Choosing the Number of Moments in Conditional Moment Restriction Models

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1 Choosg the Number of Momets Codtoal Momet Restrcto Models Stephe G. Doald Departmet of Ecoomcs Uversty of Texas Whtey Newey Departmet of Ecoomcs MIT Frst Draft: Jauary 2002 Ths draft: October 2008 Gudo Imbes Departmet of Ecoomcs UC-Berkeley Abstract Propertes of GMM estmators are sestve to the choce of strumets. Usg may strumets leads to hgh asymptotc asymptotc effcecy but ca cause hgh bas ad/or varace small samples. I ths paper we develop ad mplemet asymptotc mea square error (MSE) based crtera for strumetal varables to use for estmato of codtoal momet restrcto models. The models we cosder clude varous olear smultaeous equatos models wth ukow heteroskedastcty. We develop momet selecto crtera for the famlar two-step optmal GMM estmator (GMM), a bas corrected verso, ad geeralzed emprcal lkelhood estmators (GEL), that clude the cotuous updatg estmator (CUE) as a specal case. We also fd that the CUE has lower hgher-order varace tha the bas-corrected GMM estmator, ad that the hgher-order effcecy of other GEL estmators depeds o codtoal kurtoss of the momets. JEL Classfcato: C13, C30 Keywords: Codtoal Momet Restrctos, Geeralzed Method of Momets, Geeralzed Emprcal Lkelhood, Mea Squared Error.

2 1 Itroducto It s mportat to choose carefully the strumetal varables for estmatg codtoal momet restrcto models. Addg strumets creases asymptotc effcecy but also creases small sample bas ad/or varace. We accout for ths trade-off by usg a hgher-order asymptotc mea-square error (MSE) of the estmator to choose the strumet set. We derve the hgher-order MSE for GMM, a bas corrected verso of GMM (BGMM), ad geeralzed emprcal lkelhood (GEL). For smplcty we mpose a codtoal symmetry assumpto, that thrd codtoal momets of dsturbaces are zero, ad use a large umber of strumets approxmato. We also cosder the effect of allowg detfcato to shrk wth the sample sze at a rate slower tha 1/.The resultg MSE expressos are qute smple ad straghtforward to apply practce to choose the strumet set. The MSE crtera gve here also provde hgher order effcecy comparsos. We fd that cotuously updated GMM estmator (CUE) s hgher-order effcet relatve to BGMM. We also fd that the hgher order effcecy of the GEL estmators depeds o codtoal kurtoss, wth all GEL estmators havg the same hgher-order varace whe dsturbaces are Gaussa. Wth Gaussa dsturbaces ad homoskedastcty, Rotheberg (1996) showed that emprcal lkelhood (EL) s hgher order effcet relatve to BGMM. Our effcecy comparsos geeralze those of Rotheberg (1996) to other GEL estmators ad heteroskedastc, o Gaussa dsturbaces. These effcecy results are dfferet tha the hgher order effcecy result for EL that was show by Newey ad Smth (2004) because Newey ad Smth (2004) do ot requre that codtoal thrd momets are zero. Wthout that symmetry codto all of the estmators except for EL have addtoal bas terms that are ot corrected for here. Our MSE crtera s lke that of Nagar (1959) ad Doald ad Newey (2001), beg the MSE of leadg terms a stochastc expaso of the estmator. Ths approach s well kow to gve the same aswer as the MSE of leadg terms a Edgeworth expaso, uder sutable regularty codtos (e.g. Rotheberg, 1984). The may strumet ad [1]

3 shrkg detfcato smplfcatos seems approprate for may applcatos where there s a large umber of potetal strumetal varables ad detfcato s ot very strog. We also assume symmetry, the sese that codtoal thrd momets of the dsturbaces are zero. Ths symmetry assumpto greatly smplfes calculatos. Also, relaxg t may ot chage the results much, e.g. because the bas from asymmetry teds to be smaller tha other bas sources for large umbers of momet codtos, see Newey ad Smth (2004). Choosg momets to mmze MSE may help reduce msleadg fereces that ca occur wth may momets. For GMM, the MSE explctly accouts for a mportat bas term (e.g. see Hase et. al., 1996, ad Newey ad Smth, 2004), so choosg momets to mmze MSE avods cases where asymptotc fereces are poor due to the bas beg large relatve to the stadard devato. For GEL, the MSE explctly accouts for hgher order varace terms, so that choosg strumets to mmze MSE helps avod uderestmated varaces. However, the crtera we cosder may ot be optmal for reducg msleadg fereces. That would lead to a dfferet crtera, as recetly poted out by J, Phllps, ad Su (2007) aother cotext. The problem addressed ths paper s dfferet tha cosdered by Adrews (1996). Here the problem s how to choose amog momets kow to be vald whle Adrews (1996) s about searchg for the largest set of vald momets. Choosg amog vald momets s mportat whe there are may thought to be equally vald. Examples clude varous atural expermet studes, where multple strumets are ofte avalable, as well as tertemporal optmzato models, where all lags may serve as strumets. I Secto 2 we descrbe the estmators we cosder ad preset the crtera we develop for choosg the momets. We also compare the crtera for dfferet estmators, whch correspods to the MSE comparso for the estmators, fdg that the CUE has smaller MSE tha bas corrected GMM. I Secto 3 we gve the regularty codtos used to develop the approxmate MSE ad gve the formal results. Secto 4 shows optmalty of the crtera we propose. A small scale Mote Carlo expermet s coducted Secto 5. Cocludg remarks are offered Secto 6. [2]

4 2 The Model ad Estmators We cosder a model of codtoal momet restrctos lke Chamberla (1987). To descrbe the model let z deote a sgle observato from a..d. sequece (z 1,z 2,...), β a p 1 parameter vector, ad ρ(z,β) a scalar that ca ofte be thought of as a resdual 1. The model specfes a subvector of x, actg as codtog varables, such that for a value β 0 of the parameters E[ρ(z,β 0 ) x] =0, where E[ ] the expectato take wth respect to the dstrbuto of z. To form GMM estmators we costruct ucodtoal momet restrctos usg a vector of K codtog varables q K (x) =(q 1K (x),...,q KK (x)) 0.Letg(z, β) =ρ(z, β)q K (x). The the ucodtoal momet restrctos E[g(z,β 0 )] = 0 are satsfed. Let g (β) g(z,β), ḡ (β) P 1 g (β), ad ˆΥ(β) P 1 g (β)g (β) 0. A two-step GMM estmator s oe that satsfes, for some prelmary cosstet estmator β for β 0, ˆβ H =argm β B ḡ(β) 0 ˆΥ( β) 1 ḡ (β), (2.1) where B deotes the parameter space. For our purposes β couldbesomeothergmm 1 estmator, obtaed as the soluto to a aalogous mmzato problem wth ˆΥ( β) replaced by a dfferet weghtg matrx, such as W 0 =[ P qk (x )q K (x ) 0 /] 1. The MSE of the estmators wll deped ot oly o the umber of strumets but also o ther form. I partcular, strumetal varables that better predct the optmal strumets wll help to lower the asymptotc varace of the estmator for a gve K. Thus, for each K t s good to choose q K (x) that are the best predctors. Ofte t wll be evdet a applcato how to choose the strumets ths way. For stace, lower order approxmatg fuctos (e.g. lear ad quadratc) ofte provde the most 1 The exteso to the vector of resduals case s straghtforward. [3]

5 formato, ad so should be used frst. Also, ma terms may ofte be more mportat tha teractos. The strumets eed ot form a ested sequece. Lettg q kk (x) deped o K allows dfferet groups of strumetal varables to be used for dfferet values of K. Ideed, K flls a double role here, as the dex of the strumet set as well as the umber of strumets. We could separate these roles by havg a separate dex for the strumet set. Istead here we allow for K to ot be selected from all the tegers, ad let K fulfll both roles. Ths restrcts the sets of strumets to each have a dfferet umber of strumets, but s ofte true practce. Also, by mposg upper bouds o K we also restrct the umber of strumet sets we ca select amog, as seems mportat for the asymptotc theory. As demostrated by Newey ad Smth (2004), the correlato of the resdual wth the dervatve of the momet fucto leads to a asymptotc bas that creases learly wth K. They suggested a approach that removes ths bas (as well as other sources of bas that we wll gore for the momet). Ths estmator ca be obtaed by subtractg a estmate of the bas from the GMM estmator ad gves rse to what we refer to as the bas adjusted GMM estmator (BGMM). To descrbe t, let q = q K (x )ad ˆρ = ρ ( ˆβ H ), ŷ =[ ρ ( ˆβ H )/ β] 0, ŷ =[ŷ 1,..., ŷ ] 0, ˆΓ = q ŷ/, 0 ˆΣ = ˆΥ( ˆβ H ) 1 ˆΥ( ˆβ H ) 1ˆΓ(ˆΓ 0 ˆΥ( ˆβ H ) 1ˆΓ) 1ˆΓ 0 ˆΥ( ˆβ H ) 1. The BGMM estmator s ˆβ B = ˆβ H +(ˆΓ 0 ˆΥ( ˆβ H 1 ) 1ˆΓ) ŷ ˆρ q 0 ˆΣq. Also as show Newey ad Smth (2004) the class of Geeralzed Emprcal Lkelhood (GEL) estmators have less bas tha GMM. We follow the descrpto of these estmators gve that paper. Let s(v) be a cocave fucto wth doma that s a ope terval V cotag 0, s j (v) = j s(v)/ v j,ads j = s j (0). We mpose the ormalzatos s 1 = s 2 = 1. Defe the GEL estmator as ˆβ GEL =argm s(λ 0 g (β)) β B max λ ˆΛ (β) [4]

