Empirical Tests of Asset Pricing Models with Individual Assets: Resolving the Errors-in-Variables Bias in Risk Premium Estimation

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1 Emprcal Tess of Asse Prcng Models wh Indvdual Asses: Resolvng he Errors-n-Varables Bas n Rsk Premum Esmaon by arasmhan Jegadeesh, Joonk oh, Kunara Pukhuanhong, Rchard Roll, and Junbo Wang Sepember, 207 Absrac To aenuae an nheren errors-n-varables bas, porfolos are wdely employed o es asse prcng models; bu porfolos mgh dversfy and mask relevan rsk- or reurn-relaed feaures of ndvdual asses. We propose an nsrumenal varables approach ha allows he use of ndvdual asses ye delvers conssen esmaes of ex-pos rsk premums. Ths esmaor yelds unbased esmaes and well-specfed ess n small samples. The marke rsk premum under he CAPM and he lqudy-adjused CAPM, premums on rsk facors under he Fama-French hree- and fve-facors models and he Hou, Xue, and Zhang (205) four-facor model are all nsgnfcan afer conrollng for asse characerscs. Co-Auhor Afflaon Voce E-Mal Emory Unversy Jegadeesh@ Jegadeesh Alana GA Emory.Edu oh Pukhuanhong Roll Wang Case Wesern Reserve Unversy Cleveland OH 4406 Unversy of Mssour Columba MO 652 Calforna Insue of Technology Pasadena CA 925 Lousana Sae Unversy Baon Rouge LA Acknowledgemens Joonk.oh@ Case.Edu PukhuanhongK@ Mssour.Edu RRoll@Calech.Edu junbowang@lsu.edu For nsghful and consrucve commens, we hank Yakov Amhud, Francsco Barllas, Hank Bessembnder, Mchael Brennan, Tarun Chorda, John Cochrane, Wayne Ferson, Chrs Jones, Raymond Kan, Cheng-Few Lee, Jay Shanken, Georgos Skoulaks, Avandhar Subrahmanyam, Guofu Zhou, and semnar parcpans a CalTech, Case Wesern Reserve Unversy, Emory Unversy, KAIST, UCLA, Unversy of Melbourne, Unversy of Mssour, Unversy of ew Souh Wales, Unversy of Souh Florda, Unversy of Technology a Sydney, and Yonse Unversy, York Unversy, Fnancal Managemen Assocaon 204, and orhern Fnance Assocaon 207.

2 Emprcal Tess of Asse Prcng Models wh Indvdual Asses: Resolvng he Errors-n-Varables Bas n Rsk Premum Esmaon Absrac To aenuae an nheren errors-n-varables bas, porfolos are wdely employed o es asse prcng models; bu porfolos mgh dversfy and mask relevan rsk- or reurn-relaed feaures of ndvdual asses. We propose an nsrumenal varables approach ha allows he use of ndvdual asses ye delvers conssen esmaes of ex-pos rsk premums. Ths esmaor yelds unbased esmaes and well-specfed ess n small samples. The marke rsk premum under he CAPM and he lqudy-adjused CAPM, premums on rsk facors under he Fama-French hree- and fve-facors models and he Hou, Xue, and Zhang (205) four-facor model are all nsgnfcan afer conrollng for asse characerscs. Key Words: Rsk Premum Esmaon, Errors-n-Varables Bas, Insrumenal Varables, Indvdual Socks, Asse Prcng Models 2

3 . Inroducon A fundamenal precep of fnancal economcs s ha nvesors earn hgher average reurns by bearng sysemac rsks. Whle hs dea s well acceped, here s lle agreemen abou he denes of sysemac rsks or he magnudes of he supposed compensaons. Ths s no due o a lack of effors along wo lnes of enqury. Frs, numerous canddaes have been proposed as underlyng rsk facors. Second, emprcal effors o esmae rsk premums have a long and vared hsory. Sarng wh he sngle-facor CAPM (Sharpe, 964; Lnner, 965) and he mul-facor APT (Ross, 976), he frs lne of enqury has brough forh an abundance of rsk facor canddaes. Among ohers, hese nclude he Fama and French sze and book-o-marke facors, human capal rsk (Jagannahan and Wang, 996), producvy and capal nvesmen rsk (Cochrane, 996; Esfeld and Papankolaou, 203; Hou, Xue and Zhang, 205), dfferen componens of consumpon rsk (Leau and Ludvgson, 200; A-Sahala, Parker, and Yogo, 2004; L, Vassalou, and Xng, 2006), cash flow and dscoun rae rsks (Campbell and Vuoleenaho, 2004) and llqudy rsks (Pasor and Sambaugh, 2003; Acharya and Pedersen, 2005). The second lne of enqury has produced emprcal esmaes of rsk premums for many among, wha Cochrane (20) erms as, a zoo of rsk facors. Mos esmaon mehods have followed hose orgnally nroduced by Black, Jensen and Scholes (972), (BJS), and refned by Fama and Macbeh (973), (FM). Ther mos promnen feaure s he use of porfolos raher han ndvdual asses n esng asse prcng models. Ths has long been consdered essenal because of an error-n-varables (E) problem nheren n esmang rsk premums. The E problem s bes apprecaed by racng hrough he BJS and FM mehods. They nvolve wo-pass regressons: he frs pass s a me-seres regresson of ndvdual asse reurns on he proposed facors. Ths pass provdes esmaes of facor loadngs, wdely called beas n he fnance leraure. The second pass regresses asse reurns cross-seconally on he beas obaned from he frs-pass regresson. Snce he explanaory varables n he second pass are esmaes, raher han he rue beas, he resulng rsk premum esmaes are based and nconssen; and he drecons of he bases are unknown when here are mulple facors nvolved n he wo-pass regressons. Hereafer, we wll adop he shorhand nomenclaure Bea o mean facor sensvy or facor loadng. 3

