A DIAGNOSTIC CRITERION FOR APPROXIMATE FACTOR STRUCTURE

Transcription

1 A DIAGNOSIC CRIERION FOR APPROXIMAE FACOR SRUCURE Parck Gaglardn a, Elsa Ossola b and Olver Scalle c * Frs draf: February 4. Absrac We buld a smple dagnosc creron for approxmae facor srucure n large cross-seconal equy daases. Gven a model for asse reurns wh observable facors, he creron checks wheher he error erms are weakly cross-seconally correlaed or share a leas one unobservable common facor. I only requres compung he larges egenvalue of he emprcal cross-seconal covarance marx of he resduals of a large unbalanced panel. he panel daa model accommodaes boh me-nvaran and me-varyng facor srucures. We develop he heory for large cross-secon and me-seres dmensons. No resrcon s mposed on he relaon beween boh dmensons. he emprcal analyss runs on reurns for abou en housands US socks from January 968 o December. Among several mul-facor models proposed n he leraure, we canno selec a model wh zero facors n he errors. JEL Classfcaon: C, C3, C3, C5, C5, C58, G. Keywords: large panel, approxmae facor model, asse prcng, model selecon. a Unversy of Lugano and Swss Fnance Insue, b Unversy of Lugano, c Unversy of Geneva and Swss Fnance Insue. *Acknowledgemens: he second auhor graefully acknowledges he Swss Naonal Scence Foundaon SNSF for generously fundng her research wh a Mare Hem- Voegln fellowshp.

2 Inroducon Emprcal work n asse prcng vasly reles on lnear mul-facor models wh eher me-nvaran coeffcens uncondonal models or me-varyng coeffcens condonal models. he facor srucure s ofen based on observable varables emprcal facors and supposed o be rch enough o exrac sysemac rsks whle dosyncrac rsk s lef over o he error erm. Lnear facor models are rooed n he Arbrage Prcng heory AP, Ross 976, Chamberlan and Rohschld 983 or come from a loglnearzaon of nonlnear consumpon-based models Campbell 993. Condonal lnear facor models am a capurng he me-varyng nfluence of fnancal and macroeconomc varables n a smple seng see e.g. Shanken 99, Cochrane 996, Ferson and Schad 996, Ferson and Harvey 99, 999, Leau and Ludvgson, Pekova and Zhang 5. me varaon n rsk bases me-nvaran esmaes of alphas and beas, and herefore asse prcng es conclusons Jagannahan and Wang 996, Lewellen and Nagel 6, Boguh e al.. Ghysels 998 dscusses he pros and cons of modelng me-varyng beas. A cenral and praccal ssue s o deermne wheher here are one or more facors omed n he chosen specfcaon. Approxmae facor srucures wh nondagonal error covarance marces Chamberlan and Rohschld 983 answer he poenal emprcal msmach of exac facor srucures wh dagonal error covarance marces underlyng he orgnal AP of Ross 976. If he se of observable facors s correcly specfed, he errors are weakly cross-seconally correlaed. Gven he large menu of facors avalable n he leraure he facor zoo of Cochrane, see also Harvey, Lu and Zhu 3, we need a smple dagnosc creron o decde wheher we can feel comforable wh he chosen se of observable facors. For models wh unobservable laen facors, Connor and Korajczyk 993 were he frs o develop a es for he number of facors for large balanced panels of ndvdual sock reurns n me-nvaran models under covarance saonary and homoskedascy. Unobservable facors are esmaed by he mehod of asympoc prncpal componens developed by Connor and Korajczyk 986 see also Sock and Wason b. For heeroskedasc sengs, he recen leraure on large panels wh sac facors has exended he oolk avalable o researchers. Ba and Ng a nroduce a penalzed leas-squares sraegy o esmae he number of facors, a leas one, whou resrcons on he relaon beween he cross-seconal dmenson n and he me-seres dmenson. Caner and Han forhcomng, 4 propose an esmaor wh a group brdge penalzaon o deermne he number of unobservable facors. Onask 9,

3 looks a he behavor of he adjacen egenvalues o deermne he number of facors when n and are comparable. Ahn and Horensen 3 op for he same sraegy and cover he possbly of zero facors. Kapeanos uses subsamplng o esmae he lm dsrbuon of he adjacen egenvalues. In he spr of Lehmann and Modes 988 and Connor and Korajczyk 988, Ba and Ng 6 analyze sascs o es wheher he observable facors n me-nvaran models span he space of unobservable facors. hey do no mpose any resrcon on n and. hey fnd ha he hree facor model of Fama and French 993, FF s he mos sasfacory proxy for he unobservable facors esmaed from balanced panels of porfolo and ndvdual sock reurns. Ahn, Horensen and Wang 3 sudy a rank esmaon mehod o also check wheher me-nvaran facor models are compable wh a number of unobservable facors. For porfolo reurns, hey fnd ha he FF model exhbs a full rank bea facor loadng marx. In hs paper, we buld a smple dagnosc creron for approxmae facor srucure n large crossseconal daases. he creron checks wheher he error erms n a gven model wh observable facors are weakly cross-seconally correlaed or share a leas one common facor. I only requres compung he larges egenvalue of he emprcal cross-seconal covarance marx of he resduals of a large unbalanced panel and subracng a penalzaon erm vanshng o zero for large n and. he seps of he dagnosc are easy: compue he larges egenvalue, subrac a penaly, 3 conclude o valdy of he proposed approxmae facor srucure f he dfference s negave, or conclude o a leas one omed facor f he dfference s posve. Our heorecal conrbuon shows ha sep 3 yelds asympocally he correc model selecon. We also propose a general verson of he dagnosc creron ha deermne he number of omed common facors. We derve all properes for unbalanced panels n he seng of Connor and Korajczyk 987 o avod he survvorshp bas nheren o sudes resrced o balanced subses of avalable sock reurn daabases Brown, Goezmann, and Ross 995. he panel daa model s suffcenly general o accommodae boh me-nvaran and me-varyng facor srucures Gaglardn, Ossola, and Scalle, GOS. We develop he heory for large cross-secon and me-seres dmensons. No resrcon s mposed on he relaon beween boh dmensons. As shown below, he creron s relaed o he penalzed leas-squares approach of Ba and Ng a for model selecon wh unobservable facors. For our emprcal conrbuon, we consder he Cener for Research n Secury Prces CRSP daabase and ake he Compusa daabase o mach frm characerscs. he merged daase comprses abou en 3