6 where, ˆΛ (β) ={λ : λ 0 g (β) V,,..., }. Ths estmator cludes as a specal cases: emprcal lkelhood (EL, Q ad Lawless, 1997, ad Owe, 1988), where s(v) = l(1 v), expoetal tltg (ET, Johso, ad Spady, 1998, ad Ktamura ad Stutzer, 1997), where s(v) = exp(v), ad the cotuous updatg estmator (CUE, Hase, Heato, ad Yaro 1996), where s(v) = (1+v) 2 /2. As we wll see the MSE comparsos betwee these estmators deped o s 3, the thrd dervatve of the s fucto, where CUE : s 3 =0,ET : s 3 = 1,EL: s 3 = Istrumet Selecto Crtera The strumet selecto s based o mmzg the approxmate mea squared error (MSE)ofalearcombatoˆt 0 ˆβ of a GMM estmator or GEL estmator ˆβ, whereˆt s some vector of (estmated) lear combato coeffcets. To descrbe the crtera, some addtoal otato s requred. Let β be some prelmary estmator, ρ = ρ ( β), ỹ = ρ ( β)/ β, ad ˆΥ = ρ 2 q q/, 0 ˆΓ = q ỹ/, 0 ˆΩ =(ˆΓ 0 ˆΥ 1ˆΓ), ˆτ = ˆΩ 1ˆt, d = ˆΓ 0 ( q j qj/) 0 1 q, η =ỹ d, ˆξ j = q 0 ˆΥ 1 q j /, ˆD = ˆΓ 0 ˆΥ 1 q, j=1 ˆΛ(K) = ˆξ (ˆτ 0 ρ β ) 2, ˆΠ(K) = ˆΦ(K) = ˆξ ρ (ˆτ 0 η ), 2 ˆξ ˆτ 0 ( ˆD ρ 2 ρ β )o ˆτ 0ˆΓ 0 ˆΥ 1ˆΓˆτ The crtera for the GMM estmator, wthout a bas correcto, s S GMM (K) =ˆΠ(K) 2 / + ˆΦ(K). Also, let ˆΠ B (K) = ˆΞ(K) = ρ ρ j (ˆτ 0 η )(ˆτ 0 η j )ˆξ j 2 = tr( Q ˆΥ 1 Q ˆΥ 1 ),,j=1 {5(ˆτ 0 ˆd ) 2 ρ 4 (ˆτ 0 ˆD ) 2 }ˆξ, ˆΞ GEL (K) = [5] {3(ˆτ 0 ˆd ) 2 ρ 4 (ˆτ 0 ˆD ) 2 }ˆξ,

7 where Q = P ρ (ˆτ 0 η )q q. 0 The crtera for the BGMM ad GEL estmators are S BGMM (K) = hˆλ(k)+ˆπb (K)+ˆΞ(K) / + ˆΦ(K), S GEL (K) = hˆλ(k) ˆΠ B (K)+ˆΞ(K)+s 3ˆΞ GEL (K) / + ˆΦ(K). For each of the estmators, our proposed strumet selecto procedure s to choose K to mmze S(K). As we wll show ths wll correspod to choosg K to mmze the hgher-order MSE of the estmator. Each of the terms the crtera have a terpretato. For GMM, ˆΠ(K) 2 / s a estmate of a squared bas term from Newey ad Smth (2004). Because ˆξ s of order K ths squared bas term has order K 2 /. The ˆΦ(K) term the GMM crtera s a asymptotc varace term. Its sze s related to the asymptotc effcecy of a GMM estmator wth strumets q K (x). As K grows ths terms wll ted to shrk, reflectg the reducto asymptotc varace that accompaes usg more strumets. form of ˆΦ(K) s aalogous to a Mallows crtero, that t s a varace estmator plus a term that removes bas the varace estmator. The terms that appear S(K) for BGMM ad GEL are all varace terms. No bas terms are preset because, as dscussed Newey ad Smth (2004), uder symmetry GEL removes the GMM bas that grows wth K. As wth GMM, the ˆΦ(K) termaccoutsfor the reducto asymptotc varace that occurs from addg strumets. The other terms are hgher-order varace terms, that wll be of order K/, becauseˆξ s of order K. The sum of these terms wll geerally crease wth K, although ths eed ot happe f ˆΞ(K) s too large relatve to the other terms. As we wll dscuss below, ˆΞ(K) sa estmator of Ξ(K) = ξ (τ 0 d ) 2 {5 E(ρ 4 x )/σ 4 }. As a result f the kurtoss of ρ s too hgh the hgher-order varace of the BGMM ad GEL estmators would actually decrease as K creases. Ths pheomeo s smlar to that oted by Koeker et. al. (1994) for the exogeous lear case. I ths case the crtera could fal to be useful as a meas of choosg the umber of momet codtos, because they would mootocally decrease wth K. [6] The

8 It s terestg to compare the sze of the crtera for dfferet estmators, whch comparso parallels that of the MSE. As prevously oted, the squared bas term for GMM, whch s ˆΠ(K) 2, has the same order as K 2 /. I cotrast the hgher-order varace terms the BGMM ad GEL estmators geerally have order K/, because that s the order of ξ. Cosequetly, for large K the MSE crtera for GMM wll be larger tha the MSE crtera for BGMM ad GEL, meag the BGMM ad GEL estmators are preferred over GMM. Ths comparso parallels that Newey ad Smth (2004) ad Imbes ad Spady (2002). Oe terestg result s that for the CUE, where s 3 = 0, the MSE crtera s smaller tha t s for BGMM, because ˆΠ B (K) s postve. Thus we fd that the CUE domates the BGMM estmator, terms of hgher-order MSE,.e. CUE s hgher-order effcet relatve to BGMM. Ths result s aalogous to the hgher-order effcecy of the lmted formato maxmum lkelhood estmator relatve to the bas corrected two-stage least squares estmator that was foud by Rotheberg (1983). The comparso of the hgher-order MSE for the CUE ad the other GEL estmators depeds o the kurtoss of the resdual. Let ρ = ρ(z,β 0 )adσ 2 = E[ρ 2 x ]. For codtoally ormal ρ we have E[ρ 4 x ]=3σ 4 ad cosequetly ˆΞ GEL (K) wll coverge to zero for each K, that all the GEL estmators have the same hgher-order MSE. Whe there s excess kurtoss, wth E[ρ 4 x ] > 3σ 4, ET wll have larger MSE tha CUE, ad EL wll have larger MSE tha ET, wth these rakgs beg reversed whe E[ρ 4 x ] < 3σ 4. These comparsos parallel those of Newey ad Smth (2004) for a heteroskedastc lear model wth exogeety. The case wth o edogeety has some depedet terest. I ths settg the GMM estmator ca ofte be terpreted as usg extra momet codtos to mprove effcecy the presece of heteroskedastcty of ukow fuctoal form. Here the MSE crtera wll gve a method for choosg the umber of momets used for ths purpose. Droppg the bas terms, whch are ot preset exogeous cases, leads to crtera of [7]

9 the form S GMM (K) = ˆΞ(K)/ + ˆΦ(K) S GEL (K) = hˆξ(k)+s 3ˆΞ GEL (K) / + ˆΦ(K) Here GMM ad CUE have the same hgher-order varace, as was foud by Newey ad Smth (2002). Also, as the geeral case, these crtera ca fal to be useful f there s too much kurtoss. 3 Assumptos ad MSE Results 3.1 Basc Expaso As Doald ad Newey (2001), the MSE approxmatos are based o a decomposto of the form, t 0 ( ˆβ β 0 )( ˆβ β 0 ) 0 t = ˆQ(K)+ ˆR(K), (3.2) E( ˆQ(K) X) = t 0 Ω 1 t + S(K)+T (K), [ ˆR(K)+T (K)]/S(K) = o p (1),K,. where X =[x 1,..., x ] 0, t = plm(ˆt), Ω = P σ 2 d d 0 /, σ 2 = E[ρ 2 x ], ad d = E[ ρ (β 0 )/ β x ]. Here S(K) s part of codtoal MSE of ˆQ that depeds o K ad ˆR(K) adt (K) are remader terms that goes to zero faster tha S(K). Thus, S(K) s the MSE of the domat terms for the estmator. All calculatos are doe assumg that K creases wth. The largest terms creasg ad decreasg wth K are retaed. Compared to Doald ad Newey (2001) we have the addtoal complcato that oe of our estmators has a closed form soluto. Thus, we use the frst order codto that defes the estmator to develop approxmatos to the dfferece t 0 ( ˆβ β 0 )where remaders are cotrolled usg the smoothess of the relevat fuctos ad the fact that uder our assumptos the estmators are all root- cosstet. [8]

10 To descrbe the results, let ρ = ρ(z,β 0 ), ρ β = ρ (β 0 )/ β, η = ρ β d, q = q K (x ),κ = E[ρ 4 x ]/σ 4, Υ = σ 2 q q/, 0 Γ = X q d 0 /, τ = Ω 1 t, ξ j = qυ 0 1 q j /, E[τ 0 η ρ x ]=σ ρη, Π = ξ σ ρη, Π B = σ ρη σ ρη j ξ2 j, Λ = ξ E[(τ 0 η ) 2 x ], Ξ =,j=1 ξ (τ 0 d ) 2 (5 κ ), Ξ GEL = ξ (τ 0 d ) 2 (3 κ ), where we suppress the K argumet for otatoal coveece. The terms volvg fourth momets of the resduals are due to estmato of the weght matrx Υ 1 for the optmal GMM estmator. Ths feature dd ot arse the homoskedastc case cosdered Doald ad Newey (2001) where a optmal weght matrx depeds oly o the strumets. 3.2 Assumptos ad Results We mpose the followg fudametal codto o the data, the approxmatg fuctos q K (x) ad the dstrbuto of x: Assumpto 1 (Momets): Assume that z are..d., ad () β 0 s uque value of β B (a compact subset of R p ) satsfyg E[ρ(z,β) x ]=0; () P σ 2 d d 0 / s uformly postve defte ad fte (w.p.1.). () σ 2 s bouded ad bouded away from zero. (v) E(η ι 1 j ρ ι 2 x )=0for ay o-egatve tegers ι 1 ad ι 2 such that ι 1 + ι 2 =3. (v) E(kη k ι + ρ ι x ) s bouded for ι =6for GMM ad BGMM ad ι =8for GEL. For detfcato, ths codto oly requres that E[ρ(z,β) x ] = 0 has a uque soluto at β = β 0. Estmators wll be cosstet uder ths codto because K s allowed to grow wth, as Doald, Imbes, ad Newey (2003). Part of ths assumpto s a [9]

