Chap.5 Unit Roots and Cointegration in panels

Transcription

1 Chap.5 U Roos ad Coegrao paels 5. Iroduco Wh he growg use of cross-coury daa over me o sudy purchasg power pary, growh covergece ad eraoal R&D spllovers, he focus of pael daa ecoomercs has shfed owards sudyg he asympocs of macro paels wh large (say over ) ad large raher ha he usual asympocs of mcro paels wh large ad small (less ha ). he fac ha s allowed o crease o macro pael daa, geeraed wo srads of deas. he frs rejeced he homogeey of he regresso parameers mplc he use of a pooled regresso model favor of heerogeeous. Aoher srad of leraure appled me seres procedures o paels, worryg abou osaoary, spurous regressos ad coegrao. However, he process of dervg he lmg dsrbuo pael has may dffereces wh ha me seres, he way whch, he umber of cross-seco us, ad, he legh of he me seres, ed o fy s crucal for deermg asympoc properes of esmaors ad ess proposed for osaoary paels, see Phllps ad Moo (999). Several approaches are possble, cludg () sequeal lms oe dex, say, s fxed ad s allowed o crease o fy, gvg a ermedae lm, he by leg ed o fy subsequely. () dagoal lms whch allows he wo dexes, ad, o pass a fy alog a specfc dagoal pah he wo-dmesoal array. () jo lms, whch allows boh ad o pass o fy smulaeously whou placg specfc dagoal pah resrcos o he dvergece. esg for u roos me seres sudes s ow commo pracce amog appled researches ad has become a egral par of ecoomerc courses. However, esg for u roos paels s rese. he developme of pael u roos could be descrbed by he followg graph, Breug() Homogeeous Hadr() LLC() Frs geerao Maddala ad Wu(999) Heerogeeous Cho() Pael u roo ess IPS(3) O'Coell(998) Breug ad Das(5) Secod geerao Moo ad Perro(4) Ba ad g(4) Pesara (3) Here, we brefly descrbe hem respecvely. Cosder a followg AR() process for pael: y ρ y, + Xδ + u (5.)

2 ,,, cross-seco us or seres, ha are observed over perods,,,. he X represe he exogeous varables he model, cludg ay fxed effecs or dvdual reds, ρ are he auoregressve coeffces, ad he errors u are assumed o be muually depede dosycrac dsurbace. If ρ <, y s sad o be weakly (red-) saoary. O he oher had, f ρ he y coas a u roo. 5. Pael u roo ess assumg cross-secoal depedece (Frs geerao) 5.. es wh commo u roo process (homogeeous) Commo roo dcaes ha he ess are esmaed assumg a commo AR srucure for all of he seres. Lev, L, ad Chu (LLC), Breug, ad Hadr ess all assume ha here s a commo u roo process so ha ρ s decal across cross-secos. he frs ess employ a ull hypohess of a u roo whle he Hadr es uses a ull of o u roo. ) Lev, L, ad Chu () LLC argued ha dvdual u roo ess have lmed power agas alerave hypoheszes wh hghly persse devaos from equlbrum. hs s parcularly severe small samples. LLC sugges a more powerful pael u roo es ha performg dvdual u roo ess for each cross-seco. he ull hypohess s ha each dvdual me seres coas a u roo agas he alerave ha each me seres s saoary. he maaed hypohess s ha p Δ y ρy + θ Δ y + α d + ε m,,3 (5.), L, L m m L wh d m dcag he vecor of deermsc varables ad α m he correspodg vecor of coeffces for model,,3 ad d { } m. I parcular, d { empy se}, d { } 3,. Sce he log order p s ukow, LLC sugges a hree-sep procedure o mpleme her es. Sep. Perform separae augmeed Dckey-Fuller(ADF) regressos for each cross-seco: p Δ y ρ y + θ Δ y + α d + ε m,,3 (5.3), L, L m m L

3 he lag order p s permed o vary across dvduals. For a gve, choose a maxmum lag order p max ad he use he -sasc of ˆL θ o deerme f a smaller lag order s preferred. Oce p s deermed wo auxlary regressos are ru o ge orhogoalzed resduals: Ru Δ o y ( L p ) y Ru, Δ, L,, ad d m o ge resduals e ˆ y o y ( L p ) Δ, L,, ad d m o ge resduals vˆ, o corol for heerogeey across dvduals, sadardze hese resduals by he regresso sadard error from equao (5.3). eˆ e, ˆ σ ε ˆ σ ε s he regresso sadard error (5.3). v, vˆ, ˆ σ ε Sep. Esmae he rao of log-ru o shor-ru sadard devaos. Uder he ull hypohess of a u roo, he log-ru varace of (5.) ca be esmaed by K ˆ σ y y y y KL, L ω Δ + Δ Δ L + L K s a rucao lag ha ca be daa-depede. K mus be obaed a maer ha esures he cossecy of, ( ) ˆ σ y. For a Barle kerel, ( ) ω L K +. KL For each cross-seco, he rao of he log-ru sadard devao o he ovao sadard devao s esmaed by sˆ ˆ σ ˆ σ.he average sadard devao s y ε esmaed by Sˆ sˆ. hs mpora sasc wll be used o adjus he mea of he -sasc laer sep 3. Sep 3. Compue he pael es sascs. Pool all cross secoal ad me seres observaos o esmae e δ v + ε,, based o a oal of observaos, p s he average umber of observaos per dvdual he pael, ad p p s he average lag order for he 3

4 dvdual ADF regressos. he coveoal -sasc for esg δ s gve by ˆ δ δ SD ˆ δ ( ) ˆ δ + p v + p e, v,, SD( ˆ) ˆ v, + p δ σ ε, ˆ ( e v ) σ ˆ ε δ + p,. Compue he adjused -sasc δ ˆ δ Sσ εsd ˆ σ m ( ˆ) δ μ m (5.4) μ m ad σ m are he mea ad sadard devao adjusmes provded by able of LLC. LLC shows ha δ s asympocally dsrbued as (,). LLC sugges usg her pael u roo es for paels of moderae sze wh bewee ad 5 ad bewee 5 ad 5. Lmaos: he es crucally depeds upo he depedece assumpo across cross-secos ad s o applcable f cross-secoal correlao s prese. Secod, he assumpo ha all cross-secos have or do o have a u roo s resrcve. ) Breug () he LLC ad IPS ess requre such ha,.e., should be small eough relave o. hs meas ha boh ess may o keep omal sze well whe eher s small or s relave o. Breug() sudes he local power of LLC ad IPS es sascs agas a sequece of local aleraves. Breug fds ha he LLC ad IPS ess suffer from a dramac loss of power f dvdual-specfc reds are cluded. hs s due o he bas correco ha also removes he mea uder he sequece of local alerave. Breug suggess a es sasc ha does o employ a bas adjusme whose power s subsaally hgher ha ha of LLC or he IPS ess usg Moe Carlo expermes. he smulao resuls dcae ha he power of LLC ad IPS ess s very sesve o he specfcao of he deermsc erms. Breug () es sasc whou bas adjusme s obaed as follows. Sep s he same as LLC bu oly Δ y, L s used obag he resduals e ˆ ad vˆ,. he resduals are he adjused (as LLC) o correc for dvdual-specfc varaces. Sep, he resduals e are rasformed usg he forward orhogoalzao rasformao employed by Arellao ad Bover (995): 4

