Testing for Alpha in Linear Factor Pricing Models with a Large Number of Securities

Transcription

1 esng fo Alpa n Lnea Faco Pcng Models w a Lage umbe of Secues M. Hasem Pesaan Depamen of Economcs & USC Donsfe IE, Unesy of Souen Calfona, USA, and ny College, Cambdge akas Yamagaa DERS, Unesy of Yok and ISER, Osaka Unesy 3 Januay 8 Absac s pape consdes ess of zeo pcng eos fo e lnea faco pcng model wen e numbe of secues,, can be lage elae o e me dmenson,, of e eun sees. We focus on class of ess a ae based on Suden ess of nddual secues wc ae a numbe of adanages oe e exsng sandadsed Wald ype ess, and popose a es pocedue a allows fo non-gaussany and geneal foms of weakly coss coelaed eos. I does no eque esmaon of an neble eo coaance max, s muc fase o mplemen, and s ald een f s muc lage an. Mone Calo edence sows a e poposed es pefoms emakably well een wen = 6 and = 5;. e es s appled o monly euns on secues n e S&P 5 a e end of eac mon n eal me, usng ollng wndows of sze 6. Sascally sgn can edence agans Sape-Lnne CAPM and Fama-Fenc ee faco models ae found manly dung e ecen nancal css. Also we nd a sgn can negae coelaon beween a wele-mons mong aeage p-alues of e es and excess euns of long/so equy saeges (elae o e eun on S&P 5) oe e peod oembe 994 o June 5, suggesng a abnomal po s ae eaned dung epsodes of make ne cences. JEL Class caon: C, C5, C3, G, G Keywods: CAPM, esng fo alpa, Weak and spaal eo coss-seconal dependence, S&P 5 secues, Long/so equy saegy. e s auo acknowledges paal suppo fom e ESRC Gan o. ES/I366/.

2 Inoducon s pape s concened w esng fo e pesence of alpa n Lnea Faco Pcng Models (LFPM) suc as e capal asse pcng model (CAPM) due o Sape (964) and Lnne (965), o e Abage Pcng eoy (AP) model due o Ross (976), wen e numbe of secues,, s que lage elae o e me dmenson,, of e eun sees unde consdeaon. e Sape-Lnne CAPM model pedcs a expeced excess euns (measued elae o e sk-fee ae) on any gen secuy o a gen pofolo of secues s popoonal o e expeced excess eun on e make pofolo, w e consan of e popoonaly,, beng secuy/pofolo spec c. ee exss a lage leaue n empcal nance a ess aous mplcaons of Sape- Lnne model. Coss seconal as well as me sees ess ae been poposed and appled n many d een conexs. Usng me sees egessons, Jensen (968) was e s o popose usng sandad -sascs o es e null ypoess a e necep,, n e Odnay Leas Squaes (OLS) egesson of e excess eun of a gen secuy,, on e excess eun of e make pofolo s zeo. e es can be appled o nddual secues as well as o pofolos. Howee, wen a lage numbe of secues ae unde consdeaon, due o dependence of e eos acoss secues n e LFPM egessons, e nddual -sascs ae coelaed wc makes conollng e oeall sze of e es poblemac. Gbbons, Ross and Saken (989, GRS) popose an exac mulaae eson of e es wc deals w s poblem f e CAPM egesson eos ae Gaussan and <. s s e sandad es used n e leaue, bu s applcaon as been con ned o esng e make e cency of a elaely small numbe of pofolos, ypcally 3, usng monly euns obseed oe elaely long me peods. e use of lage as a way of ensung a <, s also lkely o ncease e possbly of sucual beaks n e s a could n un adesely a ec e pefomance of e GRS es. Recenly, ee as been a gowng body of nance leaue wc uses nddual secuy euns ae an pofolo euns fo e es of pcng eos. Ang, Lu and Scwaz (6) sow a e smalle aaon of bea esmaes fom ceang pofolos may no lead o smalle aaon of coss-secon egesson esmaes. Cemes, Hallng and Wenbaum (5) examne e pcng of bo aggegae jump and olaly sk based on nddual socks ae an pofolos. Coda, Goyal and Sanken (5) adocae e use of nddual secues o nesgae wee e souce of expeced eun aaon s fom beas o secuy-spec c caacescs. I s clealy desable o deelop ess of make e cency a can deal w a lage numbe of secues oe elaely so me peods so a e poblem of me aaons n s s somewa mgaed. I s also mpoan a suc ess ae easonably obus o non-gaussan eos, paculaly as s moe lkely a one would encoune non-nomal eos n e case of LFPM egessons fo nddual secues as compaed o egessons esmaed on pofolos compsng a lage numbe of secues. Ou of e wo man assumpons a undele e GRS es, e leaue as focussed on e mplcaons of non-nomal eos fo e GRS es, and ways of allowng fo non-nomal eos wen esng =. A eck-gaes and McDonald (989) wee amongs e s o consde e obusness of e GRS es o non-nomal eos wo, usng smulaon ecnques, Coss seconal ess of CAPM ae been consdeed by Douglas (968), Black, Jensen and Scoles (97), and Fama and Macbe (973), among oes. An ealy eew of e leaue can be found n Jensen (97), and moe ecenly n Fama and Fenc (4).

3 nd a e sze and powe of GRS es can be adesely a eced f e depaue fom nonnomaly of e eos s seous, bu conclude a e GRS es s ".. easonably obus w espec o ypcal leels of nonnomaly." (p.889). Moe ecenly, Beauleu, Dufou and Kalaf (7, BDK) and Gungo and Luge (9, GL) ae poposed ess of = a allow fo non-nomal eos, bu ean e escon <. BDK deelop an exac es wc s applcable o a wde class of non-gaussan eo dsbuons, and use Mone Calo smulaons o acee e coec sze fo e es. Gungo and Luge (9) popose wo dsbuon-fee nonpaamec sgn ess n e case of sngle faco models a allow e eo dsbuon o be non-nomal bu eque o be coss-seconally ndependen and condonally symmecally dsbued aound zeo. Ou pmay focus n s pape s on deelopmen of mulaae ess of H : = ; fo = ; ; :::;, wen >, wls allowng fo non-gaussan and weakly coss-seconally coelaed eos. e lae condon s equed fo conssen esmaon of e eo coaance max, V, wen s lage elae o. In e case of LFPM egessons w weakly cossseconally coelaed eos, conssen esmaon of V can be aceed by adape esoldng wc ses o zeo elemens of e esmao of V a ae below a gen esold. Alenaely, feasble esmaos of V can be obaned by Bayesan o classcal snkage pocedues a scale down e o -dagonal elemens of V elae o s dagonal elemens. 3 Fan, Lao and Mncea (, 3) consde conssen esmaon of V n e conex an appoxmae faco model. ey assume V s spase and popose an adape esoldng esmao of V, wc ey sow o be pose de ne w sasfacoy small sample popees. Fan, Lao and Yao (5) dee e condons unde wc sandadsed Wald ess of H can be asympocally jus ed. Gagladn, Ossola and Scalle (6) deelop wo-pass egessons of nddual sock euns, allowng me-ayng sk pema, and popose a sandadsed Wald es. Rapon, Robo and Za aon (6) popose a es of pcng eo n coss-secon egesson fo xed numbe of me sees obseaons. ey use a bas-coeced esmao of Saken (99) o sandadse e es sasc. Gungo and Luge (6) popose a smulaon based appoac fo esng pcng eos. ey clam a e es pocedue s obus agans non-nomaly and coss-seconal dependence n e eos. Amengual and Repeo (4) consde e sandadsed F-ype es sasc based on pncpal componen esmaon unde bo seal and coss-secon coelaon n e eos. In s pape we follow an alenae saegy wee we deelop a es sasc a nally gnoes e o -dagonal elemens of V and base e es of H on e aeage of e ess of =, oe = ; ; :::;. s dea was ognally poposed n e wokng pape eson of s pape (Pesaan and Yamagaa, ), ndependenly of a smla appoac followed by Gagladn, Ossola and Scalle (6; GOS). 4 Despe e smlay of e wo ess, as wll be Bossaes, Plo and Zame (7) pode a noel GMM es of CAPM wc does no eque lage, bu s desgned fo e analyss of expemenal daa on a few sky asses eld acoss a elaely lage numbe of subjecs. I s neesng o see f e appoac can be adaped o e analyss of socal obseaons of e ype consdeed n s pape. 3 ee exss a lage leaue n sascs and economecs on esmaon of g-dmensonal coaance maces wc use egulazaon ecnques suc as snkage, adape esoldng o oe dmenson-educng pocedues a mpose cean sucues on e aance max suc as spasy, o faco sucues. See, fo example, Wong, Cae and Kon (3), Ledo and Wolf (4), Huang, Lu, Pouamad, and Lu (6), Bckel and Lena (8), Fan, Fan and L (8), Ca and Lu (), Fan, Lao and Mncea (, 3), and Baley, Pesaan and Sm (7). 4 We ae gaeful o Ole Scalle fo dawng ou aenon o an eale eson of GOS (6), afe e wokng pape eson of s pape was publcly eleased n, and pesened a e Amecan Fnance Assocaon Meeng n San Dego, Januay 3. ps://q.ssn.com/confeence/repos/conf_pelmnay_pogam.cfm?con nk=afa-3-san-dego

