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 afer @ 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
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
2 Seup and Resul 2 The model and assumpons Le us consder he followng panel ARp model: y = α y, + α 2 y, 2 + + 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 2005 3
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 2002 5 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
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 2006 5
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 999 8 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
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 2007 7
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
model and l =0, 2 In hs case, he nsrumens are z 0 = c y, y, + + y,0, z 2 = c y, y, 3 + + 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 2003 9
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
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, 2 + + y, p+ y, 2 + + 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,
+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 2 + + 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
= 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
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 2 + + 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
= = = = 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, 5-27 2 Ahn, SC and P Schmd 997: Effcen Esmaon of Dynamc Panel Daa Models: Alernave Assumpons and Smplfed Esmaon, Journal of Economercs, 76, 309-32 3 Alvarez, J and M Arellano 2003: The Tme Seres and Cross-Secon Asympocs of Dynamc Panel Daa Esmaors, Economerca, 7, 2-59 4 Alvarez, J and M Arellano 2004: Robus Lkelhood Esmaon of Dynamc Panel Daa Models, mmeo 4 See also Hahn and Kuersener 2002 and Lee 2005 5
5 Anderson, TW and C Hsao 98: Esmaon of Dynamc Models wh Error Componens, Journal of he Amercan Sascal Assocaon, 76, 598-606 6 Anderson, TW and C Hsao 982: Formulaon and Esmaon of Dynamc Models Usng Panel Daa, Journal of Economercs, 8, 47-82 7 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, 277-297 0 Arellano, M and O Bover 995: Anoher Look a he Insrumenal Varable Esmaon of Error-Componen Models, Journal of Economercs, 68, 29-45 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, 409-444 2 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, 5-43 4 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, 639-657 5 Hahn, J, J Hausman, and G Kuersener 2007: Long Dfference Insrumenal Varables Esmaon for Dynamc Panel Models wh Fxed Effecs, Journal of Economercs, 27, 574-67 6 Hayakawa, K 2007: Small Sample Bas Properes of he Sysem Esmaor n Dynamc Panel Daa Models, Economcs Leers, 95, 32-38 7 Hayakawa, K 2008: On he Effec of Nonsaonary Inal Condons n Dynamc Panel Daa Models, mmeo 6
8 Holz-Eakn, D, W Newey, and H Rosen: 988 Esmang Vecor Auoregressons wh Panel Daa, Economerca, 56, 37-395 9 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, 057-22 So, BS and DW Shn 999 Recursve Mean Adjusmen n Tme Seres Inferences, Sascs and Probably Leers, 43, 65 73 23 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, 5-59 7
