Subset statistics in the linear IV regression model

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1 Subset statstcs n te lnear IV regresson model Frank Klebergen Prelmnary verson. Not to be quoted wtout permsson. Abstract We sow tat te lmtng dstrbutons of subset generalzatons of te weak nstrument robust nstrumental varable statstcs are boundedly smlar wen te remanng structural parameters are estmated usng maxmum lkelood. Tey are bounded from above by te lmtng dstrbutons wc apply wen te remanng structural parameters are welldentfed and from below by te lmtng dstrbutons wc olds wen te remanng structural parameters are completely undentfed. Te lower bound dstrbuton does not depend on nusance parameters and converges n case of Klebergen s () Lagrange multpler statstc to te lmtng dstrbuton under te g level assumpton wen te number of nstruments gets large. Te power curves of te subset statstcs are non-standard snce te subset tests converge to dentfcaton statstcs for dstant values of te parameter of nterest. Te power of a test on a well-dentfed parameter s terefore low for dstant values wen one of te remanng structural parameter s weakly dentfed and s equal to te power of a test for a dstant value of one of te remanng structural parameters. All subset results extend to statstcs tat conduct tests on te parameters of te ncluded exogenous varables. Introducton A szeable lterature currently exsts tat deals wt statstcs for te lnear nstrumental varables (IV) regresson model wose lmtng dstrbutons are robust to nstrument qualty, see e.g. Anderson and Rubn (994), Klebergen (), Morera (3) and Andrews et. al. (5). Tese robust statstcs test ypoteses tat are specfed on all structural parameters of te lnear IV regresson model. Many nterestng ypotezes are, owever, specfed on subsets of te structural parameters and/or on te parameters assocated wt te ncluded exogenous varables. Wen we replace te structural parameters tat are not specfed by te ypotess of nterest by estmators, te lmtng dstrbutons of te robust statstcs extend to tests of suc ypoteses wen a g level dentfcaton assumpton on tese remanng structural parameters olds, see e.g. Stock and Wrgt () and Klebergen (4,5). Ts g level assumpton Department of Economcs, Box B, Brown Unversty, Provdence, RI 9, Unted States and Department of Quanttatve Economcs, Unversty of Amsterdam, Roetersstraat, 8 WB Amsterdam, Te Neterlands, e-mal: Frank_Klebergen@brown.edu, omepage: ttp:// Ts researc s partly funded by te NWO researc grant Emprcal Comparson of Economc Models.

2 s rater arbtrary and ts valdty s typcally unclear. It s needed to ensure tat te parameters wose values are not specfed under te null ypotess are replaced by consstent estmators so te lmtng dstrbutons of te robust statstcs reman unaltered. Wen te g level assumpton s not satsfed, te lmtng dstrbutons are unclear. Te g level assumpton s avoded wen we test te ypoteses usng a projecton argument wc results n conservatve tests, see Dufour and Taamout (5a,5b). We sow tat wen te unspecfed parameters are estmated usng maxmum lkelood tat te lmtng dstrbutons of te robust subset statstcs are boundedly smlar (pvotal). Tey are bounded from above by te lmtng dstrbuton wc apples wen te g level assumpton olds and from below by te lmtng dstrbutons wc apply wen te unspecfed parameters are completely undentfed. Te lower bound dstrbuton does not depend on nusance parameters and converges to te lmtng dstrbuton under te g level assumpton wen te number of nstruments gets large n case of Klebergen s () Lagrange multpler (KLM) statstc. Te robust subset statstcs are tus conservatve wen we apply te lmtng dstrbutons tat old under te g level assumpton. We use te conservatve crtcal values tat result under te g level assumpton to compute power curves of te robust subset statstcs. Tese power curves sow tat te weak dentfcaton of a partcular parameter splls over to tests on any of te oter parameters. For large values of te parameter of nterest, we sow tat te robust subset statstcs correspond wt general tests of te dentfcaton of any of te structural parameters. Hence, wen a partcular (combnaton of te) structural parameter(s) s weakly dentfed, te power curves of tests on te structural parameters usng te robust subset statstcs converge to a rejecton frequency tat s well below one wen te parameter of nterest becomes large. Te qualty of te dentfcaton of te structural parameters wose values are not specfed under te null ypotess are terefore of equal mportance for te power of te tests as te dentfcaton of te ypoteszed parameters tself. Te paper s organzed as follows. In te second secton, we construct te robust statstcs for tests on subsets of te parameters. Because te subset lkelood rato statstc as no analytcal expresson, we extend Morera s (3) condtonal lkelood rato statstc to a quas-lkelood rato statstc for tests on subsets of te structural parameters. In te trd secton, we obtan te lmtng dstrbutons of te robust subset statstcs wen te remanng structural parameters are completely non-dentfed. We sow tat tese dstrbutons provde a lower bound on te lmtng dstrbutons of te robust subset statstcs wle te lmtng dstrbutons under te g level dentfcaton assumpton provde a upperbound. In te fourt secton, we analyze te sze and power of te subset statstcs and sow tat tey converge to a statstc tat tests for te dentfcaton of any of te structural parameters wen te parameter of nterest becomes large. Te fft secton llustrates some possble sapes of te p-value plots tat result from te robust subset statstcs. Te sxt secton extends te robust subset statstcs to statstcs tat conduct tests of ypotezes specfed on te parameters of te ncluded exogenous varables. It also analyzes te sze and power of suc tests. Fnally, te sevent secton concludes. We use te followng notaton trougout te paper: vec(a) stands for te (column) vectorzaton of te T n matrx A, vec(a) =(a...a n), wen A =(a...a n ).P A = A(A A) A s a projecton on te columns of te full rank matrx A and M A = I T P A s a projecton on te space ortogonal to A. Convergence n probablty s denoted by and convergence n p

3 dstrbuton by d. Subset statstcs n te Lnear IV Regresson Model We consder te lnear IV regresson model y = Xβ + Wγ + ε X = ZΠ X + V X W = ZΠ W + V W, () were y, X and W are T, T m x and T m w dmensonal matrces tat contan te endogenous varables, Z s a T k dmensonal matrx of nstruments and m = m x + m w. Te T, T m x and T m w dmensonal matrces ε, V X and V W contan te dsturbances. Te m x, m w, k m x and k m w dmensonal matrces β, γ, Π X and Π W consst of unknown parameters. We can add a set of exogenous varables to all equatons n () and te results tat we obtan next reman unaltered wen we replace all varables by te resduals tat result from a regresson on tese addtonal exogenous varables. Assumpton : old jontly: Wen te sample sze T converges to nfnty, te followng convergence results a. T X. V W ) (ε. V X. V W ) Σ, wt Σ apostvedefnte (m +) (m +)matrx p σ εε σ εx σ εw and Σ = σ Xε Σ XX Σ XW,σ εε :, σ εx = σ Xε : m x,σ εw = σ Wε : m w, σ Wε Σ WX Σ WW Σ XX : m x m x, Σ XW = Σ WX : m x m w, Σ WW : m w m w. b. T Z Z p Q, wt Q a postve defnte k k matrx. c. T Z (ε. V X. V W ) d (ψ Zε. ψ ZX. ψ ZW ), wt ψ Zε : k, ψ ZX : k m X,ψ ZW : k m w and vec(ψ Zε. ψ ZX. ψ ZW ) N (, Σ Q). Statstcs to test jont ypoteses on β and γ, lke, for example, H : β = β and γ = γ, ave been developped wose (condtonal) lmtng dstrbutons under H and Assumpton ( ) do not depend on te value of Π X and Π W, see e.g. Anderson and Rubn (949), Klebergen () and Morera (3). Tese robust statstcs can be adapted to test for ypoteses tat are specfed on a subset of te parameters, for example, H : β = β. We construct suc robust subset statstcs by usng te maxmum lkelood estmator (MLE) for te unknown value of γ, γ, wc results from te frst order condton (FOC) for a maxmum of te lkelood. Te Anderson-Rubn (AR) statstc s proportonal to te concentrated lkelood so we can obtan 3

