Bootstrap Inference in a Linear Equation Estimated by Instrumental Variables

Transcription

1 Bootstrp Inference in Liner Eqution Estimted by Instrumentl Vribles GREQAM Centre de l Vieille Chrité 2 rue de l Chrité Mrseille cedex 02, Frnce Russell Dvidson emil: RussellDvidson@mcgillc nd Jmes G McKinnon Deprtment of Economics Queen s University Kingston, Ontrio, Cnd K7L 3N6 emil: jgm@econqueensuc Deprtment of Economics McGill University Montrel, Quebec, Cnd H3A 2T7 Abstrct We study severl tests for the coefficient of the single right-hnd-side endogenous vrible in liner eqution estimted by instrumentl vribles We show tht ll the test sttistics Student s t, Anderson-Rubin, Kleibergen s K, nd likelihood rtio (LR) cn be written s functions of six rndom quntities This leds to number of interesting results bout the properties of the tests under wek-instrument symptotics We then propose severl new procedures for bootstrpping the three non-exct test sttistics nd conditionl version of the LR test These use more efficient estimtes of the prmeters of the reduced-form eqution thn existing procedures When the best of these new procedures is used, K nd conditionl LR hve excellent performnce under the null, nd LR lso performs very well However, power considertions suggest tht the conditionl LR test, bootstrpped using this new procedure when the smple size is not lrge, is probbly the method of choice This reserch ws supported, in prt, by grnts from the Socil Sciences nd Humnities Reserch Council of Cnd We re grteful to seminr prticipnts t the University of New South Wles, the University of Sydney, nd the University of Cliforni Snt Brbr, nd to three referees, for comments on erlier versions Mrch, 2007

2 1 Introduction This pper is concerned with tests for the vlue of the coefficient of the single righthnd-side endogenous vrible in liner structurl eqution estimted by instrumentl vribles We consider the Wld (or t) test, Kleibergen s (2002) K test, which cn be thought of s n LM test, nd the likelihood rtio (LR) test, s well s its conditionl vrint (Moreir, 2003), which we refer to s CLR Both symptotic nd bootstrp versions of these tests re studied, nd their reltionships to the Anderson- Rubin (AR) test (Anderson nd Rubin, 1949) re explored The nlysis llows for instruments tht my be either strong or wek The mjor theoreticl contributions of the pper depend on simple wy of writing ll the test sttistics of interest s functions of six rndom quntities This mkes it esy to understnd the properties of ll the tests under both wek nd strong instruments Our theoreticl results re supported by extensive simultions Our results lso mke it inexpensive to simulte nd bootstrp ll the test sttistics under the ssumption of normlly distributed disturbnces Although, in prctice, it is generlly preferble to use bootstrp methods bsed on resmpling residuls, our experiments suggest tht results from the prmetric bootstrp under normlity provide very good guide to the performnce of methods tht resmple residuls The pper s min prcticl contribution is to propose new procedures for bootstrpping the test sttistics we study These use more efficient estimtes of the prmeters of the reduced-form eqution thn existing procedures, nd wht seems to be the best procedure lso employs form of bis correction Using this procedure insted of more conventionl ones gretly improves the performnce under the null of ll the tests The improvement is generlly gretest for the Wld test nd lest for the K test, becuse the ltter lredy works very well in most cses Using the new procedures lso severely reduces the pprent power of the Wld test when the instruments re wek, mking its power properties much more like those of the other tests The prcticl conclusions we come to differ only modestly from those of other recent ppers, such s Andrews, Moreir, nd Stock (2006), nd Moreir, Porter, nd Surez (2004) The K test, when bootstrpped using one of our new methods, seems to be the most relible procedure under the null when the instruments re wek nd the smple size is smll However, the CLR test of Moreir (2003), which involves simultion, lso seems to be very relible when the instruments re wek nd the smple size is resonbly lrge When the CLR test is bootstrpped, it performs s well s the K test, except in very smll smples Power considertions suggest tht either the conditionl LR test or the ordinry LR test, when the ltter is bootstrpped using the sme new method, seem to be the best procedures overll In the next section, we discuss the four test sttistics nd show tht they re ll functions of six rndom quntities Then, in Section 3, we show how ll the sttistics cn be simulted very efficiently under the ssumption of normlly distributed disturbnces In Section 4, we consider the symptotic properties of the sttistics under both strong nd wek instruments In Section 5, we discuss some new nd old wys of bootstrpping the sttistics nd show how, in some cses, the properties of bootstrp 1

3 tests differ from those of symptotic tests Finlly, in Section 6, we present extensive simultion evidence on the performnce of symptotic nd bootstrp tests bsed on ll of the test sttistics 2 The Four Test Sttistics The model treted in this pper consists of just two equtions, y 1 = βy 2 + Zγ + u 1, nd (1) y 2 = Wπ + u 2 (2) Here y 1 nd y 2 re n--vectors of observtions on endogenous vribles, Z is n n k mtrix of observtions on exogenous vribles, nd W is n n l mtrix of instruments such tht S(Z) S(W ), where the nottion S(A) mens the liner spn of the columns of the mtrix A The disturbnces re ssumed to be serilly uncorrelted nd, for mny of the nlyticl results, normlly distributed We ssume tht l > k, so tht the model is either exctly identified or, more commonly, overidentified The prmeters of this model re the sclr β, the k--vector γ, the l--vector π, nd the 2 2 contemporneous covrince mtrix of the disturbnces u 1 nd u 2 : [ σ 2 Σ 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2 ] (3) Eqution (1) is the structurl eqution we re interested in, nd eqution (2) is reduced-form eqution for the second endogenous vrible y 2 We wish to test the hypothesis tht β = 0 There is no loss of generlity in considering only this null hypothesis, since we could test the hypothesis tht β = β 0 for ny nonzero β 0 by replcing the left-hnd side of (1) by y 1 β 0 y 2 Since we re not directly interested in the prmeters contined in the l--vector π, we my without loss of generlity suppose tht W = [Z W 1 ], with Z W 1 = O Notice tht W 1 cn esily be constructed by projecting the columns of W tht do not belong to S(Z) off Z The 2SLS (or IV) estimte ˆβ from (1), with instruments the columns of W, stisfies the estimting eqution y 2 P 1 (y 1 ˆβy 2 ) = 0, (4) where P 1 P W1 is the mtrix tht projects on to S(W 1 ) This follows becuse Z W 1 = O The vector û 1 of 2SLS residuls from (1) is orthogonl to Z, nd so û 1 = M Z (y 1 ˆβy 2 ) (5) We consider four test sttistics: n symptotic t sttistic on which we my bse Wld test, the Anderson-Rubin (AR) sttistic, Kleibergen s K sttistic, nd likelihood rtio (LR) sttistic 2

