A Simple Method for Estimating Betas When Factors Are Measured with Error

Transcription

1 A Smple Mehod for Esmang Beas When Facors Are Measured wh Error J. Gnger Meng * Boson College Gang Hu ** Babson College Jushan Ba *** New York Unversy Sepember 2007 Ths paper s based on a chaper of Meng s docoral dsseraon a Boson College. Meng s exremely graeful o her dsseraon commee char, Arhur Lewbel, for numerous dscussons, nsghs, gudance, and encouragemen hroughou her dsseraon research. We hank Wayne Ferson for many useful commens. We hank Marn Leau and Sydney Ludvgson for makng her daa avalable and answerng our quesons. We hank Mchelle Graham for edoral asssance. All remanng errors and omssons are our own. * Correspondng auhor. Deparmen of Economcs, Boson College, 40 Commonwealh Avenue, Chesnu Hll, MA Phone: , E-mal: mengj@bc.edu. ** Asssan Professor of Fnance, Babson College, 2 Tomasso Hall, Babson Park, MA Phone: , Fax: , E-mal: ghu@babson.edu. *** Professor of Economcs, Deparmen of Economcs, New York Unversy, 9 W 4h S, Room 84, New York, NY 002. Phone: , E-mal: jushan.ba@nyu.edu.

2 A Smple Mehod for Esmang Beas When Facors Are Measured wh Error Absrac We propose a smple mehod for esmang beas (facor loadngs) when facors are measured wh error: Ordnary Leas-squares Insrumenal Varable Esmaor (OLIVE). OLIVE s nuve and easy o mplemen. OLIVE performs well when he number of nsrumens becomes large (can be larger han he sample sze), whle he performance of convenonal nsrumenal varable mehods and wo-sep GMM becomes poor or even nfeasble. OLIVE s especally suable for esmang asse reurn beas, snce hs s ofen a large N and small T seng. Inuvely, snce all asse reurns vary ogeher wh a common se of facors, one can use nformaon conaned n oher asse reurns o mprove he bea esmae for a gven asse. We apply he mehod o reexamne Leau and Ludvgson s (200b) es of he (C)CAPM. We fnd ha n regressons where macroeconomc facors are ncluded, usng OLIVE nsead of OLS bea esmaes mproves he R-squared sgnfcanly (e.g., from 3% o 80%). More mporanly, our resuls based on OLIVE bea esmaes help o resolve wo puzzlng fndngs by Leau and Ludvgson (200b) and Jagannahan and Wang (996): frs, he sgn of he average rsk premum on he bea for he marke reurn changes from negave o posve, conssen wh he heory; second, he esmaed value of average zero-bea rae s no longer oo hgh (e.g., from 5.9% o.9% per quarer). JEL Classfcaons: C3, C30, G2. Keywords: facor model, bea esmaon, measuremen error, nsrumenal varable, many nsrumens, GMM.

3 A Smple Mehod for Esmang Beas When Facors Are Measured wh Error. Inroducon In fnancal economcs, we ofen need o esmae asse reurn beas (facor loadngs). OLS s he smples and mos wdely used mehod by boh academc researchers and praconers. However, facors, especally hose consruced usng macroeconomc daa, are known o conan large measuremen error. In addon, even when a facor s measured accuraely, may sll be dfferen from he rue underlyng facor. For example, he reurn on he sock marke ndex s perhaps measured reasonably accuraely, bu may sll conan large measuremen error n he sense ha may be an mperfec proxy for he reurn on he rue marke porfolo (Roll s (977) crque). Under hese crcumsances, he OLS bea esmaor wll be nconssen. Furhermore, n he Fama and MacBeh (973) wo-pass framework, f he frspass bea esmaes are nconssen due o measuremen error n facors, hen he second-pass rsk prema and zero-bea rae esmaes wll be nconssen as well. Insrumenal varable esmaon s he usual soluon o he measuremen error problem. Inuvely, snce all asse reurns vary ogeher wh a common se of facors, one can use nformaon conaned n oher asse reurns o mprove he bea esmae for a gven asse. Ths s ofen a large N and small T seng, because here are ypcally more asses or socks han me perods. Ideally, we would wan o use all avalable nformaon, ha s, all vald nsrumens (he oher (N-) asse reurns), bu convenonal nsrumenal varable esmaors such as 2SLS (Two- Sage Leas Squares) perform poorly when he number of nsrumens s large. Ths s smlar o he weak nsrumens problem (Hahn and Hausman (2002)). Furher, hese mehods canno accommodae more nsrumens han he sample sze. In hs paper, we propose a smple mehod for esmang beas when facors are measured wh error: Ordnary Leas-squares Insrumenal Varable Esmaor (OLIVE). OLIVE easly allows for large numbers of nsrumens (can be larger han he sample sze). I s nuve,

