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8/13/2019 Estimation With Valid and Invalid Instruments
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ANNALES DCONOMIE ET DE STATISTIQUE. N 79/80 2005
Estimation with Validand Invalid Instruments1
Jinyong HAHNand Jerry HAUSMAN 23
ABSTRACT. We demonstrate analytically that for the widely used
simultaneous equation model with one jointly endogenous variable andvalid instruments, 2SLS has smaller MSE error, up to second order, thanOLS unless the R2, or the F statistic of the reduced form equation isextremely low. We also consider the relative bias of estimators when theinstruments are invalid, i.e. the instruments are correlated with the stochasticdisturbance. Here, both 2SLS and OLS are biased in finite samples andinconsistent. We investigate conditions under which the approximatefinite sample bias or the MSE of 2SLS is smaller than the correspondingstatistics for the OLS estimator. We again find that 2SLS does better thanOLS under a wide range of conditions, which we characterize as functionsof observable statistics and one unobservable statistic.
Estimation avec instruments valides et invalides
RSUM. Nous dmontrons analytiquement que, pour le modlelargement utilis quations simultanes avec une variable conjointementendogne et des instruments valides, les DMC ont une plus petite
erreur quadratique (MSE), jusquau second ordre, que les MCO moinsque la statistique de R2 ou de Fischer de lquation en forme rduitesoit extrmement faible. Nous examinons aussi le comportement desestimateurs lorsque les instruments ne sont pas valides, cest--direlorsquils sont corrls avec la perturbation stochastique. Dans ce cas, la fois les DMC et les MCO sont biaiss et non convergents distancefinie. Nous recherchons les conditions pour lesquelles le biais distancefinie ou lerreur quadratique moyenne est plus faible pour les DMC quepour les MCO. Nous trouvons encore que les DMC se comportent mieuxque les MCO sous un large ensemble de conditions, caractrises pardes fonctions des statistiques observables et dune statistique non
observable.
J. HAHN: University of California, Los Angeles. J. HAUSMAN: Massachusetts Institute of Technology (MIT).
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26 ANNALES DCONOMIE ET DE STATISTIQUE
Introduction
While 2SLS is the most widely used estimator for simultaneous equation mod-els, OLS my o better in nite smples. Econometricins hve recognize this
possibility, n mny Monte Crlo sties were nertken in the erly yers of
econometrics to ttempt to etermine conition when OLS might o better thn2SLS. Here we emonstrte nlyticlly tht for the wiely se simltneos
eqtion moel with one jointly enogenos vrible n vli instrments, 2SLS
hs smller MSE error, p to secon orer, thn OLS nless theR2, or theFsttisticof the rece form eqtion is extremely low.
We then consier the reltive bis of estimtors when the instrments re invli,
i.e. the instrments re correlte with the stochstic istrbnce. Here, both 2SLSn OLS re bise in nite smples n inconsistent. We investigte conitions
ner which the pproximte nite smple bis or the MSE of 2SLS is smller thn
the corresponing sttistics for the OLS estimtor. We gin n tht 2SLS oesbetter thn OLS ner wie rnge of conitions, which we chrcterize s fnc-tions of observble sttistics n one nobservble sttistic.
We then present metho of sensitivity nlysis, which clcltes the mximl
symptotic bis of 2SLS ner smll violtions of the exclsion restrictions. For
given correltion between invli instrments n the error term, we erive themximl symptotic bis. We emonstrte how the mximl symptotic bis cn
be estimte in prctice.Next, we trn to inference. In the wek instrments sittion the bis in the
2SLS estimtor cretes problem, since it is bise towrs the OLS estimtor,
which is lso bise. The other problem tht rises is tht the estimte stnr
errors of the 2SLS estimtor re often mch too smll to signl the problem ofimprecise estimtes. Here we erive the bis in the estimte stnr errors for the
rst time, which trns ot to cse the problem. This erivtion lso hs implic-tions for the test of over-ientifying restrictions.
We o not srvey the wek instrments litertre. For recent srveys see STOCKet. al.[2002] and HAHNand HAUSMAN[2003].