4 From the estimting equtions (4) nd from expression (5) for the residuls û 1, it cn be shown tht the symptotic t sttistic for test of the hypothesis tht β = 0 is t = n 1/2 y 2 P 1 y 1 ( P 1 y 2 M Z y 1 y 2 P 1 y ) (6) 1 y y 2 2 P 1 y 2 It cn be seen tht the right-hnd side of (6) is homogeneous of degree zero with respect to y 1 nd lso with respect to y 2 Consequently, the distribution of the sttistic is invrint to the scles of ech of the endogenous vribles In ddition, the expression is unchnged if y 1 nd y 2 re replced by the projections M Z y 1 nd M Z y 2, since P 1 M Z = M Z P 1 = P 1, given the orthogonlity of W 1 nd Z The second fctor in the denomintor cn be rewritten s ( y 1 M Z y 1 2y 1 M Z y 2 y 2 P 1 y 1 y 2 P 1 y 2 + y 2 M Z y 2 ( ) 2 ) 1/2 y2 P 1 y 1 y 2 P 1 y 2 Thus it is not hrd to see tht the sttistic (6) depends on the dt only through the six quntities y 1 P 1 y 1, y 1 P 1 y 2, y 2 P 1 y 2, y 1 M W y 1, y 1 M W y 2, nd y 2 M W y 2 ; (7) notice tht y i M Z y j = y i (M W + P 1 )y j, for i, j = 1, 2 These six quntities, which we cn think of s sufficient sttistics, cn esily be clculted by mens of four OLS regressions on just two sets of regressors By regressing y i on Z nd W for i = 1, 2, we obtin four sets of residuls Using the fct tht P 1 y i = (M W M Z )y i, ll six quntities cn be obtined s sums of squred residuls, differences of sums of squred residuls, inner products of residul vectors, or inner products of differences of residul vectors Another wy to test hypothesis bout β is to use the fmous test sttistic of Anderson nd Rubin (1949) The Anderson-Rubin sttistic for the hypothesis tht β = β 0 cn be written s AR(β 0 ) = n l (y 1 β 0 y 2 ) P 1 (y 1 β 0 y 2 ) l k (y 1 β 0 y 2 ) M W (y 1 β 0 y 2 ) (8) Notice tht, when β 0 = β, the AR sttistic depends on the dt only through the first nd fourth of the six quntities (7) Under the normlity ssumption, this sttistic is exctly distributed s F (l k, n l) under the null hypothesis However, becuse it hs l k degrees of freedom, it hs lower power thn sttistics with only one degree of freedom when l k > 1 Kleibergen (2002) therefore proposed modifiction of the Anderson-Rubin sttistic which hs only one degree of freedom His sttistic for testing β = 0, which cn lso be interpreted s n LM sttistic, is K = (n l) y 1 P MZ W πy 1 y 1 M W y 1, (9) 3

5 which is symptoticlly distributed s χ 2 (1) under the null hypothesis tht β = 0 The mtrix P MZ W π projects orthogonlly on to the one-dimensionl subspce generted by the vector M Z W π, where π is vector of efficient estimtes of the reduced-form prmeters These estimtes will be discussed below in the context of bootstrpping Under our ssumptions, the vector M Z W π is equl to the vector W 1 π 1, where π 1 is the vector of OLS estimtes from the rtificil regression M Z y 2 = W 1 π 1 + δm Z y 1 + residuls (10) Noting tht M 1 M Z = M W, we see tht the estimte of δ from this regression is given by the FWL regression 1 M W y 2 = δm W y 1 + residuls, so tht ˆδ = y 1 M W y 2 /y 1 M W y 1 Substituting this result into (10) then shows tht W 1 π 1 = P 1 (M Z y 2 ˆδM ( Z y 1 ) = P 1 y 2 y 1 M W y ) 2 y 1 (11) y 1 M W y 1 From this, it cn be seen tht the K sttistic, like the t sttistic nd the AR sttistic, depends on the dt only through the six quntities (7) Allowing for nottionl differences nd our specil ssumptions, eqution (11) is the sme s the expression given on pge 1785 of Kleibergen (2002) The explicit expression of K in terms of these six quntities is K = (n l)(y 1 P 1 y 2 y 1 M W y 1 y 1 P 1 y 1 y 1 M W y 2 ) 2 y 2 P 1 y 2 (y 1 M W y 1 ) 3 + y 1 P 1 y 1 y 1 M W y 1 (y 1 M W y 2 ) 2 2y 1 P 1 y 2 y 1 M W y 2 (y 1 M W y 1 ) 2 (12) We see directly from this expression tht the sttistic K, like the t nd AR sttistics, is invrint to the scles of y 1 nd y 2 It is well known tht, except for n dditive constnt, the concentrted loglikelihood function for the model specified by (1), (2), nd (3) cn be written s ( n log 1 + l k ) 2 n l AR(β), (13) where AR(β) is the Anderson-Rubin sttistic (8) evluted t β Using (8) nd the fct tht M Z = M W + P 1, the loglikelihood function (13) becomes ( l(β) n MZ (y 1 y 2 β) 2 ) log 2 M W (y 1 y 2 β) 2 (14) 1 For n exposition of the FWL regression, see, for instnce, Dvidson nd McKinnon (2004, Chpter 2) 4

6 The LR sttistic for test of β = 0 is 2 ( l( ˆβ ML ) l(0) ), where ˆβ ML is the ML estimte As shown by Anderson nd Rubin (1949), the sttistic cn lso be defined in terms of the vlue of the rtio tht is the rgument of the logrithm in (14), minimized with respect to β We denote this minimized vlue by ˆκ The LR sttistic is then n(log κ 0 log ˆκ), where κ 0 is the rtio evluted t β = 0 Stndrd rguments show tht ˆκ is the smller of the two roots of the determinntl eqution y 1 (M Z κm W )y 1 y 1 (M Z κm W )y 2 y 2 (M Z κm W )y 1 y 2 (M Z κm W )y 2 = 0 Some tedious lgebr then shows tht ( LR = n log(1 + SS/n) n log 1 + SS + T T 2n where SS n y 1 P 1 y 1, y 1 M W y 1 ST T T n n 1/2 ( y 1 P 1 y 2 y 1 P 1 y 1 y 1 M W y 2 y 1 M W y 1 1 ) (SS T T )2 + 4ST 2n 2, (15) ), (16) (y 2 P 1 y 2 y 1 M W y 1 2y 1 P 1 y 2 y 1 M W y 2 + y 1 P 1 y 1 (y 1 M W y 2 ) 2 y 1 M W y 1 ), nd y 1 M W y 1 y 2 M W y 2 (y 1 M W y 2 ) 2 (17) The nottion is chosen so s to be reminiscent of tht used by Moreir (2003) in his discussion of conditionl LR test His development is different from ours in tht he ssumes for most of his nlysis tht the contemporneous disturbnce correltion mtrix Σ is known We do however discuss his conditionl pproch to the LR test in wht follows Moreir lso introduces simplified sttistic, LR 0, which is obtined by Tylor expnding the logrithms in (15) nd discrding terms of order smller thn unity s n This procedure yields LR 0 = 1 2 ( SS T T + (SS T T ) 2 + 4ST 2 ) (18) We see tht both LR nd LR 0 re invrint to the scles of y 1 nd y 2 nd depend only on the six quntities (7) Some more tedious lgebr shows tht the Kleibergen sttistic (12) cn lso be expressed in terms of the quntities ST nd T T s follows K = n l n ST 2 T T (19) 5