4 housands socks wh monhly reurns from January 968 o December. We look a ffeen emprcal facors and we buld hreen facor models popular n he emprcal fnance leraure o explan monhly equy reurns. hey dffer by he choce of he observable facors. We analyze monhly reurns usng he hree facors of FF; he fve facors of Chen, Roll and Ross 986, CRR; he hree facor of Jagannahan and Wang, 996, JW; he hree lqudy relaed facors of Pasor and Sambaugh, LIQ, plus he momenum MOM facor and he wo reurn reversal REV facors shor-erm and long-erm. We sudy me-nvaran and me-varyng versons of he facor models Shanken 99, Cochrane 996, Ferson and Schad 996, Ferson and Harvey 999. For he laer, we use boh macrovarables and frm characerscs as nsrumens Avramov and Chorda 6. Among he me-nvaran mul-facor models, we canno selec a model wh zero facors n he errors. However, we selec an approxmae facor srucure n he errors for some me-varyng specfcaons. he oulne of he paper s as follows. In Secon, we consder a general framework of condonal lnear facor model for asse reurns. In Secon 3, we presen our dagnosc creron for approxmae facor srucure. In Secon 4, we provde he dagnosc creron o deermne he number of omed facors. Secon 5 conans he emprcal resuls. In he Appendces and, we gaher he heorecal assumpons and some proofs. We use hgh-level assumpons o ge our resuls. In Appendx 3, we provde he lnk of our approach o he expecaon-maxmzaon EM algorhm proposed by Sock and Wason b. Appendx 4 ncludes he Mone Carlo smulaon resuls. We place all omed proofs n he onlne supplemenary maerals. here, we also nclude some addonal emprcal resuls and robusness checks. Condonal facor model of asse reurns In hs secon, we consder a condonal lnear facor model wh me-varyng coeffcens. We work n a mul-perod economy Hansen and Rchard 987 under an approxmae facor srucure Chamberlan and Rohschld 983 wh a connuum of asses as n GOS. Such a consrucon s close o he seng advocaed by Al-Najjar 995, 998, 999a n a sac framework wh an exac facor srucure. He dscusses several key advanages of usng a connuum economy n arbrage prcng and rsk decomposon. A key advanage s robusness of facor srucures o asse repackagng Al-Najjar 999b; see GOS for a proof. Le F, wh =,,..., be he nformaon avalable o nvesors. Whou loss of generaly, he 4

5 connuum of asses s represened by he nerval [, ]. he excess reurns R γ of asse γ [, ] a daes =,,... sasfy he condonal lnear facor model: R γ = a γ + b γ f + ε γ, where vecor f gahers he values of K observable facors a dae. he nercep a γ and facor sensves b γ are F -measurable. he error erms ε γ have mean zero and are uncorrelaed wh he facors condonally on nformaon F. Moreover, we exclude asympoc arbrage opporunes n he economy: here are no porfolos ha approxmae arbrage opporunes when he number of asses ncreases. In hs seng, GOS show ha he followng asse prcng resrcon holds: a γ = b γ ν, for almos all γ [, ], almos surely n probably, where random vecor ν R K s unque and s F -measurable. he asse prcng resrcon s equvalen o E [R γ F ] = b γ λ, where λ = ν + E [f F ] s he vecor of he condonal rsk prema. o have a workable verson of Equaons and, we defne how he condonng nformaon s generaed and how he model coeffcens depend on va smple funconal specfcaons. he condonng nformaon F conans Z and Z γ, for all γ [, ], where he vecor of lagged nsrumens Z R p s common o all socks, he vecor of lagged nsrumens Z γ R q s specfc o sock γ, and Z = {Z, Z,...}. Vecor Z may nclude he consan and pas observaons of he facors and some addonal varables such as macroeconomc varables. Vecor Z γ may nclude pas observaons of frm characerscs and sock reurns. o end up wh a lnear regresson model, we assume ha: he vecor of facor loadngs b γ s a lnear funcon of lagged nsrumens Z Shanken 99, Ferson and Harvey 99 and Z γ Avramov and Chorda 6; he vecor of rsk prema λ s a lnear funcon of lagged nsrumens Z Cochrane 996, Jagannahan and Wang 996; he condonal expecaon of f gven he nformaon F depends on Z only and s lnear as e.g. f Z follows a Vecor Auoregressve VAR model of order. o ensure ha cross-seconal lms exs and are nvaran o reorderng of he asses, we nroduce a samplng scheme as n GOS. We formalze so ha observable asses are random draws from an underlyng populaon Andrews 5. In parcular, we rely on a sample of n asses by randomly drawng..d. 5

