Exogeneity tests and estimation in IV regressions

Transcription

1 Exogeneity tests and estimation in I regressions Firmin Doko chatoka University of asmania Jean-Marie Dfor McGill University First version: May 2007 Revised: October 2007, November 2008, December 2009 his version: March 200 Comiled: Janary 27, 202, 0:02am he athors thank Atsshi Inoe, Marine Carrasco, Jan KivietandBenoitPerronforseveralseflcomments.his work was sorted by the William Dow Chair in Political Economy (McGill University), the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), thebankofcanada(researchfellowshi),agggenheim Fellowshi, a Konrad-Adenaer Fellowshi (Alexander-von-Hmboldt Fondation, Germany), the Institt de finance mathématiqe de Montréal (IFM2), the Canadian Network of Centres of Excellence [rogram on Mathematics of Information echnology and Comlex Systems (MIACS)], the Natral Sciences and Engineering Research Concil of Canada, the Social Sciences and Hmanities Research ConcilofCanada,theFondsderecherchesrlasociétéetlacltre (Qébec), and the Fonds de recherche sr la natre et les technologies(qébec),andakillamfellowshi(canada Concil for the Arts). School of Economics and Finance, University of asmania, Private Bag 85, Hobart AS 700; el: ; Fax: ; Firmin.dokotchatoka@tas.ed.a, homeage: htt:// William Dow Professor of Economics, McGill University, Centre interniversitaire de recherche en analyse des organisations (CIRANO), and Centre interniversitaire de recherche en économie qantitative (CIREQ).Mailing address: Deartment of Economics, McGill University, Leacock Bilding, Room 59, 855 Sherbrooke Street West, Montréal, Qébec H3A 27, Canada. EL: () ; FAX: () ; jean-marie.dfor@mcgill.ca.Web age: htt://

2 ABSRAC Weak identification is likely to be revalent in many economic models. Wheninstrmentsare weak, the limiting distribtions of standard test statistics - like Stdent, Wald, likelihood ratio and Lagrange mltilier criteria in strctral models - have non-standard distribtions and often deend heavily on nisance arameters. Inference rocedres robst to weak instrments have been develoed. hese robst rocedres however test hyotheses that are secified on strctral arameters. Even thogh robst rocedres solve statistical difficlties related to identification isses, alied researchers may want to first re-test the exogeneity of some regressors before inference on the arameters of interest. In linear I regression, Drbin-W-Hasman (DWH) tests are often sed as re-tests for exogeneity. Unfortnately, these tests rely on the assmtion that model arameters are identified by the available instrments. When identification is deficient or weak, the roerties of DWH tests need to be investigated. Early references that stdy the effects of weak instrments on Hasman-tye tests are not well docmented and sally focs on testing. Not mch is known abot re-test estimators based on DWH tests when I are weak. Inthisaer,werovidealargesamle analysis of the distribtion of DWH and RH tests nder both the nll hyothesis (level) and the alternative hyothesis, with or withot identification. Weshowthatnderthenllhyothesis, sal chi-sqare critical vales are alicable irresective of the resence of weak instrments, in the sense that the asymtotic critical vales obtained nder theidentificationassmtionrovide bonds when identification fails. We characterize a necessary and sfficient condition for DWH and RH tests (with fixed level) to be consistent nder the alternative of endogeneity. he latter condition atomatically holds when the rank condition for identification holds: DWH tests are consistent when identification holds. he consistency condition also holds in a wide range of cases where identification fails. Moreover, we stdy the roerties of re-test estimators where OLS or I is sed deending on the otcome of DWH exogeneity tests. We resenttheoreticalargments sggesting that OLS may be referable to I in many cases where regressorendogeneity maybean isse. We resent simlation evidence indicating that: () over a wide range cases, inclding weak instrments and moderate endogeneity, OLS erforms better than 2SLS [finding similar to Kiviet and Niemczyk (2007)]; (2) retest-estimators based on exogeneity have an excellent overall erformance. Hence, the recommendation of Gggenberger (2008) to abandon the ractice of retesting may go too far. We illstrate or theoretical reslts throgh twoemiricalalications:therelation between trade and economic growth and the widely stdied roblem of retrns to edcation. We find that exogeneity tests cannot reject the exogeneity of schooling, i.e. the I are ossibly weak in this model [Bond (995)]. However, trade share is endogenos, sggesting that the I are not too oor as showed by Dfor and aamoti (2007). Key words: Exogeneitytests;weakinstrments;retest-estimators;bias;meansqareserrors. JEL classification: C3; C2; C5; C52. i

3 . Introdction he literatre on weak instrments in linear strctral models focses on roosing statistical rocedres which are robst to instrment qality, see Anderson andrbin(949, AR-test), Dfor (997, 2003), Staiger and Stock (997), Wang and Zivot (998), Kleibergen (2002, K-test), Moreira (2003, CLR-test), Dfor (2005, 2007), Dfor and Jasiak (200), Stock, Wright and Yogo (2002), Hall, Rdebsch and Wilcox (996), Hall and Peixe (2003), Donald and Newey (200), Doko and Dfor (2008). Weak instrment robst statistics however, test hyotheses that are secified on the arameters of interest. Althogh robst rocedres revent statisticaldifficltiesrelatedtoiden- tification, alied researchers may need to check whether some regressors are exogenos before rnning inference on the arameters of interest (retesting). Exogeneity tests of the tye roosed by Drbin (954), W (973), Hasman (978), Revankar and Hartley (973) are commonly sed for this rose. Unfortnately, sch tests rely on the assmtion that model arameters are identified by the available instrments. When identification is weak, the roerties (size and ower) of exogeneity tests need to be investigated. he literatre related to weak instrment roblems on exogeneity tests is not well docmented. Early references inclde Gggenberger (200) and Hahn, Ham and Moon (200). Gggenberger (200) investigates the asymtotic size roerties of a two-stage test, where in the first stage a Hasman test is ndertaken as a retest for exogeneity of a regressor. His major finding is that the two-stage test based on DWH-tye test have arbitrary size even in large samles. In fact, when the endogeneity between the strctral and redced form errors is local to zero of order /2, where denotes the samle size, the Hasman retest statistic converges to a noncentral chi-sqared distribtion. he non centrality arameter is small when the strength of the instrments is small. In this sitation, the Hasman re-test has low ower against local deviations of the retest nll hyothesis and conseqently, with high robability, OLS-based inference is done in the second stage. However, the second stage OLS based t-statistic often takesonverylargevales ndersch local deviations. he latter cases size distortions in the two-stage test.? consider the roblem of testing the exogeneity of a sbset of exclded instrments. hey divide the exclded instrments from the strctral eqation into two comonents. he first comonent is weak bt exogenos, while the second is strong bt otentially invalid. hey thentestthevalidityofthestrongcomonent sing a modified Hasman-tye test. he test statistic roosed is valid desite the resence of the weak comonent. However, neither Gggenberger (200) nor? rovide a formal characterization of DWH-tye tests in resence of weak instrments. Frthermore, the isses related to estimation are not addressed by these aers. For examle, how do re-test estimators based on exogeneity tests behave when identification is deficient or weak? In articlar, do alternative re-test estimators based on exogeneity tests better erform (in term of bias and mean sqare error) than sal I estimators when instrments are weak? Doko and Dfor (20) rovide a finite-samle characterization of the distribtion of DWHtests nder the nll hyothesis (level) and the alternative hyotheses (ower). However, the isses