4 Wh a large number () of ndvdual asses, he E bas can be reduced by workng wh porfolos raher han ndvdual asses. Ths process begns by formng dversfed porfolos classfed by some ndvdual asse characerscs such as a bea esmaed over a prelmnary sample perod. I hen esmaes porfolo beas on he facors usng daa for a second perod. Fnally runs he cross-seconal regressons on esmaed porfolo beas usng daa for a hrd perod. BJS, Blume and Frend (973), and FM noe ha porfolos have less dosyncrac componens; so he errors-n-varables bas s reduced (and can be enrely elmnaed as grows ndefnely). Bu usng porfolos, raher han ndvdual asses, has s own shorcomngs. There s an mmedae ssue of es power snce he dmensonaly s reduced;.e., average reurns vary wh fewer explanory varables across porfolos han across ndvdual asses. Perhaps more roublng s ha dversfcaon no porfolos can mask cross-seconal phenomena n ndvdual asses ha are unrelaed o he porfolo groupng procedure. For example, advocaes of fundamenal ndexaon (Arno, Hsu and Moore, 2005) argue ha asses wh hgh marke values are overprced and vce versa, bu any porfolo groupng by an arbue oher han marke value self could dversfy away such poenal msprcng, renderng undeecable. Anoher dsqueng resul of porfolo maskng nvolves he cross-seconal relaon beween average reurns and facor exposures ( beas ). Take he sngle-facor CAPM as an llusraon (hough he same effec s a work for any lnear facor models). The cross-seconal relaon beween expeced reurns and beas holds exacly f and only f he marke ndex used for compung beas s on he mean/varance froner of he ndvdual asse unverse. Errors from he bea/reurn lne, eher posve or negave, mply ha he ndex s no on he froner. Bu f he ndvdual asses are grouped no porfolos sored by bea, any asse prcng errors across ndvdual asses no relaed o bea are unlkely o be deeced. Therefore, hs procedure could lead o a msaken nference ha he ndex s on he effcen froner. Tes porfolos are ypcally organzed by frm characerscs relaed o average reurns, e.g., sze and book-o-marke. Sorng on characerscs ha are known o predc reurns helps generae a reasonable varaon n average reurns across es asses. Bu Lewellen, agel, and Shanken (200) pon ou sorng on characerscs also mpars a srong facor srucure across es porfolos. Lewellen e al. (200) show as a resul ha even facors weakly correlaed wh 4

5 he sorng characerscs could explan he dfferences n average reurns across es porfolos, regardless of he economc mers of he heores ha underle he facors. Fnally, he sascal sgnfcance and economc magnudes of rsk premums are lkely o depend crcally on he choce of es porfolos. For example, he Fama and French sze and book-o-marke rsk facors are sgnfcanly prced when es porfolos are sored based on he correspondng characerscs, bu hey do no command sgnfcan rsk premums when es porfolos are sored only on momenum. In an effor o overcome he defcences of porfolo groupng whle avodng he E bas, we develop a new procedure o esmae rsk premums and o es her sascal sgnfcance usng ndvdual asses. Our mehod adops he nsrumenal varables echnque, a sandard economerc soluon o he E problem. We defne a parcular se of well-behaved nsrumens and hereafer refer o our approach as he mehod. To be specfc, our mehod frs esmaes beas for ndvdual asses from a poron of he observaons avalable n he daa sample. These become he ndependen varables for he second-sage cross-seconal regressons. Then, we re-esmae beas usng non-overlappng observaons, whch become he nsrumenal varables n he second-sage cross-seconal regressons. Snce we use non-overlappng observaons o esmae he ndependen and nsrumenal varables, whle reurns are only weakly auocorrelaed, f a all, he measuremen errors n bea esmaes should be vrually uncorrelaed cross-seconally wh her nsrumens. 2 The esmaor we propose s conssen for ex-pos rsk premum,.e., -conssen n Shanken (992). Snce conssency s a large sample propery, s mporan o examne he small sample performances of varous esmaors for praccal applcaons. To do so, we conduc a number of smulaon expermens. We choose smulaon parameers mached o hose n he acual daa. Smulaon resuls verfy ha he mehod produces unbased rsk premum esmaes even wh a relavely shor me-seres for bea esmaon. In conras, he sandard approach ha fs he he second-sage regressons usng OLS (hereafer we wll refer o hs sandard approach as he OLS mehod) suffers from severe E bases. For example, n smulaons wh a sngle facor model, we fnd ha he OLS esmaor, f used wh ndvdual socks, s sgnfcanly based oward zero even when beas are esmaed wh 2520 me-seres observaons. In conras, he esmaor yelds nearly unbased rsk premum esmaes when 2 Some of our emprcal ess also use sock characerscs as addonal nsrumens for beas. 5

6 only 252 me-seres observaons are avalable o esmae beas. In erms of es sze (.e., ype I error) and power (.e., ype II error), we fnd ha he convenonal -ess based on he esmaor are well specfed and hey are reasonably powerful, even n small samples. We fnd smlar resuls for he Fama-French hree-facor model. We also show analycally ha our esmaor s conssen even f beas of ndvdual socks vary over me as long as hey follow covarance saonary processes. 3 We fnd ha even wh me-varyng beas, he esmaor s unbased n small sample smulaons. Wh acual daa, we apply he mehod o examne wheher he rsk facors proposed by he CAPM, he hree-facor and fve-facor models of Fama and French (993 and 204), he q- facor asse prcng model of Hou, Xue, and Zhang (205), and he lqudy-adjused capal asse prcng model (LCAPM) of Acharya and Pedersen (2005) command posve rsk premums. These rsk facors have been successful when hey were esed wh porfolos, bu hese ess poenally suffer from he low dmensonaly problems ha Lewellen e al. (200) dscussed. In conras o he orgnal papers, when conrollng for correspondng non-β characerscs, we fnd ha none of hese facors s assocaed wh a sgnfcan rsk premum n he cross-secon of ndvdual sock reurns. Ths falure o fnd sgnfcan rsk premums s no due o he lack of es power of he mehod. Our smulaon evdence ndcaes he -ess based on he mehod provde reasonably hgh power under he alernave hypoheses ha he rue rsk premums equal he sample means of facor realzaons observed n he daa. For example, when he rue HML rsk premum equals he sample rsk premum (4.36% per year), he probably of deecng, whch s he es power, s 9.5%. In addon, when analyzng real daa, n he absence of non-β characerscs as conrol varables, we fnd some evdence ha SMB and HML beas command sgnfcan rsk premums n he cross-secon of ndvdual socks reurns. However, when we nclude correspondng non-β characerscs n he cross-seconal regressons, we fnd ha he rsk premums are no sgnfcanly dfferen from zero for any of esed beas. 3 The assumpon ha beas follow a covarance saonary process s sensble from an economc perspecve. Asse prcng models show ha expeced reurns are lnearly relaed o beas. If beas were o follow a non-saonary process, hey can go o nfny, whch would mply ha expeced reurns also go o nfny. Infne expeced reurns would no be economcally meanngful for any reasonable rsk averson parameer. 6

