GMM Estimation of the Number of Latent Factors

Transcription

1 GMM Esimain f he Number f aen Facrs Seung C. Ahn a, Marcs F. Perez b March 18, 2007 Absrac We prpse a generalized mehd f mmen (GMM) esimar f he number f laen facrs in linear facr mdels. he mehd is apprpriae fr panels a large (small) number f crsssecin bservains and a small (large) number f ime-series bservains. I is rbus heerskedasiciy and ime series aucrrelain f he idisyncraic cmpnens. All necessary prcedures are similar hree sage leas squares, s hey are cmpuainally easy use. In addiin, he mehd can be used deermine wha bservable variables are crrelaed wih he laen facrs wihu esimaing hem. Our Mne Carl experimens shw ha he prpsed esimar has gd finie-sample prperies. As an applicain f he mehd, we find ha he inernainal sck reurns are explained by ne srng glbal facr. his facr seems be highly crrelaed wih he US sck marke facrs. his resul can be inerpreed as evidence fr marke inegrain. We als find w weak facrs relaed msly wih he Eurpean and he Americas markes. a Crrespnding Auhr: Seung C. Ahn, Deparmen f Ecnmics, W.P. Carey Schl f Business, Arizna Sae Universiy, empe, AZ 85287; miniahn@asu.edu b Marcs F. Perez, Deparmen f Ecnmics, W.P. Carey Schl f Business, Arizna Sae Universiy, empe, AZ 85287; fabrici@asu.edu.

2 GMM Esimain f he Number f aen Facrs Absrac We prpse a generalized mehd f mmen (GMM) esimar f he number f laen facrs in linear facr mdels. he mehd is apprpriae fr panels a large (small) number f crsssecin bservains and a small (large) number f ime-series bservains. I is rbus heerskedasiciy and ime series aucrrelain f he idisyncraic cmpnens. All necessary prcedures are similar hree sage leas squares, s hey are cmpuainally easy use. In addiin, he mehd can be used deermine wha bservable variables are crrelaed wih he laen facrs wihu esimaing hem. Our Mne Carl experimens shw ha he prpsed esimar has gd finie-sample prperies. As an applicain f he mehd, we find ha he inernainal sck reurns are explained by ne srng glbal facr. his facr seems be highly crrelaed wih he US sck marke facrs. his resul can be inerpreed as evidence fr marke inegrain. We als find w weak facrs relaed msly wih he Eurpean and he Americas markes. Keywrds: facr mdels, GMM, number f facrs, inernainal sck marke

3 1. INRODUCION Many ecnmic and financial heries are based n linear facr mdels. A well knwn example is he Arbirage Price hery (AP, Rss, 1976), where asse reurns are generaed by a facr srucure. In he finance lieraure, he AP mdel has been exensively used analyze he prices f he sysemaic risks in he sck, mney, r fixed incme securiies markes. here are many her examples. Analyzing he daa frm G7 cunries, Gregry and Head (1999) fund ha crss-cunry variains in prduciviy and invesmen have cmmn cmpnens. Grman (1981) and ewbel (1991) fund ha if cnsumers are uiliy maximizers, heir budge shares fr individual gds r services purchased shuld be driven by a ms hree facrs. Sck and Wasn (2005) prved ha many macrecnmic variables in US are driven by a smaller number f cmmn facrs. Ahn, ee and Schmid (2007a) shwed ha he ime paern f he flucuains in individual firms echnical prduciviies can be esimaed based n a facr mdel. An excellen summary f he use f facr mdels can be fund in Campbell, and Mackinlay (1997) and als in Bai (2003). Fr any empirical sudy ha invlves facr mdels, esimain f he rue number f facrs is crucial in rder idenify and esimae he facrs. I is als impran deermine wha bservable macrecnmic and/r financial variables are relaed he unbservable facrs, in rder give an ecnmic inerpreain he mdel. We prpse a mehdlgy address hese quesins using an esimain prcedure based n GMM. Earlier empirical sudies f facr mdels were based n he maximum likelihd (M) mehd f Jöreskg (1967). Using his mehd, a researcher esimaes facr ladings and variances f idisyncraic errrs f asse reurns cncurrenly, and es fr he number f laen facrs using a likelihd-rai es. he M mehd requires quie resricive disribuinal assumpins: he idisyncraic errr erms are required be nrmal, and independenly and idenically disribued ver ime. Mre general appraches have been develped allwing fr less resricive assumpins. A cmmn mehd is cnsruc candidae facrs, repea he esimain and esing f he mdel fr differen number f facrs (), and bserve if he ess are sensiive increasing. ehman & Mdes (1988) and Cnnr & Krajczyk (1988) used his echnique analyze he US sck reurns. Success f his mehd wuld depend n he qualiy f he chsen candidae facrs. Anher apprach is use esimars f he ranks f 1

4 marices 3 (e.g., Gill and ewbel, 1992; Cragg and Dnald, 1996, 1997). A limiain f his apprach is ha i is cmpuainally burdensme, especially if he number f respnse variables analyzed is large. 4 Mre recenly, Bai and Ng (2002) have develped a general esimain mehd fr he number f facrs. heir leas squares esimain mehd is designed fr daa wih a large number f respnse variables (N) and a large number f ime series bservains (). his mehd culd prduce incnsisen esimars if eiher N r is small. Simulain resuls repred in Bai and Ng (2002) indicae ha he number f facrs is n accuraely esimaed if N r is less han 40. hus, he leas squares mehd wuld be inapprpriae fr he sudies using small ses f respnse variables. In his paper we presen an alernaive generalized mehd f mmen (GMM) esimar f he number f facrs. he advanages f his new mehd cmpared wih hse discussed abve are he fllwing. Firs, he mehd requires ha jus ne f he daa dimensins (N r ) be large; ha is, eiher he number f crss-secin r ime series bservains has be large. Several ecnmic and financial applicains invlve small crss secinal bservains. Examples are he analyses f prfli reurns, yields n bnd indexes, r cunry cmmn facrs. Secnd, he mehd prvides a way check pssible crrelains beween bservable variables (i.e., macrecnmics r financial variables) and unbservable facrs wihu esimaing facr hemselves. Using ur mehd, researchers are able give an ecnmic inerpreain he laen facrs mdel (see, Ahn, Dieckmann and Perez, 2007). hird, he mehd is cmpuainally easy implemen. All necessary prcedures are based n clsedfrm sluins, and hus, d n require nn-linear pimizain. Any sfware ha can esimae muliple equains mdels can be used. Furh, he mehd allws fr crss-secin and ime series heerskedasiciy and ime series aucrrelain f he idisyncraic cmpnens and i des n require disribuinal assumpins abu he daa generaing prcess. Our mehd is 3 If he idisyncraic errr cmpnens f he respnse variables analyzed are crss-secinally independen (exac facr mdel), he variance marix f he respnse variables (e.g., reurns) is decmpsed in a diagnal marix and a marix wih a rank equal. hus, he number f he cmmn facrs () can be fund by esimaing he rank f he difference beween he esimaes f he variance and he diagnal marices. 4 Rank f a marix can be esimaed by he wer-diagnal-upper riangular decmpsiin es (DU) develped by Gill and ewbel (1992) and Cragg and Dnald (1996). his mehd requires a Gaussian eliminain prcedure and divisin f he respnse variables in w nn-verlapping grups. he Gaussian eliminain prcedure is cmplicaed if big marices are analyzed. Alernaively, Cragg and Dnald (1997) prpse a Minimum Chi- Squared saisic (MINCHI2). his mehd is general in he sense ha i requires nly weak disribuinal assumpin abu he respnse variables and allws fr heerskedasiciy and aucrrelain. he principal prblem f MINCHI2 is ha sme nnlinear pimizain prcedures are required and he prcedures fen fail lcae sluins as shwn by Dnald, Fruna and Pipiras (2005). 2

