E cient size correct subset inference in linear instrumental variables regression

Transcription

1 E cent sze correct subset nference n lnear nstrumental varables regresson Frank Klebergen September 7 Abstract We sow tat Morera s 3) condtonal crtcal value functon for te lkelood rato statstc tat tests te structural parameter n te d lnear nstrumental varables regresson model wt one ncluded endogenous varable provdes a boundng dstrbuton for te subset lkelood rato statstc tat tests one structural parameter n an d lnear nstrumental varables regresson model wt several ncluded endogenous varables Te only adjustment concerns te usual degrees of freedom correcton for subset tests of te nvolved dstrbuted random varables Te condtonal crtcal value functon makes te subset lkelood rato test sze correct under weak dent caton of te structural parameters and e cent under strong dent caton Wen te ypoteszed value of te parameter of nterest s dstant from te true one, te subset Anderson-Rubn and lkelood rato statstcs are nvarant wt respect to te parameter of nterest and eual statstcs tat test te dent caton of all structural parameters Te value of te statstc testng a dstant value of any of te structural parameters s terefore te same All results extend to tests on te parameters of te ncluded exogenous varables Introducton For te omoscedastc lnear nstrumental varables IV) regresson model wt one ncluded endogenous varable, sze correct procedures exst to conduct tests on ts structural parameter, see e:g: Anderson and Rubn 949), Klebergen ) and Morera 3) Andrews et al Econometrcs and Statstcs Secton, Amsterdam Scool of Economcs, Unversty of Amsterdam, Roetersstraat, 8WB Amsterdam, Te Neterlands Emal: frklebergen@uvanl

2 6) sow tat te condtonal) lkelood rato statstc s optmal amongst sze correct procedures tat test a pont null ypotess aganst a two sded alternatve E cent tests of ypoteses spec ed on one structural parameter n a lnear IV regresson model wt several ncluded endogenous varables wc are sze correct under weak nstruments are, owever, stll lackng Tere are statstcs for testng ypoteses on subsets of te parameters tat are sze correct and near-optmal under weak nstruments for te untested structural parameters but wc are not e cent under strong nstruments, lke, for example, te subset Anderson-Rubn AR) statstc, see Guggenberger et al ) and Guggenberger et al 7) Tere are also statstcs tat are e cent under strong nstruments but wc are not sze correct under weak nstruments, lke, for example, te t-statstc Neter one of tese statstcs leads to con dence sets for all structural parameters, ncludng tose on te ncluded exogenous parameters, wc are vald under weak nstruments and ave mnmum lengt under strong nstruments We construct a condtonal crtcal value functon for te subset lkelood rato LR) statstc wc makes t sze correct under weak nstruments and e cent under strong nstruments Tus t allows for te constructon of optmal con dence sets tat reman vald under weak nstruments Te condtonal crtcal value functon for te subset LR statstc tat we construct s dentcal to te condtonal crtcal value functon of te LR statstc for te omoscedastc lnear IV regresson model wt one ncluded endogenous varable from Morera 3) Tat condtonal crtcal value functon depends on a condtonng statstc and two ndependent dstrbuted random varables Instead of te common spec caton of te condtonng statstc as n Morera 3), t can also be spec ed as te d erence between te sum of te two smallest) roots of te caracterstc polynomal assocated wt te lnear IV regresson model and te value of te AR statstc at te ypoteszed value of te structural parameter Ts spec caton of te condtonng statstc generalzes to te condtonng statstc of te condtonal crtcal value functon of te subset LR statstc wc conducts tests on one structural parameter wen tere are several ncluded endogenous varables Alongsde te condtonng statstc, te condtonal crtcal value functon of te subset LR statstc also as te usual degrees of freedom adjustment of one of te nvolved dstrbuted random varables wen conductng tests on subsets of parameters Wen testng a value of te structural parameter tat s dstant from te true one, te subset AR and LR statstcs no longer depend on te structural parameter tat s tested Hence, for large values of te ypoteszed parameter, te value of te subset AR and LR statstcs are te same for all structural parameters At tese values, te subset AR and LR statstcs are

3 dentcal to statstcs tat test te ypotess of a reduced rank value of te reduced form parameter matrx Te rank condton for dent caton s for te reduced form parameter matrx to ave a full rank value so at dstant values of te ypoteszed structural parameter, te subset AR and LR statstcs become dentcal to tests of te dent caton of all structural parameters For te omoscedastc lnear IV regresson model wt one ncluded endogenous varable, Andrews et al 6) sow tat te LR statstc s optmal Tey construct te power envelope for testng a pont null ypotess on te structural parameter aganst a two-sded pont alternatve Te rejecton freuences of te LR statstc usng te condtonal crtcal value functon are on te power envelope so te LR statstc s optmal Under pont ypoteses on te structural parameter, te lnear IV regresson model wt one ncluded endogenous varable s euvalent to a lnear regresson model so te power envelope can be constructed usng te Neyman-Pearson Lemma Wen te null ypotess concerns te structural parameter of one ncluded endogenous varable of several, te lnear IV regresson model no longer smpl- es to a lnear regresson model under te null ypotess We can ten no longer use te Neyman-Pearson Lemma to construct te power envelope Alternatvely we could determne te maxmal rejecton freuency under least favorable alternatve ypoteses Least favorable alternatves result wen te structural parameters of te remanng ncluded endogenous varables are not dent ed Gven te beavor of te subset AR and LR statstcs at dstant values of te ypoteszed parameter, te maxmal rejecton freuency under least favorable alternatves euals te sze of tests for te dent caton of te non-dent ed) structural parameters of te remanng endogenous varables It terefore does not provde a useful caracterzaton of e cency of sze correct subset tests n te lnear IV regresson model eter Wen all nonypoteszed structural parameters are well dent ed, testng a ypotess on te remanng structural parameter usng te subset LR statstc s euvalent to testng te structural parameter n a lnear IV regresson model wt only one ncluded endogenous varables usng te LR statstc Snce te LR statstc s optmal n tat settng, te subset LR statstc s optmal wen all non-ypoteszed structural parameters are well dent ed and sze correct n general Te optmalty results for testng te structural parameter n te omoscedastc lnear IV regresson model wt one ncluded endogenous varable ave been extended n d erent drectons Andrews 5), Montel Olea 5) and Morera and Morera 3) extend t to general covarance structures wle Montel Olea 5) and Cernozukov et al 9) analyze te admssblty of suc tests Neter one of tese extensons, owever, analyzes tests on subsets of te structural parameters 3