1 Model specification
We begin with the moel speciction with one right hn sie (RHS) jointly
enogenos vrible so tht the left hn sie (LHS) vrible epens only on
the single jointly enogenos RHS vrible. This moel speciction cconts forother RHS strictly exogenos vribles, which hve been prtille ot of the
speciction. We will ssme tht1
1 Withot loss of generlity we normlize the t sch thty2hs zero men.
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 27
(1.1)
(1.2)
where im (2) =K. Ths, the mtrixzis the mtrix of ll strictly exogenos vri-bles, n eqtion (1.1) is the rece form eqtion fory2with coefcient vector
2. We lso ssme homoscesticity:
(1.3)
We se the following nottion:
We initilly ssme the presence of vli instrments,E[z/n] = 0 n 20.
Throghot this pper, we ssme tht is xe s in BEKKER[1994].
We lso ssme:
CONDITION1:K0 asnsuch that for some0.
BEKKER [1994] introce n lterntive symptotics, where K= O(n). Bekker
symptotics re simpler version of higher (thir) orer symptotic pproxi-
mation2. The pproximtion opte here is simpler version of
secon orer symptotics, which will highlight the role of the secon orer bis invrios instrmentl vrible estimtors.
2 Estimation with valid instruments
In this section, we clclte the secon orer properties of 2SLS n OLS, n
emonstrte nlyticlly tht 2SLS hs smller bis n men sqre error (MSE),
p to secon orer, thn OLS nless theR2
, or theFsttistic of the rece formeqtion is extremely low.
2 See, e.g., HAHNand HAUSMAN[2002].
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28 ANNALES DCONOMIE ET DE STATISTIQUE
We rst chrcterize symptotic properties of 2SLS ner Conition 1. as spe-cil cse of Theorem 3 in Section 3, we obtin tht:
THEOREM1: whereV2SLS= /.
Note tht Theorem 1 preicts tht the symptotic vrince of is
eql to the sl 2SLS rst orer symptotic vrince. Theorem 1 lso preicts
tht the pproximte bis of 2SLS:
(2.1)
whereR2/vr(y2) is the theoreticl vle from the secon (rece form) eq-tion3. as conseqence, we obtin the pproximte men sqre error (MSE) of2SLS:
(2.2)
Note tht both terms in eqtion (2.2) pproch zero s increses with
incresing smple size. The rst term, bis sqre lso pproch zero moreqickly, s expecte, since 2SLS is -consistent.
We now chrcterize the symptotic properties of OLS ner Conition 1. as
specil cse of Theorem 4 in Section 3, we obtin the istribtion for the OLSestimtor:
THEOREM2:
where
Theorem 2 preicts the pproximte bis n pproximte vrince s:
(2.3)
(2.4)
3 Note tht the pproximte bis of 2SLS in eqtion (2.1) is ienticl to the well-known reslt for thesecon orer bis of 2SLS. See, e.g., ROTHENBERG[1983] or HAHNand HAUSMAN[2002, 2002b].
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 29
Ths, the pproximte MSE of OLS is
(2.5)
The inconsistency of OLS is evient from eqtion (2.5) becse while the secon
term goes to zero s nbecomes lrge, the rst term is not fnction of n.
We now compre the pproximte nite smple properties of 2SLS n OLS. We
rst compre bises:
(2.6)
where cn be interprete s the theoreticlF-sttistic from the
rst-stge rece form. Ths, if F1, 2SLS hs less bis. However the OLSvrince is less thn the 2SLS vrince so we compre the MSEs below.