4 related to estimation and the large-samle behavior of the tests are not addressed. In this aer, we consider the roblem of testing the exogeneity of inclded regressors in the strctral eqation. his roblem is qite different and more comlex than testing orthogonality restrictions of exclded instrments, as done by?. Wefocsonlarge-samleandstdy thebehavior of DWH- and RH-tye tests inclding when identification is deficient or weak (weak instrments). Frthermore, we analyze the roerties (bias and mean sqares errors) of re-test estimators based on exogeneity tests. First, we characterize the asymtotic distribtion of DWH and RH tests nder the nll hyothesis (level) and the alternative hyothesis (ower). We show that DWH- and RH-tests are asymtotically robst to weak instrments (level is controlled) and werovideanecessaryandsfficientcondition nder which the tests have no ower [similar to Doko and Dfor(20)andGggenberger (200)]. We find that exogeneity tests have no ower when all instrments are weak. Moreover, ower may exist as soon as we have one strong instrment (artial identification). Second, we characterize the asymtotic bias and mean sqare error of OLS, 2SLS and re-test estimators based on DWH and RH tests. We find that: () when identification is deficient or weak (weak instrments) and endogeneity is local to zero [i.e. the endogeneity between the strctral and redced form errors converges to zero at rate ( 2 )asthesamlesizegrows],olserforms (in terms of bias and mean sqare error) better than 2SLS [finding similar to Kiviet and Niemczyk (2007)]; (2) retest-estimators based on exogeneity tests have an excellent overall erformance comared with OLS and 2SLS estimators. herefore, the recommendation of Gggenberger (200) to abandon the ractice of retesting may go too far. We resent two Monte Carlo exeriments which confirm or theoretical reslts. he first examines the roerties (size and ower) of DWH and RH exogeneity tests. he second stdies the bias and mean sqare error of OLS, 2SLS and re-test estimators based on exogeneity tests. Or reslts indicate that: () over a wide range cases, inclding weak instrments and moderate endogeneity, ordinary least sqares estimator (OLS) erforms better than sal2slsestimator;(2) re-testesti- mators based onexogeneity tests have anexcellent overall erformance, hence more referable than OLS and I estimators. We illstrate or theoretical reslts throgh two emiricalalications: therelationbetween trade and economic growth [see, Dfor and aamoti (2007), Irwin and ervio (2002), Frankel and Romer (999), Harrison (996), Mankiw, Romer and Weil (992)] and the widely stdied roblem of retrns to edcation [Dfor and aamoti (2007), Angrist and Kreger (99), Angrist and Kreger (995), Angrist and al. (999), Mankiw et al. (992)]. he reslts indicate that exogeneity tests cannot reject the exogeneity of schooling, which sggest that instrments are ossibly weak in this model [Bond, Jaeger and Baker (995)]. However, tradeshare isendogenos,i.e., instrments are not too oor as showed in the literatre [Dfor and aamoti (2007)]. he aer is organized as follows. Section 2 formlates the model stdied. Section 3 stdies the asymtotic behavior of the tests when identification is strong or deficient (lack of identification). Section 4 examines their behavior when identification is weak (weak I). Section 5 resents the re-test estimators based on exogeneity tests and characterizes their asymtotic behavior, inclding 2

5 when identification is deficient or weak. Section 6 resents two Monte Carlo exeriments (i) the roerties (size and ower) of exogeneity; and (ii) the erformance (bias and mean sqares errors MSE) of re-test estimators. Section 7 illstrates or theoretical reslts throgh two imortant alications. We conclde in Section 8 and roofs are resented in the Aendix. 2. Framework We consider the linear strctral model: y = Yβ+ Z γ +, (2.) Y = Z Π + Z 2 Π 2 +, (2.2) where y R is a deendent variable, Y R G is a matrix of (ossibly) endogenos exlanatory variables (G ) Z R k is a matrix of exogenos variables, Z 2 R k 2 is a matrix of Is, =(,..., ) R and =[v,...,v ] R G are distrbances, β R G,γ R k, Π R k G and Π 2 R k2 G nknown coefficients. Let Z =[Z : Z 2 ] and k = k + k 2. We assme that the instrment matrix Z has fll-colmn rank and k 2 G. he sal necessary and sfficient condition for identification of this model is rank(π 2 )=G. If rank(π 2 ) <G, βis not identified and the instrments are weak. However, some comonents of β may be identified (artial identification) even if this rank condition fails. We also sose that can be regressed on yielding the following eqation: = a+ ε (2.3) where a R G is a vector of nknown coefficients, ε has mean zero, variance σ 2 ε with. Let and ncorrelated M = M Z = I Z(Z Z) Z, Z =[Z,Z 2 ], M = M Z = I Z (Z Z ) Z. (2.4) hen, M M can be exressed as M M = M Z 2 (Z 2 M Z 2 ) Z 2 M = ( ) Z 2, (2.5) where = M Z 2 Z. Let Z =[Z, ]. If we relace Z by Z in (2.27) - (2.29), then the statistics H i (i =, 2, 3), l (l =, 2, 3, 4) and RH do not change. herefore, the orthogalization between Z and has no imact on or reslts. o simlify the notations, will be sed instead of Z 2 [see for examle, eqation (2.2)]. We make the following generic assmtions on the asymtotic behavior of model variables [where B>0for a matrix B means that B is ositive definite (.d.), and refers to limits as 3

6 ]: [ ε Z Z Σ Z = [ ] ] [ ] Σ 0 ε 0 σ 2 > 0, (2.6) ε [ ] Z ε 0, (2.7) [ ] Σ Z Σ Z 2 Z > 0, (2.8) Σ Z2 Z Σ Z2 ε L S ε, Z [,, ε] L [S,S,S ε ], (2.9) vec[s,s,s ε,s ε ] N [0,Σ S ],S ε and S are ncorrelated, (2.0) [ ] [ ] [ ] S = S S 2, S = S S 2, S ε = S ε S 2ε, (2.) S N [ 0,σ 2 Σ Z ], S2 N [ 0,σ 2 Σ Z2 ], (2.2) S ε N [ 0,σ 2 ε Σ Z ], S2ε N [ 0,σ 2 ε Σ Z 2 ], (2.3) S i is a k i random vector, S i is a k i G random matrix matrix (i =, 2), Σ is G G ositive definite matrix, and σ 2 > 0. From the above assmtions, we have Z 0, [ ] [ ] Σ = [ σ 2 δ δ Σ ] > 0, (2.4) where δ = Σ a, σ 2 = a Σ a + σ 2 ε,s = S a + S ε = S (Σ δ)+s ε. (2.5) Frthermore, Z [ ε [ ] 0, Z Z Σ Z = Σ Z 0 0 Σ Z2 ] > 0, (2.6) Z [,, ε] L [ S, S, S ε ], (2.7) vec[ S, S, S ε,s ε ] N [0,Σ S], S ε and S are ncorrelated, (2.8) [ ] [ ] [ ] S S S ε S =, S =, S ε =, (2.9) S 2 S 2 S 2ε 4