7 Several papers n he leraure, ncludng Berk e al. (999), Carlson e al. (2004 and 2006), Zhang (2005), and ovy-marx (203) argue ha frm characerscs may appear o be prced because hey may serve as proxes for beas. For example, consder frms A and B ha are dencal excep for her rsk. If frm A were rsker han frm B, hen frm A would have bgger book-o-marke rao han frm B because he marke would dscoun s expeced cash flows a a bgger dscoun rae. If error-rdden beas were used n an aemp o accoun for rsk, book-omarke raos mgh appear, ncorrecly, o be explanng a leas par of he observed rsk dfference. We develop a mehod o nvesgae hs alernave explanaon. Specfcally, we allow for me-varyng beas and characerscs, and we le he characerscs ancpae fuure changes n beas. We show analycally ha hs esmaor provdes conssen rsk premum esmaes when he second-sage cross-seconal regresson employs he average reurns over a long sample perod as dependen varable whle boh beas and characerscs serve as ndependen varables. Our emprcal resuls are robus wh respec o hs modfed approach. Our paper also conrbues o a large leraure on esng asse prcng models. As he lengh of me-seres grows ndefnely, Shanken (992) shows ha he E bas becomes neglgble because he esmaon accuracy of beas mproves. He also derves an asympoc adjusmen for he FM sandard errors of he OLS mehod. Jagannahan and Wang (998) exend Shanken s asympoc analyss o he case of condonally heerogeneous errors n me-seres regresson. Shanken and Zhou (2007) and Kan, Robo and Shanken (203) exend he resul o msspecfed models. However, he evdence and analyses n hose papers manly focus on porfolos. Our paper focuses on ndvdual socks as es asses and proposes he mehod o mgae he E bas n esng asse prcng models. Usng ndvdual socks n esng asse prcng models s a recen developmen n he leraure. Km (995) correcs he E bas usng lagged beas o derve a closed-form soluon for he MLE esmaor of marke rsk premum. The soluon proposed by Km s based on he adjusmen by Thel (97). Oher mehods proposed by Lzenberger and Ramaswamy (979), Km and Skoulaks (204), and Chorda e al. (205) are smlar, producng -conssen rsk premum esmaors. To avod he E bas, Brennan e al. (998) advocae rsk-adjused reurns as dependen varable n he second-sage regressons. However, he mehod ha Brennan e al. use does no esmae he rsk premums of facors. 7

8 2. Rsk-Reurn Models and Esmaon A number of asse prcng models predc ha expeced reurns on rsky asses are lnearly relaed o her covarances wh ceran rsk facors. A general specfcaon of a K-facor asse prcng model can be wren as: () where s he expeced excess reurn on sock, s he sensvy of sock o facor k, and s he rsk premum on facor k. s he excess reurn on he zero-bea asse. If rskless borrowng and lendng are allowed, hen he zero-bea asse earns he rsk-free rae and s excess reurn s zero,.e. The CAPM predcs ha only he marke rsk s prced n he cross-secon of average reurns. Several recen papers propose mulfacor models based on emprcal evdence of devaons from he CAPM. For example, Fama and French (992) propose a hree-facor model wh sze and book-o-marke rsks as addonal prced facors. Many emprcal ess of asse prcng models employ he Fama-MacBeh (FM) wo-sage regresson procedure o evaluae wheher he beas of rsk facors are prced n he cross-secon. The frs-sage esmaes facor sensves usng he followng me-seres regressons wh T perods of daa: (2) where s he realzaon of facor k n me. The me seres esmaes of facor sensves, for facor k, are he ndependen varables n he followng second-sage cross-seconal regressons used o esmae facor rsk premums: For gven me, (3) where realzed excess reurn s he dependen varable. The sandard FM approach fs OLS regresson o esmae he parameers of Eq. (3). These OLS esmaes are based due o he E problem snce s are esmaed wh errors. To mgae such bas, he porfolos are ypcally used as es asses, raher han ndvdual asses, because porfolo beas are esmaed more precsely han ndvdual beas. 8

9 Our emprcal ess use ndvdual socks as es asses o avod he shorcomngs ha we dscussed earler when usng porfolos as es asses. We propose an nsrumenal varable esmaor o avod E-nduced bases. To descrbe our esmaor, rewre Eq. (3) as r γβˆ ξ where r s a row vecor of realzed excess reurns n monh, Βˆ s he marx conanng he un vecor and K facor loadngs, and γ s a vecor of facor rsk premums (ncludng he excess reurn of zero-bea asse). We propose he followng nsrumenal varables esmaor (): where ˆΒ and ˆΒ γˆ, ' = ( Βˆ Βˆ ') ( Βˆ r ') are he marces of nsrumenal and explanaory varables, respecvely. We esmae beas whn odd monhs and even monhs separaely. Then we use odd-monh beas as nsrumenal varables and even-monh beas as explanaory varables when monh s even and vce versa when monh s odd. 4 We use daly daa whn odd and even monhs o esmae beas so ha he measuremen errors n he nsrumenal varables and explanaory varables are no correlaed cross-seconally, bu n prncple, one could use any non-overlappng nervals o esmae nsrumenal and explanaory beas. We f he cross-seconal regressons each monh usng he esmaor. The esmaor has been wdely used n he leraure o address he E problem, and s well known ha he esmaor s conssen under mld regulary condons. In our conex, he esmaor converges o he ex-pos rsk premum even for fne T when he number of socks n he cross-secon s suffcenly large. The proposon below formally saes he - conssency 5 of he esmaor: Proposon : Suppose sock reurns follow an approxmae facor srucure wh K common facors. Under mld regulary condons, he esmaor gven by Eq. (4) s -conssen when he number of socks n he cross-secon ncreases whou bound. Proof: See Onlne Appendx E. (4) 4 The and beas are compued usng half he number of observaons ha one would use o compue OLS beas and hence hey are noser. However, hs does no affec he conssency of he esmaor. Our smulaon resuls ndcae ha he esmaor yelds unbased rsk premum esmaes even wh a farly shor me-seres for bea esmaon. 5 Shanken (992) defnes -conssency. 9

10 We can frame he esmaor as a wo-sage leas square (2SLS) cross-seconal regresson o gan he underlyng nuon. The frs-sage regresses he explanaory varables agans he nsrumenal varables. The marx of he frs-sage regresson slope coeffcens s: λˆ = ( Βˆ Βˆ ') The second-sage regresson uses he fed values from he frs-sage regresson as explanaory varables and he OLS esmaor of hs second-sage regresson s he esmaor. Afer subsung he relaon n Eq. (5) and rearrangng he erms, he second-sage regresson esmaor can be wren as: ( Βˆ Βˆ '). (5) γˆ, '= λˆ {( Βˆ Βˆ ') ( Βˆ r ')}. (6) The expresson whn braces s he OLS esmaes of he rsk premums when beas are used as regressors. These OLS esmaes are pre-mulpled by he nverse of scalng marx λˆ o adjus for he E bas. In he case of a sngle facor model, s he scalar slope coeffcen obaned from regressng he explanaory varable on he nsrumenal varable. Snce boh explanaory and ndependen varables measure rue bea wh uncorrelaed errors, s less han one. The noser he errors, he smaller, whch s he rao of he and OLS slope coeffcens; hus magnfes he OLS esmae o accoun for he E bas. also correspondngly magnfes he sandard error, and hence he -sascs would be he same for boh OLS and rsk premum esmaes n large samples. In addon, noe ha he E scalng under a sngle facor model suggess ha he mehod essenally shrnks OLS beas oward her cross-seconal mean of her nsrumens. Such shrnkage s remnscen of Vascek (973)-syle beas ha move esmaed beas owards he marke bea of. In he case of mulfacor models, he shrnkage depends on he crossseconal correlaon of bea esmaes as well. 3. Small Sample Properes of he Mehod - Smulaon Evdence To evaluae he small sample properes of he mehod, we conduc a baery of smulaons usng he parameers mached o real daa. We frs nvesgae he bas and he roomean-squared error (RMSE) of he esmaor and hen we examne he sze and power of he 0