4 te FOC from (k tmes) te AR statstc: γ (y Xβ Wγ) P Z (y Xβ Wγ) T k (y Xβ Wγ) M Z (y Xβ Wγ) AR(β γ,γ) = γ= γ = γ= γ Π ˆσ εε (β ) W (β ) Z (y Xβ W γ) =, were AR(β,γ)= (y Xβ ˆσ εε (β,γ) Wγ) P Z (y Xβ Wγ), ˆσ εε (β,γ)= (y Xβ T k Wγ) M Z (y Xβ Wγ), Π W (β )=(Z Z) Z W (y Xβ W γ) ˆσ εw (β ) ˆσ εε(β and ˆσ ) εε (β )= ˆσ εε (β, γ), ˆσ εw (β )= (y Xβ T k W γ) M Z W. Te robust subset statstcs equal te robust statstcs for testng te jont ypotess H : β = β and γ = γ wen (β,γ ) equals (β, γ). To specfy te robust subset statstcs, we decompose (Z Z) Z (y : X : W ) nto tree components tat are uncorrelated n large samples. Lemma : Wen Assumpton olds and under H : β = β, Π W (β ) and Π X (β ) = (Z Z) Z X (y Xβ W γ) ˆσ εx(β ) ˆσ εε, wt ˆσ (β ) εx (β )= (y Xβ T k W γ) M Z X, are uncorrelated wt Z (y Xβ W γ) n large samples suc tat E lm T Π T W (β δ W ) Z (y Xβ W γ) =, and E lm T Π ˆσεε(β ) T X (β δ X ) Z (y Xβ W γ) =, ˆσεε(β ) (3) were δ W and δ X are suc tat lm T T δ W Π W Z ZΠ W = C W, lm T T δ X Π XZ ZΠ X = C X wt C W and C X m W m W and m X m X matrces of constants suc tat δ W and δ X are zero n case of rrelevant or weak nstruments and one n case of strong nstruments. Proof. see te Appendx. Defnton :. Te AR statstc (tmes k) to test H : β = β reads AR(β )= ˆσ εε(β ) (y Xβ W γ) P MZ ΠW (β ) Z (y Xβ W γ). (4). Klebergen s () Lagrange multpler (KLM) statstc to test H reads, see Klebergen (4), KLM(β )= ˆσ εε (β ) (y Xβ W γ) P MZ ΠW (β ) Z Π X (β ) (y Xβ W γ). (5) 3. A J-statstc tat tests msspecfcaton under H reads, see Klebergen (4), 4. Te lkelood rato (LR) statstc to test H reads, JKLM(β )= AR(β ) KLM(β ). (6) LR(β )= AR(β ) mn β AR(β), (7) For reasons of brevty, we refran from dscussng ntermedate cases were nstead of normalzng Π W Z ZΠ W (or Π X Z ZΠ X ) by T δ W, we normalze a quadratc form wt respect to Π W Z ZΠ W by a dagonal matrx dag(t δ W,,...,T δ w,mw ) wt dfferent values of δw,,=,...,m W. Tese cases also ave no effect on te results for te robust subset statstcs. 4 ()

5 were mn β AR(β) equals te smallest root of te caracterstc polynomal: ˆΩ (y. X. W T k ) P Z (y. X. W ) =, (8) wt ˆΩ = T k (y. X. W ) M Z (y. X. W ). Te subset LR statstc (7) as no analytcal expresson wen we express t as a functon of Z (y Xβ W γ), Π X (β ) and Π W (β ),.e.te components tat are under H ndependent n large samples. By decomposng te caracterstc polynomal, we obtan an approxmaton of te subset LR statstc wt an analytcal expressson, see Klebergen (6). Teorem. A upperbound on te subset LR statstc (7) reads q MQLR(β )= AR(β ) rk(β )+ (AR(β )+rk(β )) 4(AR(β ) KLM(β )) rk(β ), (9) were rk( β ) s te smallest caracterstc root of ˆΣ MQLR (β )= ˆΣ (X : W )(X : W ).ε (X. W ) (y Xβ Z γ) ˆσ ε(x : W )(β ) ˆσ εε (β ) P Z (X. W ) (y Xβ Z γ) ˆσ ε(x : W )(β ) ˆΣ = ˆΣ (X : W )(X : W ).ε Π X (β ). Π W (β ) ˆσ εε(β ) Z Z (X : W )(X : W ).ε, Π X (β ). Π W (β ) ˆΣ (X : W )(X : W ).ε, () wt ˆσ ε(x : W ) (β )=(ˆσ εx (β ). ˆσ εw (β )) and ˆΣ (X : W )(X : W ).ε = (X. W T k ) M (Z :(y Xβ Z γ))(x. W ). Proof. see te Appendx. Unlke LR(β ) (7), MQLR(β ) (9) s an explct functon of Z (y Xβ W γ), Π X (β ) and Π W (β ). Except for te usage of te caracterstc root rk(β ), ts expresson concdes wt tat of Morera s (3) condtonal lkelood rato statstc. Tus we refer to t as MQLR(β ). Te MQLR statstc (9) s a quas-lr statstc tat preserves te man propertes of te LR statstc, tat ts condtonal dstrbuton gven rk(β ) concdes wt tat of AR(β ) wen rk(β ) s small andwttatofklm(β ) wen rk(β ) s large. We terefore nstead of LR(β ) use MQLR(β ) n te sequel of te paper. To determne te qualty of te approxmaton of LR(β ) by MQLR(β ), we analyze te dfference between LR(β ) and MQLR(β ). Proposton. a. A upperbound on te dfference between LR( β ) and MQLR( β ) s gven by rk(β ) λ mn + rk(β )+ϕ P m m = ϕ λ mn, () were λ mn =AR( β ) MQLR( β ) and ϕ : k s defned n te proof of Teorem n te Appendx. It s suc tat AR( β )= P k = ϕ and KLM( β )= P m = ϕ. 5

6 b. Te LR and MQLR statstcs are dentcal wen te FOC olds at β. Proof. see te Appendx. Te upperbound on te dfference between te LR and MQLR statstcs sows tat te approxmaton of LR(β ) by MQLR(β ) s an accurate one wen: te FOC olds at β and rk(β ) s small or large. To obtan furter nsgt nto te qualty of te approxmaton at ntermedate values of rk(β ), we computed te 95% condtonal crtcal values of te LR and MQLR statstc wen m =3for a range of values of rk(β ) and a few settngs of te larger caracterstc roots, tat nfluence te condtonal dstrbuton of te LR statstc, and te number of nstruments k. We compare tese crtcal values wt te 95% percentles of te upperbound. Te Fgures Secton n te Appendx as two Panels wc sow te 95% condtonal crtcal values tat result from te LR and MQLR statstcs, ter dfferences across dfferent values of rk(β ) and te upperbound from Proposton. Panel 5 sows te 95% condtonal crtcal values of te LR and MQLR statstcs and te 95% percentle of te upperbound from Proposton as a functon of rk(β ),.e.te smallest egenvalue. Panel 6 compares te dfferences n te 95% condtonal crtcal values of te LR and MQLR statstc wt te 95% percentle of te upperbound. Panels 5 and 6 sow tat te 95% condtonal crtcal values of te MQLR statstc are very smlar to tose of te LR statstc. Te dfference between te crtcal values s typcally small and te upperbound s a conservatve one. Only for te unrealstc settng of a rater small value of te smallest egenvalue and a large value of te second largest egenvalue s te dfference between te crtcal values close to te upperbound. Te (condtonal) lmtng dstrbutons of AR(β ),KLM(β ), JKLM(β ) and MQLR(β ) result from te ndependence of Z (y Xβ Z γ) and Π X (β ), Π W (β ) n large samples stated n Lemma and from a g level assumpton wt respect to te rank of Π W wc mples an asymptotc normal dstrbuton for Z (y Xβ Z γ), Π X (β ) and Π W (β ), see Klebergen (4). Assumpton : Tevalueoftek m w dmensonal matrx Π W s fxed and of full rank. Teorem. Under H and wen Assumptons and old, te (condtonal) lmtng dstrbutons of AR( β ), KLM( β ), JKLM( β ) and MQLR( β ) gven rk( β ) are caracterzed by. AR(β ) d ψ mx + ψ k m,. KLM(β ) d ψ mx, 3. JKLM(β ) d ψ k m, 4. MQLR(β ) rk(β ) d ψ mx + ψ k m rk(β )+q ψmx + ψ k m + rk(β ) 4ψ k m rk(β ) () were ψ mx and ψ k m are ndependent χ (m x ) and χ (k m) dstrbuted random varables. Proof. see Klebergen (4). Assumpton s a g level assumpton tat s dffcult to verfy n practce. We terefore establs te lmtng dstrbutons of te dfferent statstcs wen Assumpton fals to old,.e. wen Π W equals zero nstead of a full rank value. We sow tat te lmtng dstrbutons of te 6,