7 S 2 N [ 0,σ 2 Σ ], S2ε N [ 0,σ 2 εσ ], (2.20) where Σ Z2 = Σ Z2 Σ Z2 Z Σ Z Σ Z 2 Z. (2.2) Under assmtions (2.6) - (??), lim ˆβ = β +(Π 2Σ Π 2 + Σ ) δ (2.22) and ˆβ is consistent if and only if δ =0, irresective of the rank of Π 2. In articlar, nder local alternative considered by Gggenberger (200) [δ = δ 0 / 0 as ], ˆβ is consistent. However, [ Y β = β +[Y (M M)Y ] Y ] (M M)Y Y (M M) (M M) = β +, (2.23) so, rovided that the identification condition rank(π 2 )=G holds, Y (M M)Y Π 2Σ Z 2 Π 2 > 0, Y (M M) 0, (2.24) and lim β = β. (2.25) Nevertheless, β does not generally converge to β when rank(π 2 ) <G. his aer focses on both testing and estimation. First, we investigate the large-samle roerties of DWH and RH exogeneity tests, inclding when identification is deficient or weak (weak instrments). Second, we stdy the erformance (bias and mean sqares errors- MSE) of re-test estimators based on DWH and RH exogeneity tests, allowing for theresenceofweakinstrments. From (2.4) - (2.5), the exogeneity assmtion of Y can be exressed as H 0 : δ =0 H a : a =0. (2.26) We consider the Drbin-W-Hasman (DWH) test statistics, namely three versions of Hasmantye statistics [H i,i =, 2, 3], the for statistics roosed by W (973) [ l,l =, 2, 3, 4] and Revankar and Hartley (973, RH) test statistic. hese statistics are defined by eqations (2.27) - (2.29) below: l = κ l ( β ˆβ) Σ l ( β ˆβ), l =, 2, 3, 4; (2.27) H i = ( β ˆβ) ˆΣ i ( β ˆβ), i =, 2, 3, (2.28) 5

8 RH = κ R y ˆΣR y, (2.29) where ˆβ =(Y M Y ) Y M y is the ordinary least sqares (OLS) estimator of β, β =[Y (M M)Y ] Y (M M)y is the two-stage least sqares (2SLS) estimator of β, κ =(k 2 G)/G, κ 2 =( k 2G)/G, κ 3 = κ 4 = k G,κ R =( k k 2 G)/k 2, and Σ = σ 2 ˆ, Σ2 = σ 2 2 ˆ, Σ3 = σ 2 ˆ, Σ4 =ˆσ 2 ˆ, (2.30) ˆΣ = σ 2 ˆΩ I ˆσ2 ˆΩ LS, ˆΣ2 = σ 2 ˆ, ˆΣ3 =ˆσ 2 ˆ, (2.3) ˆΣ R = ˆσ 2 D Z 2 (Z 2D Z 2 ) Z 2D, (2.32) R ˆΩ I = Y (M M)Y, ˆΩLS = Y M Y, (2.33) ˆ = ˆΩ I ˆΩ LS, D = M M M Y, (2.34) σ 2 =(y Y β) M (y Y β)/, ˆσ 2 =(y Y ˆβ) M (y Y ˆβ)/, (2.35) σ 2 =(y Y β) (M M)(y Y β)/ = σ 2 σ 2 e, (2.36) σ 2 2 =ˆσ 2 ( β ˆβ) ˆ ( β ˆβ) =ˆσ 2 σ 2 ( β ˆβ) ˆΣ 2 ( β ˆβ), (2.37) σ 2 e =(y Y β) M(y Y β)/, ˆσ 2 R = ym X y /, (2.38) M M Y = I M Y (Y M Y ) Y M, (2.39) Note that ˆσ 2 is the OLS-based estimator of σ 2, σ 2 is the sal 2SLS-based estimator of σ 2 (both withot correction for degrees of freedom), while σ 2, σ2 2 and ˆσ2 R may be interreted as alternative I-based scaling factors. he link between W -tests and Hasman H-tests and the regression formla of these tests has been given in Doko and Dfor (20). For examle, we can observe that Σ3 = ˆΣ 2 and Σ 4 = ˆΣ 3, so 3 =(κ 3 / )H 2 and 4 =(κ 4 / )H 3. Since κ 3 / = κ 4 / as +, 3 is asymtotically eqivalent with H 2, and 4 is asymtotically eqivalent with H 3. Finite-samle distribtions for all exogeneity test statistics with ossibly weak Is and non Gassian errors are available in Doko and Dfor (20). We distingish two sets: () Π 2 = Π2 0 is fixed; and (2) Π 2 = Π2 0/, where Π2 0 =0is allowed (weak instrments). Section 3 below characterizes the limiting distribtions of the statistics nder the nll hyothesis (δ =0)andthealternativehyothesis(δ 0)whenΠ 2 is fixed (i.e. does not deend on the samle size). 3. Asymtotic behavior of exogeneity tests In this section, we characterize the asymtotic behavior of thestatisticsnderthenll(δ =0) and the alternative hyotheses (δ 0)whenarametersarefixed,sotheydonotdeendonthe samle size. We distingish two cases for the redced form arameters Π 2 : (i) Π 2 = Π2 0, with rank(π2 0) = G (strong identification); and (ii) Π 2 = Π2 0, with rank(π0 2 ) < G (artial 6

9 identification). o recover artial identification set, it will be sefl to arameterize the model as in Choi and Phillis (992): y = Y β + Y 2 β 2 + Z γ +, (3.) Y = Z Π + Z 2 Π 2 + 2, (3.2) Y 2 = Z Π , (3.3) where Π = Π S, Π 2 = Π S 2,Π 2 = Π 2 S, (3.4) Π 22 = Π 2 S 2 =0, β = S β, β 2 = S 2 β, (3.5) Y = Y S, Y 2 = Y S 2, 2 = S, 22 = S 2 (3.6) and S =[S, S 2 ] O(G), O(G) denotes the orthogonal gro of G G matrices, S 2 : G G 2 sans the nll sace of Π 2, S : G G,β : G and β 2 : G 2. he necessary and sfficient condition for identification of β is rank(π 2 )=G, (3.7) where Π 2 is a k 2 G. his can be seen easily by considering the redced form for model (3.)- (3.6) y = Z π + Z 2 π 2 + v (3.8) where π = Π β + Π 2 β 2 + γ, π 2 = Π 2 β, and v = + 2 β + 2 β 2. So, β is identified if and only if rank(π 2 )=G. It is imortant to observe that β and β 2 are linear combinations of the original coefficient β. he original coefficient β is recovered by the eqation β = S β + S 2 β 2. (3.9) Eqation (3.9) can then be sed to find the effect of artial identification on the entire vector β. Of corse, if rank(π 2 )=G (strong identification), we have S 2 β 2 =0and S = S = I G. Also, if rank(π 2 )=0(comlete non identification), we have S β =0 and S = S 2 = I G. So, the above arametrization incldes strong identification and comlete non identification sets as secial cases. We assme that β is identified bt β 2 may not (artial identification), i.e. rank(π 2 )=G, rank(π 2 ) G 2. (3.0) In articlar, if rank(π 2 )=0,β 2 is not identified at all. Note that assmtion (3.0) does not constitte a restriction of the model. If assmtion (3.0) fails, either the model is identified or 7