6 ndces γ from he populaon accordng o a probably dsrbuon G on [, ]. For any n, N, he excess reurns are R, = R γ. Smlarly, le a, = a γ and b, = b γ be he characerscs, and ε, = ε γ be he error erms. By random samplng, we ge a random coeffcen panel model e.g. Hsao 3, Chaper 6. In avalable daases, we do no observe asse reurns for all frms a all daes. hus, we accoun for he unbalanced naure of he panel hrough a collecon of ndcaor varables I,, for any asse a me. We defne I, = f he reurn of asse s observable a dae, and oherwse Connor and Korajczyk 987. hrough approprae redefnons of he regressors and coeffcens, GOS show ha we can rewre he model for Equaons and as follows: R, = x,β + ε,, 3 where he regressor x, = x,,,,, x has dmenson d = d + d and ncludes vecors x,, = vech [X ], Z, Z R d and x,, = f Z, f Z, R d wh d = pp + / + pq and d = Kp + q. he symmerc marx X = [X,k,l ] R p p s such ha X,k,l = Z,k, f k = l, and X,k,l = Z,k Z,l, oherwse, k, l =,..., p. he vecor-half operaor vech [ ] sacks he elemens of he lower rangular par of a p p marx as a p p + / vecor see Chaper n Magnus and Neudecker 7 for properes of hs marx ool. In marx noaon, for any asse, we have R = X β + ε, 4 where R and ε are vecors. Regresson 3 conans boh explanaory varables ha are common across asses scaled facors and asse-specfc regressors. I ncludes models wh me-nvaran coeffcens as a parcular case. In such a case, he regressor reduces o x =, f and s common across asses. In order o buld he dagnosc creron for he se of observable facors, we consder he followng rval models: M : he lnear regresson model 3, where he errors ε, follow an approxmae facor srucure, and M : he lnear regresson model 3, where he errors ε, sasfy a facor srucure. 6

7 hus, he error erms ε, are weakly cross-seconally dependen under model M. On he oher hand, under model M, he followng error facor srucure holds ε, = θ h + u,, 5 where he m vecor h ncludes unobservable.e., laen or hdden facors. he m vecor θ corresponds o he facor loadngs, and he number m of common facors s assumed unknown. In vecor noaon, we have: ε = Hθ + u, 6 where H s he m marx of unobservable facor values, and u s a vecor. Model M can be dsngushed from model M only f he sysemac componen Hθ n he error vecor ε s no spanned by he columns of marx X for mos asses. Oherwse, he common componen Hθ can be absorbed n he observable regressors, and we face an denfcaon ssue. herefore, we nroduce he nex assumpon under model M. Assumpon Under model M, he facor srucure n he error erm s such ha µ M X H θ θ H M X c, wh probably approachng, for a consan c >, where M X = I X X X X for any, I denoes he deny marx of dmenson, X = I X, H = I H, wh I he vecor of observably ndcaors of asse, and µ and denoe he larges egenvalue of a symmerc marx and he Hadamard produc beween marces, respecvely. In Assumpon Assumpon, vecor ˆη = M X H θ s he resdual vecor n he regresson of H θ on he explanaory varables X. hus, ˆη s he par of he sysemac componen of he error vecor, whch s no spanned by he observable regressors. Assumpon Assumpon requres ha he larges egenvalue of he cross-seconal second-order momens marx of vecors ˆη, sandardzed by her dmenson, does no vansh asympocally. By he leraure on unobservable facor models e.g., Ba and Ng a hs condon amouns o he presence of some common facors n he ˆη. For a facor model wh me nvaran 7

8 coeffcens and a balanced panel, we have X = X = X, and H = H see Appendx A..: Θ µ M X Hθ θ H Θ H M X H M X µ m µ, 7 n where Θ s he n m marx of facor loadngs and µ m s he smalles egenvalue of a m m symmerc marx. Whou loss of generaly, he unobservable facors can be seleced orhogonal o he observable regressors. hus, Assumpon Assumpon s sasfed, f marces Θ Θ n and H H,.e., he second-momens marces of he loadngs and of he facors converge o posve defne marces see Assumpons A and B n Ba and Ng a. 3 Dagnosc creron In hs secon, we provde he dagnosc creron ha checks wheher he error erms are weakly crossseconally correlaed or share a leas one common facor. o compue he creron, we esmae model 3 by ordnary leas square OLS asse by asse, and we ge esmaors ˆβ = ˆQ x, I, x, R,, for =,..., n, where ˆQ x, = I, x, x,. We ge he resduals ˆε, = R, x ˆβ,, where ˆε, s observable only f I, =. In avalable panels, he random sample sze for asse can be small, and he nverson of marx ˆQ x, can be numercally unsable. o avod unrelable esmaes of β, we { } apply a rmmng approach as n GOS. We defne χ = CN ˆQx, χ,, τ, χ,, where CN ˆQx, = µ ˆQx, /µ d ˆQx, s he condon number of marx ˆQ x,, and τ, = /. he wo sequences χ, > and χ, > dverge asympocally. he frs rmmng condon {CN ˆQx, χ, } keeps n he cross-secon only asses for whch he me seres regresson s no oo badly condoned. A oo large value of CN ˆQx, ndcaes mulcollneary problems and ll-condonng Belsley e al. 4, Greene 8. he second rmmng condon {τ, χ, } keeps n he cross-secon only asses for whch he me seres s no oo shor. We also use boh rmmng condons n he proofs of he asympoc resuls. We consder he followng dagnosc creron: ξ = µ χ ε ε gn,, 8 8

9 where he vecor ε of dmenson gahers he values ε, gn, and C n, gn,, when n,, for C n, = I,ˆε,, he penaly gn, s such ha = mn{n, }. Ba and Ng a consder several smple poenal canddaes for he penaly gn,. We ls and mplemen hem n Secon 4. In vecor ε, he unavalable resduals are replaced by zeros. he followng model selecon rule explans our choce of he dagnosc creron 8 for approxmae facor srucure n large unbalanced cross-seconal daases. Proposon Model selecon rule: Under Assumpon Assumpon and Assumpons A.-A.5, a we selec M f ξ <, snce P r ξ < M, when n, ; b we selec M f ξ >, snce P r ξ > M, when n,. Proposon Proposon characerzes an asympocally vald model selecon rule, whch reas boh models symmercally. hs s no a esng procedure snce we do no use a crcal regon based an asympoc dsrbuon and a chosen sgnfcance level. he proof of Proposon Proposon shows ha he larges egenvalue n 8 vanshes a a faser rae han he penalzaon erm under M when n and go o nfny. hs explans why we selec he frs model when ξ s negave. On he conrary, he larges egenvalue remans bounded from below away from zero under M when n and go o nfny. hs explans why we selec he second model when ξ s posve. he creron 8 can be nerpreed as he adjused gan n f ncludng a sngle addonal unobservable facor n model M. In he balanced case, where I, = for all and, we can rewre 8 as ξ = SS SS g n,, where SS = ˆε, s he sum of squared errors and SS = mn ˆε, θ h, where he mnmzaon s w.r.. he vecors H R of facor values and Θ R n of facor loadngs n a one-facor model, subjec o he normalzaon consran H H =. Indeed, he larges egenvalue µ ˆε ˆε corresponds o he dfference beween SS and SS. Furhermore, he creron ξ s equal o he dfference of he penalzed crera for zero- and one-facor models defned n Ba and Ng a appled on he resduals. Indeed, ξ = P C P C, where P C = SS, and P C = SS + g n,. Gven such an nerpreaon n erms of sums of squared errors, we can sugges anoher dagnosc creron based on a logarhmc 9