4 seen ou eson of e es pefoms muc bee een f s ey lage (aound 5; ), and we ae able o esabls s asympoc dsbuon unde muc weake condons and wou esong o g leel assumpons. We acee s by makng coecons o e numeao of e es sasc o ensue a e es s moe accuaely ceneed, and coec e denomnao of e es sasc o allow fo e e ecs of non-zeo o -dagonal elemens of e undelyng eo coaance max. 5 e coecon noles conssenly esmang (R ), wee R = j s e eo coelaon max. e esmaon of (R ) = P P j= j s subjec o e cuse of dmensonaly wc we addess by usng e mulple esng esold esmao, ~R, ecenly poposed by Baley, Pesaan and Sm (7). We sow a conssen esmaon of (R ) can be aceed unde a moe geneal spec caon of R as compaed o ess a eque a conssen esmao of e full max, R. We ae able o esabls a e esulan es s applcable moe geneally and connues o be ald fo a wde class of eo coaances, and olds een f ses fase an. e poposed es s also coeced fo small sample e ecs of non-gaussan eos, wc s of pacula mpoance n nance. We efe o s es as Jensen s es of LFPM and denoe by ^J. e es can also be ewed as a obus eson of a sandadsed Wald es, n cases wee e o -dagonal elemens of V become elaely less mpoan as. e mplemenaon of e ^J es s also compuaonally less demandng, snce does no nole esmaon of an neble g dmensonal eo coaance max. Ou assumpon egadng e spasy of V adances on Cambelan s (983) appoxmae faco model fomulaon of e asse model, wee s assumed a e lages egenalue of V (o R) s unfomly bounded n (Cambelan, 983, p.37). We elax s assumpon pand allow e maxmum column sum max nom of R o se w bu a a ae slowe an, wls conollng e oeall spasy of R by equng (R ) o be bounded n. In s way we ae able o allow fo wo ypes of coss-seconal eo dependence: one due o e pesence of weak common facos a ae no su cenly song o be deecable usng sandad esmaon ecnques, suc as pncpal componens; and anoe due o e eo dependence a ase fom neace and spll-oe e ecs. We esabls a unde e null ypoess of = ; e ^J es s asympocally dsbued as (; ) fo and jonly, so long as =, m = krk = O, < =, and (R ) s bounded n. e es s also sown o ae powe agans alenaes a ses n =. e poofs ae que noled and n some pas ae edous. Fo e pupose of clay we pode saemens of e man eoems w e assocaed assumpons n e pape, bu elegae e maemacal deals o an appendx. Small sample popees of e ^J es ae nesgaed usng Mone Calo expemens desgned spec cally o mac e coelaons, olales, and oe dsbuonal feaues (skewness and kuoss) of e esduals of Fama-Fenc ee faco egessons of nddual secues n e Sandad & Poo 5 (S&P 5) ndex. We consde e compaae es esuls fo e followng eg sample sze combnaons, = 6 and ; and = 5; ; and 5. e ^J es pefoms well fo all sample sze combnaons w sze ey close o e cosen nomnal alue of 5%, and sasfacoy powe. Compang e sze and powe of e ^J es w e GRS es n e case of expemens w = 5 < = 6, fo wc e GRS sascs can be compued, we nd a e ^J es as ge powe an e GRS es n mos expemens. s could be due o e non-nomal eos adesely a ecng e GRS es, as epoed by A eck-gaes and McDonald (989, 99). In addon, e ^J es oupefoms e es poposed by GOS as well as e feasble esons of e sandadsed Wald ess, eplacng V 5 s coecon and ow s esmaed uns ou o be ccal fo e small sample popees of e es wen e eos n e nddual eun egessons ae weakly coss coelaed. 3

5 w e ecenly deeloped esmaos of lage dmensonal aance-coaance max of Fan, Lao and Mncea (3, FLM) and Ledo-Wolf (4). e ^J es also oupefoms e smulaon-based F max es of Gungo and Luge (6) a can be mplemened wen >. e F max es s sown o be undeszed subsanally acoss e aous desgns, and as lowe powe unfomly as compaed o e ^J es. We also caed ou addonal expemens a allow fo me aaons n beas as well as eos w a mxue of weak facos and spaal auoegesse pocesses, usng muc lage alues of, namely = ; ; ; and 5; ; wls keepng a 6 and. We only consdeed e ^J es fo ese expemens, and found no majo edence of sze dsoons een fo e expemens w = 6 and = 5;. Encouaged by e sasfacoy pefomance of e ^J es, een n cases wee s muc lage an, we appled e es o monly euns on e secues n e Sandad and Poo (S&P) 5 ndex usng ollng wndows of sze 6 oe e peod Sepembe 989 o June 5. e suosp bas poblem s mnmzed by consdeng e sample of secues ncluded n e S&P 5 a e end of eac mon n eal me. We epo e ^J es sascs fo a sngle-faco and a ee Fama-Fenc faco model oe e peod 989-5, and nd sascally sgn can edence agans e Sape-Lnne CAPM and Fama-Fenc faco model only dung e ecen nancal css. Fnally, we examne f ee exss any elaonsp beween e p-alues of e ^J es and excess euns on long/so equy edge funds (elae o e eun on S&P 5). A po one would expec a eese elaonsp beween make e cency and excess euns of an nesmen saegy, w excess euns beng low dung peods of make e cency (g p-alues) and ce esa. In fac, we nd a sgn can negae coelaon beween a wele-mons mong aeage p-alues of e ^J es and excess euns of long/so equy saeges oe e peod oembe 994 o June 5, suggesng a abnomal po s ae eaned dung epsodes of make ne cences. e oulne of e es of e pape s as follows. Secon ses ou e panel daa model fo e analyss of LFPM, and e GRS es. Secon 3 poposes e ^J es fo lage panels, dees s asympoc dsbuon, and Secon 4 summases e man eoecal esuls. Secon 5 epos on small sample popees of ^J, GRS, GOS, sandadsed Wald ess and e Gungo and Luge (6) smulaon based F max es, usng Mone Calo ecnques. Secon 6 pesens e empcal applcaon. Secon 7 concludes. e poofs of e man eoems ae poded n Appendx A, and e lemmas wc ae used fo e poofs, as well as e addonal Mone Calo edence, ae poded n an onlne supplemen o s pape, a s aalable on eques. oaons We use K and c o denoe ne and small pose consans. If ff g = s any eal sequence and fg g = s a sequences of pose eal numbes, en f = O(g ), f ee exss a pose ne consan K suc a jf j =g K fo all. f = o(g ) f f =g as. If ff g = and fg g = ae bo pose sequences of eal numbes, en f = (g ) f ee exss and pose ne consans C and C, suc a nf (f =g ) C, and sup (f =g ) C. Fo a max A = (a j ), e mnmum and maxmum egenalues of max A ae denoed by mn (A) and max (A), especely, s P ace by (A), s maxmum absolue column and ow sum max noms by kak = sup j= ja P jj, and,kak = sup j ja jj, especely, s Fobenus and specal noms by kak F = p (A A), and kak = max(a = A), especely. Fo a dmensonal eco,, kk = ( ) =. 4