5 primarily designed fr exac facr mdels in which idisyncraic errr cmpnens f respnse variables are crss-secinally uncrrelaed. Hwever, even if he errrs are crss secinally crrelaed, he mehd can be used esimae he number f facrs if N is large and he respnse variables can be gruped apprpriaely (e.g., prflis). As an applicain we use ur mehdlgy analyze he inernainal sck markes cmvemens. Our empirical resuls cnclude ha ne srng glbal facr explains he cmvemen f inernainal sck markes. Ineresingly, his facr seems be crrelaed wih US sck marke facrs. his can inerpreed as evidence fr sme degree f marke inegrain. We als find evidence f w weak facrs msly relaed wih he Eurpean and Americas markes he res f he paper is rganized as fllws. Secin 2 inrduces he facr mdel we invesigae, and liss he basic assumpins we made fr he esimain. Secin 3 explains ur GMM mehd esimae he number f facrs. In secin 4, we cnsider hw he mehd culd be used fr he analysis f he mdels when he idisyncraic cmpnens are crsssecinally crrelaed. We als cnsider he cases in which sme bservable variables ha are penially crrelaed wih laen facrs. Secin 5 exhibis ur Mne Carl simulain resuls and finie-sample prperies f ur mehd. Secin 6 discusses he resuls we bain by applying he mehd he inernainal sck marke. Cncluding remarks are prvided in Secin Mdel and Assumpins We cnsider a linear mdel wih a finie number f unbservable laen cmmn facrs: r = α + β f + ε, (1) i i i i where r i is he value f he respnse variable i (= 1, 2,, N) a he ime (= 1, 2,, ), α i is an inercep, f is an 1 vecr f unbservable cmmn facrs, β i is an 1 vecr f he facr ladings fr he respnse variable i, and he ε i are he idisyncraic cmpnens f respnse variables which are crss-secinally uncrrelaed. hus, he respnse variables r i are crss-secinally crrelaed nly hrugh he cmmn facrs f. Usual facr analysis ypically applies demeaned daa wih E( r ) = 0 fr all i and. Bu we d n impse such resricins. i begin, we cnsider he cases in which N is relaively small and is large. hus, he 3

6 asympic hery we use belw applies as N fr fixed. We will cnsider laer he cases in which is large and N is small. Fr cnvenience, we adp he fllwing nain. We use r dene he vecr ha includes all he crss-secinal bservains f he respnse variable r i a ime. Similarly, r i denes he vecr including all f he ime series bservains f r i fr he respnse variable i. he vecrs ε i and (1) fr given by ε are similarly defined. Using his nain, we can sack he equains in r = α+β f + ε, (2) where α = ( α1, α2,..., α N ) and Β = ( β1, β2,..., β N ). Including he nn-zer vecr f respnsevariable-specific inerceps in he mdel, we can assume ha E( f ) = 0 wihu lss f generaliy. Since ur mehd esimae he number f facrs () is an applicain f GMM, we require a se f sufficien cndiins under which usual GMM heries apply and he number f facrs can be idenified. Fr asympics, we use and dene cnverges in p d prbabiliy and cnverges in disribuin, respecively. fllwing: he basic assumpins are he Assumpin A: he facrs in f are nn-cnsan variables wih finie mmens up he furh rder, E( f ) = 0 1 and E( f f ) =Ω f fr all, and 1 Σ 1 f f = p f Ω as, where Ω f is a finie and psiive definie marix. Assumpin B: rank( Β ) =. Assumpin C: here exiss a cnsan m (0, ), such ha fr all (wih fixed N), (C1) he errrs ε i have finie mmens up he eighh rder wih E( ε i f1, f2,..., f ) = 0 fr all i and ; (C2) E( εiε i s f1, f2,..., f ) = 0 fr all i i, s ; (C3) Σ Σ ε ε fr all i and ; 1 = 1 s= 1 E( is i ) m 4

7 Σ Λ, as, where w = ( h ε ) E( h ε ), h = 1/ 2 (C4) = 1 w d N( 0 ( N+ ) 1, ) (1, f, ε ), and Λ= p Σ Σ E w w. 1 lim s 1 1 ( = = s ) Assumpin D: e Β G be he facr lading marix crrespnding ( ) arbirarily chsen respnse variables frm r. hen, rank( Β G ) =. In Assumpin A, we assume ha he facrs are cvariance sainary; ha is, he variance marix f f, Var( f ) =Ω f, is same fr all. We adp his assumpin fr expsiry cnvenience and i can be relaxed wihu alering ur resuls. he required assumpin is ha 1 Σ 1 f f = p f (Whie, 1999). Ω as. Ms f he general mixing prcesses saisfy his cndiin Assumpin B implies ha he rue number f facrs is. Under Assumpin (C1), he facrs are weakly exgenus he idisyncraic errrs. Assumpin (C2) resrics he errr erms be crss-secinally uncrrelaed 5. hus, wih (C2), he mdel (2) is an exac facr mdel. If sme insrumenal variables crrelaed wih he facrs are bservable, we culd use hem esimae he number f facrs, even allwing he errrs be crss-secinally crrelaed. Such cases will be discussed in secin 4.2. Als, even if he errrs are crss secinally crrelaed, he mehd can be used esimae he number f facrs by gruping he respnse variables apprpriaely (e.g., prflis), as explained in secin 4.1 Assumpin (C3) indicaes ha he aucvariances f he errr erms are absluely summable, while (C4) is nhing bu a cenral limi herem. When facrs and errrs fllw general mixing prcesses, bh Assumpins (C3) and (C4) hld. 5 Alernaively, when he errrs are crss-secinally crrelaed, bu n aucrrelaed ver ime, an exac mdel can be bained by rewriing he mdel (2) as ri = Fβi+ ε, where i F = ( f1, f2,..., f ). If he errrs are serially uncrrelaed, he variance marix f ε i becmes diagnal. When is small, we can esimae by applying he mehd we discuss belw his alernaive mdel. 5

