INFERENTIAL THEORY FOR FACTOR MODELS OF LARGE DIMENSIONS. By Jushan Bai 1

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1 Economerica, Vol. 7, o. January, 2003, 35 7 IFEREIAL HEORY FOR FACOR MODELS OF LARGE DIMESIOS By Juhan Bai hi paper develop an inferenial heory for facor model of large dimenion. he principal componen eimaor i conidered becaue i i eay o compue and i aympoically equivalen o he maximum likelihood eimaor if normaliy i aumed. We derive he rae of convergence and he limiing diribuion of he eimaed facor, facor loading, and common componen. he heory i developed wihin he framework of large cro ecion and a large ime dimenion, o which claical facor analyi doe no apply. We how ha he eimaed common componen are aympoically normal wih a convergence rae equal o he minimum of he quare roo of and. he eimaed facor and heir loading are generally normal, alhough no alway o. he convergence rae of he eimaed facor and facor loading can be faer han ha of he eimaed common componen. hee reul are obained under general condiion ha allow for correlaion and heerokedaiciie in boh dimenion. Sronger reul are obained when he idioyncraic error are erially uncorrelaed and homokedaic. A neceary and ufficien condiion for coniency i derived for large bu fixed. Keyword: Approximae facor model, principal componen, common componen, large model analyi, large daa e, daa-rich environmen. inroducion Economi now have he luxury of working wih very large daa e. For example, he Penn World able conain hiry variable for more han onehundred counrie covering he powar year. he World Bank ha daa for abou wo-hundred counrie over fory year. Sae and ecoral level daa are alo widely available. Such a daa-rich environmen i due in par o echnological advance in daa collecion, and in par o he ineviable accumulaion of informaion over ime. A ueful mehod for ummarizing informaion in large daa e i facor analyi. More imporanly, many economic problem are characerized by facor model; e.g., he Arbirage Pricing heory of Ro 976. While here i a well developed inferenial heory for facor model of mall dimenion I am graeful o hree anonymou referee and a co-edior for heir conrucive commen, which led o a ignifican improvemen in he preenaion. I am alo graeful o Jame Sock for hi encouragemen in puring hi reearch and o Serena g and Richard rech for heir valuable commen. In addiion, hi paper ha benefied from commen received from he BER Summer Meeing, 200, and he Economeric Sociey Summer Meeing, Maryland, 200. Parial reul were alo repored a he BER/SF Forecaing Seminar in July, 2000 and a he ASSA Winer Meeing in ew Orlean, January, 200. Parial financial uppor from he SF Gran #SBR i acknowledged. 35

2 36 juhan bai claical, he inferenial heory for facor model of large dimenion i aben. he purpoe of hi paper i o parially fill hi void. A facor model ha he following repreenaion: X i = i F + e i where X i i he oberved daum for he ih cro ecion a ime i = = ; F i a vecor r of common facor; i i a vecor r of facor loading; and e i i he idioyncraic componen of X i. he righ-handide variable are no obervable. 2 Reader are referred o Wanbeek and Meijer 2000 for an economeric perpecive on facor model. Claical facor analyi aume a fixed, while i allowed o increae. In hi paper, we develop an inferenial heory for facor model of large dimenion, allowing boh and o increae. Example of Facor Model of Large Dimenion i Ae Pricing Model. A fundamenal aumpion of he Arbirage Pricing heory AP of Ro 976 i ha ae reurn follow a facor rucure. In hi cae, X i repreen ae i reurn in period ; F i a vecor of facor reurn; and e i i he idioyncraic reurn. hi heory leave he number of facor unpecified, hough fixed. ii Diaggregae Buine Cycle Analyi. Cyclical variaion in a counry economy could be driven by global or counry-pecific hock, a analyzed by Gregory and Head 999. Similarly, variaion a he indury level could be driven by counry-wide or indury-pecific hock, a analyzed by Forni and Reichlin 998. Facor model allow for he idenificaion of common and pecific hock, where X i i he oupu of counry indury i in period ; F i he common hock a ; and i i he expoure of counry indury i o he common hock. hi facor approach of analyzing buine cycle i gaining prominence. iii Monioring, Forecaing, and Diffuion Indice. he ae of he economy i ofen decribed by mean of a mall number of variable. In recen year, Forni, Hallin, Lippi, and Reichlin 2000b, Reichlin 2000, and Sock and Waon 998, among oher, reed ha he facor model provide an effecive way of monioring economic aciviy. hi i becaue buine cycle are defined a comovemen of economic variable, and common facor provide a naural repreenaion of hee co-movemen. Sock and Waon 998 furher demonraed ha he eimaed common facor diffuion indice can be ued o improve forecaing accuracy. iv Conumer heory. Suppoe X ih repreen he budge hare of good i for houehold h. he rank of a demand yem i he malle ineger r uch ha X ih = i G e h + ir G r e h, where e h i houehold h oal expendiure, and G j are unknown funcion. By defining he r facor a 2 I i noed ha alo repreen he number of variable and repreen he number of obervaion, an inerpreaion ued by claical facor analyi.

3 facor model of large dimenion 37 F h = G e h G r e h, he rank of he demand yem i equal o he number of facor. An imporan reul due o Gorman 98 i ha demand yem conien wih andard axiom of conumer heory canno have a rank more han hree, meaning ha houehold allocaion of reource if hey are uiliy maximizing hould be decribed by a mo hree facor. hi heory i furher generalized by Lewbel 99. Applicaion of large-dimenional facor model are rapidly increaing. Bernanke and Boivin 2000 howed ha uing he eimaed facor make i poible o incorporae large daa e ino he udy of he Fed moneary policy. Favero and Marcellino 200 examined a relaed problem for European counrie. Criadoro, Forni, Reichlin, and Veronee 200 eimaed a core inflaion index from a large number of inflaion indicaor. A i well known, facor model have been imporan in finance and are ued for performance evaluaion and rik meauremen; ee, e.g., Campbell, Lo, and Mackinlay 997, Chaper 5 and 6. More recenly, ong 2000 udied he profiabiliy of momenum rading raegie uing facor model. In all he above and in fuure applicaion, i i of inere o know when he eimaed facor can be reaed a known. ha i, under wha condiion i he eimaion error negligible? hi queion arie whenever he eimaed facor are ued a regreor and he ignificance of he facor i eed, a for example in Bernanke and Boivin 2000, and alo in he diffuion index forecaing of Sock and Waon 998. If he eimaion error i no negligible, hen he diribuional properie of he eimaed facor are needed. In addiion, i i alway ueful o conruc confidence inerval for he eimae, epecially for applicaion in which he eimae repreen economic indice. hi paper offer anwer o hee and relaed heoreical queion. 2 Limiaion of Claical Facor Analyi Le X = X X 2 X and e = e e 2 e be vecor, and le = covx be he covariance marix of X. Claical facor analyi aume ha i fixed and much maller han, and furher ha he e i are independen and idenically diribued iid over ime and are alo independen acro i. hu he variance-covariance marix of e, = Ee e, i a diagonal marix. he facor F are alo aumed o be iid and are independen of e i. Alhough no eenial, normaliy of e i i ofen aumed and maximum likelihood eimaion i ued in eimaion. In addiion, inferenial heory i baed on he baic aumpion ha he ample covariance marix = X XX X = i roo- conien for and aympoically normal; ee Anderon 984 and Lawley and Maxwell 97.