Before leving the bis comprisons, we lso consier wht hppens when we
re close to being nientie so tht where the vector ahas dimen-
sionK. Ths, the rece form coefcients re locl to zero. With
eqtion (2.1) preicts the bis of 2SLS to be
(2.7)
where a'z'za. On the other hn, eqtion (2.3) preicts the pproximte bisfor OLS to be:
(2.8)
Tking the rtio of the bises ner locl to zero symptotics:
(2.9)
From eqtion (2.9), it follows tht the bis of 2SLS is smller thn OLS s long s
Kn, conition which will slly be stise in prctice.We next compre the MSE of 2SLS to the MSE of OLS. It is convenient to intro-
ce normliztion, which will simplify theM2SLSandMOLSexpressions. Withot
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30 ANNALES DCONOMIE ET DE STATISTIQUE
loss of generlity we rescle the nits of vribles
=
= 1 so tht vr(y2) = 1/
(1-R2) n
= .4using this normliztions we n:
(2.10)
(2.11)
and
(2.12)
Which estimtor to se will epen on whether eqtion (2.12) is less thn orgreter thn nity. We cn solve for the criticl vle of 2which cses the
MSE of the 2 estimtors to be eql.5The soltion for this criticl vle hs
remrkbly simple form:
(2.13)
As nbecomes lrge the criticl vle of 2goes to zero. In ny prticlr smpleR2andFcn typiclly be ccrtely estimte from the nbise estimtes of therece form so tht only 2is nknown. While this prmeter vle is typiclly
nknown, the pplie econometricin will often hve goo (a priori) knowlegeof the possible vles of so tht she will be ble to etermine whether the criticl
vle is below the sqre of the correltion coefcient.6As we now demonstrate,
the criticl vle is often so low tht 2SLS will hve lower MSE thn OLS, evenfor sittion with reltively wek instrments or lowFsttistic.
In Tble 1 we clclte the criticl vle of (sing the bsolte vle) for rnge of vles ofR2forKof 5, 10, n 30 n for smple sizes of n= 500 nn= 1,000. Here we n tht ifR2 0.1 tht 2SLS typiclly will hve lower MSE.
Ths, except in the cse of wek instrments, which cn rise when bothR2is lown the nmber of instrments is high, 2SLS is typiclly the preferre estimtor
bse on n pproximte nite smple comprison of MSEs.
4 Strctrl eqtions re homogeneos of egree zero. Since we rescle the vrince of the stochstic
istrbnce we hve to either rescle the coefcients or rescle the nits of the vribles. We optthe ltter convention here, lthogh we cn o either.
5 The correltion prmeter is the key prmeter in simltneos eqtion nlysis becse if it is zerothe OLS estimtor is the nbise Gss-Mrkov estimtor n the rtio of MSEs in eqtion (2.12)equals 1/R2> 1, bt OLS is bise n inconsistent if the prmeter vle of is not zero.
6 The prmeter is lso estimte from the 2SLS estimtion, bt goo estimte my be ifclt tochieve in wek instrment sittion.
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 31
3 Estimation with invalid instruments
up to this point we hve ssme tht the instrments re vli so tht they reorthogonl to the stochstic istrbnce 1. However, the econometricin my not
be certin tht the instrments stisfy the orthogonlity conition. We now con-sier the sittion where the orthogonlity conition on the instrments fils so tht
E[z1/n] 0. We rst consier the tritionl lrge smple bis of 2SLS:
(3.1)
where W=z2. When we compre this expression with the nlogos expressionfor OLS
(3.2)
In generl either estimtor my be preferre on this criterion epening on circm-stnces. The nmertor of eqtion (3.1) wol likely be smller (less correl-tion in the instrment) thn the nmertor of eqtion (3.2), bt the enomintor
of eqtion (3.1) is lwys smller since R2< 1. Inee, if R2 is very smll, theOLS estimtor my o better in terms of inconsistency.
TABLE1
Critical Values of
R2 0.01 0.1 0.2 0.3 0.5 0.7 0.9
K= 5
100 ** 0.3677 0.2323 0.1863 0.1432 0.1210 0.1070
500 ** 0.1423 0.1002 0.0818 0.0634 0.0536 0.0473
1,000 0.3654 0.1002 0.0708 0.0578 0.0448 0.0378 0.0334
K= 10
100 ** ** 0.2601 0.1949 0.1455 0.1220 0.1075
500 ** 0.1445 0.1006 0.0819 0.0634 0.0536 0.0473
1,000 ** 0.1006 0.0708 0.0578 0.0448 0.0378 0.0334
K= 10
100 ** ** ** ** 0.1789 0.1339 0.1135
500 ** 0.1771 0.1050 0.0834 0.0638 0.0538 0.0474
1,000 ** 0.1049 0.0716 0.0581 0.0448 0.0379 0.0334
Notes : ** denotes no critical value of less than 1.0 exists.
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32 ANNALES DCONOMIE ET DE STATISTIQUE
In orer to gin better insight, we opt n symptotic pproximtion simi-lr to the one in the previos section, n investigte conitions ner which thepproximte nite smple bis or the MSE of 2SLS is smller thn the correspon-ing sttistics for the OLS estimtor. We gin n tht 2SLS oes better thn OLS
ner wie rnge of conitions, which we chrcterize s fnctions of observble
sttistics n one nobservble sttistic.