4 Te omoscedastc lnear IV regresson model s a fundamental model n econometrcs It provdes a stylzed settng for analyzng nference ssues wc makes t stragtforward to communcate te results As suc tere s an extensve lterature on t Ts paper provdes a furter contrbuton by solvng an mportant open problem: ow to optmally construct con dence sets wc reman vald wen nstruments are weak for all structural parameters Te lnear IV regresson model wt d errors can be extended by allowng, for example, for autocorrelaton and/or eteroscedastcty Tese extensons are emprcally relevant and wen te structural parameters are well dent ed, nference metods extend stragtforwardly Klebergen 5) sows tat te same reasonng apples to te weak nstrument robust tests on te full structural parameter vector Te extensons to tests on subsets of te parameters are, owever, far less stragtforward Tey can be obtaned for te omoscedastc lnear IV regresson model because of te algebrac structure t provdes, see also Guggenberger et al ) Ts structure s lost wen te errors are autocorrelated and/or eteroscedastc We ten bascally ave to resort to explctly analyzng te rejecton freuency of te subset tests over all possble values of te nusance parameters as, for example, n Andrews and Cen ) Unless you resort to projecton based tests, weak nstruments robust tests on subsets of te parameters for te lnear IV regresson model wt a more general error structure s terefore conceptually very d erent from a settng wt d errors It s tus mportant to determne te extent to wc t s analytcally possble to analyze te dstrbuton of tests on subsets of te parameters wle allowng for weak dent caton Snce te estmators tat are used for te non-ypoteszed structural parameters are nconsstent n suc settngs, t s from te outset unclear f any suc analytcal results can be obtaned Te paper s organzed as follows Te second secton states te subset AR and LR statstcs In te trd secton, we dscuss te bound on te condtonal crtcal value functon of te subset LR statstc Te fourt secton dscusses a smulaton experment wc sows tat te subset LR statstc wt condtonal crtcal values s sze correct Te ft secton provdes extensons to more tan two ncluded endogenous varables Te sxt secton covers te beavor of te subset AR and LR statstcs at dstant values of te ypoteszed parameter Te sevent secton deals wt te usual d omoscedastc settng to wc all results stragtforwardly extend Fnally, te egt secton concludes We use te followng notaton trougout te paper: veca) stands for te column) vectorzaton of te k n matrx A; veca) a : : : a n) for A a : : : a n ); P A AA A) A s a projecton on te columns of te full rank matrx A and M A I N P A s a projecton on te space ortogonal to A: Convergence n probablty s denoted by and convergence n p 4

5 dstrbuton by d Subset statstcs n te lnear IV regresson model We consder te lnear IV regresson model y X + W + " X Z X + V X W Z W + V W ; ) wt y and W N and N m w dmensonal matrces tat contan endogenous varables, X a N m x dmensonal matrx of exogenous or endogenous varables, Z a N k dmensonal matrx of nstruments and m m x + m w : Te spec caton of X s suc tat we allow for tests on te parameters of te ncluded exogenous varables Te N ; N m w and N m x dmensonal matrces "; V W and V X contan te dsturbances Te unknown parameters are contaned n te m x ; m w ; k m x and k m w dmensonal matrces ; ; X and W Te model stated n euaton ) s used to smplfy te exposton An extenson of te model tat s more relevant for practcal purposes arses wen we add a number of so-called ncluded control) exogenous varables, wose parameters we are not nterested n, to all euatons n ) Te results tat we obtan do not alter from suc an extenson wen we replace te expressons of te varables tat are currently n ) n te spec catons of te subset statstcs by te resduals tat result from a regresson of tem on tese addtonal ncluded exogenous varables Wen we want to test a ypotess on te parameters of te ncluded exogenous varables, we just nclude tem as elements of X: To furter smplfy te exposton, we start out as n, for example, Andrews et al 6), assumng tat te rows of u "+V W +V X ; V W and V X ; wc we ndcate by u ; VW; ; and V X; ; so u u : : : u N ) ; V W V W; : : : V W;N ) ; V X V X; : : : V X;N ) ; are d normal dstrbuted wt mean zero and known covarance matrx : We also assume tat te nstruments n Z Z : : : Z N ) are pre-determned Tese random varables are terefore uncorrelated wt te nstruments Z so: EZ " V X; V W;)) ; ; : : : ; N: ) Wen X conssts of exogenous varables, t s part of te matrx of nstruments as well so V X s n tat case eual to zero 5

6 We extend ts n Secton 7 to te usual d omoscedastc settng We are nterested n testng te subset null ypotess H : aganst H : 6 : 3) In Guggenberger et al ), te subset AR statstc for testng H s analyzed We focus on te subset LR statstc Te dstrbutons of tese statstcs for testng te jont ypotess H : and ; 4) are robust to weak nstruments, see e:g: Anderson and Rubn 949), Morera 3) and Klebergen 7) Te expressons of ter subset counterparts result wen we replace te ypoteszed value of ; ; n te expresson of tese statstcs to test te jont ypotess by te lmted nformaton maxmum lkelood LIML) estmator under H ; wc we ndcate by ~ ): We note beforeand tat our results only old wen we use te LIML estmator and do not apply wen we use te two stage least suares estmator Snce te subset LR statstc nvolves te subset AR statstc, we state bot ter expressons De nton : Te subset AR statstc tmes k) to test H : reads AR ) mn R mw y X W ) P Z y X W ) : : ) : : ) ^ "" ) y X W ~ )) P Z y X W ~ )) mn ; 5) wt ~ ) te LIMLK) estmator, ^ "" ) ~ ) ) ~ ) ; ) and mn euals te smallest root of te caracterstc polynomal I mw C A I mw ) Y X W ) P Z Y X W ) : 7) Snce we treat te reduced form covarance matrx as known, te LIML estmator s dentcal to te LIMLK estmator, see e:g: Anderson et: al: 983) C A 6) 6

7 Te subset LR statstc to test H reads wt LR ) mn mn ; 8) mn mn R mx; R mw y X W ) P Z y X W ) : : ) : : ) ; 9) wc euals te smallest root of te caracterstc polynomal y X W ) P Z y X W ) : ) Under H and wen W as a full rank value, te subset AR statstc as a k m W ) lmtng dstrbuton Ts dstrbuton provdes an upper bound on te lmtng dstrbuton of te subset AR statstc for all values of W ; see Guggenberger et al ) Alongsde te bound on te lmtng dstrbuton of te subset AR statstc, Guggenberger et al ) also sow tat te score or Lagrange multpler statstc to test H s sze dstorted Wle te subset AR statstc s sze correct under weak nstruments, t s less powerful tan optmal tests of H under strong nstruments, lke, for example, te t-statstc It s terefore mportant to ave statstcs tat test H wc are sze-correct under weak nstruments and are as powerful as te t-statstc under strong nstruments Te subset LR statstc s suc a statstc 3 Subset LR statstc Te weak nstrument robust statstcs proposed n te lterature to test H are based upon ndependently dstrbuted su cent statstcs Tese can be constructed under te jont ypotess H but not under te subset ypotess H : To obtan a weak nstrument robust nference procedure for testng H usng te subset LR statstc, we terefore proceed n tree steps: We caracterze te condtonal dstrbuton of te subset LR statstc under te jont ypotess H 4) wc depends on mm + ) condtonng statstcs de ned under H : We construct a bound on te condtonal dstrbuton of te subset LR statstc under te jont ypotess H tat depends on only m x condtonng statstcs wc are de ned under H 3 We provde an estmator for te condtonng statstcs wc can be computed under H and sow tat ts leads to a condtonal boundng dstrbuton for te subset LR statstc 7