7 Moreir (2003) demonstrtes this reltion (without the degrees-of-freedom djustment) for the cse in which the disturbnce covrince mtrix is ssumed known Finlly, is worth noting tht, except for the initil deterministic fctors, SS is equl to the Anderson-Rubin sttistic AR(0) 3 Simulting the Test Sttistics Now tht we hve expressions for the test sttistics of interest in terms of the six quntities (7), we cn explore the properties of these sttistics nd how to simulte them efficiently Our results will lso be used in the next two sections when we discuss symptotic nd bootstrp tests In view of the scle invrince tht we hve estblished for ll the sttistics, the contemporneous covrince mtrix of the disturbnces u 1 nd u 2 cn without loss of generlity be set equl to [ ] 1 ρ Σ =, (20) ρ 1 with both vrinces equl to unity Thus we cn represent the disturbnces in terms of two independent n--vectors, sy v 1 nd v 2, of independent stndrd norml elements, s follows: u 1 = v 1, u 2 = ρv 1 + rv 2, (21) where r (1 ρ 2 ) 1/2 We now show tht we cn write ll the test sttistics s functions of v 1, v 2, the exogenous vribles, nd just three prmeters With the specifiction (21), we see from (2) tht nd y 2 M W y 2 = (ρv 1 + rv 2 ) M W (ρv 1 + rv 2 ) = ρ 2 v 1 M W v 1 + r 2 v 2 M W v 2 + 2ρrv 1 M W v 2, (22) y 2 P 1 y 2 = π 1 W 1 W 1 π 1 + 2π 1 W 1 (ρv 1 + rv 2 ) + ρ 2 v 1 P 1 v 1 + r 2 v 2 P 1 v 2 + 2ρrv 1 P 1 v 2 Now let W 1 π 1 = w 1, with w 1 = 1 The squre of the prmeter is the so-clled sclr concentrtion prmeter; see Phillips (1983, p 470) nd Stock, Wright, nd Yogo (2002) Further, let w 1 v i = x i, for i = 1, 2 Clerly, x 1 nd x 2 re independent stndrd norml vribles Then Thus (23) becomes (23) π 1 W 1 W 1 π 1 = 2 nd π 1 W 1 v i = x i, i = 1, 2 (24) y 2 P 1 y 2 = 2 + 2(ρx 1 + rx 2 ) + ρ 2 v 1 P 1 v 1 + r 2 v 2 P 1 v 2 + 2ρrv 1 P 1 v 2 (25) From (1), we find tht y 1 M W y 1 = v 1 M W v 1 + 2β(ρv 1 M W v 1 + rv 1 M W v 2 ) + β 2 y 2 M W y 2 (26) 6

8 Similrly, y 1 P 1 y 1 = v 1 P 1 v 1 + 2βy 2 P 1 v 1 + β 2 y 2 P 1 y 2 = v 1 P 1 v 1 + 2β(x 1 + ρv 1 P 1 v 1 + rv 1 P 1 v 2 ) + β 2 y 2 P 1 y 2 (27) Further, from both (1) nd (2), y 1 M W y 2 = ρv 1 M W v 1 + rv 1 M W v 2 + βy 2 M W y 2, nd (28) y 1 P 1 y 2 = x 1 + ρv 1 P 1 v 1 + rv 1 P 1 v 2 + βy 2 P 1 y 2 (29) The reltions (22), (25), (26), (27), (28), nd (29) show tht the six quntities (7) cn be generted in terms of eight rndom vribles nd three prmeters The eight rndom vribles re x 1 nd x 2, long with six qudrtic forms of the sme sort s those in (7), v 1 P 1 v 1, v 1 P 1 v 2, v 2 P 1 v 2, v 1 M W v 1, v 1 M W v 2, nd v 2 M W v 2, (30) nd the three prmeters re, ρ, nd β Under the null hypothesis, of course, β = 0 Since P 1 M W = O, the first three vribles of (30) re independent of the lst three If we knew the distributions of the eight rndom vribles on which ll the sttistics depend, we could simulte them directly We now chrcterize these distributions The symmetric mtrix [ ] v1 P 1 v 1 v 1 P 1 v 2 v 2 P 1 v 1 v (31) 2 P 1 v 2 follows the Wishrt distribution W(I 2, l k), nd the mtrix [ ] v1 M W v 1 v 1 M W v 2 v 2 M W v 1 v 2 M W v 2 follows the distribution W(I 2, n l) It follows from the nlysis of the Wishrt distribution in Anderson (1984, Section 72) tht v 1 M W v 1 is equl to rndom vrible t M 11 which follows the chi-squred distribution with n l degrees of freedom, v 1 M W v 2 is the squre root of t M 11 multiplied by stndrd norml vrible z M independent of it, nd v 2 M W v 2 is zm 2 plus chi-squred vrible tm 22 with n l 1 degrees of freedom, independent of z M nd t M 11 The elements of the mtrix (31) cn, of course, be chrcterized in the sme wy However, since the elements of the mtrix re not independent of x 1 nd x 2, it is preferble to define v 2 P 1 v 2 s x 2 2 plus t P 22, v 1 P 1 v 2 s x 1 x 2 plus the squre root of t P 22 times z P, nd v 1 P 1 v 1 s x z 2 P + tp 11 Here t P 11 nd t P 22 re both chi-squred, with l k 2 nd l k 1 degrees of freedom, respectively, nd z P is stndrd norml All these vribles re mutully independent, nd they re lso independent of x 1 nd x 2 Of course, if l k 2, chi-squred vribles with zero or negtive degrees of freedom re to be set to zero, nd if l k = 0, then z P = 0 7

9 An lterntive wy to simulte the test sttistics, which does not require the normlity ssumption, is to mke use of much simplified model This model my help to provide n intuitive understnding of the results in the next two sections The simplified model is y 1 = βy 2 + u 1, (32) y 2 = w 1 + u 2, (33) where the disturbnces re generted ccording to (21) Here the n--vector w 1 S(W ) with w 1 = 1, W being, s before, n n l mtrix of instruments By normlizing w 1 in this wy, we re implicitly using wek-instrument symptotics; see Stiger nd Stock (1997) Clerly, we my choose 0 The DGPs of this simple model, which re completely chrcterized by the prmeters β, ρ, nd, cn generte the six quntities (7) so s to hve the sme distributions s those generted by ny DGP of the more complete model specified by (1), (2), nd (3) If the disturbnces re not Gussin, the distributions of the sttistics depend not only on the prmeters, ρ, nd β but lso on the vector w 1 nd the liner spn of the instruments We my suspect, however, tht this dependence is wek, nd limited simultion evidence (not reported) strongly suggests tht this is indeed the cse The distribution of the disturbnces seems to hve much greter effect on the distributions of the test sttistics thn the fetures of W 4 Asymptotic Theory To fix ides, we begin with short discussion of the conventionl symptotic theory of the tests discussed in Section 2 By conventionl, we men tht the instruments re ssumed to be strong, in sense mde explicit below Under this ssumption, the tests re ll clssicl In prticulr, Kleibergen (2002) shows tht the K sttistic is version of the Lgrnge Multiplier test The reduced-form eqution (33) of the simplified model of the previous section is written in terms of n instrumentl vrible w 1 such tht w 1 = 1 Conventionl symptotics would set w 1 2 = n nd let the prmeter be independent of the smple size Our setup is better suited to the wek-instrument symptotics of Stiger nd Stock (1997) For conventionl symptotics, we my suppose tht = n 1/2 α, for α constnt s the smple size n Under the null, β = 0 Under locl lterntives, we let β = n 1/2 b, for b constnt s n Conventionl symptotics pplied to (22), (25), (26), (27), (28), nd (29) then give y 1 P 1 y 1 = (x1 + αb) 2 + t P 11, n 1/2 y 1 P 1 y 2 = αx1 + α 2 b, n 1 y 2 P 1 y 2 = α 2, n 1 y 1 M W y 1 = 1, n 1 y 1 M W y 2 = ρ, (34) n 1 y 2 M W y 2 = 1 8