11 assocaed -es, whch we refer o as he es. 3.. Bas and RMSE of Esmaor We se he smulaon parameers o equal he correspondng parameers n he acual daa durng he sample perod of January 956 hrough December 202. The Cener for Research n Secury Prces (CRSP) value-weghed ndex provdes he marke reurn and he one-monh T- bll rae s he rsk-free rae. For each sock, a marke model regresson produces he bea and resdual reurns. Table repors he smulaon parameers. We conduc smulaons wh he cross-seconal sze of =2000 socks. We randomly generae daly reurns usng he followng procedure: ) For each sock, we randomly generae a bea and a sandard devaon of reurn resduals σ, ε from normal dsrbuons wh means and sandard devaons equal o he correspondng sample means and sandard devaons from he real daa. 6 We generae beas and σ, ε s n he begnnng of each smulaon and keep hem consan across 000 repeons. 2) For each day, we randomly generae a marke excess reurn draw from a normal dsrbuon wh mean and sandard devaon equal o he sample mean and sandard devaon from he daa. 3) For each sock and each day, we hen randomly generae resdual reurns ε, τ from ndependen normal dsrbuons wh mean zero and sandard devaon equal o he value generaed n sep (). For sock, we compue he excess reurn on day as where r MKT, s he marke excess reurns. For he frs-sage regresson n he smulaon, we esmae beas usng he followng marke model regresson wh daly excess reurns for each sock: 7 (7). (8) 6 If he random draw of s negave, we replace wh s absolue value. 7 We employ daly reurns raher han monhly reurns o oban more precse bea esmaes n he frs-sage regresson. We also expermen wh monhly daa o esmae beas. In unabulaed resuls, when T=20, 80, and 240 monhs, we fnd ha he esmaor wh monhly daa has smlar small sample properes o ha wh daly daa, whch are repored n Tables 2 and 3. All of our resuls are based on he runcaed mehod.

12 Each monh n he smulaon has 2 radng days and we use hree years of daly reurns (T=756 days) o f he me-seres regresson n Eq. (8). For he mehod, we use daly reurns from odd and even monhs durng a rollng hree-year esmaon perod o compue ndependen and nsrumenal varables, respecvely. We f he second-sage regresson wh monhly reurns, followng he common pracce n he leraure. We could have f he second-sage regresson wh daly reurns as well, bu hs would no mprove he precson of he second-sage esmaes. To see hs nuvely, compare fng one cross-seconal regresson for monh wh fng 2 separae daly regressons for he monh and averagng he daly regresson slope coeffcens over he monh. Wh he same se of frms n boh regressons and same beas for he monh, he slope coeffcen of he monhly regresson would be exacly 2 mes he average slope coeffcen of he daly regressons and he sandard error of he monhly regresson would also be 2 mes he sandard error of average daly regresson coeffcen. As a resul, boh specfcaons would yeld exacly he same - sasc for he slope coeffcen. There would be some dfferences beween he wo specfcaons f daly reurns are compounded o compue monhly reurns bu such dfferences are lkely small. We compound daly sock and facor reurns o compue correspondng monhly reurns. We f he cross-seconal regresson n Eq. (4) for each monh o esmae and. We hen roll he hree-year esmaon wndow forward by one monh and repea he esmaon procedure over 660 monhs (=55 years). Fnally, we ake he me-seres averages of and. We conduc he hree-facor model smulaons analogously, bu n addon o marke reurns and marke beas, addonal facors and beas are chosen o correspond o he Fama- French SMB and HML facors and beas. We mach he means and sandard devaons of he smulaon parameers o hose of acual daa, hen carry ou he esmaon procedure o esmae, and. Table presens he smulaon parameers n more deal. One of he ssues ha ofen arses wh esmaors s ha for any fne, here s a very small chance ha he cross-producs of ˆΒ ˆΒ and mgh be close o non-nverble; hs could resul n an unreasonably large value of parameer esmaes (see Knal, 980). To avod such a poenally ll-behaved esmaor for fne, we rea any monhly rsk premum esmae ha devaes sx sandard devaons of he correspondng facor realzaons from her sample 2

13 average as a mssng value,.e., he excluson cuoff s sx. 8 In our emprcal analyses n Secon 4, for any gven rsk facor, we adjus he excluson cuoffs o manan he chances of excluson bndng o below 3% of he number of all avalable monhs. The average dfferences beween he rsk premum esmaes and he corerspondng rue smulaon parameers over he 000 replcaons are he ex-ane bases relave o he rue rsk premums. Snce all rsk premum esmaes whn a sample are condonal on a parcular se of facor realzaons, we also repor he bases relave o he average realzed rsk premums n ha parcular sample, whch are he ex-pos bases as defned by Shanken (992). Panel A of Table 2 presens he ex-ane and ex-pos bases, as percenages of he rue marke premum. 9 The OLS esmae s based owards zero by 20% relave o he ex-ane rsk premum and by 2% relave o he ex-pos rsk premum, respecvely, whch are sascally sgnfcanly dfferen from zero, because of he E problem. In conras, he average dfferences beween esmaes and he ex-ane and ex-pos rsk premums are abou %, whch s sascally nsgnfcan. 0 The nex wo columns n Panel A presen he ex-ane and ex-pos RMSEs. The RMSE s a funcon of boh he bas and he sandard devaon of he esmaon error. The OLS esmaor has a smaller sandard devaon han he esmaor, bu he former s based, whle he laer s unbased. The ex-ane RMSE for he esmaor s slghly smaller han ha for he OLS esmaor. The ex-pos RMSE s.25 for he OLS esmaor, compared wh.080 for he esmaor. These resuls ndcae ha because of he bas, he accuracy (assessed by RMSE) of he esmaor s beer han ha of he OLS esmaor for he parameers used n our smulaons. Fgure plos he ex-ane and ex-pos bases of he and OLS esmaors as a funcon of 8 Shanken and Zhou (2007) also smlarly runcae her maxmum lkelhood esmaes of rsk premums wh porfolos as es asses o avod undue nfluene of oulers. 9 We know he ex-ane or he rue rsk premums n smulaons, bu we only observe ex-pos realzaons n pracce. Ex-pos bases measure he bases condonal on parcular facor realzaons and would lkely be more relevan n pracce alhough boh ex-ane and ex-pos measures are concepually neresng. 0 Based on he sandard errors across he,000 repeons, he -sascs of he ex-ane and ex-pos bases of OLS esmae are and -58.4, respecvely. Therefore, OLS esmaes are sgnfcanly based a any convenonal sgnfcance level. In conras, he -sascs of ex-ane and ex-pos bases of esmae are nsgnfcan a and -0.22, respecvely. The magnude of he bas n he OLS esmaor would be smaller f he rue rsk premum s smaller han wha we assume n he smulaons snce he E bas s proporonal o he magnude of he rue rsk premum. In unabulaed resuls, we fnd ha he ex-pos RMSE for he OLS esmaor would be smaller han ha for he esmaor f he rue marke rsk premum were smaller han abou 2% per annum (for comparson, he sample rsk premum s 5.8%). 3