6 Some pelmnaes and e GRS es Unde e Abage Pcng eoy (AP) of Ross (976), we ae R = + + (f f ) + u ; fo = ; ; :::; ; = ; ; :::; ; () wee, R s eun on secuy dung peod, f = (f ; f ; :::; f m ) s e m eco of facos, = ( ; ; :::; m ) s e assocaed eco of sk facos, and s zeo-bea expeced eun wc unde AP sould be equal o e sk-fee ae, s e eco of expeced coss-seconal sk pemum and f = E (f ). Seng = +, wee s e sk-fee ae, e eun egessons can be wen as wee y = R y = + f + u ; fo = ; ; :::; ; = ; ; :::; ; (), and = + ( f ): (3) o ensue a e sk fom common facos, f, canno be fully des ed we assume a a leas one of e facos s song, n e sense a sup s j s j = O(), (4) and allow fo e pesence of common unobseed weak facos n e eo em u. Spec cally we assume a u = + ; (5) wee s a k eco of unobseed common facos a ae IID(; I k ), = ( ; ; :::; k ) s e assocaed eco of faco loadngs w bounded elemens, sup ;s j s j < K. e facos ncluded n e eo pocess ae weak n e sense a e e ecs ae no pease and sasfy e condon sup s j s j = O ; w < =: (6) e dosyncac eos,, ae also allowed o be weakly coss coelaed. Spec cally, we assume a = ( ; ; ::::; ) = Q " ;, wee " ; = (" ; ; " ; ; ::::; " ; ), f" ; g ae IID pocesses oe and, w means zeo, un aances, ;" = E " ; 4 3, and sup; E(j" ; j 8+c ) K <, fo some c >. We denoe e coelaon max of by R = ;j, and noe a R = Q Q. o ensue a u = (u ; u ; :::; u ) s weakly coss-coelaed we eque a k, e numbe of weak facos, s ne, and kr k kq k kq k K. e eo spec caon n (5) s que geneal and allows fo common facos as well as newok and spaal eo coss dependence, so long as e common facos ae su cenly weak. D een ess of LFPM ae poposed n e leaue. Some eseaces ae focussed on esng =, wc ensues a e zeo-bea excess eun s zeo. Oes ae consdeed esng e escons = f, wc eque a e sk-pema on facos concde w faco means. 6 In s pape we adop a moe dec appoac and consde esng e jon ypoeses H : =, = ; ; :::; ; (7) allowng fo e mulple esng naue of e null. In e conex of e AP model, e es of = fo all can be nepeed as a es of e jon ypoeses a =, and = f. 6 See, fo example, Sanken (99). 5

7 I poes useful o sack e panel egessons n () by me sees as well as by coss secon obseaons. Sackng by me sees obseaons we ae y : = + F + u : ; (8) wee y : = (y ; y ; :::; y ), = (; ; :::; ), F = (f ; f ; :::; f ), and u : = (u ; u ; :::; u ). Sackng by coss-seconal obseaons we ae y = + Bf + u ; (9) wee y = (y ; y ; :::; y ), = ( ; ; :::; ), B = (, ; :::; ) and u = (u ; u ; :::; u ). Fo exac sample ess of LFPM, nally we assume a u s IID (; V), namely eos, u ; ae Gaussan, ae zeo means, and ae seally uncoelaed suc a E(u u j ) =, fo all, j;and 6=, w E (u u ) = V, wee V = ( j ) s an symmec pose de ne max. A non-gaussan eson of s assumpon wll be consdeed below. Sang w Jensen s (968) es of nddual s, we noe a e OLS esmao of gen by ^ = y: MF M, () F wee M F = I F (F F) F, s an e cen esmao despe e fac a V s no a dagonal max. s esul follows snce (8) s a seemngly unelaed egesson equaon (SURE) spec caon w e same se of egessos acoss all e secues. I s also easly seen a fo all = ; ; :::; ; ^ = ( + F + u MF :) M = + u F :c, () wee c = M F = M F : () Wng e aboe se of esmaes fo all n max noaon, we ae u :c u :c ^ = + B A ; u : c wee u :c = P = u c ; and c s e elemen of c. Hence ^ = + wee as befoe u = (u ; u ; :::; u ). eefoe, unde Gaussany, ^ ; M V : F u c, (3) Also n e case wee + m +, an unbased and neble esmao of V s gen by ( m ) ^V, wee ^V s e sample coaance max esmao = ^V = 6 = ^u ^u ; (4)

8 ^u = (^u ; ^u ; :::; ^u ), ^u s e OLS esdual fom e egesson of y on an necep and f. Unde Gaussany, ^u as a mulaae nomal dsbuon w zeo means, ^ and ^u ae ndependenly dsbued, and ence usng sandad esuls fom mulaae analyss follows a (see, fo example, eoem 5.. n Andeson (3)) e GRS sasc (see p.4 of GRS) GRS = ^W = m M F ^ ^V ^; (5) s dsbued exacly as a non-cenal F dsbuon w ( m) and degees of feedom, and e non-cenaly paamee = m M F V, wc s zeo unde H : =. 7 As noed n e noducon, e sngle mos mpoan lmng feaue of e GRS and oe elaed ess poposed n e leaue s e equemen a mus be lage an. o ccumen s lmaon, n applcaons of e GRS es, nddual secues ae gouped no (sub) pofolos and e GRS es s en ypcally appled o -3 pofolos oe elaely long me peods. Howee, s clealy desable o deelop ess of =, a can be appled o a ey lage numbe of nddual secues oe elaely so me peods (o mnmze e adese e ecs of sucual cange n s) wc neably lead o cases wee <. Een n cases wee <, e powe of e GRS es could be compomsed snce assumes V o be unesced, wls n e conex of e appoxmae faco model adanced n Cambelan (983), e eos ae a mos weakly coelaed, wc places escons on e o -dagonal elemens of V and s nese. As we sall see below, a es a explos escons mpled by e weak coss-seconal coelaon of e eos s lkely o ae muc bee powe popees an e GRS es a does no make use of suc escons. I s also mpoan o bea n mnd a beng a mulaae F es, e powe of e GRS es s pmaly den by e me dmenson,, wls fo e analyss of a lage numbe of asses o pofolos we need ess a ae e coec sze and ae poweful fo lage. 3 Lage ess of alpa n LFMP models o deelop lage ess of H : =, we consde e followng eson of e GRS sasc, as se ou n (5), W = ( M F ) ^ V ^; (6) wee we ae dopped e degees of feedom adjusmen em and eplaced ^V by s ue alue. W can be egaded as a Wald es sasc, and unde Gaussany and H : =, W. Snce e mean and e aance of a andom aable s and, one could consde a sandadsed Wald es sasc SW = ( M F ) ^ V ^ p : (7) Unde Gaussany and H : =, SW d (; ) as. o consuc ess of lage panels, a suable esmao of V s equed. Bu as was noed n e noducon s s possble only f we ae pepaed o mpose some escons on e sucue of V. In e case of LFPM egessons wee e eos ae a mos weakly coss-seconally coelaed, s can be 7 ong a ( + f ^ f) = ( M F ), wee f = P = f, and ^ = P = (f f)(f f), m s easly seen a (5) can be wen as e wdely used expesson of e GRS sasc, ( + f ^ f) ^ ^V ^. 7