4 easy o mplemen, and acheves beer performance n smulaons han oher nsrumenal varable esmaors such as 2SLS, B2SLS (Bas-correced Two-Sage Leas Squares), LIML (Lmed Informaon Maxmum Lkelhood), and FULLER (Fuller (977)), especally when he number of poenal nsrumens (N-) s large and he sample sze (T) s small. We show ha OLIVE s a conssen esmaor, under he assumpon ha dosyncrac errors are cross-seconally ndependen (Proposon ). Conssency s obaned when he number of asses (N) s fxed or goes o nfny. When dosyncrac errors are cross-seconally correlaed, reurns of oher asses as nsrumens are nvald n he convenonal sense because hey are correlaed wh he regresson errors. We show ha even n hs case, OLIVE bea esmaes reman conssen, provded ha N s large (Proposon 2). In a sense, we explo he large N of panel daa o arrve a a conssen esmaor. Snce convenonal GMM breaks down for N > T, and conssency n he absence of vald nsrumens requres large N, OLIVE s ably o handle large N s appealng. OLIVE can be vewed as a one-sep GMM esmaor usng he deny weghng marx. When N s larger han T, he opmal weghng marx n he GMM esmaon canno be conssenly esmaed n he usual unconsraned way. However, n our parcular seng, we are able o derve he wo-sep equaon-by-equaon GMM esmaor, as well as he jon GMM esmaor, based on he resrcons mpled by he model. Even hough he wo-sep GMM esmaor s asympocally opmal, performs worse han OLIVE n smulaons. Ths s because he wo-sep GMM esmaor has poor fne sample properes due o mprecse esmaon of he opmal weghng marx. Prevous sudes also have shown ha he wo-sep GMM esmaor whch s opmal n he asympoc sense can be severely based n fne samples of reasonable sze (e.g., Ferson and Foerser (994), Hansen, Heaon, and Yaron (996), Newey and Smh (2004), and Doran and Schmd (2006)). One-sep GMM esmaors use weghng marces ha are ndependen of esmaed parameers, whereas he effcen wo-sep GMM esmaor weghs he momen 2