7 statstcs n ts extreme settng provde a lower bound for all oter cases wle te dstrbutons from Teorem provde a upper bound. 3 Lmtng dstrbutons of subset statstcs n non-dentfed cases We construct te (condtonal) lmtng dstrbutons of te AR, KLM, JKLM and MQLR statstcs wen Π W equals zero. Lemma. Wen Π W =and Assumpton and H old, te FOC () corresponds n large samples wt γ ξ w +(ξ ε.w ξ wγ) +γ γ [ξε.w ξ wγ] =, (3) were ξ w and ξ ε.w are k and k m w dmensonal ndependently standard normal dstrbuted matrces and γ = Σ WW ( γ γ Σ WW σ Wε)σ εε.w, σ εε.w = σ εε σ εw Σ wwσ wε. Proof. see te Appendx. Te soluton of γ to te FOC n Lemma s not unque and te MLE results as te soluton tat mnmzes te AR statstc. Lemma sows tat γ, wc s a functon of te MLE γ, does not depend on any parameters. Wen Π W equals zero, te dstrbuton of γ does terefore not depend on any oter parameters as well and s a standard Caucy densty, see e.g. Marano and Sawa (97) and Pllps (989). We construct te lmtng dstrbutons of te AR, KLM, JKLM and MQLR statstcs to test H : β = β wen Π W equals zero. Teorem 3. Under Assumpton, H : β = β olds and wen Π W equals zero:. Te lmtng beavor of te AR statstc to test H : β = β s caracterzed by: AR(β ) d +γ γ [ξ ε.w ξ wγ] [ξ ε.w ξ wγ]. (4). Te lmtng beavor of te KLM statstc to test H : β = β s caracterzed by: KLM(β ) d +γ γ (ξ ε.w ξ wγ) P M [ ξ w +(ξ ε.w ξ w γ) γ +γ γ ] A (ξ ε.w ξ wγ), (5) were A s a fxed k m x dmensonal matrx. 3. Te lmtng beavor of te JKLM statstc s under H caracterzed by: JKLM(β ) d +γ γ (ξ ε.w ξ wγ) M [A : γ ξw +(ξ ε.w ξ wγ) ](ξ +γ ε.w ξ γ wγ). (6) 7

8 4. Te condtonal lmtng beavor of te MQLR statstc gven rk( β ) to test H : β = β reads MQLR(β ) rk(β ) d +γ γ [ξ ε.w ξ wγ] [ξ ε.w ξ wγ] rk(β )+ ½ +γ γ [ξ ε.w ξ wγ] [ξ ε.w ξ wγ]+rk(β ) µ ¾ # 4 +γ γ [ξ ε.w ξ wγ] M γ A : ξ w +(ξ ε.w ξ wγ) [ξ ε.w ξ +γ γ wγ] rk(β ). (7) Proof. see te Appendx. Teorem 3 sows tat te lmt beavors of AR(β ), KLM(β ), JKLM(β ) and MQLR(β ) wen Π W =do not depend on nusance parameters. Te dstrbuton functons assocated wt te lmt beavors from Teorem 3 are bounded from above by te dstrbuton functons n case of a full rank value of Π W wc result from Teorem. Ts s sown n Fgure for te KLM statstc and n Fgure for te AR statstc. Fgure sows te χ () dstrbuton functon and te lmtng dstrbuton functon of KLM(β ) for dfferent numbers of nstruments wen Π W =and m w = m x =. Fgure sows tat te χ () dstrbuton provdes a upperbound for te lmtng dstrbuton functon of KLM(β ) wen Π W =. It also sows tat te lmtng dstrbuton of KLM(β ) wen Π W = converges to a χ () wen te number of nstruments ncreases. Teorem 4. Wen Assumpton and H old and te sample sze T and te number of nstruments jontly converge to nfnty suc tat k/t, te lmtng beavor of KLM( β ) wen Π W =s caracterzed by Proof. see te Appendx. KLM(β ) d χ (m x ). (8) Teorem 4 mples tat te χ dstrbuton becomes a better approxmaton of te lmtng dstrbuton of KLM(β ) wen te number of nstruments gets large. Te number of nstruments sould, owever, not be too large compared to te sample sze because a dfferent lmtng dstrbuton of KLM(β ) results wen t s proportonal to te sample sze, see Bekker and Klebergen (3). Fgure sows te χ (k m w )/(k m w ) dstrbuton functon and te lmtng dstrbuton functon of AR(β )/(k m w ) for dfferent number of nstruments wen Π W =and m w =. Fgure sows tat te lmtng dstrbuton of AR(β ) s bounded by te χ (k m w ) dstrbuton wen Π W =. Fgure sows tat te χ (k m w ) dstrbuton s a muc more dstant upperbound for te lmtng dstrbuton of AR(β ) tan te upperbound for KLM(β ) n Fgure. Te χ approxmaton of te lmtng dstrbuton of AR(β ) wen Π W =s tus a muc more conservatve one tan for KLM(β ). Anoter mportant dfference wt KLM(β ) s tat tere s no convergence of te lmtng dstrbuton of AR(β ) towards a χ dstrbuton wen te number of nstruments gets large. 8

9 Dstrbuton Functon Dstrbuton Functon Fgure : (Lmtng) Dstrbuton functons of χ () (sold) and KLM(β ) wen Π w =,m w = m x =and k =(dotted), 5 (dased-dotted), (dased) and (ponted) Fgure : (Lmtng) Dstrbuton functons of χ (k )/(k ) and AR(β )/(k ) wen Π w =, m w = m x =and k =(dotted and dased-dotted), (sold and dased) and (sold wt trangles and sold wt plusses). 9