4 38 juhan bai hee aumpion are rericive for economic problem. Fir, he number of cro ecion i ofen larger han he number of ime period in economic daa e. Poenially imporan informaion i lo when a mall number of variable i choen o mee he mall-, requiremen. Second, he aumpion ha i aympoically normal may be appropriae under a fixed, bu i i no longer appropriae when alo end o infiniy. For example, he rank of doe no exceed min, wherea he rank of can alway be. hird, he iid aumpion and diagonaliy of he idioyncraic covariance marix, which rule ou cro-ecion correlaion, are oo rong for economic ime erie daa. Fourh, maximum likelihood eimaion i no feaible for largedimenional facor model becaue he number of parameer o be eimaed i large. Fifh, claical facor analyi can conienly eimae he facor loading i bu no he common facor F. In economic, i i ofen he common facor repreening he facor reurn, common hock, diffuion indice, ec. ha are of direc inere. 3 Recen Developmen and he Main Reul of hi Paper here i a growing lieraure ha recognize he limiaion of claical facor analyi and propoe new mehodologie. Chamberlain and Rohchild 983 inroduced he noaion of an approximae facor model o allow for a nondiagonal covariance marix. Furhermore, Chamberlain and Rohchild howed ha he principal componen mehod i equivalen o facor analyi or maximum likelihood under normaliy of he e i when increae o infiniy. Bu hey aumed a known populaion covariance marix. Connor and Korajczyk 986, 988, 993 udied he cae of an unknown covariance marix and uggeed ha when i much larger han, he facor model can be eimaed by applying he principal componen mehod o he covariance marix. Forni and Lippi 997 and Forni and Reichlin 998 conidered large dimenional dynamic facor model and uggeed differen mehod for eimaion. Forni, Hallin, Lippi, and Reichlin 2000a formulaed he dynamic principal componen mehod by exending he analyi of Brillinger 98. Some preliminary eimaion heory of large facor model ha been obained in he lieraure. Connor and Korajczyk 986 proved coniency for he eimaed facor wih fixed. For inference ha require large, hey ued a equenial limi argumen goe o infiniy fir and hen goe o infiniy. 3 Sock and Waon 999 udied he uniform coniency of eimaed facor and derived ome rae of convergence for large and large. he rae of convergence wa alo udied by Bai and g Forni e al. 2000a,c eablihed coniency and ome rae of convergence for he eimaed common componen i F for dynamic facor model. 3 Connor and Korajczyk 986 recognized he imporance of imulaneou limi heory. hey aed ha Ideally we would like o allow and o grow imulaneouly poibly wih heir raio approaching ome limi. We know of no raighforward echnique for olving hi problem and leave i for fuure endeavor. Sudying imulaneou limi i only a recen endeavor.

5 facor model of large dimenion 39 However, inferenial heory i no well underood for large-dimenional facor model. For example, limiing diribuion are no available in he lieraure. In addiion, he rae of convergence derived hu far are no he one ha would deliver a nondegenerae convergence in diribuion. In hi paper, we derive he rae of convergence and he limiing diribuion for he eimaed facor, facor loading, and common componen, eimaed by he principal componen mehod. Furhermore, he reul are derived under more general aumpion han claical facor analyi. In addiion o large and large, we allow for erial and cro-ecion dependence for he idioyncraic error; we alo allow for heerokedaiciy in boh dimenion. Under claical facor model, wih a fixed, one can conienly eimae facor loading bu no he facor; ee Anderon 984. In conra, we demonrae ha boh he facor and facor loading can be conienly eimaed up o a normalizaion for large-dimenional facor model. We alo conider he cae of large bu fixed. We how ha o eimae he facor conienly, a neceary condiion i aympoic orhogonaliy and aympoic homokedaiciy defined below. In conra, under he framework of large and large, we eablih coniency in he preence of erial correlaion and heerokedaiciy. ha i, he neceary condiion under fixed i no longer neceary when boh and are large. he re of he paper i organized a follow. Secion 2 e up he model. Secion 3 provide he aympoic heory for he eimaed facor, facor loading, and common componen. Secion 4 provide addiional reul in he abence of erial correlaion and heerokedaiciy. he cae of fixed i alo udied. Secion 5 derive conien eimaor for he covariance marice occurring in he limiing diribuion. Secion 6 repor he imulaion reul. Concluding remark are provided in Secion 7. All proof are given in he appendice. 2 eimaion and aumpion Recall ha a facor model i repreened by 2 X i = i F + e i = C i + e i where C i = i F i he common componen, and all oher variable are inroduced previouly. When i mall, he model can be ca under he ae pace eup and be eimaed by maximizing he Gauian likelihood via he Kalman filer. A increae, he ae pace and he number of parameer o be eimaed increae very quickly, rendering he eimaion problem challenging, if no impoible. Bu facor model can alo be eimaed by he mehod of principal componen. A hown by Chamberlain and Rohchild 983, he principal componen eimaor converge o he maximum likelihood eimaor when increae hough hey did no conider ampling variaion. Ye he former i much eaier o compue. hu hi paper focue on he properie of he principal componen eimaor.

6 40 juhan bai Equaion 2 can be wrien a an -dimenion ime erie wih obervaion: 3 X = F +e = 2 where X = X X 2 X, = 2, and e = e e 2, e. Alernaively, we can rewrie 2 a a -dimenion yem wih obervaion: X i = F i + e i i = 2 where X i = X i X i2 X i, F = F F 2 F, and e i = e i e i2 e i. We will alo ue he marix noaion: X = F + e where X = X X 2 X i a marix of oberved daa and e = e e 2 e i a marix of idioyncraic error. he marice r and F r are boh unknown. Our objecive i o derive he large-ample properie of he eimaed facor and heir loading when boh and are large. he mehod of principal componen minimize Vr= min F i= = X i i F 2 Concenraing ou and uing he normalizaion ha F F/ = I r an r r ideniy marix, he problem i idenical o maximizing rf XX F. he eimaed facor marix, denoed by F,i ime eigenvecor correponding o he r large eigenvalue of he marix XX, and = F F F X = F X/ are he correponding facor loading. he common componen marix F i eimaed by F. Since boh and are allowed o grow, here i a need o explain he way in which limi are aken. here are hree limi concep: equenial, pahwie, and imulaneou. A equenial limi ake one of he indice o infiniy fir and hen he oher index. Le g be a quaniy of inere. A equenial limi wih growing fir o infiniy i wrien a lim lim g. hi limi may no be he ame a ha of growing fir o infiniy. A pahwie limi reric o increae along a paricular rajecory of. hi can be wrien a lim v g v v, where v v decribe a given rajecory a v varie. If v i choen o be, hen a pahwie limi i wrien a lim g. A imulaneou limi allow o increae along all poible pah uch ha lim v min v v =. hi limi i denoed by lim g. he preen paper mainly conider imulaneou limi heory. oe ha he exience of a imulaneou limi implie he exience of equenial and pahwie limi and he limi are he ame, bu he convere i no rue. When g i a