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 RMA4 05 0 00 0467 0493 0500 0499 0499 0498 0497 0497 05 0 500 0493 0499 050 0500 0500 0500 0499 0500 05 5 00 0472 0495 0499 0498 0498 0498 0498 0499 05 5 300 0490 0499 050 0500 0500 0499 0499 0500 05 20 00 0476 0498 0500 0500 0500 0499 0499 0499 05 20 200 0488 0499 0500 0500 0500 0500 0500 0500 05 50 00 0482 0499 0500 0500 0500 0500 0500 0500 05 00 00 0484 0500 0500 0500 0500 0500 0500 0500 09 0 00 069 0805 0906 089 0894 0889 0884 0882 09 0 500 0827 0878 0903 0897 0897 0899 0898 0898 09 5 00 0767 0862 0902 0896 0896 0897 0896 0897 09 5 300 0833 0885 0902 0899 0899 0899 0900 0900 09 20 00 0802 088 0900 0899 0899 0898 0898 0898 09 20 200 0837 0892 0902 0900 0900 0900 0900 0899 09 50 00 0860 0897 0899 0900 0900 0899 0899 0899 09 00 00 0875 0899 0900 0900 0900 0900 0900 0900 α T N bα LEV bαrma bα LEV IQR bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 05 0 00 0075 0084 07 0088 0086 0092 003 08 05 0 500 0034 0038 0050 0039 0039 004 0046 0052 05 5 00 0050 0055 0082 0057 0056 0057 0062 0067 05 5 300 0030 0032 0048 0033 0032 0034 0035 0038 05 20 00 0040 0043 0069 0044 0044 0044 0046 0047 05 20 200 0028 0030 0048 0030 003 003 0032 0033 05 50 00 0020 002 0038 002 002 002 002 002 05 00 00 003 004 0025 004 004 004 004 004 09 0 00 056 0222 0407 029 0278 0289 034 0367 09 0 500 0086 04 074 027 025 030 039 059 09 5 00 0085 07 0237 043 039 037 042 050 09 5 300 0058 0073 036 0082 0079 0078 008 0086 09 20 00 0059 0078 076 0088 0087 0086 0088 0090 09 20 200 0045 0057 020 0064 0062 0062 0062 0065 09 50 00 008 002 0066 0023 0023 0023 0023 0023 09 00 00 0008 000 0037 00 00 00 00 00 α T N bα LEV bαrma bα LEV MAE bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 05 0 00 0043 0042 0058 0044 0043 0046 005 0059 05 0 500 008 009 0025 009 0020 0020 0023 0026 05 5 00 0032 0028 0040 0028 0028 0028 003 0033 05 5 300 006 006 0024 006 006 007 007 009 05 20 00 0027 0022 0034 0022 0022 0022 0023 0024 05 20 200 006 005 0024 005 005 005 006 007 05 50 00 008 000 009 000 000 00 00 00 05 00 00 006 0007 002 0007 0007 0007 0007 0007 09 0 00 0209 03 0203 046 039 046 058 085 09 0 500 0075 0059 0087 0064 0063 0065 0070 0080 09 5 00 033 0064 08 0072 0070 0069 007 0075 09 5 300 0067 0037 0067 004 0039 0039 0040 0043 09 20 00 0098 0040 0087 0044 0043 0043 0044 0045 09 20 200 0063 0029 0060 0032 003 003 003 0033 09 50 00 0040 00 0032 002 00 002 002 00 09 00 00 0025 0005 008 0005 0005 0005 0005 0005 8
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 RMA4 0 00 06 0565 0586 0599 0598 0597 0598 0597 0598 0 00-0 -07-008 -00-002 -002-003 -004-002 0 500 06 0593 0598 0600 0600 0600 0600 0600 0599 0 500-0 -004-002 -00-00 -00-00 -00-00 5 00 06 0577 0595 060 0600 0600 0600 0600 0599 5 00-0 -04-003 -00-00 -00-00 -00-00 5 300 06 059 0598 0600 0600 0600 0600 0600 0599 5 300-0 -005-00 -000-000 -000-000 -000-000 20 00 06 0590 0599 060 0600 0600 0600 0600 0600 20 00-0 -006-00 -000-000 -000-000 -000-000 20 200 06 0590 0599 060 0600 0600 0600 0600 0600 20 200-0 -006-00 -000-000 -000-000 -000-000 50 00 06 0588 0600 060 060 060 0600 060 060 50 00-0 -00-00 -000-000 -000-000 -000-000 00 00 06 0590 0600 0600 0600 0600 0600 0600 0600 00 00-0 -009-000 -00-000 -000-000 -000-000 T N α α 2 bα LEV bαrma bα LEV IQR bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 0 00 06 0067 0077 00 0079 0079 0084 0094 0 0 00-0 005 0056 0064 0058 0059 0063 0068 0076 0 500 06 0032 0035 0046 0036 0036 0038 0042 0050 0 500-0 0024 0026 0029 0027 0026 0029 003 0035 5 00 06 0046 0050 0070 0050 0050 005 0054 0059 5 00-0 0038 004 0050 0043 0043 0045 0047 005 5 300 06 0028 0029 004 0030 0030 003 0032 0034 5 300-0 0023 0024 0029 0024 0025 0026 0027 0029 20 00 06 0027 0029 004 0029 0029 0029 0030 003 20 00-0 0024 0025 003 0026 0026 0026 0027 0028 20 200 06 0027 0029 004 0029 0029 0029 