11 restrcto that the thrd momets are zero. Ths greatly smplfes the MSE calculatos. The last codto s a restrcto o the momets that s used to cotrol the remader terms the MSE expaso. The codto s more restrctve for GEL whch has a more complcated expaso volvg more terms ad hgher momets. The ext assumpto cocers the propertes of the dervatves of the momet fuctos. Specfcally, order to cotrol the remader terms we wll requre certa smoothess codtos so that Taylor seres expasos ca be used ad so that we ca boud the remader terms such expasos. Assumpto 2 (Expaso): Assume that ρ(z,β) s at least fve tmes cotuously dfferetable a eghborhood N of β 0, wth dervatves that are all domated absolute value by the radom varable b wth E(b 2 ) < for GMM ad BGMM ad E(b 5 ) < for GEL. Ths assumpto s used to cotrol remader terms ad has as a mplcato that for stace, sup k ( / β 0 ) ρ(z,β)k <b β N It should be oted that the lear case oly the frst dervatve eeds to be bouded sce all other dervatves would be zero. It s also terestg to ote that although we allow for oleartes the MSE calculatos, they do ot have a mpact o the domat terms the MSE. The codto s stroger for GEL reflectg the more complcated remader term. Our ext assumpto cocers the strumets represeted by the vector q K (x ). Assumpto 3 (Approxmato): () There s ζ(k) such that for each K there s a osgular costat matrx B such that q K (x) =Bp K (x) for all x the support of x ad sup x X k q K (x)k ζ(k) ad E[ q K (x) q K (x) 0 ] has smallest egevalue that s bouded away from zero, ad K ζ(k) CK for some fte costat C.() For each K there exsts a sequece of costats π K ad π K such that E(kd q 0 π K k 2 ) 0 ad ζ(k) 2 E(kd /σ 2 q 0 π K k2 ) 0 as K. [10]

12 The frst part of the assumpto gves a boud o the orm of the bass fuctos, ad s used extesvely the MSE dervatos to boud remader terms. The secod part of the assumpto mples that d ad d /σ 2 be approxmated by lear combatos of q. Because σ 2 s bouded ad bouded away from zero, t s easly see that for the same coeffcets π K, kd /σ σ q π Kk 2 σ 2 kd /σ 2 q 0 π K k 2 so that d /σ ca be approxmated by a lear combato of σ q. Ideed the varace part of the MSE measures the mea squared error the ft of ths regresso. Sce ζ(k) the approxmato codto for d /σ 2 s slghtly stroger tha for d. Ths s to cotrol varous remader terms where d /σ eeds to be approxmated uform maer. Sce may cases oe ca show that the expectatos () are bouded by K 2α where α depeds o the smoothess of the fucto d /σ 2, the codto ca be met by assumg that d /σ 2 smooth fucto of x. s a suffcetly We wll assume that the prelmary estmator β used to costruct the weght matrx s a GMM estmator s tself a GMM estmator wth weghtg matrx that may ot be optmal. where we do ot requre ether optmal weghtg or that the umber of momets crease. I other words we let β solve, m g (β) 0 W0 g (β), g (β) =(1/) β q(x )ρ (β) for some K vector of fuctos q(x )adsome K K matrx W 0 whch potetally could be I K or t could be radom as would be the case f more tha oe terato were used to obta the GMM estmator. We make the followg assumpto regardg ths prelmary estmator. Assumpto 4 (Prelmary Estmator): : Assume () β p β 0 () there exst some o-stochastc matrx W 0 such that W p 0 W 0 0 adwecawrte β = P β φ ρ + o p ( 1/2 ), φ = ( Γ 0 W 0 Γ) 1 ΓW 0 q wth Γ = P q(x )d / ad E( ρ 2 φ ) φ0 < Note that the assumpto requres that we just use some root- cosstet ad asymptotcally ormally dstrbuted estmator. The asymptotc varace of the prelmary [11]

13 estmator wll be, p lm(( Γ 0 W 0 Γ) 1 ΓW 0 ΥW 0 Γ(( Γ 0 W 0 Γ) 1 ), Υ = q(x ) q(x ) 0 σ 2 / ad f the prelmary estmator uses optmal weghtg we ca show that ths cocdes wth p lm Ω provded that K crease wth a way that the assumptos of Doald, Imbes ad Newey (2003) are satsfed. Also ote that for the GMM estmator we ca wrte, ˆβ = β φ ρ + o p ( 1/2 ), φ = Ω 1 d /σ 2 The covarace betwee the(ormalzed)prelmary estmator ad the GMM estmator s the, 1 X φ φ σ 2 = Ω 1 a fact that wll be used the MSE dervatos to show that the MSE for BGMM does ot deped o the prelmary estmator. Fally we use Assumpto 6 of Doald, Imbes ad Newey (2003) for the GEL class of estmators. Assumpto 5 (GEL): s(v) s at least fve tmes cotuously dfferetable ad cocave o ts doma, whch s a ope terval cotag the org, s 1 (0) = 1, s 2 (v) = 1 ads j (v) s bouded a eghborhood of v =0for j =1,...5. The followg three propostos gve the MSE results for the three estmators cosdered ths paper. The results are proved Appedx A ad use a expaso that s provded Appedx B. Proposto 1: For GMM uder Assumptos 1-4, f w.p.a.1 as, Π ck for some c>0, K, ad ζ(k) p K/ 0 the the approxmate MSE for t 0 ( ˆβ H β 0 ) s gve by, S H (K) =Π 2 / + τ 0 (Ω Γ 0 Υ 1 Γ)τ [12]

14 Proposto 2: For BGMM uder Assumptos 1-4, the codto that w.p.a.1 as, Λ + Π B + Ξ 1 ck for some c>0, ad assumg that K wth ζ(k) 2p K/ 0 the approxmate MSE for t 0 ( ˆβ B β 0 ) s gve by, S B (K) =[Λ + Π B + Ξ] / + τ 0 (Ω Γ 0 Υ 1 Γ)τ Proposto 3: ForGEL,fAssumptos1-3,5aresatsfed, w.p.a.1 as, {Λ Π B + Ξ + s 3 Ξ GEL } ck, K, ad ζ(k) 2 K 2 / 0 the approxmate MSE for t 0 ( ˆβ GEL β 0 ) s gve by, S GEL (K) =[Λ Π B + Ξ + s 3 Ξ GEL ] / + τ 0 (Ω Γ 0 Υ 1 Γ)τ For comparso purposes, ad to help terpret the formulas, t s useful to cosder the homoskedastc case. Let σ 2 = E[ρ 2 ], σ ηρ = E[τ 0 η ρ ], σ ηη = E[(τ 0 η ) 2 ], κ = E[ρ 4 ]/σ 4, Q = q( 0 q j qj) 0 1 q (K) = σ 2 { (τ 0 d ) 2 τ 0 d q( 0 q q) 0 1 τ 0 d q }/, The we have the followg expressos uder homoskedastcty, S H (K) = (σ ρη /σ 2 ) 2 K 2 / + (K), j=1 S B (K) = (σ ηη /σ 2 + σ 2 ηρ/σ 4 )K/ + σ 2 (5 κ) X (τ 0 d ) 2 Q / + (K), S GEL (K) = (σ ηη /σ 2 σ 2 ηρ/σ 4 )K/ + σ 2 [(5 κ)+s 3 (3 κ)] X (τ 0 d ) 2 Q / + (K). For GMM, the MSE s the same as that preseted Doald ad Newey (2001) for 2SLS, whch s the same as Nagar (1959) for large umbers of momets. The leadg K/ term the MSE of BGMM s the same as the MSE of the bas-corrected 2SLS estmator, but [13]

15 there s also a addtoal term, where (5 κ) appears, that s due to the presece of the estmated weghtg matrx. Ths term s also preset for GMM, but s domated by the K 2 / bas term, ad so does ot appear our large K approxmate MSE. As log as κ<5, ths addtoal term adds to the MSE of the estmator, represetg a pealty for usg a heteroskedastcty robust weghtg matrx. Whe κ>5, usg the heteroskedastcty robust weghtg matrx lowers the MSE, a pheomeo that was cosdered Koeker et. al. (1994). For GEL the leadg K/ term s the same as for LIML, ad s smaller tha the correspodg term for BGMM. Ths comparso s detcal to that for 2SLS ad LIML, adrepresetsaeffcecy mprovemet from usg GEL. For the CUE (or ay other estmator where s 3 = 0) the addtoal term s the same for BGMM ad CUE, so that CUE has smaller MSE. The comparso amog GEL estmators depeds o the kurtoss κ. For Gaussa ρ(z, β 0 ), κ =3, ad the MSE of all the GEL estmators s the same. For κ>3, the MSE of EL s greater tha ET whch s greater tha CUE, wth the order reversed for κ<3. For Gaussa dsturbaces the relatoshps betwee the asymptotc MSE of LIML, BGMM, ad EL were reported by Rotheberg (1996), though expressos were ot gve. Whe there s heteroskedastcty, the comparso betwee estmators s exactly aalogous to that for homoskedastcty, except that the results for LIML ad B2SLS o loger apply. I partcular, CUE has smaller MSE tha BGMM, ad BGMM ad all GEL estmators have smaller MSE tha GMM for large eough K. Sce the comparsos are so smlar, ad sce may of them were also dscussed the last Secto, we omt them for brevty. 4 Mote Carlo Expermets I ths secto we exame the performace of the dfferet estmators ad momet selecto crtera the cotext of a small scale Mote Carlo expermet based o the setup Hah ad Hausma (2002) that was also used Doald ad Newey (2001). [14]

