24)

(25)

4.

Proofs

Proof

of

Theorem

1. Itsufficestoestablish (16).

To

doso, itsufficesto

show

that

LI^AMs

(5, t) isproportionalto

D"

(s, i).

Now,

/s(5|0,0)/T(i|0,0)

exp

(-\^ [lis

-

/?^f

-

||s||-

+

\\t

-

(1

-

bH) vrf

-

jjtf]")

exp

^J!Mll\

exp [y\,\(is

+

(1

-

6/3) tj'yr) ,

where ||-|| signifiestheEucfideannorm, A^r

=

tt'tt, andtt

=

A^^^^tt.

Becausetthasunit length, itfollowsfrom Muirhead (1982,

Theorem

7.4.1)that

Lir'^Ms {^s,t)

=

fs{s\P,7r)fT{t\P,7r) /s(5|0,0)/r(t|0,0)

dW^,jsif3,7r)

exp Fi -,-V'f3Q{S:t)V0

dw{P,X),

as wastobeshown.

(We

are grateful toarefereeforsuggesting the useofMuirhead

(1982,

Theorem

7.4.1).)

Proof

of

Theorem

3.

The

resultisobviousif

Q

(s, t)

=

0, sosuppose

Q

(5, t) 7^ 0.

^^Details arcprovidedinanAppendix,availablefromtheauthorsuponrequest.

An

inspection of

theproofgiven thereshowsthat the distributionalassumptiononN^/^Bnismadeforconcrcteness

insofar as theprecedingrepresentations arc validwhenever N^/'^Bn hasa(fixed)distributionwhose

Admissible

Invariant

Similar

Tests

12

The

proofwill

make

useofthefact [e.g.,

Andrews

and Ploberger(1995,

Lemma

2)] that

<

C

<

C

<

oo,where

C

=

inf.

A[;k/2;z/i]

exp (y^) max(2,1) ^" ''

By

construction,

C

=

sup^ ,Fr[;k/2-z/4]

exp(v^)

C" {s,t) exp I ;^— ) 0^1

N

=

expI

-

—

2 7;',-7V'i3Qis,t)ri0

T

4 dwr (/?,A) d$(/?)

where<J>(•) isthestandardnormalcdf. Usingthis representation, therelation (21)

and monotonicityofoi^i[;/c/2; ],

£"'c^^«'~(s,t)exp TV ^F^ d$(/3)

'24

^ '

<

oFx

<

Cexp\VNCLR*is,t)\

,

from whichitfollowsimmediatelythat

1

limr

^A^

N

Iq^jTwclr.n _(5^ij

₊

_<

_CLR*[s,t)

On

the otherhand, for any

<

e

<

CLR*

{s,t)

, (26) £"'"«.'^(s,t)exp(

_y

)

=

3^1 (^,t) -;jr?aQ(s,f)r?^ )i^i d«i>(/?) d$(/3)

>

exp[\/A(CL/?*(s,f)-e:

xCmaxlN

CLR*

{s,ty,1] -(fc-l)/4 rf$(/3) '.(s,i)

Admissible Invariant Similar Tests

13

where

B,{s,

=

_{/3: ^Jf]],Q{s,t)fi0

>

CLW

(.s,t)

-

e],

thefirstinequahty usespositivity ofqFi [;k/2\•],andthelastinequalityuses (21)and

monotonicity ofo-^i[;k/2\].

The

integral j^ . ^,

d^

(/?) isstrictly positive because

$

(•) hasfullsupport,so

lim/v^c

nv

log/:"'"«''^(s,i)

+

A^

>

CLR*{s,t)-£.

Lettingetendtozerointhe displayedinequality,

we

obtainan inequalitywhich can

be combinedwith (26) to yield (22) .

Indeed, because

sup(y_(,)'g/^CLi?*(s,t)

<

oo and inf^^/^/j'^/,^ / (i$(/?)

>

for anycompactsetA'

C

M?^, theresult (22) canbestrengthenedas follows:

hm

N-^oo^^^V(s>,t')'eK

7n

]C"-'^^'''^ {s,t)

+

N'

Admissible Invariant

Similar

Tests

14

References

Anderson,

T.

W.,

and

H.

Rubin

(1949): "EstimationoftheParametersofaSingle

Equation in a Complete Set of Stochastic Equations," Annals ofMathematical

Statistics,20,46-63.

Andrews,

D.

W.

K. (1996): "Admissibihtyofthe LikelihoodRatio Test

When

the

Parameter SpaceisRestrictedundertheAlternative," Econometrica,64,705-718.

