Exogeneity tests, incomplete models, weak identification and non-Gaussian distributions: invariance and finite-sample distributional theory

0.31 MB·

by  Firmin Doko Tchatoka

Exogeneity tests, incomplete models, weak identification
and non-Gaussian distributions:
7 invariance and finite-sample distributional theory∗
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n FirminDokoTchatoka† Jean-MarieDufour‡
a TheUniversityofAdelaide McGillUniversity
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December 2016
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X ∗TheauthorsthankNazmulAhsan,MarineCarrasco,AtsushiInoue,JanKiviet,VinhNguyen,BenoitPerron,Pas-
r caleValéry,andHuiJunZhangforseveralusefulcomments. ThisworkwassupportedbytheWilliamDowChairin
a
PoliticalEconomy(McGillUniversity),theBankofCanada(ResearchFellowship),theToulouseSchoolofEconomics
(Pierre-de-FermatChairofexcellence),theUniversitadCarlosIIIdeMadrid(BancoSantanderdeMadridChairofexcel-
lence),aGuggenheimFellowship,aKonrad-AdenauerFellowship(Alexander-von-HumboldtFoundation,Germany),the
CanadianNetworkofCentresofExcellence[programonMathematicsofInformationTechnologyandComplexSystems
(MITACS)], the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities
ResearchCouncilofCanada,andtheFondsderecherchesurlasociétéetlaculture(Québec).
†SchoolofEconomics,TheUniversityofAdelaide,10PulteneyStreet,Adelaide,SA5005. Tel: [phone];
e-mail:fi[email protected]. Homepage: http://www.adelaide.edu.au/directory/firmin.dokotchatoka
‡WilliamDow Professor of Economics, McGill University, Centreinteruniversitaire de recherche en analyse des
organisations(CIRANO),andCentreinteruniversitairederechercheenéconomiequantitative(CIREQ).Mailingaddress:
Department of Economics, McGill University, Leacock Building, Room 414, 855 Sherbrooke Street West, Montréal,
QuébecH3A2T7,Canada. TEL:[phone]ext.09156;FAX:[phone];e-mail:jean-marie.dufour@
mcgill.ca.Webpage:http://www.jeanmariedufour.com
ABSTRACT
Westudythedistribution ofDurbin-Wu-Hausman (DWH)andRevankar-Hartley (RH)testsforex-
ogeneity from a finite-sample viewpoint, under the null and alternative hypotheses. We consider
linearstructuralmodelswithpossiblynon-Gaussian errors,wherestructuralparametersmaynotbe
identifiedandwherereducedformscanbeincompletelyspecified(ornonparametric). Onlevelcon-
trol,wecharacterize thenulldistributions ofalltheteststatistics. Throughconditioning andinvari-
ancearguments, weshowthatthesedistributions donotinvolve nuisance parameters. Inparticular,
thisapplies toseveralteststatistics forwhichnofinite-sample distributional theoryisyetavailable,
such as the standard statistic proposed by Hausman (1978). The distributions of the test statistics
maybenon-standard –socorrections tousual asymptotic critical values areneeded –butthechar-
acterizations are sufficiently explicit to yield finite-sample (Monte-Carlo) tests of the exogeneity
hypothesis. The procedures so obtained are robust to weak identification, missing instruments or
misspecified reduced forms, and can easily be adapted to allow for parametric non-Gaussian error
distributions. We give a general invariance result (block triangular invariance) for exogeneity test
statistics. This property yields a convenient exogeneity canonical form and a parsimonious reduc-
tionoftheparameters onwhichpowerdepends. Intheextremecasewherenostructural parameter
isidentified, thedistributions under thealternative hypothesis andthenullhypothesis areidentical,
so the power function is flat, for all the exogeneity statistics. However, as soon as identification
does not fail completely, this phenomenon typically disappears. We present simulation evidence
which confirms the finite-sample theory. The theoretical results are illustrated with two empirical
examples: the relation between trade and economic growth, and the widely studied problem of the
returnofeducation toearnings.