16 The basc model used s of the form, y = γy + ρ (4.3) Y = X 0 π + η for =1,..., ad the momet fuctos take the form (for K strumets), X 1 X 2 g (γ) =. (y γy ) X K where we are terested methods for determg how may of the X j should be used to costruct the momet fuctos. Because of the varace of the estmators to the value of γ we set γ = 0 ad for dfferet specfcatos of π we geerate artfcal radom samples uder the assumptos that µµ ρ E ( ρ η η ) = Σ = µ 1 c c 1 ad X N(0,I K) where K s the maxmal umber of strumets cosdered. As show Hah ad Hausma (2002) the specfcato mples a theoretcal frst stage R-squared that s of the form, R 2 f = π0 π π 0 π +1 (4.4) We cosder oe of the models that was cosdered Doald ad Newey (2001) where, µ πk 2 = c( K) 1 k 4 for k =1,.., K +1 K where the costat c( K) schosesothatπ 0 π = Rf 2/(1 R2 f ). I ths model all the strumets are relevat but they have coeffcets that are declg. Ths represets a stuato where oe has pror formato that suggests that certa strumets are more mportat tha others ad the strumets have bee raked accordgly. I ths model all of the potetal K momet codtos should be used for the estmators to be asymptotcally effcet. Note also, that our setup LIML ad 2SLS are also asymptotcally effcet estmators provded that we evetually use all of the strumets X j. [15]

17 Ideed the expermets we compute ot oly GMM, BGMM, ET, EL ad CUE (the last three beg members of the GEL class) but we also exame the performace of 2SLS ad LIML alog wth the strumet selecto methods proposed Doald ad Newey (2001). Ths allows us to gauge the small sample cost of ot mposg heteroskedastcty. As Doald ad Newey (2001) we report for each of the seve dfferet estmators, summary statstcs for the verso that uses all avalable strumets or momet codtos plus the summary statstcs for the estmators based o a set of momet codtos or strumets that were chose usg the respectve momet or strumet selecto crtero. For each model expermets were coducted wth the specfcatos for sample szes of = 200 ad = Whe the sample sze s 200 we set Rf 2 =0.1, K =10ad performed 500 replcatos, whle the larger sample sze we set Rf 2 =0.1, K =20ad we performed 200 replcatos (due to tme costrats). Both of these choces reflect the farly commo stuato where there may be a relatvely small amout of correlato betwee the strumets ad the edogeous varable (see Stager ad Stock (1997) ad Stock ad Wrght (2000) as well as the fact that wth larger data sets emprcal researchers are more wllg to use more momet codtos to mprove effcecy. For each of these cases we cosder c {.1,.5,.9}, though for brevty we wll oly report results for c =.5. I addto we cosder the mpact of havg excess kurtoss, whch as oted above has dfferetal effect o the hgher order MSE across the dfferet estmators. Thedstrbutowecosdersthatof µ ρ η = e µ ρ η, µ ρ η N(0, Σ),e logstc(0,1). where e s depedet of ρ ad η ad s dstrbuted as a logstc radom varable wth mea zero ad varace equal to oe. Gve ths partcular setup we wll have that (ρ,η ) are jotly dstrbuted wth mea zero ad a covarace matrx equal to Σ, ada coeffcet of kurtoss of approxmately κ =12.6. Wth two dfferet models, two dfferet dstrbutos for the errors, ad three dfferet choces for resdual correlatos there are a total of 12 specfcatos for each sample sze. [16]

18 The estmator that uses all momets or strumets s dcated by the suffx -all whle the estmator that uses a umber of momet codtos as chose by the respectve momet or strumet selecto crtero s dcated by -op. Therefore, for stace, GMM-all ad GMM-op are the two step estmator that uses all of the momet codtos ad the momet codtos the mmze the estmated MSE crtero respectvely. The prelmary estmates of the objects that appear the crtera were each case based o a umber of momet codtos that was optmal wth respect to cross valdato the frst stage. As Doald ad Newey (2001) we preset robust measures of cetral tedecy ad dsperso. We computed the meda bas (Med. Bas) for each estmator, the meda of the absolute devatos (MAD) of the estmator from the true value of γ =0 ad examed dsperso through the dfferece betwee the 0.1 ad 0.9 quatle (Dec. Rge) the dstrbuto of each estmator. We also examed statstcal ferece by computg the coverage rate for 95% cofdece tervals as well as the rejecto rate for a overdetfcato test ( cases where overdetfyg restrctos are preset) usg the test statstc correspodg to the estmator ad a sgfcace level of 5%. I addto we report some summary statstcs cocerg the choces of K the expermets, cludg the modal choce of K f oe used the actual MSE to choose K. There was very lttle dsperso ths varable across replcatos ad geerally the optmal K wth the true crtero was equal to the same value most f ot all replcatos. I cases where there was some dsperso t was usually ether beg some cases o ether sde of the mode. To dcate such cases we use + ad -, so that for stace 3+ meas that the mode was 3 but that there were some cases where 4 was optmal. The otato 3++ meas that the mode was 3 but that a good proporto of the replcatos had 4asbegoptmal. Tables I-VI ad X-XV cotas the summary statstcs for the estmators for =200 ad = 800 respectvely, whle Tables VII-IX ad XVI-XVIII cota the summary statstcs for the chose umber of momets across the replcatos. I geeral the results are ecouragg for all the estmators. As expected the GEL ad LIML estmators are less dspersed whe the optmal umber of momets s used, whle for GMM ad 2SLS [17]

19 the use of the crtero reduces the bas that occurs whe there s a hgh degree of covarace betwee the resduals. The mprovemets for the GEL estmators are more marked whe the there s a low to moderate degree of covarace. It s oteworthy that such stuatos there s also a dramatc mprovemet the qualty of ferece as dcated by the coverage rates for the cofdece terval. As far as testg the overdetfyg restrctos oly whe there s a hgh degree of covarace s there ay problem wth testg these restrctos. Ths occurs wth most of the estmators the small sample wth a hgh covarace ad wth GMM ad TSLS the large sample wth a hgh covarace. It also seems that usg the crtera does ot really help fxg ay of these problems. There are a umber of thgs to ote about the results for ˆK. Frst, the estmated crtera gve values for ˆK that are ofte ear the values that mmze the true crtero, suggestg that the estmated crtero s a good approxmato to the true crtero. It also oteworthy that, as oe would expect, the crtera suggest use of a small umber of momets for GMM ad 2SLS whe there s a hgh error covarace ad for the GEL estmators whe there s a low covarace. For BGMM the optmal umber s qute stable as the covarace creases. I the larger sample the optmal umber decreases as the covarace creases, but s slghtly larger whe the resduals have fat tals compared to the stuato where they do ot. Amog the GEL estmators creasg the covarace ad havg fat taled errors has the most dramatc mpact o CUE as oe would expect gve the crtera. Cocerg the effect of excess kurtoss, t does appear that the mprovemet from usg the crtera s more otceable for EL, whch s most sestve to havg fat taled errors. There also was some evdece that gog from ormal to fat taled errors helped CUE more tha the other estmators, as suggested the theory, although ths led to a lower mprovemet from usg the momet selecto crtero. [18]

20 5 Cocluso I ths paper we have developed approxmate MSE crtera for momet selecto for a varety of estmators codtoal momet cotexts. We foud that the CUE has smaller MSE tha the bas corrected GMM estmator. I addto we proposed data based methods for estmatg the approxmate MSE, so that practce the umber of momets ca be selected by mmzg these crtera. The crtera seemed to perform adequately a small scale smulato exercse. The preset paper has cosdered a restrctve evromet whch the data are cosdered a radom sample. It would be useful to exted the results two drectos. The frst would be to the dyamc pael data case. I that stuato there wll typcally be dfferet sets of strumets avalable for each resdual comg from sequetal momet restrctos. It would also be useful to exted the results to a purely tme seres cotext where oe would eed to deal wth seral correlato. Kuersteer (2002) has derved terestg results ths drecto. [19]

21 Appedx A: MSE Dervato Results Throughout the Appedx repeated use of the Cauchy Schwarz (CS), Markov (M) ad Tragle equaltes s made. We let k.k deote the usual matrx orm. The followg Maxmum Egevalue (ME) equalty s also used repeatedly, ka 0 BCk λ max (BB) kakkck for a square symmetrc matrx B ad coformable matrces A ad C. Forsmplctyof otato ad wthout loss of geeralty we assume that the true value of the coeffcets are all zero ad we oly perform the calculato forthecasewheretheresoeparameter ( addto to the auxlary parameters λ). Because hgher order dervatves are requred for the MSE calculatos (eve though they do ot appear the fal result) we use the followg otato: for j =0, 1,..., 4welet, Γ j = 1 X Γ j, Γ j = 1 X Γ j (0), ˆΓ j = 1 X Γ j ( ˆβ) Γ j = q E( j+1 β j+1 ρ (0) x ), Γ j (β) =q j+1 β j+1 ρ (β) η j = j+1 β j+1 ρ (0) E( j+1 β j+1 ρ (0) x ) ad Γ j deotes Γ j evaluated at some pot(s) lyg betwee the respectve estmator ad ts true value. Hece Γ 0 correspods to Γ the text. I addto we assume as Doald, Imbes ad Newey (2003) (hereafter DIN) that q has bee ormalzed so that kq k <Cζ(K) ade(q q)=i 0 K so that, Ã! 2 1 X kq k 2 1 X = O(K), λ max ( q q 0 δ )=O(1), f δ <C< λ m ( 1 X q q 0 δ ) > 0, for 0 <c<δ <C< where here ad elsewhere we let c deote a geerc small costat ad C ageerclarge costat. TheMSEarebasedoaexpasothatscotaedAppedxB.The remader term for GEL s dealt wth a techcal appedx that s avalable o request. [20]