10 #nstr. \ stat. KLM(β ) MQLR(β ) AR(β ) JKLM(β ) SLS(β ) Table : Observed sze (n percentages) of te dfferent statstcs tat test H wen Π w =usng te 95% asymptotc sgnfcance level. Te condtonal lmtng dstrbuton of MQLR(β ) gven rk(β ) wen Π W = beaves smlar to tat of AR(β ) and KLM(β ) snce t s just a functon of tese statstcs gven te value of rk(β ). We terefore, and because of ts dependence on rk(β ), refran from sowng ts dstrbuton functon. Snce JKLM(β ) s a functon of AR(β ) and KLM(β ) as well, we also refran from sowng te dstrbuton functon of JKLM(β ). Fgures and sow tat te lmtng dstrbuton functons of KLM(β ) and AR(β ) wen Π W =are bounded by te lmtng dstrbutons of tese statstcs under a full rank value of Π W. Teorem 5 states tat te lmtng dstrbutons of KLM(β ), JKLM(β ), MQLR(β ) and AR(β ) are n general bounded by te lmtng dstrbutons under a full rank value of Π W and tat te lmtng dstrbutons under Π W =provde a lowerbound on tese dstrbutons. Teorem 5. Te (condtonal) lmtng dstrbutons of AR( β ), KLM( β ), JKLM( β ) and MQLR( β ) under a full rank value of Π W provde a upperbound on te (condtonal) lmtng dstrbutons for general values of Π W wle te (condtonal) lmtng dstrbutons under a zero value of Π W provde a lowerbound. Proof. see te Appendx. Teorem 5 sows tat te (condtonal) lmtng dstrbutons of AR(β ), KLM(β ), JKLM(β ) and MQLR(β ) areboundedlysmlar. TecrtcalvaluesofAR(β ), KLM(β ), JKLM(β ) and MQLR(β ) tat result from te (condtonal) lmtng dstrbutons of AR(β ), KLM(β ), JKLM(β ) and MQLR(β ) n Teorem can terefore be appled n general, so even for (almost) lower rank values of Π W, snce te sze of tese tests s at most equal to te sze under a full rank value of Π W. Usage of te crtcal values from Teorem tus results n tests tat are conservatve. 4 Sze and Power We conduct a sze and power comparson of te dfferent statstcs to analyze te nfluence of te qualty of te dentfcaton of γ for tests on β. We terefore conduct a smulaton experment usng () wt m x = m w =,γ=,t=5and vec(ε. V X. V W ) N(, Σ I T ). Te nstruments Z are generated from a N(,I k I T ) dstrbuton. We compute te rejecton frequency of testng te ypotess H : β =usng te AR-statstc (4), KLM-statstc (5), JKLM-statstc (6), MQLR-statstc (9), a combnaton of te KLM and JKLM statstcs and te two stage least squares (SLS) t-statstc, to wc we refer as SLS(β ). Te number of smulatons tat we conduct equals 5.

11 KLM(β ) MQLR(β ) JKLM(β ) CJKLM(β ) AR(β ) SLS(β ) Fg Fg Fg Fg Fg Fg Fg Fg Fg Fg Fg Fg Table : Sze of te dfferent statstcs n percentages tat test H at te 95% sgnfcance level. We control for te dentfcaton of β and γ by specfyng Π X and Π W n accordance wt a prespecfed value of te matrx generalsaton of te concentraton parameter, see e.g. Pllps (983) and Rotenberg (984). We terefore analyze te sze and power of tests on β for dfferent values ΣXX Σ XW Σ WX Σ WW of Θ =(Z Z) (Π X. Π W )Ω XW, wt Ω XW =, wose quadratc form consttutes te matrx concentraton parameter. We specfy Θ suc tat only ts frst two rows ave non-zero elements. Observed sze wen γ s not dentfed. We frst analyze te sze of te dfferent statstcs for conductng tests on β wen γ s completely undentfed so Π W =. We terefore specfy Σ and Θ suc tat Σ equals te dentty matrx and Θ =5, Θ = Θ = Θ =. Table contans te observed sze of te dfferent statstcs wen we test H at te 95% asymptotc (condtonal) sgnfcance level tat results from Teorem. Table confrms Fgures, and Teorem 4. It sows tat KLM(β ), JKLM(β ), MQLR(β ) and AR(β ) are conservatve tests wen we use te crtcal values tat result from applyng te (condtonal) lmtng dstrbutons from Teorem. Table also confrms te convergence of te asymptotc dstrbuton of KLM(β ) wen Π W = towards a χ dstrbuton wen te number of nstruments gets large as stated n Teorem 4 and sown n Fgure. Snce KLM(β )=MQLR(β )=AR(β ) wen k =, te sze of tese statstcs concdes wen k = m =and te model s exactly dentfed suc tat JKLM(β ) s not defned. Te sze of te SLS t-statstc n Table sows tat te lmtng dstrbuton of te SLS t- statstc s conservatve wen Π W =and Σ equals te dentty matrx. Ts result s specfc for te dentty covarance matrx case and, as we sow later, does not apply to general specfcatons oftecovarancematrx. Power and sze for varyng levels of dentfcaton. We conduct a power comparson of te dfferent statstcs to analyze te nfluence of te dentfcaton of γ on tests for te value of β. Except for te specfcaton of te covarance matrx Σ, we use te above specfcaton of te model parameters. Te covarance matrx Σ s specfed suc tat σ εε = σ XX = σ WW =,

12 One mnus p-value One mnus p-value One mnus p-value One mnus p-value σ Xε = σ εx =.9, σ Wε = σ εw =.8 and σ XW = σ WX =.6 and te number of nstruments equals, k =. Panel : Power curves of AR(β ) (das-dotted), KLM(β ) (dased), JKLM(β ) (ponts), MQLR(β ) (sold), CJKLM(sold-plusses) and SLS(β ) (dotted) for testng H : β = β β Fgure.: Strongly dentfed β and γ : Fgure.: Strongly dentfed β and weakly Θ = Θ =. dentfed γ : Θ =, Θ = β β Fgure.3: Weakly dentfed β and strongly Fgure.3: Weakly dentfed β and γ : dentfed γ : Θ =3, Θ =. Θ = Θ =3. SnceteKLM-statstcsproportonaltoaquadratcformoftedervatveofteARstatstc, t s equal to zero at (local) mnma, maxma and saddle ponts of te AR statstc,.e. were te FOC olds. Ts affects te power of te KLM statstc, see e.g. Klebergen (6). We terefore also compute te power of testng H usng a combnaton of te KLM and JKLM statstcs were we apply a 96% sgnfcance level for te KLM statstc and a 99% sgnfcance level for te JKLM statstc so te sze of te combned test procedure equals 5% snce te KLM and JKLM statstcs converge to ndependent random varables under H.TecombnedKLM, JKLM test procedure s ndcated by CJKLM.

13 Panel sows te power curves for dfferent values of te matrx concentraton parameter Θ wt Θ = Θ =and Table sows te observed szes wen we test at te 95% sgnfcance level. Te value of Θ n Fgure. s suc tat bot β and γ are well dentfed. Hence all statstcs ave nce saped power curves and te AR statstc s te least powerful statstc because of te larger degrees of freedom parameter of ts lmtng dstrbuton. Te power of JKLM(β ) s rater low snce t tests te ypotess of overdentfcaton wc s satsfed for all te dfferent values of β. Table sows tat te SLS-statstc already as consderable sze dstorton n ts well dentfed settng. Te value of Θ n Fgure. s suc tat γ s weakly dentfed and β s well dentfed. Fgure. sows tat te weak dentfcaton of γ as large consequences for especally te power of tests on β. Te MQLR statstc s te most powerful statstc n Fgure.. As sown n Table, except for te SLS t-statstc, te sze of te tests remans almost unaltered by te weak dentfcaton of γ but te power s strongly affected. Fgure.3 as a value of Θ tat makes β weakly dentfed and γ strongly dentfed. Agan te MQLR statstc s te most powerful statstc but te power of te KLM statstc s comparable. Table3sowstatteszedstortons of all statstcs, except te SLS t-statstc, s rater small. Te sze of te SLS t-statstc s completely spurous. Te specfcaton of Θ s suc tat all parameters are weakly dentfed n Fgure.4. Te power of all statstcs s terefore rater low and none of te statstcs clearly domnates te oters. Because of te low degree of dentfcaton, Table sows tat te AR statstc s rater underszed wc corresponds wt Table. Te sze of te SLS t-statstc n Table s agan completely spurous. Te specfcaton of te covarance matrx Σ n Panel s suc tat tere are spll-overs between te dentfcaton of β and γ tat results from Θ. It s terefore dffcult to determne te nfluence of te weak dentfcaton of γ onteszeandpoweroftestsonβ. To analyze te nfluence of te weak dentfcaton of γ on te power of tests on β n an solated manner, we equate te covarance matrx Σ to te dentty matrx. Table and Panel sow te resultng sze and power for tests on β. Table sows tat KLM(β ), JKLM(β ), CJKLM(β ), MQLR(β ) and AR(β ) are underszed wen γ s weakly dentfed wc s n accordance wt Table and Teorem 5. Te values of Θ n Fgure. and. are dentcal but KLM(β ), JKLM(β ), CJKLM(β ), MQLR(β ) and AR(β ) are only underszed n Fgure. and not n Fgure.. Ts results because of te dfferent values of Σ tat are used for Fgures. and. suc tat Π W s small n Fgure. but szeable n Fgure.. Te power curves n Panel sow tat SLS(β ) s te most power ful statstc for testng H. Because of te absence of correlaton between te dfferent endogenous varables, SLS(β ) s sze correct. Te prevous Fgures, owever, sow tat SLS(β ) s often severely sze-dstorted n cases wen any correlaton s present wc makes ts results dffcult to trust. Among te statstcs tat reman sze-correct wen dentfcaton s weak, MQLR(β ) s te most powerful statstc for testng H. Te power of MQLR(β ) exceeds tat of AR(β ) for values of β tat are relatvely close to zero but s remarkably smlar to tat of AR(β ) for more dstant values of β. Ts argument olds n a reversed manner wt respect to KLM(β ). Te beavor of te power curve of MQLR(β ) tus resembles tat of KLM(β ) close to zero and tat of AR(β ) for more dstant values of β. 3