8 3 Subset LR statstc under H : Te subset LR statstc conssts of two components, :e: te subset AR statstc and te smallest root mn ) Teorems and state tem as functons of te ndependent su cent statstcs de ned under H : For reasons of brevty, we ntally focus only on te case of one structural parameter tat s tested and one wc s left unrestrcted so m x m w : We later extend ts to more unrestrcted structural parameters Teorem rst states te ndependent su cent statstcs de ned under H and tereafter expresses te subset AR statstc as a functon of tem Teorem states te smallest caracterstc root mn as a functon of te ndependent su cent statstcs Teorem Under H : ; ; te ndependent su cent statstcs: ; ) Z Z) Z y W X ) "" ; ) Z Z) Z W X) y W X ) "V "" V V:" ; ) wc are N; I k ) and NZ Z) W X ) V V:" ; I mk) dstrbuted random varables wt "" V " "V V V I mx I mw C A I mx I mw C A ; ) "" : ; V " "V : m ; V V : m m and V V:" V V V " "V "" ; can be used to specfy te dstrbuton of te subset AR statstc tat tests H : as AR ) +g g ; ) ; ) g ; ) ; ) g ' s ' s ) 4 + )s mn gr mw 3) 8

9 were ; ) ; ) I mw ; ) ; ) N; I mw ) ' [ I mx ; ) ; )] I mx [ I mx ; ) ; )] ; ) ; ) N; I mx ) ; )? ; ) N; I k m ) s ; ) ; ) 4) wt '; and ndependently dstrbuted, ; )? s a k k m) dmensonal ortonormal matrx wc s ortogonal to ; ) : ; )? ; ) and ; )? ; )? I k m ; "" : ; V " "V : m ; V V : m m and V V:" V V V " "V "" : Proof see te Appendx and Morera 3) Teorem Under H : ; ; te smallest caracterstc root mn ) euals mn mn br mx; gr mw +b b+g g ; ) and s dentcal to te smallest root of te caracterstc polynomal: I m+ ; ) b g ; ) ; ) g b ; 5) + S S S 6) wt S dags max; s mn); s max s mn; a dagonal matrx tat contans te two egenvalues of ; ) ; ) n descendng order and so and are m and k vectors ; ) ; )) ; ) ; ); 7) Proof see te Appendx and Klebergen 7) m dmensonal ndependent standard normal dstrbuted random Te closed form expresson for te dstrbuton of te subset AR statstc n Teorem results snce t s te smallest root of a second order polynomal Te smallest root n Teorem results from a trd order polynomal so we only provde t n an mplct manner Teorems and state te dstrbutons of te subset AR statstc and te smallest root mn as functons of te ndependent su cent statstcs ; ) and ; ) ) wc are de ned under 9

10 H 3 Snce ; ) and ; ) are ndependent, we use te condtonal dstrbutons of te subset AR statstc and te smallest root mn gven te realzed value of a functon of) ; ) : ^ ; ); see Morera 3) Teorems and sow tat tese furter smplfy so we can use te condtonal dstrbutons of te subset AR statstc gven te realzed value of s ; ^s ; and te condtonal dstrbuton of mn gven te realzed values of s mn and s max : ^s mn; ^s max: Ts makes te total number of condtonng statstcs eual to tree Teorem 3 sows tat tese tree condtonng statstcs are an nvertble functon of ^ ; ) ^ ; ): Teorem 3 also sows ow, gven ^ ; ) ^ ; ); we can construct '; ) from ; wc s a standard normal dstrbuted random vector, and vce versa Snce bot and are standard normal dstrbuted random vectors, tey consttute te random components n te condtonal dstrbuton of te subset LR statstc under H gven te realzed value ^ ; ) ^ ; ): Teorem 3 Under H : ; ; te condtonal dstrbuton of te subset LR statstc tat tests H : gven te realzed value of ; ) ; ); ^ ; ) ^ ; ); can be spec ed as LR ) ' ^s ' ^s ) 4 + )^s mn ; 8) were mn results from 6) usng te realzed value of S Te relatonsp between '; ; ^s ) used n Teorem and ; ^s mn; ^s max) from Teorem s caracterzed by ' ^s ^ ; ) ^ ; ) I mw VS V I mw VS VS I mx V VS I mx I mx SV I mw VS V cos^) wt V sn^) ^smax cos^) ^s mn sn^) r ' cos^) ^s VS V C A ' + S V max + cos^) ^s max + cos^)^s max sn^)^s mn [cos^)] ^s max+[sn^)] ^s mn sn^) + cos^) ^s mn ^smax r sn^)) ^s + cos^)) max ^s mn I mx I mx VS V I mx sn^) ^s C A, sn^) ^s mn + sn^)^s max cos^)^s sn^)) + cos^)) mn ^s max ^s mn 9) sn^) ; : te matrx of ortonormal egenvectors of ^ cos^) ; ) ^ ; ) mn 3 see Morera 3) and Andrews et: al: 6) for a proof tat ; ) and ; ) are su cent statstcs for te parameters under H wc tey reman to be under H :

11 Proof It results from te sngular value decomposton, ^ ; ) USV ; wt U and V k m and m m dmensonal ortonormal matrces, e U U I m, V V I m ; and te dagonal m m matrx S contanng te m non-negatve sngular values ^s : : : ^s m ) n decreasng order on te man dagonal, tat U ; ): Te remanng part results from usng te sngular value decomposton for te expressons n Teorems and Te condtonal dstrbuton of te subset LR statstcs s a functon of tree condtonng statstcs none of wc s de ned under H To obtan a workable bound of t, we rst reduce te number of condtonng statstcs for wc we tereafter provde estmators wc are de ned under H : 3 Bound on subset LR statstc wt one condtonng statstc Te condtonal dstrbuton of te subset LR statstc depends n an mplct manner on ts condtonng statstcs Ts makes t ard to sow tat t s a monotone functon of any or several) of tem wc would make t stragtforward to obtan a bound on t In order to construct suc a bound, we terefore start out to sow tat te two elements tat comprse te subset LR statstc are monotone functons of some of) ter condtonng statstcs Teorem 4 Te condtonal dstrbutons of te subset AR statstc and mn gven ^s ; ^s mn; ^s max) are respectvely non-decreasng functons of ^s and ^s max: Proof see te Appendx Teorem 4 mples tat te condtonal dstrbutons of te subset AR statstc and mn are bounded by ter condtonal) dstrbutons tat result for te smallest and largest feasble values of te realzed value of ter condtonng statstcs ^s and ^s max resp Gven te realzed value of s mn; ^s mn; bot ^s and ^s max can be n nte wle ter lower bounds are eual to ^s mn:

12 Teorem 5 Gven te realzed value of s mn : ^s mn; te condtonal dstrbuton of te subset AR statstc s bounded accordng to AR low js ^s mn) ARjs ^s mn) ' ^s mn ' ^s mn ) AR )js ^s ) + AR up ARjs ) k m w ) 4 + )^s mn ) and te condtonal dstrbuton of mn s bounded accordng to low js mn ^s mn) mn js mn ^s mn; s max ^s mn) ^s mn ^s mn 4 ^s mn mn js mn ^s mn; s max ^s max) + + ^s mn + + ^s mn 4 ^s mn mn js mn ^s mn; s max ) up js mn ^s mn): ) Proof see te Appendx Snce ^s mn ^s ^s max 4 ; te bounds on te condtonal dstrbuton of te subset AR statstc are rater wde but tey are sarp for large values of ^s mn: Bot te lower and upper bound of te condtonal dstrbuton of mn are non-decreasng functons of ^s mn and are eual wen ^s mn euals zero and for large values of ^s mn n wc case tey bot eual : It mples tat tey are tgt wc can be furter ver ed by conductng a mean-value expanson of te lower bound Te bounds are tgt snce te condtonal dstrbuton of mn gven s mn ^s mn; s max ^s max) prmarly depends on ^s mn and muc less so on ^s max as one would expect from te smallest caracterstc root) Te condtonal dstrbuton of te subset LR statstc stated n Teorem 3 depends on tree condtonng statstcs wc are all de ned under H : Te tree condtonng statstcs result from te tree d erent elements of te estmator of te concentraton matrx ^ ; ) ^ ; ): Ts estmator provdes an ndependent estmate of te dent caton strengt of te two parameters restrcted under H : Under H ; tere s only one restrcted parameter so ts dent caton strengt can be represented by one condtonng statstc Te smallest caracterstc root of ^ ; ) ^ ; ) s re ected by ^s mn: Snce t re ects te mnmal den- 4 Snce ^s ^ ; ) ^ ; ) ; ^s s bounded by te smallest and largest caracterstc roots of ^ ; ) ^ ; ) so ^s mn ^s ^s max:

13 t caton strengt of any combnaton of te parameters n H, we use t as te condtonng statstc n a boundng functon of te condtonal dstrbuton of te subset LR statstc gven ^ ; ) ^ ; ): Te boundng functon ten results as te d erence between te upper boundng functons of te subset AR statstc and mn stated n Teorem 5 It s obtaned by notng tat ^s max ^s sn^) ^s [cos^)] mn ; ) so wen ^s goes o to n nty, cos^) 6 ; ^s max goes o to n nty as well Oter settngs of te d erent condtonng statstcs do not result n an upper bound For example, consder sn^) ; ^s ^s mn so ^s max ^s mn; wc results from applyng l Hôptal s rule to ) Snce te subset AR statstc, wc consttutes te rst component of te subset LR statstc n 8), s an ncreasng functon of ^s ; we obtan a lower bound on te subset AR statstc gven ^s mn so te resultng settng for te subset LR statstc s more akn to a lower bound tan an upper bound De nton We denote te condtonal dstrbuton of te subset LR statstc gven ^s ; ^s mn; ^s max) tat results from Teorem 3 wen cos ^) 6 ; ^s and ^s max go o to n nty, so ' and ; by CLR ) : 5 CLR )js mn ^s mn) lm ^s ; ^s LR max) ) + ^s mn ^s mn ) 4 ^s mn : 3) We use CLR ) de ned n 3) as a condtonal bound gven ^s mn for te condtonal dstrbuton of LR ) gven ^s mn; ^s max; ^s ): It euals te d erence between te upper bounds on AR ) and mn stated n Teorem 4 wt eual to : Te d erence between te upper bounds of two statstcs not necessarly provdes an upper bound on te d erence between te two statstcs Here t does snce te upper bound on te subset AR statstc as a lot of slackness wen mn s close to ts lower bound To prove ts, we specfy te condtonal dstrbuton of te subset LR statstc as LR ) CLR ) D ); 4) 5 Te expresson of CLR ) s dentcal to tat of Morera s 3) condtonal lkelood rato statstc wc explans te acronym 3

14 wt D ) AR up AR ) + mn + + ^s mn + + ^s mn ) 4 ^s mn : 5) and analyze te propertes of te condtonal approxmaton error D ) gven ^s mn over te range of values of ^s max and ^s ^): We note tat only negatve values of D ) can lead to sze dstortons so we only focus on worst case settngs of te condtonng statstcs ŝ ; ^s mn; ^s max) tat lead to suc negatve values Teorem 6 Under H ; te condtonal dstrbuton of CLR ) gven s mn ^s mn provdes an upper bound for te condtonal dstrbuton of LR ) gven s mn ^s mn; s max ^s max; s ^s ) snce te approxmaton error D ) s non-negatve for all values of ^s mn; s max; ^s ): Proof see te Appendx Teorem 6 s proven usng approxmatons to te d erent components of D ): Tese approxmatons are analyzed over te range of values ^s mn; s max; ^s ) can take For none of tese do we nd tat D ) s negatve Corollary Under H ; te rejecton freuency of a -) % sgn cance test of H usng te subset LR test wt condtonal crtcal values from CLR ) gven ^s mn s less tan or eual to % Wle te condtonal crtcal value functon makes te subset LR test of H sze correct, t s nfeasble snce te condtonng statstc ^s mn s de ned under H : We next construct a feasble estmator for ^s mn under H wc s suc tat te resultng condtonal crtcal value functon makes te subset LR statstc a sze correct test of H : 33 Condtonng statstc under H To motvate our estmator of ^s mn under H ; we start out from te caracterstc polynomal n 6) wc s wen, m w m x ; a trd order polynomal: max ) ) mn ) 3 a + a a 3 ; 6) 4