5 e + + e e e +, +, Also, v, v, v,, wh ercep ad red v, v, v,, wh ercep, o red v,, wh o ercep or red he las sep s o ru he pooled regresso ad oba he -sasc for : dsrbuo. e ρv + ε, H ρ whch has he lm a sadard (,) 3) Hadr () Hadr() derves a resdual-based Lagrage mulpler (LM) es he ull hypohess s ha here s o u roo ay of he seres he pael agas he alerave of a u roo he pael. I parcular, Hadr() cosders he followg wo models: y r + ε ad y r + β+ ε,, ;,, (5.5) r r, u + s a radom walk. ε II (, σ ε ) ad u II (, σ u ) muually depede ormals ha IID across ad over. Usg back subsuo, model becomes y r + β+ u + ε r + β+ v (5.6) s s are v u + ε. he saoary hypohess s smply s s H σ, whch case : u v ε. he LM sasc s gve by LM S ˆ σ ε (5.7) S ˆ εs s are paral sum of OLS resduals ˆs ε from (5.6) ad σ ˆε s a cosse esmae of σ ε uder he ull hypohess H. A possble caddae s 5

6 ˆ ˆ ε ε. σ Hadr () suggesed a alerave LM es ha allows for Heeroskedascy across, say σ ε. hs s fac LM S ˆ σ ε (5.8) he es sasc s gve by Z ( LM ξ ) ς ad s asympocally dsrbued as ad (,), ξ ad 6 ς, oherwse. 63 ς f he model oly clude a cosa, ad 45 ξ es wh dvdual u roo process (heerogeeous) ) Im, Pesara ad Sh es (IPS 3) he LLC es s resrcve he sese ha requres ρ o be homogeeous across. As Maddala(999) pos ou, he ull may be fe for esg covergece growh amog coures, bu he alerave resrcs every coury o coverge a he same rae. Im e al.(3) (IPS) allow for a heerogeeous coeffce of y, ad propose a alerave esg procedure based o averagg dvdual u roo es sascs. IPS sugges a average of he ADF ess whe u s serally correlaed wh dffere seral correlao properes across cross-secoal us,.e., he model gve (5.3). he ull hypohess s ha each seres he pael coas a u roo,.e., H : ρ for all ad he alerave hypohess allows for some (bu o all) of he dvdual seres o have u roos,.e., ρ < for,,, H : ρ for +,, Formally, requres he fraco of he dvdual me seres ha are saoary o be ozero,.e., lm ( ) δ < δ. hs codo s ecessary for he cossecy of he pael u roo es. he IPS -bar sasc s defed as he average of he dvdual ADF sascs as ρ ρ s he dvdual -sasc for esg H : ρ for all I (5.3). I case he lag 6

7 order s always zero ( p for all ), IPS provde smulaed crcal values for for dffere umber of cross-seco, seres legh ad Dckey-Fuller regressos coag erceps oly or erceps ad lear reds. I he geeral case he lag order p may be ozero for some cross-secos, IPS show ha a properly sadardzed has a asympoc (,) dsrbuo. Sarg from he well-kow resul me seres ha for a fxed WZdWZ ρ W Z (5.9) as, W( r) dr deoes a Weer egral wh he argume r suppressed (5.9), IPS assume ha are IID ad have fe mea ad varace. he E ρ var ρ as by he Ldeberg-Levy ceral lm heorem. Hece (,) (5.) IPS E ρ var ρ (,) (5.) as followed by sequeally. he values of E ρ ad var ρ have bee compued by IPS va smulaos for dffere values of ad p s. ) Fsher-ADF ad Fsher-PP Le G be a u roo es sasc for he h group (5.) ad assume ha as he me seres observaos for he h group, G G G s a odegeerae radom varable. Le.e., p F( G ), () p be he asympoc p-value of a u roo es for cross-seco, F s he dsrbuo fuco of he radom varable Maddala ad Wu (999) ad Cho () proposed a Fsher-ype es: G. 7

8 χ ( ) (5.) P l p whch combes he p-values from u roo ess for each cross-seco o es for u roo pael daa. Cho() proposes wo oher es sascs besdes Fsher s verse ch-square es sasc P. he frs s he verse ormal es Z Φ ( p ) (5.3) Φ s he sadard ormal cumulave dsrbuo fuco. Sce p, Φ ( ) s a (,) p radom varable ad as he secod s he log es p l p for all, ml 5+ 4 for all, Z (,). p L l p (5.4) has he logsc dsrbuo wh mea ad varace π 3. As ( + ) ( 5+ ) 35 4 m π Whe s large, Cho () proposed a modfed P es, P p followed by ( l ) (, ) ( ) (5.5) m he dsrbuo of he Z sasc s vara o fe, ad Z (,). as p 3 p by ad he. Also, he dsrbuo of ml l (,) π he LIdeberg-Levy ceral lm heorem as ad he. herefore, Z ad ml ca be used whou modfcao for fe. 5. Pael u roo ess allowg for cross-secoal depedece (Secod geerao) 5.. Aoher approach o coemporaeous correlao esg Pesara (4) suggess a smple es of dsurbace cross-seco depedece (CD) ha s applcable o a varey of pael models cludg saoary ad u roo dyamc heerogeeous paels wh shor ad large. he proposed es s based o a average 8

9 of parwse correlao coeffces of OLS resduals from he dvdual regresso he pael raher ha her squares as he Breusch-Paga LM es: CD ˆ ρ (5.6) ( ) j j + ˆ ρ j ee j ( e ) ( e j ), wh e deog OLS resduals based o observaos for each,,. Moe Carlo expermes show ha he sadard Breusch-Paga LM es performs badly for > paels, as Pesara s CD es performs well eve for small ad large. 5.. Secod geerao pael u roo ess ) he SUR Mehod Accordg o a seemgly urelaed regresso sysem, O coell (998) suggess o esmae he sysem by usg a GLS esmaor (see also Flores, Joro, Preumo ad Szarfarz, 999). Le ˆ Σ ˆˆ εε deoe he sample covarace marx of he resdual vecor. he GLS- sasc s gve by gls ( ) Δy Σˆ y Σˆ y y (5.7) y s he vecor of demeaed varables. Harvey ad Baes (3) derve he lmg dsrbuo of ( ) gls for a fxed ad as, ad abulae s asympoc dsrbuo for varous values of. Breug ad Das (5) show ha f y y y s used o demea he varables ad f followed by, he he GLS -sasc possesses a sadard ormal lmg dsrbuo. he GLS approach cao be used f < as hs case he esmaed covarace marx ˆΣ s sgular. Furhermore, Moe Carlo smulaos sugges ha for reasoable sze properes of he GLS es, mus be subsaally larger ha (e.g. Breug ad Das, 5) Maddala ad Wu (999) ad Chag (4) have suggesed a boosrap procedure ha mproves he sze properes of he GLS es. A alerave approach based o pael correced sadard errors (PCSE) s cosdered by Josso (5) ad Breug ad Das (5). I he model wh weak depedece, f s followed by he robus sasc 9