14 One mnus p-value One mnus p-value One mnus p-value One mnus p-value One mnus p-value One mnus p-value Panel : Power curves of AR(β ) (dased-dotted), KLM(β ) (dased), MQLR(β ) (sold), JKLM(β ) (ponts), CJKLM(sold wt plusses) and SLS(β ) (dotted) for testng H : β = β β Fgure.: Θ =, Θ =3. Fgure.: Θ =3, Θ = β β Fgure.3: Θ =, Θ =5. Fgure.4: Θ =5, Θ = β β Fgure.5: Θ =, Θ =7. Fgure.6: Θ =7, Θ =. 4

15 Te level of dentfcaton of β and γ s reversed n te two columns of Panel. In te leftandsde column, te dentfcaton of γ s worse tan of β and vce versa n te rgt-andsde column. Table terefore sows tat te statstcs are somewat underszed n te left-andsde column wle tey are sze correct n te rgt-andsde column. Besdes te sze ssue, te power curves n te left and rgt-andsde columns of Panel are remarkably smlar for dstant values of β. Tey only dffer around te ypoteszed value of te parameter. Ts ndcates tat te statstcs beave n a systematc manner for dstant values of β. Ts s stated n Teorem 6. Teorem 6. Wen m X =, Assumpton olds and for tests of H : β = β wt a value of β tat dffers substantally from te true value:. TeAR-statstcAR(β ) s equal to te smallest egenvalue of ˆΩ XW (X. W ) P Z (X. W )ˆΩ XW wc s a statstc tat tests for a reduced rank value of (Π X. Π W ), ˆΩ XW = (X. W T k ) P Z (X. W ).. Te egenvalues of ˆΣMQLR (β ) tat are used to obtan rk( β ) correspond for large numbers of observatons wt te egenvalues of ψ ε.(x : W ). Θ (X : W ) + Ψ (X : W ) V ψ ε.(x : W ). Θ (X : W ) + Ψ (X : W ) V, (9) were (Z Z) Z ε (X. W )Ω σxε XW σ Wε σ εε.(x : W ) ψ ε.(x : W ) = Q [ψ Zε ψ (ZX : ZW) d Ω σxε XW σ Wε ]σ εε.(x : W ), (Z Z) (Π X. Π W )Ω XW Θ (X : W ) and (Z Z) Z (V X. V W )Ω p XW p Ψ (X : W ) = Q ψ (ZX : ZW) Ω XW, and V s a m m w matrx tat contans te egenvectors of te largest m w egenvalues of Ω XW (X. W ) P Z (X. W )Ω XW,σ εε.(x : W ) = σ εε σ Xε σ Wε Ω σxε XW σ Wε. 3. For large numbers of observatons, te χ (k m w ) dstrbuton provdes a upperbound on te dstrbuton of rk( β ). Proof. see te Appendx. Teorem 6 sows tat te power of te AR statstc equals te rejecton frequency of a rank test wen te value of β gets large. Te rank test to wc te AR statstc converges s dentcal for all structural parameters. Hence, te power of te AR statstc for dscrmnatng dstant values of any structural parameter s dentcal. Ts explans te equalty of te rejecton frequences of te AR statstc for dstant values of β n te left and rgt-andsde fgures of Panel 3. Te MQLR statstc conssts of AR(β ), KLM(β ) and rk(β ). Teorem 6 sows tat rk(β ) s bounded by a χ (k m w ) dstrbuted random varable for values of β tat are dstant from tetruevalue. Tsmplesarelatvelysmallvalueofrk(β ) so MQLR(β ) beaves smlar to AR(β ) for dstant values of β. Snce bot te value were rk(β ) and AR(β ) converge to are 5

16 One mnus p-value One mnus p-value te same for all structural parameters, te power of MQLR(β ) s te same for all structural parameters at dstant values and smlar to tat of AR(β ). Ts corresponds wt te Fgures n Panel. Te dentfcaton of β and γ s governed by te matrx concentraton parameter Θ. Besdes avng values tat especally dentfy β and/or γ, te matrx concentraton parameter can also be suc tat lnear combnatons of β and γ are strong or weakly dentfed. To analyze te nfluence of te strong/weak dentfcaton of combnatons of β and γ on tests for β, we specfed te value of Θ suc tat t s close to a reduced rank one. We used te prevous non-dagonal specfcaton of Σ to furter dsperse te dentfcaton of combnatons of β and γ. Table and Panel 3 sows te sze and power of tests for β wen te value of Θ s close to a reduced rank one wc s revealed by te egenvalues of Θ Θ. Except for te SLS t-statstc, te sze of te statstcs s close to 5%. Te weak dentfcaton of a lnear combnaton of γ and β s suc tat te power of all statstcs s rater low. Fgures 3. and 3. sow tat te MQLR(β ) s te most powerful statstc. Panel 3: Power curves of AR(β ) (dased-dotted), KLM(β ) (dased), MQLR(β ) (sold), JKLM(β ) (ponts), CJKLM(sold wt plusses) and SLS(β ) (dotted) for testng H : β = β β Fgure.: Strongly dentfed β and weakly Fgure.: Weakly dentfed β and strongly dentfed γ : Θ =, Θ =5, Θ =5, dentfed γ : Θ =5, Θ =, Θ =5, Θ =5, Egenvalues Θ Θ : 3.65, 7. Θ =5, Egenvalues Θ Θ : 3.65, 7. 5 Confdence Sets Teorem 6 sows tat tests on dfferent parameters become dentcal wen te parameters of nterest get large. Its consequences for te power curves n Panels -3 are clearly vsble and t as smlar mplcatons for te confdence sets of te structural parameters. We terefore use te prevously dscussed data generatng process to compute some (one mnus te) p-value plots wc allow us to obtan te confdence set of a specfc parameter. Tep-value plots are constructed by nvertng te values of te statstcs tat test H : β = β for a range of values of 6