8 Assumpin D implies ha all facrs ( ) influence all pssible subses f respnse variables. In rder mivae Assumpin D, le us pariin he respnse variables in r in w arbirary grups. g α + Β f + ε = = + Β + =, (3) g g g ( P 1) ( P 1) ( P ) ( 1) ( P 1) r α f ε z z z ( N 1) z ( N 1) ( N ) ( 1) ( N 1) α + Β f + ε ( Q 1) ( Q 1) ( Q ) ( 1) ( Q 1) such ha P+ Q= N, P>, and Q>. hen, Assumpin D, wih Assumpins A-C, implies ha: z g g z g rank[ E(( z α )( g α ) )] = rank[ E( z ( g α ) )] = rank [ Β Ω fβ ] =. (4) g Based n his bservain, we prpse esimae by esimaing he rank f E[ z ( g α ) ]. Clearly, Assumpin D is srnger han Assumpin B. Many f he mehds ppularly used fr facr analysis d n require Assumpin D since under Assumpins A-C, where Ψ is he N E[( r α)( r α) ] =ΒΩ Β +Ψ, f N diagnal marix f he variances f ε i. he M esimain f Jöreskg (1967) and he Minimum Chi-Squared saisic (MINCHI2) f Cragg and Dnald (1997) esimae based n esimaes f Β and Ψ. Bu use f hese mehds is smewha limied. he legiimacy f he M mehd requires sme srng disribuinal assumpins n daa such as nrmaliy. Use f MINCHI2 des n require such srng disribuinal assumpins, bu i fen suffers frm he cmpuainal difficuly f esimaing Ψ. Adping Assumpin D, we n lnger need esimae Ψ. g E( z ( g α ) ). I suffices esimae he rank f he mmen marix Assumpin D requires ha ms f he respnse variables shuld depend n all f he facrs in f. see why, suppse ha r mre respnse variables in g depend n nly a subse f f ; ha is, he facr ladings f many ( r mre) respnse variables crrespnding a subse f facrs are zers. Fr such cases, Assumpin D is vilaed depending n he pariins f g and z. We will cnsider such cases laer. he rank cndiin (4) can be cnvered a mmen cndiin ha can be used in 6

9 GMM. Accrding Assumpin D, here mus exis a P ( P ) marix Ξ= ( Ξ 1, Ξ 2 ) f full clumn, where Ξ 1 is a ( P ) ( P ) square inverible marix, such ha g Β Ξ= 0 ( P ). hus, under Assumpins A-D, we have E ( Ξ g α ) = E ( g α ) Ξ = E ε Ξ = 0 z z z g g Ξ ( Q+ 1) ( P ), (5) g where α Ξ α is a ( P ) 1 vecr. Assumpin C(2), which resrics he mdel (2) be Ξ an exac ne, is crucial fr his mmen cndiin. Fr fuure use, define θ = vec[( α Ξ, Ξ 2 ) ]. g Clearly, Ξ is n unique, since fr any cnfrmable square marix A, ( ΞA ) Β = 0. here are many pssible resricins we can impse avid his under-idenificain prblem. Amng hem, we use he resricin Ξ 1 = I P, while leaving Ξ 2 unresriced. Amng he P ( P ) marices saisfying his resricin, Ξ is he unique P ( P ) marix f full clumn ha is rhgnal g Β GMM ESIMAION OF HE NUMBER OF AEN FACORS In his secin we presen he GMM mehd fr esimain he number f facrs. Firs, given he assumpins explained befre, we cnsruc he mmen cndiins ha will be used in he esimain. e us dene by he number f facrs we use fr esimain, which culd be differen frm. Given, we pariin g in α + Β f + ε = =, (6) P y y y y P P P, (( P ) 1) (( P ) 1) (( P ) ) ( 1) (( P ) 1) g x x x ( P 1) x α + Β f + ε, ( 1) ( 1) ( ) ( 1) ( 1) where = 0, 1, 2,, P 1. Wih his nain, define he fllwing mmen funcin: 1 P m ( b ) = I P [ y ( I P (1, x )) b ] z i, (7) 6 Specifically, ( ) g g 1 Ξ 2 =Β1 Β 2 where g g g 1 2 g Β = [ Β, Β ], and Β 2 is a square inverible marix. 7

10 where b is a ( P )( + 1) 1 vecr f unknwn parameers. Observe ha he mmen funcin (7) is linear in b. Als ne ha he mmen funcin (7) is he ne implied by a muliple equain mdel wih ( P ) differen dependen variables ( P y ), wih cmmn regressrs ( x ) and cmmn insrumenal variables ( z ). hus, he mmen funcin (7) can be easily impsed in GMM using any sfware ha can handle hree-sage leas squares. he inuiin behind mmen funcin (7) cmes frm he fac ha i is linked mmen cndiin (5). see why, le H = ( I, S ). be a P ( P ) marix wih a ( P ) P P unresriced parameer marix SP ; and le ap be a ( P ) 1unresriced parameer vecr. By cnsrucin, H is a full-clumn marix. Furhermre, i can be shwn: 1 m ( b ) = vec ( H g ap ) z. (8) hus, he mmen cndiin (5) implies ha under Assumpins A-D, when E[ m ( b )] = 0 if and nly if b =, = θ. ha is, ur mmen cndiins will hld jus a he rue value f he parameers, and if and nly if he rue number f facrs ( ) was used in he esimain. Nw, we explain hw use he mmen funcin cnsisenly esimae he facrs. Fr given, cnsider he fllwing minimizain prblem: =, (9) 1 min b c ( ( ), ) ( ) [ ( )] ( ) b W d b W d b where d b = Σ m b is he sample mean f he mmen funcins m ( b ), and 1 ( ) = 1 ( ) he weighing marix W ( ) is ( P ) Q ( P ) Q psiive-definie marix wih a nnschasic and finie prbabiliy limi, say W ( ). e b ˆ dene he GMM esimar ˆ minimizing c ( b W ( ), ) ; and use b dene he GMM esimar minimizing c ( b W ( ), ) (i.e. a he rue number f facrs). e Wɶ ( ) be a cnsisen esimar f ( ) lim Var d ( b ). he esimar Wɶ ( ) can be bained by using he mehd f Whie (1980) if daa are serially uncrrelaed, and he mehds f Newey and Wes (1987) r Andrews (1991) if daa are serially crrelaed. We nw dene by b ɶ he pimal GMM 8

