A Simple Efficient Instrumental Variable Estimator for Panel AR(p) Models When Both N and T are Large

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1 A Smple Effcen Insrumenal Varable Esmaor for Panel ARp Models When Boh N and T are Large Kazuhko Hayakawa Deparmen of Economcs, Hosubash Unversy JSPS Research Fellow Frs Draf: May 2007 Ths verson: February 9, 2008 Absrac In hs paper, we show ha for panel ARp models, an nsrumenal varable esmaor wh nsrumens devaed from pas means has he same asympoc dsrbuon as he nfeasble opmal esmaor when boh N and T, he dmensons of he cross secon and he me seres, are large If we assume ha he errors are normally dsrbued, he asympoc varance of he proposed esmaor s shown o aan he lower bound when boh N and T are large A smulaon sudy s conduced o assess he esmaor Keywords: panel ARp models, he opmal nsrumens, nsrumens devaed from pas means JEL classfcaon: C3, C23 E-mal : em032@ yahoocojp Remove he space The auhor s deeply graeful o wo anonymous referees, Kaddour Hadr, Cheng Hsao, Naoo Kunomo, Ej Kurozum, Kosuke Oya, Donggyu Sul, Taku Yamamoo, and he parcpans of he 4h Inernaonal Conference of Panel Daa a Xamen Unversy, he Fall meeng of Japanese Economc Assocaon a Nhon Unversy and Hosubash Conference on Economercs 2007 for helpful commens I also acknowledge Ryo Oku who posed a queson ha nspred hs paper Ths research benefed from he JSPS fellowshp All he remanng errors are mne

2 Inroducon Snce he work of Anderson and Hsao 98, 982, nsrumenal varables have been wdely used for he esmaon of dynamc panel daa models However, snce he esmaor s no generally effcen, Holz-Eakn, Newey, and Rosen 988 and Arellano and Bond 99 proposed o use he generalzed mehod of momens esmaor o mprove effcency The esmaor has subsequenly been refned n a number of sudes, ncludng Arellano and Bover 995, Ahn and Schmd 995, 997 and Blundell and Bond 998 However, alhough he esmaor s generally more effcen han he esmaor, s well known ha he esmaor s more based han he esmaor n fne sample In hs paper, we focus on he esmaor and address he effcency problem of he esmaor Specfcally, we show ha, for panel ARp models, a smple one-sep esmaor usng nsrumens devaed from pas means has he same asympoc dsrbuon as he nfeasble opmal esmaor derved by Arellano 2003b when boh N and T are large If normaly s assumed on he errors, he proposed esmaor s shown o be asympocally effcen Compared o he exsng esmaors, here are wo advanages n he proposed esmaor The frs s ha alhough he WG and esmaors are conssen only when T and N s large, respecvely, he proposed esmaor s conssen under large N and fxed T, fxed N and large T, or large N and large T asympocs Ths mples ha he proposed esmaor can be used for large N and small T, small N and large T, or large N and large T panel daa The second advanage s ha he proposed esmaor s more effcen han Anderson and Hsao s 98 esmaor, and as effcen as he WG and esmaors when boh N and T are large Smulaon resuls reveal ha he proposed esmaor s almos unbased, and he dfference n dspersons beween he feasble opmal esmaor and he proposed esmaor s small when T s large The remander of hs paper s organzed as follows Secon 2 provdes he seup and he man resul Secon 3 presens a Mone Carlo smulaon and assess he heorecal resul Fnally, Secon 4 concludes A word on noaon For a vecor x and a marx A, we defne x 2 = x x and A 2 = ra A where r denoes he race operaor Recen papers ha dscuss he esmaor are Arellano 2003b and Hahn, Hausman, and Kuersener 2007, proposng wo-sep effcen esmaors and he long dfference esmaor, respecvely 2