10 ROLS y Δy y Σˆ y (5.8) Is asympocally sadard ormally dsrbued (Breug ad Das, 5). If s assumed ha he cross correlao s due o commo facors, he he larges egevalue of he error covarace marx, Σ, s O ( ) p ad he robus PCSE approach breaks dow. Specfcally, Breug ad Das (8) showed ha hs case ROLS s dsrbued as he ordary Dckey-Fuller es appled o he frs prcpal compoe. ) he mehods of Commo Facor Cross-seco depedece ca arse due o a varey of facors, such as omed observed commo facors, spaal spll over effecs, uobserved commo facors, or geeral resdual erdepedece ha could rema eve whe all he observed ad uobserved commo effecs are ake o accou. Dyamc facor models have bee used o capure cross-seco correlao. Moo ad Perro (4c) cosder he followg model: y α + y (5.9) y ρ y + ε, ε are uobservable error erms wh a facor srucure ad α are fxed effecs. ε s geeraed by M uobservable radom facors f ad dosycrac shocks e as follows: ε Λ f + e (5.) Λ are oradom facor loadg coeffce vecors ad he umber of facors M s ukow. Each ε coas he commo radom facor f, geerag he correlao amog he cross-secoal us of ε ad y. he exe of he correlao s deermed by he facor loadg coeffces Λ,.e., E( y y ) E( f f ) Λ Λ. Moo j j ad Perro rea he facors as usace parameers ad sugges poolg defacored daa o cosruc a u roo es. Le Q Λ be he marx projecg oo he space orhogoal o he facor loadgs. he defacored daa s YQ Λ ad he defacoreed resduals eq Λ o loger have cross-secoal depedece, Y s a marx whose -h colum

11 coas he observaos for cross-secoal u. Le σ be he varace of e, e, w be he log-ru varace of e ad λ e, be e, he oe-sded log ru varace of e. Also, σ e, w e ad λ e be her cross-secoal averages, ad φ be he cross-secoal average of w 4 e,. he pooled bas-correlae 4 e esmae of ρ s + ˆ ρ pool ( Λ ) r ( Y Q Y ) r Y Q Y Λ λ e (5.) Y s he marx of lagged daa. Moo ad Perro sugges wo sascs o es H : ρ for all,, M agas he alerave hypohess H A: ρ < for some. hese are ad a + ( ˆ ρ pool ) φ (5.) w 4 e 4 e we + ( ˆ b ρ pool ) r( Y QΛY ) φ (5.3) 4 e hese ess have a sadard (,) lmg dsrbuo ad ed o fy such ha. Moo ad Perro also show ha esmag he facors by prcpal compoes ad replacg w e ad φ 4 e by cosse esmaes leads o feasble sascs wh he same lmg dsrbuo. Ba ad g (4) cosder he followg dyamc facor model: y α +Λ f + y y ρ y + ε, (5.4) hey es separaely he saoary of he facors ad he dosycrac compoe. o do so, hey oba cosse esmaes of he facors regardless of wheher resduals are saoary or o. hey accomplsh hs by esmag facors o frs-dffereced daa ad cumulag hese esmaed facors. Ba ad g sugges poolg resuls from dvdual ADF ess o he esmaed defacored daa by combg p-values as Maddala ad Wu (999) ad Cho (): P c eˆ c l pe ˆ () d 4 (,) (5.5)

12 c p () s he p-value of he ADF es (whou ay deermsc compoe) o he eˆ esmaed dosycrac shock for cross-seco. Pesara (3) suggess a smpler way of geg rd of cross-secoal depedece ha esmag he facor loadg. Hs mehod s based o augmeg he usual ADF regresso wh he lagged cross-secoal mea ad s frs dfferece o capure he cross-secoal depedece ha arses hrough a sgle facor model. hs s called he cross-secoally augmeed Dckey-Fuller (CADF) es. hs smple CADF regresso s Δ y α + ρ y + d y + d Δ y + ε (5.6), y s he average a me of all observaos. he presece of he lagged cross-secoal average ad s frs dfferece accous for he cross-secoal depedece hrough a facor srucure. If here s seral correlao he error erm or he facor, he regresso mus be augmeed as usual he uvarae case, bu lagged frs-dffereces of boh y ad y mus be added, whch leads o p p α ρ, j+ j k, k ε j k (5.7) Δ y + y + d y + d Δ y + c Δ y + he degree of augmeao ca be chose by a formao crero or sequeal esg. Afer rug hs CADF regresso for each u he pael, Pesara averages he -sascs o he lagged value (called CADF ) o oba he CIPS sasc CIPS CADF (, ) (5.8) he jo asympoc lm of he CIPS sasc s osadard ad crcal values are provded for varous choces of ad. he -ess based o hs regresso should be devod of Λ f he lm ad herefore free of cross-secoal depedece. he lmg dsrbuo of hese ess s dffere from he Dckey-Fuller dsrbuo due o he presece of he cross-secoal average of he lagged level. Pesara uses a rucaed verso of he IPS es ha avods he problem of mome calculao. ha s (, ) (, ) (, ) (, ) (, ) (, ) (, ) K < < K K K K K (5.9) Pesara suggess ha K 6.9 ad K.6, he values of K ad K deped o he deermed red he CADF regresso, Model (5.7) cludes ercep oly.

13 5.3 Coegrao Aalyss Pael Daa 5.3. Resdual-Based Approachs o Pael Coegrao Lke he pael u roo ess, pael coegrao ess ca be movaed by he search for more powerful ess ha hose obaed by applyg dvdual me seres coegrao ess. he laer ess are kow o have low power, especally for shor ad shor spa of he daa.. Kao ess Kao (999) preseed wo ypes of coegrao ess pael daa, he DF ad ADF ypes ess. Cosder he pael regresso model y x β + z γ + e (5.3) y ad x are () I ad ocoegraed. For z { μ }, Kao(999) proposed DF ad ADF-ype u roo ess for e as a es for he ull of o coegrao. he DF-ype ess ca be calculaed from he fxed effecs resduals eˆ ρeˆ + v (5.3), e ˆ ˆ y x β ad y y y, x x x. I order o es he ull hypohess of o coegrao, he ull ca be wre as H : ρ. he OLS esmae of ρ ad he -sasc are gve as ad ˆ ρ ee ˆˆ, eˆ ρ ( ˆ ρ ) s e eˆ, ( 5. )... 3 s ( ˆ ˆ ˆ ) e e ρe,. Kao proposed he followg four DF ype ess: ( ) ˆ ρ + 3 DF ρ. DF ρ