10 ransform as n Corollary of Ba and Ng a. he second dagnosc creron s ˇξ = ln χ ˆε, ln χ ˆε, µ χ ˆε ˆε gn,. 9 In he balanced case, we ge ˇξ = lnss /SS gn, and s equal o he dfference of IC and IC crera n Ba and Ng a. he followng proposon saes he model selecon rule based on ˇξ. Proposon he model selecon rule s he same as n Proposon wh ˇξ subsued for ξ. he recen leraure on he properes of he wo-pass regressons for fxed n and large shows ha he presence of useless facors Kan and Zhang 999a,b, Gospodnov, Kan and Robo 4 or weak facor loadngs Klebergen 9 does no affec he asympoc dsrbuonal properes of facor loadng esmaes, bu alers he ones of he rsk prema esmaes. Useless facors have zero loadngs, and weak loadngs drf o zero a rae /. he vanshng rae of he larges egenvalue of he emprcal crossseconal covarance marx of he resduals does no change f we face useless facors or weak facor loadngs n he observable facors under M. he same remark apples under M. Hence he selecon rule remans he same snce he probably of akng he rgh decson sll approaches. If we have a number of useless facors or weak facor loadngs srcly lower han he number m of he omed facors under M, hs does no mpac he asympoc rae of he dagnosc creron f Assumpon Assumpon holds. If we only have useless facors n he omed facors under M, we face an denfcaon ssue. Assumpon Assumpon s no sasfed. We canno dsngush such a specfcaon from M snce corresponds o a parcular approxmae facor srucure. Agan he selecon rule remans he same snce he probably of akng he rgh decson sll approaches. Fnally, le us sudy he case of only weak facor loadngs under M. We consder a smplfed seng: R, = x,β + ɛ, where ɛ, = θ h + u, has only one facor wh a weak facor loadng, namely m = and θ = θ / γ wh γ >. Le us assume ha µ M X H H M X θ s bounded from below away from zero see Assumpon Assumpon and bounded from above. By he properes of he egenvalues of a scalar mulple of a marx, we deduce ha c / γ µ M X H H M X θ c / γ, for some consans

11 c, c such ha c c >. Hence, by smlar argumens as n he proof of Proposon Proposon, we ge: c γ gn, + O p C + χ ξ c γ gn, + O p C + χ, where we defne χ = χ 4, χ,. o conclude M, we need ha C + χ and he penaly gn, vansh a a faser rae han γ, namely C + χ = o γ and gn, = o γ. o conclude M, we need ha gn, s he domnan erm, namely γ = o gn, and C + χ = o gn,. As an example, le us ake gn, = log and n = γ wh γ >, and assume ha he rmmng s such ha χ = olog. hen, we conclude M f γ < / and M f γ > /. hs means ha deecng a weak facor loadng srucure s dffcul f gamma s no suffcenly small. he facor loadng should drf o zero no oo fas o conclude M. Oherwse, we canno dsngush asympocally from an approxmae facor srucure. 4 Deermnng he number of facors In he prevous secon, we have suded a dagnosc creron o check wheher he error erms are weakly cross-seconally correlaed or share a leas one unobservable common facor. hs secon ams a answerng: do we have one, wo, or more omed facors? he desgn of he dagnosc creron o check wheher he error erms share exacly k unobservable common facors or share a leas k + unobservable common facors follows he same mechancs. We consder he followng rval models: M k : he lnear regresson model 3, where he errors ε, sasfy a facor srucure wh exacly k unobservable facors, and M k : he lnear regresson model 3, where he errors ε, sasfy a facor srucure wh a leas k + unobservable facors. he followng assumpon consss of denfcaon assumpons smlar o Assumpon Assumpon, bu for exacly k unobservable facors and a leas k + unobservable facors n he error erms.

12 Assumpon a Under model M k, he facor srucure n he error erm s such ha µ k M X H θ θ H M X c, wh probably approachng, for a consan c >. b Under model M k, he facor srucure n he error erm s such ha µ k+ M X H θ θ H M X c, wh probably approachng, for a consan c >. For a facor model wh me nvaran coeffcens and a balanced panel, Assumpon Assumpon s sasfed f marces Θ Θ n and H M X H converge o posve defne marces snce we deduce as n Secon from he nequales n Wang and Zhang 99: Θ µ k M X Hθ θ H Θ H M X H M X µ m µ k. n he dagnosc creron explos he kh larges egenvalue of he emprcal cross-seconal covarance marx of he resduals: ξk = µ k+ χ ε ε gn,. As dscussed n Ahn and Horensen 3 see also Onask 3, we can rewre n he balanced case as ξk = SS k SS k+ gn, where SS k equals he sample mean of he squared resduals from he me seres regressons of ndvdual response varables ˆε, on he frs k prncpal componens of ˆε ˆε. he creron ξk s equal o he dfference of he penalzed crera for k and k + - facor models defned n Ba and Ng a appled on he resduals. Indeed, ξk = P Ck P Ck +, where P Ck = SS k + kgn,, and P Ck + = SS k+ + k + gn,. o deermne he number of unobservable facors, we choose he mnmum k such ha ξk <. Graphcally, we can buld a penalzed scree plo where we dsplay he penalzed egenvalues assocaed wh each facor n descendng order versus he number of he facor, and use he x-axs for he cu-off pon. he followng model selecon rule exends Proposon Proposon o deermne he number of facors.