Keywords: Exogeneity; Durbin-Wu-Hausman test; weak instrument; incomplete model; non-
Gaussian;weakidentification;identificationrobust;finite-sampletheory;pivotal;invariance;Monte
Carlotest;power.
JEL classification: C3;C12;C15;C52.
i
Contents
ListofDefinitions,Assumptions,PropositionsandTheorems iii
1. Introduction 1
2. Framework 3
3. Exogeneitytests 6
3.1. Teststatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2. Regression-based formulations ofexogeneity statistics . . . . . . . . . . . . . . 9
4. Incompletemodelsandpivotalproperties 11
4.1. Distributions ofteststatistics underexogeneity . . . . . . . . . . . . . . . . . . 11
4.2. ExactMonteCarloexogeneity tests . . . . . . . . . . . . . . . . . . . . . . . . 12
5. Block-triangular invarianceandexogeneity canonicalform 14
6. Power 15
7. Simulationexperiment 19
7.1. Sizeandpowerwiththeusualcriticalvalues . . . . . . . . . . . . . . . . . . . 19
7.2. PerformanceoftheexactMonteCarlotests . . . . . . . . . . . . . . . . . . . . 20
8. Empiricalillustrations 29
8.1. Tradeandgrowth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.2. Education andearnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
9. Conclusion 31
A. WuandHausmanteststatistics 33
B. Regression interpretation ofDWHteststatistics 34
C. Proofs 37
ii
List of Tables
1 Sizeandpowerofexogeneity testswithGaussianerrorsatnominallevel5% . . . 21
2 SizeandPowerofexogeneity testswitht(3)errorsatnominallevel5% . . . . . 23
3 SizeandpowerofexactMonteCarlotestswithGaussianerrorsatnominallevel5% 25
4 SizeandpowerofexactMonteCarlotestswitht(3)errorsatnominallevel5% . . 27
5 Exogeneity intradeandgrowthmodel . . . . . . . . . . . . . . . . . . . . . . 30
6 Exogeneity ineducation andearningmodel . . . . . . . . . . . . . . . . . . . . 31
List of Definitions, Assumptions, Propositions and Theorems
Assumption2.0 : Conditional scalemodelforthestructural errordistribution . . . . . . . 4
Assumption2.0 : Conditional mutualindependence ofeandV . . . . . . . . . . . . . . 5
Assumption2.0 : Homoskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Assumption2.0 : Orthogonality betweeneandV . . . . . . . . . . . . . . . . . . . . . 6
Assumption2.0 : Endogeneity-parameter distributional separability . . . . . . . . . . . . 6
Assumption2.0 : Reduced-form linearseparability forY . . . . . . . . . . . . . . . . . 6
Proposition 4.1 : Quadratic-form representations ofexogeneity statistics . . . . . . . . . 11
Theorem4.2 : Nulldistributions ofexogeneity statistics . . . . . . . . . . . . . . . . . . 11
Proposition 5.1 : Block-triangular invariance ofexogeneity tests . . . . . . . . . . . . . 14
Theorem6.1 : Exogeneity testdistributions underthealternative hypothesis . . . . . . . 15
Theorem6.2 : Invariance-based distributions ofexogeneity statistics . . . . . . . . . . . 16
Theorem 6.3 : Invariance-based distributions of exogeneity statistics components with
Gaussianerrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Theorem6.4 : Doublynoncentral distributions forexogeneity statistics . . . . . . . . . . 18
LemmaA.1 : Differenceofmatrixinverses . . . . . . . . . . . . . . . . . . . . . . . . . 33
LemmaC.1 : Propertiesofexogeneity statistics components . . . . . . . . . . . . . . . . 37
ProofofLemmaC.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ProofofProposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
LemmaC.2 : Propertiesofweightingmatricesinexogeneity statistics . . . . . . . . . . . 40
ProofofLemmaC.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
ProofofTheorem4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ProofofProposition 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ProofofTheorem6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ProofofTheorem6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ProofofTheorem6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ProofofCorollary6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
iii
1. Introduction
The literature on weak instruments is now considerable and has often focused on inference for the
coefficients of endogenous variables in so-called “instrumental-variable regressions” (or “IV re-
gressions”); seethereviewsofStock,WrightandYogo(2002), Dufour(2003),AndrewsandStock
(2007), andPoskitt andSkeels(2012). Although research ontests forexogeneity inIVregressions
isconsiderable, mostofthesestudieseitherdealwithcaseswhereinstrumental variablesarestrong
(thus leaving out issues related to weak instruments), or focus on the asymptotic properties of ex-
ogeneity tests.1 To the best of our knowledge, there is no study on the finite-sample performance
of exogeneity tests when IVs can be arbitrary weak, when the errors may follow a non-Gaussian
distribution, orwhenthereducedformisincompletelyspecified. Thelatterfeatureisespeciallyim-
portant to avoid losing the validity of the test procedure when important instruments are “left-out”
whenapplying anexogeneity test,ashappenseasilyforsomecommon“identification-robust” tests
onmodelstructural coefficients[seeDufourandTaamouti(2007)].