22 Dervatves that are used the expaso are also avalable a techcal appedx that s avalable o request. ProofofProposto1: I dervg the MSE for GMM we smplfy otato ad refer to the estmator as ˆβ as dstct from the prelmary estmator. Sce we eed the results for BGMM we expad the estmator ad dsplay all terms that are eeded to perform the calculatos Proposto 2. Terms that wll ot be eeded for GMM ad BGMM are those that are o(k 2 / 3/2 )ado(k/ 3/2 ) respectvely. Now for GMM we have from Newey ad Smth (2004) (hereafter NS) we have that GMM ca be wrtte as the soluto to the Frst Order Codtos, where by DIN 1 X β ρ ( ˆβ)q ˆλ 0 = 0 1 X q ρ ( ˆβ)+ ˆΥ( β)ˆλ = 0 ˆλ = O( p K/) ad ˆβ = O(1/ ). Usg Appedx B ad the parttoed verse formula appled to µ 0 Γ M = 0 Γ 0 Υ 1 = µ Ω 1 Ω 1 Γ 0 0Υ 1 Υ 1 Γ 0 Ω 1 where Ω = Γ 0 0Υ 1 Γ 0 ad Σ = Υ 1 Υ 1 Γ 0 Ω 1 Γ 0 0Υ 1 we have that, µ µ Ω M 1 m = 1 Γ 0 0Υ 1 ḡ T β = 1 Σḡ T1 λ Note that we have kω 1 k = O(1), kυ 1 Γ 0 k = O(1) by ME kγ 0 k = O(1) ad λ max (Υ 1 )= O(1) ad fally, λ max (Σ) λ max (Υ 1 )=O(1). Smlarly by Appedx B ad the techcal appedx, M 1 ( ˆM M)θ = µ T β 3 T λ 3 = = M X 1 j Γ0 0 ³³ Ω 1 Γ 0 ˆλ Ω 1 Γ 0 0Υ 1 ˆΥ( β) Υ ˆλ + Γ0 Γ 0 ˆβ Υ 1 Γ 0 Ω Γ 1 0 ³ ³ 0 Γ 0 ˆλ Σ Γ 0 Γ 0 ˆβ + ˆΥ( β) Υ ˆλ µ µ T β 2 O(K/)+O(K/)+O(1/) T2 λ = O(K/)+O( K/)+O(ζ(K)K/) µ µ Ω 1 0 ˆβΓ θ j A j θ/2 = 1ˆλ (1/2)Ω 1 Γ 0 0Υ 1 ˆβ2 Γ 1 O(1/) = Υ 1 Γ 0 Ω 1 0 ˆβΓ 1ˆλ (1/2)Σ ˆβ2 Γ 1 O(1/) [21] Σ

23 M 1 X j θ j (Â j A j )θ = = µ 0 Ω 1 ˆβ Γ 1 Γ 1 ˆλ (1/2)Ω 1 Γ 0 0Υ Γ 1 ˆβ2 1 Γ 1 Υ 1 Γ 0 Ω 1 0 ˆβ Γ 1 Γ 1 ˆλ Γ (1/2)Σ ˆβ2 1 Γ 1 µ µ T β 41 + T β 42 O(K/ T41 λ + T42 λ = 3/2 )+O( K/ 3/2 ) O(K/ 3/2 )+O( K/ 3/2 ) Next we have that, µ T β 5 T5 λ = M X X µ Ω 1 θ j θ k B jk θ/6 = (1/6) 1 3 ˆβ 2 Γ 0 2ˆλ + Ω 1 Γ 0 0Υ 1 ˆβ3 Γ 2 Υ 1 Γ j k 0 Ω 1 3 ˆβ 2 Γ 0 2ˆλ + Σ ˆβ 3 Γ 2 = O( ˆβ 2 µ kγ 2 k ˆλ ) O( ˆβ O( K/ 3 = 3/2 ) kγ 2 k) O(1/ 3/2 ) by CS, the results for ˆβ ad ˆλ, Assumpto 3, ad the codtos o the elemets of M 1.Nextwehave, X X ³ θ j θ k ˆB jk B jk θ/6 = µ Γ 0 3 ˆβ2 (1/6) 2 Γ 2 ˆλ ˆβ Γ 3 2 Γ 2 j k R β,k = µ O ˆβ 2 Γ 2 Γ 2 ˆλ O( ˆβ 3 Γ2 Γ 2 ) = µ O(K/ 2 ) O( K/ 2 ) R β,k The last term s (1/24) tmes, Ã X X X ˆβ4 ³ Γ 0 θ j θ k θ l Ĉ jkl θ = ˆλ ˆβ 3 Γ ˆλ! 3 j k l Γ ˆβ 3 4 O( ˆβ 3 ˆλ ( ˆβ Γ 4 + Γ 3 ) = µ O Γ 3 ˆβ 4 = O( K/ 2 ) Therefore, = O(K/ 2 )ad = O(K/ 2 ) by CS ad ME ad the codto o the elemets of M 1. Here we have used Γ 0 0Υ 1 Γ0 Γ 0 = O(1/ )ad ³ p Γ 0 0Υ 1 ˆΥ( β) Υ = O( K/ + ζ(k)/ )=O(ζ(K)/ )whchfollowfromdin Lemma A4. Note that by Assumpto 2 we ca wrte, ˆΥ( β) Υ = ˆΥ v1 +2 βυ ρη +2 β ˆΥ v2 +2 β ˆΥ v3 + β 2 ˆΥr = ˆΥ v1 +2 βυ ρη + ˆR Υ,K [22]

24 where, ˆΥ vj = 1 X q q v 0 j for j =1, 2, 3 E(v j x )=0 v 1 = ρ 2 σ 2, v 2 = d ρ, v 3 =(η 0 ρ σ ρη (x )) Υ ρη = 1 X q q 0 σ ρη (x ), ˆΥ r = 1 X q q 0 r ad E(v j x )=0,E(v 2 j x ) <CAssumpto 1.E(kr k x ) <Cby Assumpto 2. Hece, we have that, ˆΥ p vj = O(ζ(K) K/), ˆRΥ,K = O ³(ζ(K) K + K)/ λ max (Υ ρη Υ ρη ) = O(1), λ max ( ˆΥ r ˆΥr )=O(1) ˆΥ vj Σḡ = O(ζ(K) K/) forj =1, 3 ˆΥ vj Σ Γ 0 Γ 0 = O(ζ(K) K/) forj =1, 3 ˆΥ v2 Σḡ = O(ζ(K)K/), ˆΥ v2 Σ Γ 0 Γ 0 = O(ζ(K)K/) Γ 0 Υ 1 ˆΥ vj Σ = O( p K/), Γ 0 Υ 1 Υ ρη Σ = O(1), Γ 0 Υ 1 ˆΥ r Σ = O(1) Γ 0 Γ 0 = O( p K/), Γ 0 0 Γ 0 Υ 1 Γ 0 = O(1/ ) wth the last fact followg from M ad Assumptos 1 ad 3. For the ˆλ terms from the above expaso t follows that, ˆλ = Σḡ Υ 1 Γ 0 Ω 1 Γ 0 Γ 0 0 ˆλ Σ Γ 0 Γ 0 ˆβ Σ ³ ˆΥ( β) Υ ˆλ + R λ 1 where R 1 λ = O(1/) uder the codto o K for GMM ad hece for BGMM also. Repeated substtuto ad usg the facts that by CS ad the results for ˆλ ad Γ 0 Γ 0 ad the fact that from the above expaso ˆβ = Ω 1 Γ 0 0Υ 1 ḡ+r β 1 wth = O(K/), R β 1 ˆλ = Σḡ + Υ 1 Γ 0 Ω 1 Γ0 Γ 0 0 Σḡ + Σ Γ0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ +Σ ˆΥ vj Σḡ +2 βσυ ρη Σḡ + R2, λ Υ 1 Γ 0 Ω Γ Γ 0 Σḡ = O(K/) Σ Γ 0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ = O( K/) Σ ˆΥ vj Σḡ = O(ζ(K) K/), 2 βσυ ρη Σḡ = O( K/), = O(ζ(K) K/)O(ζ(K) p K/) R λ 2 [23]

25 where R 2 λ = o( K/) uder the codto o K for BGMM ad R 2 λ = O(1/ )o(ζ(k)k/) = O(1/ )o(k 2 /) uder the codto o K for GMM. Take the lead term the expaso for ˆβ, T β 2 ad substtute for ˆλ ad apply CS, ad M to get that, Ω Γ Γ 0 ˆλ = Γ Ω Γ 0 Σḡ + Ω Γ Γ 0 Υ 1 Γ 0 Ω Γ Γ 0 Σḡ +Ω Γ Γ 0 Σ Γ 0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ +Ω Γ Γ 0 Σ ˆΥ v1 Σḡ +2 βω Γ Γ 0 ΣΥ ρη Σḡ + R β 2 R β 2 = Ω Γ Γ 0 R2 λ Here R β 2 = O( p K/)o( K/) =O(1/ )o(k/) uder the codto for BGMM, ad Ω 1 Γ0 0 Γ 0 Σḡ = O(K/) Ω Γ Γ 0 Σ Γ 0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ = O(1/ )O(K/) Ω Γ Γ 0 Υ 1 Γ 0 Ω Γ Γ 0 Σḡ = O(1/ )O(K/) Ω Γ Γ 0 Σ ˆΥ v1 Σḡ = O(1/ )O(ζ(K)K/) 2 βω Γ Γ 0 ΣΥ ρη Σḡ = O(1/ )O(K/) Uder the codto o K for GMM we have, Ω Γ Γ 0 ˆλ Γ = Ω Γ 0 Σḡ + Ω Γ Γ 0 Σ ˆΥ v1 Σḡ + R β 3 where = O(1/ )o(k 2 /). R β 3 Now cosder the secod term T β 2 gve by Ω 1 Γ 0 0Υ 1 ³ ˆΥ( β) Υ ˆλ. Usg the facts above we ca wrte, ³ Ω 1 Γ 0 0Υ 1 ˆΥ( β) Υ = Ω 1 Γ 0 0Υ 1 ˆΥ v1 2 βω 1 Γ 0 0Υ 1 Υ ρη 2 βω 1 Γ 0 0Υ 1 ˆΥ v2 + o( K/) Ω 1 Γ 0 0Υ 1 ˆΥ v1 = O( p K/), 2 βω 1 Γ 0 0Υ 1 Υ ρη = O(1/ ), 2 βω 1 Γ 0 0Υ 1 ˆΥ v2 = O( K/) [24]