15 wt, resultng from Teorem : a + + s mn + s max tr Y X W ) P Z Y X W )) mn + + max a s mn + s max) + s mns max + s max + s mn a 3 s mns max mn max ; 7) and were mn max are te tree roots of te caracterstc polynomal n 6) We next factor out te largest root max to specfy te trd order polynomal as te product of a rst and second order polynomal: 3 a + a a 3 max ) b + b ) ; 8) wt b + + s mn + s max max b s mns max max : 9) We obtan our estmator for te condtonng statstc ^s mn from te second order polynomal In order to so, we use tat max provdes an estmator of s max + : Teorem 7 Under H ; te largest root max s suc tat max s max + + s max + ) + ; 3) wt s max s max + and Omaxs 4 max + ) ; s mn s 4 max)) ; were Oa) ndcates tat te respectve element s proportonal to a: Proof see te Appendx Teorem 7 sows tat max s an estmator of s max + wc gets more precse wen s max ncreases We use t to purge s max + from te expresson of b : wt d b d + s mn; 3) s + ) : 3) max Snce s non-negatve, te statstc d n 3) s bounded from above by a k ) dstrbuted random varable Teorem 4 sows tat under H ; te subset AR statstc s also bounded from above by a k ) dstrbuted random varable We terefore use te subset AR statstc 5

16 as an estmator for d n 3) to obtan te estmator for te condtonng statstc ^s mn tat s feasble under H : ~s mn b AR ) tr Y X W ) P Z Y X W )) max AR ) smallest caracterstc root of Y X W ) P Z Y X W ))+ second smallest caracterstc root of Y X W ) P Z Y X W )) AR ): 33) We use ~s mn as te condtonng statstc for te condtonal boundng dstrbuton CLR ) gven tat s mn ~s mn 3) Te condtonng statstc ~s mn n 33) estmates s mn wt error so t s mportant to determne te propertes of ts estmaton error Teorem 8 Under H ; te estmator of te condtonng statstc ~s mn can be spec ed as: ~s mn s mn + g; 34) wt g + ' ' +s + ) s max + ) + e; 35) and were e O Proof see te Appendx ' ; ) M ; ) ) ; ) ' + ) ; ) ; ) ) ): Te common element n te upper) boundng dstrbutons of te statstc d and te subset AR statstc s te k ) dstrbuted random varable : It mples tat te d erence between tese two statstcs, wc consttutes te estmaton error n ~s mn; conssts of: Te d erence between two possbly correlated ) dstrbuted random varables: ; 36) wt tat part of ; ) tat s spanned by te egenvectors of te smallest sngular value of ; ) and tat part of ; ) tat s spanned by ; ) I mx : Te d erence between te devatons of d and AR ) from ter boundng k ) dstrbuted random varables: ' ' +s + ) s max + ) + e: 37) 6

17 Snce s s smaller tan or eual to s max; ts error s largely non-negatve and becomes neglgble wen s and s max get large Snce s as a non-central dstrbuton wt k degrees of freedom ndependent of '; and ; and a smlar argument apples to s max; ; and ; te combned e ect of te components n 37) s small, snce every element s at most of te order of magntude of one and a decreasng functon of s and s max: Te same argument apples to 36) as well Corollary Te estmaton error for estmatng s mn by ~s mn s bounded and decreasng wt te strengt of dent caton of Te dervatve of CLR ) gven s mn wt respect to s mn CLR )js mn s ) + p +s +s ) +4 < ; 38) wc s constructed n Lemma n te Appendx; s suc tat CLR ) s not senstve to te value of s mn: Tus small errors n te estmaton of s mn just lead to a small cange n te condtonal crtcal values gven ~s mn wt lttle e ect on te sze of te subset LR test under H : Corollary and 38) mply tat te estmaton error n ~s mn as just a mnor e ect on te sze of te subset LR test under H : We next provde a more detaled dscusson of te e ect of te estmaton error n ~s mn on te sze of te subset LR test Under H ; te condtonng statstc s mn s ndependent of ; ) wle te components of te estmaton error g n 36) and 37) are not We terefore analyze te propertes of te estmaton error n ~s mn and ts e ect wen usng ~s mn for te approxmaton of te condtonal dstrbuton of te subset LR statstc 3) One part of te estmaton error results from te devaton of te dstrbuton of te subset AR statstc from ts boundng k ) dstrbuton We terefore assess te two fold e ect tat ts devaton as: one drectly on te subset LR statstc troug te subset AR statstc and one on te approxmate condtonal dstrbuton troug ts e ect on ~s mn: We analyze te e ect of te estmaton error n ~s mn on te approxmate condtonal dstrbuton of te subset LR statstc for four d erent cases: Strong dent caton of and : Bot and are well dent ed, so s mn s large and s s mn) s large as well Ts mples tat bot components of te subset LR statstc are at ter upperbounds stated n Teorem 4 so te condtonal dstrbuton of te subset LR statstc corresponds wt tat of CLR ): Snce bot s and s max are large, te estmaton error s: g : 39) 7

18 Te proof of Teorem 8 sows te expressons of te covarance between and wc, snce bot s mn and s max are large, can not be large Te estmaton error s terefore O p ): Te dervatve of te approxmate condtonal dstrbuton of te subset LR statstc wt respect to s mn goes to zero wen s mn gets large Hence, snce s mn s large, te estmaton error n ~s mn as no e ect on te accuracy of te approxmaton of te condtonal dstrbuton of te subset LR statstc Strong dent caton of ; weak dent caton of : Snce s weakly dent ed s mn s small but s s large because s strongly dent ed and so s terefore s max: Snce bot s and s max are large, bot components of te subset LR statstc are at ter upperbounds stated n Teorem 4 wc mples tat te condtonal dstrbuton of te subset LR statstc euals tat of CLR ): Also snce s and s max are large, te estmaton error n ~s mn s just g : 4) Because s mn s small and s s large, Teorem 3 sows tat cos) s close to one wle sn) s close to zero Ts mples tat s approxmately eual to so g s small Te estmaton error does terefore not lead to sze dstortons wen usng te approxmaton of te condtonal dstrbuton of te subset LR statstc 3 Weak dent caton of ; strong dent caton of : s weakly dentfed so s mn and s are small wle s max s large snce s strongly dent ed Snce s max s large, mn s at ts upperbound up : Te d erence between te condtonal dstrbuton of te subset LR statstc and te condtonal boundng dstrbuton of CLR ) ten solely results from te d erence between te upper bound on te dstrbuton of te subset AR statstc, AR up, and ts condtonal dstrbuton Wen usng condtonal crtcal values from CLR ) gven s mn for te subset LR test, t s conservatve We, owever, use ~s mn nstead of s mn wt estmaton error g : g + ' ' + ) ; ) ; ) ) + ) + e; 4) wc, snce t ncreases te estmate of te condtonng statstc ~s mn; reduces te condtonal crtcal values Te last part of 4) results from te subset AR statstc Snce te condtonal crtcal values of CLR ) gven s mn make te subset LR statstc test conservatve for ts settng, te decrease of te condtonal crtcal values does not lead to over-rejectons Ts olds snce te reducton of te subset AR statstc compared to ts boundng k ) dstrbuton exceeds te decrease of te condtonal dstrbuton of CLR ) gven ~s mn nstead of s mn: Te latter results snce te dervatve of te condtonal dstrbuton of CLR ) gven s mn wt 8