To o symptotic pproximtions we nee to specify the correltion of the instr-ment with the stochstic istrbnce in the strctrl eqtion (1.1). We se loclspeciction:
(3.3)
We ssme tht (e1, 2) is homoscestic n zero men normlly istribte withcovrince mtrix:
Throghot this section, we lso ssme tht is xe.
First, we erive the symptotic istribtion of the 2SLS estimtor with loclly
invli instrments7:
THEOREM3:Under Condition 1,
The rst term in the nmertor of the men rises from filre of the orthogonl-ity conition. The secon term is the sl nite smple bis term n it ecreses
with the smple size. The vrince contines to be V2SLSner instrment invli-ity becse of the locl eprtre in eqtion (3.3) similr to HAUSMAN ([1978],
p. 1256).
We se Theorem 3 to clclte the pproximte bis of the 2SLS estimtor with
invli instrments is:
(3.4)
where we se the previos normliztions n set using
Theorem 3 we n the pproximte MSE of 2SLS to be:
(3.5)
7 See appenix for proof of Theorem 3.
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 33
We now erive the symptotic istribtion of the OLS estimtor with loclly
invli instrments8:
THEOREM4:Under Condition 1,
The istribtion is centere ron the sl OLS bis, s before, n the nmer-tor of the men of the istribtion rises from the instrment invliity. agin,
the vrince contines to be VOLS
ner instrment invliity becse of the locleprtre in eqtion (3.3). using Theorem 4 n the previos normliztions, we
n the pproximte MSE of OLS to be:
(3.6)
The rst term in prentheses is the sl simltneos eqtion bis of OLS tht
oes not ecrese with the smple size.We now compre the bis of 2SLS ner instrment invliity with the bis of
OLS given similr circmstnces. We re-write the bis of OLS sing the norml-iztion:
(3.7)
as before, we tke the rtio of (3.4) n (3.7):
(3.8)
The rtio of the bises is homogeneos of egree zero in the correltion coefcient
, so we cn simplify terms. We plot the rtio of the bises in Figre 1 for the cse
of n= 100 nK= 5 n = 0.1.We n tht the 2SLS bis is less thn the OLS bis if:
(3.9)
8 See appenix for proof of Theorem 4.
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34 ANNALES DCONOMIE ET DE STATISTIQUE
Eqtion (3.9) is very esy to interpret. We clclte criticl in Figre 2, n
note tht it increse qite rpily, so tht the bis of 2SLS with invli instrments
remins less thn the bis of OLS so long sFexcees 1.0 by smll mont. Thestrightforwr reltionship of eqtion (3.9) llows for n esy interprettion on
which the econometricin my well hve some priori knowlege.
Note tht the common empiricl ning tht the 2SLS coefcient is lrger thn
the OLS coefcient cn rise becse of the OLS bis when the instrments re
vli or becse of n improper instrment. Ths, even if the instrment is lmostncorrelte so tht sbstntil bis cn still rise becseR2is oftenqite smll in the wek instrments sittion. Ths, compring eqtion (3.4) to
the bis of OLS in eqtion (3.7), the empiricl ning tht the 2SLS estimte
increses compre to the OLS estimte my inicte tht the instrment is not
orthogonl to the stochstic istrbnce. The reslting bis cn be sbstntil.Inee, it col excee the OLS bis, leing to n increse in the estimte 2SLS
coefcient over the estimte OLS coefcient.
Retrning to the generl sittion n sing the normliztions the rtio of the
MSEs is
(3.10)
No strightforwr conition cn be erive where the rtio is less thn one. In
orer to gin some insight, we clclte the rtio (3.10) for vrios vles ofR2and xing = 0.1,K= 5, n n= 100. The rtio (3.10) is below 1.0 except in thesituation whereR2becomes qite smll (s with wek instrments) and becomessmll (which ecreses the OLS bis).