11 esimar f θ ha minimizes c ( b Wɶ ( ), ). Using his nain, he fllwing resul esablishes ha he mmen cndiins n (7) can be used esimae he number f facrs. Prpsiin 1: Under Assumpins A-D, fr any W ( ), c ( ˆ b W ( ), ) p fr any < and ( ˆ c b W ( ), ) d ϒ, where ϒ is a weighed average f independen χ 2 (1) randm variables. In addiin, c bɶ Wɶ χ P Q. 2 ( ( ), ) d [( )( )] he prf f Prpsiin 1 is given in he appendix. he disribuin f c ( bˆ W ( ), ) is generally unknwn, bu he resuls frm Prpsiin 1 are sufficien derive he esimain mehds fr he number f facrs. We can frmulae he mdel (2) assuming differen number f facrs (i.e. differen values f ) and hen, use he c saisics selec he mdel ha has he bes fi. w appraches have been prpsed in he lieraure fr mdel selecin. he firs ne uses a sequenial hyphesis esing apprach, and he secnd is based n mdel selecin crierin. We can apply hese w appraches esimae he number f facrs. Our sequenial esing apprach is based in he asympic disribuin f c ( bɶ Wɶ ( ), ) saisic, which is simply he veridenifying resricin es saisic (Hansen, 1982). Using his apprach, we firs frmulae he facr mdel (2) assuming ha he rue number f facrs is equal ne ( = 1). hen we esimae b by GMM, cmpue he veridenifying resricin saisic, and es he hyphesis f = 1 agains he alernaive hyphesis f > 1. By prpsiin 1, if is greaer han ne, he saisic diverges infiniy in large sample. hus, we can expec ha he es is likely rejec he hyphesis f = 1, if he sample size is reasnably large. If he hyphesis is rejeced, we will frmulae he mdel (2) wih = 2, and cmpue he veridenifying resricin saisic es he null hyphesis f = 2 agains he alernaive f > 2. We cninue his prcedure unil he null hyphesis is n rejeced. his sequenial prcedure can yield a cnsisen esimar f 0 if an apprpriae adjusmen is made he significance level used fr he es. he adjusmen is necessary because ype 1 errrs are accumulaed as he es cninues. Cragg and Dnald (1997) shw ha he significance level α shuld be adjused such ha α 0, and lg α / 0 as. 9

12 he mdel secin crierin mehd has been used exensively in deermining he rder f ARMA prcesses in ime series analysis, specifically by Hannan and Quinn (1979), Hannan (1980,1981), Akinsn (1981), and Nishii (1988). Cragg and Dnald (1997) use his mehd esimae he ranks f marices. funcin: Fllwing hese sudies, we define he fllwing crierin MS = c bˆ W f g, (10) 1 ( ) ( ( ), ) ( ) ( ) where f ( ) and g( ) are predefined funcins f (he number f bservains) and (he number f facrs), respecively. Wih apprpriae chices f f ( ) and g( ), a cnsisen esimae f can be bained by minimizing he crierin funcin MS ( ). here are many pssible chices f f ( ) and g( ). One cmmnly used crierin is: Schwarz Crierin (BIC): f ( ) = ln( ), and g( ) = ( P )( Q ). In BIC, g( ) is simply he degrees f veridenifying resricins in he mmen cndiin E[ m ( b )] = 0. Wih (10) and BIC, we bain he fllwing resul: Prpsiin 2: e ˆ be he minimizer f MS ( ) wih BIC. hen, ˆ. p he prf f Prpsiin 2 is given in he appendix 7. Observe ha Prpsiin 2 hlds even if he pimal GMM esimar is n used. One impran advanage f he crierin mehd ver he sequenial mehd is ha i des n require use f he pimal GMM esimar. In he GMM lieraure, many sudies have shwn ha pimal GMM esimars fen have pr finie-sample prperies, especially when daa are aucrrelaed r/and many mmen funcins are used (see, fr example, Alngi and Segal, 1996; Andersen and Sørensen, 1996; and Chrisian and den Haan, 1996). One f he main reasns fr his prblem is ha fr such cases, he pimal weighing marix, ɶ 1 is prly esimaed. Given his prblem, in [ W ( )] 7 he prf is an exensin f a resul frm Ahn, ee and Schmid (2007b). hey have sudied a panel daa mdel wih laen cmpnens f facr srucure. hey develped a GMM mehd esimae he mdel and he number f facrs in he laen cmpnens wih BIC. heir resuls are easily exended ur facr mdel. Ineresed readers may refer he paper. 10

13 pracice, he selecin crierin mehd appears be an aracive alernaive he sequenial mehd. In ur Mne Carl simulains (secin 5) we cmpare he perfrmance f he BIC crierin wih he fllwing crierins : Akaike Infrmain (AIC): f ( ) = l, and g( ) = 2( P )( Q ) Schwarz Crierin 2 (BIC2): f ( ) = lg( ), and g( ) = ( P )( Q ). Schwarz Crierin 3 (BIC3): f ( ) = ln( ), and g( ) = ( P )( Q+ 1). I can be shwn ha Prpsiin 2 hlds using he mdificains he Schwarz Crierin labeled BIC2 and BIC3. AIC crierin is cmmnly used, bu i leads incnsisen esimain f he number f facrs he sequenial esing and mdel selecin crierin mehds can cnsisenly esimae if Assumpin D hlds. Hwever, as we have discussed abve, he assumpin wuld be vilaed if sme facrs influence nly a subse f he respnse variables. When he assumpin des n hld, ur mehds end underesimae he number f facrs. see why, cnsider he fllwing alernaive assumpin: Assumpin D : ( z x Β Ω Β ) =, and rank[ Β Ω ( Β ) ] =. z g rank f f In he appendix (emma A.1) we shwn ha when =, a unique vecr θ exiss such ha E m [ ( )] b ˆ and θ = 0. e W b ɶ be he minimizers f ɶ ( ) be a cnsisen esimar f ( lim ( ) ) Var d θ c b W and ( ( ), ) c b Wɶ ( ( ), ) by replacing Assumpin D by D, we bain he fllwing resuls: ; and le, respecively. hen, Prpsiin 3: Under Assumpins A-C and D, fr any chice f p ˆ ( ( ), ) d < and ( ) W, c ( bˆ W ( ), ). In cnras, c b W ϒ, where ϒ is a weighed average f independen 2 χ (1) randm variables. In addiin, c ( bɶ Wɶ ( ), ) d 2 χ [( P )( Q )]. 11

14 Since he pariin f g and z is arbirary, he rank f z g Β Ω fβ culd change depending n he chice f g and z if Assumpin D des n hld. hus, Prpsiin 3 indicaes ha when Assumpin D is vilaed, he esimaed number f facrs culd be sensiive he pariin used in esimain. As a reamen his prblem, we prpse ry many differen pariins esimae he number f facrs. We can ry a subse f all pssible pariins, r sme randmly generaed pariins. Our simulain exercises shw ha using he frequency able f he esimaes frm a sufficienly large number f differen pariins, we can bain an accurae esimae he crrec number f facrs. he esimar we prpse is he number which is ms fen esimaed frm he esimain wih differen pariins. Frm nw n, we will refer his esimar as highes frequency (HS) esimar. 4. Exensins In his secin, we cnsider he w cases which he GMM mehdlgy develped in he previus secin can be generalized Apprximae Facr Mdels Chamberlain and Rhschild (1983) prpse an apprximae facr mdel es he Arbirage Price hery. his mdel differs frm he exac facr mdel since i allws idisyncraic cmpnens be crss-secinally crrelaed. Assumpin C implies ha Var( ε ) Ψ is diagnal. In cnras, he apprximae facr mdel allws Ψ be nn-diagnal, alhugh he crrelains amng he errrs in ε are resriced be mild. Chamberlain and Rhschild (1983) have shwn ha fr an apprximae mdel wih facrs, he firs eigenvalues f he variance marix f he respnse variables diverge infiniy as N, while her eigenvalues remain bunded. Based n his finding, hey sugges esimaing by cuning he number f larger eigenvalues f he variance marix f respnse variables. Bai and Ng (2002) prpses a mre elabraed saisical mehd. hese w mehds are apprpriae fr he daa wih bh large N and. Hwever, hey may n be apprpriae fr he daa wih small N (see Brwn, 1989; Bai and Ng, 2002). 12