In this paper, we investigate the finite-sample properties (size and power) of exogeneity tests
of the type proposed by Durbin (1954), Wu (1973), Hausman (1978), and Revankar and Hartley
(1973), henceforth DWH and RH tests, allowing for: (a) the possibility of identification failure
(weak instruments); (b) model errors with non-Gaussian distributions, including heavy-tailed dis-
tributions which may lack moments (such as the Cauchy distribution); and (c) incomplete reduced
forms(e.g.,situationswhereimportantinstrumentsaremissingorleftout)andarbitraryheterogene-
ityinthereducedformsofpotentially endogenous explanatory variables.
As pointed out early by Wu (1973), a number of economic hypotheses can be formulated in
terms of independence (or “exogeneity”) between stochastic explanatory variables and the distur-
banceterminanequation. Theseinclude, forexample,thepermanentincomehypothesis, expected
profitmaximization, andrecursiveness hypotheses insimultaneous equations. Exogeneity (or“pre-
determination”) assumptions can also affect the “causal interpretation” of model coefficients [see
Simon (1953), Engle, Hendry and Richard (1982), Angrist and Pischke (2009), Pearl (2009)], and
eventually thechoiceofestimation method.
Toachievetheabovegoals,weconsiderageneralsetupwhichallowsfornon-Gaussiandistribu-
tionsandarbitraryheterogeneity inreduced-form errors. Undertheassumptionthatthedistribution
ofthestructuralerrors(givenIVs)isspecifieduptoanunknownfactor(whichmaydependonIVs),
we show that exact exogeneity tests can be obtained from all DWH and RH statistics [including
Hausman (1978) statistic] through the Monte Carlo test (MCT) method [see Dufour (2006)]. The
null distributions of the test statistics typically depend on specific instrument values, so “critical
1See,forexample,Durbin(1954),Wu(1973,1974,1983a,1983b),RevankarandHartley(1973),Farebrother(1976),
Hausman(1978),Revankar(1978),Dufour(1979,1987),Hwang(1980,1985),KariyaandHodoshima(1980),Hausman
andTaylor(1981),SpencerandBerk(1981),NakamuraandNakamura(1981,1985),Engle(1982),Holly(1982,1983b,
1983a), Holly and Monfort (1983), Reynolds (1982), Smith (1983, 1984, 1985, 1994), Thurman (1986), Rivers and
Vuong(1988),SmithandPesaran(1990),Ruud(1984,2000),Newey(1985a,1985b),DavidsonandMackinnon(1985,
1985, 1989, 1990, 1993), Meepagala (1992), Wong (1996, 1997)„ Ahn (1997), Staiger and Stock (1997), Hahn and
Hausman(2002),Baum,SchafferandStillman(2003),KivietandNiemczyk(2006,2007),BlundellandHorowitz(2007),
Guggenberger (2010), Hahn,HamandMoon(2010), JeongandYoon(2010), ChmelarovaandHill(2010),Kivietand
Pleus(2012),LeeandOkui(2012),Kiviet(2013),Wooldridge(2014,2015),Caetano(2015),DokoTchatoka(2015a),
Kabaila,MainzerandFarchione(2015),andLochnerandMoretti(2015).