7 facor model of large dimenion 4 funcion of or alone, i limi wih repec o or i auomaically a imulaneou limi. Due o he naure of our problem, hi paper alo conider imulaneou limi wih rericion. For example, we may need o conider he limi of g for and aifying / 0; ha i, lim v g v v over all pah uch ha lim v min v v =and lim v v / v = 0. Le A=rA A /2 denoe he norm of marix A. hroughou, we le F 0 be he r vecor of rue facor and 0 i be he rue loading, wih F 0 and 0 being he correponding marice. he following aumpion are ued in Bai and g 2002 o eimae he number of facor conienly: Aumpion A Facor: EF 04 M< and = F 0F p F ome r r poiive definie marix F. for Aumpion B Facor Loading: i <, and 0 / 0 for ome r r poiive definie marix. Aumpion C ime and Cro-Secion Dependence and Heerokedaiciy: here exi a poiive conan M< uch ha for all and :. Ee i = 0, Ee i 8 M. 2. Ee e / = E i= e ie i =, M for all, and = = M 3. Ee i e j = ij wih ij ij for ome ij and for all. In addiion, ij M i= j= 4. Ee i e j = ij and i= j= = = ij M. 5. For every, E /2 i= e ie i Ee i e i 4 M. Aumpion D Weak dependence beween facor and idioyncraic error: E F 0 e 2 i M i= = Aumpion A i more general han ha of claical facor analyi in which he facor F are i.i.d. Here we allow F o be dynamic uch ha ALF =. However, we do no allow he dynamic o ener ino X i direcly, o ha he relaionhip beween X i and F i ill aic. For more general dynamic facor model, reader are referred o Forni e al. 2000a. Aumpion B enure ha each facor ha a nonrivial conribuion o he variance of X. We only conider nonrandom facor loading for impliciy. Our reul ill hold when he are random, provided hey are independen of he facor and idioyncraic

8 42 juhan bai error, and E i 4 M. Aumpion C allow for limied ime erie and cro ecion dependence in he idioyncraic componen. Heerokedaiciie in boh he ime and cro-ecion dimenion are alo allowed. Under aionariy in he ime dimenion, =, hough he condiion i no neceary. Given Aumpion C, he remaining aumpion in C are aified if he e i are independen for all i and. Correlaion in he idioyncraic componen allow he model o have an approximae facor rucure. I i more general han a ric facor model which aume e i i uncorrelaed acro i, he framework on which he AP heory of Ro 976 wa baed. hu, he reul o be developed alo apply o ric facor model. When he facor and idioyncraic error are independen a andard aumpion for convenional facor model, Aumpion D i implied by Aumpion A and C. Independence i no required for D o be rue. For example, if e i = i F wih i being independen of F and i aifying Aumpion C, hen Aumpion D hold. 4 Chamberlain and Rohchild 983 defined an approximae facor model a having bounded eigenvalue for he covariance marix = Ee e.ife i aionary wih Ee i e j = ij, hen from marix heory, he large eigenvalue of i bounded by max i j= ij. hu if we aume j= ij M for all i and all, which implie Aumpion C3, hen 3 will be an approximae facor model in he ene of Chamberlain and Rohchild. Since we alo allow for nonaionariy e.g., heerokedaiciy in he ime dimenion, our model i more general han approximae facor model. hroughou hi paper, he number of facor r i aumed fixed a and grow. Many economic problem do ugge hi kind of facor rucure. For example, he AP heory of Ro 976 aume an unknown bu fixed r. he Capial Ae Pricing Model of Sharpe 964 and Linner 965 implie one facor in he preence of a rik-free ae and wo facor oherwie, irrepecive of he number of ae. he rank heory of conumer demand yem of Gorman 98 and Lewbel 99 implie no more han hree facor if conumer are uiliy maximizing, regardle of he number of conumpion good. In general, larger daa e may conain more facor. Bu hi may no have any pracical conequence ince large daa e allow u o eimae more facor. Suppoe one i inereed in eing he AP heory for Aian and he U.S. financial marke. Becaue of marke egmenaion and culural and poliical difference, he facor explaining ae reurn in Aian marke are differen from hoe in he U.S. marke. hu, he number of facor i larger if daa from he wo marke are combined. he oal number of facor, alhough increaed, i ill fixed a a new level, regardle of he number of Aian and U.S. ae if AP heory prevail in boh marke. Alernaively, wo eparae e of AP can be conduced, one for Aian marke, one for he U.S.; each individual daa e 4 While hee aumpion ake a imilar forma a in claical facor analyi in ha aumpion were made on he uobervable quaniie uch a F and e daa generaing proce aumpion, i would be beer o make aumpion in erm of he obervable X i. Aumpion characerized by obervable variable may have he advanage of, a lea in principle, having he aumpion verified by he daa. We plan o purue hi in fuure reearch.

9 facor model of large dimenion 43 i large enough o perform uch a e and he number of facor i fixed in each marke. he poin i ha uch an increae in r ha no pracical conequence, and i i covered by he heory. Mahemaically, a porion of he cro-ecion uni may have zero facor loading, which i permied by he model aumpion. I hould be poined ou ha r can be a ricly increaing funcion of or. hi cae i no covered by our heory and i lef for fuure reearch. 3 aympoic heory Aumpion A D are ufficien for conienly eimaing he number of facor r a well a he facor hemelve and heir loading. By analyzing he aiical properie of Vk a a funcion of k, Bai and g 2002 howed ha he number of facor r can be eimaed conienly by minimizing he following crierion: + ICk = logv k + k log + ha i, for k>r, le ˆk = arg min 0 k k ICk. hen Pˆk = r, a. hi coniency reul doe no impoe any rericion beween and, excep min. Bai and g alo howed ha AIC wih penaly 2/ and BIC wih penaly log / do no yield conien eimaor. In hi paper, we hall aume a known r and focu on he limiing diribuion of he eimaed facor and facor loading. heir aympoic diribuion are no affeced when he number of facor i unknown and i eimaed. 5 Addiional aumpion are needed o derive heir limiing diribuion: Aumpion E Weak Dependence: here exi M< uch ha for all and, and for every and every i :. = M. 2. k= ki M. hi aumpion renghen C2 and C3, repecively, and i ill reaonable. For example, in he cae of independence over ime, = 0 for. hen Aumpion E i equivalen o / i= Ee2 i M for all and, which i implied by C. Under cro-ecion independence, E2 i equivalen o Ee i 2 M. hu under ime erie and cro-ecion independence, El and E2 are equivalen and are implied by C. 5 hi follow becaue P F x = P F x ˆk = r+p F x ˆk r. Bu P F x ˆk r Pˆk r = o. hu P F x = P F x ˆk = r+ o = P F xˆk = rpˆk = r+ o = P F xˆk = r+ o becaue Pˆk = r. In ummary, P F x = P F xˆk = r+ o.