14 he number of days (=T) n he rollng wndow o esmae he marke beas wh =2000 socks under he sngle-facor CAPM. The vercal axs repors he ex-ane and ex-pos bases as percenages of he rue marke rsk premum. The bas of he OLS esmaor s farly large, -44% for T=252 days. 2 The magnude of he bas s greaer han 5% even for T=2520 days, or 0 years. In conras, he bas s farly close o zero for he esmaor even for T=252 days, or year. Panel B of Table 2 presens he resuls for he Fama-French hree-facor model. The E problem always bases OLS rsk premum esmaes owards zero n unvarae regressons, bu n heory he bas could be n any drecon n mulvarae regressons. The resuls n Panel B ndcae ha he OLS rsk premum esmaes for he Fama-French hree-facor model are all based owards zero. For example, he ex-ane bases of he OLS esmaes are -54.4% and % for SMB and HML, respecvely. We fnd ha all ex-ane and ex-pos bases of he OLS esmaes are sgnfcanly dfferen from zero. In conras, he magnudes of he bases of esmaes are all less han 2.%, and hese bases are sascally ndsngushable from zero Sze and Power of Tes Our ess follow he Fama-MacBeh approach o es wheher he rsk premums assocaed wh varous common facors are relably dfferen from zero. For example, n he case of a sngle facor model, he es sasc s defned as:, (9) where s he me-seres average of monhly rsk premum esmaes and s he correspondng Fama-MacBeh sandard error (FMSE). 4 To examne he small sample properes of he -sasc n Eq. (9) under he null hypoheses, we follow he same seps as above o generae smulaed daa, bu we se all rue rsk premums equal o zero. We hen examne he percenage of repeons (ou of 000 oal 2 Snce our smulaon assumes 2 days per monh, T=252 days corresponds o one year. 3 Based on he sandard errors across he,000 repeons, he -sascs of he ex-ane and ex-pos bases of OLS esmae for he hree Fama-French facors are smaller han -45, and hence hghly sgnfcan. In conras, he - sascs of ex-ane and ex-pos bases of esmaes for he hree Fama-French facors range from - o 0, and hey are all nsgnfcan sascally. 4 An earler verson of our paper analycally derved he asympoc dsrbuon of he esmaor, whch could also be employed n our emprcal ess. However, we use he Fama-MacBeh sandard errors because hey are farly sraghforward o compue and more commonly used n he leraure. Snce he monhly esmaes are serally uncorrelaed, he usual nuon behnd he FM approach goes hrough. 4

15 repeons) when he -sascs are posvely sgnfcan a he varous levels (one-sded) usng crcal values based on he sandard normal dsrbuon. Panel A of Table 3 presens he es szes of he ess under he CAPM and he Fama- French hree-facor model for =2000 socks, respecvely. The resuls ndcae ha he ess are well specfed when T=756 days (=hree years of daly daa) are used for rollng bea esmaon. For example, he es szes for all rsk premums a he 5% sgnfcance level are beween 5.0% and 5.2% and hose a he 0% sgnfcance level are beween 9.8% and 0.2%. In unrepored resuls, we fnd ha he dsrbuon of he es sasc becomes closer o he heorecal dsrbuon as we ncrease T. We also fnd smlar resuls n smulaons wh =500 socks. We now nvesgae he power of he ess o rejec he null hypoheses when he alernave hypoheses are rue. To evaluae he power, we modfy he smulaon expermens by addng rsk premums equal o he average rsk premums ha we observe from real daa. All he oher smulaon parameers are he same as n he smulaons under he null hypoheses. We fx he sze of ess a he 5% sgnfcance level. Panel B of Table 3 shows ha he power of he es o rejec he null hypohess under he sngle-facor CAPM s 85.6%. Under he Fama-French hree-facor model, we fnd ha he frequency of rejecon of he null of zero marke rsk premum s 83.8% and ha of zero HML rsk premum s 9.5%. The es power s somewha weaker n deecng he posve SMB rsk premum bu s sll 5.8%. We also fnd ha n 99.6% of he smulaons, a leas one of he rsk premums for he Fama-French hree facors s sgnfcanly dfferen from zero. For comparson, Table 3 also presens he power of OLS ess. Under he CAPM, we rejec he null hypohess ha he marke rsk premum equals zero n 84.2% of he smulaons wh OLS ess, compared wh 85.6% wh he ess. Alhough, he OLS esmaes are based owards zero, he OLS ess are almos as powerful as he ess because of smaller sandard errors. We fnd smlar power resuls for he Fama-French hree-facor model as well alhough he OLS ess are generally less powerful han he ess. We also examne he power of he and OLS as he rue marke rsk premum vares (under he CAPM.) Fgure 2 presens hese resuls. If he marke rsk premum equals 2.9% per annum, whch s 50% of he ex-pos rsk premum observed from real daa, hen he power of he ess s abou 40% and he power of he OLS es s slghly smaller. Therefore, low power 5

16 would be a concern durng a sample perod when he rue rsk premum s farly small Tme-varyng beas Our smulaons so far assume ha beas are consan over me. Appendx A proves ha he esmaor provdes a conssen esmae of ex-pos rsk premum even wh me-varyng beas and rsk premums. We also conduc smulaons o nvesgae he small sample properes of he esmaor and assocaed ess wh me-varyng beas. When beas follow AR() processes, we fnd ha he small sample properes of he esmaor and he sze and power of he ess are smlar o wha we repor wh consan beas n Tables 2 and Rsk Premum Esmaes for Seleced Asse Prcng Models Ths secon employs he mehod o esmae he premums for rsk facors proposed by promnen asse prcng models. 4.. Daa We oban sock reurn, radng volume, and marke capalzaon daa from CRSP and fnancal saemen daa from COMPUSTAT for he sample perod of January 956 hrough December 202. We nclude all common socks (CRSP share codes of 0 or ). 5 The sample for monh excludes all socks prced below $ or socks wh marke capalzaons less han $,000,000 a he end of monh -. Snce daly reurns are used o esmae beas, we resrc he sample o socks wh reurns n monh, wh a leas 200 days of reurn daa durng each of he hree years pror o monh. 6 Table 4 presens summary sascs for he ncluded socks. A oal of 4,058 dsnc socks ener he sample a dfferen pons n me; 2,425 socks are presen n an average monh The CAPM and he Fama-French Three-Facor Model Ths secon frs ess wheher esmaed rsk premums under he CAPM and he Fama- French hree-facor models are sgnfcanly dfferen from zero usng he mehod wh ndvdual socks. I hen esmaes he rsk premums afer conrollng for sock characerscs. Early emprcal ess of he CAPM by Fama and MacBeh (973) and ohers fnd srong 5 We exclude Amercan deposory receps (ADRs), shares of benefcal neres, Amercan Trus componens, close-end funds, preferred socks, and real esae nvesmen russ (REITs). 6 We fnd smlar resuls of asse prcng ess when he sample ncludes all socks wh a leas 00 or 50 reurn observaons per year nsead of 200 observaons. 6