10 Using these results, it is esy to check tht, for β = 0, the sttistics t 2, K, nd LR, given by (6), (12), nd (15), respectively, re ll symptoticlly equl to (x 1 + αb) 2 They hve common symptotic distribution of χ 2 with one degree of freedom nd noncentrlity prmeter α 2 b 2 = 2 β 2 Incidentlly, we cn lso see tht the Anderson- Rubin sttistic AR(0), s given by (8), is symptoticlly equl to (x 1 +αb) 2 +z 2 P +tp 11, with l k degrees of freedom nd the sme noncentrlity prmeter Thus AR(0) is symptoticlly equl to the sme noncentrl χ 2 (1) rndom vrible s the other three sttistics, plus n independent centrl χ 2 (l k 1) rndom vrible We now turn to the more interesting cse of wek-instrument symptotics, for which is kept constnt s n The three right-hnd results of (34) re unchnged, but the left-hnd ones hve to be replced by the following equtions, which involve no symptotic pproximtion, but hold even in finite smples: y 1 P 1 y 1 = x zp 2 + t P 11, y 1 P 1 y 2 = x 1 + ρ ( x zp 2 + t P ) ( 11 + r x 1 x 2 + z P t P 22 y 2 P 1 y 2 = 2 + 2(ρx 1 + rx 2 ) + ρ 2( x zp 2 + t P ) 11 ) + r 2 (x t P 22) + 2ρr (x 1 x 2 + z P t P 22 ), nd (35) Since the Anderson-Rubin sttistic AR(0) is exctly pivotl for the model we re studying, its distribution under the null tht β = 0 depends neither on nor on ρ Since the quntity SS in (16) is equl to AR(0) except for degrees-of-freedom fctors, it too is exctly pivotl Its symptotic distribution under wek-instrument symptotics is tht of y 1 P 1 y 1 Thus, s we see from the first line of (35), which follows the centrl χ 2 (l k) distribution SS = x z 2 P + t P 11, (36) Although Kleibergen s K sttistic is not exctly pivotl, it is symptoticlly pivotl under both wek-instrument nd strong-instrument symptotics From (12), nd using (34) nd (35), we cn see, fter some lgebr, tht, under wek-instrument symptotics nd under the null, K = (y 1 P 1 y 2 ρy 1 P 1 y 1 ) 2 y 2 P 1 y 2 2ρy 1 P 1 y 2 + ρ 2 y 1 P 1 y 1 = ( x1 + r(x 1 x 2 + z P t P 22 ) ) rx 2 + r 2 (x tp 22 ) = ( ) x1 ( + rx 2 ) + rz P t P 2 22 ( + rx 2 ) 2 + r 2 t P (37) 22 Although the lst expression bove depends on nd ρ, it is in fct just chi-squred vrible with one degree of freedom To see this, rgue conditionlly on ll rndom vribles except x 1 nd z P, reclling tht ll the rndom vribles in the expression re mutully independent The numertor is the squre of liner combintion of the stndrd norml vribles x 1 nd z P, nd the denomintor is the conditionl vrince 9

11 of this liner combintion Thus the conditionl symptotic distribution of K is χ 2 1, nd so lso its unconditionl distribution As Kleibergen (2002) remrks, this implies tht K is symptoticlly pivotl in ll configurtions of the instruments, including tht in which = 0 nd the instruments re completely invlid For the LR sttistic, we cn write down expressions symptoticlly equl to the quntities ST nd T T in (16) First, from (17), we hve It is then strightforwrd to check tht ST = /n 2 = 1 ρ 2 1 ( x (1 ρ 2 ) 1/2 1 + r ( ) x 1 x 2 + z p t ) P 22, nd T T = 1 1 ρ 2 ( 2 + 2rx 2 + r 2 (x t P 22) ) (38) Comprison with (37) then shows tht, in ccordnce with (19), K = ST 2 /T T It is cler from (38) nd (36) tht SS nd T T re symptoticlly independent, since the former depends only on the rndom vribles x 1, z P, nd t P 11, while the ltter depends only on x 2 nd t P 22 The discussion bsed on (37) shows tht, conditionl on T T, ST is distributed s T T times stndrd norml vrible nd tht K is symptoticlly distributed s χ 2 1 Even though SS nd ST re not conditionlly independent, the vribles ST nd SS ST 2 /T T re so symptoticlly This follows becuse, conditionlly on x 2 nd t P 22, the normlly distributed vrible x 1 ( + rx 2 ) + rz P t P 22 is liner combintion of the stndrd norml vribles x 1 nd z P tht prtilly constitute the symptoticlly chi-squred vrible SS These properties led Moreir (2003) to suggest tht the distribution of the sttistics LR nd LR 0, which re deterministic functions of SS, ST, nd T T, conditionl on given vlue of T T, sy tt, cn be estimted by simultion experiment in which ST nd SS ST 2 /T T re generted s independent vribles distributed respectively s N(0, tt) nd χ 2 l k 1 The vrible SS is then generted by combining these two vribles, replcing T T by tt Such n experiment hs bootstrp interprettion tht we develop in the next section It my be helpful to mke explicit the link between the quntities SS, ST, nd T T, defined in (16), nd the vectors S nd T used in Andrews, Moreir, nd Stock (2006) For the simplified model given by (32) nd (33), these vectors cn be expressed s S = W v 1 nd T = 1 r W (y 2 ρy 1 ) It is strightforwrd to check tht, with these definitions, S S, S T, nd T T re wht the expressions SS, ST, nd T T would become if the three qudrtic forms in the right pnel of (34) were replced by their symptotic limits 10