26 so that usg the above expaso for ˆλ ad repeated use of CS ad M, ³ Ω 1 Γ 0 0Υ 1 ˆΥ( β) Υ ˆλ = Ω 1 Γ 0 0Υ 1 ˆΥv1 Σḡ +2 βω 1 Γ 0 0Υ 1 Υ ρη Σḡ +2 βω 1 Γ 0 0Υ 1 ˆΥv2 Σḡ Ω 1 Γ 0 0Υ 1 ˆΥv1 Υ 1 Γ 0 Ω 1 Γ0 0 Γ 0 Σḡ 2 βω 1 Γ 0 0Υ 1 Υ ρη Υ 1 Γ 0 Ω 1 Γ0 0 Γ 0 Σḡ Ω 1 Γ 0 0Υ 1 ˆΥ v1 Σ ˆΥ v1 Σḡ 2 βω 1 Γ 0 0Υ 1 Υ ρη Σ ˆΥ v1 Σḡ 2 βω 1 Γ 0 0Υ 1 ˆΥ v1 ΣΥ ρη Σḡ + R β R β = O( p K/)O( R 2 λ ) usg the fact that also uder the Assumpto 1, Ω 1 Γ 0 0Υ 1 ˆΥ v1 Σ Γ 0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ = O(1/ )O( K/) Here we have that, Ω 1 Γ 0 0Υ 1 ˆΥ v1 Σḡ = O(1/ )O( p K/) 2 βω 1 Γ 0 0Υ 1 Υ ρη Σḡ = O(1/ )O(1/ ) 2 βω 1 Γ 0 0Υ 1 ˆΥ v2 Σḡ = O(1/ )O(K/) Ω 1 Γ 0 0Υ 1 ˆΥv1 Υ 1 Γ 0 Ω 1 Γ0 0 Γ 0 Σḡ = O(1/ )O(K/) 2 βω 1 Γ 0 0Υ 1 Υ ρη Υ 1 Γ 0 Ω Γ Γ 0 Σḡ = O(1/ )O(K/) Ω 1 Γ 0 0Υ 1 ˆΥv1 Σ ˆΥ v1 Σḡ = O( p K/)O(ζ(K) K/)=O(1/ )O(ζ(K)K/) 2 βγ 0 0Υ 1 Υ ρη Σ ˆΥ v1 Σḡ = O(1/ )O(ζ(K) K/) 2 βω 1 Γ 0 0Υ 1 ˆΥv1 ΣΥ ρη Σḡ = O(1/ )O(K/) ad uder the codto o K for BGMM R β = O(1/ )o(k/). So uder the codtos for GMM we have that R β = O(1/ )o(k 2 /) ad that by the above results, ³ Ω 1 Γ 0 0Υ 1 ˆΥ( β) Υ ˆλ = Ω 1 Γ 0 0Υ 1 ˆΥ v1 Σḡ +2 βω 1 Γ 0 0Υ 1 Υ ρη Σḡ Ω 1 Γ 0 0Υ 1 ˆΥ v1 Σ ˆΥ v1 Σḡ 2 βω 1 Γ 0 0Υ 1 ˆΥ v1 ΣΥ ρη Σḡ + R β 3 [25]

27 wth R β 3 = O(1/ )o(k 2 /). Now take the thrd term T β 2 ad usg the above results ad the fact that Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 ˆβ O(1/) wehavethat ˆβ = Ω 1 Γ 0 0Υ 1 ḡ Ω 1 Γ 0 Γ 0 0 Σḡ + o(k/) so, Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 ˆβ = Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ uder the codtos for BGMM ad +Ω 1 Γ 0 0Υ 1 Γ 0 Γ 0 Ω 1 Γ 0 Γ 0 0 Σḡ + O(1/ )o(k/) Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 ˆβ = Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ + O(1/ )o(k 2 /) uder the codtos for GMM where, Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ = O(1/ )O(1/ ) Ω 1 Γ 0 0Υ 1 Γ0 Γ 0 Ω 1 Γ0 0 Γ 0 Σḡ = O(1/ )O(K/) Next we have for T β 3 by the above expaso for ˆλ, Γ 0 1ˆλ = Γ 0 1Σḡ Γ 0 1Υ 1 Γ 0 Ω 1 Γ0 Γ 0 0 Σḡ + Γ 0 1Σ Γ0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡ +Γ 0 1Σ ˆΥ v1 Σḡ +2 βγ 0 1ΣΥ ρη Σḡ + Γ 0 1R2, λ By showg that Γ 0 1Σ ˆΥ v1 Σḡ = O( K/) adkγ 0 1ΣΥ ρη Σḡk = O(1/ )wecashow that uder the codto for BGMM Γ 0 1ˆλ = Γ 0 1Σḡ Γ 0 1Υ 1 Γ 0 Ω 1 Γ0 Γ 0 0 Σḡ + o(k/) so that, after substtutg for ˆβ we have, Ω 1 0 ˆβΓ 1ˆλ (1/2) ˆβ2 Ω 1 Γ 0 0Υ 1 Γ 1 = Ω 1 Γ 0 1ΣḡΩ 1 Γ 0 0Υ 1 ḡ Γ 0 1Υ 1 Γ 0 Ω 1 Γ0 0 Γ 0 ΣḡΩ 1 Γ 0 0Υ 1 ḡ +Ω 1 Γ 0 1ΣḡΩ Γ Γ 0 Σḡ (1/2)Ω 1 Γ 0 0Υ 1 Γ 1 Ω 1 Γ 0 0Υ 1 ḡ 2 Ω 1 Γ 0 0Υ 1 Γ 1 Ω 1 Γ 0 0Υ 1 ḡω Γ Γ 0 Σḡ + O(1/ )o(k/) [26]

28 where Ω 1 Γ 0 1ΣḡΩ 1 Γ 0 0Υ 1 ḡ = O(1/ )O(1/ ) Γ 0 1Υ 1 Γ 0 Ω 1 Γ 0 Γ 0 0 ΣḡΩ 1 Γ 0 0Υ 1 ḡ = O(1/ )O(K/) Ω 1 Γ 0 1ΣḡΩ 1 Γ 0 Γ 0 0 Σḡ = O(1/ )O(K/) (1/2)Ω 1 Γ 0 0Υ 1 Γ 1 Ω 1 Γ 0 0Υ 1 ḡ 2 = O(1/) Ω 1 Γ 0 0Υ 1 Γ 1 Ω 1 Γ 0 0Υ 1 ḡω 1 Γ 0 Γ 0 0 Σḡ = O(1/ )O(K/) For the term T β 4 we have, T β 4 =(1/2)2 ˆβΩ 1 ( Γ 1 Γ 1 ) 0ˆλ + O(1/ )o(k/) = Ω 1 ( Γ 1 Γ 1 ) 0 ΣḡΩ 1 Γ 0 0Υ 1 ḡ + O(1/ )o(k/) Ω 1 ( Γ 1 Γ 1 ) 0 Σḡ Ω 1 Γ 0 0Υ 1 ḡ = O(1/ )O(K/) FallyweotethatwehaveT β 5 = O(1/ )o(k/) sothatwehaveowfoudall terms that are the rght order. For GMM let γ K, = K 2 / + K, wth K, = Ω Ω. The by, ³ ζ(k)k/ 3/2 + 2 K/ +1/ = o(k 2 /) ³ ζ(k)k/ 3/2 + K/ +1/ K/ = O(ζ(K)/ )O(K 2 /) =o(k 2 /) so wth h = Γ 0 0Υ 1 ḡ, we ca wrte, ˆβH = Ω 1 (h + T H 1 T H 4 4X Tj H + Z H ) j=1 = O(K/ ), T 2 H = O( p K/), T 3 H = O(ζ(K)K/), Z H = o(k 2 /) = O(1/ ) Usg the expaso, Ω 1 = Ω 1 + Ω 1 (Ω Ω) Ω 1 + O( 2 K,) ad otg that uder the codto o K, T1 H = o(1) the we ca wrte, Ã ³ 2 ˆβH = Ω 1 hh 0 Ω 1 + Ω 1 T1 H T1 H +2 [27] 4X j=1 ht H j! Ω 1 + o(γ K, )

29 Ω 1 E(hh 0 )Ω 1 = Ω 1 = Ω 1 + Ω 1 (Ω Ω) Ω 1 + O( 2 K,) wth E ht H 1 = E(hT H 2 ) = 0 by the thrd momet codto. Next, E Ã! 2 T1 H T1 H 1 X = E(ρ η 0 )ξ + o(γ K, ) adbythethrdmometcodto, E(T1 H h) = E(Γ 0 0Υ 1 ˆΥ v1 Σḡh) = 1 X (κ 1)d 2 ξ + o(γ K, ) = O(K/) =o(γ K, ) E(T H 3 h) = E( Γ0 Γ 0 0 Σ ˆΥ v1 Σḡh)+E(Γ 0 0Υ 1 ˆΥv1 Σ ˆΥ v1 Σḡh) Therefore we have the result, 2E( βγ 0 0Υ 1 Υ ρη Σ ˆΥ v1 Σḡ) = 1 X (κ 1)d 2 ξ + o(γ K, ) E(tΩ 1 ³h 2 +2hT 1 h + T 1 h 2 +2hT 2 h Ω 1 t) = Ω 1 + Π 2 / + Ω Γ 0 0Υ 1 Γ 0 + o(γk, ) usg the fact that Ω Ω 1 = I +(Ω Ω) Ω 1 + O( 2 K,) adthefactthatτ = Ω t ProofofProposto2:ForBGMMwehavetheaddtoalterm, (ˆΓ 0 ˆΥ 1ˆΓ 0 0 ) 1 ˆΓ 0 ˆΣĝ 0 / 2 = ˆΩ 1 ˆΓ 0 ˆΣĝ 0 / 2 ˆΓ 0 = ˆΓ 0 /, ˆΓ 0 = q ŷ,ŷ =[ ρ ( ˆβ H )/ β] 0,ĝ = q ρ ( ˆβ H ) ˆΣ = ˆΥ 1 ˆΥ 1ˆΓ 0 ˆΩ 1ˆΓ 0 0 ˆΥ 1,ˆΩ = ˆΓ 0 0 ˆΥ 1ˆΓ 0 Frst we have by Assumpto 3, lettg Γ 0 = q E( ρ (0)/ β), Γ 0 (0) = q ρ (0)/ β ˆΓ 0 = Γ 0 +(Γ 0 (0) Γ 0 )+ ˆβΓ 1 + ˆβ(Γ 1 (0) Γ 1 )+³ ˆβ 2 Γ 2 ĝ = g + Γ 0 ˆβ +(Γ0 (0) Γ 0 ) ˆβ ³ 2 + ˆβ Γ 1 [28]