3 2 Seup and Resul 2 The model and assumpons Le us consder he followng panel ARp model: y = α y, + α 2 y, α p y, p + η + v = α x + η + v =,, N, =,, T where α =α,, α p, x =y,,, y, p, v has zero mean gven by η,y, p,, y, and p s fxed and known 2 For convenence, we assume ha y,0,,y, p are observed can be wren n a companon form as x,+ = Πx + d η + v 2 where d =, 0,, 0 of dmenson p and Π s he p p marx gven by α α p Π = I p O p where I k s an deny marx of order k and O k l s a k l marx of zeros We make he followng assumpons, whch are par of he assumpons made by Lee 2005 Assumpon {v } =,, T, =,, N are d over and and ndependen of η and x, wh Ev =0, varv =σv 2 and fne fourh order momen {η } =,, N are d over wh Eη =0and varη =ση 2 Assumpon 2 The nal observaons sasfy x =I p Π d η + w 0 where w 0 = j=0 Πj v, j d Assumpon 3 de I p Πz 0for all z Assumpon 4 Le m j, =Π j d v, j For all,, and for any r,, r 4 {, 2, p}, j,,j 4 =0 cum r,,r 4 m j,,m j2,,m j3,,m j4, < 2 The problem how o choose p s exensvely dscussed by Lee

4 Unlke Lee 2005, we do no need o mpose he asympoc relave rao beween N and T Assumpons and 2 are sandard ones n he leraure 3 Alhough Assumpon 2 can be relaxed o nonsaonary nal condons, we do no pursue hs here for he purpose of smplcy However, he man resul of hs paper s expeced o hold snce he nal condons are neglgble when T s large and snce we do no use momen condons ha rely on saonary nal condons as Blundell and Bond 998 do Assumpon 3 s he sably condon, and Assumpon 4 s necessary o use he cenral lm heorem for double ndexed processes 4 Under Assumpons 2 and 3, x can be wren as x,+ =I p Π d η + w where w = Π j v, j d j=0 To remove he ndvdual effecs, η, we use he forward orhogonal devaon FOD snce he errors ransformed by he FOD are serally uncorrelaed and homoskedasc f he orgnal errors are 5 Specfcally, he model o be esmaed s gven by y = α x + v =,, N, =,, T 3 where y = c y y,+ + + y T /, x = c x x,+ + + x T /, v = c v v,+ + + v T /, and c 2 =/ + 22 The nsrumenal varable esmaors The nfeasble opmal nsrumens Followng Arellano 2003a, b, he nfeasble opmal esmaor n a large N and small T conex akes he followng form: α OP T N T N T = h x h y ÂOP T T = α + bop = = = = where h = Ex y and y =y,,, y, p α OP T s an deal esmaor snce s conssen and asympocally effcen when N s large and T s fxed 3 See Alvarez and Arellano 2003 for he AR case 4 See Phllps and Moon 999 and Hahn and Kuersener Noe ha akng a frs dfference nduces a seral correlaon n he errors, and hs correlaon changes he form of he opmal nsrumens defned n he nex secon 4

5 However, he drawback of hs esmaor s ha s nfeasble snce he opmal nsrumens h s unknown A sandard approach o oban a feasble opmal esmaor s o use a sample lnear projecon of h, whch s gven by N N ĥ = x y y y y = = In hs case, he feasble opmal esmaor s equvalen o he esmaor usng y as nsrumens: T α LEV T = X M LEV X X M LEV y 4 = where X =x,, x N, M LEV = Z LEV and y =y,, y N However, one problem of α LEV = Z LEV Z LEV Z LEV, Z LEV =y,, y N, s ha f N and T ncrease a he same rae, he esmae of h s asympocally based see Arellano 2003a, p70 Ths causes a bas n α LEV In fac, for he case of p =, Alvarez and Arellano 2003 show ha α LEV has a bas of he order O/N 6 Thus, n hs paper, we propose an alernave approach Insead of esmang he opmal nsrumens, we propose o use an observable varable ha has he same srucure as he opmal nsrumens, h Hence, we need o nvesgae he srucure of h Arellano 2003b shows ha, under he assumpon ha Eμ y concdes wh he lnear projecon, he nfeasble opmal nsrumens can be rewren n he followng form: Ex y = c I p ΠI p Π T I p Π x ι p Eμ y = c I p ΠI p Π T I p Π { w, + ι p μ Eμ y } 5 = c I p O w, + O p 6 where he second equaly comes from he fac ha x = ι p μ + w,, μ = η / α ι p, and he hrd equaly s proved n Lemma A see Appendx From 5 and 6, we fnd ha he ndvdual effec μ s demeaned n 5, when s large, w, s he domnang erm n 6 Our nex ask s o fnd an observable varable ha has he same srucure as 6 6 Also see Bun and Kve

