DETERMINING THE NUMBER OFFACTORS IN APPROXIMATE FACTOR MODELS. By Jushan Bai and Serena Ng 1

Transcription

1 Economerica, Vol. 70, o. 1 January, 2002, DEERMIIG HE UMBER OFFACORS I APPROXIMAE FACOR MODELS By Jushan Bai and Serena g 1 In his paper we develop some economeric heory for facor models of large dimensions. he focus is he deerminaion of he number of facors r, which is an unresolved issue in he rapidly growing lieraure on mulifacor models. We firs esablish he convergence rae for he facor esimaes ha will allow for consisen esimaion of r. We hen propose some panel crieria and show ha he number of facors can be consisenly esimaed using he crieria. he heory is developed under he framework of large crosssecions and large ime dimensions. o resricion is imposed on he relaion beween and. Simulaions show ha he proposed crieria have good finie sample properies in many configuraions of he panel daa encounered in pracice. Keywords: Facor analysis, asse pricing, principal componens, model selecion. 1 inroducion he idea ha variaions in a large number of economic variables can be modeled by a small number of reference variables is appealing and is used in many economic analyses. For example, asse reurns are ofen modeled as a funcion of a small number of facors. Sock and Wason 1989 used one reference variable o model he comovemens of four main macroeconomic aggregaes. Cross-counry variaions are also found o have common componens; see Gregory and Head 1999 and Forni, Hallin, Lippi, and Reichlin 2000b. More recenly, Sock and Wason 1999 showed ha he forecas error of a large number of macroeconomic variables can be reduced by including diffusion indexes, or facors, in srucural as well as nonsrucural forecasing models. In demand analysis, Engel curves can be expressed in erms of a finie number of facors. Lewbel 1991 showed ha if a demand sysem has one common facor, budge shares should be independen of he level of income. In such a case, he number of facors is an objec of economic ineres since if more han one facor is found, homoheic preferences can be rejeced. Facor analysis also provides a convenien way o sudy he aggregae implicaions of microeconomic behavior, as shown in Forni and Lippi Cenral o boh he heoreical and he empirical validiy of facor models is he correc specificaion of he number of facors. o dae, his crucial parameer 1 We hank hree anonymous referees for heir very consrucive commens, which led o a much improved presenaion. he firs auhor acknowledges financial suppor from he aional Science Foundaion under Gran SBR We would like o hank paricipans in he economerics seminars a Harvard-MI, Cornell Universiy, he Universiy of Rocheser, and he Universiy of Pennsylvania for help suggesions and commens. Remaining errors are our own. 191

2 192 jushan bai and serena ng is ofen assumed raher han deermined by he daa. 2 his paper develops a formal saisical procedure ha can consisenly esimae he number of facors from observed daa. We demonsrae ha he penaly for overfiing mus be a funcion of boh and he cross-secion dimension and he ime dimension, respecively in order o consisenly esimae he number of facors. Consequenly he usual AIC and BIC, which are funcions of or alone, do no work when boh dimensions of he panel are large. Our heory is developed under he assumpion ha boh and converge o infiniy. his flexibiliy is of empirical relevance because he ime dimension of daases relevan o facor analysis, alhough small relaive o he cross-secion dimension, is oo large o jusify he assumpion of a fixed. A small number of papers in he lieraure have also considered he problem of deermining he number of facors, bu he presen analysis differs from hese works in imporan ways. Lewbel 1991 and Donald 1997 used he rank of a marix o es for he number of facors, bu hese heories assume eiher or is fixed. Cragg and Donald 1997 considered he use of informaion crieria when he facors are funcions of a se of observable explanaory variables, bu he daa sill have a fixed dimension. For large dimensional panels, Connor and Korajczyk 1993 developed a es for he number of facors in asse reurns, bu heir es is derived under sequenial limi asympoics, i.e., converges o infiniy wih a fixed and hen converges o infiniy. Furhermore, because heir es is based on a comparison of variances over differen ime periods, covariance saionariy and homoskedasiciy are no only echnical assumpions, bu are crucial for he validiy of heir es. Under he assumpion ha for fixed, Forni and Reichlin 1998 suggesed a graphical approach o idenify he number of facors, bu no heory is available. Assuming wih /, Sock and Wason 1998 showed ha a modificaion o he BIC can be used o selec he number of facors opimal for forecasing a single series. heir crierion is resricive no only because i requires, bu also because here can be facors ha are pervasive for a se of daa and ye have no predicive abiliy for an individual daa series. hus, heir rule may no be appropriae ouside of he forecasing framework. Forni, Hallin, Lippi, and Reichlin 2000a suggesed a mulivariae varian of he AIC bu neiher he heoreical nor he empirical properies of he crierion are known. We se up he deerminaion of facors as a model selecion problem. In consequence, he proposed crieria depend on he usual rade-off beween good fi and parsimony. However, he problem is nonsandard no only because accoun needs o be aken of he sample size in boh he cross-secion and he imeseries dimensions, bu also because he facors are no observed. he heory we developed does no rely on sequenial limis, nor does i impose any resricions beween and. he resuls hold under heeroskedasiciy in boh he ime and 2 Lehmann and Modes 1988, for example, esed he AP for 5, 10, and 15 facors. Sock and Wason 1989 assumed here is one facor underlying he coinciden index. Ghysels and g 1998 esed he affine erm srucure model assuming wo facors.