19 respect to s mn exceeds mnus one Hence, usage of te condtonal crtcal values of CLR ) gven ~s mn make te subset LR test conservatve for ts settng Weak dent caton of and strong dent caton of covers te parameter settng for wc Guggenberger et al ) sow tat te subset score statstc from Klebergen ) for testng H s sze dstorted Ts sze dstorton occurs for values of W and X wc are suc tat W X wt X relatvely large so s well dent ed and a small scalar so s weakly dent ed Tese settngs tus do not lead to sze dstorton for te subset LR test wen usng te condtonal crtcal values tat result from CLR ) gven ~s mn: 4 Weak dent caton of and : Bot s mn and s max are small and so s terefore s : Te proof of Teorem 6 n te Appendx sows tat te error of approxmatng te subset LR statstc by CLR ) gven s mn s non-negatve for ts settng Usage of te condtonal crtcal values tat result from CLR ) gven s mn would ten make te subset LR test conservatve Wen we use ~s mn nstead of s mn; te estmaton error g s ten suc tat bot te boundng dstrbutons of d and te subset AR statstc devate from ter k bounds so te estmaton error contans all components of 35) devaton of te boundng dstrbuton of te subset AR statstc from a k ) dstrbuted lower Te twofold e ect of te ) dstrbuton s now dmnsed snce ts contrbuton to te estmator of te condtonng statstc ~s mn s largely o set by te devaton of te boundng dstrbuton of d from a k Hence, v v + ) ; ) ; ) ) + ' ') ) dstrbuton s max + ) + e ; 4) s small Also te oter component of g s typcally small snce and are gly correlated wen bot and are weakly dent ed Ts all mples tat ~s mn s close to s mn so te subset LR test remans conservatve wen we use condtonal crtcal values from CLR ) gven ~s mn nstead of CLR ) gven s mn: Summarzng, we observe no sze dstorton for any of te above settngs wen usng te subset LR test to test H wt condtonal crtcal values from CLR ) gven ~s mn: It s nterestng to note tat wen non-negatve estmaton errors n ~s mn occur, wc result wen s weakly dent ed, te subset LR test usng crtcal values from CLR ) gven s mn s conservatve wc o sets any sze dstortons wc mgt occur because of te larger crtcal values tat result from CLR ) gven ~s mn: 9

20 Spec caton of condtonng statstc s dentcal to te one wt ncluded endogenous varable For te lnear IV regresson model wt one ncluded endogenous varable: y X + " X Z X + V X ; 43) te AR statstc tmes k) for testng H reads AR ) ^ "" ) y X ) P Z y X ); 44) wt ^ "" ) and te known) reduced form covarance matrx, Te LR statstc for testng H euals te AR statstc mnus ts mnmal value over : Y Y XY Y X XX LR ) AR ) mn AR): 45) Ts mnmal value euals te smallest root of te uadratc polynomal: : wt a tr Y X) P Z Y X)) AR ) + s a s [AR ) LM )] LM ) Y X ^ "" ) ) P Z X ~ ) y X ) s ~ X ) Z Z ~ X )^ XX:" ) ~ X ) Z Z) Z X y X ) ^ X" ) ^ "" ) a + a ; 46) Z Z) Z y X) ^ and ^ XX:" ) X" ) XX ^ "" ) ; ^X" ) XY XX : Under H ; te LR statstc as a condtonal dstrbuton gven te realzed value of s wc s dentcal to 3) wt s mn eual to s and a k ) dstrbuted random varable, see Morera 3) Te statstc a n 47) does not depend on : For a gven value of AR ); we can terefore 47)

21 stragtforwardly recover s from a : s tr Y X) P Z Y X)) AR ) smallest caracterstc root of Y X) P Z Y X))+ second smallest caracterstc root of Y X) P Z Y X)) AR ); 48) wc sows tat te spec caton of te condtonng statstc for te condtonal dstrbuton of te condtonal lkelood rato statstc for te lnear IV regresson model wt one ncluded endogenous varable s dentcal to ~s mn n 33) 4 Smulaton experment To sow te adeuacy of usage of condtonal crtcal values tat result from CLR ) gven ~s mn for testng H usng LR ); we conduct a smulaton experment Before we do so, we rst state some nvarance propertes wc allow us to obtan general results by just usng a small number of nusance parameters Teorem 9 Under H ; te subset LR statstc only depends on te su cent statstcs ; ) and ; ) wc are de ned under H and ndependently normal dstrbuted wt means resp zero and Z Z) W X ) V V:" Proof see te Appendx and dentty covarance matrces Teorem 9 sows tat under H ; Z Z) W X ) V V:" s te only parameter of te IV regresson model tat a ects te subset LR statstc Te number of nusance) parameters were te subset LR statstc depends on s terefore eual to km: We furter reduce ts number Teorem Under H ; te dependence of te dstrbuton of te subset LR statstc on te parameters of te lnear IV regresson model s fully captured by te mm + ) parameters of te matrx concentraton parameter: V V:" W X ) Z Z W X ) V V:" R R ; 49) wt R an ortonormal m m matrx and a dagonal m m matrx tat contans te caracterstc roots

22 Proof see te Appendx In our smulaton experment we use two ncluded endogenous varables so m : We also use te spec catons for R and : R cos) sn) sn) ; ; cos) : 5) Wt tese tree parameters: ; and ; we can generate any value of te matrx concentraton parameter and terefore also every dstrbuton of te subset LR statstc In our smulaton experment, we compute te rejecton freuences of testng H usng te subset AR and LR statstcs for a range of values of ; ; and k: Ts range s cosen suc tat: < ; ; ; 5) and we use values of k from two to one undred For every parameter, we use fty d erent values on an eudstant grd and ve tousand smulatons to compute te rejecton freuency Maxmal rejecton freuency over te number of nstruments Fgure sows te maxmal rejecton freuency of testng H at te 95% sgn cance level usng te subset AR and LR statstcs over te d erent values of ; ; ) as a functon of te number of nstruments We use te crtcal value functon for te subset AR statstc and te condtonal crtcal values of CLR ) gven ~s mn for te subset LR statstc Fgure sows tat bot statstcs are sze correct for all numbers of nstruments