6 Insrumens devaed from pas means We consder nsrumens z l =z l,z 2 l z l = c y, y, l + + y, p+ + p l z 2 l = c y, 2 y, 2 l + + y, p+ + p l 2 z p l = c y, p y, p l + + y, p+ l,, z p l as follows: where l 0 s fxed 7 Snce z l s devaed from pas means, can be seen as a modfcaon of he recursve mean adjusmen RMA mehod by So and Shn Now, we show ha z l mees he above wo requremens For he frs requremen, s sraghforward o show ha he ndvdual effecs are demeaned snce z l s devaed from pas means For he second requremen, we show n Appendx ha z l can be wren as z l = c w, + O p 7 Thus, comparng 6 and 7, we fnd ha unobservable h and observable z l have he same srucure, e, demeanng ndvdual effecs, w, s domnang The esmaor usng z l as nsrumens s gven by α RMAl N T N T = z l x z l y 8 = 0 = 0 = α + = ÂRMAl brmal where 0 = 2 for l =0, and 0 = l + for l 2 The followng proposon esablshes he asympoc equvalence of he nfeasble opmal esmaor, α OP T, and he proposed esmaor α RMAl n he sense ha boh esmaors have he same asympoc dsrbuon Proposon Le Assumpons, 2, and 3 hold Then, for fxed l 0, as boh N and T end o nfny, he nfeasble opmal esmaor α OP T and he feasble esmaor α RMAl are conssen If we furher assume ha Assumpon 4 holds, hen, as boh N and T end o nfny, we have d α α N 0,σv 2 Ew, w, 9 7 For he choce of l, we consder l =0,, 4 n smulaon sudes see Secon 3 8 The case l = corresponds o he orgnal RMA mehod = 6

7 where α denoes α OP T and α RMAl Noe ha he asympoc varance σv 2 as he whn groups WG esmaor derved by Lee 2005 Ew, w, s of he same form Remark For he case of p =, Alvarez and Arellano 2003 show ha α LEV and he WG esmaor, α WG, has he followng asympoc dsrbuon: α LEV α WG α N + α α T + α d N 0, α 2, 0 d N 0, α 2 Also, from Proposon, we have α RMAl α d N 0, α 2 2 Comparng 0, and 2, we fnd ha alhough all esmaors have he same asympoc varance, α LEV and α WG have asympoc bases of he order O/N and O/T, respecvely, whle he dsrbuon of α RMAl α s cenered a zero Ths s because α LEV and α WG suffer from he many nsrumens problem and ncdenal parameer problem, respecvely, whle α RMAl does no 9 Remark 2 Hahn and Kuersener 2002 show ha f we furher assume normaly on v, hen σv 2 Ew, w, s equal o he lower bound under large N and T asympocs 0 Hence, α RMAl s an effcen esmaor under large N and T asympocs whou an asympoc bas when v s normally dsrbued Remark 3 Anoher feaure of α RMAl s ha snce he ndvdual effecs are compleely elmnaed from boh he model and nsrumens under saonary nal condons, he performance of α RMAl s no affeced by he varance rao of he ndvdual effecs o he dsurbances alhough he ypcal esmaors usng nsrumens n levels are Remark 4 Alhough we use large N and T asympocs n dervng he properes, conssency and asympoc normaly are also obaned under large N and fxed T, or fxed N and large T asympocs Especally, under fxed N and large T asympocs, he same expresson as 9 s obaned Ths s n marked conras o he esmaor where large N s requred Furhermore, alhough he esmaor can be compued only when T N, he proposed esmaor can be compued for any N and T 9 Ths nerpreaon was suggesed by a referee 0 Noe ha he ARp model can be wren as he VAR model 2 See Bun and Kve 2006, Hayakawa 2007, and Bun and Wndmejer