30 Usg, ³ ³ ˆΩ 1 = Ω 1 + Ω ³Ω 1 ˆΩ Ω 1 + Ω 1 Ω ˆΩ ˆΩ 1 Ω ˆΩ Ω 1 ad, ˆΩ Ω = ˆΓ 0 ˆΥ 1ˆΓ 0 0 ˆΓ 0 0Υ 1ˆΓ 0 + ˆΓ 0 0Υ 1ˆΓ 0 Γ 0 0Υ 1 Γ 0 ˆΓ 0 ˆΥ 1ˆΓ 0 0 ˆΓ 0 0Υ 1ˆΓ ³ 1 ³ 0 = Γ 0 0 ˆΥ 1 Υ Γ 0 + ³ˆΓ 1 0 Γ 0 0 ˆΥ 1 Υ ³ ˆΥ 1 Υ 1 ³ˆΓ ³ 0 Γ 0 + ³ˆΓ 0 Γ 0 0 ˆΥ 1 Υ 1 ³ˆΓ 0 Γ 0 +Γ 0 0 Γ 0 wth ˆΓ 0 Γ 0 = Γ 0 Γ 0 + ˆβ Γ 1 + ˆβ 2 Γ 1 = Γ 0 Γ 0 + ˆβΓ 1 + ˆβ( Γ 1 Γ 1 )+ ˆβ 2 Γ 1 we ca wrte, ³ 1 ³ ³ ˆΥ 1 Υ = Υ ³Υ 1 ˆΥ Υ 1 + Υ 1 Υ ˆΥ ˆΥ 1 Υ ˆΥ ³ 1 Γ 0 0 ˆΥ 1 Υ Γ 0 = Γ 00Υ ³ 1 Υ ˆΥ Υ 1 Γ 0 ³ +Γ 0 0Υ ³Υ 1 ˆΥ ˆΥ 1 Υ ˆΥ Υ 1 Γ 0 ³ Γ 0 0Υ ³Υ 1 ˆΥ ˆΥ 1 Υ ˆΥ Υ 1 Γ 0 = O(K/) Γ 0 0Υ ³Υ 1 ˆΥ Υ 1 Γ 0 = Γ 0 0Υ 1 ˆΥv1 Υ 1 Γ 0 Γ 0 0Υ 1 ˆΥv1 Υ 1 Γ 0 = O(1/ ) ˆβΓ 0 0Υ 1 Υ ρη Υ 1 Γ 0 + O(1/) ˆβΓ 0 0Υ 1 Υ ρη Υ 1 Γ 0 = O(1/ ) ³ˆΓ 0 Γ 0 0 ³ ˆΥ 1 Υ 1 Γ 0 = O(ζ(K) K/) ³ˆΓ 0 Γ 0 0 ³ ˆΥ 1 Υ 1 ³ˆΓ 0 Γ 0 = o(k/) Υ 1 ˆΓ 0 0Υ 1ˆΓ 0 Γ 0 0Υ 1 Γ 0 = ³ˆΓ 0 Γ 0 0 Υ 1 Γ 0 + Γ 0 0Υ ³ˆΓ 1 0 Γ 0 + ³ˆΓ0 Γ 0 0 Υ 1 ³ˆΓ0 Γ 0 ³ˆΓ 0 Γ 0 0 Υ 1 Γ 0 = Γ 0 Γ 0 0 Υ 1 Γ 0 + ˆβΓ 1 Υ 1 Γ 0 + O(1/) ³ˆΓ 0 Γ 0 0 Υ 1 ³ˆΓ 0 Γ 0 = O(K/) [29]

31 so that, ˆΩ Ω = Γ 0 0Υ 1 ˆΥ v1 Υ 1 Γ 0 +2ˆβΓ 0 0Υ 1 Υ ρη Υ 1 Γ 0 Γ 0 Γ 0 0 Υ 1 Γ 0 ˆβΓ 1 Υ 1 Γ 0 Γ 0 0Υ Γ 1 0 Γ 0 0 ˆβΓ 0 Υ 1 Γ 1 + O(ζ(K) K/) ˆΩ Ω = O(1/ ) Now, ˆΓ 0 ˆΣĝ 0 / 2 = µ µ 1 1 ˆΓ 0 0Σĝ 2 / 2 + ˆΓ 0 0(ˆΣ Σ)ĝ 2 / 2 ad, ˆΓ 0 Σĝ / 2 = Γ 0 0Σg / 2 + Γ 0 0ΣΓ 0 ˆβ/ 2 + (Γ 0 (0) Γ 0 ) 0 Σg / ˆβ (Γ 0 (0) Γ 0 ) 0 Σ (Γ 0 (0) Γ 0 ) ˆβ/ 2 (Γ 1 (0) Γ 1 ) 0 Σg / 2 + O(1/ )o(k/) wth, Γ 0 0Σg / 2 = O(1/ )O(ζ(K) K/) Γ 0 0ΣΓ 0 ˆβ/ 2 = O(1/ )O(K/) (Γ 0 (0) Γ 0 ) 0 Σg / 2 = O(1/ )O(K/ ) (Γ 0 (0) Γ 0 ) 0 Σ (Γ 0 (0) Γ 0 ) ˆβ/ 2 = O(1/ )O(K/) ˆβ (Γ 1 (0) Γ 1 ) 0 Σg / 2 = O(1/ )O(K/) [30]

32 Next = ˆΓ 0 0(ˆΣ Σ)ĝ / 2 + ˆΓ 0 0 ³ ˆΥ 1 Υ 1 ĝ / 2 + ˆΓ 0 0 ˆΓ 0 0 ³ ˆΥ 1ˆΓ 0 Ω 1ˆΓ 0 0 ˆΥ 1 Υ 1 Γ 0 Ω 1 Γ 0 0Υ 1 ĝ / 2 ³Υ 1 Γ 0 ˆΩ 1ˆΓ 00 ˆΥ 1 ˆΥ 1ˆΓ 00 0 Ω 1ˆΓ ˆΥ 1 ĝ / 2 For the frst term usg the above expasos for ˆΥ 1 Υ 1 ad for Υ ˆΥ Proposto 1, we ca show that usg T, followed by CS ad the ME ad the codtos o K, for BGMM we have ˆΓ 0 0Υ ³Υ 1 ˆΥ Υ 1 ĝ / 2 = Γ 0 0Υ 1 ˆΥv1 Υ 1 g / 2 wth, 2 ˆβ (Γ 0 (0) Γ 0 ) 0 Υ 1 ˆΥv1 Υ 1 g / 2 (Γ 0 (0) Γ 0 ) 0 Υ 1 Υ ρη Υ 1 g / 2 + O(1/ )o(k/) Γ 0 0Υ 1 ˆΥ v1 Υ 1 g / 2 = (1/ )O(ζ(K)K/) (Γ 0 (0) Γ 0 ) 0 Υ 1 ˆΥv1 Υ 1 g / 2 = (1/ )O(ζ(K)K/) 2 ˆβ (Γ 0 (0) Γ 0 ) 0 Υ 1 Υ ρη Υ 1 g / 2 = O(1/ )O(K/) where the secod part of the frst term satsfes (usg smlar argumets), ³ ˆΓ 0 0Υ ³Υ 1 ˆΥ ˆΥ 1 Υ ˆΥ Υ 1 ĝ / 2 = O(1/ )o(k/) For the remag terms we ca show usg smlar argumets that they are each O(1/ )o(k/). So altogether P ˆΓ 0 ˆΣĝ / 2 = O(K/ )sothatbythecodtook ad, O(K/ ³ ) O(ζ(K) K/)+O( K, ) = O(γ K, ) [31]

33 we have that ˆΩ 1 ˆΓ 0 ˆΣĝ 0 / 2 = Ω 1 Γ 0 0Σg / 2 + Ω 1 Γ 0 2 0ΣΓ 0 ˆβ/ +Ω 1 (Γ 0 (0) Γ 0 ) 0 Σg / 2 + Ω 1 (Γ 0 (0) Γ 0 ) 0 Σ (Γ 0 (0) Γ 0 ) ˆβ/ 2 +Ω 1 ˆβ (Γ 1 (0) Γ 1 ) 0 Σg / 2 Ω 1 Γ 0 0Υ 1 ˆΥ v1 Υ 1 g / 2 Ω ˆβΩ (Γ 0 (0) Γ 0 ) 0 Υ 1 ˆΥv1 Υ 1 g / 2 (Γ 0 (0) Γ 0 ) 0 Υ 1 Υ ρη Υ 1 g / 2 +Ω 1 Γ 0 0Υ 1 ˆΥ v1 Υ 1 Γ 0 Ω 1 Γ 0 0Σg / 2 +2 ˆβΩ 1 Γ 0 0Υ 1 Υ ρη Υ 1 Γ 0 Ω 1 Γ 0 0Σg / 2 Ω ³Γ 1 0 0Υ Γ Γ 0 + Γ 0 Γ 0 Υ 1 Γ Ω 1 Γ 0 0Σg / 2 ˆβΩ 1 Γ 0 0Υ 1 Γ 1 + Γ 1 Υ 1 Γ Ω 1 Γ 0 0Σg / 2 + O(1/ )o(k/) Now combg terms here wth the terms that are ot O(1/ )o(k/) fromgmm ad usg the facts that for j =0, 1, Γ j Γ j 0 Σḡ + Γ 0 0Σg / 2 = Γ 0 0Σg j / 2 = O(1/ )O( p K/) ad the fact that ˆβ = Ω 1 Γ 0 0Υ 1 ḡ + o(1/ )havethatforstace, 6=j ( Γ 1 Γ 1 ) 0 ΣḡΩ 1 Γ 0 0Υ 1 ḡ + ˆβ (Γ 1 (0) Γ 1 ) 0 Σg / 2 = O(1/ )O( K/) Ths also occurs by combg terms from GMM ad BGMM. The for BGMM we have [32]