Andrews,

D.

W.

K.,

M.

J.

Moreira,

and

J.H.

Stock

(2004): "Optimal

Invari-ant Similar TestsforInstrumental Variables Regression,"

NBER

TechnicalWorking

PaperNo.299.

Andrews,

D.

W.

K.,

and

W.

Ploberger

(1995): "AdmissibihtyoftheLikehhood

Ratio Test

When

a Nuisance Parameter is Present

Only

under the Altenative,"

Annals ofStatistics,23, 1609-1629.

Andrews,

D.

W.

K.,

and

J.H.

Stock

(2006): "Inferencewith

Weak

Instruments,"

AdvancesinEconomicsandEconometrics, Theory andApplications: NinthWorld CongressoftheEconometricSociety,forthcoming.

Chamberlain,

G. (2006): "DecisionTheory Applied toanInstrumental Variables

Model," Econometrica, forthcoming.

DuFOUR,

J.-M. (1997):

"Some

Impossibihty

Theorems

inEconometricswith

Appli-cationstoStructuraland

Dynamic

Models," Econometrica,65, 1365-1387.

(2003): "Identification,

Weak

Instruments, and Statistical Infei'ence in

Econometrics," Canadian Journal ofEconomics, 36, 767-808.

Ferguson,

T. S. (1967): MathematicalStatistics:

A

Decision Theoretic Approach.

New

York:

Academic

Press.

Hahn,

J.,

and

J.

Hausman

(2003):

"Weak

Instruments: Diagnosis and Curesin

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,

E. L.,

and

J. P.

Romano

(2005): TestingStatisticalHypotheses.Third

Edition.

New

York: Springer.

Moreira, M.

J. (2001): "Testswith Correct SizeWlienInstruments

Can Be

Admissible

Invariant Similar Tests

15

(2003):

"A

Conditional LikelihoodRatio TestforStructuralModels," Econo-metnca, 71, 1027-1048.

MuiRHEAD,

R. J. (1982): Aspects ofMultivariate Statistical Theory.

New

York: Wiley.

Staiger, D.,

and

J. H.

Stock

(1997): "Instrumental Variables Estimation with

Weak

Instruments," Econometrica, 65, 557-586.

Stock,

J. H., J. H.

Wright,

and

M.

Yogo

(2002):

"A

Surveyof

Weak

Instru-ments and

Weak

Identification in Generalized

Method

ofMoments," Journal of

Admissible

Invariant Similar Tests

16

5.

Appendix:

Omitted Proofs

This

Appendix

providesa proofof (24)andaderivation of (11)

—

(12)

.

5.1.

Proof

of (24).

To

motivate tlieresult, noticethatif

wlm,n

issuch thatits

supportconsistsofallpairs (/3,A) for which \0\

<

N'^^^ andXripij^

=

N, then the

large A^ behavioroftheintegrand in (13) isreminiscentofthebehaviorof

exp TV dFi

kN

2'T

{t't

+

2ps't)

Because

maX|^l<;v->/3 \/t't

+

2(3s't

=

^Vt

+

2N-y^\s't\

reasoningsimilar tothatleadingto

Theorem

3 thereforesuggeststhat |s'^|/Vt'tcan

berepresentedasthelimit as A^

—

>ooofa suitably normalized versionof0"^m,n(^^ i^

As

in Section 3.3, let

Wlm,n

denote the cdfof [B^,

N/

^Jv'bn^Ibn] ' where the

distribution of

N^^^Bn

isuniformon[—1,1].

Then

, .

C"^'''" {sj) exp ) oi^i A: 1

—

exp

N\

7V1/3 )Fi |-_Af-i/3_Ar-i/3] d'WLM,N(/?,A) ' 2' ₄

The

integrandcanbe writtenas

0-^1

where, for

some

finiteconstant

R

(depending onlyon sandt),

k

N

--{t't

+

2l3s't

+

RN[^\s,t))

VpVis

d/3.

SUP|/3|<w-i/3 \RniP;S,t)\=SUP|^|<^-i/3

i'0Q{s,t)v0 Nv'pVp

Admissible

Invariant Similar Tests

17

As

a consequence,bymonotonicityofqFi[;/c/2;•]and thedefiningpropertyof

C

(appearinginthe proofof

Theorem

3),

£Wi^M,N (^^ ^)gj.p A^

<

4f

^ J\-\/Vn,i/Vn]

/

„A

k

N

2'T

[t't

+

2(5s't

+

7V-2/^i?)

dp

-

SUP|^|<yv-i/3 o-f^i

=

0^1

24V

/

<

Cexp

^N\t't

+

2N'^l^\s't\+N-^I^R

,

from which itfollowsthat

7V-V6

<

A^^i/*^

_logC

₊

_N^/\/t't

₊

_27V-1/3|5'f I

+

7V-2/3i?