8 3 Mone Carlo Smulaon In hs secon, we compare α RMAl wh oher esmaors by Mone Carlo smulaon We consder AR and AR2 models v and η are drawn from N0, ndependenly We consder he cases of T, N = 0, 00, 0, 500, 5, 00, 5, 300, 20, 00, 20, 200, 50, 00, and 00, 00 For he AR model, we se α =05, 09, and for he AR2 model, we se α,α 2 =06, 0, 06, 03 We generae T + p + 50 observaons for each and dscard he frs 50 perods o dmnsh he effec of nal condons We compue he medan Medan, he nerquarle range IQR, and he medan absolue error MAE The number of replcaons s 5000 for all cases We consder he and esmaors usng nsrumens n levels or devaed from pas means The esmaor usng y as nsrumens s defned as 4 The esmaor usng z as nsrumens s defned by 2 α RMA = where M RMA α RMA T = Z RMA X M RMA X =2 Z RMA Z RMA Z RMA does no share he problem wh αlev T X M RMA y =2, and Z RMA =z,, z N ha he number of nsrumens s oo large In fac, he number of nsrumens used n α RMA s OT, whle ha n α LEV s OT 2 For he proposed esmaors, we consder α RMA0,, α RMA4 as defned by 8 Also, for he purpose of comparson, we consder an esmaor usng x as nsrumens as follows: α LEV = N T x x = = N T x y = = Noe ha α LEV s no exacly he same esmaor as he one by Anderson and Hsao 98, 982 snce hey used he frs-dfference o remove he ndvdual effecs from he model The smulaon resuls for AR and AR2 model are provded n Tables and 2, respecvely For he choce of l, we fnd ha, n erms of MAE, α RMA performs bes n many cases To descrbe he nuon behnd hs resul, we consder he AR 2 The reason why we consder he esmaor usng z as nsrumens s ha, n erms of MAE, he esmaor may perform beer han he esmaor snce he esmaor s more effcen han he esmaor under large N and fxed T asympocs Also, he reason why we choose l =s ha he esmaor wh l = performs bes as wll be shown 8

9 model and l =0, 2 In hs case, he nsrumens are z 0 = c y, y, + + y,0, z 2 = c y, y, y,0 2 For he case of l = 2, we fnd ha y, 2 s no used and hs causes an effcency loss The same resul apples o he case l 2 For he case of l = 0, alhough z 0 uses all nformaon, y, nduces an addonal correlaon and make he second erm larger alhough s order s O/ We frs consder he AR case We fnd from Table ha, n erms of he bas, he esmaors, α LEV and α RMAl, have lle bas for all cases, whle he esmaors have non-neglgble bas when α =09 Especally, α LEV has large bas for all cases However, wh regard o he IQR, α LEV has he smalles dsperson and α LEV has he larges dsperson Also, we fnd ha he dfferences n he IQR of α LEV, αrma and αrmal become que small when T s as large as 50 Ths resul s conssen wh Proposon where α LEV and α RMAl are shown o have he same asympoc varance when N and T are large Also, asympoc varances n 0 and sugges ha for gven N and T, he dsperson of α RMA and α RMAl becomes smaller as α grows However, he smulaon resuls do no show such a endency when T 50 Ths may be due o he weak denfcaon problem as α approaches uny 3 I s of neres how much effcency of he proposed esmaor s los compared o he nfeasble opmal esmaor Lookng a he able, we fnd ha he nfeasble opmal esmaor s slghly less effcen han α LEV, whch s a feasble opmal esmaor Alhough he proposed esmaors are less effcen han he nfeasble opmal esmaors, he dfference becomes neglgble as T ges larger For he medan absolue error, we fnd ha α RMA beween α RMA has he smalles MAE n many cases However, he dfference n he MAE and αrmal s farly small Nex, we dscuss he resuls for he AR2 case The esmaors are vrually medan unbased and α LEV has he larges bas In erms of he IQR, α LEV has he smalles dsperson n all cases Also, we fnd ha he dfference n he IQR beween α LEV, αrma, and αrmal becomes small when T s large For α OP T,we fnd ha performs well for he case α,α 2 =06, 0 However, for he case α,α 2 =06, 03, α OP T does no work well and he reason s unclear Therefore, s dffcul o nvesgae how much effcency s los n he proposed esmaor In erms of he MAE, alhough α RMA performs bes n many cases, he dfference 3 Noe ha smlar resuls are also repored n Alvarez and Arellano