17 One mnus p-value One mnus p-value One mnus p-value One mnus p-value One mnus p-value One mnus p-value β usng te (condtonal) lmtng dstrbutons tat result from Teorem. Panel 4: One mnus p-value plots of AR (das-dotted), KLM (dased), MQLR (sold) JKLM (ponts) and SLS (dotted) for testng β and γ, k =, Θ = Θ = γ β Fgure 4.: Θ =, Θ =. Fgure 4.: Θ =, Θ = γ β Fgure 4.3: Θ =3, Θ =. Fgure 4.4: Θ =3, Θ = γ β Fgure 4.5: Θ =5, Θ =. Fgure 4.6: Θ =5, Θ =. 7

18 One mnus p-value One mnus p-value One mnus p-value One mnus p-value One mnus p-value One mnus p-value Panel 5: One mnus p-value plots of AR (das-dotted), KLM (dased), MQLR (sold) JKLM (ponts) and SLS (dotted) for testng β and γ, k =, Θ = Θ = γ β Fgure 5.: Θ =, Θ =. Fgure 5.: Θ =, Θ = γ β Fgure 5.3: Θ =3, Θ =. Fgure 5.3: Θ =3, Θ = γ β Fgure 5.5: Θ =5, Θ =. Fgure 5.6: Θ =5, Θ =. 8

19 Panel 4 contans te one mnus p-value plots for a data generatng process tat s dentcal to tat of Panel. Te Fgures n Panel 4 are suc tat te Fgures on te left-andsde contan te p-value plot of tests on γ wle te Fgures on te rgt-andsde contan p-value plots of tests on β. Te data set used to compute te p-value plot of β and γ stesameandonlydffers over te rows of Panel 4. Panel 4 sows tat tests on β and γ dfferaroundtetruevalueofβ () and γ () but are dentcal at dstant values. Ts s exactly n lne wt Teorem 6. It sows tat even wen β s well dentfed, confdence sets of β are unbounded wen γ s weakly dentfed. Te odd beavor of te p-value plot of KLM(β ) results snce t s equal to zero wen te FOC olds. Fgures 4., 4.4 and 4.6 terefore sow tat KLM(β ) s equal to zero wen AR(β ) s maxmal. We note tat te p-value plots of KLM(β ), MQLR(β ) and SLS(β ) are equal to zero at resp. te MLE and for SLS(β ), te SLS estmator, but ts s not vsble n all of te Fgures n Panel 4 because of te specfed grd for β. Te data generatng process tat s used to construct Panel 5 s dentcal to tat of Panel. Because of te presence of correlaton, a lnear combnaton of β and γ s weakly dentfed n te Fgures n te top two rows of Panel 5 suc tat te p-value plots do not converge to one. Te resultng 95% confdence sets of β are terefore unbounded for tese Fgures. For dstant values of β and γ, Panel 5 sows agan tat te statstcs tat conduct tests on β or γ become dentcal. Panels 4 and 5 sow tat te dstngusng features of te subsets statstcs sown for te power curves,.e. tat tey do not converge to one wen te parameters of nterest gets large and statstcs tat test ypoteses on dfferent parameter become dentcal for dstant values of te parameter of nterest, approprately extend to confdence sets. 6 Tests on te parameters of exogenous varables Te subset statstcs extend to tests on te parameters of te exogenous varables tat are ncluded n te structural equaton. Te expressons of KLM(β ), JKLM(β ), AR(β ) and MCLR(β ) reman almost unaltered wen X s exogenous and s spanned by te matrx of nstruments. Te lnear IV regresson model ten reads y = Xβ + Wγ + ε W = XΠ WX + ZΠ WZ + V W, () were (X. Z) s te T (k + m x ) dmensonal matrx of nstruments and Π XW and Π ZW are m x m w and k m w matrces of parameters. All oter parameters are dentcal to tose defned for (). We are nterested n testng H : β = β and we adapt te expressons of te statstcs from Defnton to accomodate tests of ts ypotess. Defnton :. Te AR statstc (tmes k) to test H : β = β reads AR(β )= ˆσ εε(β ) (y Xβ W γ) P M Z ΠW (β ) Z (y Xβ W γ), () 9

20 wt Z =(X. Z), Π W (β )=( Z Z) Z W (y Xβ W γ) ˆσ εw (β ) ˆσ εε and ˆσ (β ) εε (β )= (y T k Xβ W γ) M Z (y Xβ W γ), ˆσ εw (β )= (y Xβ T k W γ) M Z W and γ te MLE of γ gven tat β = β.. Te KLM statstc to test H reads, KLM(β )= ˆσ εε(β ) (y Xβ W γ) P M Z ΠW (β ) X (y Xβ W γ), () snce Π X (β )=( Z Z) Z X (y Xβ W γ) ˆσ εx(β ) ˆσ εε(β ) (y Xβ T k W γ) M Z X =. 3. A J-statstc tat tests msspecfcaton under H reads, =( Z Z) Z X = I m x as ˆσεX (β )= JKLM(β )= AR(β ) KLM(β ). (3) 4. A quas lkelood rato statstc based on Morera s (3) lkelood rato statstc to test H reads, q MQLR(β )= AR(β ) rk(β )+ (AR(β )+rk(β )) 4(AR(β ) KLM(β )) rk(β ), (4) were rk( β ) s te smallest egenvalue of ˆΣ MQLR = ˆΣ WW.ε W (y Xβ Z γ) ˆσ εw (β ) PMXZ ˆσ εε W (y Xβ (β ) Z γ) ˆσ εw (β ) ˆσ εε (β ) ˆΣ WW.ε. wt ˆσ εw (β )= T k (y Xβ W γ) M Z W, ˆΣ WW = T k W M Z W, ˆΣ WW.ε = ˆΣ WW ˆσ εw (β ) ˆσ εw (β ) ˆσ εε (β ). Except for MQLR(β ), all statstcs n Defnton are drect extensons of tose n Defnton wenwenotetat Π X (β )= I mx, wen X belongs to te set of nstruments. Te alteraton of te expresson of ˆΣ MQLR for MLR(β ) partly results from M ZX = and snce only te nstruments Z dentfy γ. Under a full rank value of Π WZ, te (condtonal) lmtng dstrbutons of te statstcs n Defnton are dentcal to tose n Teorem wen k sequalto k + m x. Alongsde Teorem, Teorems 3-5 apply to te statstcs from Teorem as well. Teorem 7. Te (condtonal) lmtng dstrbutons of AR( β ), KLM( β ), JKLM( β ) and MQLR( β ) n Defnton are bounded from above by te lmtng dstrbuton under a full rank value of Π WZ and from below by te lmtng dstrbuton under a zero value of Π WZ. Proof. results from Teorem Sze and power propertes To llustrate te beavor of te exogenous varable statstcs from Defnton, we analyze ter sze and power propertes. We terefore conduct a smulaton experment usng () wt T = 5, m w = m x =and k =9so te total number of nstruments equals k + m x =. All

21 One mnus p-value One mnus p-value One mnus p-value nstruments are ndependently generated from N(,I T ) dstrbutons and vec(ε. V W ) s generated from a N(, Σ I T ) dstrbuton. Te number of smulatons equals 5. Panel 6: Power curves of AR(β ) (dased-dotted), KLM(β ) (dased), MQLR(β ) (sold), JKLM(β ) (ponts), CJKLM(sold wt plusses) and SLS(β ) (dotted) for testng H : β = β Fgure 6.: Θ WZ, =3 Fgure 6.: Θ WZ, = β Fgure 6.3: Θ WZ, = β Te data generatng process for te power curves n Panel 6 as Π WX Σ = I mw+. Te specfcaton of Θ WZ =(Z M X Z) Π WZ Σ W =,γ=and n Panel 6 s suc tat ts frst element Θ WZ, s unequal to zero and all remanng elements of Θ WZ are equal to zero. Table 3 sows te observed sze of te dfferent statstcs wen we test at te 95% sgnfcance level. Te parameters of te data generatng process used for Panel 6 are specfed suc tat β s not partly dentfed by te parameters n te equaton of W snce Π XW =and σ εw =. Panel 6 s tus comparable to Panel wose data generatng process s specfed n a smlar manner. Te resultng power curves and observed szes terefore closely resemble tose n Panel and Table. Table 3 sows tat te statstcs are conservatve wen te dentfcaton s rater low, wc s n accordance wt Teorem 7.