10 absoltely not. Both sets are secial cases of (3.)-(3.6) and will be recovered by or reslts. where From the above arametrization, the 2SLS estimator of β and β 2 are defined by β = (Y EY ) Y Ey, β2 =(Y 2JY 2 ) Y 2Jy, (3.) E = M M (M M)Y 2 [Y 2 (M M)Y 2 ] Y 2 (M M), J = M M (M M)Y [Y (M M)Y ] Y (M M). (3.2) hroghot the aer, the following definitions and notations will be sed: σ 2 = σ 2 + S 2Σ (Σ Π 0 + S 2 )Ψ Σ Ψ (Σ Π 0 + S 2 ) Σ S2, (3.3) Σ A Σ A ( S 2 )=Σ (Σ Z2 Π 0 + S 2 )Ψ (Σ Π 0 + S 2 ) Σ, (3.4) Σ = Σ Z2 Σ Z2 Π 0 2 (Π0 2 Σ Π Σ ) Π 0 2 Σ, (3.5) Ψ =(Σ Π 0 + S 2 ) Σ (Σ Π 0 + S 2 ), (3.6) =(Σ Π 0 + S 2 )Ψ (Σ Π 0 + S 2 ), = I k2 Σ /2 Σ /2, (3.7) λ = σ 2 ε σ 2 = σ2 2 = σ2,σ2 3 = σ2 ε, σ2 2 = σ2 4 = σ2 ε, σ2 3 = σ2, (3.8) σ 2 =(Σ /2 S2 a + Σ /2 S 2ε ) (Σ /2 S2 a + Σ /2 S2ε ), (3.9) σ 2 = σ2 2δ Ψ (Σ Π 0 + S 2 ) Σ S2 + S 2Σ (Σ Π 0 + S 2 )Ψ Σ Ψ (Σ Π 0 + S 2 ) Σ S2, (3.20) a S 2 Σ /2 µ = σ 2 ε a Π 0 Π 0 a = δ Σ Π 0 Π 0 Σ δ, (3.2) Σ /2 S2 a = σ 2 a S 2 (Σ Σ Z 2 Z Σ ) S 2 a, (3.22) 2 ε Σ 0 A = Σ S2 ( S 2 Σ S2 ) S 2 Σ, (3.23) σ 2 = σ 2 + N BS 2(Π 0 2 Σ Π Σ ) N B S 2, (3.24) σ 2 0 = σ2 ε + S 2ε Σ S2 ( S 2 Σ S2 ) Σ ( S 2 Σ S2 ) S 2 Σ S2ε σ 2 ε. (3.25) Finally, for a random variable ζ whose distribtion deends on the samle size, the notation ζ L + means that P [ζ >x] as,foranyx. We will now characterize the behavior of the tests nder the nll hyothesis H 0 (section 3.) and the alternative (section 3.2). 3.. Asymtotic distribtions nder the nll hyothesis his sbsection describes the asymtotic behavior of DWH and RH tests nder the nll hyothesis δ =0, inclding when identification is deficient. heorem 3. below shows that all exogeneity tests are valid (level is controlled) even if arameters are not identifiable. 8

11 heorem 3. ASYMPOIC DISRIBUIONS UNDER H 0. Sose the assmtions (2.) - (2.3) and (2.6) - (2.3) hold, and let δ =0. If rank(π2 0 )=G, then If rank(π2 0 ) G, then H i L χ 2 (G), i=, 2, 3, (3.26) L F (G, k2 G), 2 L G χ2 (G), l L χ 2 (G), l=3, 4, (3.27) RH L k 2 χ 2 (k 2 ). (3.28) L H i σ 2 N B S 2 AS 2N B χ 2 (G), i=, 2, H 3 L χ 2 (G), (3.29) L F (G, k2 G), 2 L G χ2 (G), L 4 χ 2 L (G), 3 σ 2 N B S 2 AS 2N B χ 2 (G), (3.30) RH L k 2 χ 2 (k 2 ), (3.3) where N B = S 2a + B S 2 S 2 [Σ Π 2 (Π 2Σ Π 2 ) Π 2] S 2ε N B S2 N [ S 2a, σ 2 εb ], B = S 2 S 2 [Σ Π 2 (Π 2 Σ Π 2 ) Π 2 ] S 2 S 2, A = S 2 BS 2 = S 2 [Σ Π 2 (Π 2Σ Π 2 ) Π 2] S 2, σ 2 is defined by (3.24), a= Σ δ and S 2 is defined in (3.6) - (3.9). In the above theorem, since δ = 0 if and only if a = 0,wefirstnotethatN B S 2 N [ 0,σ 2 ε B ]. Second, when identification is strong, the asymtotic nll distribtion of all exogeneity tests is free of nisance arameters (as exected). When identification fails, the asymtotic nll distribtion of, 2, 4 and H 3 is still ivotal. However, the nll distribtion of 3, H and H 2 is asymtotically bonded by a central chi-sqare with G degrees of freedom. Overall, sal chisqare critical vales are alicable irresective of the resence weak instrments, in the sense that the asymtotic critical vales obtained nder the identification assmtion rovide bonds when identification fails [similar to Doko and Dfor (20)]. We now stdy the roerties of the tests nder the alternative hyothesis δ 0. 9

12 3.2. Asymtotic ower We distingish two cases for the characterization of the ower of the tests. (i) he arameter reresenting the level of endogeneity δ is fixed and different from zero; (ii) the endogeneity is local to zero, i.e, δ converges to zero at rate 2 heorem 3.2 below resents the reslts for δ fixed. as the samle size increases [δ = δ 0 /, δ 0 is given]. heorem 3.2 NECESSARY AND SUFFICIEN CONDIION FOR CONSISENCY. Sose the assmtions (2.) - (2.3) and (2.6) - (2.3) hold. If Π 2 = Π2 0 is fixed, the necessary and sfficient conditions nder which DWH and RH exogeneity tests are consistent is Π2 0 a 0, where a = Σ δ. More recisely, for i =, 2, 3 and l =, 2, 3, 4, if and only if Π 0 2 a 0. H i L +, l L +, RH L +, (3.32) heorem 3.2 above rovides the necessary and sfficient condition for consistency of all DWH and RH exogeneity tests when Π 2 is fixed. he reslt shows that exogeneity tests can detect an exogeneity roblem even if not all model arameters are identified, rovided artial identification holds. In articlar, we have the following reslt when model arameters are identified(strong instrments). heorem 3.3 CONSISENCY OF EXOGENEIY ESS. Sose the assmtions (2.) - (2.3) and (2.6) - (2.3) hold. If rank(π2 0 )=G, thenalldwhandrhexogeneitytestsareconsistent. Clearly, exogeneity tests may be inconsistent only when identification is deficient. When identification is strong, the tests always detect an endogeneity roblem. We can now show the following reslt concerning the asymtotic behavior of the tests when Π 0 2 a =0. Corollary 3.4 ASYMPOIC POWER WHEN Π 2 a =0. Sosetheassmtions (2.) - (2.3) and (2.6) - (2.3) hold and let Π 2 = Π2 0 fixed. If Π 2a =0, and rank(π2 0 )=G,then H i L χ 2 (G), i=, 2, 3, (3.33) L F (G, k2 G), 2 L G χ2 (G), l L χ 2 (G), l=3, 4, (3.34) If Π 2 a =0, and rank(π 0 2 ) G,then RH L k 2 χ 2 (k 2 ). (3.35) L H i σ 2 N BS 2AS 2 N B χ 2 L (G), i=, 2, H 3 χ 2 (G), (3.36) L F (G, k2 G), 2 L G χ2 (G), 0