15 While ur mehd is designed fr exac facr mdels wih small N, i culd be used esimae sme apprximae facr mdels. Fr example, cnsider a mdel in which he respnse variables in r are caegrized in a finie number ( M ) f grups (e.g., prflis). Each f he grups, indexed by G 1, G 2,, G M, cnains M NG variables, such ha Σ 1NG j j= i = N, and fr all j= 1,..., M, NG / N a fr sme psiive number j j respnse variables are generaed by he fllwing prcesses: gl g lc l j, i j, i j, i j, i j, j, i a j, as N. Suppse ha he r = α + ( β ) f + ( β ) f + u, (11) where i indexes individuals, j= 1,..., M indexes individual grups, he variables in gl f are he glbal facrs ha influence all f he respnse variables in differen grups, he variables in f are he lcal facrs ha are crrelaed wih he variables in grup j, bu n wih hse in lc j, lc lc her grups (e.g., E( f j, f j, ) = 0, fr j j ), he α, are inercep erms, and he vecrs β, lc and β j, i are he ladings f he crrespnding facrs. he u, are idisyncraic errrs. Apprximae facr mdels resric he crss-secin crrelains in he errr erms be mild. Fr example, Bai and Ng (2002) impse he fllwing resricin, which we name Apprximae Assumpin (AA): j i j i gl j i Assumpin AA: e τ ii, s = E( uiui s ), where u i and u i are he errr erms frm he same r differen grups, and and s are ime indexes. hen, ( ) 1 N = 1 Σ Σ τ, τ fr ii ii sme τ ii and fr all, and N 1 Σ Σ τ M fr sme psiive number M, fr all N. N N i= 1 i = 1 ii e u = ( NG ) Σ u. hen, Assumpin AA warrans ha N,, 1 j, j i G j j, i 1 E( u u ) = Σ Σ u u 0 ; (12-1) j, j, i G j i G j j, i j, i NG j NG j 1 1 Σ u u = Σ Σ Σ u u NG NG = 1 j, j, = 1 i G j i G j j, i j, i p j j Nw, cnsider he fllwing grup-mean equains f (11): gl g lc lc gl g j, j j j j, j, j j j, 0. (12-2) r = α + ( β ) f + ( β ) f + u = α + ( β ) f + ε, (13) 13

16 where he symbls wih verhead bar are defined similarly u j,. By (12-1)-(12-2) and he fac lc ha he variables in f j, are grup-specific, we can shw ha ε j, are asympically uncrrelaed ver differen grups. ha is, we can rea he equains in (13) as an exac facr mdel if N and NG j are sufficienly large. hus, using ur mehd, we culd esimae he number f he glbal facrs by esimaing he rank f gl gl gl gl Β = lim N ( β1, β2,..., βn ) GMM Esimain wih Observable Insrumens When sme bservable variables are penially crrelaed wih he laen facrs, we culd use hem esimae, r es hw many f hem are indeed crrelaed wih he facrs. his es will allw he researcher give ecnmics meaning he laen facrs and es if he variables prpsed by ecnmic r financial mdels are in fac crrelaed wih he laen facrs. We firs cnsider hw esimae. e s be he K 1 vecr f insrumens which saisfies fllwing assumpin: Assumpin D : rank[ E( s f )] = < K and E( s ) = 0. ε N K Under Assumpin D, here mus be a N ( N ) marix f full clumn, E 1 ( r ) 0 α Ξ = ( K+ 1) ( N ) s Ξ, such ha. (14) hus, we can esimae using he same mehd discussed in secin 3. Our mehds apply as we use r fr g and s fr z. When bservable insrumens are n available, we need pariin respnse variables in w grups use a grup f respnse variables as he insrumens fr laen facrs. Bu fr he respnse variables in a grup be legiimae insrumens, he errr erms in ε shuld be crss-secinally uncrrelaed. When uside insrumens are bservable, we d n need pariin he respnse variables. In addiins, he errr erms are allwed be crss-secinally crrelaed as lng as he insrumens are n crrelaed wih hem. 14

17 In cases in which he number f facrs is already knwn, r esimaed by he mehds discussed in secin 3, we can es by GMM hw many f he facrs are crrelaed wih he bservable insrumenal variables in s. If sme facrs are n crrelaed wih s, i shuld be he case ha rank[ E( s f )] = <. Fr his case, by he same mehd we used in secin 3, we can shw ha he GMM mehds based n he mmen cndiin (14) esimae, n. 5. MONE CARO SIMUAION 5.1. Daa Generain he fundain f ur Mne Carl exercises is he fllwing he hree-facr mdel: r = α + β f + β f + β f + ε = α + c + c + c + ε, (15) i i i1 1 i2 2 i3 3 i i 1, i 2, i 3, i i where he f k ( k = 1, 2, 3) are he cmmn facrs f he mdel. Our benchmark mdel is he hree-facr mdel f Fama and French (1993): EMR (excess marke reurn), SMB, and HM. 8 We generae randmly β ik and f k mach he mmens f he Fama-French daa. ha is, we generae daa such ha he mmens f c k, i mach he cunerpars frm he daa ha Fama and French (1993) used. A he sample means f he esimaed beas ( β 1, β 2, and β 3) fr he 25 size and bk--marke prflis, he esimaed variances f he Fama-French cmmn cmpnens are he fllwing: var( β EMR) = 21.72; var( β SMB) = 4.50; var( β HM) = w ypes f idisyncraic errr cmpnens are used. Firs, we generae he errrs which are crss-secinally heerskedasic, bu n aucrrelaed. Specifically, he errrs are FF FF drawn frm N (0, σ i ), where he σ i are he variances f he residuals frm he ime-series FF regressins f (15) fr each i. he values f σ i are beween 1.21 and 3.78, wih he average f hus, he variances f he firs and secnd cmmn cmpnens a he means f beas are mre han wice as grea as he average variance f he idisyncraic cmpnens, while he 8 he Fama-French facrs are cnsruced using he 6 value-weigh prflis frmed n size and bk--marke. SMB (Small Minus Big) is he average reurn n he hree small prflis minus he average reurn n he hree big prflis. HM (High Minus w) is he average reurn n he w value prflis minus he average reurn n he w grwh prflis. EMR is he excess reurn n he marke: he value-weigh reurns n all NYSE, AMEX, and NASDAQ scks (frm CRSP) minus he ne-mnh reasury bill rae (frm Ibbsn Assciaes). See Fama and French (1993) fr a cmplee descripin f he facr reurns. 15