3 approximae facor models 193 he cross-secion dimensions. he resuls also hold under weak serial and crosssecion dependence. Simulaions show ha he crieria have good finie sample properies. he res of he paper is organized as follows. Secion 2 ses up he preliminaries and inroduces noaion and assumpions. Esimaion of he facors is considered in Secion 3 and he esimaion of he number of facors is sudied in Secion 4. Specific crieria are considered in Secion 5 and heir finie sample properies are considered in Secion 6, along wih an empirical applicaion o asse reurns. Concluding remarks are provided in Secion 7. All he proofs are given in he Appendix. 2 facor models Le X i be he observed daa for he ih cross-secion uni a ime, for i = 1, and = 1. Consider he following model: 1 X i = i F + e i where F is a vecor of common facors, i is a vecor of facor loadings associaed wih F, and e i is he idiosyncraic componen of X i. he produc i F is called he common componen of X i. Equaion 1 is hen he facor represenaion of he daa. oe ha he facors, heir loadings, as well as he idiosyncraic errors are no observable. Facor analysis allows for dimension reducion and is a useful saisical ool. Many economic analyses fi naurally ino he framework given by Arbirage pricing heory. In he finance lieraure, he arbirage pricing heory AP of Ross 1976 assumes ha a small number of facors can be used o explain a large number of asse reurns. In his case, X i represens he reurn of asse i a ime F represens he vecor of facor reurns, and e i is he idiosyncraic componen of reurns. Alhough analyical convenience makes i appealing o assume one facor, here is growing evidence agains he adequacy of a single facor in explaining asse reurns. 3 he shifing ineres owards use of mulifacor models ineviably calls for a formal procedure o deermine he number of facors. he analysis o follow allows he number of facors o be deermined even when and are boh large. his is especially suied for financial applicaions when daa are widely available for a large number of asses over an increasingly long span. Once he number of facors is deermined, he facor reurns F can also be consisenly esimaed up o an inverible ransformaion. 2. he rank of a demand sysem. Le p be a price vecor for J goods and services, e h be oal spending on he J goods by household h. Consumer heory posulaes ha Marshallian demand for good j by consumer h is X jh = g j p e h. Le w jh = X jh /e h be he budge share for household h on he jh good. he 3 Cochrane 1999 sressed ha financial economiss now recognize ha here are muliple sources of risk, or facors, ha give rise o high reurns. Backus, Forsei, Mozumdar, and Wu 1997 made similar conclusions in he conex of he marke for foreign asses.

4 194 jushan bai and serena ng rank of a demand sysem holding prices fixed is he smalles ineger r such ha w j e = j1 G 1 e+ + jr G r e. Demand sysems are of he form 1 where he r facors, common across goods, are F h = G 1 e h G r e h. When he number of households, H, converges o infiniy wih a fixed JG 1 e G r e can be esimaed simulaneously, such as by nonparameric mehods developed in Donald his approach will no work when he number of goods, J, also converges o infiniy. However, he heory o be developed in his paper will sill provide a consisen esimaion of r and wihou he need for nonparameric esimaion of he G funcions. Once he rank of he demand sysem is deermined, he nonparameric funcions evaluaed a e h allow F h o be consisenly esimable up o a ransformaion. hen funcions G 1 e G r e may be recovered also up o a marix ransformaion from F h h = 1Hvia nonparameric esimaion. 3. Forecasing wih diffusion indices. Sock and Wason 1998, 1999 considered forecasing inflaion wih diffusion indices facors consruced from a large number of macroeconomic series. he underlying premise is ha hese series may be driven by a small number of unobservable facors. Consider he forecasing equaion for a scalar series y +1 = F + W + he variables W are observable. Alhough we do no observe F, we observe X i i = 1. Suppose X i bears relaion wih F as in 1. In he presen conex, we inerpre 1 as he reduced-form represenaion of X i in erms of he unobservable facors. We can firs esimae F from 1. Denoe i by F.We can hen regress y on F 1 and W 1 o obain he coefficiens ˆ and ˆ, from which a forecas ŷ +1 = ˆ F + ˆW can be formed. Sock and Wason 1998, 1999 showed ha his approach of forecasing ouperforms many compeing forecasing mehods. Bu as poined ou earlier, he dimension of F in Sock and Wason 1998, 1999 was deermined using a crierion ha minimizes he mean squared forecas errors of y. his may no be he same as he number of facors underlying X i, which is he focus of his paper. 21 oaion and Preliminaries Le F 00 i, and r denoe he rue common facors, he facor loadings, and he rue number of facors, respecively. oe ha F 0 is r dimensional. We assume ha r does no depend on or. A a given, we have 2 X = 0 F 0 + e 1 r r 1 1 where X = X 1 X 2 X 0 = , and e = e 1 e 2 e. Our objecive is o deermine he rue number of facors, r. In classical