14 ad * DF ρ DF * 3 ˆ σ v + ˆ σ ( ˆ ρ ) ρ 36 ˆ σ ˆ σ + 4 v 4 ov 6 ˆ σ v ˆ σ ov ˆ σ 3 ˆ σ σ ov v + ˆ ˆ v σ ov ov Σˆ Σˆ Σ ˆ ad σˆv yy yx xx Ωˆ Ωˆ Ω ˆ, ad ˆΣ s esmaor of log ru σˆov yy yx xx covarace of ( y, x ) ξ Δ Δ, ˆΩ s he esmaor of coemporaeous covarace of ξ ( Δy, Δ x ). Whle DF ρ ad DF are based o he srog exogeey of he regressors ad dsurbaces, * DF ρ ad * DF are for he coegrao wh edogeous relaoshp bewee regressors ad dsurbaces. For he ADF es, we ca ru he followg regresso: P, j, j j ( ) eˆ ρeˆ + δ Δ eˆ + v wh he ull hypohess of o coegrao, he ADF es sascs ca be cosruced as: ADF ADF + 6 ˆ σ v ˆ σ ov ˆ σ 3 ˆ σ σ ov v + ˆ ˆ v σov... ( 5.34) Where * DF ρ, ADF s he -sasc of ρ (5.33). he asympoc dsrbuos of DF ρ, DF, DF ad ADF coverge o a sadard ormal dsrbuo (,) * by sequeal lm heory.. Pedro ess Pedro (999, 4) also proposed several ess for he ull hypohess of coegrao a pael daa model ha allows for cosderable heerogeey. Pedro cosdered he followg ype of regresso: ( ) y α + δ + X β + e for a me seres pael of observables y ad X for members,, over me

15 perods,,, X s a m-dmesoal colum vecor for each member ad β s a m-dmesoal colum vecor for each member. he varables y ad X are assumed o be I(), for each member of he pael, ad uder he ull of o coegrao he resdual e wll also be I(). he parameers α ad δ allow for possbly of member specfc fxed effecs ad deermsc reds, respecvely. he slope coeffces β are also permed o vary by dvdual, so ha geeral he coegrag vecors may be heerogeeous across members of he pael. he DF-ype ess ad ADF-ype ess ca be calculaed from he fxed effecs resduals eˆ ρ eˆ + v, P eˆ ρeˆ + ϕ Δ eˆ + v..., j, j j he ull hypoheses for coegraed ess are: H : ρ ; H : ρ ρ < (,,, ) Ad H : ρ ; : H ρ < (,,, ) o sudy he dsrbuoal properes of above ess, Pedro descrbed he DGP erms of he paroed vecor Z ( Y, X ) geeraed as Z Z, ξ such ha he rue process Z s +, for (, ) Y X ξ ξ ξ. We he assume ha for each member he followg codo holds wh regard o he me seres dmeso. sasfes Assumpo 5.3. (Ivarace Prcple) he process (, ) Y X ξ ξ ξ [ ] r ξ B ( Ω ) covergece ad ( ), for each member as. Where sgfes weak B Ω s vecor Browa moo wh asympoc covarace such ha he m m lower dagoal block Ω > ad he ( ) Ω B Ω are ake o be defed o he same probably space for all. he above assumpo saes ha he sadard fucoal CL s assumed o hold dvdually for each member seres as grows large. he ( m ) ( m ) + + asympoc 3

16 covarace marx s gve by Ω lm E ξ ξ Ad ca be decomposed as ( )... 6 ( ) Ω Ω +Γ +Γ Where Ω ad Γ represe he coemporaeous ad dyamc covarace, respecvely of ξ for a gve. Assumpo 5.3. (Cross-Secoal Idepedece) he dvdual processes are assumed o be..d. cross-secoally, so ha ( ) E ξξ for all s,, j. More geerally, he asympoc log-ru varace marx for a pael of sze dagoal posve defe marx such ha dag ( Ω Ω ) js,,. s block Pedro s ess ca be classfed o wo caegores. he frs se (wh dmeso) s smlar o he ess dscussed above, ad volves averagg es sascs for coegrao he me seres across cross-seco. For he secod se (bewee dmeso), he averagg s doe peces so ha he lmg dsrbuos are based o lms of pecewse umeraor ad deomaor erms. he basc approach boh cases s o frs esmae he hypoheszed coegrao relaoshp separaely for each member of he pael ad he pool he resulg resduals whe cosrucg he pael ess for he ull of o coegrao. Specfcally, he frs sep, oe ca esmae he proposed coegrag regresso for each dvdual member of he pael he form of (5.35), cludg dosycrac erceps or reds as he parcular model warras, o oba he correspodg resduals e ˆ. I he secod sep, he way whch he esmaed resduals are pooled wll dffer amog he varous sascs, whch are defed as follows. Pael varace rao sasc: ˆ V, ( ) Z Lˆ A Lˆ e 5.38 Pael-rho sasc: ˆ ( ˆ ρ ) λ e, Δ, Z A A ( eˆˆ e ˆ λ ) ( 5. ) Pael- sasc: 4

17 / ˆ σ ( ˆ λ ) Z A A / e,, ( eˆˆ e ˆ λ ).( 5.4)... σ Δ... Group-rho sasc: Z ˆ ρ A A λ e, Δee, λ Group- sasc: ( ˆ ) ( ˆ ˆˆ )......( 5.4) / Z ˆ ˆ ˆ,, 4 σ e Δee K Where ˆ μ ˆ ˆ ˆ e ρe,, ˆ λ w ˆ ˆ, ( ˆ ˆ λ )....( 5. ) for some choce of lag wdow μ μ sk s s s+ w sk s, + K Sˆ, ˆ μ σ Sˆ + ˆ λ, σ, ad ˆ σ Lˆ Lˆ, ˆ ( ˆ ˆ ˆ L ˆ ) / Ω Ω Ω Ω such ha Ω ˆ s a cosse esmaor of Ω. he frs hree sascs are based o poolg he daa across he wh group of he pael; he ex wo sascs are cosruced by poolg he daa alog he bewee group of he pael. Pedro (999) derved asympoc dsrbuos ad crcal values for several resdual-based ess of he ull of o coegrao paels here are mulple regressors. χ K μ V K K (, ) ( as.., )...( 5. 43) 3/ / / χ ( Z ˆ, Z ˆ, ˆ, ˆ, V ˆ ρ Z Z Z ) ρ seq for each of he K,, 5 sascs of χ, he values of μ K ad V K ca be foud from he able Pedro (999), whch deped o wheher he model cludes esmaed fxed effecs ˆ α, esmaed fxed effecs ad esmaed reds ˆ α, ˆ δ. hus, o es he ull of o coegrao, oe smply compues he value of he sasc so ha s he form of (5.43) above based o he values of μ K ad V K from 5