17 suppor for he CAPM. However, several subsequen papers fnd ha marke beas are no prced afer conrollng for oher characerscs. For nsance, Jegadeesh (992) and Fama and French (993) conclude ha he marke rsk premum s no relably dfferen from zero afer conrollng for frm sze. The nably of he CAPM o accoun for any of he cross-seconal dfferences n average reurns renvgoraes he search for alernave asse prcng models. The arbrage prcng heory proposed by Ross (976) provdes he general framework for mul-facor asse prcng models. The Fama-French hree-facor model s perhaps he mos wdely used, whch denfes sze and book-o-marke rsk facors n addon o he marke facor. We employ ndvdual socks as es asses n he asse prcng ess and avod he low dmensonaly problem nheren n he ess ha use characerscs-sored porfolos. We use daly rollng wndows from monh -36 o monh - o esmae beas for monh. In unabulaed ess, we fnd smlar es resuls when we esmae beas wh daly rollng wndows over he pas 2, 24, and, 60 monhs. To accoun for non-synchronous radng effecs, bea esmaon s supplemened wh a oneday lead and lag of he ndependen varables (Dmson, 979). For example, he followng regresson esmaes marke beas under he CAPM: for frm and day, (0). We esmae odd- and even-monh beas separaely usng reurns on days belongng o odd and even monhs, respecvely. Because of he non-synchronous radng adjusmen n Eq. (0), he frs and he las days of each monh are excluded o avod any poenal bases due o overlap. 7 An analogous mulvarae regresson esmaes he hree beas under he Fama-French hreefacor model. For each sock and monh, SIZE s he naural logarhm of marke capalzaon a he end of he prevous monh. BM s he book value dvded by he marke value where book value s he sum of book equy value plus deferred axes and creds mnus he book value of preferred sock. We compue cross-seconal correlaons beween each par of frm-specfc varables each monh 7 We fnd almos dencal resuls whle ncludng he frs and las days of each monh. Also, he resuls are qualavely smlar when no adjusmen s used for non-synchronous radng. 7

18 and Table 5 presens he average cross-seconal correlaons among beas and characerscs. The CAPM bea esmaed usng he marke model exhbs negave correlaon wh boh SIZE and BM. In he Fama-French hree-facor model, he correlaons beween marke bea and he SMB and HML beas are posve. The correlaon beween SIZE and SMB beas s negave, and he correlaon beween BM and HML beas s posve. For comparson, Table 5 also presens he average cross-seconal correlaons for 25 Fama- French SIZE and BM sored porfolos ha he leraure ypcally uses as ess asses. For each porfolo and each monh, we compue SIZE and BM as he value-weghed averages across all socks ha belong o he porfolo. The magnudes of correlaons among porfolo beas and characerscs are much larger; beween he SMB bea and SIZE s -.97 and beween he HML bea and BM s.88. Table 6 presens he rsk premum esmaes usng he mehod and ndvdual socks as es asses. We frs es he CAPM usng beas esmaed from he unvarae regresson. The marke rsk premum esmae s -.246%, whch s no relably dfferen from zero (Column ().) Therefore, here s no emprcal suppor for he CAPM wh ndvdual socks. For he Fama-French hree-facor model, he beas come from mulvarae me-seres regressons wh all hree facors. In Column (2), he marke rsk premum esmae s -.288% and nsgnfcan and he SMB and HML rsk premums are.30% and.344%, respecvely. The rsk premums of SMB and HML beas are sascally sgnfcan a he 5% level. The sgnfcance of SMB and HML rsk premum esmaes suggess ha hese facor rsks may be prced n he cross-secon, bu s also possble ha hese sgnfcan esmaes mgh be due o an omed varable bas because he second-sage cross-seconal regressons n Column (2) do no nclude SIZE and BM, he characerscs ha underle SMB and HML facors, as conrol varables. To examne hs possbly, we nclude SIZE and BM as addonal ndependen varables n he second-sage cross-seconal regressons. Under he CAPM, n Column (3), he slope coeffcens of SIZE and BM are -.20% and.96%, respecvely, and boh are sascally sgnfcan a he % level. The marke rsk premum esmae s -.090%, whch s sll no sgnfcanly dfferen from zero. Under he Fama-French hree-facor model n Column (4), none of he rsk premums s sgnfcan a he 5% level n he presence of SIZE and BM, ncludng he prevously sgnfcan SMB and HML rsk premums. In conras, SIZE and BM reman hghly sgnfcan. We also fnd smlar resuls when we use he logarhm of BM 8

19 (logbm) nsead of BM n Column (5). Table 6 also presens he es resuls usng OLS regresson esmaes. As n he ess, we fnd ha he SMB and HML rsk premums are sascally sgnfcan when we do no use SIZE and BM as conrol varables. However, hey become nsgnfcan when SIZE and BM are ncluded. The OLS es resuls are smlar o wha we fnd wh he rsk premum esmaes usng he mehod and hey also ndcae ha he facor rsks under he CAPM or Fama-French hreefacor model are no prced n he cross-secon of ndvdual sock reurns. Table 6 also repors he resuls on wo roughly equal subperods. The facor rsk premum esmaes are nsgnfcan n he boh subperods when SIZE and BM characerscs are ncluded. The slope coeffcens of SIZE and BM are sgnfcan n he boh subperods a convenonal levels. For he wo subperods, he rsk premum esmaes are smlar o her OLS counerpars excep for he SMB rsk premum n he frs subperod under he Fama-French hree-facor model; see Columns (2) and (7). Gven ha he mehod works very well n smulaons, here are several possble nerpreaons concernng hese emprcal resuls. Frs, somehng n he real daa compromses he mehod;.e., somehng ha s mssng from he smulaons. For example, alhough our smulaon evdence ndcaes ha he ess are reasonably powerful o deec posve rsk premums under he CAPM and Fama-French hree-facor models, hey mgh no be n he real daa. Ths nerpreaon does no seem convncng due o he followng observaon: Whou conrollng for SIZE and BM, Panel A n Table 6 fnds ha he SMB and HML rsk premums are sgnfcan. I s also possble ha SIZE and BM measure he rue fuure SMB and HML beas beer han he SMB and HML beas esmaed from pas daa under he Fama-French hree-facor model. Consequenly, he sgnfcan slope coeffcens on SIZE and BM mgh acually represen he rsk premums for SMB and HML facors. We evaluae hs possbly n greaer deal n Secon 5, and we fnd weak suppor for hs alernave explanaon The Fama-French Fve-Facor Model ovy-marx (203) and Aharon, Grundy, and Zeng (203) among ohers fnd ha sock reurns are sgnfcanly relaed o profably and nvesmen afer conrollng for Fama and French s hree facors. Fama and French (204) propose he followng fve-facor model ha adds wo 9