9 aceed by adape esoldng wc ses o zeo elemens of V a ae su cenly small, o by use of snkage ype esmaos a pu a subsanal amoun of weg on e dagonal elemens of e snkage esmao of V. Fan, Lao and Mncea (, 3) consde conssen esmaon of V n e conex of an appoxmae faco model. ey assume V s spase and popose an adape esold esmao, ^V P OE, wc ey sow o be pose de ne w sasfacoy small sample popees. We efe o e feasble sandadzed Wald es sasc eplacng V w ^V P OE as SW P OE es. Anoe canddae s e snkage esmao of V poposed by Ledo-Wolf (4), wc we denoe by ^V LW, and efe o e assocaed sandadsed Wald sasc as SW LW. Suc "plug-n" appoaces ae subjec o wo mpoan socomngs. Fs, een f V can be esmaed conssenly, e es mg pefom pooly n e case of non-gaussan eos. oce a e sandadsaon of e Wald sasc s caed ou assumng Gaussany. Fue, conssen esmaon of V n e Fobenus nom sense sll eques o se fase an, and n pacce esold esmaos of V ae no guaaneed o be neble fo ne samples wee >>. 3. A ^J es fo lage secues o aod some of e aboe menoned lmaons of e plug-n pocedue, we aod usng an esmao of V alogee and base ou poposed es on dagonal elemens of V, namely e dagonal max, D = dag( ; ; :::; ), w = E (u ), ae an e full coaance max. Spec cally, we consde e sasc and s feasble counepa gen by W d = ( M F ) ^ D ^ = ( M F ) ^W d = ( M F ) ^ ^D ^ = M F ^ ; (8) ^ ; (9) ^ wee ^ = ^u :^u : =, and e degees of feedom = m s noduced o coec fo small sample bas of e es. 8 e nfeasble sasc, W d, can also be wen as W d = z ; () wee I s en easly seen a z = ^ ( M F )=. () ^W d = ; () wee denoes e sandad -ao of n e OLS egesson of y on an necep and f ; namely = ^ ( M F ) ^ : (3) 8 Only secues w ^ > ae ncluded n ^W d. 8

10 As w e panel esng saegy deeloped n Im e al. (3), a sandadzed eson of ^W d, de ned by (9), can now be consdeed: = ^Wd E ^Wd, (4) V a ^Wd wee E ^Wd = E ; (5) P V a ^Wd = V a = P V a P + P = j= Co ; j. (6) Unde Gaussany, e nddual sascs ae dencally dsbued as Suden w degees of feedom, and we ae (assumng = m > 4) E( ) =, V a( ) = ( ) 4. (7) Usng (5), (6) and (7), e sandadzed sasc (4) can now be wen as J = ^Wd E ^Wd = = P = q ( ) ; (8) V a ^Wd + 4 wee and = P P = j= Co ; j ; (9) Co( ; j) = Co( ; j)=[v a( )V a( j)] = : o make e J es opeaonal, we need o pode a lage conssen esmao of. Second, we need o sow a, despe e fac a J es s sandadsed assumng as a sandad dsbuon, e es wll connue o ae sasfacoy small sample pefomance een f suc an assumpon does no old due o e non-gaussany of e undelyng eos. Moe fomally, n wa follows we elax e Gaussany assumpon and assume a u = Q", wee Q s an neble max, " = (" ; " ; :::; " ), and f" g s an IID pocess oe and, w means zeo and un aances, and fo some c >, E(j" j 8+c ) exss, fo all and. en E (u u ) = V = ( j ) = QQ ; and V s an symmec pose de ne max, w mn (V) c >. We allow fo coss-seconal eo eeoskedascy, bu assume a e eos ae omoskedasc oe me. s assumpon can be elaxed by eplacng e assumpon of eo ndependence by a suable mangale d eence assumpon. s exenson wll no be aemped n s pape Spasy condons on eo coelaon max As noed aleady, we adance on e leaue by allowng V = ( j ) o be appoxmaely spase. Equalenly, we de ne spasy n ems of e elemens of e coelaon max R = j, wee j = j = = = jj. We consde e followng wo condons m = max j = O( ), w < =; (3) P j= 9 We conduced an expemen w GARCH(,) eo and e edence suppos ou clam. e esuls ae epoed n able M6 of e onlne supplemen, wc s aalable upon eques. 9

11 and R = j = O () : (3) j= Unde (3), m s allowed o se w, bu a a slowe ae an =. Fo example, consde e case wee condon (3) apples o e s p ows of R (w p xed), and e es of e p ows of R ae absolue summable, namely j = O, fo = ; ; :::; p; j= j = O(), fo = p + ; p + ; :::;. j= en, snce j j, eadly follows a R = p p j= j + j + j= =p+ j= j =p+ j= j O(p ) + ( p)o() = O(), fo < =: Anoe mpoan case coeed by ou spasy assumpon s wen u as e weak faco sucue gen by (5), w e faco loadngs,, sasfyng (6). Denong e coelaon max of e dosyncac eos, = ( ; ; :::; ) by R = ;j, and assumng a kr k < K; (3) we ae R = O(). I s now easly seen a condons (3) and (3) ae also sas ed unde s se up. Denong e coelaon max of u = (u ; u ; ::::; u ) by R = j we ae = j = ~ ; ;jj ~ j + ;j; (33) jj wee ~ = = = = = ( + ; ) =. Snce j P k s= j~ sj ~js + ;j, en m = krk = max j= k s= k sup j~ s j max ;s j~ s j ~js + max ;j j= ~js + kr k : j=

12 Snce sup ;s j~ s j sup ;s j s j, and sup s j= ~js sups j= js = O( ), and by assumpon kr k < K, e condon (3) s me f. Also, (nong a sup ;s j~ s j ) R = k s;s = j= j= k j~ s j ~js + ;j s= k j~ s j ~js + s= k j~ s j j~ s j + k + k sup s s= j= k j~ s j ~js + R s= j~ s j + R j s j + R : eefoe, unde condons (6) and (3), (R ) s bounded n f < =: Remak Ou assumpon of appoxmae spasy allows fo a su cenly g degee of coss eo coelaons, wc s mpoan fo e analyss of nancal daa, wee s no guaaneed a ncluson of common facos n e eun egessons wll oally elmnae weak eo coelaons due o spaal and/o wn seco eo coelaons. I s mpoan a bo faco and spaal ype eo coelaons, epesenng song and weak foms of nedependences, ae aken no accoun wen esng fo alpa. By allowng e eo em o nclude weak facos, one only need o focus on den caon of song facos o be ncluded n f, wc can be aceed by usng make facos o pncpal componens of nddual euns. e eo assocaed w e esmaon of song facos s lkely o be neglgble fo and su cenly lage. In e pesen pape we absac fom suc esmaon eos and condon ou analyss on gen alues of f. 3.3 on-gaussany Fo e dscusson of e e ecs of non-gaussany on e J es below, s conenen o noduce e followng scaled eo = u = =, (34) so a fo eac, as zeo mean and un aance. In e case wee e eos ae non-gaussan e skewness and excess kuoss of u ; ae gen by ; = E( 3 ) and ; = E( 4 ) 3, especely, a could d e acoss. oe a unde non-gaussan eos, s no longe Suden dsbued and E( ) and V ( ) need no be e same acoss, due o e eeogeney of ; and ; oe. Usng a slgly exended eson of e Laplace appoxmaon of momens of e ao of quadac foms by Lebeman (994), we ae able o dee e followng appoxmaons of E( ) and V a( ): E = + O 3= ; (35) oe also a e conssency of e plug-n pocedue poposed by Fan, Lao and Mncea (, 3) eques a song common facos ae emoed befoe esmaon of e eo coaance max, V. See Lemma n e onlne supplemen o e pape, wc s aalable upon eques.