13 L 4 χ 2 L (G), 3 σ 2 N B S 2 AS 2N B χ 2 (G), (3.37) RH L k 2 χ 2 (k 2 ), (3.38) where σ 2, N B, S 2 and A are defined in heorem 3.. When rank(π2 0)=G, Π0 2 a =0if and only if δ =0. Hence, the nll hyothesis is satisfied. Since identification is strong, all DWH and RH statistics are ivotal. However, when rank(π2 0) G, Π2 0 a =0does not entails that δ =0. he reslts of the above corollary indicate that when identification is deficient and Π2 0 a =0, the asymtotic distribtion of the statistics is the same nder the nll hyothesis (δ =0) andthealternativehyothesis(δ 0). Conseqently, exogeneity tests have no asymtotic ower in this case. We now characterize the asymtotic distribtions of the statistics tests when the endogeneity is local to zero (δ = δ 0 / )andrank(π2 0 )=G (strong identification). he reslts are resented in the following theorem. heorem 3.5 ASYMPOIC POWER WHEN ENDOGENEIY IS LOCAL O ZERO. Sose that the assmtions (2.) - (2.3) and (2.6) - (2.3) hold, and let δ = δ 0 /. We have: H i L χ 2 (G, µ δ0 ),i=, 2, 3, (3.39) L L F (G, k2 G; µ δ0 ), 2 G χ2 L (G, µ δ0 ), l χ 2 (G, µ δ0 ),l=3, 4, (3.40) RH L k 2 χ 2 (k 2,ν δ0 ), (3.4) if rank(π2 0 )=G, where and µ δ0 = σ 2 δ 0(Π2 0 Σ Z2 Π2 0 + Σ ) Π (Π0 2 Σ Z2 Π2 0 + Σ ) δ 0, ν δ0 = σ 2 δ 0 (Π0 2 Σ Π2 0 + Σ ) Π 0 2 Σ Σ Σ Z2 Π 0 2 (Π0 2 Σ Π Σ ) δ 0 (3.42) L H i σ 2 N B S 2 AS 2N B χ 2 L (G), i=, 2, H 3 χ 2 (G), (3.43) L F (G, k2 G), 2 L G χ2 (G), L 4 χ 2 L (G), 3 σ 2 N B S 2 AS 2N B χ 2 (G), (3.44)

14 RH L k 2 χ 2 (k 2 ), (3.45) where σ 2, N B, S 2 and A are defined in heorem 3.. First, wenote that when identification is strong, all exogeneity tests have non zero ower against local alternatives. However, the tests are not consistent whenever δ = δ 0 / 0, as +. If δ 0 0, the distribtions of all exogeneity tests are non central chi-sqares, where the non centrality arameters are given in (3.42). Second, when identification is deficient, the distribtion of the tests remain the same as when δ =0. In this case, all tests have no ower against local alternatives. So, OLS rocedre is sed with a high robability in the second stageifonesesatwo-staget-test based on a DWH or RH re-tests. Unlike Gggenberger (200), wewillseeinsection5thatthisis agoodnewintheviewointofestimation.infact,whenidentification is deficient and endogeneity local to zero, OLS estimator is referable to 2SLS. Since re-test estimators behave like OLS in this case, they are also referable to 2SLS. Clearly, the ractice ofre-testingsholdnotbeabandoned, as recommended by Gggenberger (200). We now focs on weak instrments set. 4. Asymtotic behavior of exogeneity tests when I are asymtotically weak In this section, we focs on weak instrments set and characterize the behavior of DWH and RH tests nder the nll hyothesis (δ =0) andthealternativehyothesis(δ 0). Weak instrments are characterized as in Staiger and Stock (997), i.e. Π 2 = Π2 0/ where Π2 0 is a k 2 G constant matrix and Π2 0 =0is allowed. he sbsection 4. stdies the roerties of the tests nder the nll hyothesis. 4.. Asymtotic distribtions nder the nll hyothesis Following Staiger and Stock (997), weak instrments are characterized by the local to zero condition for the redced form matrix Π 2 : Π 2 = Π2 0 /, (4.) where Π2 0 is a k 2 G constant matrix and Π2 0 =0is allowed. heorem 4. below shows that all exogeneity are valid when instrments are weak. heorem 4. ASYMPOIC DISRIBUION UNDER H 0 WHEN ASYMPOICALLY I ARE WEAK. (Π 0 2 Sose that the assmtions (2.) - (2.3) and (2.6) - (2.3) hold. If Π 2 = Π 0 2 / =0is allowed), thennderthenllhyothesisδ =0, all DWH and RH tests are valid (level 2

15 is controlled). In articlar, we have L H i S σ 2 2 Σ A S 2 χ 2 L (G), i=, 2, H 3 χ 2 (G), (4.2) L L F (G, k2 G), 2 G χ2 L (G), 4 χ 2 L (G), 3 S σ 2 2 Σ A S 2 χ 2 (G), (4.3) RH L k 2 χ 2 (k 2 ), (4.4) S 2 is defined in (2.7) - (2.20), σ 2 and Σ A are defined in (3.3) - (3.25). We observe that when identification is weak (weak Is), the statistics, 2, 4 and H 3 are asymtotically ivotal nder the nll hyothesis (δ =0). However, the asymtotic distribtions of 3, H and H 2 deend on model arameters, bt are bonded by a central chi sqare with G degrees of freedom. Hence, 3, H and H 2 are conservative [similar to Doko and Dfor (20)]. We now stdy the roerties of the tests nder the alternativehyothesisδ Asymtotic ower We will now examine the roerties of exogeneity tests nder the alternative hyothesis δ 0.he following theorem resents the reslts. heorem 4.2 ASYMPOIC POWER WHEN INSRUMENS ARE ASYMPOICALLY WEAK. Sose that the assmtions (2.) - (2.3) and (2.6) - (2.3) hold. If Π 2 = Π 0 2 / (Π 0 2 = 0 is allowed), then, for i =, 2, 3 and l =, 2, 3, 4, we have L H i σ 2 (Π2 0 a Σ S2ε ) Z (Π a Σ S2ε ),i=, 2, 3, (4.5) i l L κ l σ 2 (Π2 0 a Σ S2ε ) Z 2 (Π2 0 a Σ S2ε ),l=, 2, 3, 4, (4.6) l RH L k 2 σ 2 ( S 2ε Σ Z2 Π2 0 a) Σ ( S 2ε Σ Z2 Π2 0 a) χ 2 (k 2,µ ε k R ), (4.7) 2 where µ R = a Π 0 2 Σ Π 0 2 a, κ =(k 2 G)/G, κ 2 =/G and κ 3 = κ 4 =. Frthermore, we have, H i S L 2 σ 2 (Π2 0 a Σ S2ε ) Z (Π a Σ S2ε ) S χ 2 (G, µ 2 ),i=, 2, (4.8) i H 3 S 2 L χ 2 (G, µ ), S 2 L F (G, k2 G; µ,λ ), (4.9) 2 S 2 L G χ2 (G, µ ), 4 S 2 L χ 2 (G, µ ), (4.0) 3