5 condons by a conssen esmae of her covarance marx. Ths weghng marx s consruced usng an nal conssen esmae of he parameers n he model. Wyhowsk (998) performs smulaons for he dynamc panel daa model ha show he GMM esmaor performs que well f he rue opmal weghng marx s used. Mehods o correc he bas problem nclude, for example, usng a subse of he momen condons and normal quas-mle. Oher soluons o hs problem use hgher order expansons o consruc weghng marx esmaors, or use generalzed emprcal lkelhood (G.E.L.) esmaors as n Newey and Smh (2004). In anoher recen paper, Doran and Schmd (2006) sugges usng prncpal componens of he weghng marx. Gven he dffculy n esmang he opmal weghng marx, especally when N s large, usng deny weghng marx becomes an nuve opon. OLIVE can be vewed as a GMM esmaor usng he deny weghng marx. Fama and MacBeh s (973) wo-pass mehod can be modfed by usng OLIVE nsead of OLS o esmae beas n he frs-pass. As an emprcal applcaon, we reexamne Leau and Ludvgson s (200b) es of he (C)CAPM usng hs modfed Fama-MacBeh mehod. Leau and Ludvgson s facor cay has been found o have srong forecasng power for excess reurns on aggregae sock marke ndces. The facor cay s he conegrang resdual beween log consumpon c, log asse wealh a, and log labor ncome y. Macroeconomc varables usually conan large measuremen error. We fnd ha n regressons where macroeconomc facors are ncluded, usng OLIVE nsead of OLS mproves he R 2 sgnfcanly (e.g., from 3% o 80%). More mporanly, our resuls based on OLIVE bea esmaes help o resolve wo puzzlng fndngs by Leau and Ludvgson (200b). As menoned earler, f we use OLS when facors are measured wh error, boh he frs-pass bea esmaes and he second-pass rsk prema and zero-bea rae esmaes wll be nconssen. On he oher hand, snce OLIVE bea esmaes are conssen even when facors conan measuremen error, he rsk prema and zero-bea rae can be conssenly esmaed n he second-pass f OLIVE s used n he frs-pass o esmae beas. 3

6 Frs, Leau and Ludvgson (200b) sae ha [a] problem wh hs model, however, s ha here s a negave average rsk prce on he bea for he value-weghed reurn. They also noe ha Jagannahan and Wang (996) repor a smlar fndng for he sgns of he rsk prces on he marke and human capal beas (page 259). When we use OLIVE bea esmaes, hs problem goes away. Usng OLIVE nsead of OLS esmaon n he frs-pass changes he sgn of he average rsk premum on he bea for he value-weghed marke ndex from negave o posve, whch s n accordance wh he heory. Second, n Leau and Ludvgson (200b), he esmaed value of he average zero-bea rae s large. As he auhors observed, hs fndng s no uncommon n sudes ha use macroeconomc facors. They noe ha Jagannahan and Wang (996) also fnd he esmaed zero-bea raes o be large. Leau and Ludvgson (200b) noe ha samplng error could be one of he poenal explanaons for hs puzzlng fndng. However, hey conclude ha [p]rocedures for dscrmnang he samplng error explanaon for hese large esmaes of he zero-bea rae from ohers are no obvous, and s developmen s lef o fuure research (page 260). We fnd ha, when OLIVE bea esmaes from he frs-pass are used, he esmaed value of he average zero-bea rae n he second-pass s no longer oo hgh (e.g., from 5.9% o.9% per quarer). Our resuls sugges ha measuremen error n facors s he cause of hs problem. Samplng error s a second-order ssue; becomes neglgble as he sample sze T becomes large. Unlke samplng error, he measuremen error problem does no dmnsh as he sample sze T becomes large, because OLS produces nconssen bea esmaes. When macroeconomc facors wh measuremen error are ncluded n he model, OLIVE can provde more precse bea esmaes n he frs-pass, whch lead o more precse esmaes of he zero-bea rae n he second-pass. In conras, makes almos no dfference wheher we use OLIVE or OLS o esmae beas for he Fama-French hree-facor model, where he facors may conan lle measuremen error as hey are consruced from sock reurns. Overall, our resuls from hs emprcal 4