1
values” should also depend on the latter. Despite this, the MCT procedure automatically controls
the level irrespective of this complication, and thus avoids the need to compute critical values. Of
course, asusual,thenullhypothesis isinterpreted hereastheconjunction ofallmodelassumptions
(including “distributional” ones)withtheexogeneity restriction.
The finite-sample tests built in this way are also robust to weak instruments, in the sense that
they never over-reject the null hypothesis of exogeneity even when IVs are weak. This entails
that size control is feasible in finite samples for all DWH and RH tests [including the Hausman
(1978) test]. Allexogeneity tests considered can also be described as identification-robust in finite
samples. TheseconclusionsstandincontrastwithonesreachedbyStaigerandStock(1997,Section
D)whoargue –following alocalasymptotic theory –thatsize adjustment maynotbefeasible due
to the presence of nuisance parameters in the asymptotic distribution. Of course, this underscores
the fundamental difference between afinite-sample theory and an asymptotic approximation, even
whenthelatteris“improved”.
Moreimportantly, weshowthattheproposed MonteCarlotestprocedure remainsvalidevenif
theright-hand-side (possibly)endogenousregressorsareheterogenousandthereduced-form model
is incompletely specified (missing instruments). Because of the latter property, we say that the
DWHandRHtestsarerobusttoincomplete reduced forms. Forexample,robustness toincomplete
reducedformsisrelevantinmacroeconomicmodelswithstructuralbreaksinthereducedform: this
showsthat exogeneity tests remain applicable without knowledge ofbreak dates. Insuch contexts,
inference on the structural form may be more reliable than inference on the reduced form. This is
of great practical interest, for example, in inference based on IV regressions and DSGE models.
For further discussion of this issue, see Dufour and Taamouti (2007), Dufour, Khalaf and Kichian
(2013)andDokoTchatoka(2015b).
We study analytically the power of the tests and identify the crucial parameters of the power
function. In order to do this, we first prove a general invariance property (block triangular invari-
ance) for exogeneity test statistics – a result of separate interest, e.g. to study how nuisance pa-
rametersmayaffectthedistributions ofexogeneity teststatistics. Thisproperty yields aconvenient
exogeneitycanonicalformandaparsimoniousreductionoftheparametersonwhichpowerdepends.
Inparticular, wegiveconditions underwhichexogeneity testshavenopower,andconditions under
which they have power. We show formally that the tests have little power when instruments are
weak. Inparticular, thepowerofthetests cannot exceed the nominal levelifallstructural parame-
tersarecompletely unidentified. Nevertheless, powermayexistassoonasoneinstrumentisstrong
(partialidentification).
We present a Monte Carlo experiment which confirms our theoretical findings. In particular,
simulation results confirm that the MCT versions of all exogeneity statistics considered allow one
to control test size perfectly, while usual critical values (under a Gaussian error assumption) are
either exact or conservative. The conservative property is visible in particular when the two-stage-
least-squares (2SLS) estimator of the structural error variance is used in covariance matrices. In
suchcases, theMCTversionofthetestsallowssizablepowergains.
Theresults arealso illustrated through twoempirical examples: the relation between trade and
economicgrowth,andthewidelystudiedproblem ofthereturnofeducation toearnings.
The paper is organized as follows. Section 2 formulates the model studied, and Section 3 de-
2
scribes the exogeneity test statistics, including a number of alternative formulations (e.g., linear-
regression-based interpretations) which may have different analytical and numerical features. In
Section 4, we give general characterizations of the finite-sample distributions of the test statistics
andshowhowtheycanbeimplementedasMonteCarlotests,witheitherGaussianornon-Gaussian
errors. In Section 5, we give the general block-triangular invariance result and describe the as-
sociated exogeneity canonical representation. Power is discussed in Section 6. The simulation
experiment is presented in Section 7, and the empirical illustration in Section 8. We conclude in
Section 9. Additional details on the formulation of the different test statistics and the proofs are
supplied inAppendix.