16 3 S L κ 3 2 σ 2 (Π2 0 a Σ S2ε ) Z 2 (Π2 0 a Σ S2ε ) S 2 χ 2 (G, µ ), (4.) 3 S 2ε, S 2 are defined in (2.7) - (2.20), σ 2 i,i=, 2, 3 σ2 l,l=, 2, 3, 4, and,µ,λ, are defined in (3.3) - (3.25). From the above theorem, we note that when identification is weak, exogeneity tests do not converge nder the alternative hyothesis δ 0.he asymtotic distribtion of the statistics converge to finite non-degenerate distribtions. Frthermore, the conditional limiting distribtions of H 3, 2, 4 and RH given S 2 are noncentral chi-sqare distribtions while follows a doble noncentral F -distribtion. However, H, H 2, and 3 are bonded ward by a non central chi sqare distribtion with G degrees of freedom and non centrality arameter µ. his sggests that exogeneity tests can have non zero ower even in resence of weak identification, rovided the non central arameters in the above theorem are different from zero. So, we can then characterize in heorem 4.3 below, the necessary and sfficient condition nder which exogeneity tests have no ower when identification is weak. heorem 4.3 NECESSARY AND SUFFICIEN CONDIIONS FOR NO POWER. Under the assmtion of heorem 4.2, the ower of DWH and RH tests does not exceed the nominal levelsifandonly if Π 0 2 a =0. More recisely, we have nder Π0 2 a =0 L H i S σ 2 2ε Σ0 S A 2ε χ 2 L (G), i=, 2, H 3 χ 2 (G), (4.2) 0 L L F (G, k2 G), 2 G χ2 L (G), 4 χ 2 (G), (4.3) L 3 S σ 2 2ε Σ0 S A 2ε χ 2 (G), RH L χ 2 (k 2 ), (4.4) 0 k 2 where σ 2 0,Σ0 A are defined in (3.3) - (3.25) and S 2ε in (2.7) - (2.20). Observe that when Π2 0 a =0, the non centrality arameters in heorem 4.2 vanish so that the statistics H 3, 2, 4 and RH have central chi-sqare limiting distribtions while is asymtotically distribted as a Fisher with (k 2 G, G) degrees of freedom. Frthermore, H, H 2 and 3 are bonded by a central chi-sqare distribtion with G degrees of freedom. herefore, the asymtotic ower of H 3, 2, 4, and RH eqals the nominal levels while those of H, H 2 and 3 cannot exceed the nominal level [similar to Doko and Dfor (20)]. Section 5 below stdies the asymtotic behavior of re-test estimators based on DWH and RH tests. 5. Pre-test estimators based on exogeneity tests An imortant and ratical roblem in econometrics consists in sing DWH-and RH-tye tests to retest the exogeneity of regressors to decide whether one shold aly ordinary least sqares or instrmental variables methods for satistical inference. Althogh this ractice seems to be revalent in alied research, some athors, inclding Gggenberger (200), have shown that the two-stage 4

17 t-test rocedre based on DWH-and RH-tests is nreliable from theviewointofsizecontrolwhen endogeneity is local to zero of order /2 and the instrments are weak. In both cases, exogeneity tests are inconsistent and the two-stage t-test rocedre may be arbitrary size distorted. his is showed by Gggenberger (200), sing some configrations of model arameters. Gggenberger (200) sggests to se a 2SLS based t-test when instrments are strong and the identification-robst rocedres [Anderson and Rbin (949, AR-test), Kleibergen (2002,K-test),Moreira(2003,CLRtest), rojection-based techniqes, see Dfor (997, 2003), Dfor (2005, 2007), slit-samle methods, see Dfor and Jasiak (200)] when there are weak. his sggests that the ractice of retesting of the regressors shold be abandoned. However, it is not clear how behave the re-test estimators when instrments are weak. he framework of Gggenberger (200) focses in testing and does not deal with estimation. he main objective of this section is to stdy the behavior of re-test estimators based on exogeneity tests, inclding whenidentificationisdeficientorweak (weak instrments). We consider eight re-test estimators associated to DWH and RHre-tests defined by eqations (5.) - (5.3) below: ˆβ Hi = ˆβ[H i c Hi, ξ]+ β[h i >c Hi, ξ],i =, 2, 3, (5.) ˆβ l = ˆβ[ l c l, ξ]+ β[ i >c l, ξ],l =, 2, 3, 4, (5.2) ˆβ RH = ˆβ[RH c RH, ξ ]+ β[rh >c RH, ξ ], (5.3) (5.4) where ˆβ and β are the OLS and 2SLS estimators of β, [.] is the indicator fnction and c Hi, ξ, i =, 2, 3, c l, ξ, l =, 2, 3, 4, and c RH, ξ are the sal ξ qantile of the standard distribtions of DWH and RH statistics resectively. It is imortant to observe that the re-test estimators defined by (5.)-(5.3) are convex combinations of OLSand 2SLSestimators. he weight allocated to each estimator is determined by the otcome of the nderline re-test in the first stage. fixed. Lemma 5. below characterizes the robability limit of OLS and 2SLS estimators when Π 2 is Lemma 5. LIMI ALUES OF OLS AND 2SLS ESIMAORS. Sose theassmtions (2.) - (2.3) and (2.6) - (2.3) hold. If Π 2 is fixed, then lim ˆβ = β LS = β +(Π 2 Σ Π 2 + Σ ) δ, (5.5) β 2 = β 2 + N B, (5.6) lim lim β = β, lim β = β I = β + S 2 N B (5.7) 5