12 The results (35) re independent of b This shows tht, for locl lterntives of the sort used in conventionl symptotic theory, no test sttistic tht depends only on the six quntities (7) cn hve symptotic power greter thn symptotic size under wek-instrument symptotics However, if insted we consider fixed lterntives, with prmeter β independent of n, then the expressions do depend on β For nottionl ese, denote the three right-hnd sides in (35) by Y 11, Y 12, nd Y 22, respectively Then it cn be seen tht the wek-instrument results become y 1 P 1 y 1 = Y βY 12 + β 2 Y 22, y 1 P 1 y 2 = Y 12 + βy 22, y 2 P 1 y 2 = Y 22, n 1 y 1 M W y 1 = 1 + 2βρ + β 2, n 1 y 1 M W y 2 = ρ + β, (39) n 1 y 2 M W y 2 = 1 Notice tht, if we specilize the bove results, letting β be O(n 1/2 ) nd letting be O(n 1/2 ), then we obtin the conventionl strong-instrument results (34) We hve not written down the wek-instrument symptotic expression for the Wld t sttistic given in (6), becuse it is complicted nd not very illuminting Suffice it to sy tht it depends nontrivilly on the prmeters nd ρ, s does its distribution Consequently, the sttistic t is not symptoticlly pivotl Indeed, in the terminology of Dufour (1997), it is not even boundedly pivotl, by which we men tht rejection probbilities of tests bsed on it cnnot be bounded wy from one We will see this explicitly in moment The estimting equtions (4) imply tht the IV estimte of β is ˆβ = y 1 P 1 y 2 /y 2 P 1 y 2 Under wek-instrument symptotics, we see from (39) tht ˆβ = β + Y 12 /Y 22 (40) Since E(Y 12 /Y 22 ) 0, it follows tht ˆβ is bised nd inconsistent The squre of the t sttistic (6) cn be seen to be symptoticlly equl to Y 22 (Y 12 + βy 22 ) 2 Y ρY 12Y 22 + Y 2 12 under wek-instrument symptotics Observe tht this expression is of the order of β 2 s β Thus, for fixed nd ρ, the distributions of t 2 for β = 0 nd β 0 cn be rbitrrily fr prt For β = 0, however, the distribution of t 2, for nd ρ sufficiently close to 0 nd 1, respectively, is lso rbitrrily fr from tht with fixed 0 nd ρ 1 It is this fct tht leds to the filure of t 2 to be boundedly pivotl Let nd r = (1 ρ 2 ) 1/2 be treted s smll quntities, nd then expnd the denomintor of expression (41) through the second order in smll quntities Note tht, to this order, ρ = 1 r 2 /2 Then we see tht, to the desired order, Y 12 = v 1 P 1 v 1 + (x 1 + rv 1 P 2 v 2 ) 1 2 r 2 v 1 P 1 v 1, nd Y 22 = v 1 P 1 v 1 + 2(x 1 + rv 1 P 1 v 2 ) + ( 2 + 2rx 2 + r 2 v 2 P 1 v r 2 v 1 P 1 v 1 ) 11 (41)

13 Consequently, to the order t which we re working, Y ρY 12 Y 22 + Y 2 12 = r 2 (v 1 P 1 v 1 ) 2 To leding order, the numertor of (41) for β = 0 is just (v 1 P 1 v 1 ) 3, nd so, in the neighborhood of = 0 nd r = 0, we hve from (41) tht t 2 = v1 P 1 v 1 /r 2 (42) The numertor here is just chi-squred vrible, but the denomintor cn be rbitrrily close to zero Thus the distribution of t 2 cn be moved rbitrrily fr wy from ny finite distribution by letting tend to zero nd ρ tend to one The points in the prmeter spce t which = 0 nd ρ = ±1, which implies tht r = 0, re points t which β is completely unidentified To see this, consider the DGP from model (32) nd (33) tht corresponds to these prmeter vlues The DGP cn be written s y 2 = v 1, y 1 = (1 ± β)y 2 (43) It follows from (40) tht ˆβ = 1 ± β This is not surprising, since the second eqution in (43) fits perfectly This fct then ccounts for the t sttistic tending to infinity All the other tests hve power tht does not tend to 1 when β under wekinstrument symptotics For K, some lgebr shows tht K = ( Y12 ρy 11 + β(y 22 Y 11 ) + β 2 (ρy 22 Y 12 ) ) 2 D(1 + 2βρ + β 2, (44) ) where D Y 22 2ρY 12 +ρ 2 Y 11 +2β ( ρy 22 (1+ρ 2 )Y 12 +ρy 11 ) +β 2 (Y 11 2ρY 12 +ρ 2 Y 22 ) (45) As β, the complicted expression (44) tends to the much simpler limit of (ρy 22 Y 12 ) 2 Y 11 2ρY 12 + ρ 2 Y 22 (46) Thus, unlike t 2, the K sttistic does not become unbounded s β Consequently, under wek-instrument symptotics, the test bsed on K is inconsistent for ny nonzero β, in the sense tht the rejection probbility does not tend to 1 however lrge the smple size A similr result holds for the AR test It is esy to see tht SS = Y βY 12 + β 2 Y βρ + β 2 β Y 22, (47) which does not depend on β Thus, since AR is proportionl to SS, we see tht the symptotic distribution of AR does not depend on β under wek-instrument symptotics 12

14 Similr results lso hold for the LR nd LR 0 sttistics By n nlysis like the one tht produced (47), we hve tht ST = Y 12 ρy 11 + β(y 22 Y 11 ) + β 2 (ρy 22 Y 12 ) r(1 + 2βρ + β 2 ) T T = D r 2 (1 + 2βρ + β 2 ) β Y 11 2ρY 12 + ρ 2 Y 22 r 2, where D is given by (45) From (18), therefore, β ρy 22 Y 12, nd r 1 ( LR 0 β 2r 2 Y 22 (1 2ρ 2 ) + 2ρY 12 Y 11 + ( Y Y 11 Y 22 (1 2ρ 2 ) 4ρY 11 Y 12 + Y Y12 2 ) 1/2 4ρY 12 Y 22 ) The inconsistency of the LR 0 test follows from the fct tht this rndom vrible hs bounded distribution This is true for the LR test s well, but we will spre reders the detils We sw bove tht, when = 0 nd ρ = 1, the prmeter β is unidentified We expect, therefore, tht test sttistic for the hypothesis tht β = 0 would hve the sme distribution whtever the vlue of β This turns out to be the cse for the K sttistic If one computes the limit of expression (44) for = 0, r 0, the limiting expression is just (v 1 P 1 v 2 ) 2 /v 2 P 1 v 2, independently of the vlue of β Presumbly more complicted clcultion would show tht the sme is true for LR nd LR 0 The result tht the AR, K, nd LR tests re inconsistent under wek-instrument symptotics ppers to contrdict some of the principl results of Andrews, Moreir, nd Stock (2006) The reson for this pprent contrdiction is tht we hve mde different, nd in our view more resonble, ssumption bout the covrince mtrix of the disturbnces We ssume tht the mtrix Σ, defined in (3) s the covrince mtrix of the disturbnces in the structurl eqution (1) nd the reduced form eqution (2), remins constnt s β vries In contrst, Andrews, Moreir, nd Stock (2006) ssumes tht the covrince mtrix of the reduced form disturbnces does so In terms of our prmetriztion, the covrince mtrix of the disturbnces in the two reduced form equtions is [ σ ρβσ 1 σ 2 + β 2 σ 2 2 ρσ 1 σ 2 + βσ 2 2 ρσ 1 σ 2 + βσ 2 2 σ 2 2 ] (48) This expression depends on β in nontrivil wy In order for it to remin constnt s β chnges, both ρ nd σ 1 must be llowed to vry Thus the ssumption tht (48) is fixed, which ws mde by Andrews, Moreir, nd Stock (2006), implies tht (3) cnnot remin constnt s β A little lgebr shows tht, s β ± with the covrince mtrix (48) held fixed, the prmeters β nd ρ of the observtionlly equivlent DGP of the model given by 13