5 approximae facor models 195 facor analysis e.g., Anderson 1984, is assumed fixed, he facors are independen of he errors e, and he covariance of e is diagonal. ormalizing he covariance marix of F o be an ideniy marix, we have = 0 0 +, where and are he covariance marices of X and e, respecively. Under hese assumpions, a roo- consisen and asympoically normal esimaor of, say, he sample covariance marix = 1/ X XX X can be obained. he essenials of classical facor analysis carry over o he case of large bu fixed since he problem can be urned ino a problem, as noed by Connor and Korajczyk 1993 and ohers. Inference on r under classical assumpions can, in heory, be based on he eigenvalues of since a characerisic of a panel of daa ha has an r facor represenaion is ha he firs r larges populaion eigenvalues of he covariance of X diverge as increases o infiniy, bu he r + 1h eigenvalue is bounded; see Chamberlain and Rohschild Bu i can be shown ha all nonzero sample eigenvalues no jus he firs r of he marix increase wih, and a es based on he sample eigenvalues is hus no feasible. A likelihood raio es can also, in heory, be used o selec he number of facors if, in addiion, normaliy of e is assumed. Bu as found by Dhrymes, Friend, and Gluekin 1984, he number of saisically significan facors deermined by he likelihood raio es increases wih even if he rue number of facors is fixed. Oher mehods have also been developed o esimae he number of facors assuming he size of one dimension is fixed. Bu Mone Carlo simulaions in Cragg and Donald 1997 show ha hese mehods end o perform poorly for moderaely large and. he fundamenal problem is ha he heory developed for classical facor models does no apply when boh and. his is because consisen esimaion of wheher i is an or a marix is no a well defined problem. For example, when >, he rank of is no more han, whereas he rank of can always be. ew heories are hus required o analyze large dimensional facor models. In his paper, we develop asympoic resuls for consisen esimaion of he number of facors when and. Our resuls complemen he sparse bu growing lieraure on large dimensional facor analysis. Forni and Lippi 2000 and Forni e al. 2000a obained general resuls for dynamic facor models, while Sock and Wason 1998 provided some asympoic resuls in he conex of forecasing. As in hese papers, we allow for cross-secion and serial dependence. In addiion, we also allow for heeroskedasiciy in e and some weak dependence beween he facors and he errors. hese laer generalizaions are new in our analysis. Evidenly, our assumpions are more general han hose used when he sample size is fixed in one dimension. Le X i be a 1 vecor of ime-series observaions for he ih cross-secion uni. For a given i, we have 3 X i = F 0 0 i + e i 1 r r 1 1

6 196 jushan bai and serena ng where X i = X i1 X i2 X i F 0 = F 0 1 F0 2 F0, and e i = e i1 e i2 e i. For he panel of daa X = X 1 X, we have 4 X = F e r r wih e = e 1 e. Le ra denoe he race of A. he norm of he marix A is hen A = ra A 1/2. he following assumpions are made: Assumpion A Facors: EF 0 4 < and 1 F 0 F 0 F as for some posiive definie marix F. Assumpion B Facor Loadings: i <, and 0 0 / D 0 as for some r r posiive definie marix D. Assumpion C ime and Cross-Secion Dependence and Heeroskedasiciy: here exiss a posiive consan M<, such ha for all and, 1. Ee i = 0Ee i 8 M; 2. Ee s e / = E 1 e ise i = s s s M for all s, and 1 s M; 3. Ee i e j = ij wih ij ij for some ij and for all ; in addiion, 1 j=1 ij M; 4. Ee i e js = ij s and 1 j=1 ij s M; 5. for every s E 1/2 e ise i Ee is e i 4 M. Assumpion D Weak Dependence beween Facors and Idiosyncraic Errors: 1 E 1 F 0 e i 2 M Assumpion A is sandard for facor models. Assumpion B ensures ha each facor has a nonrivial conribuion o he variance of X. We only consider nonrandom facor loadings for simpliciy. Our resuls sill hold when he i are random, provided hey are independen of he facors and idiosyncraic errors, and E i 4 M. Assumpion C allows for limied ime-series and cross-secion dependence in he idiosyncraic componen. Heeroskedasiciy in boh he ime and cross-secion dimensions is also allowed. Under saionariy in he ime dimension, s = s, hough he condiion is no necessary. Given Assumpion C1, he remaining assumpions in C are easily saisfied if he e i are independen for all i and. he allowance for some correlaion in he idiosyncraic componens ses up he model o have an approximae facor srucure. I is more general han a sric facor model, which assumes e i is uncorrelaed across i, he framework in which he AP heory of Ross 1976 is based. hus, he resuls o be developed will also apply o sric facor models. When he facors