13 Proposon 3 Model selecon rule: under Assumpons..., a we selec M k f ξk <, snce P r[ξk < M k], when n, such ha n = O ; b we selec M k f ξk >, snce P r[ξk > M k], when n,. In Proposon Proposon 3 par a, we need he addonal consran n = O on he relave rae of he cross-seconal dmenson w.r.. he me seres dmenson. he conrbuon µ k+ H θ θ H = O p / max{n, } comng from he k omed facors Lemma... n he appendx does no domnae asympocally he conrbuon µ ũ ũ = C n, under M k when n = O. We do no need such an addonal resrcon n a balanced panel snce µ k+ H θ θ H = f we have exacly k facors. hs exemplfes a key dfference beween he asympocs for balanced and unbalanced panels, and he proporonal asympocs used n Onask 9, or Ahn and Horensen 3. hose papers rely on he asympoc dsrbuon of he egenvalues of large dmensonal sample covarances marces when n/ n c > as n. he condon n = O agrees wh he large n, small case ha we face n he emprcal applcaon en housands ndvdual socks monored over fory-fve years of monhly reurns. he proof of Proposon Proposon 3 s also more complcaed han he proof of Proposon Proposon. he proof of he laer drecly explos he equaly beween he larges value of a symmerc marx and s operaor norm, he rangular nequaly of he marx norm, and s upper bound gven by he Frobenus norm. We need addonal argumens based on Weyl nequales heorem 4.3. n Horn and Johnson 985 when we look a he k + h egenvalue. 5 Emprcal resuls 5. Facor models and daa descrpon We consder ffeen non-repeve emprcal facors as n Ahn, Horensen and Wang 3. he hree facor of Fama and French 993 are he monhly excess reurn on CRSP NYSE/AMEX/Nasdaq value-weghed marke porfolo over he rsk free rae r m,, he monhly reurns on zero-nvesmen facor-mmckng porfolos for sze, book-o-marke, denoed by r smb, and r hml, respecvely. he monhly reurns on porfolo for momenum s denoed by r mom,. wo reversal facors are monhly reurns on porfolo for shor r sr, 3

14 and long erm r lr. We have downloaded he me seres of hese facors from he webse of Kenneh French. We consder he fve facors of Chen, Roll and Ross 986 avalable from Laura Xaole Lu s webpage. he monhly CRR facors are he growh rae of ndusral producon mp, he unexpeced nflaon u, he erm spread us, proxed by he dfference beween yelds on -year reasury and 3-monh -bll, and he defaul prema upr, proxed by he yeld dfference beween Moody s Baa-raed and Aaa-raed corporae bonds. Moreover, we consder he hree lqudy-relaed facors of Pasor and Sambaugh ha concern of he monhly lqudy level al, raded lqudy l and he nnovaon n aggregae lqudy l. We have downloaded he LIQ facors from he webse of Lubos Pasor. Fnally, we buld he monhly growh rae of labor ncome lab from he Bureau of Economc Analyss s webpage. We proxy he rsk free rae wh he monhly 3-day -bll begnnng-of-monh yeld. o accoun for me-varyng coeffcens, we use wo condonal specfcaons based on wo common varables and a frm-level varable. We ake he nsrumens Z =, Z, where bvarae vecor Z ncludes eher he erm spread and he defaul spread, or he monhly 3-day -bll and he dvdend yelds. We ake a scalar Z, correspondng o he book-o-marke equy of frm. We refer o Avramov and Chorda 6 for convncng heorecal and emprcal argumens n favor of he chosen condonal specfcaon. he parsmony explans why we have no ncluded e.g. he sze of frm as an addonal sock specfc nsrumen. able repors he hreen lnear facor models ha we esmae n order o compued he dagnosc crera. For each model, we specfy he emprcal facors nvolved and he number K of observable facors. We look a facor models popular n he emprcal fnance. We also consder nesed models bul from he ffeen emprcal facors. We compue he frm characerscs from Compusa as n he appendx of Fama and French 8. he CRSP daabase provdes he monhly sock reurns daa and we exclude fnancal frms Sandard Indusral Classfcaon Codes beween 6 and 6999 as n Fama and French 8. he daase afer machng CRSP and Compusa conens comprses n =, 44 socks, and covers he perod from January 968 o December wh = 58 monhs. 4

15 5. Dagnosc resuls In hs secon, we compue he dagnosc crera n Equaons 8 and 9 assumng me-nvaran and me-varyng specfcaons of he lnear facor models n able. In order o compue he crera we need o defne he specfcaon for he penaly g n,. Ba and Ng a propose hree choces for he penaly funcon n Equaon 8, leadng o he followng crera:. ξ = µ χ n + ε ε ˆσ ln ; n +. ξ = µ χ ε ε 3. ξ 3 = µ χ ε ε ˆσ n + ˆσ ln C C ln C ;, where ˆσ = χ ε,,, and ε,, s he fed resdual of he me-varyng lnear facor model bul on he FF, MOM, REV observable facors and a laen facor. Furhermore, we defne he specfcaon for he penaly gn, for he logarhmc dagnosc creron n Equaon 9. We ge he followng logarhmc crera:. ˇξ = ln χ ε, ln χ ε, µ χ ε ε. ˇξ = ln χ ε, ln χ ε, µ χ ε ε 3. ˇξ3 = ln χ ε, ln χ ε, µ χ ε ε n + ln ; n + n + ln C ; ln C C In order o ensure ha all seres have a common scale of measuremen, each me-seres s demeaned and sandardzed o have un varance before o compung he egenvalues see Pena and Poncela 6. In order o compue he dagnosc crera, we esmae me-nvaran and me-varyng facor models. We fx χ, = 5 as advocaed by Greene 8, and χ, = 546/ for he me-nvaran esmaon and χ, = and χ, = 546/6 for he me-varyng esmaon. In able, we repor he sze of rmmed cross-seconal dmenson n χ ha comes from he rmmng procedure appled n he esmaon approach., 5