15 (32) nd (33) long with (21) tend to ±1 nd 1 respectively Thus the omnipresent denomintor 1 + 2βρ + β 2 tends to zero in either of these limits But it is cler from (39) tht this mens tht the estimte of σ1 2 from the full model (1) nd (2) tends to zero Bsed on the bove remrk, our view is tht it is more resonble tht (3) should remin constnt thn tht (48) should The prmeter ρ is much more interesting prmeter thn the correltion in (48) Even when ρ = 0, in which cse the OLS estimtor of β is consistent, the correltion between the two reduced form disturbnces tends to ±1 s β ± Thus we believe tht the ltter correltion is not sensible quntity to hold fixed In ny cse, it is rther disturbing tht something s seemingly innocuous s the prmetriztion of the covrince mtrix of the disturbnces cn hve profound consequences for the nlysis of power when the instruments re wek 5 Bootstrpping the Test Sttistics There re severl wys to bootstrp the non-exct test sttistics tht we hve been discussing (Wld, K, nd LR) In this section, we discuss five different prmetric bootstrp procedures, three of which re new We obtin number of interesting theoreticl results We lso briefly discuss how to convert the new procedures into semiprmetric bootstrps In the next section, we will see tht two of the new procedures, nd one of them in prticulr, perform extremely well when used with ll three of the test sttistics For ll the sttistics, we perform B bootstrp simultions nd clculte the bootstrp P vlue s ˆp (ˆτ) = 1 B I(τj > ˆτ), (49) B j=1 where ˆτ denotes the ctul test sttistic nd τj the j th bootstrp smple denotes the sttistic clculted using Dufour (1997) mkes it cler tht bootstrpping is not in generl cure for the difficulties ssocited with the Wld sttistic t However, since the Wld sttistic is still frequently used in prctice when there is no dnger of wek instruments, it is interesting to look t the performnce of the bootstrpped Wld test when instruments re strong When they re wek, we confirm Dufour s result bout the ineffectiveness of bootstrpping In the context of our new bootstrp methods, this mnifests itself in n lmost complete loss of power, for resons tht we nlyze Since the K sttistic is symptoticlly pivotl under wek-instrument symptotics, it should respond well to bootstrpping, t lest under the null The LR sttistic is not symptoticlly pivotl, but, s shown by Moreir (2003), conditionl LR test gives the symptoticlly pivotl sttistic we cll CLR As we explin below, the implementtion of this conditionl likelihood rtio test is in fct form of bootstrpping, so tht bootstrpping the conditionl test is sort of double bootstrp, nd thus much more computtionlly intensive thn the other bootstrp tests we consider 14

16 For ny bootstrpping procedure, the first tsk, nd usully the most importnt one, is to choose suitble bootstrp DGP; see Dvidson nd McKinnon (2006) An obvious but importnt point is tht the bootstrp DGP must be ble to hndle both of the endogenous vribles, tht is, y 1 nd y 2 A strightforwrd, conventionl pproch is to estimte the prmeters β, γ, π, σ 1, σ 2, nd ρ of the model specified by (1), (2), nd (3), nd then to generte simulted dt using these equtions with the estimted prmeters However, the conventionl pproch estimtes more prmeters thn it needs to The bootstrp DGP should tke dvntge of the fct tht the simple model specified by (32) nd (33) cn generte sttistics with the sme distributions s those generted by the full model Eqution (32) becomes especilly simple when the null hypothesis is imposed: It sys simply tht y 1 = u 1 If this pproch is used, then only the prmeters nd ρ need to be estimted In order to estimte, we my substitute n estimte of π into the definition (24) with n pproprite scling fctor to tke ccount of the fct tht is defined for DGPs with unit disturbnce vrinces Since we hve ssumed up to now tht the disturbnces of our model re Gussin, it is pproprite to use prmetric bootstrp in which the disturbnces re normlly distributed In prctice, however, investigtors will often be reluctnt to mke this ssumption At the end of this section, we therefore discuss semiprmetric bootstrp techniques tht resmple the residuls In the next section, we will see tht the performnce of the prmetric nd semiprmetric bootstrps ppers to be very similr when the disturbnces re in fct normlly distributed We investigte five different wys of estimting the prmeters ρ nd To estimte ρ, we just need residuls from equtions (32) nd (33), or, in the generl cse, (1) nd (2) To estimte, we need estimtes of the vector π 1 from the reduced-form eqution (33), or from (2), long with residuls from tht eqution If ü 1 nd ü 2 denote the two residul vectors, nd π 1 denotes the estimte of π 1, then our estimtes re ü 1 ü 2 ρ =, nd (50) (ü 1 ü 1 ü 2 ü 2 ) 1/2 ä = n π 1 W 1 W 1 π 1 /ü 2 ü 2 (51) Existing methods, nd the new ones tht we propose, use vrious estimtes of π 1 nd vrious residul vectors The simplest wy to estimte ρ nd is probbly to use the restricted residuls ũ 1 = M Z y 1 = M W y 1 + P 1 y 1, which, in the cse of the simple model, re just equl to y 1, long with the OLS estimtes ˆπ 1 nd OLS residuls û 2 from the FWL regression M Z y 2 = W 1 π 1 + u 2 (52) 15

17 We cll this widely-used method the RI bootstrp, for Restricted, Inefficient It cn be expected to work much better thn the pirs bootstrp tht ws studied by Freedmn (1983) nd better thn other prmetric procedures tht do not impose the null hypothesis As the nme implies, the problem with the RI bootstrp is tht ˆπ 1 is not n efficient estimtor Tht is why Kleibergen (2002) did not use ˆπ 1 in constructing the K sttistic Insted, he used the estimtes π 1 from eqution (10) It cn be shown tht these estimtes re symptoticlly equivlent to the ones tht would be obtined by using 3SLS or FIML on the system consisting of equtions (1) nd (2) The estimted vector of disturbnces from eqution (10) is not the vector of OLS residuls but rther the vector ũ 2 = M Z y 2 W 1 π Insted of eqution (10), it my be more convenient to run the regression y 2 = W 1 π 1 + Zπ 2 + δm Z y 1 + residuls (53) This is just the reduced form eqution ugmented by the residuls from restricted estimtion of the structurl eqution Becuse of the orthogonlity between Z nd W 1, the vector ũ 2 is equl to the vector of OLS residuls from regression (53) plus ˆδM Z y 1 We cll the bootstrp tht uses ũ 1, π 1, nd ũ 2 the RE bootstrp, for Restricted, Efficient Two other bootstrp methods do not impose the restriction tht β = 0 when estimting ρ nd For the purposes of testing, it is bd ide not to impose this restriction, s we rgued in Dvidson nd McKinnon (1999) However, it is quite inconvenient to impose restrictions when constructing bootstrp confidence intervls, nd since confidence intervls re implicitly obtined by inverting tests, it is of interest to see how much hrm is done by not imposing the restriction The UI bootstrp, for Unrestricted, Inefficient, uses the unrestricted residuls û 1 from IV estimtion of (1), long with the estimtes ˆπ 1 nd residuls û 2 from OLS estimtion of (2) The UE bootstrp, for Unrestricted, Efficient, lso uses û 1, but the other quntities come from the rtificil regression M Z y 2 = W 1 π 1 + δû 1 + residuls, (54) which is similr to regression (10) Of course, regression nlogous to (53) could be used insted of (54) A fifth bootstrp method will be proposed fter we hve obtined some results on which it depends It is possible to write the estimtes of nd ρ used by ll four of these bootstrp schemes s functions solely of the six quntities (7) This mkes it possible to progrm the bootstrp very efficiently Becuse mny of the functions re quite complicted, we will spre reders most of the detils However, we need the following results for the RE bootstrp: ρ = y 1 M W y 2 + y 1 M W y 2 y y 1 P 1 y 1 1 M W y 1 ( (y1 ) ( ( ) 2 y1 M W y ) ), (55) 1/2 2 M W y 1 + y 1 P 1 y 1 y 2 M W y 2 + y y 1 P 1 y 1 1 M W y 1 16

18 nd ã 2 = y 2 P 1 y 2 2y 1 P 1 y 2 y 1 M W y 2 y 1 M W y 1 + y 1 P 1 y 1 y 2 M W y 2 + y 1 P 1 y 1 ( y1 M W y 2 y 1 M W y 1 ( y1 M W y 2 ) 2 y 1 M W y 1 ) 2 (56) Dvidson nd McKinnon (1999) show tht the size distortion of bootstrp tests my be reduced by use of bootstrp DGP tht is symptoticlly independent of the sttistic tht is bootstrpped In generl, this is true only for bootstrp DGPs tht re bsed on efficient estimtors Thus it mkes sense to use the efficient estimtor π 1 rther thn the inefficient estimtor ˆπ 1 in order to estimte, nd, vi the reducedform residuls, ρ Either restricted or unrestricted residuls from (1) cn be used s the extr regressor in estimting π 1 without interfering with the desired symptotic independence, but generl considertions of efficiency suggest tht restricted residuls re the better choice Thus we would expect tht, when conventionl symptotics yield good pproximtion, the best choice for bootstrp DGP is RE Under wek-instrument symptotics, things re rther different We use the results of (35) nd the right-hnd results of (34) to see tht, with dt generted by the model (32) nd (33) under the null hypothesis, σ 2 1 = 1, σ 2 2 = 1, nd ρ σ 1 σ 2 = ρ Thus the RE bootstrp estimtor ρ, s defined by (55), is consistent estimtor, s is the estimtor used by the RI bootstrp It cn be checked tht this result does not hold for ny of the estimtors tht use unrestricted residuls from eqution (32), since they depend on the inconsistent IV estimte of β; recll (40) The wek-instrument symptotic version of (56) under the null cn be seen to be ã 2 = 2 + 2rx 2 + r 2 (x t P 22) (57) Unless r = 0, then, ã 2 is inconsistent It is lso bised, the bis being equl to r 2 (l k) It seems plusible, therefore, tht the bis-corrected estimtor ã 2 BC mx ( 0, ã 2 (l k)(1 ρ 2 ) ) (58) my be better for the purposes of defining the bootstrp DGP Thus we consider fifth bootstrp method, REC, for Restricted, Efficient, Corrected It differs from RE in tht it uses ã BC insted of ã This hs the effect of reducing the R 2 of the reduced-form eqution in the bootstrp DGP For the purposes of n nlysis of power, it is necessry to look t the properties of the estimtes ρ nd ã 2 under the lterntive, tht is, for nonzero β From (39), we see tht σ 2 1 = 1 + 2βρ + β 2, σ 2 2 = 1, nd ρ σ 1 σ 2 = ρ + β, 17

19 from which we find tht ρ = ρ + β (1 + 2βρ + β 2 ) 1/2 As β, then, we see tht ρ 1, for ll vlues of nd ρ For the rte of convergence, it is better to reson in terms of the prmeter r (1 ρ 2 ) 1/2 We hve Thus r = O p (β 1 ) s β r 2 = 1 ρ 2 = 1 (ρ + β) βρ + β 2 = r βρ + β 2 The clcultion for ã 2 is little more involved From (56) nd (39), we find tht ( ) 2 ã 2 2(ρ + β) = Y βρ + β 2 (Y ρ + β 12 + βy 22 ) βρ + β 2 (Y βY 12 + β 2 Y 22) 1 ( = (1 + βρ)y22 (1 + 2βρ + β 2 ) 2 2(ρ + β)(1 + βρ)y 12 + (ρ + β) 2 ) Y 11 Clerly, this expression is of the order of β 2 in probbility s β, so tht ã 0, gin for ll nd ρ In fct, it is cler tht ã = O p (β 1 ) s β, from which we conclude tht ã nd r tend to zero t the sme rte s β, s in the clcultion tht led to (42) These results cn be understood intuitively by considering (32) nd (33) Estimtion of ρ uses residuls which for tht model re just the vector y 1 = βy 2 + v 1 For lrge β, this residul vector is lmost colliner with y 2, nd so lso with the residul vector ũ 2 The estimted correltion coefficient therefore tends to 1 Similrly, when y 1 is introduced s n extr regressor for the estimtion of, it is highly colliner with the dependent vrible nd explins lmost ll of it, leving no pprent explntory power for the wek instruments For lrge β then, the RE bootstrp DGP is chrcterized by prmeters nd ρ close to 0 nd 1, respectively As we sw ner the end of the lst section, t this point in the prmeter spce β is unidentified, nd the Wld sttistic hs n unbounded distribution These fcts need not be worrisome for the bootstrpping of sttistics tht re symptoticlly pivotl with wek instruments, but they men tht the bootstrp version of the Wld test, like the Kleibergen nd LR tests, is inconsistent, hving probbility of rejecting the null hypothesis tht does not tend to one s β To see this, we mke use of expression (42) to see tht the distribution of the Wld sttistic t 2, for nd r smll nd of the sme order nd β = 0, is of order r 2 For lrge β, therefore, the distribution of the bootstrp Wld sttistic, under the null, is of order r 2, which we hve just seen is the sme order s β 2 But the distribution of the Wld sttistic itself for lrge β is lso of order β 2, unlike the K nd LR sttistics Although the distribution of the ctul sttistic t 2 for lrge β nd tht of the bootstrp sttistic (t ) 2 re not the sme, nd re unbounded, the distributions of t 2 /β 2 nd (t ) 2 /β 2 re of order unity in probbility, nd, hving support on the whole rel line, 18