18 he able II Pedro (999) ad compares hese o he approprae als of he ormal dsrbuo. Uder he alerave hypohess, he pael varace sasc dverges o posve fy, ad cosequely he rgh al of he ormal dsrbuo s used o rejec he ull hypohess. Cosequely, for he pael varace sasc, large posve values mply ha he ull of o coegrao s rejeced. For each of he oher four es sascs, hese dverge o egave fy uder he alerave hypohess, ad cosequely he lef al of he ormal dsrbuo s used o rejec he ull hypohess. hus, for ay of hese laer ess, large egave values mply he ull of o coegrao s rejeced ess for Mulple Coegrao I s also possble o adap Johase s (995) mulvarae es based o a VAR represeao of he varables. Le ( r) Λ deoe he cross-seco specfc lkelhood-rao ( race ) sasc of he hypohess ha here are (a mos) r saoary lear combaos he coegraed VAR sysem gve by Y (,, y y k ). Followg he u roo es proposed IPS (3), Larsso e al. () suggesed he sadardzed LR-bar sasc Λ( r) E λ( r) Λ ( r) (5.44) Var λ ( r) o es he ull hypohess ha r agas he alerave ha a mos r r. Usg a sequeal lm heory ca be show ha Λ ( r) s asympocally sadard ormally dsrbued. Asympoc values of E λ ( r) ad Var λ ( ) r are abulaed Larsso e al. () for he model whou deermsc erms ad Breug (5) for models wh a cosa ad a lear me red. Ulke he resdual-based ess, he LR-bar es allows for he possbly of mulple coegrao relaos he pael Pael Coegrao ess allowg for cross-secoal depedece:he Durb-Hausma ess Weserlud proposes wo ew pael coegrao ess ha ca be appled uder very geeral codos, ad ha are show by smulaoo be more powerful ha oher exsg ess. ) Model ad Assumpo We beg by assumg ha he Fsher equao holds so ha α + βπ + z (5.45) π δπ, + w (5.46) π s he acual rae of flao observed a me perod for coury, s he 6

19 ex pos omal eres rae o a omal bod. We have argued above ha, alhough he omal eres rae ca geeral be vewed as osaoary, seems reasoable o perm flao o be saoary.herefore, we do o mpose ay a pror resrcos o he value ake by δ Ifδ, he flao s osaoary, as f δ, flao s saoary. he dsurbace z s assumed o obey he followg se of equaos ha allow for cross-secoal depedece hrough he use of commo facors: z λ F + e (5.47) F ρ F + u (5.48) j j j j e φ e + v (5.49) F s a k-dmesoal vecor of commo facors F wh j, k, ad λ s a j coformable vecor of facor loadgs. By assumg ha ρ j < for all j, we esure ha F s saoary, whch mples ha he order of egrao of he compose regresso error z depeds oly o he egraedess of he dosycrac dsurbace e. hus, hs daa-geerag process, he relaoshp (5.45) s coegraed f φ < ad s spurous f φ. oe ha, sce s assumed o be osaoary, φ < mples boh ha π s osaoary ad ha s coegraed wh. ex, we lay ou he key assumpos eeded for developg he ew ess. Assumpo (error process) (a) v ad w are mea zero for all ad ; (b) E( v v ) ad ( ) kj E v w for all, k, ad j; (c) ( ) kj (d) ( v ) ϖ < ad ( w ) var E w w for all k, ad j; kj var Ω s posve defe. For he asympoc heory, he followg codo s also requred. Assumpo (varace prcple). he paral sum processes of v ad w sasfy a varace prcple. I 7

20 [ r ] parcular, v ϖ W ( r) as for each, ( ) Browa moo defed o he u erval r [,]. W r s a sadard Fally, o be able o hadle he commo facors, he followg codos are assumed o hold. Assumpo 3 (commo facor). (a) E( u ) ad var ( ) u < ; (b) u s depede of v ad w for all ad ; (c) λλ Σ as, Σ s posve defe; (d) ρ j < for all j. ) es Cosruco Our objecve s o es wheher ad π are coegraed or o by ferrg wheher e s saoary or o. A aural approach o do hs s o employ he Ba ad g (4) approach, whch amous o frs esmag (5.45) s frs dfferece form by OLS ad he o esmae he commo facors by applyg he mehod of prcpal compoes o he resulg resduals. A es of he ull hypohess of o coegrao ca he be mplemeed as a u roo es of he recumulaed sum of he defacored ad frs dffereaed resduals. We beg by akg frs dffereces, whch case (5.47) becomes Δ z λ Δ F +Δ e. hus, had Δ z bee kow, we could have esmaed λ ad Δ F drecly by he mehod of prcpal compoes. However, Δ z s o kow, ad we mus herefore apply prcpal compoes o s OLS esmae sead, whch ca be wre as ˆ β s obaed by regressg Δ z Δ ˆ β Δ π, (5.5) ˆ Δ o Δ π. Le λ, Δ F ad Δ ẑ be K, ( ) K ad ( ) marces of sacked observaos o λ, Δ F ad zˆ Δ, respecvely. he prcpal compoes esmaor Δ ˆF of Δ F ca be obaed by compug mes he egevecors correspodg o he K larges egevalues of 8

21 marx ΔΔ zˆ z ˆ. he correspodg marx of esmaed facor loadgs he ( ) ( ) s gve by ˆ λ F ˆ z ˆ ( ) Δ Δ. Gve ˆ λ ad Δ Fˆ, he defacored ad frs dffereaed resduals ca be recovered as whch ca be recumulaed o oba Δ eˆ Δzˆ λ ˆ Δ Fˆ (5.5) eˆ Δeˆj. j As show he Appedx, e ˆ s a cosse esmae of coegrao es ca be mplemeed usg (5.43) wh e ˆ place of e, whch suggess ha he e. I oher words, esg he ull hypohess of o coegrao s asympocally equvale o esg wheher φ he followg auoregresso: eˆ φ eˆ + error. (5.5) A hs po, s useful o le deoe he umber of us for whch he o coegrao resrco φ s o be esed. hs umber ca be equal o bu ca also be a subse. he po of havg wo ses for he cross-seco s o hghlgh he fac ha eve f < accuracy wll be gaed by usg all us he esmao of he commo facors. I wha follows, we shall propose wo ew pael coegrao ess ha are based o applyg he Durb Hausma prcple o (5.5) (see Cho, 994). he frs, he pael es, s cosruced uder he maaed assumpo ha φ φ for all, whle he secod, he group mea es, s o. Boh ess are composed of wo esmaors of φ ha have dffere probably lms uder he alerave hypohess of coegrao bu share he propery of cossecy uder he ull of o coegrao. I parcular, le OLS esmaor of φ (5.45), ad le ˆ φ deoe s pooled couerpar. he correspodg dvdual ad pooled srumeal varable (IV) esmaors of φ deoe he φ, deoed φ ad φ, respecvely, are obaed by smply srumeg ˆ e wh e ˆ. As show by Cho (994), he IV esmaors are cosse uder he ull hypohess bu are cosse uder he alerave. O he oher had, he OLS esmaors are cosse boh uder he ull ad alerave hypoheses (see Phllps ad Oulars, 99). he IV ad OLS esmaors ca hus be used o cosruc he Durb Hausma ess. 9