7 applcaon valdae he use of OLIVE o help mprove bea esmaon when facors are measured wh error. Our emprcal resuls suppor and srenghen Leau and Ludvgson (200b) and Jagannahan and Wang (996) by resolvng some of her puzzlng fndngs. Our fndngs are also conssen wh he heme n Ferson, Sarkssan, and Smn (2007) ha he (C)CAPMs mgh work beer han prevously recognzed n he leraure. Many exsng emprcal asse prcng models mplcly assume ha macroeconomc varables are measured whou error, for example, Chen, Roll, and Ross (986). Prevous sudes have noed he measuremen error problem n hs conex (see, e.g., Ferson and Harvey (999)). Connor and Korajczyk (99) develop and apply a procedure smlar o 2SLS. They frs regress macroeconomc varables on sascal facors obaned from Asympoc Prncpal Componens (APC, developed n Connor and Korajczyk (986)), and hen use fed values nsead of orgnal macroeconomc varables n asse prcng ess. They fnd ha mos of he varaon n macroeconomc varables s measuremen error. Ther mehod can poenally overcome he measuremen error problem n macroeconomc varables. However, snce he fed values are lnear combnaons of sascal facors, hey do no conan any more nformaon beyond sascal facors, whch lack clear economc nerpreaons. We, Lee, and Chen (99) also noe he presence of errors-n-varables problem n facors. They use he sandard economerc reamen: nsrumenal varables approach (IV or 2SLS). Boh her facors and nsrumens are sze-based porfolos. Even f here s measuremen error n sze-based porfolo reurns, he problem would no be solved by usng oher sze-based porfolo reurns as nsrumens. As one would expec, hey fnd exremely hgh frs-sage R 2 s. Ths means her IV resuls wll be very smlar o OLS resuls, and ndeed ha s wha hey fnd. The res of he paper s organzed as follows. In Secon 2, we descrbe he model seup and nroduce he smple esmaor, OLIVE. We also oulne oher convenonal IV esmaors. In Secon 3, as a heorecal exenson, we derve he wo-sep equaon-by-equaon GMM esmaor and he jon GMM esmaor. We conduc a Mone Carlo smulaon sudy o compare 5

8 he above esmaors n Secon 4. In Secon 5, we reexamne some resuls n Leau and Ludvgson (200b) as an emprcal applcaon of OLIVE. Secon 6 concludes. 2. Esmaon Framework 2.. Model Seup To descrbe he model, we begn by assumng ha asse reurns are generaed by a lnear mul-facor model: y = x *' B + e, () where =,, N, =,, T, y s asse s reurn a me, x * s an M vecor of rue facors a me, and β s an M vecor of facor loadngs for asse. However, he rue facors x * are observed wh error: x = x * + v, (2) where v s an M vecor of measuremen error. Ths s smlar o he seup n Connor and Korajczyk (99) and Wansbeek and Mejer (2000). Usng (2), we can rewre () as: where ε = e v ' B. y = x ' B + ε, (3) We canno use OLS o esmae β equaon-by-equaon, even hough x s observable, because he error erm ε s correlaed wh he observable facors x due o he measuremen error v. For a fxed asse, rewre (3) as Y = XB + ε, (4) where Y s a T vecor of asse reurns, X [ ι,( x,... x T )'] s a T ( M + ) marx of observable facors (ι s a T vecor of s), and B s an (M+) vecor of facor loadngs. As noed before, OLS produces nconssen esmaes of facor loadngs: 6

9 Le + N OLS B = ( X ' X) X ' Y. (5) Y [ Y,... Y, Y,..., Y ] be a T ( N ) marx of all asse reurns excludng he h asse. Then Y - can serve as nsrumenal varables. Le Z [, ι Y ], mulply boh sdes of equaon (4) by Z o oban: Z ' Y = Z ' XB + Z ' ε. (6) I can be shown ha he usual IV or 2SLS s equvalen o runnng Feasble GLS on (6); ha s, 2 SLS B X Z Z Z Z X X Z Z Z Z Y = ( ' ( ' ) ' ) ' ( ' ) '. (7) The dea of 2SLS s frs o projec he regressors (X) ono he space of nsrumens ( Z ), and hen o regress he dependen varables ( Y ) on fed values of regressors nsead of regressors hemselves. I s well known ha wo-saged leas squares (2SLS) esmaors may perform poorly when he nsrumens are weak or when number of nsrumens s large. In hs case 2SLS ends o suffer from subsanal small sample bases OLIVE The movaon behnd our approach begns wh he fac ha 2SLS only works when N (number of nsrumens) s much smaller han T (sample sze), whch s no he case for mos fnance applcaons. To llusrae he problem, magne he case where N = T. Then he fed values are he same as orgnal regressors, and 2SLS becomes he same as OLS. Ths problem of 2SLS s relaed o he weak nsrumens leraure n economercs, whch has grown rapdly n recen years; see for example Hahn and Hausman (2002). We propose o esmae facor loadngs B by smply runnng OLS on equaon (6). We call Ordnary Leas-squares Insrumenal Varable Esmaor (OLIVE): 7