7 approximae facor models 197 and idiosyncraic errors are independen a sandard assumpion for convenional facor models, Assumpion D is implied by Assumpions A and C. Independence is no required for D o be rue. For example, suppose ha e i = i F wih i being independen of F and i saisfies Assumpion C; hen Assumpion D holds. Finally, he developmens proceed assuming ha he panel is balanced. We also noe ha he model being analyzed is saic, in he sense ha X i has a conemporaneous relaionship wih he facors. he analysis of dynamic models is beyond he scope of his paper. For a facor model o be an approximae facor model in he sense of Chamberlain and Rohschild 1983, he larges eigenvalue and hence all of he eigenvalues of he covariance marix = Ee e mus be bounded. oe ha Chamberlain and Rohschild focused on he cross-secion behavior of he model and did no make explici assumpions abou he ime-series behavior of he model. Our framework allows for serial correlaion and heeroskedasiciy and is more general han heir seup. Bu if we assume e is saionary wih Ee i e j = ij, hen from marix heory, he larges eigenvalue of is bounded by max i j=1 ij. hus if we assume j=1 ij M for all i and all, which implies Assumpion C3, hen 2 will be an approximae facor model in he sense of Chamberlain and Rohschild. 3 esimaion of he common facors When is small, facor models are ofen expressed in sae space form, normaliy is assumed, and he parameers are esimaed by maximum likelihood. For example, Sock and Wason 1989 used = 4 variables o esimae one facor, he coinciden index. he drawback is ha because he number of parameers increases wih, 4 compuaional difficulies make i necessary o abandon informaion on many series even hough hey are available. We esimae common facors in large panels by he mehod of asympoic principal componens. 5 he number of facors ha can be esimaed by his nonparameric mehod is min, much larger han permied by esimaion of sae space models. Bu o deermine which of hese facors are saisically imporan, i is necessary o firs esablish consisency of all he esimaed common facors when boh and are large. We sar wih an arbirary number kk <min. he superscrip in k i and F k signifies he allowance of k facors in he esimaion. Esimaes of k and F k are obained by solving he opimizaion problem Vk= min 1 Xi k F k i F k 2 4 Gregory, Head, and Raynauld 1997 esimaed a world facor and seven counry specific facors from oupu, consumpion, and invesmen for each of he G7 counries. he exercise involved esimaion of 92 parameers and perhaps sreched he sae-space model o is limi. 5 he mehod of asympoic principal componens was sudied by Connor and Korajzcyk 1986 and Connor and Korajzcyk 1988 for fixed. Forni e al. 2000a and Sock and Wason 1998 considered he mehod for large.

8 198 jushan bai and serena ng subjec o he normalizaion of eiher k k / = I k or F k F k / = I k. If we concenrae ou k and use he normalizaion ha F k F k / = I k, he opimizaion problem is idenical o maximizing rf k XX F k. he esimaed facor marix, denoed by F k,is imes he eigenvecors corresponding o he k larges eigenvalues of he marix XX. Given F k k = F k F k 1 k F X = F k X/ is he corresponding marix of facor loadings. he soluion o he above minimizaion problem is no unique, even hough he sum of squared residuals Vk is unique. Anoher soluion is given by F k k, where k is consruced as imes he eigenvecors corresponding o he k larges eigenvalues of he marix X X. he normalizaion ha k k / = I k implies F k = X k /. he second se of calculaions is compuaionally less cosly when >, while he firs is less inensive when <. 6 Define F k = F k F k F k / 1/2 a rescaled esimaor of he facors. he following heorem summarizes he asympoic properies of he esimaed facors. heorem 1: For any fixed k 1, here exiss a r k marix H k rankh k = mink r, and C = min, such ha 1 C 2 k 5 F 2 = O p 1 wih Because he rue facors F 0 can only be idenified up o scale, wha is being considered is a roaion of F 0. he heorem esablishes ha he ime average of he squared deviaions beween he esimaed facors and hose ha lie in he rue facor space vanish as. he rae of convergence is deermined by he smaller of or, and hus depends on he panel srucure. Under he addiional assumpion ha s 2 M for all and, he resul 7 6 C 2 F 2 = O p 1 for each can be obained. eiher heorem 1 nor 6 implies uniform convergence in. Uniform convergence is considered by Sock and Wason hese auhors obained a much slower convergence rae han C 2, and heir resul requires. An imporan insigh of his paper is ha, o consisenly esimae he number of facors, neiher 6 nor uniform convergence is required. I is he average convergence rae of heorem 1 ha is essenial. However, 6 could be useful for saisical analysis on he esimaed facors and is hus a resul of independen ineres. 6 A more deailed accoun of compuaion issues, including how o deal wih unbalanced panels, is given in Sock and Wason he proof is acually simpler han ha of heorem 1 and is hus omied o avoid repeiion.