10 44 juhan bai Aumpion F Momen and Cenral Limi heorem: here exi an M< uch ha for all and :. for each, E 2 F 0 e ke k Ee k e k M = k= 2. he r r marix aifie E = k= 3. for each, a, i= F 0 k e k 0 i e d i 0 2 M where = lim / i= j= 0 i j Ee ie j ; 4. for each i, a, = F 0 e d i 0 i where i = p lim / = = EF 0 F e ie i. Aumpion F i no ringen becaue he um in F and F2 involve zero mean random variable. he la wo aumpion are imply cenral limi heorem, which are aified by variou mixing procee. Aumpion G: he eigenvalue of he r r marix F are diinc. he marice F and are defined in Aumpion A and B. Aumpion G guaranee a unique limi for F F 0 /, which appear in he limiing diribuion. Oherwie, i limi can only be deermined up o orhogonal ranformaion. A imilar aumpion i made in claical facor analyi; ee Anderon 963. oe ha hi aumpion i no needed for deermining he number of facor. For example, in Bai and g 2002, he number of facor i deermined baed on he um of quared reidual Vr, which depend on he projecion marix P F. Projecion marice are invarian o orhogonal ranformaion. Alo, Aumpion G i no required for udying he limiing diribuion of he eimaed common componen. he reaon i ha he common componen are idenifiable. In he following analyi, we will ue he fac ha for poiive definie marice A and B, he eigenvalue of AB, BA, and A /2 BA /2 are he ame. Propoiion : Under Aumpion A D and G, F F 0 plim = Q

11 facor model of large dimenion 45 he marix Q i inverible and i given by Q = V /2 /2, where V = diagv v 2 v r v >v 2 > >v r > 0 are he eigenvalue of /2 F /2, and i he correponding eigenvecor marix uch ha = I r. he proof i provided in Appendix A. Under Aumpion A G, we hall eablih aympoic normaliy for he principal componen eimaor. Aympoic heory for he principal componen eimaor exi only in he claical framework. For example, Anderon 963 howed aympoic normaliy of he eimaed principal componen for large and fixed. Claical facor analyi alway ar wih he baic aumpion ha here exi a roo- conien and aympoically normal eimaor for he underlying covariance marix of X auming i fixed. he framework for claical facor analyi doe no exend o iuaion conidered in hi paper. hi i becaue conien eimaion of he covariance marix of X i no a well defined problem when and increae imulaneouly o infiniy. hu, our analyi i necearily differen from he claical approach. 3 Limiing Diribuion of Eimaed Facor A noed earlier, F 0 and 0 are no eparaely idenifiable. However, hey can be eimaed up o an inverible r r marix ranformaion. A hown in he Appendix, for he principal componen eimaor F, here exi an inverible marix H whoe dependence on will be uppreed for noaional impliciy uch ha F i an eimaor of F 0 H and i an eimaor of 0 H.In addiion, F i an eimaor of F 0, he common componen. I i clear ha he common componen are idenifiable. Furhermore, knowing F 0 H i a good a knowing F 0 for many purpoe. For example, in regreion analyi, uing F 0 a he regreor will give he ame prediced value a uing F 0 H a he regreor. Becaue F 0 and F 0 H pan he ame pace, eing he ignificance of F 0 in a regreion model conaining F 0 a regreor i he ame a eing he ignificance of F 0 H. For he ame reaon, he porfolio-evaluaion meauremen of Connor and Korajczyk 986 give valid reul wheher F 0 or F 0 H i ued. heorem : Under Aumpion A G, a, we have: i if / 0, hen for each, F H F 0 = V F F 0 0 i e i + o p i= d 0V Q Q V where V i a diagonal marix coniing of he fir r eigenvalue of /XX in decreaing order, V and Q are defined in Propoiion, and i defined in F3; ii if lim inf / >0, hen F H F 0 = Op

12 46 juhan bai he dominan cae i par i; ha i, aympoic normaliy generally hold. Applied reearcher hould feel comforable uing he normal approximaion for he eimaed facor. Par ii i ueful for heoreical purpoe when a convergence rae i needed. he heorem ay ha he convergence rae i min. When he facor loading 0 i i = 2 are all known, F 0 can be eimaed by he cro-ecion lea quare mehod and he rae of convergence i. he rae of convergence min reflec he fac ha facor loading are unknown and are eimaed. Under ronger aumpion, however, he roo- convergence rae i ill achievable ee Secion 4. Aympoic normaliy of heorem i achieved by he cenral limi heorem a. hu large i required for hi heorem. Addiional commen:. Alhough rericion beween and are needed, he heorem i no a equenial limi reul bu a imulaneou one. In addiion, he heorem hold no only for a paricular relaionhip beween and, bu alo for many combinaion of and. he rericion / 0 i no rong. hu aympoic normaliy i he more prevalen iuaion for empirical applicaion. he reul permi imulaneou inference for large and large. For example, he porfolio-performance meauremen of Connor and Korajczyk 986 can be obained wihou recoure o a equenial limi argumen. See foonoe he rae of convergence implied by hi heorem i ueful in regreion analyi or in a forecaing equaion involving eimaed regreor uch a Y + = F 0 + W + u + = 2 where Y and W are obervable, bu F 0 i no. However, F 0 can be replaced by F. A crude calculaion how ha he eimaion error in F can be ignored a long a F = H F 0 + o p /2 wih H having a full rank. hi will be rue if / 0 by heorem becaue F H F 0 = O p / min. A more careful bu elemenary calculaion how ha he eimaion error i negligible if F F F 0 H/ = o p /2. Lemma B.3 in he Appendix how ha F F F 0 H/ = o p 2, where 2 = min. hi implie ha he eimaion error in F i negligible if / 0. hu for large, F 0 can be reaed a known. Sock and Waon 998, 999 conidered uch a framework of forecaing. Given he rae of convergence for F, i i eay o how ha he foreca Y + = ˆ F + ˆW i -conien for he condiional mean EY + F 0 W, auming he condiional mean of u + i zero. Furhermore, in conrucing confidence inerval for he foreca, he eimaion effec of F can be ignored provided ha i large. If / 0 doe no hold, hen he limiing diribuion of he foreca will alo depend on he limiing diribuion of F ; he confidence inerval mu reflec hi eimaion effec. heorem allow u o accoun for hi effec when conrucing confidence inerval for he foreca.