16 In some me-varyng specfcaons, we ncur n emprcal mulcollneary problems due o he correlaons whn he vecor of regressors x,, ha nvolves cross produc of facors f and nsrumens Z e.g., n he JW and CRR models, and he large dmenson of vecor x, e.g., he number of parameer o esmae s larger han 4 n models -3. For he me-nvaran specfcaons of -3 models, we plo he values of he dagnosc crera ξ, ξ and ξ 3 n Fgure, and ˇξ, ˇξ and ˇξ 3 n Fgure. For he me-varyng specfcaons, Fgures 3 and 4 plo he values of he dagnosc crera compued by usng he common nsrumens. Fgures 5 and 6 plo he resuls by usng he second se of common nsrumens. Snce he penaly funcon s proporonal o ln, he numercal value of crera ξ s and ˇξ s, wh s =,, 3, are no oo much dfferen from each oher. For he majory of he models, he concluson abou he selecon model s he same boh for he dagnosc creron n equaon 8, ha for he logarhmc dagnosc creron n equaon 9. In parcular, we canno selec a me-nvaran model wh zero facors n he errors. We conclude for an approxmae facor srucure n he error erms when we esmae he me-varyng lnear facor models based on FF and REV facors. In general, focusng on nesed models, when he number of facor ncreases he dagnosc crera decreases. Fnally, n many cases, he dagnosc crera s smaller for he me-varyng specfcaons han for he me-nvaran models. In ables 3-6, we compare he descrpve sascs of four measures of mssng facor mpac: he esmaed me-seres coeffcen of deermnaon ˆρ = ESS,- where ESS =, I, ˆR, ˆR wh SS ˆR, = ˆβ x, and ˆR = I, ˆR,, and SS = I, R, R, wh R = I, R, ; he esmaed adjused R defned by ˆρ ad, = ˆρ RSS d ; he dosyncrac rsk IdV ol =, wh RSS = I,ˆε ESS,; v he sysemac rsk SysRsk =, for he me-nvaran and mevaryng specfcaons. We consder hose esmaes as measures of mssng facor mpac see Ang, Lu and Schwarz 8. he me-seres adjused coeffcen of deermnaon end o be a b larger n he me-varyng model han n he me-nvaran specfcaons. he ˆρ, ˆρ ad, and SysRsk adm large values for he models ha nroduced he FF, MOM and/or REV facors n her specfcaon. For hese lnear specfcaons, we observe ha he dagnosc crera ξ and ˇξ adms small values. 6

17 5.3 he number of facors In hs secon, we compue he dagnosc crera 9 ha explo he k-h larges egenvalue of he emprcal cross-seconal covarance marx of he errors. We compue he dagnosc crera for he frs fve egenvalues, and we use he penaly funcon g n, defned n he prevous secon. In parcular, we compue he followng crera:. ξ k = µ k+ χ ε ε. ξ k = µ k+ χ ε ε 3. ξ 3 k = µ k+ χ ε ε ˆσ n + ˆσ n + ˆσ ln C C ln ; n + ln C ; wh k =,..., 5. For each lnear facor specfcaon, we buld a penalzed scree plo. Fgures 7 and 8 show he resuls for he me-nvaran specfcaons. We observe ha dagnosc crera change sgns when we consder he me-nvaran specfcaons based on he FF facors. In parcular, he dagnosc crera becomes negave when k = 4 for he FF and Carhar 997 models. he number of unobservable common facors k s 3 for he me-nvaran model ha accouns for more han 8 observable facors e.g., models -3. However, he hree FF facors alone do no explan he excess reurns for socks. Le us consder he resuls for he me-varyng specfcaons n Fgures 9 and. In boh he fgures, he cu-off pon s smaller han for he me-nvaran specfcaons. hus, he me-varyng specfcaons capure more properes of excess reurns han he correspondng me-nvaran models. Indeed, he number of omed facors s smaller for he me-varyng models han for he me-nvaran cases. Moreover, he se of common nsrumens nvolvng he monhly 3-day -bll and he dvdend yelds seems o capure n a beer way he characerscs of reurns of ndvdual socks., 7

18 able : Lnear facor models Model Emprcal facors K CAPM r m, FF model r m,, r smb,, r hml, 3 3 LIQ model al, l, l 3 4 JW model r m,, lab, upr 3 5 MOM and REV facors r mom,, r sr, r lr 3 6 Carhar 997 model r m,, r smb,, r hml,, r mom, 4 7 CRR model mp, u, de us upr 5 8 FF and REV facors r m,, r smb,, r hml,, r sr, r lr 5 9 FF and JW facors r m,, r smb,, r hml,, lab, upr 5 FF, MOM and REV facors r m,, r smb,, r hml,, r mom,, r sr, r lr 6 FF and CRR facors FF, CRR and JW facors 3 FF, MOM, REV, CRR and JW facors r m,, r smb,, r hml,, mp, u, de us upr 8 r m,, r smb,, r hml,, mp, u, de, us upr, lab 9 r m,, r smb,, r hml,, r mom,, r sr, r lr, mp, u, de us upr, lab he able lss he lnear facor models ha we esmae n order o compue he dagnosc crera. For each model, he emprcal facors nvolved are specfed. K s he number of observable facors. FF, CRR, MOM, REV, LIQ and JW, respecvely, refer o he hree Fama-French facors, he fve Chen-Roll-Ross macroeconomc facors, he momenum facor, he reversal facors, he hree lqudy facors of Pasor and Sambaugh, and he hree Jagannahan and Wang 996 facors. 8