10 beween α RMA and α RMAl s que small The smulaon resuls sugges ha, n erms of he bas and MAE, he and esmaors usng he proposed nsrumens perform beer han he commonly used esmaor, α, even when T s as large as 0 4 Concluson In hs paper, we showed ha he nfeasble opmal esmaor and he esmaor usng nsrumens devaed from pas means are asympocally equvalen n he sense ha boh esmaors have he same asympoc dsrbuon when boh N and T are large We furher showed ha f we assume normaly on he errors, he proposed esmaor s asympocally effcen when boh N and T are large Smulaon resuls demonsraed ha n erms of he bas and medan absolue error, he new esmaor ouperforms he and esmaors usng nsrumens n levels, whch are commonly used n he leraure Lasly, we noe some possble exensons Alhough we consdered an ARp model wh d errors, s of grea neres o nvesgae wheher he resuls obaned n hs paper apply o more general models and errors, say, models ha nclude addonal regressors besdes he lagged dependen varables Arellano, 2003b and/or heeroskedasc errors Alvarez and Arellano, 2004 Also, may be neresng o apply Oku s 2006 mehod, e, a procedure o selec he number of momen condons so as o mnmze he MSE of he esmaors, o mprove he / esmaors usng nsrumens devaed from pas means Bu hese asks are lef for fuure research Appendx Lemma A Le Assumpons, 2, and 3 hold Then, h and z l can be wren as a h = Ex y = c I p ΠI p Π T I p Π w, + g = c I O w, + O p b z l = c w, + g 3 = c w, + O p 4 0

11 where g = ι p μ +φκ p R κ p φ α ι p v, + + v, +κ pr r +φ { α ι p 2 + κ pr κ p } { } Φ v, l + + Φ l v d + Φ l w 0 + q g = μ G p ι p I p G p,5 l, Φ j = Π 0 + Π + + Π j =I p Π I p Π j y, + y, y, p+ y, y, p+ q = y, p+ p G p = dag l + p, p 2 l + p 2,, and κ p, R, and r are defned laer Proof of Lemma A a Frs, noe ha under he assumpon ha Eμ y concdes wh he lnear projecon, we have Ex y = c I p ΠI p Π T I p Π φ ι +p x ι V y p +φι +p V ι +p where φ = σμ 2/σ2 v, V = σv 2E y μ ι +p y μ ι +p, μ = η / α ι p, and σμ 2 = varμ In he followng, we derve formulas of ι +p V y and ι +p V ι +p Followng Whle 95 and Wse 955, le us defne he + p + p marx U as follows O+p 2 I +p 2 U = O O +p 2 Then, we have U 2 O+p 3 2 I +p 3 = U p = O 2 2 O p O p p O 2 +p 3 I O p Usng hese expressons, y, U 3 O+p 4 3 I +p 4 = O 3 3 O 3 +p 4, U p O p I = can be wren as y = α Uy + α 2 U 2 y + + α p U p y O p p O p,