20 they overlp Thus the probbility of rejection of the null by the bootstrp test does not tend to 1 however lrge β my be This conclusion, which is borne out by the simultion experiments of the next section, bers some discussion In Horowitz nd Svin (2000), it is pointed out tht, unless one is working with pivotl sttistics, it is not in generl possible to define n empiriclly relevnt definition of the power of test tht does not hve true level equl to its nominl level They conclude tht the best mesure in prctice is the rejection probbility of well-constructed bootstrp test In Dvidson nd McKinnon (2006b), we point out tht, even for well-constructed bootstrp tests, mbiguity remins in generl Only when the bootstrp DGP is symptoticlly independent of the symptoticlly pivotl sttistic being bootstrpped cn level djustment be performed unmbiguously on the bsis of the DGP in the null hypothesis of which the prmeters re the probbility limits of the estimtors used to define the bootstrp DGP This result, s proved, pplies only to the prmetric bootstrp, nd, more importntly here, to cses in which these estimtors hve nonrndom probbility limits But, s we hve seen, tht is not the cse here It seems therefore tht there is no theoreticlly stisfying mesure of the power of tests for which the bootstrp DGP is symptoticlly nonrndom It is therefore pointless to try to refine our erlier result for the Wld test, whereby we lern merely tht its bootstrp version is inconsistent Becuse the Wld sttistic is not boundedly pivotl, test bsed on it hs size equl to one, s shown by Dufour (1997) Dufour lso drws the conclusion tht no Wldtype confidence set bsed on sttistic tht is not t lest boundedly pivotl cn be vlid, whether or not the confidence set is constructed by bootstrpping If, however, insted of using conventionl bootstrp confidence set, we invert the bootstrp Wld test to obtin confidence set tht contins ll prmeter vlues which re not rejected by bootstrp Wld test, we my well obtin confidence sets with level of less thn one, since, on ccount of the inconsistency of the bootstrp test, unbounded confidence sets cn rise with positive probbility We mentioned erlier tht Moreir s conditionl LR test hs bootstrp interprettion We my consider the vrible T T defined in (16) s rndom vrible on which bootstrp distribution is conditioned In fct, s cn be seen from (38) nd (57), T T is equivlent under wek-instrument symptotics to ã 2 /r 2 Given vlue for T T, Moreir shows tht the sttistics LR nd LR 0 re symptoticlly pivotl Thus, rther thn estimting both nd ρ nd using the estimtes to generte bootstrp versions of the six sufficient sttistics (7), we cn evlute T T nd then generte bootstrp versions of the two (conditionlly) sufficient sttistics SS nd ST bsed on their symptotic conditionl distributions, s discussed erlier From this we obtin conditionl empiricl distribution which is used to compute P vlue for the CLR test in the usul wy This procedure is not quite rel bootstrp, lthough it is lmost s computtionlly intensive s fully prmetric bootstrp bsed on simulting the six quntities, becuse the bootstrp conditionl distributions of SS nd ST re not known ex- 19

21 ctly Insted, they re pproximted on the bsis of the distributions when the contemporneous covrince mtrix is known Of course, it is possible to bootstrp the CLR test, but this mounts to form of double bootstrp, which is expensive nd requires some cre to implement correctly Suppose we perform s simultions to clculte CLR test P vlue ˆp Since this P vlue is 1/s times the number of simultions for which the simulted LR sttistic, conditionl on T T, exceeds the ctul LR sttistic, it cn tke on only s + 1 different vlues Now imgine bootstrpping this entire procedure If we lso use s simultions to compute ech bootstrp sttistic p j, then p j cn only tke on the sme s + 1 vlues s ˆp Thus the ctul nd bootstrp sttistics will be equl with probbility pproximtely 1/(s + 1) To void this problem, we either need to mke s very lrge or use different number of simultions, sy s, for the bootstrp sttistics The bootstrp P vlue is computed by modified version of (49), with I(p j < ˆp) replcing I(τj > ˆτ) Some remrks concerning bootstrp vlidity re in order t this point If the sttistic tht is bootstrpped is symptoticlly pivotl, then the bootstrp is vlid symptoticlly in the sense tht the difference between the bootstrp distribution nd the distribution under the true DGP, provided the ltter stisfies the null hypothesis, converges to zero s the smple size tends to infinity; see, mong mny others, Dvidson nd McKinnon (2006) The bootstrp provides higher-order refinements if, in ddition, the bootstrp DGP consistently estimtes the true DGP under the null; see Bern (1988) Yet nother level of refinement cn be ttined if the sttistic bootstrpped is symptoticlly independent of the bootstrp DGP; see Dvidson nd McKinnon (1999) All of these requirements re stisfied by ny of the sttistics considered here under strong-instrument symptotics if the RE or REC bootstrp is used Besides the AR sttistic, only the K sttistic nd CLR test P vlue re symptoticlly pivotl with wek instruments, nd so it is only for them tht we cn conclude without further do tht the bootstrp is vlid with wek-instrument symptotics However, even if the sttistic bootstrpped is not symptoticlly pivotl, the bootstrp my still be vlid if the bootstrp DGP consistently estimtes the true DGP under the null But with wek instruments this is true of no conceivble bootstrp method, since there is no consistent estimte of the prmeter Consequently, however lrge the smple size my be, the bootstrp distributions of the Wld sttistic nd the LR sttistic re different from their true distributions Although our discussion hs focused on prmetric bootstrp procedures, we do not necessrily recommend tht they should be used in prctice, since the ssumption of Gussin disturbnces my often be uncomfortbly strong All of the prmetric bootstrp procedures tht we hve discussed hve semiprmetric nlogs which do not require tht the disturbnces should be normlly distributed We will discuss only the RE nd REC bootstrps, prtly becuse they re new, prtly becuse they will be seen in the next section to work very well, nd prtly becuse it will be obvious how to construct semiprmetric nlogs of the other procedures 20

22 For the semiprmetric RE bootstrp, we first estimte eqution (1) under the null hypothesis to obtin restricted residuls M Z y 1 We then run regression (10) or, equivlently, regression (53) The residul vector ũ 2 tht we wnt to resmple from is the vector of residuls from the regression plus ˆδM Z y 1 The bootstrp DGP is then y 1 = u 1 y 2 = W 1 π 1 + u 2, [u 1 u 2] EDF[ũ 1 ũ 2 ] (59) Thus the bootstrp disturbnces re resmpled from the joint EDF of the two residul vectors This preserves the smple correltion between them For the REC bootstrp, we need to use different set of fitted vlues in the reducedform eqution To do so, we first compute ã 2 using either the formul (56) or, more conveniently, ã 2 = π 1 W 1 W 1 π 1, ũ 2 ũ 2 /n which is the squre of (51) evluted t the pproprite vlues of π 1 nd u 2 Then we clculte ã 2 BC from (58) The bootstrp DGP is lmost the sme s (59), except tht the fitted vlues W 1 π 1 re replced by ã BC /ã times W 1 π 1 This reduces the length of the vector of fitted vlues somewht The fitted vlues ctully shrink to zero in the extreme cse in which ã BC = 0 The quntity ã 2 is very closely relted to the test sttistic for wek instruments recently proposed by Stock nd Yogo (2005) When it is lrge, the instruments re lmost certinly not wek, nd even symptotic inference should be resonbly relible When it is very smll, however, mny tests re likely to overreject severely, nd those tht do not re likely to be seriously lcking in power There is evidence on these points in the next section 6 Simultion Evidence In this section, we report the results of lrge number of simultion experiments with dt generted by the simplified model (32) nd (33) All experiments hd 100,000 replictions for ech set of prmeter vlues In most cses, we used fully prmetric bootstrp DGPs, but we lso investigted the semiprmetric ones discussed t the end of the preceding section In most of the experiments, n = 50 A different choice for n, but for the sme vlue(s) of, would hve hd only modest effect on the results for most of the bootstrp tests Mny of the symptotic tests re quite sensitive to n, however, nd we lter provide some evidence on this point For the bse cses, we consider every combintion of = 2 nd = 8 with ρ = 01 nd ρ = 09 The limiting R 2 of the reduced-form regression (33) is 2 /(n + 2 ) Thus, when = 2, the instruments re very wek, nd when = 8, they re modertely strong When ρ = 01, there is not much correltion between the structurl nd reduced-form disturbnces, nd when ρ = 09, there is gret del of correltion 21