10 OLIVE B X ZZ X X ZZ Y = ( ' ' ) ' '. (8) Proposon. Under he assumpon ha dosyncrac errors e are cross-seconally ndependen, hen for eher fxed N or N gong o nfny, he OLIVE esmaor s T conssen and asympocally normal. See Appendx A for a proof of Proposon. Proposon reles on he assumpon of vald nsrumens. Tha s, e j s uncorrelaed wh e ( j ). However, f he dosyncrac errors are also cross-seconally correlaed, none of he nsrumens wll be vald n he convenonal sense. For example, f he objecve s o esmae B, by equaon (3), ε = e B v. When e s correlaed wh e j, ' y j wll be correlaed wh ε. Thus y j wll no be a vald nsrumen. However, we can sll esablsh he conssency of he OLIVE, provded ha he cross-seconal correlaon s no oo srong and N s large. To hs end, le ( ) γ = E ee. We assume j j N γ j C < (9) j = for each. Ths condon s analogous o he sum of auocovarances beng bounded n he me seres conex, a requremen for a me seres beng weakly correlaed. Ba (2003) shows ha he condon mples (3) beng an approxmae facor model of Chamberlan and Rohschld (983). Proposon 2. Under he assumpon of weak cross-seconal correlaon for he dosyncrac errors as saed n (9), f T / N 0, hen he OLIVE esmaor s T conssen and asympocally normal. A proof of Proposon 2 s provded n Appendx B. Mere conssency would only requre / N 0. I s he T conssency and asympoc normaly ha requre T / N 0. Noe ha under fxed N, all IV esmaors dscussed n he nex secon ncludng 8

11 OLIVE (usng y j as nsrumens) wll be nconssen due o he lack of vald nsrumens. In a sense, we explo he large N of panel daa o arrve a a conssen esmaor. Far from beng a nusance, large N s clearly benefcal. In vew ha convenonal GMM breaks down for N and conssency n he absence of vald nsrumens requres large N, OLIVE s ably o handle large N s appealng. > T Le OLIVE ε = Y XB and σ ' =, T M ε ε he varance-covarance marx of 2 OLIVE B s a ( M + ) ( M + ) marx: 2 Ω = σ ( X ' Z Z ' X) ( X ' Z Z ' Z Z ' X)( X ' Z Z ' X). (0) The above esmaon s done for each =,, N. Wh he B obaned for each, we can esmae x * usng a cross-secon regresson based on equaon (). Ths s done for each =,, T. Gven x *, he esmaed rsk prema can be recovered as n Black, Jensen, and Scholes (972) (see also Campbell, Lo, and MacKnlay (997, Chaper 6)). The above seup also allows us o es he valdy of he mul-facor models. When he nsrumenal varables are Z [, ι Y ], he consan regressor ι self s an nsrumenal varable. The es for he consan coeffcen s beng zero s = a Ω, where Ω s he frs dagonal elemen of he nverse marx Ω. There s an alernave mehod for esmang he rue facors,.e., he mehod of Connor and Korajczyk (99). They frs regress he observed facors on APC esmaed sascal facors and use he fed values as esmaes of he rue facors (roae observed facors ono sascal facors). They fnd he R-squared o be que small, and hey nerpre hs as evdence for much measuremen error n he observed facors. APC should have good performance heorecally and emprcally. However, he sascal facors usng he prncple-componens 9