19 able : rmmed cross-seconal dmensons n χ and number of parameer o esmae d Model me-nvaran spec. me-varyng spec. n χ d n χ n χ CAPM,4 3 5,46,66 FF model,4 4,476,476 3 LIQ model,4 3,393,8 4 JW model 7, MOM and REV facors,4 4,568,47 6 Carhar 997 model,4 5 4,,354 7 CRR model 7, FF and REV facors, ,88,76 9 FF and JW facors 5, FF, MOM and REV facors 7, ,7 96 FF and CRR facors 6, FF, CRR and JW facors 6, FF, MOM, REV, CRR and JW facors 5, For each lnear facor model, he able repors he rmmed cross-seconal dmenson n χ ha comes from he esmaon procedure. For he me-varyng specfcaons, he dmenson of vecor x,, denoed by d, s also specfed. For he me-nvaran specfcaons, he number of regressors corresponds o he number of observable facors K see able. 9

20 able 3: Summary sascs of ˆρ, ˆρ ad,, IdV ol and SysRsk for he me-nvaran specfcaons Model ˆρ Mn Quanle 5% Medan Mean Quanle 75% Max Sd ˆρad, Mn Quanle 5% Medan Mean Quanle 75% Max Sd he able conans he descrpve sascs cross-seconal mnmum, 5% and 75% quanles, medan, mean, maxmum and sandard devaon of he esmaed coeffcen of deermnaon ˆρ, he esmaed adjused coeffcens of deermnaon ˆρ ad, me-nvaran lnear facor models. for he

21 able 4: Summary sascs of ˆρ, ˆρ ad,, IdV ol and SysRsk for he me-nvaran specfcaons Model IdV ol Mn Quanle 5% Medan Mean Quanle 75% Max Sd SysRsk Mn Quanle 5% Medan Mean Quanle 75% Max Sd he able conans he descrpve sascs cross-seconal mnmum, 5% and 75% quanles, medan, mean, maxmum and sandard devaon of he dosyncrac rsks IdV ol, and he sysemac rsks SysRsk for he me-nvaran lnear facor models.

22 able 5: Summary sascs of ˆρ, ˆρ ad,, IdV ol and SysRsk for he me-varyng specfcaons Model ˆρ Mn Quanle 5% Medan Mean Quanle 75% Max Sd ˆρ ad, Mn Quanle 5% Medan Mean Quanle 75% Max Sd IdV ol Mn Quanle 5% Medan Mean Quanle 75% Max Sd SysRsk Mn Quanle 5% Medan Mean Quanle 75% Max Sd he able conans he descrpve sascs cross-seconal mnmum, 5% and 75% quanles, medan, mean, maxmum and sandard devaon of he esmaed coeffcen of deermnaon ˆρ, he esmaed adjused coeffcens of deermnaon ˆρ ad,, he dosyncrac rsks IdV ol, and he sysemac rsks SysRsk for he me-varyng lnear facor models esmaed by usng he erm spread and he defaul spread as common nsrumens.

23 able 6: Summary sascs of ˆρ, ˆρ ad,, IdV ol and SysRsk for he me-varyng specfcaons Model ˆρ Mn Quanle 5% Medan Mean Quanle 75% Max Sd ˆρ ad, Mn Quanle 5% Medan Mean Quanle 75% Max Sd IdV ol Mn Quanle 5% Medan Mean Quanle 75% Max Sd SysRsk Mn Quanle 5% Medan Mean Quanle 75% Max Sd he able conans he descrpve sascs cross-seconal mnmum, 5% and 75% quanles, medan, mean, maxmum and sandard devaon of he esmaed coeffcen of deermnaon ˆρ, he esmaed adjused coeffcens of deermnaon ˆρ ad,, he dosyncrac rsks IdV ol, and he sysemac rsks SysRsk for he me-varyng lnear facor models esmaed by usng he monhly 3-day -bll and he dvdend yelds as common nsrumens.

24 Fgure : Values of he dagnosc crera ξ, ξ and ξ 3 for he me-nvaran models Dagnosc creron Facor model he fgure plos he values of he dagnosc crera ξ red crcle, ξ green plus sgn and ξ 3 blue cross for he me-nvaran specfcaons. We also repor he zero axs red dashed horzonal lne. Fgure : Esmaed values of he dagnosc crera ˇξ, ˇξ and ˇξ 3 for he me-nvaran models Dagnosc creron Facor model he fgure plos he values of he logarhmc dagnosc crera ˇξ red crcle, ˇξ green plus sgn and ˇξ 3 blue cross for he me-nvaran specfcaons. We also repor he zero axs red dashed horzonal lne. 4

25 Fgure 3: Values of he dagnosc crera ξ, ξ and ξ 3 for he me-varyng models.. Dagnosc creron Facor model he fgure plos he values of he dagnosc crera ξ red crcle, ξ green plus sgn and ξ 3 blue cross for he me-varyng specfcaons when Z ncludes defaul and erm spreads. he dagnosc crera canno be compued for he JW, CRR, 9, -3 models due o he mulcollneary problems. We also repor he zero axs red dashed horzonal lne. Fgure 4: Values of he dagnosc crera ˇξ, ˇξ and ˇξ 3 for he me-varyng models Dagnosc creron Facor model he fgure plos he values of he logarhmc dagnosc crera ˇξ red crcle, ˇξ green plus sgn and ˇξ 3 blue cross for he me-varyng specfcaons when Z ncludes defaul and erm spreads. he logarhmc dagnosc crera canno be compued for he JW, CRR, 9, -3 models due o he mulcollneary problems. We also repor he zero axs red dashed horzonal lne. 5