23 Rjecton freuency Fgure Maxmal rejecton freuences of subset AR dased) and subset LR sold) statstcs wen testng te 95% sgn cance level for d erent numbers of nstruments Number of nstruments Maxmal rejecton freuences as functon of te caracterstc roots of te matrx concentraton parameter To furter llustrate te sze propertes of te subset AR and LR tests, we compute te maxmal rejecton freuences over as a functon of ; ) for k 5; ; ; 5 and : Tese are sown n Panels -5 All panels are n lne wt Fgure 5 and sow no sze dstorton of eter te subset AR or subset LR tests Te panels sow tat bot tests are conservatve at small values of bot and : 3

24 Rejecton freuency Rejecton freuency Rejecton freuency Rejecton freuency Panel Maxmal rejecton freuency over for d erent values of ; ) for k 5: λ 4 4 λ λ 4 4 λ 6 8 Fgure subset AR statstc Fgure subset LR statstc Panel Maxmal rejecton freuency over for d erent values of ; ) for k : λ 4 4 λ λ 4 4 λ 6 8 Fgure subset AR statstc Fgure subset LR statstc 4

25 Rejecton freuency Rejecton freuency Rejecton freuency Rejecton freuency Panel 3 Maxmal rejecton freuency over for d erent values of ; ) for k : λ 4 4 λ λ 4 4 λ 6 8 Fgure 3: subset AR statstc Fgure 3 subset LR statstc Panel 4 Maxmal rejecton freuency over for d erent values of ; ) for k 5: λ 4 4 λ λ 4 4 λ 6 8 Fgure 4 subset AR statstc Fgure 4 subset LR statstc 5

26 Rejecton freuency Rejecton freuency Panel 5 Maxmal rejecton freuency over for d erent values of ; ) for k : λ 4 4 λ λ 4 4 λ 6 8 Fgure 5 subset AR statstc Fgure 5 subset LR statstc To sow te prevously referred to sze dstorton of te subset score statstc, Panels 6 and 7 sow te rejecton freuency of te subset LM statstc for testng H : Tese gures agan sow te maxmal rejecton freuency over as a functon of ; ): Tey clearly sow te ncreasng sze dstorton wen k gets larger wc occurs for settngs were W X wt X szeable and small so W s small and tangent to X : Te mpled value of s terefore of reduced rank so eter or s eual to zero 6

27 Rejecton freuency Rejecton freuency Rejecton freuency Rejecton freuency Panel 6 Maxmal rejecton freuency over as functon of ; ) for subset LM statstc λ 4 4 λ λ 4 4 λ 6 8 Fgure 6 k Fgure 6 k Panel 7 Maxmal rejecton freuency over as functon of ; ) for subset LM statstc λ 4 4 λ λ 4 4 λ 6 8 Fgure 63 k 5 Fgure 64 k 5 More ncluded endogenous varables Teorems,, 4 and 5 extend to more non-ypoteszed structural parameters, :e: settngs were m W exceeds one Teorem 3 can be generalzed as well to sow te relatonsp be- 7

28 tween te condtonng statstc of te subset AR statstc under H and te sngular values of ; ) ; ) for values of m larger tan two Combnng tese results, Corollary, wc states tat CLR ) gven ^s mn provdes a bound on te condtonal dstrbuton of te subset LR statstc, extends to values of m larger tan two Teorem 6 states te maxmal error of ts bound by runnng troug te d erent settngs of te condtonng statstcs Snce te number of condtonng statstcs s larger, we refran from extendng Teorem 6 to settngs of m larger tan two For te estmator of te condtonng statstc, Teorem 7 s extended n te Appendx to cover te sum of te largest m caracterstc roots of ) wen m exceeds two wle te bound on te subset AR statstc s extended n Lemma n te Appendx Hence, te estmator of te condtonng statstc ~s mn smallest caracterstc root Y X W ) P Z Y X W ))+ second smallest caracterstc root Y X W ) P Z Y X W )) AR ); 5) apples to tests of H : for any number of addtonal ncluded endogenous varable and so does te bound on te condtonal dstrbuton of te subset LR statstc stated n Corollary Range of values of te estmator of te condtonng statstc Te estmator of te condtonng statstc n 5) s a functon of te subset AR statstc Before we determne some propertes of ~s mn; we terefore rst analyze te beavor of te realzed value of te jont AR statstc tat tests H : ; as a functon of ) : Teorem Te realzed value of te jont AR statstc tat tests H : ; wt ) : AR H ) "") y ~ X) P Z y ~ X); s a functon of tat as a mnmum, maxmum and m ) saddle ponts Te values of te AR statstc at tese statonarty ponts are eual to resp te smallest, largest and, f m exceeds one, te second up to m-t root of te caracterstc polynomal ) Proof see te Appendx Teorem mples tat n a lnear IV regresson model wt one ncluded endogenous varable, te AR statstc as one mnmum and one maxmum wle n lnear IV models wt more ncluded endogenous varables, te AR statstc also as m 8 ) saddle ponts Saddle

29 ponts are statonary ponts at wc te Hessan s postve de nte n a number of drectons and negatve de nte n te remanng drectons Te saddle pont wt te lowest value of te jont AR statstc terefore results from maxmzng n one drecton and mnmzng n all oter m ) drectons Te subset AR statstc tat tests H results from mnmzng te jont AR statstc over at : Te maxmal value of te subset AR statstc s terefore smaller tan or eual to te smallest value of te jont AR statstc over te d erent saddle ponts snce t results from constraned optmzaton because of te orderng of te varables were you optmze over) Wen m ; te optmzaton s unconstraned, snce no mnmzaton s nvolved, so te maxmal value of te subset AR statstc s eual to te second smallest caracterstc root wc s n tat case also te largest caracterstc root Corollary 3 Te maxmal value of te subset AR statstc s less tan or eual to te second smallest caracterstc root of ): max AR) second smallest root Y X W ) P Z Y X W )): 53) Corollary 4 Te mnmal value of te condtonng statstc s larger tan or eual to te smallest caracterstc root of ): mn ~s mn smallest root Y X W ) P Z Y X W )): 54) Corollary 4 sows tat te beavor of te condtonng statstc as a functon of for larger values of m s smlar to tat wen m : 6 Testng at dstant values An mportant applcaton of subset tests s to construct con dence sets Con dence sets result from specfyng a grd of values of and computng te subset statstc for eac value of on te grd 6 Te ) % con dence set ten conssts of all values of on te grd for wc te subset test s less tan ts % crtcal value Tese con dence sets sow tat te subset LR statstc tat tests H : at a value of tat s dstant from te true one s dentcal to te subset LR statstc tat tests H : at a value of tat s dstant 6 Te con dence sets tat result from te subset tests can not yet) be constructed usng te e cent procedures developed by Dufour and Taamout 3) for te AR statstc and Mkuseva 7) for te LR statstc snce tese apply to tests on all structural parameters 9