13 and V a = ( ) ( 4) + O : (36) Subsung (35) and (36) no (4) we ae e followng non-gaussan eson of J, de ned by (8): P = J p= + O 3 = ( ) + O (, ( 4) ) + wee s de ned by (9). Wen e numeao of e J sasc s eplaced by P = ( ), wc s e ypcal mean adjusmen employed by Fan e al. (5) and Gagladn e al. (6), fo example, en e ode of e asympoc eo of e numeao of suc es sascs becomes p =. s s one of e easons wy ou poposed es pefoms bee an e ones poposed n e leaue, especally n cases wee >>, and ee ae sgn can ( ) depaues fom Gaussany. e asympoc eo of usng fo V a( ( 4) ) unde non-gaussany n e J es s O( ), wc s small fo su cenly lage. 3.4 Allowng fo eo coss-seconal dependence A second mpoan d eence beween e J es and e oe ess poposed n e leaue s e ncluson of n e denomnao of e es sasc o ake accoun of eo coelaons. As wll be sown moe fomally below, e lmng popey of s goened by e spasy of V, and s gen by 3 ( ) ; (37) as and ; so long as = ; and < =, wee = P P = j= ( ) j. (38) s known as e aeage pa-wse squaed coelaon coe cen and plays a key ole n ess of eo coss-seconal coelaons n panel egessons. See, fo example, Beusc and Pagan (98) and Pesaan, Ulla and Yamagaa (8). o see e elaonsp beween and e spasy of V, we noe a R = + P = P j= j = + ( ) ; wc n ew of (37) jus es eplacng + by (R ) fo and su cenly lage so long as = ; and < =. eefoe, gnong can lead o seous sze dsoons een fo lage and panels wen e eos ae coss-coelaed and (R ) does no end o zeo, snce e denomnao of J wll be unde-esmaed. e sze dsoon wll be pesen een f we mpose songe spasy condons on V, fo example, by equng m o be bounded n. I s, eefoe, mpoan a (o ) s eplaced by a suable esmao. One possble way of esmang would be o use sample coelaon coe cens, ^ j, de ned as Small sample edence on e e cacy of usng = P ^ j = ^ j =^ = ^ = jj ; (39) n able M3 of e onlne supplemen, wc s aalable upon eques. 3 (37) follows fom Lemma 8 n e Onlne Supplemen wc s aalable on eques. oe = P s epoed

14 wee ^ j = P = ^u ^u j, and ^u s e esduals fom e OLS egesson of y on G = ( ; F). Howee, suc an esmao s lkely o pefom pooly n cases wee s lage elae o, and some fom of esoldng s equed, as dscussed n e leaue on esmaon of lage coaance maces. 4 Hee we consde e applcaon of e mulple esng (M) appoac o egulasaon of lage coaance maces ecenly poposed by Baley Pesaan and Sm (7, BPS). Howee, BPS esabls e esuls fo y y, wls we need o apply e esoldng appoac o ^u. Second BPS consde exac spasy condons on e eo coaance max, wls we allow fo muc moe geneal spasy condons. We exend BPS s analyss o addess bo of ese ssues. 56 e mulple esng (M ) esmao of j, denoed by ~ j ; s gen by wee = m, ~ j = ^ j I p ^ j > cp () ; (4) c p () = p ; (4) f() p s e nomnal p-alue ( < p < ), and f() =, = c d d, wee c d, and d ae ne pose consans. Usng (4), e mulple esng esmao of s gen by ~ ; = P P = j= ( ) ~ j: (4) Unde e spasy condons (3) and (3), can be sown a ( ) ~ ; n pobably and n l -nom so long as =, (o equalenly f d > =) as and ; jonly, and f ( d) > ( ) ' max, (43), fo some small >, wee ' max + ;" wee ;" = E " 4 ; 3, " ; s e elemen of e eo eco " ; = Q, w = ( ; ; ::::; ). 7 e ccal alue funcon, c p () ; depends on e nomnal leel of sgn cance, p, and e coce of, subjec o condon (43). e es esuls ae unlkely o be sense o e coce of p, oe e conenonal alues n e ange of o pe cen. 8 d deemnes e elae expanson ae of and. e alue of ' depends on e degee of dependence of e eos een f ey ae uncoelaed. In e case wee e eos, " ;, ae Gaussan ;" = and ', and s su cen o se = d. Howee, n e non-gaussan case, and gen e edence poded by Longn and Solnk () and Ang, Cen and ng (6) on e degee of nonlnea dependence of asse euns, ge alues of mg be equed. In smulaons and empcal execses o be epoed below we se f () =, wc s equalen o seng =, wc could be oo low n cases wee s lage elae o. 9 4 See, fo example, Ca and Lu (), Fan e al. (3), Baley Pesaan and Sm (7), among oes. 5 Oe esoldng esmaos of V poposed n e leaue can also be used. e e cacy of usng e esmao ~ ; oe oe esmaos n small samples s nesgaed and e esuls ae summased n able M n e Onlne Supplemen (aalable on eques). 6 Gagladn, Ossola and Scalle (6) employ Bckel and Lena (8) esoldng (BL). e ne sample edence n BPS sows a e M esmao unfomly oupefom e BL n all e desgns consdeed n BPS. 7 See eoem 4 n Secon 4 and s poof n Appendx A. 8 In e Mone Calo expemens epoed below, we se p = %. 9 e obusness of e J a es agans non-gaussan and nonlnea eo dependence s nesgaed and epoed n able 4. ese esuls ae geneally suppoe of seng =. 3

15 Accodngly, we popose e followng feasble eson of e J sasc P ^J = = q ( ) ; (44) ( 4) + ( )~ ; wee s e -ao fo esng =, de ned by (3), = m, and ~ ; s gen by (4). e ^J es s obus o non-gaussan eos and allows fo a elaely g degee of eo coss-seconal dependence. In wa follows we pode a fomal saemen of e condons unde wc ^J ends o a nomal dsbuon. 3.5 e Gagladn e al (6, GOS) es GOS popose e followng sasc fo esng e ypoess of zeo pcng eo (GPS, p.8-9) GOS = P = ( ) q ; (45) + ( )^ BL wee ^ BL s an esmao of based on Bckel and Lena (8, BL) esold esmao of j. As noed n e Inoducon GOS s closely elaed o e ^J es sasc, bu also d es fom n a numbe of mpoan especs. Fs, GOS does no employ e degees of feedom adjusmen fo e sandadsaon of, wc we ae sown wll pode moe accuae nomal appoxmaon especally wen s muc lage an. Second, fo e esmaon of lage aance-coaance max, e edence n BPS sows a e M esmao oupefoms e BL esmao almos unfomly n e expemens, and ou use of M esmao of uns ou o yeld muc bee esuls. d, e BL esmaon eques coss-aldaon, wc can be compuaonally fa moe cosly an e M esmaon. Fnally, we dee lmng dsbuon of e ^J es sasc unde pme assumpons w faly geneal eo coaance sucue, wle GOS place g leel assumpon of asympoc nomaly of e es sasc (see e Assumpon A.5) o only consde a esce eo coaance sucue (see e Appendx F). We belee a ou eo spec caon s ald moe geneally n empcal asse pcng leaue wee no all facos can be den ed and esmaed, and n consequence one needs o allow fo a muc wde degee of eo coss coelaons o ake accoun of weak unobseed e ecs. 3.6 Suosp bas Fnally, s mpoan a e applcaon of e ^J es s no subjec o e suosp bas. e GRS ype ess of alpa consdes a elaely small numbe of pofolos oe a elaely lage me peod o acee su cen powe. By makng use of pofolos ae an nddual secues e GRS es s less lkely o su e fom suosp bas. By compason ess suc as e ^J es can su e fom e suosp bas due o e fac a ey ae appled o nddual secues decly and oban powe fom nceases n as well as fom. o deal w e suosp bas we popose a e ^J es s appled ecusely o secues a ae been adng fo a leas me peods (days o mons) a any gen me. e se of secues ncluded n e ^J es aes oe me and dynamcally akes accoun of ex and eny of secues n e make. e numbe of secues,, used n e es a any pon of Fo moe deals, see Secon M. of e onlne supplemen. See Assumpons BD.-3 n GOS. 4