FIGURE1
Ratio of 2SLS Bias to OLS Bias with Invalid InstrumentsN = 100, K = 5, = 0.1
R2
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 35
In or comprisons of 2SLS with OLS, two sorces of bis rise. The rst sorceof bis is from the se of estimte prmeters, in eqtion (1.2), in forming
the instrments. This sorce of bis isppers s the smple becomes lrge. The
secon sorce of bis is from the se of invli instrments, 0 in eqtion (3.3).This sorce of bis oes not ispper sfciently fst with the smple size to
cse 2SLS to be consistent. an interesting qestion wol be bot how the com-prison of IV to OLS wol chnge if the rst sorce of bis were eliminte. We
cn eliminte this sorce of bis (to secon orer) by sing the Ngr estimtor
We erive the symptotic istribtion of the Ngr estimtor with loclly invliinstruments9:
THEOREM5:Under Condition 1,
9 See appenix for proof of Theorem 5.
FIGURE2
Critical Values for Alphan = 100 and K = 5, 10, 30
R2
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36 ANNALES DCONOMIE ET DE STATISTIQUE
Ths to compre the MSE of the Ngr estimtor to the MSE of the 2SLS estimtor
with invli instrments, we see tht the vrince of the two estimtors is the sme,bt tht the bis iffers s expline bove. However, when we compre the bis
sqre of 2SLS from eqtion (3.4) with the Ngr estimtor we n tht
(3.11)
Eqtion (3.11) cn be less thn or greter thn zero. Ths, we cnnot concle thtsing the Ngr estimtor to compre with OLS wol mke the comprison more
fvorble to n IV estimtor.
4 Sensitivity analysis
CARD[2001] iscsses possible concerns tht the instrments my be invli iniscssing the empiricl litertre tht estimtes the retrn to itionl ection.
The se of instrmentl vribles in this sittion begn with GRILICHES [1977]well known pper. To investigte the possible effect of invli instrments, we
consier the speciction:
(4.1)
Note tht we hve e z to the error which cses the instrments to beinvli.10We erive the mximl symptotic bis for smll violtion of the excl-sion restriction, whereis the correltion betweenz
iand so that2is theR2 of
betweenziand We n the mximl symptotic bis11
to be:
THEOREM6:
Note tht the mximl symptotic bis cn be consistently estimte by
(4.2)
10 IMBENS[2003] consiers the qestion of sensitivity nlysis, bt not in the context of instrmentlvribles.
11 See appenix for proof of Theorem 6.
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 37
The mximl symptotic bis in (4.2) cn be se to conct bon nlysis.
The bon cn often be qite lrge in wek instrment sittions. This bon
cn sometimes conict with the bon proce by MANSKIS[1990, 2003] non-
prmetric pproch, since the Mnski pproch oes not llow for errors invribles.
5 Bias in estimated standard errors
We hve previosly iscsse the bises in the 2SLS estimtor in eqtion (2.1)
n Theorem 1. In the wek instrments sittion this bis my be qite lrge.
a frther problem rises in tht the 2SLS estimtor is bise in the sme irec-tion s the OLS estimtor s eqtion (2.4) n Theorem 2 emonstrte. Ths,
HAUSMAN [1978] speciction type test will be bise towrs not rejecting the
nll hypothesis of lck of orthogonlity between 1and 2in eqtions (1.1) n(1.2). However, nother problem hs been recognize in the wek instrments sit-tion. The estimte stnr errors for the 2SLS estimtor re ownwr bise,
sometimes leing to the mistken inference tht the 2SLS estimte re mch more
precise thn they ctlly re. From nlysis bse on rst orer symptotics the
sl conclsion wol be tht with wek instrments tht the reporte stnrerror of the 2SLS estimtor wol be sfciently lrge to signl the ning tht
so mch ncertinty exists with the estimte tht it wol not be of mch se.
However, reserchers hve fon tht, to the contrry, often the 2SLS estimtor in
the presence of wek instrments les to resonbly smll stnr error. Ths,
the resercher my be nwre of the wek instrments problem. The sorce of
the problem of smll reporte stnr errors of the 2SLS estimtor hs not been
iscsse in the litertre. Here we erive the sorce of the problem n offer
possible pproch to xing it.
The vrince of 2SLS is erive in Theorem 1 n tkes the sl form of
V2SLS= -1where = 'z'z/nis ssme to be xe. Now is not ifclt toestimte since nbise estimte of follow from OLS on eqtion (1.2). Ths,
the downward bias in the estimated 2SLS standard errors must arise from a down-ward biased estimate of
. We now erive the bis. The intition follows from the
fct tht 2SLS is bise towrs the OLS estimtor, which minimizes Thus,
we n tht the bis of the 2SLS estimtor of cretes bis in the 2SLS estimte
of
. We n the bis to be:
THEOREM7:
Note tht the leing term in the bis clcltion of Theorem 5 is 2 times the bis of
the 2SLS estimtor from eqtion (2.1). as either the nmber of instrments grows
or the covrince between the strctrl n rece term stochstic istrbnces
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38 ANNALES DCONOMIE ET DE STATISTIQUE
becomes lrge, the bis in the estimtion of
will lso become lrge. We now
pply the normliztion tht we se bove to n:
(5.1)
The bis cn be qite sbstntil s emonstrte by eqtion (5.1). The nl termin eqtion (4.2) will typiclly be smll so tht it cn be ignore. Eqtion (5.1)
emonstrtes tht the ownwr bis cn be sbstntil; in Monte-Crlo12resultsreporte in Tble 2, we n tht for R2= .01 n = 0.9 tht the men bis of
the 2SLS estimte of the vrince vries from -70% to -80% s K, the number ofinstrments, increses from 5 to 30. Ths, we note tht the bis in the estimtion
even when K= 5 cn be qite lrge. This ning explins the reslt tht when
wek instrments re present, the estimte stnr errors of 2SLS cn pper to
be ner those of OLS n smll enogh to llow the resercher to mke concl-sions bot the likely tre prmeter vle. However, with wek instrments these
conclsions col be erroneos becse of the sbstntil bis in the estimte
standard error of the 2SLS estimator13.We now consier the ning tht the often se test of over ientifying restric-
tions (OId test) rejects too often when wek instrments re present, i.e. thectl size of the test is consierbly lrger thn the nominl size. The OId test
cn be qite importnt since it tests the economic theory emboie in the moel
s iscss by e.g. HAUSMAN[1983]. In the wek instrment sittion it my hve
increse importnce given the sbstntil bis in the 2SLS estimtor n the lrgeMSE tht we clcltion in eqtions (3.4) n (3.5). We my write the OId test
as14:
(5.2)
Wis istribte s chi-sqre with K - 1 egrees of freeom ner conventionl
symptotics. From eqtion (5.2), we see tht ownwr bise of
cn le tosbstntil over-rejection n n pwr bise size of the OId test. Ths, correct-ing for this problem cn hve n importnt effect on test reslts.
12 The Monte-Crlo esign is the sme s in HAHN-HAUSMAN[2002].13 We note recent evelopment on the correctly size conence intervls of , incling KLEIBERGEN
[2002], my be of importnce in etecting these problems. The new conence intervls my besbject to power problem. See, e.g., ANDREWS, MOREIRAand STOCK[2004].
14 See, e.g., HAUSMAN[1983].
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40 ANNALES DCONOMIE ET DE STATISTIQUE
6 Conclusions
We erive secon orer pproximtions for the bis n MSE of 2SLS (n
the Ngr estimtor) with both vli n invli instrments. The erivtion for
invli instrments is new, to the best of or knowlege. We n tht sbstntil
nite smple bis cn occr when wek instrments exist which rises when the
R2of the rece form regression is low, the nmber of instrments is high, or the
correltion between the strctrl n rece form stochstic terms is high.
We then compre the bis n MSE of 2SLS with OLS. The OLS estimtor is
bise n inconsistent, bt its smller vrince my mke it preferble to 2SLS
in wek instrments sittion. We etermine strightforwr n esily checke
conitions ner which 2SLS hs smller bis thn OLS. These bis conitionscrry over, in lrge prt, to the MSE comprisons becse chnges in the bis
term re qite importnt in chnges in the MSE term given typicl smple sizes of
n= 100 or lrger. We n tht 2SLS is generlly the preferre estimtor. However,
the econometricin cn se or formle to check the expecte performnce of
2SLS n OLS in given sittion given some a prioriknowlege bot likely
prmeter vles.
We lso n tht the estimte stnr errors for the 2SLS estimtor re own-wr bise, sometimes leing to the mistken inference tht the 2SLS estimte
re mch more precise thn they ctlly re. Sch bis explins why the ctl
size of the often se test of over ientifying restrictions (OId test) is consierblylrger thn the nominl size.
References
ANDREWSD.W.K., MOREIRAM.J.and STOCKJ.H.(2004). Optiml Invrint Similr Testsfor Instrmentl Vribles Regression , unpublished manuscript.
ANGRISTJ. and KRUEGERA. (1991). does Complsory School attennce affectSchooling n Ernings , Quarterly Journal of Economics, 106, pp. 979-1014.
BEKKERP.A. (1994). alterntive approximtions to the distribtions of Instrmentl
Vrible Estimtors ,Econometrica, 92, pp. 657-681.CARDD.(2001). Estimting the Retrn to Schooling ,Econometrica, pp. 1127-1152.
GRILICHESZ.(1957). Speciction Bis in Estimtes of Proction Fnction , Journalof Farm Economics, 38, pp. 8-20.
GRILICHESZ. (1977). Estimting the Retrns to Schooling: Some EconometricProblems ,Econometrica,45, pp. 1-22.
HAHNJ. and HAUSMANJ. (2002). a New Speciction Test for the Vliity ofInstrmentl Vribles ,Econometrica, 70, pp. 163-189.
HAHNJ.and HAUSMANJ.(2002b). Notes on Bis in Estimtors for Simltneos EqtionMoels ,Economics Letters,75, pp. 237-241.
HAHNJ.and HAUSMANJ.(2003). Wek Instrments: dignosis n Cres in Empiricl
Econometrics ,American Economic Review.HAHNJ., HAUSMANJ., KUIERSTEINERG. (2004). Estimtion with Wek Instrments:
accrcy of Higher Orer Bis n MSE approximtions , Econometrics Journal, 7,pp. 272-306.
HAUSMANJ.A. (1978). Speciction Tests in Econometrics , Econometrica, 46,pp. 1251-1271.
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 41
HAUSMANJ.A.(1983). Speciction n Estimtion of Simltneos Eqtion Moels ,in Z. Griliches n M. Intriligtor (es.),Handbook of Econometrics, Vol. 1, amsterm:North Holln.
HAUSMANJ.(2001). Mismesre Vribles in Econometric anlysis: Problems from
the Right n Problems from the Left ,Journal of Economic Perspectives, 2001.IMBENSG. (2003). Sensivity to Exogeneity assmptions in Progrm Evltion ,
American Economic Review, 93, pp. 126-132.
KLEIBERGENF. (2002). Pivotl Sttistics for Testing Strctrl Prmeters inIV Regression ,Econometrica, 70, pp. 1781-1803.
MANSKIC. (1990). Nonprmetric Bons on Tretment Effects ,American EconomicReview Papers and Proceedings,80, pp. 319-323.
MANSKIC. (2003). Prtil Ientiction of Probbility distribtions , New York:Springer-Verlag.
ROTHENBERGT.J. (1983). asymptotic Properties of Some Estimtors in StrctrlMoels , in Studies in Econometrics, Time Series, and Multivariate Statistics.
STAIGERD.,STOCKJ.H.(1997). IV Regression with Wek Instrments ,Econometrica,65, pp. 557-586.STOCKJ.H., WRIGHTJ., YOGOM. (2002). a Srvey of Wek Instrments n Wek
Ientiction in GMM ,Journal of Business and Economic Statistics, 20, pp. 518-529.