12 +η ι O p + v O p + O r where v =v,,, v,, and y 0 α y, α 2 y, 2 α p y, p+ r = y, p+3 α y, p+2 α 2 y, p+ y, p+2 α y, p+ y, p+ Snce y s saonary and s condonal mean gven η s μ = η / α ι p, I +p Δ ỹ ι = η μ I +p Δι +p + + = O p v O p = ṽ + r + O r κ p μ v O p Op where ỹ = y μ ι +p,δ= α U + α 2 U α p U p, and ι α ι p α α 2 α p 2 α p α I +p Δ ι +p = α 2 α p 2 ι α ι p = κ p α Then, follows ha V = I +p Δ I O O R I +p Δ r where R = σ 2 v E r κ p μ r κ p μ Therefore, we have ι +p V ι +p = α ι p 2 + κ pr κ p, ι +p V y = α ι p η + v, + + v, +κ p R r b Snce x, l =y, l,, y, p l, we have z l y, z 2 l y, 2 z l = = c z p l y, p y, l + +y, p+ l+p y, l 2 + +y, p+ l+p 2 y, l p + +y, p+ l 2

13 = c y, y, 2 y, p I G p = c { x I G p x, l + + x + q l Snce x = μ ι p + w,, afer some algebra, we ge y, l + +y 0 +y, + +y, p+ l y, l 2 + +y, +y, 2 + +y, p+ l y, l p + +y, p+ l } x, l + + x = lμ ι p +Φ v, l + + Φ l v d + Φ l w 0 Thus, we ge 3 To prove 4, we have o show ha g s O p / However, snce I p G p = O, v, l,, v are ndependen random varables, and p s fxed, he second erm n 5 s O p / For he frs erm, snce p s fxed, Eμ G p ι p 2 = σμ 2 p j= p j/ l + p j2 = O/, he resul follows Lemma B Le Assumpons, 2, and 3 hold Then, Eg w, and E g w, are O/ Proof of Lemma B Frs, noe ha Eμ w, =O p Nex, snce p s fxed, we have E v, + + v,p w, = σv 2 d Ip Π I p Π p = O, E κ p R r w, = O, 2 E Φ v, l + + Φ l v d w, = σ 2 v Φ j d d Π j j= = O E Φ l w,0 + q w, = O The second resul holds snce all he elemens are of dmenson p orp p Then, he resul follows from he fac ha he denomnaors of g and g are O Nex, we derve he asympoc properes of he esmaors Noe ha esmaors α OP T and α RMAl can be wren as α α = b T where  denoes ÂOP, ÂRMAl, and b OP T RMAl denoes b, and b The asympoc behavor of  and b are gven n he followng lemma 3

14 Lemma C Le Assumpons, 2, and 3 hold nfny, Then, as boh N and T end o a b  OP T,  RMAl bop T, brmal p E w, w,, p 0 If we furher assume ha Assumpon 4 holds, hen as boh N and T end o nfny, c T bop, b RMAl d N 0,σ 2 ve w, w, Proof of Lemma C To derve he resuls, we use he followng decomposon: x = Ψ w, c ṽ T, Ψ = c I p ΠΦ T, ṽ T = Φ T v + Φ 2 v,t Φ v,t d a: Frs, we consder ÂOP T Usng Lemma A, B, and he above decomposon, we have E ÂOP T = T = T T E h x = T = I p O E w, w, + O E w, w, The las convergence comes from T T = O/ = Olog T/T 0 ÂOP T var s shown o end o zero as follows: {ÂOP } T T { } var vec = 2 var vec h x = O 0 N For α RMAl, we have E ÂRMAl = T T = = { } Ew, w, I p + O + O Ew, w, ÂRMAl var s shown o end o zero n a smlar way o ÂOP T b,c: Frs, we consder b OP T bop T = N T = = h v 4

15 = = = = N T = = N T = = N T = = N T = = c I p O w, + g v c w, v + o p + w, v w, v,+ + + v T w, v + o p + o p Then, usng he cenral lm heorem of Phllps and Moon 999, we have 4 T d bop N 0,σvE 2 w, w, s obaned n a smlar way RMAl b, The resul for b RMAl From c, s sraghforward o show ha b OP T p 0 Proof of Proposon Usng Lemma C, he resuls are easly obaned References Ahn, SC and P Schmd 995: Effcen Esmaon of Models for Dynamc Panel Daa, Journal of Economercs, 68, Ahn, SC and P Schmd 997: Effcen Esmaon of Dynamc Panel Daa Models: Alernave Assumpons and Smplfed Esmaon, Journal of Economercs, 76, Alvarez, J and M Arellano 2003: The Tme Seres and Cross-Secon Asympocs of Dynamc Panel Daa Esmaors, Economerca, 7, Alvarez, J and M Arellano 2004: Robus Lkelhood Esmaon of Dynamc Panel Daa Models, mmeo 4 See also Hahn and Kuersener 2002 and Lee