22 I so dog, cosder he followg kerel esmaor: ϖˆ M ˆ ˆ vv j j M M + j+ j, v ˆ s he OLS resdual obaed from (5.45) ad M s a badwdh parameer ha deermes how may auocovaraces of v ˆ o esmae he kerel. As dcaed he Appedx, he quay ˆ ω s a cosse esmae of ω, he log-ru varace of v. he correspodg coemporaeous varace esmae s deoed by σ. Gve hese esmaes, we ca cosruc he wo varace raos ( ) Sˆ ϖˆ ˆ σ, Sˆ ϖˆ ˆ σ ad 4 ϖˆ ad ˆ σ ˆ σ ˆ ϖ he Durb Hausma es sascs ca ow be obaed as ( ) φ φ ad ( ) ˆ ˆ DH ˆ p S e DH Sˆ ˆ eˆ g. (5.53) φ φ oe ha whle he pael sasc, deoed DH p, s cosruced by summg he dvdual erms before mulplyg hem ogeher, he group mea sasc, deoed DH g, s cosruced by frs mulplyg he varous erms ad he summg. he mporace of hs dsco les he formulao of he alerave hypohess. For he pael es, he ull ad alerave hypoheses are formulaed as H : φ for all p,, versus H : φ φ ad φ < for all. Hece, hs case, we are effec presumg a commo value for he auoregressve parameer boh uder he ull ad alerave hypoheses. hus, f hs assumpo holds, a rejeco of he ull should be ake as evdece favor of coegrao for all us. By coras, for he group mea es, H s esed versus he alerave ha g H : φ < for a leas some. hus, hs case, we are o presumg a commo value for he auoregressve parameer ad, as a cosequece, a rejeco of he ull cao be ake o sugges ha all us are coegraed. Isead, a rejeco should be erpreed as provdg evdece favor of rejecg he ull hypohess for a leas some of he cross-secoal us.

23 3)Asympoc Dsrbuo he Durb-Hausma ess are based o he esmaed dosycrac error erm e ˆ,ad are herefore asympocally depede of he commo facors. As showed Weserlud s Appedx,uder he ull ad Assumpos -3, each of dvdual group mea sascs coverges o ( ( ) ) as, B W r dr he fac ha B does o deped o he commo facors s a drec cosequece of he defacorg, whch asympocally removes he commo compoes from he lmg dsrbuo of he dvdual ess. he followg heorem shows ha he effecs of he commo facors s asympocally eglgble,ad he asympocally ormal DH g ad DH p are deed heorem(asympoc dsrbuo). Uder he ull hypohess H ad Assumpos -3(c), for δ ad δ <, as, wh ad ( ) DHg E B Var B (, ( )), ( ) ( ) 4 DHp E C E C Var C (, ( )) C s he verse of B. 5.4 Pooled Mea Group Esmao of osaoary Heerogeeous Paels 5.4. he Model Specfcao he asympocs of large, large dyamc paels are dffere from he asympocs of radoal large, small dyamc paels. Small pael esmao usually reles o fxed- or radom-effecs esmaors, or a combao of fxed-effecs esmaors ad srumeal-varable esmaors, such W as he Arellao ad Bod (99) geeralzed mehod-of-momes esmaor. hese mehods requre poolg dvdual groups ad allowg oly he erceps o dffer across he groups. Oe of he ceral fdgs from he large, large leraure, however, s ha he assumpo of homogeey of slope parameers s ofe approprae. Wh he crease me observaos here large, large dyamc paels, osaoary s also a cocer. Rece papers by Pesara, Sh, ad Smh (997, 999) offer wo mpora ew echques o esmae osaoary dyamc paels whch he parameers are heerogeeous across groups: he mea-group (MG) ad pooled

24 mea-group (PMG) esmaors. he MG esmaor (see Pesara ad Smh 995) reles o esmag me-seres regressos ad averagg he coeffces, as he PMG esmaor (see Pesara, Sh, ad Smh 997, 999) reles o a combao of poolg ad averagg of coeffces. Assume a auoregressve dsrbuve lag (ARDL) dyamc pael specfcao of he form p y λ y + δ X + μ + ε q (5.55) j, j j, j j j he umber of groups,,..., ; he umber of perods,,..., ; X s a k vecor of explaaory varables; δ are he k coeffce vecors; λ j are scalars; μ s he group-specfc effec ad ( ) ε var σ. mus be large eough such ha he model ca be fed for each group separaely. me reds ad oher fxed regressors may be cluded. If he varables (5.55) are, for example, I ( ) ad coegraed, he he error erm s a I ( ) process for all. A prcpal feaure of coegraed varables s her resposveess o ay devao from log-ru dyamcs of he varables he sysem are flueced by he devao from equlbrum. hus s commo o reparameerze (5.55) o he error correco equao p φ ( λ j j) p q * * φ (, θ ) λj, j δ j, j μ ε j j (5.56) Δ y y X + Δ y + Δ X + +, θ q δ j ( λ k), j k * λj p m j+ λ m j,,..., p, ad * j q δ δ j,,..., q. m j+ m he parameer φ s he error-correcg speed of adjusme erm. f φ, he here would be o evdece for log-ru relaoshp. hs parameer s expeced o be sgfcaly egave uder he pror assumpo ha he varables show a reur o a log-ru equlbrum. Of parcular mporace s he vecor θ, whch coas he log-ru relaoshps bewee he varables. he rece leraure o dyamc heerogeeous pael esmao whch boh ad are large suggess several approaches o he esmao of (5.56). O oe exreme, a fxed-effecs (FE) esmao approach could be used whch he me-seres daa for each group are pooled ad oly he erceps are allowed o dffer across groups. If he slope coeffces are fac o decal, however, he he FE approach produces

25 cosse ad poeally msleadg resuls. O he oher exreme, he model could be fed separaely for each group, ad a smple arhmec average of he coeffces could be calculaed. hs s he MG esmaor proposed by Pesara ad Smh (995). Wh hs esmaor, he erceps, slope coeffces, ad error varaces are all allowed o dffer across groups. More recely, Pesara, Sh, ad Smh (997, 999) have proposed a PMG esmaor ha combes boh poolg ad averagg. hs ermedae esmaor allows he ercep, shor-ru coeffces, ad error varaces o dffer across he groups (as would he MG esmaor) bu cosras he log-ru coeffces o be equal across groups (as would he FE esmaor). Uder hs assumpo, relao (5.56) ca be wre more compacly as Δ y φ ξ ( θ) + Wκ + ε,,..., (5.57) ξ( θ) y, Xθ,,..., s he error correco compoe, (,,...,, +,,,,...,, +, ) W Δy Δy ΔX ΔX Δ X ι ad p q κ ( λ,..., λ, δ, δ,..., * * * *, p * δ, ) q, μ he Model Esmao ad Iferece Sce (5.57) s olear he parameers, Pesara, Sh, ad Smh (999) develop a maxmum lkelhood mehod o esmae he parameers. Expressg he lkelhood as he produc of each cross-seco's lkelhood ad akg he log yelds l ϕ πσ Δy φ ξ θ H Δy φ ξ θ ( ) ( ( )) (5.58) ( ) l ( ) σ H I W W W W, (,, ), ( ) ( ) (,,..., ) σ σ σ σ. ϕ θ φ σ φ φ, φ,..., φ, ad Maxmum lkelhood (ML) esmao of he log-ru coeffces, θ, ad group-specfc error-correco coeffces, φ, ca be compued by maxmzg (5.58) wh respec o ϕ. hese ML esmaors are ermed he PMG esmaors o hghlgh boh he poolg mpled by homogeey resrcos o he log-ru coeffces ad he averagg across groups used o oba meas of he esmaed error-correco coeffces ad he oher shor-ru parameers of he model. he PMG esmaors ca be compued by he famlar ewo-raphso algorhm, whch makes use of boh he frs ad secod dervaves. Aleravely, hey ca be compued by a back-subsuo algorhm ha makes use of oly he frs dervaves of (5.58). I hs case, seg he frs dervaves of he coceraed log-lkelhood fuco wh respec o ϕ o yelds he followg relaos ˆ θ, ˆ φ, ad solved eravely: ˆ σ, whch eed o be 3