34 that for γ K, = K/ + K,, T B 1 T B 4 ˆβB = Ω 1 (h + = O( p K/), T 2 B = O(ζ(K) K/) T 5 B 5X Tj B + Z B ) j=1 = O(ζ(K)K/), T 3 B = O(1/ ) = O(K/), Z B = o(γ K, ) the uder the codto o K as wth GMM, ³ 2 ˆβB = Ω 1 hh 0 Ω 1 + Ω 1 (T1 B T1 B + 5X 2Tj B h)ω 1 + o(γ K, ) Now dog the calculatos we get E(hT B 3 )=O by Assumpto 1(v). The we have for E(T B 1 T B 1 ) the terms, E( E(Γ 0 0Υ 1 ˆΥ v1 Σḡ j=1 Γ 0 0Σg j / 2 ) = 1 X E(η 0)ξ 2 + X,j Γ 0 0Σg j / 2 ) = o(γ K, ) 6=j 6=j E(Γ 0 0Υ 1 ˆΥv1 Σḡḡ 0 Σ ˆΥ v1 Υ 1 Γ) = 1 X (κ 1)d 2 ξ + o(γ K, ) For the term 2T1 B h we have by Assumpto 1(v) 2E(Γ 0 0Υ 1 ˆΥv1 Σḡh) = 2 1 X (κ 1)d 2 ξ + o(γ K, ) Next for 2T B 2 h we have, E(ρ η 0 )E(ρ j η 0j )ξ 2 j + o(γ K, ) 2E(Γ 0 0Υ 1 ˆΥv1 Σ ˆΥ v1 Σḡh) = o(γ K, ) 2E( Γ 0 0 Γ 0 Σ ˆΥ v1 Σḡh 0 ) = o(γ K, ) 4E( βγ 0 0Υ 1 Υ ρη Σ ˆΥ v1 Σḡh 0 ) = o(γ K, ) 2E( (Γ 0 (0) Γ 0 ) 0 Υ 1 ˆΥ v1 Υ 1 g / 2 h 0 ) = o(γ K, ) 2E( Γ 0 0Υ 1 ˆΥ v1 Υ 1 g / 2 h 0 ) = o(γ K, ) [33]

35 For the terms 2T B 4 h we have, 2E( Γ 0 0Σg / 2 h 0 )= 2 1 X d 2 ξ + o(γ K, ) Next we must deal wth the may terms 2T4 B h. Wehaveforthetermscomgfrom Proposto 1, 4E( βγ 0 0Υ 1 ˆΥ v2 Σḡh 0 ) = 4 1 X d 2 ξ + o(γ K, ) 4E( βγ 0 0Υ 1 Υ ρη Υ 1 Γ 0 Ω Γ Γ 0 Σḡh 0 ) = 4Γ 0 0Υ 1 Υ ρη Υ 1 Γ 1 X E(ρ η 0 )ξ +o(γ K, ) 4E( βγ 0 0Υ 1 ˆΥ v1 ΣΥ ρη Σḡh 0 ) = o(γ K, ) 2E( Γ 0 0 Γ 0 Σ Γ 0 Γ 0 Ω 1 Γ 0 0Υ 1 ḡh 0 ) = 2 1 X E(η 0)ξ 2 + o(γ K, ) 4E( β Γ 0 0 Γ 0 ΣΥ ρη Σḡh 0 ) = 1 X E(ρ η 0 )E(ρ j η j )ξj 2 + o(γ K, ) 2E(Γ 0 1ΣḡΩ 1 Γ0 0 Γ 0 Σḡh 0 ) = o(γ K, ) ad the terms comg from the expaso of the bas adjustmet factor,,j 2E( 2E( Γ 0 0ΣΓ 0 ˆβ/ 2 h 0 ) = 2 1 X d 2 ξ (Γ 0 (0) Γ 0 ) Σ (Γ 0 (0) Γ 0 ) ˆβ/ 2 h 0 ) = 2 1 X E(η 2 )ξ + o(γ K, ) 4E( ˆβ (Γ 0 (0) Γ 0 ) 0 Υ 1 Υ ρη Υ 1 g / 2 h 0 ) = 4 1 X E(ρ η 0 )E(ρ j η j )ξj 2 + o(γ K, ),j 4E( ˆβΓ 0 0Υ 1 Υ ρη Υ 1 Γ 0 Ω 1 = 4Γ 0 0Υ 1 Υ ρη Υ 1 Γ 0 1 (Γ 0 (0) Γ 0 ) 0 Σg / 2 h) X E(ρ η 0 )ξ + o(γ K, ) The collect all terms use the defto of τ to get the result. [34]

36 ProofofProposto3: For ease of otato assume that ˆβ the GEL estmator has populato zero, so that both ˆβ ad ˆλ have populato value zero. Also ote that uder the codtos of the proposto γ K, C(K/ + K, )sothattermstheexpaso ca be dropped f they are o(k/).the the FOC gve by, P m (ˆθ)/ = 0 where, µ m (θ) = s 1 (λ 0 Γ0 (β) g (β)) 0 λ g (β) adwehavebyappedxbthatm ad M 1 are the same as for GMM. Usg Appedx Bwehave µ T β 1 T λ 1 M 1 m = For M 1 ³ ˆM M ˆθ we have terms, µ Ω 1 Γ 0 0Υ 1 ḡ Σḡ = µ O(1/ ) O( p K/) T β 2 = Ω Γ Γ 0 ˆλ Ω 1 Γ 0 0Υ 1 Υ vˆλ Ω 1 Γ 0 0Υ Γ 1 0 Γ 0 ˆβ = O(K/)+O( K/)+O(1/) T2 λ = Υ 1 Γ 0 Ω 1 Γ0 0 ³ Γ 0 ˆλ Σ Γ0 Γ 0 ˆβ + Υvˆλ = O(K/)+O( K/)+O(ζ(K) K/) usg CS ad argumets smlar to Proposto 1. For the term (1/2)M 1 P j ˆθ j A j ˆθ we have smlarly, ³ T β 3 = Ω 1 ˆβΓ 0 1ˆλ + ˆλ0 Υ ρηˆλ Ω 1 Γ 00Υ ³ 1 (1/2) ˆβ 2 Γ 1 +2ˆβΥ ρηˆλ = O(1/)+O(K/)+O(1/)+O(1/) ³ ³ T3 λ = Υ 1 Γ 0 Ω 1 ˆβΓ 0 1ˆλ + ˆλ0 Υ ρηˆλ Σ (1/2) ˆβ 2 Γ 1 +2ˆβΥ ρηˆλ = O(1/)+O(K/)+O(1/)+O( K/) [35]

37 ad from M P 1 ˆθ j j ³Âj A j ˆθ/2 wehaveterms, T β 4 = Ω 1 0 ³ ˆβ Γ 1 Γ 1 ˆλ + Ω 1ˆλ0 ˆΥ v2 + ˆΥ v3 ˆλ (1/2)Ω 1 Γ 0 0Υ Γ 1 ˆβ2 1 Γ 1 ³ 2Ω 1 Γ 0 0Υ 1 ˆβ ˆΥ v2 + ˆΥ v3 ˆλ +(s3 /2)Ω 1 Γ 0 0Υ X 1 ˆλ j ˆΥ j v4ˆλ j = O(K/ 3/2 )+O(K/)O(ζ(K) p K/)+O(1/ 3/2 )+O(K/ 3/2 ) +O(K/)O( Kζ(K)/ ) T4 λ = Υ 1 Γ 0 Ω 1 (3/2) ˆβ Γ 0 ³ 1 Γ 1 ˆλ +Υ 1 Γ 0 Ω 1ˆλ0 ˆΥ v2 + ˆΥ v3 Γ ˆλ Σ(1/2) ˆβ2 1 Γ 1 Σ ˆβ ³ ˆΥ v2 + ˆΥ v3 ˆλ (s3 /2)Σ X ˆλ j ˆΥ j v4ˆλ j = O(K/ 3/2 )+O(K/)O(ζ(K) p K/)+O( K/ 3/2 )+O(ζ(K) K/ 3/2 ) +O(ζ(K)K/ 3/2 )+O(K/)O(ζ(K) 2p K/) = o(1/) whereweusetheotato ˆΥ j v4 = P q q ρ 3 q j /. Note that for T β 4 the order of the last term follows from, 1 X ˆλ j Γ 0 0Υ 1 ˆΥ j v4ˆλ ˆλ ˆλ 0 1 X D q q 0 ρ 3 ˆλ 2 1 X D q q 0 ρ 3 j O(K/)O( Kζ(K)/ ) by, 1 X 2 E( D q q 0 ρ 3 ) 1 X D 2 2 E(ρ 6 ) kq k 4 = O(Kζ(K) 2 /) whle for T4 λ t follows from use of T, CS, the result for ˆλ ad the fact that, X E( ˆΥ j v4 2 ) 1 λ max(υ) X kq 2 k 2 E(ρ 6 ) X q j 2 q 0 Υ 1 q j j 1 Cλ max(υ) X kq k 2 qυ 0 1 q / kq k 2 = O(Kζ(K) 4 /) [36]