9 approximae facor models esimaing he number of facors Suppose for he momen ha we observe all poenially informaive facors bu no he facor loadings. hen he problem is simply o choose k facors ha bes capure he variaions in X and esimae he corresponding facor loadings. Since he model is linear and he facors are observed, i can be esimaed by applying ordinary leas squares o each equaion. his is hen a classical model selecion problem. A model wih k + 1 facors can fi no worse han a model wih k facors, bu efficiency is los as more facor loadings are being esimaed. Le F k be a marix of k facors, and VkF k 1 = min Xi k i F k 2 be he sum of squared residuals divided by from ime-series regressions of X i on he k facors for all i. hen a loss funcion VkF k + kg, where g is he penaly for overfiing, can be used o deermine k. Because he esimaion of i is classical, i can be shown ha he BIC wih g = ln / can consisenly esimae r. On he oher hand, he AIC wih g = 2/ may choose k>r even in large samples. he resul is he same as in Geweke and Meese 1981 derived for = 1 because when he facors are observed, he penaly facor does no need o ake ino accoun he sample size in he cross-secion dimension. Our main resul is o show ha his will no longer be rue when he facors have o be esimaed, and even he BIC will no always consisenly esimae r. Wihou loss of generaliy, we le 7 Vk F k 1 = min Xi k i F k 2 denoe he sum of squared residuals divided by when k facors are esimaed. his sum of squared residuals does no depend on which esimae of F is used because hey span he same vecor space. ha is, Vk F k = Vk F k = Vk F k. We wan o find penaly funcions, g, such ha crieria of he form PCk = Vk F k + kg can consisenly esimae r. Lekmax be a bounded ineger such ha r kmax. heorem 2: Suppose ha Assumpions A D hold and ha he k facors are esimaed by principal componens. Le ˆk = arg min0 k kmax PCk. hen lim Probˆk = r = 1 if i g 0 and ii C 2 g as,, where C = min. Condiions i and ii are necessary in he sense ha if one of he condiions is violaed, hen here will exis a facor model saisfying Assumpions A D, and

10 200 jushan bai and serena ng ye he number of facors canno be consisenly esimaed. However, condiions i and ii are no always required o obain a consisen esimae of r. A formal proof of heorem 2 is provided in he Appendix. he crucial elemen in consisen esimaion of r is a penaly facor ha vanishes a an appropriae rae such ha under and overparameerized models will no be chosen. An implicaion of heorem 2 is he following: Corollary 1: Under he Assumpions of heorem 2, he class of crieria defined by ICk = lnv k F k + kg will also consisenly esimae r. oe ha Vk F k is simply he average residual variance when k facors are assumed for each cross-secion uni. he IC crieria hus resemble informaion crieria frequenly used in ime-series analysis, wih he imporan difference ha he penaly here depends on boh and. hus far, i has been assumed ha he common facors are esimaed by he mehod of principle componens. Forni and Reichlin 1998 and Forni e al. 2000a sudied alernaive esimaion mehods. However he proof of heorem 2 mainly uses he fac ha F saisfies heorem 1, and does no rely on principal componens per se. We have he following corollary: Corollary 2: Le Ĝk be an arbirary esimaor of F 0. Suppose here exiss a marix H k such ha rank H k = mink r, and for some C 2 C2, 8 C 2 1 Ĝ k H k F 0 2 = O p 1 hen heorem 2 sill holds wih F k replaced by Ĝk and C replaced by C. he sequence of consans C 2 does no need o equal C2 = min. heorem 2 holds for any esimaion mehod ha yields esimaors Ĝ saisfying 8. 8 aurally, he penaly would hen depend on C 2, he convergence rae for Ĝ. 5 he PC p and he IC p In his secion, we assume ha he mehod of principal componens is used o esimae he facors and propose specific formulaions of g o be used in 8 We are graeful for a referee whose quesion led o he resuls repored here.