13 facor model of large dimenion he covariance marix of he limiing diribuion depend on he correlaion rucure in he cro-ecion dimenion. If e i i independen acro i, hen 4 = lim i= 2 i 0 i i If, in addiion, i 2 = j 2 = 2 for all ij, we have = 2. In Secion 5, we dicu conien eimaion of Q Q. ex, we preen a uniform coniency reul for he eimaed facor. Propoiion 2: Under Aumpion A E, max F H F 0 = O p /2 + O p / /2 hi lemma give an upper bound on he maximum deviaion of he eimaed facor from he rue one up o a ranformaion. he bound i no he harpe poible becaue he proof eenially ue he argumen ha max F H F 0 = F H F 0. oe ha if lim inf /2 c>0, hen he maximum deviaion i O p /2, which i a very rong reul. In addiion, if i i aumed ha max F 0 =O p e.g., when F i ricly aionary and i momen generaing funcion exi, = log, hen i can be hown ha, for = min, max F H F 0 = O p /2 + O p + O p / /2 32 Limiing Diribuion of Eimaed Facor Loading he previou ecion how ha F i an eimae of F 0 H. ow we how ha i an eimae of 0 H. ha i, i i an eimae of H 0 i for every i. he eimaed loading are ued in Lehmann and Mode 988 o conruc variou porfolio. heorem 2: Under Aumpion A G, a : i if / 0, hen for each i, i H 0 F F 0 i = V 0 d 0Q i Q = F 0 e i + o p where V i defined in heorem, Q i given in Propoiion, and i in F4; ii if lim inf / >0, hen i H 0 i = Op

14 48 juhan bai he dominan cae i par i, aympoic normaliy. Par ii i of heoreical inere when a convergence rae i needed. hi rae i min. When he facor F 0 = 2 are all obervable, 0 i can be eimaed by a ime erie regreion wih he ih cro-ecion uni, and he rae of convergence i. he new rae min i due o he fac ha F 0 are no obervable and are eimaed. 33 Limiing Diribuion of Eimaed Common Componen he limi heory of eimaed common componen can be derived from he previou wo heorem. oe ha Ci 0 = F 0 i and C i = F i. heorem 3: Under Aumpion A F, a, we have for each i and, V i + /2 W d 0 i Ci Ci 0 where V i = i 0 i, W i = F F i F F 0, and,, F, and i are all defined earlier. Boh V i and W i can be replaced by heir conien eimaor. A remarkable feaure of heorem 3 i ha no rericion on he relaionhip beween and i required. he eimaed common componen are alway aympoically normal. he convergence rae i = min. o ee hi, heorem 3 can be rewrien a 5 2 Ci C i 0 V i + 2 W i /2 d 0 he denominaor i bounded boh above and below. hu he rae of convergence i min, which i he be rae poible. When F 0 i obervable, he be rae for i i. When 0 i obervable, he be rae for F i. I follow ha when boh are eimaed, he be rae for F i i he minimum of and. heorem 3 ha wo pecial cae: a if / 0, hen Ci Ci 0 d 0V i ; b if / 0, hen C i Ci 0 0W d i. Bu heorem 3 doe no require a limi for / or /. 4 aionary idioyncraic error In he previou ecion, he rae of convergence for F i hown o be min.if i fixed, i implie ha F i no conien. he reul eem o be in conflic wih ha of Connor and Korajczyk 986, who howed ha he eimaor F i conien under fixed. hi inquiry lead u o he dicovery of

15 facor model of large dimenion 49 a neceary and ufficien condiion for coniency under fixed. Connor and Korajczyk impoed he following aumpion: 6 e i e i 0 and i= e 2 i 2 for all a i= We hall call he fir condiion aympoic orhogonaliy and he econd condiion aympoic homokedaiciy 2 no depending on. hey eablihed coniency under aumpion 6. hi aumpion appear o be reaonable for ae reurn and i commonly ued in he finance lieraure, e.g., Campbell, Lo, and Mackinlay 997. For many economic variable, however, one of he condiion could be eaily violaed. We how ha aumpion 6 i alo neceary under fixed. heorem 4: Aume Aumpion A G hold. Under a fixed, a neceary and ufficien condiion for coniency i aympoic orhogonaliy and aympoic homokedaiciy. he implicaion i ha, for fixed, conien eimaion i no poible in he preence of erial correlaion and heerokedaiciy. In conra, under large, we can ill obain conien eimaion. hi reul highligh he imporance of he large-dimenional framework. ex, we how ha our previou reul can alo be renghened under homokedaiciy and no erial correlaion. Aumpion H Ee i e i = 0 if, Ee 2 i = 2 i, and Ee ie j = ij, for all, i, and j. Le 2 = / i= i 2, which i a bounded equence by Aumpion C2. Le V be he diagonal marix coniing of he fir r large eigenvalue of he marix /XX p. Lemma A.3 Appendix A how V V, a poiive definie marix. Define D = V V 2 p ; hen D I r,a and go o infiniy. Define H = HD. heorem 5: Under Aumpion A H, a, we have F H F 0 d 0V Q Q V where = plim 0 / and = Ee e = ij. oe ha cro-ecion correlaion and cro-ecion heerokedaiciy are ill allowed. hu he reul i for approximae facor model. hi heorem doe no require any rericion on he relaionhip beween and excep ha hey boh go o infiniy. he rae of convergence hold even for fixed, bu he limiing diribuion i differen.

16 50 juhan bai If cro-ecion independence and cro-ecion homokedaiciy are aumed, hen heorem 2 par i alo hold wihou any rericion on he relaionhip beween and. However, cro-ecion homokedaiciy i unduly rericive. Aumpion H doe no improve he reul of heorem 3, which already offer he be rae of convergence. 5 eimaing covariance marice In hi ecion, we derive conien eimaor of he aympoic variancecovariance marice ha appear in heorem 3. a Covariance marix of eimaed facor. hi covariance marix depend on he cro-ecion correlaion of he idioyncraic error. Becaue he order of cro-ecional correlaion i unknown, a HAC-ype eimaor ee ewey and We 987 i no feaible. hu we will aume cro-ecion independence for e i i = 2. he aympoic covariance of F i given by = V Q Q V, where i defined in 4. ha i, F = plimv F 0 2 F F i 0 i i V i= hi marix involve he produc F 0, which can be replaced by i eimae F. A conien eimaor of he covariance marix i hen given by F F F F 7 = V i= ẽ 2 i i i V = V i= ẽ 2 i i i V where ẽ i = X i i F. oe ha F F/ = I. b Covariance marix of eimaed facor loading. he aympoic covariance marix of i i given by ee heorem 2 i = Q i Q Le i be he HAC eimaor of ewey and We 987, conruced wih he erie F ẽ i = 2. ha i, i = D 0i + q v= v q + Dvi + D vi where D vi = / F =v+ ẽ i ẽ i v F v and q goe o infiniy a goe o infiniy wih q/ /4 0. One can alo ue oher HAC eimaor uch a Andrew 99 daa dependen mehod wih quadraic pecral kernel. While a HAC eimaor baed on F 0 e i he rue facor and rue idioyncraic error eimae i, a HAC eimaor baed on F ẽ i i direcly eimaing i becaue F eimae H F 0. he coniency of i i proved in he Appendix.