18 where 0 if rank(π 2 )=G, N B = S 2 a + B S 2 S 2 [Σ Π (Π 2 2 Σ Π 2 ) Π 2 ] S 2ε if rank(π 2 ) G, (5.8) N B S 2 N [ S 2 a, σ2 εb ], B = S 2 S 2 [Σ Π 2 (Π 2 Σ Π 2 ) Π 2 ] S 2 S 2, S 2 is defined in (3.6) - (3.9) and a = Σ δ. We make the following observations concerning Lemma 5.. (i) From (5.7)-(5.8), we have lim β = β + a + S 2 NB 0 = β + S 2 NB 0 (5.9) 0 if rank(π 2 )=G, where β = β+a and NB 0 = B S 2 S 2 [Σ Π 2 (Π 2 Σ Π 2 ) Π 2 ] S 2ε if rank(π 2 ) G. Frthermore, by sing the generalization of matrix inversion lemma [see ylavsky and Sohie (986, Eqation (d))], we have so that (Σ + Π 2Σ Π 2 ) = Σ (Π 2Σ Π 2 + Σ ) δ = Σ hs (5.5) can be written as lim (ii) If Π 2 a =0, we have Σ (I + Π 2Σ Π 2 Σ ) Π 2Σ Π 2 Σ (5.0) δ Σ (I + Π 2Σ Π 2 Σ ) Π 2Σ Π 2 Σ δ = a Σ (I + Π 2 Σ Π 2 Σ ) Π 2 Σ Π 2 a. ˆβ = β Σ (I + Π 2 Σ Π 2 Σ ) Π 2 Σ Π 2 a. (5.) lim ˆβ = β = β + a, lim β = β + S 2 NB, 0 (5.2) AMSE(ˆβ) = lim [MSE(ˆβ)] = a = δ Σ 2 δ, (5.3) AMSE(ˆβ) = lim [MSE( β)] = a + S 2 NB 0 = a + S 2 NB 0 +2a S 2 NB 0 = δ Σ 2 δ + N B 0 2 S 2NB 0 +2a S 2 NB 0, (5.4) 6

19 where AMSE(ˆθ) is the asymtotic mean sqare error of ˆθ {ˆβ, β}. Hence OLS is always consistent nder the hyothesis of exogeneity (δ =0), bt 2SLS may not rovided identification is deficient [rank(π 2 ) <G]. get Sose that rank(π 2 )=G. hen,wehaveπ 2 a =0if and only if a =0. By sing (5.2), we lim ˆβ = β, lim β = β, (5.5) AMSE(ˆβ) = AMSE( β) =0. (5.6) Both OLS and 2SLS estimators are consistent if strong is identification (as exected). Sose now that rank(π 2 ) <G(i.e. identification is deficient or weak). Since Π 2 a =0 a =0, if endogeneity is resent (a 0), OLS converges to a sedo vale β = β + a while 2SLS converges to β ls a non degenerate random variable. More interestingly, the sedo vale β is observationally eqivalent to the tre vale β. o see this latter oint, consider eqations (2.)- (2.3). From (2.2) and (2.3), we have = Y Z Π Z 2 Π 2 and = a+ ε. Sbstitting these exressions in (2.) gives y = Yβ + Z 2 Π 2 a + Z γ + ε, (5.7) where β = β + a and γ = γ + Π a. If Π 2 a =0, (5.7)becomes y = Yβ + Z γ + ε, (5.8) and ˆβ = ˆβ. Clearly, the sedo vale β is observationally eqivalent to the tre vale β. his means that when identification fails, nlike 2SLS estimator, theinconsistencyofolsestimatoris not too roblematic as one shold think. Now, define AMSE OLS (β ) = lim ˆβ β and AMSE I (β )=lim β β. (5.9) If Π 2 a =0, then we have AMSE OLS (β ) = 0, AMSE I (β )= S 2 N 0 B > 0. (5.20) Hence, OLS is referable to 2SLS if identification is deficient. Of corse, (5.7)-(5.20) remain valid if Π 2 =0(comlete non identification of β). (iii) If Π 2 a 0, then both OLS and 2SLS estimators are biased and their resective asymtotic biases and mean sqare errors (centered at β )aregivenby lim (ˆβ β ) = Σ (I + Π 2Σ Z2 Π 2 Σ ) Π 2Σ Z2 Π 2 a, AMSE OLS (β ) = a Ca, (5.2) 7

20 where C = Π 2 Σ Π 2 (I + Π 2 Σ Π 2 Σ ) Σ 2 (I + Π 2 Σ Π 2 Σ ) Π 2 Σ Π 2 and lim ( β β ) = S 2 NB 0,AMSE OLS(β )= S 2 NB 0. (5.22) So, nlike 2SLS, we observe that the asymtotic bias and mean sqare error of OLS, centered at β, deend on the degree of endogeneity a. Frthermore, since C 0, AMSE OLS (β ) is a nondecreasing and nbonded fnction of a. his sggests that if Π 2 a 0and endogeneity is large, 2SLS is referable to OLS. (iv) Finally, we note that β is still consistent even if identification is deficient or weak, while β 2 is consistent only when I are strong. Hence, the inconsistency of β comes from β 2. We can state a similar lemma concerning the behavior of OLS and 2SLS estimators when instrments are asymtotically weak [Π 2 = Π2 0/ ]. he reslts are resented in Lemma 5.2 below. Lemma 5.2 LIMI ALUES OF OLS AND 2SLS ESIMAORS WHEN INSRUMENS ARE ASYMPOICALLY WEAK. Sose that the assmtions (2.) - (2.3) and (2.6) - (2.3) hold. If Π 2 = Π2 0/,whereΠ2 0 is a k 2 G constant matrix (Π2 0 =0is allowed), then where Ψ = (Σ Π2 0 + S 2 )Σ (Σ Π2 0 + S 2 ), N Ψ W Σ Z2 Π2 0a), N Ψ W S 2 N[ Ψ (Σ Π2 0 + S 2 ) Σ So, we see that the observations in Lemma 5.-(ii) still hold. lim ˆβ = β, (5.23) lim β = β + NΨ W, (5.24) = Ψ (Σ Π2 0 + S 2 ) Σ ( S 2ε ] and β = β + a. Σ Z2 Π 0 2 a, σ2 ε Ψ We can now rove the above reslts on the behavior of re-testestimatorsdefinedin(5.)-(5.3). heorem 5.3 LIMI ALUES OF PRE-ES ESIMAORS. Sose theassmtions (2.) - (2.3) and (2.6) - (2.3) hold. If Π 2 is fixed, then lim (ˆβ W β ) = W Σ (I + Π 2 Σ Π 2 Σ ) Π 2 Σ Π 2 a + ( W )S 2 NB 0, (5.25) where S 2 N 0 B is defined by (5.8). If Π 2 = Π 0 2 /,then lim (ˆβ W β ) = ( W )NΨ W, (5.26) where N W Ψ is defined in Lemma 5.2 and W = lim P [W c W, ξ] (5.27) 8