0030 003 20 200-0 0024 0025 003 0026 0026 0026 0027 0028 50 00 06 0020 002 003 002 002 002 0022 0022 50 00-0 009 0020 0026 0020 0020 0020 002 002 00 00 06 004 004 0020 004 004 004 004 004 00 00-0 004 004 009 004 004 004 004 004 T N α α 2 bα LEV bαrma bα LEV MAE bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 0 00 06 0043 0040 0050 0040 0040 0042 0047 0055 0 00-0 0028 0029 0032 0029 0029 0032 0034 0038 0 500 06 006 007 0023 008 008 009 002 0025 0 500-0 002 003 004 003 003 004 006 008 5 00 06 0029 0025 0035 0025 0025 0026 0027 0030 5 00-0 002 002 0025 002 002 0022 0023 0025 5 300 06 005 005 002 005 005 005 006 007 5 300-0 002 002 004 002 002 003 004 005 20 00 06 005 004 002 005 005 005 005 005 20 00-0 003 003 005 003 003 003 004 004 20 200 06 005 004 002 005 005 005 005 005 20 200-0 003 003 005 003 003 003 004 004 50 00 06 004 00 005 00 00 00 00 00 50 00-0 002 000 003 000 000 000 000 000 00 00 06 00 0007 000 0007 0007 0007 0007 0007 00 00-0 000 0007 000 0007 0007 0007 0007 0007 9
T N α α 2 bα LEV bαrma Table 2 Con bα LEV Medan bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 0 00 06 0357 0430 0600 0583 0583 0586 0580 0577 0 00 03 096 0225 0298 0285 0287 0286 0284 0285 0 500 06 0482 054 0597 0595 0595 0593 0596 0593 0 500 03 0250 0274 0299 0297 0297 0296 0297 0296 5 00 06 0445 0524 0608 0596 0597 0597 0597 0597 5 00 03 0227 0263 0303 0295 0295 0296 0296 0295 5 300 06 0508 0566 0602 0600 0600 0600 0600 0600 5 300 03 0258 0284 0300 0299 0299 0299 0299 0299 20 00 06 052 0575 0605 0600 0600 0600 0600 0600 20 00 03 026 0288 030 0299 0299 0298 0299 0299 20 200 06 052 0575 0605 0600 0600 0600 0600 0600 20 200 03 026 0288 030 0299 0299 0298 0299 0299 50 00 06 056 0596 060 0600 0600 060 0600 060 50 00 03 0275 0296 0300 0299 0299 0299 0299 0299 00 00 06 0579 0599 0599 0600 0600 0600 0600 0600 00 00 03 0284 0299 0299 0300 0300 0300 0300 0300 T N α α 2 bα LEV bαrma bα LEV IQR bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 0 00 06 06 0225 0474 0365 0353 0373 0390 0465 0 00 03 0073 0098 0207 060 057 065 070 092 0 500 06 00 029 0222 06 057 06 078 0200 0 500 03 0044 0056 0092 0069 0067 0070 0074 0083 5 00 06 0093 026 0292 066 06 062 069 083 5 00 03 0050 0062 040 0083 0082 0082 0085 0090 5 300 06 0066 0083 066 0098 0095 0097 0099 007 5 300 03 0032 004 0076 0047 0047 0048 0049 005 20 00 06 0052 0063 04 0074 0072 0072 0075 0075 20 00 03 0030 0036 007 0040 0039 0040 0040 004 20 200 06 0052 0063 04 0074 0072 0072 0075 0075 20 200 03 0030 0036 007 0040 0039 0040 0040 004 50 00 06 0024 0026 0066 0028 0028 0028 0028 0028 50 00 03 0020 002 0044 0023 0023 0023 0023 0023 00 00 06 004 005 0034 005 005 005 005 005 00 00 03 003 004 0027 004 004 004 004 005 T N α α 2 bα LEV bαrma bα LEV MAE bα RMA0 bα RMA bα RMA2 bα RMA3 bα RMA4 0 00 06 0243 076 0236 084 079 087 096 0233 0 00 03 004 0079 004 008 0079 0083 0087 0099 0 500 06 08 0078 00 008 0078 008 0089 00 0 500 03 0050 0034 0046 0035 0034 0035 0037 0042 5 00 06 055 0084 046 0083 0080 008 0084 0092 5 00 03 0073 0042 0070 004 004 004 0043 0045 5 300 06 0092 0047 0083 0049 0047 0048 0049 0053 5 300 03 0042 0023 0038 0024 0023 0024 0024 0025 20 00 06 0079 0036 0070 0037 0036 0036 0037 0037 20 00 03 0039 009 0035 0020 0020 0020 0020 002 20 200 06 0079 0036 0070 0037 0036 0036 0037 0037 20 200 03 0039 009 0035 0020 0020 0020 0020 002 50 00 06 0039 004 0033 004 004 004 004 004 50 00 03 0025 00 0022 00 00 00 00 002 00 00 06 002 0007 006 0007 0007 0007 0007 0007 00 00 03 006 0007 003 0007 0007 0007 0007 0007 20