Throughout the paper, I stands for the identity matrix of order m. For any full-column-rank
m
T×mmatrixA, P¯[A]=A(A′A)−1A′istheprojectionmatrixonthespacespannedbythecolumnsof
A,andM¯ [A]=I −P¯[A].Forarbitrary m×mmatrices AandB,thenotation A>0meansthat Ais
T
positivedefinite(p.d.),A≥0meansAispositivesemidefinite(p.s.d.),andA≤BmeansB−A≥0.
Finally,kAkistheEuclidiannormofavectorormatrix,i.e.,kAk=[tr(A′A)]12.
2. Framework
Weconsiderastructural modeloftheform:
y=Yb +X g +u, (2.1)
1
Y =g(X ,X ,X ,V,P¯), (2.2)
1 2 3
where (2.1) is a linear structural equation, y∈RT is a vector of observations on a dependent vari-
able,Y ∈RT×G is a matrix of observations on (possibly) endogenous explanatory variables which
are determined by equation (2.2), X1 ∈RT×k1 is a matrix of observations on exogenous variables
included in the structural equation (2.1), X2 ∈RT×k2 and X3 ∈RT×k3 are matrices of observations
onexogenousvariablesexcludedfromthestructuralequation, u=(u ,...,u )′∈RT isavectorof
1 T
structural disturbances, V =[V ,...,V ]′ ∈RT×G is a matrix of random disturbances, b ∈RG and
1 T
g ∈Rk1 are vectors of unknown fixed structural coefficients, and P¯ is a matrix of fixed (typically
unknown)coefficients. Wesuppose G≥1,k ≥0,k ≥0,k ≥0,anddenote:
1 2 3
X =[X ,X ]=[x ,...,x ]′, X¯ =[X ,X ,X ]=[x¯ ,...,x¯ ]′, (2.3)
1 2 1 T 1 2 3 1 T
Y¯ =[Y,X ], Z=[Y,X ,X ]=[z ,...,z ]′, Z¯ =[Y,X ,X ,X ]=[z¯ ,...,z¯ ]′, (2.4)
1 1 2 1 T 1 2 3 1 T
U =[u,V]=[U ,...,U ]′ . (2.5)
1 T
Equation(2.2)usuallyrepresentsareduced-form equationforY. Theformofthefunctiong(·)may
benonlinearorunspecified, somodel(2.2)canbeviewedas“nonparametric” or“semiparametric”.
The inclusion of X in this setup allows for Y to depend on exogenous variables not used by the
3
exogeneity tests. Thisassumption iscrucial, because itcharacterizes the factthatweconsider here
“incomplete models” where the reduced form for Y may not be specified and involves unknown
exogenous variables. Itiswellknownthat several “identification-robust” tests forb [such asthose
proposed by Kleibergen (2002) and Moreira (2003)] are not robust to allowing a general reduced
3
formforY suchastheonein(2.2);seeDufourandTaamouti(2007)andDokoTchatoka(2015b).
Wealsomakethefollowingrankassumption onthematrices[Y,X]and P¯[X]Y,X :
1
[Y,X]and P¯[X]Y,X havefull-column rankwithprobability one(cond(cid:2)itional on(cid:3)X). (2.6)
1
This (fairly sta(cid:2)ndard) con(cid:3)dition ensures that the matrices X, M¯ [X ]Y and M¯[X]Y have full column
1
rank, hence the unicity of the least-squares (LS)estimates when each column ofY is regressed on
X, as well as the existence of a unique two-stage-least-squares (2SLS) estimate for b and g based
onX astheinstrumentmatrix. Clearly,(2.6)holdswhenX hasfullcolumnrankandtheconditional
distribution ofY givenX isabsolutely continuous (withrespecttotheLebesgue measure).