17 facor model of large dimenion 5 c Covariance marix of eimaed common componen. Le V i = i ẽ 2 i i i i i= F Ŵ i = F F F F i F F i F be he eimaor of V i and W i, repecively. heorem 6: Aume Aumpion A G and cro-ecional independence. A,,, i, V i, and Ŵi are conien for, i, V i, and W i, repecively. One major poin of hi heorem i ha all limiing covariance are eaily eimable. 6 mone carlo imulaion We ue imulaion o ae he adequacy of he aympoic reul in approximaing he finie ample diribuion of F i, and C i. o conerve pace, we only repor he reul for F and C i becaue i hare imilar properie o F. We conider combinaion of = 50 00, and = Daa are generaed according o X i = i F 0 + e i, where 0 i, F 0, and e i are i.i.d. 0 for all i and. he number of facor, r, i one. hi give he daa marix X. he eimaed facor F i ime he eigenvecor correponding o he large eigenvalue of XX. Given F, we have = X F/ and C = F. he repored reul are baed on 2000 repeiion. o demonrae ha F i eimaing a ranformaion of F 0 = 2, we compue he correlaion coefficien beween F = and F 0 =. Le l be he correlaion coefficien for he lh repeiion wih given. able I repor hi coefficien averaged over L = 2000 repeiion; ha i, /L L l= l. he correlaion coefficien can be conidered a a meaure of coniency for all. I i clear from he able ha he eimaion preciion increae a grow. Wih = 000, he eimaed facor can be effecively reaed a he rue one. We nex conider aympoic diribuion. he main aympoic reul are 8 F H F 0 d 0 ABLE I Average Correlaion Coefficien beween F 0 = and F = = 25 = 50 = 00 = 000 = =

18 52 juhan bai where = V Q Q V, and V i + /2 W i Ci C 0 d i 0 where V i and W i are defined in heorem 3. We define andardized eimae a f = /2 F H F 0 and c i = V i + Ŵi /2 Ci C 0 i where, V i, and Ŵi are he eimaed aympoic variance given in Secion 5. ha i, he andardizaion i baed on he heoreical mean and heoreical variance raher han he ample mean and ample variance from Mone Carlo repeiion. he andardized eimae hould be approximaely 0 if he aympoic heory i adequae. We nex compue he ample mean and ample andard deviaion d from Mone Carlo repeiion of andardized eimae. For example, for f, he ample mean i defined a f = /L L l= f l and he ample andard deviaion i he quare roo of /L L l= f l f 2, where l refer o he lh repeiion. he ample mean and ample andard deviaion for c i are imilarly defined. We only repor he aiic for = /2 and i = /2, where a i he large ineger maller han or equal o a; oher value of i give imilar reul and hu are no repored. he fir four row of able II are for f and he la four row are for c i. In general, he ample mean are cloe o zero and he andard deviaion are cloe o. Larger generae beer reul han maller. If we ue he woided normal criical value.96 o e he hypohei ha f ha a zero mean wih known variance of, hen he null hypohei i rejeced marginally for ju wo cae: = and = Bu if he eimaed andard deviaion i ued, he wo-ided aiic doe no rejec he zero-mean hypohei for all cae. A for he common componen, c i, all cae excep = 25 rongly poin o zero mean and uni variance. hi i conien wih he aympoic heory. ABLE II Sample Mean and Sandard Deviaion of f and c i = 25 = 50 = 00 = 000 f = 50 mean = 00 mean f = 50 d = 00 d c i = 50 mean = 00 mean c i = 50 d = 00 d

19 facor model of large dimenion 53 We alo preen graphically he above andardized eimae f and c i. Figure diplay he hiogram of f for = 50 and Figure 2 for = 00. he hiogram i caled o be a deniy funcion; he um of he bar heigh ime he bar lengh i equal o. Scaling make for a meaningful comparion wih he andard normal deniy. he laer i overlayed on he hiogram. Figure 3 and Figure 4 how he hiogram of c i for = 50 and = 00, repecively, wih he deniy of 0 again overlayed. I appear ha aympoic heory provide a very good approximaion o he finie ample diribuion. For a given, he larger i, he beer i he approximaion. In addiion, F doe appear o be conien becaue he hiogram i for he eimae F H F 0 divided by i variance. he hiogram ay wihin he ame range for he andardized F a grow from 25 o 000. Similarly, he convergence rae for C i i min for he ame reaon. In general, he limied Mone Carlo imulaion lend uppor o he heory. Figure. Hiogram of eimaed facor = 50. hee hiogram are for he andardized eimae f i.e., F HF 0 divided by he eimaed aympoic andard deviaion. he andard normal deniy funcion i uperimpoed on he hiogram.

20 54 juhan bai Figure 2. Hiogram of eimaed facor = 00. hee hiogram are for he andardized eimae f i.e., F HF 0 divided by he eimaed aympoic andard deviaion. he andard normal deniy funcion i uperimpoed on he hiogram. Finally, we conruc confidence inerval for he rue facor proce. hi i ueful becaue he facor repreen economic indice in variou empirical applicaion. A graphical mehod i ued o illurae he reul. For hi purpoe, no Mone Carlo repeiion i neceary; only one draw a ingle daa e X i needed. he cae of r = i conidered for impliciy. A mall = 20 i ued o avoid a crowded graphical diplay for large, he reuling graph are difficul o read becaue of oo many poin. he value for are 25, 50, 00, 000, repecively. Since F i an eimae of H F 0, he 95% confidence inerval for H F 0, according o 8, i /2 F 96 /2 F / /2 = 2 Becaue F 0 i known in a imulaed environmen, i i beer o ranform he confidence inerval o hoe of F 0 raher han H F 0 and ee if he reuling confidence inerval conain he rue F 0. We can eaily roae F oward F 0 by

21 facor model of large dimenion 55 Figure 3. Hiogram of eimaed common componen = 50. hee hiogram are for he andardized eimae c i i.e., min C i C 0 i divided by he eimaed aympoic andard deviaion. he andard normal deniy funcion i uperimpoed on he hiogram. he regreion 6 F 0 = F+ error. Le ˆ be he lea quare eimae of. hen he 95% confidence inerval for F 0 i L U = /2 ˆ F 96 ˆ /2 /2 ˆ F + 96 ˆ /2 = 2 Figure 5 diplay he confidence inerval L U and he rue facor F 0 = 2. he middle curve i F 0. I i clear ha he larger i, he narrower are he confidence inerval. In mo cae, he confidence inerval conain he rue value of F 0.For = 000, he confidence inerval collape o he rue 6 In claical facor analyi, roaing he eimaed facor i an imporan par of he analyi. hi paricular roaion regreion i no feaible in pracice becaue F 0 i no obervable. he idea here i o how ha F i an eimae of a ranformaion of F 0 and ha he confidence inerval before he roaion i for he ranformed variable F 0 H.