26 ˆ φ ˆ X φ HX X H y y ˆ σ ˆ σ ˆ θ ( ˆ Δ φ, ) (5.59) ( ) ˆ φ ˆ ξ H ˆ ξ ˆ ξ H Δ y,,..., (5.6) ( ) ( ) σ Δy ˆˆ φξ H Δy ˆˆ φξ,,..., (5.6) ˆ ˆ ˆ ( ) ξ y, X θ. Sarg wh a al esmae of θ, say θ, esmaes of ad σ ca be compued usg (5.6) ad (5.6), whch ca he be subsued ˆ φ (5.59) o oba a ew esmae of θ, say ˆ θ ( ), ad so o ul covergece s acheved. Uder some regular codos, he MLE of he shor-ru coeffces φ ad σ he dyamc heerogeeous pael daa model (5.57) are θ s cosse, amely ˆ θ θ op ( ) cosse ad he MLE of, ˆ φ φ ( ) ad ˆ σ σ () Furhermore, for fxed ad as, he MLE of (, ) has he mxure-ormal dsrbuo Dψ dag( Ik, I) o p { } a ( ˆ ), ( ) Dψ ψ ψ M I ψ ad ( ) ψ θ φ o p. asympocally (5.6) I ψ s he radom formao marx. More specfcally, he pooled MLE ˆ θ, defed by (5.59), has he followg large asympoc dsrbuo: XX ( ˆ a φ θ θ) M, R XX (5.63) σ R,,,...,, are he radom probably lms defed by XHX RXX. Oce he pooled MLE of he log-ru parameers, ˆ θ, s successfully compued, he shor-ru coeffces, cludg he group-specfc error-correco coeffces, φ ca be cossely esmaed by rug he dvdual ordary leas squares (OLS) regressos of Δy o ( ξ, W),,,...,, ˆ ξ ˆ y, X θ. he covarace marx of he 4

27 MLEs, ( ˆ, ˆ,..., ˆ ˆ ˆ,,..., ) θ φ φ κ κ, s he cossely esmaed by he verse of ˆ ˆ ˆ ˆ φ ˆ ˆ ˆ X X φx ξ φx ξ φxw φx W ˆ σ ˆ ˆ ˆ ˆ σ σ σ σ ˆˆ ξξ ˆ ξ W ˆ σ ˆ σ ˆˆ ξξ ˆ ξ W ˆ σ ˆ σ WW ˆ σ WW ˆ σ 5

28 5.5 Specfcao ad Esmao of Spaal Pael Daa Models 5.5. Foudaos: Spaal auocorrelao he formal frame work used for he sascal aalyss of spaal auocorrelao s a so-called spaal sochasc process, or a colleco of radom varables Y, dexed by locao, { Y, D} (5.64) he dex se D s eher a couous surface or a fe se of dscree locaos. he spaal auocorrelao ca be formally expressed by he mome codo CovY (, Y) EY (, Y) EY ( ) EY ( ) for j (5.65) j j j, j refer o dvdual observaos (locaos) ad Y ( Y ) s he value of a radom j varable of eres a ha locao. he mos ofe used approach o formally express spaal auocorrelao s hrough he specfcao of a fucoal form for he spaal sochasc process (5.54) ha relaes he value of a radom varable a a gve locao o s value a oher locao. For example,for a vecor of d radom varables Y, observed across apace, ad vecor of d radom error ε, a smulaeous spaal auoregressve(sar) process s defed as ( Y μ) ρw( Y μ) + ε or ( Y ) ( I W) μ ρ ε (5.66) μ s he (cosa) mea of Y, s a vecor of oes, ρ s he spaal auoregressve parameer, ad he marx W s he spaal weghs marx, specfes whch of he oher locaos he sysem affec he value a ha locao. he spaal weghs crucally deped o he defo of a eghborhood se for each observao. A spaal lag for Y a he follows as or marx form, as [ WY ] sce for each he marx elemes w Y (5.67) j,..., j j WY (5.68) w j are oly ozero for hose j s ( s s he eghborhood se), oly he machg Y j are cluded he lag. For ease of erpreao, he elemes of he spaal weghs marx are ypcally row-sadardzed, such ha for each, w, so he spaal lag may be erpreed as weghed j j average of he eghbors. For he SAR srucure (5.66), he varace marx for Y s a fuco of wo parameers, he varace σ ad he spaal coeffce ρ :

29 [ μ μ ] σ [ ρ ρ ] Var( Y ) E ( Y )( Y ) ( I W )( I W ) (5.69) hs marx mples ha shocks a ay locao affec all oher locaos, hrough a so-called spaal mulpler effec (or, global eraco). A major dsco bewee processes space compared o me doma s ha eve wh d dsurbaces ε, he dagoal elemes (5.69) are o cosa. Furhermore, he heeroscedascy depeds o he eghborhood srucure embedded he spaal wegh marx W. Whe specfyg he spaal depedece bewee observaos, he model may corporae a spaal auoregressve process he dsurbace, or he model may coa a spaally auoregressve depede varable. he frs model s kow as he spaal error model ad he secod as he spaal lag model he Fxed Effecs Spaal Error ad Spaal Lag Model he radoal fxed effecs model exeded o cluded spaal error auocorrelao ca be specfed as Y Xβ + μ+ φ φ ρwφ + ε E( ε ) E( εε ) σ I (5.7) μ μ μ (,... ),ad he radoal model exeded wh a spaally lagged depede varable reads as Y ρwy + X β + μ+ ε E( ε ) E( εε ) σ I (5.7) W deoes a spaal wegh marx descrbg he spaal arrageme of he spaal us, s assumed ha W s a marx of kow cosas. I he spaal error specfcao, he properes of he dsurbace srucure have bee chaged, ρ s usually called he spaal auocorrelao coeffce; as he spaal lag specfcao, he umber of explaaory varables has creased by oe, ρ s referred o as he spaal auoregressve coeffce. he spaal ecoomerc leraure has show ha ordary leas squares (OLS) esmao s approprae for models corporag spaal effecs. I he case of spaal error auocorrelao, he OLS esmaor of he respose parameers remas ubased, bu loses he effcecy propery. I he case whe he specfcao coas a spaally lagged depede varable, he OLS esmaor of he respose parameers o oly loses he propery of beg ubased bu also s cosse. ()MLE Isead of esmag he demeaed equao by OLS, ca also be esmaed by maxmum lkelhood (ML). he oly dfferece s ha ML esmaors do o make correcos for degrees of freedom. he log-lkelhood fuco correspodg o he demeaed equao corporag spaal error auocorrelao s l( πσ ) + l I ρw, ( ) ( ) ee e I ρw Y Y X X β σ