11 approximae facor models 201 pracice. Le ˆ 2 be a consisen esimae of 1 Ee i 2. Consider he following crieria: + PC p1 k = Vk F k + k ˆ 2 + PC p2 k = Vk F k + k ˆ 2 PC p3 k = Vk F k + k ˆ 2 ln C 2 C 2 ln + ln C 2 Since Vk F k = 1 ˆ i 2, where ˆ i 2 = ê iêi/, he crieria generalize he C p crierion of Mallows 1973 developed for selecion of models in sric imeseries or cross-secion conexs o a panel daa seing. For his reason, we refer o hese saisics as Panel C p PC p crieria. Like he C p crierion, ˆ 2 provides he proper scaling o he penaly erm. In applicaions, i can be replaced by Vkmax F kmax. he proposed penaly funcions are based on he sample size in he smaller of he wo dimensions. All hree crieria saisfy condiions i and ii of heorem 2 since C 2 + / 0as. However, in finie samples, C 2 + /. Hence, he hree crieria, alhough asympoically equivalen, will have differen properies in finie samples. 9 Corollary 1 leads o consideraion of he following hree crieria: 9 IC p1 k = lnv k F + k + k IC p2 k = lnv k F + k + k IC p3 k = lnv k F ln C 2 k + k C 2 ln + ln C 2 he main advanage of hese hree panel informaion crieria IC p is ha hey do no depend on he choice of kmax hrough ˆ 2, which could be desirable in pracice. he scaling by ˆ 2 is implicily performed by he logarihmic ransformaion of Vk F k and hus no required in he penaly erm. he proposed crieria differ from he convenional C p and informaion crieria used in ime-series analysis in ha g is a funcion of boh and.o undersand why he penaly mus be specified as a funcion of he sample size in 9 oe ha PC p1 and PC p2, and likewise, IC p1 and IC p2, apply specifically o he principal componens esimaor because C 2 = min is used in deriving hem. For alernaive esimaors saisfying Corollary 2, crieria PC p3 and IC p3 are sill applicable wih C replaced by C.

12 202 jushan bai and serena ng boh dimensions, consider he following: 2 AIC 1 k = Vk F k + k ˆ 2 ln BIC 1 k = Vk F k + k ˆ 2 AIC 2 k = Vk F k + k ˆ 2 2 BIC 2 k = Vk F k + k ˆ 2 ln AIC 3 k = Vk F k + k ˆ k BIC 3 k = Vk F k + k ˆ 2 + kln he penaly facors in AIC 1 and BIC 1 are sandard in ime-series applicaions. Alhough g 0as AIC 1 fails he second condiion of heorem 2 for all and. When and log /, he BIC 1 also fails condiion ii of heorem 2. hus we expec he AIC 1 will no work for all and, while he BIC 1 will no work for small relaive o. By analogy, AIC 2 also fails he condiions of heorem 2, while BIC 2 will work only if. he nex wo crieria, AIC 3 and BIC 3, ake ino accoun he panel naure of he problem. he wo specificaions of g reflec firs, ha he effecive number of observaions is, and second, ha he oal number of parameers being esimaed is k + k. I is easy o see ha AIC 3 fails he second condiion of heorem 2. While he BIC 3 saisfies his condiion, g does no always vanish. For example, if = exp, hen g 1 and he firs condiion of heorem 2 will no be saisfied. Similarly, g does no vanish when = exp. herefore BIC 3 may perform well for some bu no all configuraions of he daa. In conras, he proposed crieria saisfy boh condiions saed in heorem 2. 6 simulaions and an empirical applicaion We firs simulae daa from he following model: r X i = ij F j + e i j=1 = c i + e i where he facors are r marices of 01 variables, and he facor loadings are 01 variaes. Hence, he common componen of X i, denoed by c i, has variance r. Resuls wih ij uniformly disribued are similar and will no