16 me, ; depends on e coce of, and declnes as s nceased. I s clealy mpoan a a balance s suck beween and. Snce e ^J es s applcable een f s muc lage an, and gen a e powe of e ^J es ses bo n and, en s adsable o se suc a mn ( )= s su cenly small. s pocedue s followed n e empcal applcaon dscussed n Secon 6 below, wee we se = 6 and end up w n e ange [464; 487], gng mn ( )= = :. 4 Summay of e man eoecal esuls In s secon we pode e ls of assumpons and a fomal saemen of e eoems fo e sze and powe of e poposed ^J. Fs, we sae e assumpons fo esablsng e esuls. Assumpon : e m eco of common obseed facos, f, n e eun egessons, (), ae dsbued ndependenly of e eos, u fo all, and. e numbe of facos, m, s xed, and e facos can be song n e sense a sup s j s j = O( ); (46) and sasfy f f K < ; fo all. e (m + ) (m + ) max G G; w G = ( ; F) ; s a pose de ne max fo all, and as, and M F >, wee M F = I F (F F) F. Assumpon : e eos, u, n (), ae e followng mxed weak-faco spaal epesenaon u = + ; fo = ; ; :::; ; = ; ; :::; ; (47) wee = ( ; ; :::; k ) s a k eco of faco loadngs, = ( ; ; :::; k ) s a k eco of unobseed common facos and ae e dosyncac componens. () e unobseed facos, ae seally ndependen and e k elemens ae ndependen of eac oe, suc a s IID(; I k ), ; = E (s) 4 3, and sup s; E s 8+c < K, fo some c >. e faco loadngs, s fo s = ; ; :::; k, ae bounded, sup ;s j s j < K, and e facos,, ae weak n e sense a sup s j s j = O, w < =: (48) () Fo any and j, e pas of ealzaons, ; j ; ; j ; :::; ; j ; ae ndependen daws fom a common baae dsbuon w mean E ( ) = ; V a ( ) = ;, < c < ; K, and e coaance E j = ;j. Wng e eo faco spec caon, (47), n max noaon we ae u = + ; (49) wee u = (u ; u ; :::; u ), = ( ; ; :::; ), and = ( ; ; :::; ). Unde Assumpon, and denong E ( ) = V = ( ;j ), we ae E (u u ) = + V = V = ( j ); w j = j + ;j : (5) We now make e followng fue assumpon. 5

17 Assumpon 3: e coaance maces V and V de ned by (5) ae symmec, pose de ne maces w mn (V) mn (V ) c, " = (" ; " ; ::::; " ) = Q u, and " ; = (" ; ; " ; ; ::::; " ; ) = Q, (5) wee Q and Q ae e Colesky facos of V and V, especely. Max Q s ow and column bounded n e sense a kq k < K, and kq k < K. (5) f" g and f" ; g ae IID pocesses oe and, w means zeo, un aances, ;" = E " 4 ; 3, and sup ; E(j" j 8+c ) K <, and sup ; E(j" ; j 8+c ) K <, fo some c >. Remak e aboe assumpons allow e euns on nddual secues o be songly cossseconally coelaed oug e obseed facos, f, and allow fo weak eo coss-coelaons once e e ecs of song facos ae emoed. Suc esdual nedependences could ase due o spaal o oe newok ype spll-oe e ecs no capued by e obseed common facos. Remak 3 Unde condon (5) kv k Q Q kq k kq k < K = O(); (53) neeeless due o e weak facos we ae kvk = sup j j j j = O ; and allows e oeall eo aance max, V, o be appoxmaely spase, n conas o e leaue a eques kvk < K. e elaxaon of e spasy condon on V s paculaly mpoan n nance wee secuy euns could be a eced by weak unobseed facos. Usng pncpal componens does no esole e poblem snce, pncpal componens pode conssen esmaes of e facos (up o a oaon max) only f e facos ae song. Remak 4 e g-ode momen condons n Assumpon 3 allow us o elax e Gaussany assumpon wls a e same me ensung a ou es s applcable een f s muc lage an. Remak 5 Assumpons () and 3 ensue a e sample coss coelaon coe cens of e esduals, ^ j, ae an Edgewo expanson wc s needed fo conssen esmaon of, de ned by (38). Fo fue deals see Baley e al (7). Ou man eoecal esuls ae se ou n e followng eoems. e poofs of ese eoems ae poded n Appendx A, and necessay lemmas fo e poofs ae gen n e onlne supplemen aalable upon eques. eoem Consde e eun egessons, (), and e sasc P z de ned by (). Suppose a Assumpons -3 old, and (R ) s bounded n, wee R = j, j = E( j ), and = u = = s e sandadzed eo of e eun egesson equaon (). en, unde H : = ; n () fo all ; q = = z d (; ); as and ; jonly, (54) 6

18 wee w = lm R = + lm ( ) ; = P P = j= ( ) j. (55) eoem Consde e egesson model (), and e sascs P z and P, wc ae de ned by () and (), especely. Suppose a Assumpons -3 old. en, unde e null ypoess, H : = fo all, S = = z p ; as and jonly, so long as =, < =, wee s de ned by (48). eoem 3 Consde e egesson model (), and suppose a Assumpons -3 old. en, unde H : = ; fo all ; J = q = P ( ) 4 [ + ( ) ] d (; ) ; (56) so long as = ; and < =; as and ; jonly, wee, and ae de ned by (3), (55) and (48), especely, w = m. eoem 4 Le wee ~ ; = P P = j= ( ) ~ j, (57) ~ j = ^ j I p ^ j > cp () ; (58) j = E( j ), = u = = s e sandadzed eo of e eun egesson equaon (), = m, ^ j s de ned by (39) c p () = p ; (59) f() p s e nomnal p-alue ( < p < ), and f() = and = c d d, wee c d, and d ae ne pose consans. Suppose a Assumpons -3 old and j = O(): (6) P ;j= en ( )E ~ ;, as and, wc mples ( ) ~ ; p, f = = d ( d), (o f d > =), and f > ' ( ) max, fo some small >, wee, ' max + ;" and ;" = E " 4 ; 3 (Assumpon 3). 7

19 eoem 5 Consde e panel egesson model () n asse euns, and suppose a Assumpons -3 old. Consde e sasc P ^J = = q ; (6) ( ) ( 4) + ( )~ ; wee s gen by (3), = m, ~ ; s de ned by (57), usng e esold c p () gen by (59), w p ( < p < ), f() =, = c d d, wee c d, and d ae ne pose ( d) consans, > ', ( ) max, fo some small >, wee ' max + ;" and ;" = E " ; 4 3. en, unde H : = fo all ; ^J d (; ) ; (6) f =, as and, jonly. Fo e powe of e ^J es, we consde e local alenaes H a : = & =4 = ; w j& j <, fo all : (63) eoem 6 Consde e panel egesson model () n asse euns, and suppose a condons of eoem 5 apply. en, unde e local alenaes, H, de ned by (63), ^J d = p ; ; (64) wee = lm P & =. Remak 7 s eoem esablses a e ^J es s conssen (n e sense a s powe ends o uny), f >, wc s sas ed f lm P & >. I s also neesng o noe a e powe of e ^J es nceases unfomly w and, n conas o e powe of e GRS es a ses w, only. e ^J es as powe een f P does no ncease P w, so long as nceases su cenly slowly as compaed o. o see s, le =, and noe a unde e local alenaes, (63), and seng = d, we ae P = P & = d =, o P & = +d =. Hence, e poposed es wll be conssen so long as + d =. e case of = s of pacula nees snce does no eque e numbe of secues w non-zeo alpas o se w fo e es o ae powe. 5 Small sample edence based on Mone Calo expemens We examne e ne sample popey of e ^J es by Mone Calo expemens, and compae s pefomance o a numbe of exsng ess. Fo compason, we consde e GRS es, e GOS es, and e feasble esons of e sandadsed Wald ess, SW P OE and SW LW, wc ae dscussed n Secon 3. We also consde e F max es ecenly poposed by Gungo and Luge (6, GL). ey popose basng a es of H : = on e smulaed dsbuon of F max = max F, wee F s a sandad F -sasc fo esng = n e OLS egesson of y on an necep and f. e smulaons ae caed ou by esdual esamplng allowng 8