26 Fgure 5: Values of he dagnosc crera ξ, ξ and ξ 3 for he me-varyng models 7 x Dagnosc creron Facor model he fgure plos he values of he dagnosc crera ξ red crcle, ξ green plus sgn and ξ 3 blue cross for he me-varyng specfcaons when Z ncludes one-monh -Bll and dvdend yeld. he dagnosc crera canno be compued for he JW, CRR, 9, -3 models due o he mulcollneary problems. We also repor he zero axs red dashed horzonal lne. Fgure 6: Values of he dagnosc crera ˇξ, ˇξ and ˇξ 3 for he me-varyng models.4.. Dagnosc creron Facor model he fgure plos he values of he logarhmc dagnosc crera ˇξ red crcle, ˇξ green plus sgn and ˇξ 3 blue cross for he me-varyng specfcaons when Z ncludes one-monh -Bll and dvdend yeld. he logarhmc dagnosc crera canno be compued for he JW, CRR, 9, -3 models due o he mulcollneary problems. We also repor he zero axs red dashed horzonal lne. 6

27 Fgure 7: Values of crera ξ k for he me-nvaran models CAPM FF model 4 x x Dagnosc creron.5.5 Dagnosc creron k-h egenvalues k-h egenvalues 3 LIQ model 4 JW model x Dagnosc creron..8.6 Dagnosc creron k-h egenvalues k-h egenvalues 5 MOM and REV facors 6 Carhar 997 model x Dagnosc creron..8.6 Dagnosc creron k-h egenvalues k-h egenvalues he fgure plos he values of he dagnosc crera ξ kred crcle, ξ k green plus sgn and ξ 3 k blue cross wh k =,..., 5, for he me-nvaran specfcaons -6. We also repor he zero axs red dashed horzonal lne.

28 Fgure 8: Values of crera ξ k for he me-nvaran models 7 CRR model 8 FF and REV facors.8 3 x Dagnosc creron Dagnosc creron k-h egenvalues k-h egenvalues 9 FF and JW facors FF, MOM and REV facors 3 x 3 3 x Dagnosc creron.5 Dagnosc creron k-h egenvalues k-h egenvalues FF and CRR facors FF, CRR, and JW facors 3 x 3.5 x 3.5 Dagnosc creron.5.5 Dagnosc creron k-h egenvalues k-h egenvalues 3 FF, MOM, REV, CRR and JW facors x 3.5 Dagnosc creron k-h egenvalues he fgure plos he values of he dagnosc crera ξ kred crcle, ξ k green plus sgn and ξ 3 k blue cross wh k =,..., 5, for he me-nvaran specfcaons 7-3. We also repor he zero axs red dashed horzonal lne.

29 Fgure 9: Values of crera ξ k for he me-varyng models CAPM FF model 3 x 3 x Dagnosc creron.5.5 Dagnosc creron k-h egenvalues k-h egenvalues 3 LIQ model 5 MOM and REV facors x 3 x Dagnosc creron 6 4 Dagnosc creron k-h egenvalues k-h egenvalues 6 Carhar 997 model 8 FF and REV facors.5 x 3.5 x 3 Dagnosc creron.5 Dagnosc creron k-h egenvalues k-h egenvalues FF, MOM and REV facors x 3.5 Dagnosc creron k-h egenvalues he fgure plos he values of he dagnosc crera ξ kred crcle, ξ k green plus sgn and ξ 3 k blue cross wh k =,..., 5, for he me-varyng specfcaons when Z ncludes defaul and erm spreads. We also repor he zero axs red dashed horzonal lne.

30 Fgure : Values of crera ξ k for he me-varyng models CAPM FF model x 3 4 x 4.5 Dagnosc creron.5 Dagnosc creron k-h egenvalues k-h egenvalues 3 LIQ model 5 MOM and REV facors 6 x 3 7 x Dagnosc creron 3 Dagnosc creron k-h egenvalues k-h egenvalues 6 Carhar 997 model 8 FF and REV facors x 4 x 3 Dagnosc creron Dagnosc creron k-h egenvalues k-h egenvalues FF, MOM and REV facors x 3 Dagnosc creron k-h egenvalues he fgure plos he values of he dagnosc crera ξ kred crcle, ξ k green plus sgn and ξ 3 k blue cross wh k =,..., 5, for he me-varyng specfcaons when Z ncludes one-monh -Bll and dvdend yeld. We also repor he zero axs red dashed horzonal lne.

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36 Appendx Regulary condons In hs Appendx, we ls and commen addonal assumpons used o derve he proofs n Appendx. Assumpon A. here exss a consan M > such ha, for all n, N, we have: a E [ ] n ε, ε j, ε,3 ε j,3 x,, x j,, γ, γ j M; b,j n,j,, 3,, 3, 4 E [ ε, ε, ε j,3 ε j,4 x,, x j,, γ, γ j ] M. Assumpon A. he error erms ε, are ε, = u, under model M, and ε, = θ h + u, under model M, where he u, are such ha for a consan M > and for all n, N we have: [ ] a E u, u h j,, γ, γ j M;,j b E [ ] u, u j, h, x,, x j,, γ, γ j M;,j, c n E [u, u j, u, u j, ] M;,j, d E [ ] n u, u j, u,3 u j,3 h, x,, x j,, γ, γ j M; e,j n,j,, 3,, 3, 4 f E [ ε 4,] M, for all n and. E [ u, u, u j,3 u j,4 h, x,, x j,, γ, γ j ] M. Assumpon A.3 here exss a consan M > such ha x, M, P -a.s., for any and. Assumpon A.4 a here exss a consan M > such ha h M, P -a.s., for all. Moreover, b θ < M, for all. Assumpon A.5 he rmmng consans χ, and χ, are such ha χ, χ, = o g n,. Assumpon A.6 Under model M, Ẽ = ε,..., ε,..., ε N = E / E E / 3N, where ε = I ε, and under model M, Ũ = ũ,..., ũ,..., ũ N, where ũ = I u, such ha E = e, and E / and E / 3N are he symmerc square roos of and N N posve semdefne marces E and E 3N, respecvely, he e are ndependen and dencally dsrbued..d. random varables wh zero mean 36