-

N^/^VVt

R

2yM

wherethelastinequalityusesconcavityof y^.

To

obtainaninequality intheoppositedirection,

we

distinguishbetweenthecases

Admissible

Invariant Similar

Tests

18 7V-V6 A^i/6 A^

I—

I—

)Fi [_/V-l/3,iV-V3] k Nri'f^Q{s,t)r]p 2' 4 VpVf3

dp

>

N-^/^ oFi

¥

O

{N-"^)

,

whichcompletes the proofof (24) in the casewheret

—

Q.

On

the other hand, ifi7^ 0, thenit follows from monotonicityand positivity of

oFi[;k/2]•] that £t.^LM,N(5^^)exp

N

7V1/3 Ari/3 iF, l-iV-i/3,Ar-V3] fc TV

2'T

[t't

+

2(5s't

-

N-^I^R) d/3 5F1 [o,yv-i/3]

.L!l^t't

+

2p\s't\-N-~/m)

dp

for every

A

>

iV

=

inf{

A

:t't

-

2N~^'^\s't\

-

N'^^^R

>

0}. Therefore, for any

<

£

<

1 and any

N

>

K,

jru>LM,M (5^ ^)gj-p

N

iVi/3 0-^1 [(l~£)W-V3,iV-l/3]

k_N

2'T

't't

+

2(3\s't\-N-''/'^R)

dp

^

9"^f/3e[(l-e)A'-i/3,/v-i/3] o-f^l

^j{t't

+

2P\s't\-N-'/m)

3F1 ;-;

-

(i'i

+

2(1

-

e) A-i/^_\s't\

-

iV^/^^R)

>

-C\Nt't-N'i^R

'/_/vi/37^-('-''/'_exp

_N\

_t't

₊

₂

Admissible

Invariant Similar

Tests

19

wherethe first inequahty usespositivity ofo^^i [;k/'2\•] and the lastinequalityuses

thedefiningpropertyofC_ (appearingin theproofof

Theorem

3) and monotonicity

„f[.]-(fc-l)/4_

By

implication,

Ar-i/6

>

iV"i/6

N

r-

I—

log{Cel2)

- ^^-^

log(TVt'i

-

N^I^'R)

+N^'\lt't

+

2(1

-

e)yV^V3\s't\

-

N-ym

-

N^^^VFt

(1-£)\s't\

-

N~^/'^R/2

'ft

+

max

[2 (1

-

e)A^-V3\s't\

-

N-^m,

O]

>

{l-e)\s't\/^t

+

0(N-^^HogN).

+

0{N-^/^\ogN)

The

proofof (24) can

now

be completed by letting e tend tozero in thepreceding

display.

5.2.

Derivation

of(11)

—

(12).

The

testbasedon(p{S,T)isa-similarifandonly if

/ I (p{s,t)fs{s\0,n)fT{t\0,n)dsdt

=

a

Vtt.

Because

T

iscompleteunder

Hq

:(3

=

0, thepreceding condition holds ifandonlyif

/ (f){s,t)fs{s\0,7r)ds

=

a

./«'

for almost everyt G M.'^. Furthermore, fs(s|0,7r) does not depend on tt,so thetest

basedon 0(5, T) isa-similarifand onlyif

/

((>{s,t)fsis\0,0)ds

=

a

Admissible Invariant Similar Tests

20

Using(theFubinitheorem and) the representation(9),

we

canwrite (8) as

{s,t) fs{s\P,1T)fTit\(3,7T)dW{f3,Tv) dsdt.

To

maximizethisexpression subjecttothe condition that thetestbasedon (5,T)is

a-similar,

we

canproceedona"tby

f

basisandconsidertheproblemofmaximizing

subjectto the condition

/5(5|/?,^)/T(t|/?,7r)diy(/?,7r) ds

By

the

Neyman-Pearson

lemma,thismaximizationproblemissolvedby

[s,t-a)

= l\LR

{s,t)>KYi^{t;a:

where

~w

,,, J^,,,fs{s\P,n)fT{t\P,n)dW(A7r)

—

w

and k^u{t;a)isthe1

-

a

quantileofthedistribution of

LR

(2^.,t).Finally,because

/r(t|0,0) ispositive

we

obtain theequality