16 5 Anderson, TW and C Hsao 98: Esmaon of Dynamc Models wh Error Componens, Journal of he Amercan Sascal Assocaon, 76, Anderson, TW and C Hsao 982: Formulaon and Esmaon of Dynamc Models Usng Panel Daa, Journal of Economercs, 8, Arellano, M 2003a: Panel Daa Economercs, Oxford Unversy Press, New York 8 Arellano, M 2003b: Modellng Opmal Insrumenal Varables for Dynamc Panel Daa Models, Workng Paper No 030, CEMFI, Madrd 9 Arellano, M and SR Bond 99: Some Tess of Specfcaon for Panel Daa: Mone Carlo Evdence and an Applcaon o Employmen Equaons, Revew of Economc Sudes, 58, Arellano, M and O Bover 995: Anoher Look a he Insrumenal Varable Esmaon of Error-Componen Models, Journal of Economercs, 68, Bun, MJG and JF Kve 2006: The Effecs of Dynamc Feedbacks on LS and MM Esmaor Accuracy n Panel Daa Models, Journal of Economercs, 32, Bun, MJG and F Wndmejer 2007: The Weak Insrumen Problem of he Sysem Esmaor n Dynamc Panel Daa Models, Unversy of Brsol, Dscusson Paper No 07/595 3 Blundell, R and S Bond 998: Inal Condons and Momen Resrcons n Dynamc Panel Daa Models, Journal of Economercs, 87, Hahn, J and G Kuersener 2002: Asympocally Unbased Inference for a Dynamc Panel Model wh Fxed Effecs When Boh n and T are Large, Economerca, 70, Hahn, J, J Hausman, and G Kuersener 2007: Long Dfference Insrumenal Varables Esmaon for Dynamc Panel Models wh Fxed Effecs, Journal of Economercs, 27, Hayakawa, K 2007: Small Sample Bas Properes of he Sysem Esmaor n Dynamc Panel Daa Models, Economcs Leers, 95, Hayakawa, K 2008: On he Effec of Nonsaonary Inal Condons n Dynamc Panel Daa Models, mmeo 6

17 8 Holz-Eakn, D, W Newey, and H Rosen: 988 Esmang Vecor Auoregressons wh Panel Daa, Economerca, 56, Lee, Y 2005: A General Approach o Bas Correcon n Dynamc Panels under Tme Seres Msspecfcaon, mmeo 20 Oku, R 2006 The Opmal Choce of Momens n Dynamc Panel Daa Models, mmeo 2 Phllps, PCB and H R Moon: 999 Lnear Regresson Lm Theory for Nonsaonary Panel Daa, Economerca, 67, So, BS and DW Shn 999 Recursve Mean Adjusmen n Tme Seres Inferences, Sascs and Probably Leers, 43, Whle, P 95: Hypohess Tesng n Tme-Seres Analyss, Almqvs and Wksell, Upsala 24 Wse, J 955: The Auocorrelaon Funcon and he Specral Densy Funcon, Bomerka, 42,

18 Table : Smulaon resuls for he AR model α T N bα LEV bαrma bα LEV Medan bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA α T N bα LEV bαrma bα LEV IQR bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA α T N bα LEV bαrma bα LEV MAE bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA

19 Table 2: Smulaon resuls for an AR2 model T N α α 2 bα LEV bαrma Medan bα LEV bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA T N α α 2 bα LEV bαrma bα LEV IQR bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA T N α α 2 bα LEV bαrma bα LEV MAE bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA

20 T N α α 2 bα LEV bαrma Table 2 Con bα LEV Medan bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA T N α α 2 bα LEV bαrma bα LEV IQR bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA T N α α 2 bα LEV bαrma bα LEV MAE bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA