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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