20 new facors o capure hese effecs: () where and are he beas wh respec o marke, sze, booko-marke, profably, and nvesmen facors, and and are he correspondng rsk premums. The RMW facor s based on he dfference beween he reurns on dversfed porfolos of socks wh robus and weak operang profably and he CMA facor s based on he dfference beween he reurns on dversfed porfolos of he socks wh conservave and aggressve nvesmen. We oban he daly daa for he fve facors from Ken French s webse. As n Fama and French (204), he sample perod for he asse prcng ess n hs subsecon s from January 964 hrough December 202. Panel A of Table 7 presens he asse prcng es resuls for he Fama-French fve-facor model. For he enre sample perod, we fnd ha he rsk premum esmaes for he fve facors are nsgnfcan a he 5% level regardless of conrollng for he correspondng characerscs: SIZE, BM, operang profably (OP), and nvesmen (IV). 8 Specfcally, smlar o Panel A of Table 6, he SMB and HML rsk premum esmaes have he hghes -sascs whou conrollng for he four characerscs (Column (3).) Bu hey become farly weak when hose four characerscs are ncluded (Column (6).) In conras, he slope coeffcens of SIZE, BM, OP, and IV are hghly sgnfcan wh her usual sgns (Column (6).) For he wo subperods whose lenghs are roughly equal, we fnd qualavely smlar resuls as well The q-facor Asse Prcng Model Cochrane (99) and Lu, Whed and Zhang (2009) presen producon-based asse prcng models n whch producvy shocks are ed o he changes n he nvesmen opporuny se, whch s conssen wh Meron s (973) ICAPM framework. Snce he shocks o producvy are dffcul o measure accuraely, Hou, Xue, and Zhang (205) (henceforh HXZ) propose an 8 We follow Fama and French (204) o consruc OP and IV. As descrbed n French s daa lbrary: OP for June of year s annual revenues mnus cos of goods sold, neres expense, and sellng, general, and admnsrave expenses dvded by book equy for he las fscal year end n -. The slope coeffcen n a unvarae regresson of sock reurns agans hs measure of OP s nsgnfcan, as n column (4) of Table 7. ovy-marx (203) defnes gross prof as revenue mnus cos of goods, scaled by curren perod oal asses. In unabulaed resuls, we fnd sgnfcanly posve slope coeffcen n he unvarae regresson wh ovy-marx s defnon of OP. Apparenly; he defnon of OP affecs he unvarae relaon. Because we use Fama-French facors, we follow her defnon. 20

21 emprcal q-facor model where an nvesmen facor and a ROE facor capure producvy shocks. Ther asse prcng model s specfed as: (2) where and are he beas wh respec o marke, sze, nvesmen, and ROE facors, respecvely, and, and are he correspondng rsk premums. The nvesmen facor capures he level of nvesmens and he ROE facor capures he reurn on nvesmens,.e., profably. The nvesmen facor s consruced as he reurn dfference beween frms wh low and hgh levels of nvesmens and he ROE facor s consruced as he reurn dfference beween frms wh hgh and low levels of profably. Inuvely, he nvesmens and raes of reurn on nvesmens are lkely o reflec he sensvy o unancpaed producvy shocks, and hese facors are supposed o capure he prce mpac of such shocks. HXZ argue ha her facors beer explan he cross-seconal reurn dfferences across porfolos based on varous anomales, e.g., BM, SIZE, momenum, and earnngs surprse han he Fama-French hree-facor model and he Carhar four-facor model. HXZ use a varey of dfferen es porfolos for her asse prcng ess. For nsance, her es of SIZE and BM uses he 25 Fama-French SIZE and BM sored porfolos, her es of momenum uses 0 porfolos based on momenum, and her es of he sandardzed earnngs surprses (SUE) uses 0 SUE sored porfolos. Snce all he ess employ seleced ses of ess porfolos, hey are also subjec o he low dmensonaly problems. We examne wheher he HXZ facors are prced n he cross-secon of ndvdual sock reurns. We oban daly HXZ facors from HXZ. Table 8 presens he es resuls of he q-facor asse prcng model usng he mehod. To faclae comparson, hs able uses he sample perod of January 972 hrough December 202, whch HXZ use n her emprcal ess. Columns () o (4) repor he rsk premum esmaes for each of he four beas under he HXZ model n unvarae regressons. one of rsk premums are sascally sgnfcan a convenonal levels. In Columns (6) and (7), when esng each of he nvesmen and ROE facor beas wh her correspondng characerscs, we fnd ha only he slope coeffcens of he characerscs are relably dfferen from zero. Column (5) presens he rsk premum esmaes for all four beas under he HXZ model ogeher, none of whch are relably dfferen from zero. In Column (8), when addng he hree 2

22 characerscs as conrol varables: SIZE, OP, and IV, o he HXZ model, we fnd ha he slope coeffcens of hese characerscs are all sascally sgnfcan wh usual sgns, whle none of rsk premums under he HXZ model s relably dfferen from zero. Therefore, for he enre sample perod, we fnd no emprcal suppor for he HXZ model when usng ndvdual socks as es asses. For he subperods whose lenghs are equal, we oban qualavely smlar resuls of asse prcng ess as well Lqudy-Adjused CAPM ex, we examne he lqudy-adjused capal asse prcng model (LCAPM) proposed by Acharya and Pedersen (2005), whch accouns for he mpac of llqudy-based radng frcons on asse prcng. 9 Accordng o he LCAPM, he level of llqudy and he covarances of reurn and llqudy nnovaon wh he marke-wde reurn and llqudy nnovaon affec expeced reurn. The uncondonal expeced reurn n excess of he rsk-free rae under he LCAPM s defned as: where, (3) c, s he llqudy cos, he rsk premum s he marke excess reurn mnus aggregae llqudy cos (.e., ), and he beas are, (4),,. The erm s he reward for frm-specfc llqudy level, whch s he compensaon for holdng an llqud asse as n Amhud and Mendelson (986). Acharya and Pedersen defne llqudy-adjused ne bea as:. (5) 9 Several oher papers, e.g., Pasor and Sambaugh (2003), also propose models where a sock s reurn sensvy o marke-wde (l)lqudy s prced n he cross-secon. Snce we do no have daly Pasor and Sambaugh s lqudy facor, we do no examne her model here. 22