18 variance f he hird cmmn cmpnen (1.29) is smaller. We define he signal nise rai (SNR) f a cmmn cmpnen ( c k, i ) as he rai f he variances f he cmmn cmpnen and he idisyncraic errr cmpnen. In ur simulain, he SNRs are apprximaely 10.8, 2.2, and 0.65 fr cmmn cmpnens 1, 2, and 3, respecively. Secnd, we generae he errr erms frm a simple AR (1) prcess: εi = ρε i i, 1+ vi. Using he residuals frm he ime-series regressins f (15), we esimae he parameersρ and esimae var( v i ) such ha var( εi ) = σ. he errrs generaed by his way are crss-secinal FF i heerskedasic and serially crrelaed ver ime. i 5.2. Esimain f he number f facrs Using daa generaed using hree facrs as defined in secin 5.1, we esimae he number f facrs using he sequenial hyphesis esing and he mdel selecin crierin mehds. We perfrm ur simulains wih six differen cmbinains f and N: = 500 and 1000; and N = 12, 15, and 25. he values f N and are chsen be clse he sample sizes ms fen used in he finance lieraure. Fr each cmbinain, we cnsider w cases: he cases wih aucrrelaed (AR(1)) and serially uncrrelaed idisyncraic errrs. We randmly divide he N prflis in w grups fr each simulain (grups g and z in he nain f secin 2). As described in secin 3 he prflis in grup z will ur insrumens in he esimain (Q). Based n unrepred simulain resuls, we recmmend using arund half f he respnse variables (N/2) as insrumens 9. We repr resuls including 6, 8, and 12 prflis in grup insrumens fr he daa wih N = 12, 15, and 25, respecively. Resuls using her number f insrumens are available frm he auhrs upn reques Resuls are n significanly sensible he number f insrumens used as lng as hey are clse N/2 10 Many sudies find ha he GMM esimars cmpued wih many insrumens and small daa are fen biased (see, fr example, Andersen and Sørensen, 1996) Using nly a small subse f he available mmen cndiins is n a sluin eiher. Andersen and Sørensen (1996) shwed ha using few mmen cndiins are as bad as esimars using many cndiins. his resul indicaes ha here is a rade-ff beween infrmainal gain and finie-sample bias caused by using mre mmen cndiins. 16

19 As discussed in secin 3, bain cnsisen esimaes by using he sequenial es mehd, we need adjus he significance level ( α ) depending n he sample size (). We use α = /. his funcin is chsen such ha α 500 = ,000 differen ses f randmly generaed prfli reurns are used fr simulains. he weighing marix used in GMM is Newey-Wes (1987). Ne ha his weighing marix cllapses he heerskedasiciy-rbus weighing marix f Whie (1980) when bandwidh is equal zer. We esimae he number f facrs fllwing he sequenial hyphesis esing and mdel selecin crierin mehds fr he 1000 simulains. able 1 summarizes he resuls when sequenial hyphesis esing mehd is used. When he idisyncraic errrs are n aucrrelaed and we use Whie weighing marix, we esimae crrecly he number f facrs arund 80-85% f he ime. As expeced, he esimaes becme mre accurae as increases. We bain less accurae resuls when N = 25 and = 500 where he esimaed numbers f facrs are hree fr 78.6% f he imes. here is a prbabiliy f ver esimaing he number f facrs f 15-20% using sequenial hyphesis esing. Fr he cases f aucrrelaed errrs, we bain he similar resuls even if he Whie weighing marix (which is n pimal) is used. As expeced resuls imprve when he Newey- Wes marix wih bandwidh = 3 12 is used, excep fr ne case. Fr he simulaed daa wih N = 25 and = 500, he sequenial es wih bandwidh = 3 predics zer facrs fr 100% f he imes. Our simulains resuls frm he sequenial hyphesis esing mehd sugges ha larger samples are required analyze a large number f prflis ( N 15 ) when idisyncraic errrs are aucrrelaed. As in he case f n aucrrelaed errrs, he mehd ver esimaes he number f facrs arund 15% f he ime. We repr in able 2 resuls fr he esimain f he number f facrs using he mdel selecin crierin mehd. We use he fur differen penaly funcins defined in secin 3: BIC1 11 In unrepred experimens, we als have ried many her significance levels, bu he resuls d n shw remarkable changes. 12 In unrepred experimens we use her differen chices f bandwidh. Resuls and main cnclusins d n change. he aumaic bandwidh selecin mehds by Andrews (1991) r Newey and Wes (1994) chse he values f bandwidh greaer han 8 fr ur simulaed daa, bu wih he values greaer han 8, ur es resuls ge wrse. he ess wih bandwidh f hree perfrmed beer. 17

20 BIC2 BIC3 and AIC. Resuls shw ha he mdel selecin crierin mehd is cnsisenly beer han he sequenial es mehd in esimaing he number f facrs, fr all f he differen cmbinains f and N. Fr he case f n aucrrelaed errrs (Panel A), BIC1 crierin is he ms accurae fr N= 12 and 15, esimaing crrecly hree facrs 90-95% f he ime. Fr he case f N=25, BIC1 is als he bes crierin fr =1000, bu fr he case =500 i esimaes zer facrs 100% f he imes. BIC2 and BIC3 have a higher prbabiliy f ver esimain especially in small samples (12-15%). Fr he case f N=25, =500, BIC2 and BIC3 are he mre accurae esimaing he crrec number f facrs arund 86-88% f he ime. As prediced by he hery, AIC crierin perfrm wrse cmpared wih any BIC crierins. Panel B f able 2 includes esimain resuls when errrs are aucrrelaed. Resuls are very similar Panel A, s we can cnclude ha he mehd is rbus ime series aucrrelain. We jus repr resuls using he Whie cvariance marix (which is asympically n an pimal chice). As shwed in secin 3, he mdel selecin crierin prcedure des n require use f he pimal weighing marix. In unrepred experimens we use Newey-Wes cvariance marix, and in general resuls deerirae specially fr he cases wih N = 15 and 25. I appears ha he Newey-Wes esimar becmes less reliable when N is large. I may be s because he number f parameers in he weighing marix rapidly increases wih N 13. check he rbusness f ur resuls, we nw invesigae if esimain resuls differ if we change he prflis included n each pariin grup ( g and z ). accmplish his bjecive we generae ne se f prfli reurns, and hen, we randmly creae 100 differen pariins. Fr each pariin, we esimae he number f facrs. experimens wih = 500 are presened in able he resuls frm he Resuls are very similar he nes repred in ables 1 and 2. Mdel selecin crierin esimaes he number f facrs mre accuraely in all cases. Sequenial hyphesis esing leads veresimae he number f facrs specially fr N= 15 and 25. Mdel selecin crierin is mre rbus changing in pariins. 13 We als invesigaed he perfrmance f her nn pimal weighing marixes. Fr example he case f a blck diagnal weighing marices, r equivalenly he esimain f he sysem equain by equain. he resuls are similar in magniude and cmpuainally faser specially when N= In he fllwing ables we d n repr resuls fr =1000 in rder save space and since resuls imprve as increases. he main cnclusins are n alered when =1000. Resuls are available upn reques. 18