A common additional maintained hypothesis in this context consists in assuming that g(·) is a
linearequation oftheform
Y =X P +X P +V =XP +V (2.7)
1 1 2 2
whereP 1∈Rk1×GandP 2∈Rk2×Garematricesofunknownreduced-formcoefficients. Inthiscase,
thereducedformforyis
y=X p +X p +v (2.8)
1 1 2 2
wherep =g +P b ,p =P b ,andv=u+Vb . WhentheerrorsuandV havemeanzero(though
1 1 2 2
thisassumption mayalsobereplacedbyanother“location assumption”, suchaszeromedians), the
usualnecessary andsufficientcondition foridentification ofthismodelis
rank(P )=G. (2.9)
2
IfP =0, theinstrumentsX areirrelevant,andb iscompletelyunidentified. If1≤rank(P )<G,
2 2 2
b isnotidentifiable,butsomelinearcombinations oftheelementsofb areidentifiable [seeDufour
and Hsiao (2008) and Doko Tchatoka (2015b)]. If P is close not to have full column rank [e.g.,
2
if some eigenvalues of P ′P are close to zero], some linear combinations of b are ill-determined
2 2
by the data, a situation often called “weak identification” in this type of setup [see Dufour (2003),
AndrewsandStock(2007)].
Westudyhere, from afinite-sample viewpoint, thesizeandpowerproperties oftheexogeneity
testsofthetypeproposed byDurbin(1954), Wu(1973), Hausman(1978), andRevankarandHart-
ley(1973) forassessing theexogeneity ofY in(2.1)-(2.7)when: (a)instruments maybeweak;(b)
[u,V] may not follow a Gaussian distribution [e.g., heavy-tailed distributions which may lack mo-
ments (such as the Cauchy distribution) are allowed]; and (c) the usual reduced-form specification
(2.7) is misspecified, andY follows the more general model (2.2) which allows for omitted instru-
ments,anunspecified nonlinearformandheterogeneity. Toachievethis,weconsiderthefollowing
distributional assumptionsonmodeldisturbances (whereP[·]referstotherelevantprobabilitymea-
sure).
Assumption2.1 CONDITIONAL SCALE MODEL FOR THE STRUCTURAL ERROR DISTRIBUTION.
ForsomefixedvectorainRG,wehave:
u=Va+e, (2.10)
4
e=(e ,...,e )′=s (X¯)e , (2.11)
1 T 1
where s (X¯) is a (possibly random) function of X¯ such that P[s (X¯)6=0|X¯]=1, and the condi-
1 1
tionaldistribution ofe givenX¯ iscompletelyspecified.
Assumption2.2 CONDITIONAL MUTUAL INDEPENDENCE OF e ANDV. V and e are indepen-
dent,conditional onX¯.
Intheaboveassumptions, possibledependence betweenuandV isparameterized bya,whilee
isindependentofV (conditionalonX¯),ands (X¯)isanarbitrary(possiblyrandom)scaleparameter
1
which may depend on X¯ (except for the non-degeneracy condition P[s (X¯)6=0|X¯]=1). So we
1
call a the “endogeneity parameter” of the model. Assumption 2.1 is quite general and allows for
heterogeneityinthedistributionsofthereduced-formdisturbancesV,t=1,...,T.Inparticular,the
t
rows ofV need not be identically distributed or independent. Further, non-Gaussian distributions
are covered, including heavy-tailed distributions which may lack second moments (such as the
Cauchy distribution). In such cases, s (X¯)2 does not represent a variance. Since the scale factor
1
maybe random, wecan have s (X¯)=s¯(X¯,V,e).Ofcourse, these conditions hold when u=se ,
1
where s is an unknown positive constant and e is independent of X with a completely specified
distribution. Inthiscontext, thestandard Gaussian assumption isobtained bytaking: e ∼N[0,I ].
T
The distributions of e and s may also depend on a subset of X¯, such as X =[X ,X ]. Note also
1 1 2
theparameteraisnotpresumedtobeidentifiable, andemaynotbeindependent ofV –thoughthis
wouldbeareasonable additional assumptiontoconsider inthepresentcontext.