22 56 juhan bai Figure 4. Hiogram of eimaed common componen = 00. hee hiogram are for he andardized eimae c i i.e., min C i C 0 i divided by he eimaed aympoic andard deviaion. he andard normal deniy funcion i uperimpoed on he hiogram. value. hi implie ha here exi a ranformaion of F uch ha i i almo idenical o F 0 when i large. 7 concluding remark hi paper udie facor model under a nonandard eing: large cro ecion and a large ime dimenion. Such large-dimenional facor model have received increaing aenion in he recen economic lieraure. hi paper conider eimaing he model by he principal componen mehod, which i feaible and raighforward o implemen. We derive ome inferenial heory concerning he eimaor, including rae of convergence and limiing diribuion. In conra o claical facor analyi, we are able o eimae conienly boh he facor and heir loading, no ju he laer. In addiion, our reul are obained under very general condiion ha allow for cro-ecional and erial

23 facor model of large dimenion 57 Figure 5. Confidence inerval for F 0 =. hee are he 95% confidence inerval dahed line for he rue facor proce F 0 = 220 when he number of cro ecion varie from 25 o 000. he middle curve olid line repreen he rue facor proce. correlaion and heerokedaiciie. We alo idenify a neceary and ufficien condiion for coniency under fixed. Many iue remain o be inveigaed. he fir i he qualiy of approximaion ha he aympoic diribuion provide o he finie ample diribuion. Alhough our mall Mone Carlo imulaion how adequae approximaion, a horough inveigaion i ill needed o acce he aympoic heory. In paricular, i i ueful o documen under wha condiion he aympoic heory would fail and wha are he characeriic of finie ample diribuion. he ak i hen o develop an alernaive aympoic heory ha can beer capure he known properie of he finie ample diribuion. hi approach o aympoic analyi wa ued by Bekker 994 o derive a new aympoic heory for inrumenalvariable eimaor, where he number of inrumen increae wih he ample ize; alo ee Hahn and Kureiner hi framework can be ueful for facor analyi becaue he daa marix ha wo growing dimenion and ome of our aympoic analyi reric he way in which and can grow. In addiion,

24 58 juhan bai Bekker approach migh be ueful when he number of facor r alo increae wih and. Aympoic heory for an increaing r i no examined in hi paper and remain o be udied. Anoher fruiful area of reearch would be empirical applicaion of he heoreical reul derived in hi paper. Our reul how ha i i no neceary o divide a large ample ino mall ubample in order o conform o a fixed requiremen, a i done in he exiing lieraure. A large ime dimenion deliver conien eimae even under heerokedaiciy and erial correlaion. In conra, conien eimaion i no guaraneed under fixed. Many exiing applicaion ha employed claical facor model can be reexamined uing new daa e wih large dimenion. I would alo be inereing o examine common cycle and co-movemen in he world economy along he line of Forni e al. 2000b and Gregory and Head 999. Recenly, Granger 200 and oher called for large-model analyi o be on he forefron of he economeric reearch agenda. We believe ha large-model analyi will become increaingly imporan. Daa e will naurally expand a daa collecion, orage, and dieminaion become more efficien and le coly. Alo, he increaing inerconnecedne in he world economy mean ha variable acro differen counrie may have igher linkage han ever before. hee fac, in combinaion wih modern compuing power, make large-model analyi more perinen and feaible. We hope ha he reearch of hi paper will encourage furher developmen in he analyi of large model. Deparmen of Economic, ew York Univeriy, 269 Mercer S., 7h Floor, ew York, Y 0003 U.S.A.; juhan.bai@nyu.edu Manucrip received February, 200; final reviion received May, APPEDIX A: Proof of heorem A defined in he main ex, le V be he r r diagonal marix of he fir r large eigenvalue of / XX in decreaing order. By he definiion of eigenvecor and eigenvalue, we have / XX F = FV or / XX FV = F.LeH = 0 / F F/V be an r r marix and = min. Aumpion A and B ogeher wih F F/ = I and Lemma A.3 below imply ha H=O p. heorem i baed on he ideniy alo ee Bai and g 2002: A. where F H F 0 = V F + F + F + F = = = = A.2 = e e = F e / = F e / o analyze each erm above, we need he following lemma.

25 facor model of large dimenion 59 Lemma A.: Under Aumpion A D, 2 F H F 0 2 = O p = Proof: See heorem of Bai and g oe ha hey ued V F V H F 0 2 a he ummand. Becaue V converge o a poiive definie marix ee Lemma A.3 below, i follow ha V =O p and he lemma i implied by heorem of Bai and g. Q.E.D. Lemma A.2: Under Aumpion A F, we have a F = = O p ; b F = = O p ; c F = = O p ; d F = = O p. Proof: Conider par a. By adding and ubracing erm, = = F = = = F H F 0 + H F 0 = F H F 0 + H F 0 ow / F 0 = = O p / ince E F 0 max E F 0 M +/4 by Aumpion A and E. Conider he fir erm: /2 /2 F H F 0 F H F = = = = = which i O p O = O p by Lemma A. and Aumpion E. Conider par b: = = F = = F H F 0 + H F 0 For he fir erm, /2 /2 F H F 0 F H F Furhermore, 2 = [ e e = = [ = = = ] 2 = /2 i= = = [ e e Ee e ] 2 = ] 2 e i e i Ee i e i = O p

26 60 juhan bai hu he fir erm i O p O p ex, = F 0 = = i= F 0 e ie i Ee i e i = O p by Aumpion F. hu = Conider par c: = F = O p O p + O p F = = oe F 0 = = F 0 = = F H F 0 + H F 0 F F H F 0 = = O p k= ke k = O p. he fir erm i /2 F H F 0 2 = /2 2 he fir expreion i O p / by Lemma A.. For he econd expreion, = 2 = = F e / 2 e / 2 F 0 2 = O p ince / = F 0 2 = O p, and e / 2 = O p. hu, c i O p /. Finally for par d, = F = = = = Conider he fir erm F H F 0 = F F = = e / = F e 0 / F 0 = F H F 0 e 0 F 0 + e 0 F 0 = H F 0 e 0 F 0 /2 F H F 0 2 = = = O p O p O p = O p following argumen analogou o hoe of par c. For he econd erm of d, F 0 e 0 F 0 = F 0 e k k F 0 = O p = = k= e 0 2 /2 F 0 by Aumpion F2. hu, H = F 0 e 0 F 0 = O p / and d i O p /. he proof of Lemma A.2 i complee. Q.E.D.