21 his experimen cnfirms ha he GMM esimain resuls culd change depending n he pariin we use. he es resuls are mre sensiive he pariins used in GMM when N is large. Als ne ha he rue number f facrs (hree) is always esimaed wih he highes frequency. Given hese resuls we sugges ha he esimain shuld be repeaed using many randmly pariined daa. Our experimens sugges ha he number f facrs can be mre accuraely esimaed if 100 differen pariins are used fr esimain. 15 he esimar we prpse is he highes frequency (HS) esimar, which is simply he number f facrs ms fen esimaed frm 100 differen pariins. In rder invesigae he finie-sample prperies f his esimar, we perfrm he fllwing experimen: We generae 1,000 differen ses f prfli reurns. Fr each daa se, we esimae he number f facrs using 100 randmly creaed pariins and cmpue he HS esimae. he resuls (n repred) shw ha he HS esimaes 3 facrs 100% f he ime, fr N = 12, 15, and 25, wheher r n idisyncraic errrs are aucrrelaed r n. he las par f ur simulain exercises ries evaluae he perfrmance f ur mehds when a facr explains a very small prprin f he al variain f he respnse variable. We will call such facr a weak facr. In ur simulain, he variance f he cmmn cmpnen ( c, = β f ) assciaed a weak facr will be small cmpared wih he variance k i ik k f he idisyncraic cmpnen. In rder wrds, a weak facr is a facr wih a lw signal nise rai (SNR). As described in secin 5.1, ur daa was generaed using as a benchmark he hree-facr mdel f Fama and French (1992), where he SNRs f hree facrs are 10.8, 2.2, and 0.65, respecively. generae he daa wih ne weak facr, we reduce he SNR f he secnd cmmn cmpnen (SMB) and increase he ne f he firs cmmn cmpnen (EMR). We d s because we wish generae daa such ha he al variains in he respnse variables explained by he hree facrs and he variains in idisyncraic errrs remain cnsan. In his experimen, we reduce he SNR f he secnd cmmn cmpnen hree differen values: 1.0, 0.50, and Ne ha SNR f he hird cmmn cmpnen is keep cnsan a value smaller han 1. As we have dne befre, we generae ne se f prfli reurns, and hen, we randmly creae 100 differen pariins. Fr each pariin, we esimae he number f facrs. 15 We als perfrmed he same experimen using all pssible pariins f prfli reurns in w grups. Resuls d differ significanly wih he nes presened jus using 100 randm specificains f he grups. 19

22 Resuls are presened in able 4. Since he mdel selecin crierin mehd appears be superir he sequenial mehd, we nly repr resuls frm he frmer mehd. When he SNR f he secnd cmmn cmpnen is equal 1 (Panels A), mdel selecin crierin mehd esimaes ms repeaedly hree facrs wih any penaly crierins fr N= 12, 15 and 25. he nly excepin is he case BIC1 when N=25 which cnsisenly esimaes 0 facrs fr all values f SNR, N and in able 4. Fr he case f BIC1 he highes frequencies are n larger ha 56%. he secnd highes relaive frequencies fr BIC1 crrespnd w facrs, wih values beween 42%-46%. Fr he case f BIC2 highes frequencies (crrespnding hree facrs) are bigger han 80%, bu BIC2 veresimaes he facrs 8-9% f he ime. hese resuls sugges ha BIC1 is very successful esimaing srng facrs wih very small prbabiliy f veresimain. In he her hand BIC2 is ms successful esimaing weak facrs, bu has higher changes ver esimae he rue number f facrs. In any case HS esimar will predic 3 facrs wih all crierins and fr any daa specificain when facrs are srng. When he SNR drps 0.5, similar endency is bserved. BIC1 esimaes msly w facrs, capuring he weak facr jus 12-15% f he ime. BIC3 behaves very similar han BIC1. BIC2 sill esimaing hree facr wih he highes frequency. AIC crierin is he wrse since ver esimaes he facrs beween 15-26% f he ime. he secnd highes frequency is eiher 2 r 3 facrs fr all BIC crierins (excep BIC1 N=25). Finally wih SNR equal 0.25 all crierins esimae 2 facrs wih he highes frequency and he secnd highes frequency crrespnds 3 facrs. Again BIC2 seems capure he hird facr wih a higher frequency. While i is smewha arbirary, given hese resuls, we define he facr wih SNR smaller han r equal 0.25 as a weak facr. Our simulain resuls shw ha when a weak facr is presen in daa, ur HS esimar underesimaes he rue number f facrs. ha is, he HS esimar has nly limied pwer deec he facrs wih SNR smaller han r equal Hwever, he rue number f facrs ( = 3) is esimaed wih he secnd highes frequency. Als, we can cnclude ha esimain resuls using BIC1 are differen cmpared wih BIC2 when he facr becmes weaker. he HS esimar appears be a reliable esimar unless weak facrs are presen in daa. In rder cnfirm his fac, we carry u he same experimen we cnduc befre: we generae ne se f prfli reurns, and hen, we randmly creae 100 differen pariins. Fr 20

23 each pariin, we esimae he number f facrs by he HS mehd. We repea his experimen fr 1,000 generaed samples wih fur differen SNRs f he secnd cmmn cmpnen: 1.0, 0.5 and Resuls are presened in able 5. When SNR is equal 1 using BIC2, BIC3 and AIC we esimaed 3 facrs alms 100% f he imes fr all daa specificains. BIC1 underesimaes he number f facrs 35% f he imes. his is cnsisen wih he resuls in able 4, since he secnd highes frequency was as high as 46% percen fr BIC1. As SNR decreases, ur mehd esimaes hree facrs less fen and w facrs mre fen. Fr example, using BIC2 if SNR = 0.5 and N = 12, hree facrs are esimaed wih he highes frequency (73.0%), bu we als esimae w facrs 27.0% f he ime. When N = 12 and SNR = 0.25, we esimae w facrs 97.0% f he ime wih BIC2, and he hree facrs jus 3% f he ime. Resuls using BIC1 and BIC3 are even mre exreme, since HS esimar predics 2 facrs 100% f he ime even fr signal nise equal hus, he HS esimar is dwnward biased in he presence f weak facrs. king he secnd highes frequency i is impran, since i can be viewed as an evidence f a weak facr. Hwever, he secnd highes frequency fr BIC2 shuld be used jus if is greaer ha 15-20%, since as shwed in ables 2 and 3 BIC2 can veresimae he number f facrs arund 10-15% f he ime. he main resuls frm ur simulain culd be summarized as fllws. Firs, he mdel selecin crierin mehd appears be superir he sequenial esing mehd. Anher advanage f using he frmer mehd is ha i des n require use f pimal weighing marix. BIC crierin perfrm beer ha AIC as prediced by hery, and beween hem BIC1 is he ms accurae esimaing srng facrs. Secnd, using ur mehd, researchers shuld esimae he number f facrs wih differen specificains f w grups. 100 randmly generaed pariins appear be enugh bain a reliable esimae. he HS esimar bained frm 100 randm pariins perfrms quie well if here is n weak facr under any crierin, and in his case all BIC crierins esimae he same number f facrs wih alms equal frequency. hird, in he presence f weak facrs he HS mehd wih BIC1 may lead underesimain. BIC2 ends capure weaker facrs, hen resuls using bh crierins shuld be cmpared. If resuls frm BIC1 and BIC2 are significanly differen, hen he secnd highes frequency higher han 15% culd be viewed as an evidence f a weak facr. In such cases, if esimaed number wih he secnd highes frequency is greaer han he HS esimae, hen ha number is likely be he crrec number f facrs. 21