13 approximae facor models 203 be repored. Our base case assumes ha he idiosyncraic componen has he same variance as he common componen i.e. = r. We consider hiry configuraions of he daa. he firs five simulae plausible asse pricing applicaions wih five years of monhly daa = 60 on 100 o 2000 asse reurns. We hen increase o 100. Configuraions wih = 60 = 100 and 200 are plausible sizes of daases for secors, saes, regions, and counries. Oher configuraions are considered o assess he general properies of he proposed crieria. All compuaions were performed using Malab Version 5.3. Repored in ables I o III are he averages of ˆk over 1000 replicaions, for r = 1 3, and 5 respecively, assuming ha e i is homoskedasic 01. For all cases, he maximum number of facors, kmax, is se o Prior o compuaion of he eigenvecors, each series is demeaned and sandardized o have uni variance. Of he hree PC p crieria ha saisfy heorem 2, PC p3 is less robus han PC p1 and PC p2 when or is small. he IC p crieria generally have properies very similar o he PC p crieria. he erm / + provides a small sample correcion o he asympoic convergence rae of C 2 and has he effec of adjusing he penaly upwards. he simulaions show his adjusmen o be desirable. When min is 40 or larger, he proposed ess give precise esimaes of he number of facors. Since our heory is based on large and,iis no surprising ha for very small or, he proposed crieria are inadequae. Resuls repored in he las five rows of each able indicae ha he IC p crieria end o underparameerize, while he PC p end o overparameerize, bu he problem is sill less severe han he AIC and he BIC, which we now consider. he AIC and BIC s ha are funcions of only or have he endency o choose oo many facors. he AIC 3 performs somewha beer han AIC 1 and AIC 2, bu sill ends o overparameerize. A firs glance, he BIC 3 appears o perform well. Alhough BIC 3 resembles PC p2, he former penalizes an exra facor more heavily since ln > ln C 2. As can be seen from ables II and III, he BIC 3 ends o underesimae r, and he problem becomes more severe as r increases. able IV relaxes he assumpion of homoskedasiciy. Insead, we le e i = ei 1 for odd, and e i = ei 1 +e2 i for even, where e1 i and e2 i are independen 01. hus, he variance in he even periods is wice as large as he odd periods. Wihou loss of generaliy, we only repor resuls for r = 5. PC p1 PC p2 IC p1, and IC p2 coninue o selec he rue number of facors very accuraely and dominae he remaining crieria considered. We hen vary he variance of he idiosyncraic errors relaive o he common componen. When <r, he variance of he common componen is relaively large. o surprisingly, he proposed crieria give precise esimaes of r. he resuls will no be repored wihou loss of generaliy. able V considers he case = 2r. Since he variance of he idiosyncraic componen is larger han he 10 In ime-series analysis, a rule such as 8 in[ /100 1/4 ] considered in Schwer 1989 is someimes used o se kmax, bu no such guide is available for panel analysis. Unil furher resuls are available, a rule ha replaces in Schwer s rule by min could be considered.

14 204 jushan bai and serena ng ABLE I DGP: X i = r j=1 ij F j + e i ; r = 1; = 1. PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC oes: able I able VIII repor he esimaed number of facors ˆk averaged over 1000 simulaions. he rue number of facors is r and kmax = 8. When he average of ˆk is an ineger, he corresponding sandard error is zero. In he few cases when he averaged ˆk over replicaions is no an ineger, he sandard errors are no larger han.6. In view of he precision of he esimaes in he majoriy of cases, he sandard errors in he simulaions are no repored. he las five rows of each able are for models of small dimensions eiher or is small. common componen, one migh expec he common facors o be esimaed wih less precision. Indeed, IC p1 and IC p2 underesimae r when min < 60, bu he crieria sill selec values of k ha are very close o r for oher configuraions of he daa. he models considered hus far have idiosyncraic errors ha are uncorrelaed across unis and across ime. For hese sric facor models, he preferred crieria are PC p1 PC p2 IC 1, and IC 2. I should be emphasized ha he resuls repored are he averages of ˆk over 1000 simulaions. We do no repor he sandard deviaions of hese averages because hey are idenically zero excep for a few

15 approximae facor models 205 ABLE II DGP: X i = r j=1 ij F j + e i ; r = 3; = 3. PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC cases for which he average iself is no an ineger. Even for hese laer cases, he sandard deviaions do no exceed 0.6. We nex modify he assumpion on he idiosyncraic errors o allow for serial and cross-secion correlaion. hese errors are generaed from he process e i = e i 1 + v i + J j 0j= J v i j he case of pure serial correlaion obains when he cross-secion correlaion parameer is zero. Since for each i, he uncondiional variance of e i is 1/1 2, he more persisen are he idiosyncraic errors, he larger are heir variances relaive o he common facors, and he precision of he esimaes can be expeced o fall. However, even wih = 5, able VI shows ha he esimaes provided by he proposed crieria are sill very good. he case of pure cross-

16 206 jushan bai and serena ng ABLE III DGP: X i = r j=1 ij F j + e i ; r = 5; = 5. PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC secion dependence obains wih = 0. As in Chamberlain and Rohschild 1983, our heory permis some degree of cross-secion correlaion. Given he assumed process for e i, he amoun of cross correlaion depends on he number of unis ha are cross correlaed 2J, as well as he magniude of he pairwise correlaion. We se o.2 and J o max / Effecively, when 200, 10 percen of he unis are cross correlaed, and when >200 20/ of he sample is cross correlaed. As he resuls in able VII indicae, he proposed crieria sill give very good esimaes of r and coninue o do so for small variaions in and J. able VIII repors resuls ha allow for boh serial and cross-secion correlaion. he variance of he idiosyncraic errors is now 1 + 2J 2 /1 2 imes larger han he variance of he common componen. While his reduces he precision of he esimaes somewha, he resuls generally confirm ha a small degree of correlaion in he idiosyncraic errors will no affec he properies of