22 KLM(β ) MQLR(β ) JKLM(β ) CJKLM(β ) AR(β ) SLS(β ) Fg Fg Fg Fg Fg Fg Table 3: Sze of te dfferent statstcs n percentages tat test H at te 95% sgnfcance level. Panel 6 sows tat te rejecton frequences converge to a constant unequal to one for dstant values of β wen te dentfcaton of γ s rater weak. Ts ndcates tat Teorem 6 extends to tests on subsets of te parameters. Teorem 8. Wen m X =, Assumpton olds, X s exogenous and for tests of H : β = β wt a value of β tat dffers substantally from te true value:. TeAR-statstcAR(β ) s equal to te smallest egenvalue of ˆΣ WW W P MX ZW ˆΣ WW wc s a statstc tat tests for a reduced rank value of Π WZ, ˆΣ WW = T k W P ZW.. Te egenvalues of ˆΣMQLR (β ) tat are used to obtan rk( β ) correspond for large numbers of observatons wt te egenvalues of ψ ε.w. (Θ WZ + Ψ W ) V ψ ε.w. (Θ WZ + Ψ W ) V, (5) were (Z M X Z) Z M X ε W Σ WW σ Wε σ εε.w d ψ ε.w, (Z M X Z) Π WZ Σ WW p Θ ZW and (Z M X Z) Z M X V W Σ WW Ψ W, and V s a m m w matrx tat contans te egenvectors of te largest m w egenvalues of Σ p WW W P MX ZW Σ WW,σ εε.w = σ εε σ εw Σ WW σ Wε. 3. For large numbers of observatons, te χ (k m w ) dstrbuton provdes a upperbound on te dstrbuton of rk( β ). Proof. follows from te proof of Teorem 6. Teorem 8 explans te convergence of te rejecton frequences n Panel 6 and mples tat te beavor of MQLR(β ) s smlar to tat of AR(β ) for dstant values of β. Identcal to te prevous Panels, SLS(β ) s te most powerful statstc n Panel 6 wle Table 3 sows tat t also as lttle sze dstorton. Ts results because σ εw =. For non-zero values of σ εw, te sze-dstorton s often substantal. Te parameter settngs for Panel 7 are suc tat β s partally dentfed by te parameters n te equaton of W snce Π XW =and σ εw =.8. All remanng parameters are dentcal to tose n Panel 6. Because of te partal dentfcaton, Table 3 sows tat te statstcs are no

23 One mnus p-value One mnus p-value One mnus p-value longer conservatve wen Θ WZ, s small. Because of te non-zero value of σ εw, SLS(β ) s now severly sze dstorted wen Θ WZ, s small. Altoug te small value of Θ WZ, does not affect te sze of te tests from Defnton, t stll strongly nfluences te power. Panel 7 sows tat te power curves do not converge to one wen Θ WZ, s small wc s n accordance wt Teorem 8. Panel 7: Power curves of AR(β ) (dased-dotted), KLM(β ) (dased), MQLR(β ) (sold), JKLM(β ) (ponts), CJKLM(sold wt plusses) and SLS(β ) (dotted) for testng H : β = β Fgure 7.: Θ WZ, =3 Fgure 7.: Θ WZ, = β β Fgure 7.3: Θ WZ, =7 7 Conclusons Te lmtng dstrbutons of te robust subset nstrumental varable statstcs tat result under a g level dentfcaton assumpton on te remanng structural parameters provde a upperbound on te lmtng dstrbuton of tese statstcs n general. Lower bounds result from te lmtng dstrbutons under complete dentfcaton falure of te remanng parameters. For dstant 3

24 values of te parameter of nterest, te subset nstrumental varable statstcs correspond wt dentfcaton statstcs. Even f te parameter of nterest s well-dentfed, te power of tests on t do terefore not necessarly converge to one wen te ypoteszed value of nterest gets large. Te subset AR statstc s less conservatve tan te projecton based AR statstc from Dufour and Taamout (5a,b). Ts results snce te degrees of freedom parameter of ts lmtng dstrbuton s smaller tan tat of te projecton based AR statstc wle te latter s also based on te mnmal value of te AR statstc gven tat H olds. Appendx ProofofLemma. Because of te FOC: Π ˆσ εε (β ) W (β ) Z (y Xβ W γ) =, t automatcally follows tat wt Π W (β )=Π W + T ( Z Z T ) T Z V W ˆε ˆσ εw (β ) ˆσ εε (β ) E lm T Π T W (β δ W ) Z ˆσεε(β ˆε, wt ˆε = y Xβ ) W γ = ε W ( γ γ ), s uncorrelated n large samples so ˆσεε Zˆε (β ) =, were δ W s suc tat lm T T δ W Π W Z ZΠ W = C W wt C W a m W m W matrx of constants so δ W =n case of rrelevant or weak nstruments and δ W =n case of strong nstruments. To sow tat Z ˆε and Π X (β )=(Z Z) Z X ˆε ˆσ εx(β ) ˆσ εε are uncorrelated n large samples, (β ) we use tat W = ZΠ W + V W = Z Π W (β )+M Z V W P Zˆε ˆσ εw (β ) ˆσ εε (β ) wc enables us to caracterze te covarance between X and ˆε = M Z Π W (β )ˆε by E lm T T X M Z ΠW (β )ˆε = E lm T T X + V X ) M Z Π W (β ) n (ε W ( γ γ )) o = E lm T T (ZΠ X + V X ) M Z Π W (β ) n ε Z Π W (β )+M Z V W P Zˆε ˆσ εw (β ) ˆσ εε(β ) W ( γ γ ) o = E lm T T VXM Z ΠW (β ) ε V XM Z ΠW (β ) n Z Π W (β )+M Z V W P Zˆε ˆσ εw (β ) ˆσ εε (β ) ( γ γ ) o = E lm T V T X M Z Π W (β ) ε V X M ZV W ( γ γ )+VX P M Zˆε ˆσ εw (β ) Z ΠW (β ) ˆσ εε (β ) ½ ) = E lm T V T XM Z ΠW (β ) ε V XM Z V W ( γ γ )+VXP MZ ΠW (β ) Z ε P ¾ ˆσεW (β ) = ˆσ εε(β ) ) n o = E lm T V T X M Z Π W (β ) ε V X M ZV W ( γ γ )+VX P M Z ΠW (β ) Z ˆσ ε εw (β )( γ γ ) ˆσ εε (β ) ˆσ εw (β )( γ γ ) n = E lm T V T XM Z ΠW (β ) ε V XM Z V W ( γ γ )+ o VXP MZ ΠW (β ) Z ˆσ ε εw ( γ γ ) ( γ γ) ˆΣ WW ( γ γ ) ˆσ εε 3ˆσ εw ( γ γ )+( γ γ) ˆΣ WW ( γ γ ) = E lm T T X ε V X M ZV W ( γ γ )} = E [lm T ˆσ Xε (β )], 4