30 from te true one and te same olds true for te subset AR statstc Teorem Wen m x ; Assumpton olds and for tests of H : for values of tat are dstant from te true value: a Te subset AR statstc AR ) euals te smallest egenvalue of XW X W ) P Z X W ) XW ; wt XW XX W X XW W W : b Te subset LR statstc euals LR ) mn mn ; 55) wt mn te smallest egenvalue of XW X W ) P Z X W ) XW and mn te smallest egenvalue of ) c Te condtonng statstc ^s mn euals ^s mn smallest caracterstc root Y X W ) P Z Y X W ))+ second smallest caracterstc root Y X W ) P Z Y X W )) smallest caracterstc root XW X W ) P Z X W )): 56) Proof see te Appendx Teorem sows tat te expressons of te subset AR and LR statstcs at values of tat are dstant from te true value do not depend on : Hence, te same value of te statstcs result wen we use tem to test for a dstant value of any element of : Te weak dent caton of one structural parameter terefore carres over to all te oter structural parameters Hence, wen te power for testng one of te structural parameters s low because of ts weak dent caton, t s low for all oter structural parameters as well Te smallest egenvalue of XW X W ) P Z X W ) XW s dentcal to Anderson s 95) canoncal correlaton reduced rank statstc wc s te lkelood rato statstc under omoscedastc normal dsturbances tat tests te ypotess H r : rank W X ) m w + m x ; see Anderson 95) Tus Teorem sows tat te subset AR statstc s eual to a reduced rank statstc tat tests for a reduced rank value of W X ) at values of tat are dstant from te true one Snce te dent caton condton for and s tat W X ) as a full rank value, te subset AR statstc at dstant values of s dentcal to a test for te dent caton of and : 3

31 7 Weak nstrument settng For ease of exposton, we ave assumed sofar tat te nstruments are pre-determned and u and V are jontly normal dstrbuted wt mean zero and a known value of te reduced form) covarance matrx Our results extend stragtforwardly to d errors, nstruments tat are possbly) random and an unknown covarance matrx : Te analogues of te subset AR and LR statstcs n De nton for an unknown value of are obtaned by replacng n tese expressons by te estmator: ^ N k y X W ) M Z y X W ); 57) wc s a consstent estmator of under te outlned condtons, ^ p : We next specfy te parameter space for te null data generatng processes Assumpton Te parameter space under H s suc tat: f f ; g : ; W ; X ); R mw ; W R kmw ; X R kmx ; F : EjjT jj + ) < M; for T f" ; V ; Z ; Z " ; Z V ; " V g; EZ " ) ; EZ V ) ; EvecZ " V ))vecz" V )) ) E" V ) " V )) EZ Z)) B C A 9 > C A ; >; for some > ; M < ; Q EZ Z ) postve de nte and R m+)m+) postve de nte symmetrc Assumpton s a common parameter space assumpton, see eg ), Andrews and Guggenberger 9) and Guggenberger et al ) 58) Andrews and Ceng To determne te asymptotc sze of te subset LR test, we analyze parameter seuences n wc lead to te spec caton of te model for a sample of N d observatons as y n X n + W n n + " n X n Z n X;n + V X;n W n Z n W;n + V W;n ; 59) wt y n : n ; X n : n m x ; W n : n m w ; Z n : n k; " n : n ; V X;n : n m x ; V W;n : n m w ; 3

32 : m x ; n : m w ; X;n : k m x ; W;n : k m w : Te rows of " n V X;n V W;n Z n ) are d dstrbuted wt dstrbuton F n : Te mean of te rows of " n V X;n V W;n Z n ) euals zero and ter covarance matrx s n "";n V ";n "V;n V V;n : 6) Tese seuences are assumed to allow for a sngular value decomposton, see e:g: Golub and Van Loan 989), of te normalzed reduced form parameter matrx Assumpton Te sngular value decomposton of n) Z nz n ) W;n X;n ) V V:";n tat results from a seuence n n ; W;n ; X;n ; F n ) of null data generatng processes n as a sngular value decomposton: n) Z nz n ) W;n X;n ) V V:";n H nt n R n R km ; 6) were H n and R n are k k and m m dmensonal ortonormal matrces and T n a k n rectangular matrx wt te m sngular values n decreasng order) on te man dagonal, wt a well de ned lmt Teorem 3 states tat te subset LR test s sze correct for weak nstrument settngs Teorem 3 Under Assumptons and, te asymptotc sze of te subset LR test of H wt sgn cance level : AsySz LR, lm sup n sup Pr LRn ) > CLR js mn ~s mn;n) ; 6) were LR n ) s te subset LR statstc for a sample of sze n and CLR js mn ~s mn) s te ) % uantle of te condtonal dstrbuton of CLR ) gven tat s mn ~s mn; s eual to for < < : Proof see te Appendx Eualty of te rejecton freuency of te subset LR test and te sgn cance level occurs wen s well dent ed Wen becomes less well dent ed, te subset LR test, dentcal to te subset AR test, becomes conservatve 3

33 8 Conclusons Inference usng te LR statstc to test a ypotess on one structural parameter n te omoscedastc lnear IV regresson model extends stragtforwardly from one ncluded endogenous varable to several Te rst and foremost extenson s tat of te condtonal crtcal value functon Te condtonal crtcal value functon of te LR statstc n te lnear IV regresson model wt one ncluded endogenous varable from Morera 3) extends wt te usual degrees of freedom adjustments of te nvolved dstrbuted random varables to te subset LR statstc tat tests a ypotess on te structural parameter of one of several ncluded endogenous varables n a lnear IV regresson model wt multple ncluded endogenous varables Te expresson of te condtonng statstc nvolved n te condtonal crtcal value functon also remans unaltered Ts spec caton of te condtonal crtcal value functon and ts condtonng statstc makes te LR statstc for testng ypoteses on one structural parameter sze correct A second mportant property of te condtonal crtcal value functon s optmalty of te resultng subset LR test under strong dent caton of all untested structural parameters Wen all untested structural parameters are well dent ed, te subset LR test becomes dentcal to te LR test n te lnear IV regresson model wt one ncluded endogenous varable for wc Andrews et al 6) sow tat te LR test s optmal under weak and strong dent caton of te ypoteszed structural parameter Establsng optmalty wle allowng for any knd of dent caton strengt for te untested parameters s complcated snce te usual optmalty crtera are often no longer sensble In Guggenberger et al 7), condtonal crtcal values for te subset AR statstc are constructed wc make t nearly optmal under weak nstruments for te untested structural parameters but not so under strong nstruments 33