27 facor model of large dimenion 6 Lemma A.3: Aume Aumpion A D hold. A : p i F F XX = V V F F 0 0 F F p ii V where V i he diagonal marix coniing of he eigenvalue of F. Proof: From / XX F = FV and F F/ = I, we have F / XX F = V. he re i implicily proved by Sock and Waon 999. he deail are omied. Q.E.D. Becaue V i poiive definie, he lemma ay ha F F/ i of full rank for all large and and hu i inverible. We alo noe ha Lemma A.3ii how ha a quadraic form of F F/ ha a limi. Bu hi doe no guaranee F F/ ielf ha a limi unle Aumpion G i made. In wha follow, an eigenvecor marix of W refer o he marix whoe column are he eigenvecor of W wih uni lengh and he ih column correpond o he ih large eigenvalue. Proof of Propoiion : Muliply he ideniy / XX F FV on boh ide by 0 / /2 F o obain: 0 /2 XX F F 0 /2 F F = V Expanding XX wih X = F 0 + e, we can rewrie he above a 0 /2 F F 0 0 F F 0 /2 F F A.3 + d = V where 0 /2 [ ] F F 0 d = e F/+ F e 0 F F/+ F ee F/ =o p he o p i implied by Lemma A.2. Le 0 /2 F F 0 0 /2 B = and A.4 0 /2 F F R = hen we can rewrie A.3 a [ B + d R] R = R V hu each column of R, hough no of lengh, i an eigenvecor of he marix B +d R.Le V be a diagonal marix coniing of he diagonal elemen of R R. Denoe = R V /2 o ha each column of ha a uni lengh, and we have [ B + d R] = V hu /2 F /2 i he eigenvecor marix of B + d R. oe ha B + d R converge o B = by Aumpion A and B and d = o p. Becaue he eigenvalue of B are diinc by Aumpion G, he eigenvalue of B + d R will alo be diinc for large and large by he

28 62 juhan bai coninuiy of eigenvalue. hi implie ha he eigenvecor marix of B +d R i unique excep ha each column can be replaced by he negaive of ielf. In addiion, he kh column of R ee A.4 depend on F only hrough he kh column of F k = 2r. hu he ign of each column in R and hu in = R V /2 i implicily deermined by he ign of each column in F. hu, given he column ign of F, i uniquely deermined. By he eigenvecor perurbaion heory which require he diincne of eigenvalue; ee Franklin 968, here exi a unique eigenvecor marix of B = /2 F /2 uch ha =o p. From we have F F 0 = F F p /2 V /2 /2 V /2 by Aumpion B and by V p V in view of Lemma A.3ii. Q.E.D. Proof of heorem : Cae : / 0. By A. and Lemma A.2, we have A.5 F H F 0 = O p + O p + O p + O p he limiing diribuion i deermined by he hird erm of he righ-hand ide of A. becaue i i he dominan erm. Uing he definiion of, A.6 F H F 0 = V F F 0 e i i + o p = i= d ow / i= 0e i i 0 by Aumpion F3. ogeher wih Propoiion and Lemma A.3, we have F H F d 0V Q Q V a aed. Cae 2: If lim inf / >0, hen he fir and he hird erm of A. are he dominan erm. We have F H F 0 = Op + O p / = O p in view of lim up / / <. Q.E.D. Proof of Propoiion 2: We conider each erm on he righ-hand ide of A.. For he fir erm, A.7 max F /2 = = F 2 /2 max /2 2 = he above i O p /2 following from F = 2 = O p and = 2 M for ome M < uniformly in. he remaining hree erm of A. are each O p / /2 uniformly in. o ee hi, le = F =. I uffice o how max 2 = O p /. Bu Bai and g 2002 proved ha = 2 = O p / hey ued noaion b inead of 2. Bai and g alo obained he ame reul for he hird and he fourh erm. Q.E.D.

29 facor model of large dimenion 63 APPEDIX B: Proof of heorem 2 o prove heorem 2, we need ome preliminary reul. Lemma B.: Under Aumpion A F, we have F F 0 H e i = O p 2 Proof: From he ideniy A., we have = F H F 0 [ ei = V We begin wih I, which can be rewrien a I = 2 = = he fir erm i bounded by /2 2 = = F e i F e i + 2 = = = V I + II + III + IV = = = = = = F H F 0 e i + 2 H F 0 e i = /2 F H F 0 2 = = 2 e 2 i = F e i ] F e i /2 = /2 O p Op where he O p follow from Ee 2 M and i = = 2 M by Lemma i of Bai and g he expeced value of he econd erm of I i bounded by ignore H 2 E F 0 2 /2 Ee 2 i /2 M = O = = by Aumpion C2. hu I i O p /2.ForII, we rewrie i a II = 2 F H F 0 e i + 2 H F 0 e i = = = = = = = he econd erm i O p / by Aumpion F. o ee hi, he econd erm can be wrien a / / z = e i wih z = / F 0e = k= ke k Ee k e k. By F, Ez 2 <M. hu Ez e i EZ 2 Ee 2 i /2 M. hi implie / z = e i = O p. For he fir erm, we have 2 F H F 0 e i = = /2 2 /2 F H F 0 2 e i = = Bu = e i = e k e k Ee k e k e i = O p /2 = k=

30 64 juhan bai So he fir erm i O p hu II = O p /. For III, we rewrie i a III = 2 = O p = O p = = he fir erm i bounded by /2 F H F 0 2 F H F 0 e i + 2 H F 0 e i = = = 2 /2 e i = = O p O p becaue e i = F k e k e i = = k= which i O p /2. he econd erm of III can be wrien a H F 0 F B. k e k e i = = k= which i O p /2 if cro-ecion independence hold for he e. Under weak cro-ecional dependence a in Aumpion E2, he above i O p /2 + O p. hi follow from e k e i = e k e i ki + ki, where ki = Ee k e i. We have / = k= ki / k= ki = O by E2, where ki ki. In ummary, III i O p / + O. he proof of IV i imilar o ha of III. hu I + II + III + IV = O p + O p + O p + O p = O p 2 Lemma B.2: Under Aumpion A F, he r r marix F F 0 H F 0 = O p 2. Proof: Uing he ideniy A., we have [ F H F 0 F = V 2 = = = F F + 2 = = + 2 F F = V I + II + III + IV = = + 2 = = F F F F ] erm I i O p /2. he proof i he ame a ha of I of Lemma B.. ex, II = 2 = = F H F 0 F + 2 H F 0 = = F QED