20 fo coss-seconal coelaons and coss-seconal eeoskedascy usng wld boosaps. GL employ a bounds esng appoac o allow fo unconsdeed nusance paamees, wc could esul n ang nconcluse es oucomes. Compuaonal deals of e aboe ess ae gen n Secon M. of e onlne supplemen aalable on eques. 5. Mone Calo desgns and expemens We consde e followng daa geneang pocess (DGP) = + m ` f` + u ; = ; ; ::; ; = ; ; :::;, (65) `= and calbae s paamees o closely mac e man feaues of e me sees obseaons on nddual euns and e ee Fama-Fenc facos (make faco, HML and SMB) used n e leaue on ess of make e cency. 3 e Mone Calo (MC) desgn s also nended o mac e models used fo e empcal applcaons a follow. Accodngly, we se m = 3 and geneae e facos as wee ` IID(; ) and 4 f` = :53 + :6f`; + p ` ` ; fo ` = ; (Make faco); f` = :9 + :9f`; + p ` ` ; fo ` = ; (HML); f` = :9 + :5f`; + p ` ` ; fo ` = 3; (SMB); ` = :89 + :85`; + : `;, fo ` = ; Make; ` = :6 + :74`; + :9 `;, fo ` = ; HML; ` = :8 + :76`; + :5 `;, fo ` = 3; SMB. e aboe pocesses ae geneaed oe e peod = 49; 48; ::::; ; ; ::::; w f`; 5 = and `; 5 = fo ` = ; ; 3. Obseaons = ; ; :::; ae used n e MC expemens. o capue e man feaues of e nddual asse euns and e coss coelaons, we geneae e dosyncac eos, u = (u ; u ; :::; u ), accodng o u = Q", wee " = (" ; " ; :::; " ), and Q = D = P w D = dag( ; ; :::; ), = V a(u ), and P beng a Colesky faco of coelaon max of u, R, wc s an max used o calbae e coss coelaon of euns. Fo eac, " s geneaed suc a u exbs skewness and kuoss wc s ypcal of nddual secuy euns. o s end, R s geneaed as R = I + bb B ; (66) wee b = (b ; b ; ::::; b ) ; and B = dag(b). e coelaon max R also ases fom e sngle faco model, u = + = ;, w s IID(; ); and s IID(; ), and b = = =, We also consdeed wo dsbuon-fee sgn ess of =, poposed by Gungo and Luge (9). ese ess, efeed o as SS and W S ess, ae ald fo sngle faco models w eos a ae condonally symmec aound zeo, bu ey do allow fo non-nomal eos, ae elaely easy o compue, and ae applcable een wen >. e esuls of ese smulaons ae epoed n able M4 of e Onlne Supplemen. ese ess ae also oupefomed by e ^J es. 3 SMB sands fo "small make capalzaon mnus bg" and HML fo "g book-o-make ao mnus low". See Fama and Fenc (993), and Appendx C fo fue deals and daa souces. 4 e esmaes used n e geneaon of e facos and e olales ae compued usng monly obseaons oe e peod Apl Sepembe. 9

21 wee = + ;. o geneae d een degees of eo coss-seconal dependence, we daw e s and e las (< ) elemens of b as Unfom(:7; :9), and se e emanng mddle elemens o. We se = b c; wee bac s e lages nege pa of A. Usng, ou assumpon m = o( = ) can be expessed by m = w < =. In ou expemens, we consde e alues of exponens = =4; =, and 3=5. e case of no eo coss-seconal dependence s obaned wen =, and e eo coss-seconal dependence s weak wen < =. e case of = 3=5 s ncluded o see ow e ^J es pefoms wen coss-seconal eo coelaons ae ge an e esold alue of = allowed by e eoy. o sae space, we om epong e esuls fo e case wee = as ey ae qualaely smla o e case w = =4. e pesen desgn focusses on e weak faco eo coelaons and assumes e dosyncac eos,, ae coss-seconally uncoelaed. A moe geneal desgn a allows fo bo foms of eo coelaons wll be consdeed below. Recenly, Fan, Lao and Yao (5; FLY) ae deed e condons unde wc e lmng nomal dsbuon of SW P OE wll be asympocally jus ed. Unde e assumpons e SW P OE es allows fo >. Howee, FLY s assumpons ae muc moe esce an ous. 5 Fo example, FLY do no coe cases wee =4 < =. Wen = =4, FLY eque a = O ( ln() ), fo some > : us, wen = =4, so long as ses slgly fase an, e SW P OE es s asympocally jus ed. On e oe and, ^J d (; ) so long as = d w d > =3 wen = =4. eefoe, e ^J es s expeced o pode bee ne sample appoxmaon an e SW P OE es, especally wen s lage an and/o wen eo coss-coelaon s no ey weak. e smulaon esuls a follow seem o suppo ese eoecal nsgs. 6 o calbae e aance, skewness and kuoss of e smulaed euns, we used esmaed alues of ese measues based on esduals of Fama-Fenc egessons fo eac secuy oe e esmaon wndows =Sepembe 989,..., Sepembe, usng sample szes equal o = 6 mons. Spec cally, fo eac = ; ; :::; we un e Fama-Fenc egessons ; f; = ^ + ^ ; ( m; f; ) + ^ ; SMB + ^ 3 HML + ^u ;, = ; ; :::; 6; a e end of eac mon =Sepembe 989,..., Sepembe, and compued ^ ; = ^m ;, ^ ;; = ^m 3; = ^m 3= ; and ^ ; = ^m 4; = ^m ; 3 w ^m s; = (6) P 6 = ^u ; ^u ; s, and ^u; = (6) P 6 = ^u ;: We ended up w 6,8 d een alues of ^ ;, ^ ;; and ^ ;; esmaed fo aound 476 secues oe 65 d een esmaon wndows. We dscaded esmaes a led below e.5% and aboe e 97.5% quanles o aod e calbaed alues beng domnaed by exeme oules. e same pocedue was appled o e esmaed faco loadngs, ^`:. e means and medans of ^ ;, ^ ;;, ^ ;; and ^`; fo ` = ; ; 3; and e :5% and 97:5% quanles ae summazed n able. As can be seen fom ese esuls ee s a consdeable degee of eeogeney n esmaes of e faco loadngs and n e measues of deaons, skewness and kuoss, acoss secues and sample peods. e deals of e pocedue o geneae e non-nomal and coss-coelaed eos ae descbed n Appendx B. o esmae sze of e ess, we se = fo all. o nesgae powe, we geneaed as IID(; ) fo = ; ; :::; w = b c; = fo = + ; + ; :::;. We consdeed e alues = :8; :9; :, bu e powe ended up o be ey g een fo 5 In addon o some egulay condons, FLY eque Assumpon A.. wc de nes e eson of "spaseness": Suppose = (log ) = o ( ) fo some > ; and () mn j6= j j j >> p (log ) = ; () a leas one of e followng cases olds: (a) D = P P = j= I ( j 6= ) = O( = ) and = O o; (b) = (ln ) D = O () and m = O (). en ey sow a SW P OE d (; ) ;as ; jonly (see Poposon 4. of FLY). 6 s may also explan wy FLY es su es fom sze-dsoon as dscussed by Baley, Pesaan and Yamagaa n Fan, Lao and Mncea (3), wee s allowed o ncease w xed.