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42 ANNALES DCONOMIE ET DE STATISTIQUE
APPENDIX
Bekker asymptotic distribution of 2SLS, OLS, andnagar under misspecification
Sppose tht
where
Following is the Lemm reproce from HAHNand HAUSMAN[2001]:
LEMMA:Let Assume that and thatis xedat .Let and We then have
where anddenote symmetric3 3 matrices such that
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 43
and
REMARK: The s in the Lemm correspon to the rece form. It wol beconvenient to rewrite the bove with strctrl form prmeters. Becse
we cn see tht
LEMMA:Suppose that Then we have
PROOF:Sppose tht = 0. using the previos Lemm, we obtin
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44 ANNALES DCONOMIE ET DE STATISTIQUE
and
Therefore, sing delt metho, we obtin the following:
where we se the fct tht
Becse we cn see tht
and
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 45
Asymptotic distribution of 2SLS under misspecification
Note that
Bt
so that
It follows that
Asymptotic distribution of OLS under misspecification
Note that
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46 ANNALES DCONOMIE ET DE STATISTIQUE
Bt
so that
It follows that
Asymptotic distribution of nagar undermisspecification
Note that
Bt
so that
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 47
It follows that
Sensitivity analysis
Consier moel with one enogenos regressor where other incle exog-enos vribles re prtille ot. The moel tkes the form where
denote the vilble instrment szi, n write the rst stge regression s
2SLS estimtor is obviosly given by
where
Wht is the property of bif the exclsion restriction is in fct violte? In orer
to implement violtion exclsion restriction, we add a little noise to i, n consier
a new model
where
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48 ANNALES DCONOMIE ET DE STATISTIQUE
Let
and
We wol like to exmine the mximl symptotic bis |b2SLS()| for smll viol-tion of exclsion restriction, i.e., the violtion sch tht the correltion between
and is some small number . We rge tht
provies sch mesre of sensitivity. Here, denotes theR2in the rst stge.
It cn be shown tht
where
Note that
which is mximize when . We therefore focs on the type of violtion schthat = for some sclr . Withot loss of generlity, we will write
Note tht the popltionR2in the regression of * onz, which is eql to the
sqre of the correltion between and is equal to
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 49
and
We cn solve 2s fnction of 2,and obtain
Now, note tht the popltionR2in the rst stge is equal to
which cn be solve for 'as
We therefore obtain
or
We note that cn be pproximte by the empiricl conterprt
Digression: robustness of 2SLS
In generl, we estimte by
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50 ANNALES DCONOMIE ET DE STATISTIQUE
n the conterprt ner smll misspeciction is
so that
Note that
and
Inste of eling with normliztion involving the weight mtrix , it isconvenient to se ssme tht =I. We then hve
and
REMARK:If there is only one instrument, then Therefore, small
inictes tht 2SLS is sensitive to misspeciction.
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 51
REMARK:If there re mltiple components in , n if the rst component of is
smll reltive to other components of , then wol be smll, i.e., 2SLS
is not very sensitive to the violtion of the exclsion restriction in zi,1.
REMARK:Note that
and
Therefore, 2SLS is the most robst estimtor mong the clss of IV estimtors
bA.
Higher order bias of
Or moel is given by
where (i, u
i) is homoscestic n norml. We consier the 2SLS
n the relte estimtor for the vrince of i:
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52 ANNALES DCONOMIE ET DE STATISTIQUE
We hve the following chrcteriztion of
where
LEMMA:
for
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 53
PROOF:Note tht 2SLS is specil cse of the k-clss estimtor
for
and is the eigenvle. Note tht 2SLS correspons to a= 0 n b= 0. Theresult follows from DONALDandNEWEY[1998].
We therefore obtain
Lemm:
PROOF: We hve
Becse
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54 ANNALES DCONOMIE ET DE STATISTIQUE
and
we obtain
Now, note that
We therefore obtain
and
It follows that
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 55
or
assme tht we cn ignore the term in Lemm expnsion in clcltion
of expecttion. We then obtin.
where
This reslt cn be prove in the following wy. From the immeitely preceing
lemma,we hve
Becse expecte vles of the terms in the secon line re zero, it sf-
ces to consier the in the thir line. First, we note that
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56 ANNALES DCONOMIE ET DE STATISTIQUE
from which we obtin
Secon, we note tht
e to symmetry. Thir, we note tht
from which we obtin
We therefore obtain
REMARK:In orer to nerstn this reslt, imgine conter-fctl sittion
where the rst orer symptotic pproximtion for is exct, i.e.,
write
We wol then hve
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ESTIMaTION WITH VaLId aNd INVaLId INSTRuMENTS 57
and
Therefore, our result implies that the approximate mean of is smaller by
than would be expected out of rst order asymptotic approximation.
Remark: or reslt cn be nerstoo from ifferent perspective. Note tht the
pproximte bis of 2SLS is eql to
Roghly speking, 2SLS is bise towr OLS, which minimizes
with respect to b. If the 2SLS is close to the OLS
then we shol expect
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