25 were for: te fourt equaton, we use tat E lm T T Π X Z M Z Π W (β ) ε so E lm T T Π XZ M Z ΠW (β ) P Zˆε =as well. =,E lm T T Π X Z M Z V W = te fft equaton, we use tat M Z Π W (β ) Z Π W (β )=,M Z Π W (β ) M Z = M Z and M Z Π W (β ) P Z = P MZ ΠW (β ) Z. te sxt equaton, we recurrently substtute te expresson for ˆε. te sevent equaton, we use tat P ˆσεW (β ) = ( γ γ ˆσ εε (β ) ) = ˆσ εw (β )( γ γ ) ˆσ εε (β ) ˆσ εw. For ts (β )( γ γ ) to old ˆσ εw (β ) ( γ γ ˆσ εε (β ) ) < wc olds true snce ˆεˆε = T T ˆε M Z ΠW (β )ˆε s fnte wc mples tat P ˆσεW (β ) = ( γ γ ˆσ εε(β ) ) s fnte as well. te egt equaton, we use tat ˆσ εw (β )=ˆσ εw ( γ γ ) ˆΣ WW and ˆσ εε (β )=ˆσ εε ˆσ εw ( γ γ )+( γ γ) ˆΣWW ( γ γ ), wt ˆσ εε = T k ε M Z ε, ˆσ εw = T k ε M Z W and ˆΣ WW = M T k Z W. for te nnet equaton, we note tat E lm T lm T V T XP MZ ΠW (β ) Z ε V T XP Z ΠW (β ) ε =and E. We also note tat ˆσ εε, ˆσ Wε, ˆΣ WW and ˆσ εε are uncorrelated wt V T XP MZ ΠW (β ) Z ε because tey result from projectng on spaces tat are ortogonal to M Z ΠW (β ) Z. for te tent equaton, we note tat E lm T V T k XP Z ε =and E lm T V T k XP Z W W =. Te above sows tat M Z Π W (β ) X ˆε ˆσ εx(β ) ˆσ εε and M (β ) Z Π W (β )ˆε ˆσ εx(β ) ˆσ εε are uncorrelated suc (β ) tat E lm T Π T X (β δ X ) Z ˆε =, were δ X s suc tat lm T T δ X Π XZ ZΠ X = C X wt C X a m X m X matrx of constants. ProofofTeorem. Te LR statstc to test H reads LR(β )=AR(β ) mn β AR(β). Te value of AR(β) s obtaned by mnmzng over γ so mn β AR(β) can also be specfed as mn β AR(β) = mn β,γ T k (y Xβ Wγ) M Z (y Xβ Wγ) (y Xβ Wγ) P Z (y Xβ Wγ), wc equals te smallest root of te caracterstc polynomal λˆω (y. X. W ) P Z (y. X. W ) =, 5

26 wt ˆΩ = (y. X. W T k ) M Z (y. X. W ). Te roots of te caracterstc polynomal do not alter wen we pre- and post-multply by a trangular matrx wt ones on te dagonal: β I mx λˆω (y. X. W ) P Z (y. X. W ) β I mx = γ I mw γ I mw λˆσ(β ) (y. X. W ) P Z (y. X. W ) =. wt ˆΣ(β )= β I mx ˆΩ µ β I mx ˆσ εε (β = ) ˆσ ε(x : W ) (β ), ˆσ γ I mw γ I (X : W )ε (β ) ˆΣ (X : W )(X : W ) mw ˆσ εε (β ):, ˆσ ε(x : W ) (β )=ˆσ ε(x : W ) (β ) : m, ˆΣ (X : W )(X : W ) : m m. We decompose ˆΣ(β ) as ˆΣ(β ) = ˆΣ(β ) ˆΣ(β ), ˆΣ(β ) = ˆσ εε(β ) ˆσ εε (β ) ˆσ ε(x : W ) (β )ˆΣ (X : W )(X : W ).ε ˆΣ (X : W )(X : W ).ε wt ˆΣ (X : W )(X : W ).ε = (X. W T k ) M (Z : (y Xβ W γ))(x. W ), suc tat ˆΣ(β ) ˆΣ(β )ˆΣ(β ) = I k(m+) and we can specfy te caracterstc polynomal as λi m+ ˆΣ(β ) (y. X. W ) P Z (y. X. W )ˆΣ(β ) = µ λi m+ (Z Z) Z (y Xβ Z γ). (X. W ) (y Xβ ˆσεε(β ) Z γ) ˆσ ε(x : W )(β ) ˆσ εε ˆΣ (β ) (X : W )(X : W ).ε µ (Z Z) Z (y Xβ Z γ). (X. W ) (y Xβ ˆσεε (β ) Z γ) ˆσ ε(x : W )(β ) ˆσ εε ˆΣ (β ) (X : W )(X : W ).ε =. Wen we conduct a sngular value decomposton, see e.g. Golub and van Loan (989), (Z Z) Z (X W) (y Xβ Z γ) ˆσ ε(x : W )(β ) ˆΣ (X : W )(X : W ).ε = USV, ˆσ εε (β ) were U : k k, U U = I k, V : m m, V V = I m and S s a dagonal k m dmensonal matrx wt te sngular values n decreasng order on te man dagonal, we can specfy te caracterstc polynomal as, see Klebergen (6), µ λi m+ µη. USV η. USV = µ η λi m+ η η USV VS U η VS SV = µ µ µ λi η m+. U Uη η US V S U η S S. V = µ ϕ λi m+ ϕ ϕ S S ϕ S S =, 6,

27 wt η =(Z Z) Z (y Xβ Z γ),ϕ= Uη. Snce U s an ortonormal matrx, ts expresson sows ˆσεε(β ) tat te roots of te caracterstc polynomal only depend on te sngular values wc equal te square roots of te egenvalues of Σ (X : W )(X : W ).ε (X. W ) (y Xβ Z γ) σ ε(x : W )(β, γ) σ εε(β, γ) Σ (X : W )(X : W ).ε. (X. W ) (y Xβ Z γ) σ ε(x : W )(β, γ) σ εε (β, γ) P Z Usng te propertes of te determnant, te caracterstc polynomal λi m+ can be specfed as µ f(λ, s,...,s mm) = ϕ λi m+ ϕ ϕ S S ϕ S S = Q m j= (λ s jj)[λ ϕ ϕ] P m Q = s ϕ m j=,j6= (λ s jj) = Q m j= (λ s jj) λ ϕ ϕ P m =, s ϕ λ s ϕ ϕ ϕ S S ϕ S S wt ϕ =(ϕ...ϕ k ) and s >...>s mm are te m dagonal elements of S. Te (m+)-t order polynomal f(λ, s,...,s mm) as m +roots. Snce f(,s,...,s mm) = ( ) P m+ k Q m Q=m+ ϕ j= s jj f(s mm,s,...,s mm) = ( ) m ϕ m m j= Q s jj f(s m m,s,...,s mm) = ( ) m ϕ m m j= s jj. f(s,s,...,s mm) = ϕ Q m j= s jj, te polynomal f(λ, s,...,s mm) alters sgn between and s mm,s mm and s m m, etc. Tus te smallest root of f(λ, s,...,s mm) les between and s mm, te second smallest root les between s mm and s m m, etc. and te largest root exceeds s because f(λ, s,...,s mm) s postve at nfnte values of λ snce s s fnte valued. Te roots of te polynomal f(λ, s,...,s mm) ave no analytcal expresson snce m>. We terefore approxmate te smallest root of te polynomal f(λ, s,...,s mm) by te smallest root tat results by restrctng s,...,s m m to te smallest root, s mm : f(λ, s mm,...,s mm) = Q m j= (λ s mm) λ ϕ ϕ P m s mmϕ = λ s mm P = (λ s mm) m [(λ ϕ ϕ)(λ s mm) s m mm = ϕ ]. Te P smallest root of f(λ, s mm,...,s mm) equals te smallest root of (λ ϕ ϕ)(λ s mm) s m mm = ϕ wc s a quadratc polynomal so t as an analytcal expresson of ts smallest root: q λ mn = = ϕ ϕ + s mm AR(β )+rk(β ) P (ϕ ϕ + s mm) 4s k mm =m+ ϕ q (AR(β )+rk(β )) 4(AR(β ) KLM(β )) rk(β ), 7