12 mehod lack clear economc nerpreaons. In conras, noe ha esmaed facors x * usng OLIVE has he same nerpreaons as x, he observable facors. Thus he esmaed rsk prema also have economc nerpreaons Oher IV Esmaors We compare he performance of OLIVE wh OLS and several well known IV esmaors: 2SLS, LIML, B2SLS, as well as FULLER. OLS s o be consdered as a benchmark. 2SLS s he mos wdely used IV esmaor. I has fne sample bas ha depends on he number of nsrumens used (K) and nversely on he R 2 of he frs-sage regresson (Hahn and Hausman (2002)). The hgher-order mean bas of 2SLS s proporonal o he number of nsrumens K. However 2SLS can have smaller hgher-order mean squared error (MSE) han LIML usng he second-order approxmaons when he number of nsrumens s no oo large. LIML s known no o have fne sample momens of any order. LIML s also known o be medan unbased o second order and o be admssble for medan unbased esmaors (Rohenberg (983)). The hgher-order mean bas for LIML does no depend on K. B2SLS denoes a bas adjused verson of 2SLS. The formulae for hese esmaors are as follows: Le P ZZZ Z = ( ' ) ' be he dempoen projecon marx, M I P =, W [ Y, X], Z = ( y, y2,, y, y+,, yn), hen: β OLS β 2SLS = ( X ' X) X ' Y = ( X ' PX) X ' PY β LIML = ( X '( P λm) X) X '( P λm) Y β B2SLS = ( X '( P λm) X) X '( P λm) Y. () β FULLER = ( X '( P λm) X) X '( P λm) Y β = ( X ' ZZ' X) X ' ZZ' Y OLIVE 0

13 For he above equaons, 2SLS, LIML, B2SLS, and FULLER can all be regarded as κ- X ' PY κ X ' MY class esmaors gven by X ' PX κ X ' MX. Forκ = 0, we ge 2SLS. Forκ = λ, whch s he smalles egenvalue of he marxw ' PW( W ' MW), we oban LIML. Forκ = λ, whch equals K 2, we oban B2SLS. For κ = λ, whch equals α λ, we oban FULLER. T T K Followng Hahn, Hausman, and Kuersener (2004), we consder he choce of α o be eher or 4 n our smulaon sudes laer (Secon 4). The choce of α = s advocaed by Davdson and McKnnon (993), whch has he smalles bas, whle α = 4 has a nonzero hgher mean bas, bu a smaller MSE accordng o calculaon based on Rohenberg s (983) analyss. 3. Effcen Two-Sep GMM Wha makes OLIVE appealng s s ease of use. Snce OLIVE s a GMM esmaor when seng he weghng marx o an deny marx, s naural o ry o mprove he effcency of he esmaor by usng he opmal weghng marx. Tradonal unconsraned GMM wll break down when N>T (he esmaed weghng marx s no nverble). We wll derve he heorecal weghng marx, whch depends on far fewer number of parameers. Replacng he unknown parameers by her esmaed counerpars wll resul n an esmaed heorecal weghng marx, whch s nverble even for N>T. In Secon 3.., we descrbe he equaon-by-equaon GMM esmaon, and n Secon 3.2., we descrbe he jon GMM esmaon. 3.. Equaon-by-Equaon GMM Consder esmang B for equaon. By defnon, ε = y x ' B = e v ' B. For every j ( j ), y j can serve as an nsrumen. Le u = y ε = ( x *' B + e )( e v ' B). j j j j

14 Under he assumpon ha e, e j, v, and x * are muually ndependen, he momen condons, or orhogonaly condons, wll be sasfed a he rue value of ( ) B : Eu ( j ) = E yj y x ' B = 0. (2) Each of he (N-) momen equaons corresponds o a sample momen, and we wre hese (N-) sample momens as: u ( B) = u ( B). (3) T j T = j Le u( B ) be defned by sackng uj ( B ) over j. For a gven weghng marx W, he equaon-by-equaon GMM s esmaed by mnmzng: mn u( B) ' W u( B) B. For each,, le u be he (N-) vecor by sackng u j over j. The opmal weghng marx s W = E( uu'). Gven he above funconal form for u j, W can be parameerzed n erms of var( e ) and B for each, var( v ), and var( x*) = var( x) var( v). We now derve he expresson of W. The (j, k)h elemen of W ( j, k ) s gven by: Eu ( u ) = E y ( e v' B)( e B' v) y j k j k ( var( ) ' var( ) ) = E y e + B v B y j k * * ( j ' j )( ' k k )( var( ) 'var( ) ) = E B x + e x B + e e + B v B * E ( Bj ' var( x ) Bk δ jk var( ej) )( var( e) B'var( v) B) * ( Bj 'var( x ) Bk δ jk var( ej ))( var( e ) + B ' var( v ) B ) = + + = +, (4) where δ jk = f j = k, and zero oherwse. In he las equaly, B j s are assumed non random coeffcens. Le For example, suppose =, hen he above covarance marx s smply he followng. 2