24 6. EMPIRICA APICAION We use he develped mehdlgy analyze he inernainal sck reurn cmvemens. In he las years markes have becme mre inegraed a a wrld level hrugh increased capial and rade inegrain. inear facrs mdels are a generalized mehdlgy analyze cmvemens in inernainal sck markes, given heir parsimny and pracical naure. Sme examples are he wrld CAPM, he Fama-French 1998 hree facr mdel, he wrld AP and he Hesn and Ruwenhrs (1994) mdel. he main differences beween hese mdels are hw many facrs are included, and hw he facrs are esimaed r cnsruced. Our empirical applicain has w main bjecives. Firs, esimae he number facrs ha explains he inernainal sck marke. Secnd, in an effr es marke inegrain, we wan measure he level f crrelain beween US sck marke facrs and he wrld laen facrs. Bh quesins can be answered using he GMM mehdlgy prpsed in his paper. A deailed and careful sudy f he perfrmance f differen facr mdels in inernainal sck reurns can be fund in Bekaera, Hdricka and Zhang 2005 (BHZ). Based n cunry prflis hey analyze he explanary pwer f he differen facrs mdels. hey cnclude ha an AP mdel accmmdaing glbal and lcal facrs, bes fis he cvariance srucure. Als a facr mdel ha embeds bh glbal and reginal Fama-French (1998) facrs cmes prey clse in perfrmance. he sudy f BHZ, as several hers, includes nly infrmain f sck markes f develped cunries. his is bvius given daa availabiliy and qualiy and he research quesin hey ry answer. he use f such daa ses, leads include a large number f Eurpean cunries in he sample, and a very few number frm he Americas and Africa. In his case any facr mdel esimain will exacerbae he effec f reginal (Eurpean) facrs. Wih his mivain we prpse use daa n sck markes keeping he reginal prprins cnsan. ha is, we ry include he same number f cunries frm every regin. he GMM mehdlgy prpsed in his paper is very adequae in his case since very few bservains can be bained frm regins like Africa, Suh America r he Middle Eas. We divide he wrld in fur regins: Americas, Eurpe, Asia and Africa-Middle Eas. Our daa se includes reurns f sck marke indexes fr he fur regins as fllws: 22

25 Americas: US (S&P500), Mexic (IPC), and Brazil (Bvespa). Eurpe: UK(FSE), Germany (DAX), and Swizerland (SMI). Asia: Japan (Nikkei225), Hng Kng (Hang Seng) and Krea (Kspi). Africa-Middle Eas: Israel (ASE100) and Suh Africa (Dw Jnes ians 30). 16. We use 420 weekly excess 17 reurn bservains f he cunry indexes frm January 2000 January he validiy f using index daa represen he marke perfrmance individual cunries can quesinable. Differences in mehdlgies and number f cmpanies included in each index can lead esimain prblems. We acknwledge his fac, bu since ur bjecive is n esimae facrs f risk premiums, we believe ha ur index daa can be safely used as a gd prxy fr he cunry prflis. Als, given he illusraive characer f ur empirical applicain, he presen sudy can by exended wih a richer daa se. Descripive saisics f ur daa is presened is able 6. Crrelains f sck reurns wihin cunries in he same regin are high. Als, crrelains beween develped cunries seem be high. Mean and variances als differ acrss cunries and regins. Resuls f esimain f he number f facrs are presened in able 7. We use he mdel selecin mehd and repr resuls using he penaly crierins BIC1 and BIC2. As explained befre, we firs pariin he N respnse variables in w randmly seleced grups f N/2 elemens. We esimae he number f facrs and repea he esimain fr 100 differen randm pariins. We repr he relaive frequency (percenage) fr each number f facrs. Panel A shws esimain resuls including all 11 cunries in r sample. BIC1 predics 1 facr wih mre he 90%. BIC2 predics 2 facrs 34% f he ime and 3 facrs 46% f he ime. As we shwed in simulains BIC2 is able capure weaker facrs, and BIC1 is he ms cnsisen idenifying srng facrs. We can cnclude hen ha ne srng facr explains he inernainal sck marke reurns. here is als evidence f w weak facrs in sense ha hey jus explain an small fracin f he variance f he respnse variable. he weak facrs can als be inerpreed as lcal r regin specific facrs. In rder furher analyze his pssibiliy we esimae he number f facrs fr differen sub-ses f he 16 All daa cmes frm Yah Finance excep Dw Jnes ians 30 bained frm Dw Jnes web sie. 17 he risk free ineres ra is he ne mnh reasury bill rae frm K French nline daa library. 23

26 respnse variable. he inuiin fr he fllwing analyses is ha, if we remve sme regin r cunry frm he sample and he weak facr becmes srnger (in he sense ha is ms frequenly esimaed), hen his facr is n relaed wih he excluded regin. Similarly if afer we exclude sme regin frm he sample, he weak facr becmes weaker (i.e. i is esimaed wih lwer frequency), hen his weak facr is highly relaed wih he excluded regin. Panel B includes resuls excluding US sck marke index frm he sample. Resuls are very similar cmpared wih Panel A, s i appears ha he weak facrs are n specific US marke. Similar experimen is presened in Panel C, bu nw we exclude he 3 indexes f he regin Americas. BIC1 and BIC2 nw esimae 2 facrs wih he highes frequency, implying ha remving Americas frm he sample increase he explanary pwer f ne f he weak facrs. We cnclude ha he firs weak facr is n relaed he regin Americas. In he her hand, Panel C als shws ha he hree facrs are esimaed wih smaller frequency nw frm 46% in Panel A 28% in Panel C using BIC2. his implies ha he secnd weak facr becmes weaker, implying ha i is relaed he regin Americas. When we exclude Eurpe (Panel D) ne f he weak facrs ally disappear, since we d n esimae 3 facrs under any crierin. his is clear evidence ha ne he weak facrs is msly a Eurpean facr. Panel E include resuls excluding Asia, resuls are n much differen cmpared wih Panel A. Finally when remve Africa frm he sample (Panel F) we esimae hree facrs mre fen, which implies ha he weak facr is n relaed wih Africa. Summarizing we find ne srng glbal facr. Als here is evidence f a w weak facrs ne relaed msly wih Eurpean markes and he her wih he Americas marke. he secnd par f ur sudy analyzes he relain f US marke facrs wih he glbal facrs. Under marke inegrain i is inuiive hink ha US marke facrs can be crrelaed in sme degree wih he glbal facrs. We d n aemp esimae neiher he number f facrs nr he facrs fr he US sck marke. We raher use he ms widely used prxy fr he US sck marke facrs, he Fama-French benchmark facrs. Descripive saisics fr weekly bservains f EMR (excess marke reurn), SMB, and HM are presened in able As explained in secin 4.2 if we use bservable insrumens (in his case EMR, SMB, and HM 18 Daa fr Fama-French facrs cmes frm K. French nline daa library. 24