21 and W {Hi, l, RH}, i=, 2, 3, l=, 2, 3, 4. We make the following remarks: (i) when Π 2 is fixed, if frther Π 2 a =0, we have lim (ˆβ W β ) = ( W )S 2 NB 0 S 2NB 0, (5.28) AMSE W (β ) = ( W ) 2 AMSE I (β ) AMSE I (β ). (5.29) In articlar, when identification is deficient, the two-stage estimator is referable to 2SLS. If Π 2 a 0, we have W =0(consistency of DWH and RH tests) and lim (ˆβ W β ) = S 2 NB, 0 (5.30) AMSE W (β ) = AMSE I (β ). (5.3) So, re-test estimators based on exogeneity tests behave like 2SLS. Since 2SLS is referable to OLS when Π 2 a 0and endogeneity is large, re-test estimators estimators are also referable to OLS in this cases; (ii) if Π 2 = Π2 0/ (instrments are asymtotically weak), the reslts are similar to Π 2 a =0. hs, re-test estimators based on exogeneity tests are referable to 2SLS. Overall, re-test estimators based on exogeneity have an excellent erformance comared to OLS and 2SLS estimators. Section 6 below resents the Monte Carlo exeriment. 6. Monte Carlo exeriment In this section, we erform two Monte Carlo exeriments. he first exeriment stdy the effects of weak Is on DWH and RH tests. In this exeriment, we consider threeset:()strongidentification of model arameters; (2) artial identification; and (3) weak identification. he second exeriment analyzes the erformance (bias and MSE) of the re-test estimators based on DWH and RH exogeneity tests. he framework of this exeriment is similar to Gggenberger (200). 6.. Size and ower of DWH and RH tests Consider the two endogenos variables model described by the followingdatageneratingrocess: y = Y β + Y 2 β 2 +, (Y,Y 2 )=(Z 2 Π 2,Z 2 Π 22 )+(, 2 ), (6.32) 9

22 where Z 2 is a k 2 matrix of instrments sch that Z 2t follow i.i.d N(0, I k2 ) for t =,...,, Π 2 and Π 22 are vectors of dimension k 2.Weassmethat = a+ ε = a + 2 a 2 + ε, (6.33) where a and a 2 are 2 vectors and ε is indeendent with =(, 2 ), and 2 are vectors. hrogh this exeriment, and ε are drawn as ( t, 2t ) i.i.d N ( 0, [ 0 0 ]) and ε t i.i.d N (0, ), for all t =,...,. (6.34) he above set allows s to take into accont sitations where β =(β,β 2 ) is artially identified. In articlar, if Π 2 =0and det(π 22 Π 22) 0, the instrments Z 2 cannot identify β. However, β 2 is identified. We define Π 2 = η C 0, Π 22 = η 2 C, (6.35) where η and η 2 take the vale 0 (design of comlete non identification),.0 (design of weak identification) or.5 (design of strong identification), [C 0,C ] is a k 2 2 matrix obtained by taking the first two colmns of the identity matrix of order k 2. he nmber of instrments k 2 varies in {5, 0, 20} and the tre vale of β is set at β 0 =(2, 5). It is worthwhile to note that when η and η 2 belong to {0,.0}, the instrments Z 2 are weak and both ordinary least sqares and two stage least sqares estimators of β in (6.32) are biased and inconsistent nless a = a 2 =0. he simlations are rn the samle =500, and the nmber of relications is N =0, 000. he endogeneity a is chosen sch that a =(a,a 2 ) {( 20, 0), ( 5, 5), (0, 0), (.5,.2), (00, 00) }. (6.36) From the above set, the exogeneity hyothesis for Y is exressed as H 0 : a =(a,a 2 ) =(0, 0). (6.37) he nominal level of the tests is 5%. Foreachvaleofthevectora, we comte the emirical rejection robability of exogeneity test statistics. When a =0,therejectionfreqenciesarethe emirical levels of the tests. However, if a 0, the rejection freqencies reresent the ower of the tests. he reslts are resented in able below. In the first colmn of the table, we reort the statistics while in the second colmn, we reort the vales of k 2 (nmber of exclded instrments). Finally in the other colmns, we reort for each vale of the endogeneity a and instrment qalities η and η 2, the rejection freqencies at nominal level 5%. First, we note that all exogeneity tests are valid whether the instrmentsarestrongorweak.in articlar,, 2, 4, H 3 and RH control the level while 3, H and H 2 are conservative when Is are weak. However, 3, H and H 2 do not exhibit this roblem when identification is strong [see 20

23 the colmn (a,a 2 ) =(0, 0) in able below]. Second, all exogeneity tests have a low ower when both β and β 2 are not identified even in large-samle Nevertheless, when at least one comonent of β is identified [able (contined)], all exogeneity tests exhibit ower. 2

24 able. Power of exogeneity tests at nominal level 5%; G =2, =500 (a,a 2 ) =( 20, 0) (a,a 2 ) =( 5, 5) (a,a 2 ) =(0, 0) (a,a 2 ) =(.5,.2) (a,a 2 ) =(00, 00) k 2 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 =0 η 2 = H H H RH H H H RH H H H RH

25 able (contined). Power of exogeneity tests at nominal level 5%; G =2, =500 (a,a 2 ) =( 20, 0) (a,a 2 ) =( 5, 5) (a,a 2 ) =(0, 0) (a,a 2 ) =(.5,.2) (a,a 2 ) =(00, 00) k 2 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η =0 η =.0 η =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 =.5 η 2 = H H H RH H H H RH H H H RH

26 6.2. Performance of OLS, 2SLS and two-stage estimators Consider now a single simltaneos eqations system described by the following DGP: y = Yβ+, Y = Z 2 Π 2 +, (6.38) where y and Y are random vectors (G =),Z 2 is a k 2 matrix of instrments sch that i.i.d Z 2t N(0, I k2 ),t=,...,, and Π 2 is a vector of dimension k 2 with Π 2 = µ 2 Z 2 C C, where C is a k 2 vector of ones and µ 2 is a concentration arameter. As in Gggenberger (200), we cover several vales of µ 2 : µ 2 {0; 3; 200; 63;, 000;, 000, 000} where the vales of µ 2 less than 63 corresond to those in Hansen, Hasman and Newey (2008). In or framework, small vales of µ 2 (say µ 2 63 )deictedcaseswheretheiareweaksothatthe arameter of interest β is not identified or weakly identified. he correlation between and is set at ρ {0,.05,.,.5,.6,.95} and the tre vale of β eqals. We take k 2 =5instrments,so, both 2SLS and OLS estimators have finite moments. he samle size is =500and the nmber of relications is N =0, 000. hereslts areresented inables 2-3above. In the first colmn of the tables, we reort the different estimators while inthe second, we reort the concentration arameters µ 2 which reresents the qality of the I. Finally, the other colmns reort the correlation ρ between the errors and (ossibly) endogenos regressors. Or major findings can be smmarized into two oints: () over awiderangecases,inclding weak I and moderate endogeneity, OLS erforms better than 2SLS [finding similar to Kiviet and Niemczyk (2007)]; (2) retest-estimators based on exogeneity have an excellent overall erformance comared with sal I estimator. his sggests that the ractice of retesting based on exogeneity tests is not to bad (at least in the viewoint of estimation) as claimed by Gggenberger (200). he choices of k 2 =0, 20 lead to the same conclsions. 24

27 able 2. Relative bias of OLS and two-stage estimators comared with 2SLS for β =0 Estimators µ 2,ρ MCO Pre-tests two-stage