22 = :8. eefoe, we only epo powe esmaes fo = :8. All combnaons of = 6; and = 5; ; ; 5 (and ;, ;, 5; fo e ^J es) ae consdeed. All ess ae conduced a a 5% sgn cance leel. Expemens ae based on R = ; eplcaons. 5. Sze and powe able epos e sze and powe of e GRS, ^J, GOS, SW P OE, SW LW and F max ess of Gungo and Luge (6), n e case of models w ee facos, unde aous degees of coss-seconal eo coelaons, as measued by e exponen,. Fs, consde Panel A of able wc deals w e case wee e eos ae nomally dsbued bu coss-seconally weakly dependen w = =4. 7 e GRS es wen applcable (namely wen > ) beng an exac es, as e coec sze. e empcal sze of e ^J es s also ey close o e 5% nomnal leel fo all combnaons of and. Een wen = 5, e sze of e ^J es les n e ange 5.% o 5.3% fo d een alues of. In conas, bo GOS and SW P OE ess gossly oe-ejec e null ypoess, and e degee of e oe-ejecon becomes moe seous as nceases fo a gen. Fo example, wen = 6, nceasng fom 5 o 5; e sze of e GOS es ses fom.% o 6.% and a of e SW P OE es ses fom 8.3% o 53.%. In lne w e dscusson n Secon 3.4, e sze dsoon s mgaed wen nceases. Fo = 6 and = 5 e sze of e GOS es and e SW P OE es ae.% and 8.3% bu ey fall o 8.3% and.% wen = and = 5, especely. e sze popees of e SW LW es ae ey smla o ose of e SW P OE es. e sze of e F max es ends o be subsanally smalle an e nomnal leel fo all combnaons of and (s s n lne w e epoed esuls n Gungo and Luge, 6). e ejecon fequences ange beween.% and.%. Fuemoe, nconcluse es oucomes ae obseed moe ofen, angng beween.7% and 4.6% of e oucomes. 8 e powe of e ^J es s subsanally ge an a of e GRS es. Fo example, fo = 6 and = 5 e powe of e GRS es s 5.% as compaed o 65.9% fo e ^J es, aloug bo ess ae smla szes (4.6% fo e GRS es and 7.4% fo e ^J es). s s n lne w ou dscusson a e end of Secon, and e ecs e fac a GRS assumes an abay degee of coss-seconal eo coelaons and us eles on a lage me dmenson o acee a easonably g powe. In conas, e powe of e ^J es s den lagely by e cossseconal dmenson. s can be seen clealy fom e abulaed esuls. Keepng xed a 5, and nceasng fom 6 o nceases e powe of e GRS es fom 5.% o 69.%, wls e powe of e ^J es (fo example) ses fom 65.9% o 87.4%. I s neesng a een n s case (w muc lage an ) e ^J es sll as subsanally ge powe an e GRS es, w compaable ype I eos. e powe compason of e GOS, SW P OE and SW LW w oe ess seem nappopae, gen e lage sze-dsoons. Hang sad s, s peaps emakable a e powe of e ^J es s compaable o e unadjused powe of e GOS, SW P OE and SW LW ess. e powe of e ^J es unfomly domnaes a of e F max es fo all expemens. e low powe of e F max es s paally explaned by e lage popoons of nconcluse esuls. Fo = 6, beween 9.3% and 45.5% of nconcluse esuls ae obseed fo d een. Fo =, e popoon of nconcluse esuls ends o declne as nceases. Fo example, nceasng fom 5 o 5 lowes e fequences of nconcluse esuls of e F max es fom 39.% o 9.%. 7 In lne w ou eoecal ndngs (see Secon ), e esuls of coss-seconally ndependen case (w = ) s qualaely smla o e case wee = =4. 8 e fequences of nconcluse oucomes fo e F max es fo d een combnaons of and ae epoed n able M of e Onlne Supplemen.

23 Consde now e case wee e eos ae nomally dsbued and coss-seconally elaely songly dependen. Fs le us dscuss e esuls wen = =. e ^J es seems que obus o coss-seconal eo coelaons, w s sze fallng n e ange 5.% o 6.6%. e sze of e ^J es fo = 5 and = 6 s 6.4%, and s powe s 53.6%, wc sll exceed e powe of e GRS es, wc s.7%. Bu, as expeced, nceasng fom 6 o esuls n e powe of e GRS es o se o 84.9%, wc magnally beas e powe of e ^J es a 8.3%. e sze dsoon of e GOS es becomes moe ponounced as e alue of s nceased. Fo = =4, e sze of e GOS es nceases fom.% o 5.4% wen s nceased fom =4 o = n e case of e sample combnaons = and = 6, and nceases fom 6.% o 9.% f we consde e lage sample of = 5 and = 6. I s also neesng a wen = 5, nceasng fom 6 o does no mpoe e sze dsoon of GOS and SW P OE ess, wc amoun o.5% and 3.3%, especely. Wen = 3=5 > =, ou of all e ess consdeed, only e GRS es s ald so long as <, and ndeed as e coec sze n suc cases. Howee, neesngly, e sze of e ^J es s also close o s nomnal leel (a 5.5%-7.%) een fo suc a g alue of. s seems o be due o e ncluson of ( )~ ; n e denomnao of e ^J sasc. We now consde e empcally mos elean case wee e eos ae non-nomal as well as beng coss-seconally coelaed. e e ecs of non-nomal eos on e ess ae documened n Panel B of able. Consde s e case wee e eos ae non-nomal and coss-seconally weakly coelaed ( = =4). We see a e sze of e GRS es s adly a eced by e ypes of depaues fom Gaussany obseed n e egesson esduals. e obusness of e GRS es o non-nomal eos of e ype encouneed n pacce as also been documened by A eck-gaes and McDonald (989). As o be expeced fom e eoecal dscussons, e ^J es s easonably obus o non-gaussan eos, and exb only a ey mld endency of oe-ejecng e null ypoess, een fo elaely lage. Fo example, wen = 6, fo = 5,,, and 5, e szes of e ^J es ae 6:5%, 6:9%, 5:9%, and 6:6%, especely. e oe-ejecon of e GOS es and e SW P OE es ends o be somewa magn ed by non-nomaly. e e ecs of non-nomaly upon e sze of e SW LW s less obous. e sze of e F max es s agan muc smalle an e nomnal leel, bu on aeage slgly ge an a unde nomal eos. Fo example, e aeage of e sze of e F max es fo all e combnaons of (; ) s.4% unde nomal eos, bu unde non-nomal eos s.5%. Also, on aeage e ncdence of nconcluse oucomes fo e F max es s slgly ge unde non-nomal eos. Fo example, e aeage of e fequences of e nconcluse oucomes fo all e combnaons of (; ) s 3.7% unde nomal eos, bu nceases o 4:3% unde non-nomal. Unde non-nomal eos, e ^J es connues o manan s powe supeoy oe e GRS and e F max ess. Wen = = and 3=5 e sze of e ^J es s easonably conolled and les n e ange 6.%-7.9%. e powe compasons dscussed fo e weakly coss-seconally uncoelaed case ( = =4) also cay oe o e pesen se of expemens w e muc ge degees of eo coss-seconal coelaons ( = = and 3=5). We also caed ou addonal expemens w muc lage alues of, namely = ; ; ; and 5; ; wls keepng a 6 and. We only consdeed e ^J es fo ese expemens, as s unlkely a oe ess consdeed, gen e elaely poo pefomance fo alues of 5, would pefom bee an e ^J es. e esuls ae summased n able 3. As can be seen, e sze s sasfacoly conolled w good powe popees, only sowng modeae oe-ejecon unde non-gaussany fo = 6, and fo elaely song eo coss coelaons. Fo example, fo = 5;, wen = 6 w non-nomal eos, e sze of e ^J es fo = =4; = and 3=5 ae 7.8%, 9.5% and 9.3%, weeas, by nceasng