30 ad wh a spaally lagged depede varable, (5.7) πσ + ρ ρ β l( ) l I W, ( )( ) ( ) ee e I W Y Y X X σ (5.73) Y ( Y,, Y ) ad X ( X,, X ). A erave wo-sage procedure ca be used o maxmze he log-lkelhood fuco of he frs model, ad a smple wo-sage procedure s avalable for he secod model (Asel 988,8-8). Asel ad Hudak (99) gve srucos o how o mpleme hese procedures commercal ecoomerc sofware. Oe may also use Spacesa or he MALAB roues of spaal error model (SEM) ad spaal lag model (SAR), whch are freely dowloadable from LeSage s Web se a www. spaal-ecoomercs.com. Alhough hese roues are wre for spaal cross seco, hey ca easly be geeralzed o spaal pael models. ()IV for Fxed Effecs Spaal Lag Model he edogeey of he spaally lagged depede varable suggess a sraghforward srumeal varables sraegy whch he spaally lagged (exogeous) explaaory varables WX are used as srumes (Keleja ad Robso, 993; Keleja ad Prucha, 998). hs apples drecly o he spaal lag he pooled model, he srumes would be ( I ) W X (wh X as a sacked ( K ) marx, excludg he cosa erm). We should use he wh-rasformed varables he model wh fxed effec. (3)GMM for Fxed Effecs Spaal Error Model I he sgle cross-seco, a cosse esmaor ca be cosruced from a se of mome codos o he error erms, as demosraed he Keleja-Prucha geeralzed momes (KPGM) esmaor (Keleja ad Prucha, 999). hese codos ca be readly exeded o he pooled or fxed effec model, by replacg he sgle equao spaal weghs by her pooled couerpars ( I W ) ad usg he wh-rasformed varables he model wh fxed effec. he po of deparure s he sacked vecor of SAR errors: ( I W). vecors, ad ε IID (, σ I ) φ ρ φ+ ε, boh φ ad ε are he hree KPGM mome codos (Keleja ad Prucha, 999, p. 54) pera o he dosycrac error vecor ε. Exedg hem o he pooled seg yelds: E ε ε σ E ε I W I W ε σ r WW ( )( ) ( ) 3

31 E ( I W) ε ε r s he marx race operaor ad use s made of r ( I W W ) r ( W W ) ad ( W ).he esmaor s made operaoal by subsug ( I W) r I ε φ ρ φ ad replacg φ by he regresso resduals. he resul s a sysem of hree equaos ρ, ρ ad σ.whch ca be solved by olear leas squares (for echcal deals, see Keleja ad Prucha, 999). Uder some farly geeral regulary codos, subsug he cosse esmaor for ρ o he spaal FGLS wll yeld a cosse esmaor for β he Radom Effecs Spaal Error Model Kapoor e al. (7) roduce geeralzaos of he GM procedure Keleja ad Prucha (999) o pael daa models volvg a frs order spaally auoregressve dsurbace erm, whose ovaos have a error compoe srucure. I parcular, hey roduce hree GM esmaors whch correspod o alerave weghg schemes for he momes ad derve he large sample properes whe s fxed ad. her specfcaos are such ha he model s dsurbaces are poeally boh spaally ad me-wse auocorrelaed, as well as heeroskedasc. Also hey defe a feasble geeralzed leas squares (FGLS) esmaor for he model s regresso parameers. hs FGLS esmaor s based o a spaal couerpar o he Cochrae Orcu rasformao, as well as rasformaos ulzed he esmao of classcal error compoe models. Balag e al. (7) exeded he GM procedure Kapoor e al. (7) o he ubalaced pael daa case. More specfcally, we assume ha each me perod,,, he DGP s: y X β + φ, φ ρwφ + ε ρ < Sackg he observaos we have y Xβ + φ (5.74) ( I W) φ ρ φ + ε (5.75) We assume furhermore he followg error compoe srucure for he ovao vecor ε : ( I ) ε μ+ υ (5.76) μ represes he vecor of u specfc error compoe, ad υ coas he error compoes ha vary over boh he cross-secoal us ad me perods. he 4

32 error compoes are assumed o sasfy: μ IID(, σ μ ) ad υ IID(, συ ). he acual esmao of he parameers of he model (5.74)-(5.76) s performed hree seps. Sep: I he frs sep we esmae he regresso model (5.74) usg ordary leas squares o oba ( ) dsurbaces ˆ φ y Xβ. βˆols X X Xy ad ge a cosse esmaor of he ˆOLS Aleravely, Balag e al. (7) ru fxed effecs o model (5.74) o oba he β s cosse esmaes ad hey fd alhough he magudes for some esmaes chage, he resuls for he sascally sgfca esmaes are bascally he same he OLS esmaes. Sep: We esmae he spaal auoregressve parameer ρ ad he varace compoes σ ad σ (or equvalely υ μ συ ad σ σ + σ ) erms of he υ μ resduals obaed va he frs sep ad he geeralzed momes procedure suggesed he paper. Defg φ ( I W) φ, φ ( I W) φ, ε ( I W) P mome codos: ε, Q I I I, Kapoor e al. (7) sugges a GM esmaor based o he followg sx ( ε ε ( ) ) συ ε ε ( ) συ ( ε ε ( ) ) ( ε ε ) σ ( ε ε ) σ ( ) ( ε ε ) E Q ( ) ( ) E Q r WW E Q E P E P r WW E P ad (5.77) From (5.75), ε φ ρφ ad ε φ ρφ subsug hese expressos (5.77) we oba a sysem of sx equaos volvg he secod momes ofφ,φ ad φ.he GM esmaor of σ υ, σ ad ρ s he soluo of he sample couerpar of he sx equaos (5.77). Kapoor e al. (7) sugges hree GM esmaors. he frs volves oly he frs hree momes (5.77) whch do o volve σ ad yeld esmaes of σ υ ad ρ. he 5

33 fourh mome codo s he used o solve for σ gve esmaes of σ υ ad ρ. Kapoor e al. (7) gve he codos eeded for he cossecy of hs esmaor as. he secod GM esmaor s based upo weghg he mome equaos by he verse of a properly ormalzed varace covarace marx of he sample momes evaluaed a he rue parameer values. A smple verso of hs weghg marx s derved uder ormaly of he dsurbaces. he hrd GM esmaor s movaed by compuaoal cosderaos ad replaces a compoe of he weghg marx for he secod GM esmaor by a dey marx. Kapoor e al. (7) perform Moe Carlo expermes comparg MLE ad hese hree GM esmao mehods. hey fd ha o average, he RMSE of ML ad her weghed GM esmaors are que smlar. However, he frs uweghed GM esmaor has a RMSE ha s 7% o 4% larger ha ha of he weghed GM esmaors. sep3: I hs sep he regresso model (5.74)-(5.75) s reesmaed erms of a feasble GLS esmaor. 6