17 approximae facor models 207 ABLE IV DGP: X i = r j=1 ij F j + e i ; e i = e 1 + i e 2 i = 1 for Even, = 0 for Odd; r = 5; = 5. PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC he esimaes. However, i will generally be rue ha for he proposed crieria o be as precise in approximae as in sric facor models, has o be fairly large relaive o J canno be oo large, and he errors canno be oo persisen as required by heory. I is also noeworhy ha he BIC 3 has very good properies in he presence of cross-secion correlaions see ables VII and VIII and he crierion can be useful in pracice even hough i does no saisfy all he condiions of heorem Applicaion o Asse Reurns Facor models for asse reurns are exensively sudied in he finance lieraure. An excellen summary on mulifacor asse pricing models can be found in Campbell, Lo, and Mackinlay wo basic approaches are employed. One

18 208 jushan bai and serena ng ABLE V DGP: X i = r j=1 ij F j + e i ; r = 5; = r 2. PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC is saisical facor analysis of unobservable facors, and he oher is regression analysis on observable facors. For he firs approach, mos sudies use grouped daa porfolios in order o saisfy he small resricion imposed by classical facor analysis, wih excepions such as Connor and Korajczyk he second approach uses macroeconomic and financial marke variables ha are hough o capure sysemaic risks as observable facors. Wih he mehod developed in his paper, we can esimae he number of facors for he broad U.S. sock marke, wihou he need o group he daa, or wihou being specific abou which observed series are good proxies for sysemaic risks. Monhly daa beween are available for he reurns of 8436 socks raded on he ew York Sock Exchange, AMEX, and ASDAQ. he daa include all lived socks on he las rading day of 1998 and are obained from he CRSP daa base. Of hese, reurns for 4883 firms are available for each

19 approximae facor models 209 ABLE VI DGP: X i = r j=1 ij F j + e i ; e i = e i 1 + v i + J v j= Jj 0 i j; r = 5; = 5, = 5, = 0, J = 0. PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC of he 60 monhs. We use he proposed crieria o deermine he number of facors. We ransform he daa so ha each series is mean zero. For his balanced panel wih = 60 = 4883 and kmax = 15, he recommended crieria, namely, PC p1 PC p2 IC p1, and IC p2, all sugges he presence of wo facors. 7 concluding remarks In his paper, we propose crieria for he selecion of facors in large dimensional panels. he main appeal of our resuls is ha hey are developed under he assumpion ha and are hus appropriae for many daases ypically used in macroeconomic analysis. Some degree of correlaion in he errors is also allowed. he crieria should be useful in applicaions in which he number of facors has radiionally been assumed raher han deermined by he daa.

20 210 jushan bai and serena ng ABLE VII DGP: X i = r j=1 ij F j + e i ; e i = e i 1 + v i + J v j= Jj 0 i j; r = 5; = 5, = 00, = 20, J = max / PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC Our discussion has focused on balanced panels. However, as discussed in Rubin and hayer 1982 and Sock and Wason 1998, an ieraive EM algorihm can be used o handle missing daa. he idea is o replace X i by is value as prediced by he parameers obained from he las ieraion when X i is no observed. hus, if i j and F j are esimaed values of i and F from he jh ieraion, le Xi j 1 = X i if X i is observed, and Xi j 1 = ij 1 F j 1 oherwise. We hen minimize V k wih respec o Fj and j, where V k = 1 X i j 1 k i jf kj2. Essenially, eigenvalues are compued for he marix X j 1X j 1. his process is ieraed unil convergence is achieved. Many issues in facor analysis awai furher research. Excep for some resuls derived for classical facor models, lile is known abou he limiing disribuion

21 approximae facor models 211 ABLE VIII DGP: X i = r j=1 ij F j + e i ; e i = e i 1 + v i + J v j= Jj 0 i j; r = 5; = 5, = 050, = 20, J = max / PC p1 PC p2 PC p3 IC p1 IC p2 IC p3 AIC 1 BIC 1 AIC 2 BIC 2 AIC 3 BIC of he esimaed common facors and common componens i.e., ˆ F i. Bu using heorem 1, i may be possible o obain hese limiing disribuions. For example, he rae of convergence of F derived in his paper could be used o examine he saisical propery of he forecas ŷ +1 in Sock and Wason s framework. I would be useful o show ha ŷ +1 is no only a consisen bu a consisen esimaor of y +1, condiional on he informaion up o ime provided ha is of no smaller order of magniude han. Addiional asympoic resuls are currenly being invesigaed by he auhors. he foregoing analysis has assumed a saic relaionship beween he observed daa and he facors. Our model allows F o be a dependen process, e.g, ALF =, where AL is a polynomial marix of he lag operaor. However, we do no consider he case in which he dynamics ener ino X direcly. If he