In this context, we consider the hypothesis thatY can be treated as independent of u in (2.1),
deemed the (strict) exogeneity of Y with respect to u, so no simultaneity bias would show up if
(2.1)isestimated byleastsquares. UndertheAssumptions 2.1and2.2,a=0isclearly asufficient
conditionforuandetobeindependent. Further,assoonasV hasfullcolumnrankwithprobability
one,a=0isalsonecessary forthelatterindependence property. Thisleadsonetotest:
H : a=0. (2.12)
0
We stress here that “exogeneity” may depend on a set of conditioning variables (X¯), though of
course we can have cases where it does not depend on X¯ or holds unconditionally. The setup we
consider inthispaperallowsforbothpossibilities.
Beforewemovetodescribetestsofexogeneity,itwillbeusefultostudyhowH canbereinter-
0
preted in the more familiar language of covariance hypotheses, provided standard second-moment
assumptions aremade.
Assumption2.3 HOMOSKEDASTICITY. ThevectorsUt =[ut,Vt′]′,t=1,...,T,havezeromeans
andthesame(finite) nonsingular covariance matrix:
s 2 s ′
E[UU′|X¯]=S = u Vu >0, t =1,...,T. (2.13)
t t s S
Vu V
(cid:20) (cid:21)
wheres 2,s andS maydependonX¯.
u Vu V
5
Assumption2.4 ORTHOGONALITY BETWEEN e AND V. E[Vtet|X¯] = 0, E[et|X¯] = 0 and
E[e2|X¯]=s 2,fort =1,...,T.
t e
Undertheaboveassumptions, thereduced-form disturbances
W =[v ,V′]′=[u +V′b ,V′]′, t =1,...,T, (2.14)
t t t t t t
alsohaveanonsingular covariance matrix(conditional onX¯),
s 2+b ′S b +2b ′s b ′S +s ′
W = u V Vu V Vu . (2.15)
S b +s S
V Vu V
(cid:20) (cid:21)
Inthiscontext,theexogeneity hypothesis ofY canbeformulatedas
H :s =0. (2.16)
0 Vu
Further,
s =S a, s 2=s 2+a′S a=s 2+s ′ S −1s , (2.17)
Vu V u e V e Vu V Vu
so s = 0 ⇔ a = 0, and the exogeneity of Y can be assessed by testing whether a = 0. Note,
Vu
however,thatAssumptions 2.3and2.4willnotbeneededfortheresultspresented inthispaper.
Inordertostudy thepowerofexogeneity tests, itwillbeuseful toconsider thefollowingsepa-
rabilityassumptions.
Assumption2.5 ENDOGENEITY-PARAMETER DISTRIBUTIONAL SEPARABILITY. P¯ is not re-
stricted by a, and the conditional distribution of [V,e] given X¯ does not depend on the parameter
a.
Assumption2.6 REDUCED-FORM LINEAR SEPARABILITY FORY. Y satisfies theequation
Y =g(X ,X ,X ,P¯)+V. (2.18)
1 2 3
Assumption 2.5 means that the distributions ofV and e do not depend on the endogeneity pa-
rameter a, while Assumption 2.6 means thatV can be linearly separated from g(X ,X ,X ,P¯) in
1 2 3
(2.2).
3. Exogeneity tests
We consider the four statistics proposed by Wu (1973) [T ,l =1,2,3,4], the statistic proposed by
l
Hausman (1978) [H ] as well as some variants [H ,H ] occasionally considered in the literature
1 2 3
[see, forexample, Hahnetal. (2010)], and the testsuggested byRevankar and Hartley (1973, RH)
[R]. These statistics can be formulated in two alternative ways: (1) as Wald-type statistics for
the difference between the two-stage least squares (2SLS) and the ordinary least squares (OLS)
estimatorsofb inequation(2.1),wheredifferentstatistics areobtained bychangingthecovariance
matrix; or (2) a F-type significance test on the coefficients of an “extended” version of (2.1), so
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