23 The LCAPM mples ha he lnear relaon beween rsk and reurn apples for he lqudyadjused marke bea, bu no for he sandard marke bea under he CAPM. The LCAPM also mples ha he lneary beween rsk and reurn apples o excess reurns ne of frm-specfc llqudy cos (. Acharya and Pedersen es he LCAPM usng wo ses of es porfolos sored on he average and sandard devaon of llqudy. They sor socks based on Amhud (2002) llqudy measures durng each year and form 25 value-weghed llqudy porfolos for he subsequen year. They also form 25 value-weghed (llqudy) porfolos smlarly by sorng based on he sandard devaon of llqudy. We examne he correlaons beween and he value-weghed averages of SIZE and BM for hose porfolos used by Acharya and Pedersen. The average cross-seconal correlaons beween wh SIZE for llqudy and (llqudy) porfolos are -.96 and -.97, and hose wh BM are.7 and.74, respecvely. Such hgh correlaons beween lqudy-adjused marke bea,.e.,, and SIZE sugges ha would be parcularly hard o deermne emprcally wheher average reurns dffer across es porfolos due o SIZE or llqudyadjused marke beas. Ths suaon parallels ha n Chan and Chen (988) who use 20 szesored porfolos as es asses and fnd srong suppor for he sandard CAPM. The correlaons beween he sandard marke bea and SIZE across Chan and Chen s es porfolos range from o over dfferen sample perods, and he correspondng correlaons n he cases of llqudy and (llqudy) porfolos are whn hs range. Jegadeesh (992) shows ha when es porfolos are consruced so ha SIZE and sandard marke bea have low correlaons (n absolue value), he marke bea s no prced and ha he sgnfcan marke rsk premum found across sze-sored porfolos s due o he hgh correlaon (n absolue value) beween SIZE and marke bea. To avod such ambguy, we employ he mehod wh ndvdual socks o nvesgae wheher under he LCAPM s prced n he cross-secon. To faclae comparson, we follow he same procedure as n Acharya and Pedersen (2005) n all oher respecs. Because of he dfferences n he marke srucures of he YSE/AMEX and ASDAQ, he radng volumes repored n hese wo markes are no comparable and hence ASDAQ socks are excluded for hs es. In addon o exsng screenng crera, followng Acharya and Pedersen, we exclude socks ha do no rade for a leas 00 days per year, whch can suppress nosy llqudy 23

24 measures. Acharya and Pedersen defne he llqudy cos as follows: 20, (6), (7) where s he reurn on day n monh, s he dollar volume (n mllons) and s he monh - value of $ nvesed n he marke porfolo as of he end of July 962. Eq. (6) s based on Amhud s (2002) llqudy measure. Acharya and Pedersen use Eq. (7) as a measure of llqudy cos where s used o adjus for nflaon and he llqudy cos s capped a 30% o avod an obvously unreasonable value for. The marke-wde llqudy cos s he value-weghed average of ndvdual llqudy coss usng marke capalzaon n monh -. As n Acharya and Pedersen (2005), we esmae he nnovaons n llqudy coss usng AR models and hen esmae each ndvdual componen of beas n Eq. (5) usng a me-seres GMM approach and Dmson-ype correcons. 2 We hen f he followng cross-seconal regresson each monh : where c, s he average llqudy for sock n monh. 22. (8) The esmaor n monh s: γˆ ' ( Ψˆ, ˆ ' Ψ ), Ψˆ, r ', where ˆΨ, s even, ˆΨ when monh s odd and s ˆΨ odd, when monh s even, and ˆΨ even, 3 marx wh un vecor as he frs row,, and esmaed even-monh LMKT beas for socks as he second and hrd rows, respecvely. We esmae he even-monh LMKT beas usng daly daa n even monhs n he rollng esmaon wndow of monh -36 o monh -. ˆΨ odd, Analogous o ˆΨ even, esmaed usng all daly daa n odd monhs. 20 Acharya and Pedersen use llqudy coss a monhly frequency bu we use hem a daly frequency. 2 Appendx B presens he AR models ha we use o esmae expeced and unexpeced componens of llqudy. 22 As n Acharya and Pedersen (2005), 30% cappng s appled afer akng monhly average. 24

25 For he esmaor n monh +, we move he hree-year rollng esmaon wndow forward by one monh. We repea hs esmaon procedure unl all avalable observaons are exhaused. Then compung he average rsk premum esmae under he LCAPM and s sandard error s conduced n he same way as he oher asse prcng models esed above. Table 9 presens he rsk premum esmaes for he LMKT beas when usng ndvdual socks as es asses. The slope coeffcen on he Amhud llqudy measure s.220%, and s posve and hghly sgnfcan. However, he rsk premum esmaes for are.50% and.085%, respecvely, whou and wh conrollng for Amhud llqudy. These premums are no relably dfferen from zero. These resuls ndcae ha frm-specfc llqudy, whch s a frm characersc, s posvely relaed o average reurns, bu lqudy-adjused marke bea, whch s he sysemac rsk under he LCAPM, does no command a rsk premum. Table 9 also shows ha he lqudy-adjused marke bea s no prced n eher of subperods, whle Amhud llqudy affecs average reurns sgnfcanly n boh subperods. In comparson, Acharya and Pedersen (2005) repor a lqudy-adjused marke rsk premum esmae of abou 2.5% per monh usng he value-weghed ndex (see Panel B of Table 5 n Acharya and Pedersen, 2005), whch s abou 30% per year. 23 The equy rsk premum puzzle leraure argues ha even an annual rsk premum of abou 6% observed n he daa s hard o jusfy wh realsc levels of rsk averson, and larger rsk premums would be harder o jusfy. The large rsk premum esmae obaned wh es porfolos seems lkely o be he resul of hgh correlaons beween and porfolos characerscs raher han a rue depcon of he compensaon for a sysemac rsk. 5. Addonal Tess Ths secon examnes he robusness of our fndngs o a number of varaons n he es specfcaon and evaluaes he srengh of he nsrumens. 5.. Do Characerscs Proxy for True Beas? Our resuls ndcae ha many of he sysemac rsk facors proposed n he leraure do no command a premum afer conrollng for sock characerscs. However, s possble ha 23 The lqudy-adjused marke rsk premum equals he marke rsk premum mnus expeced marke-wde llqudy cos and hence s smaller han he unadjused marke rsk premum. 25