15 B2 B, (5) BN 3 Λ = hen he (N-) by (N-) covarance marx W s gven by: ( var( ) ' var( ) )( var( *) ' ) W = e + B v B Λ x Λ +Ω, (6) where Ω s a dagonal marx of dmenson (N-), ha s Noe ha ( var( e ) B 'var( v ) B ) Ω = dag(var( e ),, var( e )). (7) 2 + s a scalar, whch s he varance of he OLS resdual N T ε, hus can be esmaed by ε 2. T = For a general, he formula for W becomes: ( var( ) 'var( ) )( var( *) ' ) W = e + B v B Λ x Λ +Ω. (8) The analycal expresson for he nverse of W s: W = * ( Λ var( x ) Λ +Ω ) var( e ) + B 'var( v ) B (( x ) ) * var( ) ' ' Ω Ω Λ +Λ Ω Λ Λ Ω = var( ε ) (9) The esmaon procedure s hen as follows. Frs we use OLIVE o oban, for each asse, B and ε = y x ' B, whch equals an esmae of e v ' B. The denomnaor of W s compued by he sample varance of ε. Second, gven B, we run cross-seconal regresson o oban x * for each, and hen esmae var( x *). Also, gven x *, we can esmae e = y x * B, so ha var( e ) are compued for each. 3

16 Thrd, we use he above esmaes o consruc a conssen esmae of E( uu '), and use ha o do wo-sep GMM. For each asse, here s an (N-) ( N-) weghng marx W. The esmae of bea s: B ( ' ' ) ' = X ZW Z X X Z W Z ' Y. (20) The choce of W s opmal n he sense ha leads o he smalles asympoc varance marx for he GMM esmae. However, a number of papers have found ha GMM esmaors usng all of he avalable momen condons may have poor fne sample properes n hghly denfed models. Wh many momen condons, he opmal weghng marx s poorly esmaed. The problem becomes more severe when many of he momen condons (mplc nsrumens) are weak. The poor fne sample performance of he esmaes has wo aspecs, as noed by Doran and Schmd (2006). Frs, he esmaes may be serously based. Ths s generally beleved o be a resul of correlaon beween he esmaed weghng marx W and he sample momen condons n equaon (3). Second, he asympoc varance expresson may serously undersae he fne sample varance of he esmaes, so ha he esmaes are spurously precse Jon GMM B B B2 B N = ', ',, ' '. Le u In hs subsecon, we dscuss jon GMM esmaon of ( ) be he vecor wh elemens j u for all, j pars ( j ). The opmal GMM weghng marx, E( uu '), s dffcul o esmae n he usual unconsraned way because he number of momen condons, N(N-), can be much larger han T. Under our model specfcaon, however, E( uu ') can also be parameerzed n erms of var( e ), B, var( v ), and var( x *). 4

17 The N(N-) by N(N-) weghng marx W can be paroned no N 2 block marces, each beng (N-) by (N-). We denoe hese block marces W = E( u u '), for all, h =,, h h N. The block dagonal marx W corresponds o he equaon-by-equaon weghng marx W, as derved n equaon (4) n he prevous subsecon. In shor, he ( jk, ) h elemen of he block dagonal marx W (denoed as w jk ) s: w = E( u u ) = E y ( e v ' B)( e B ' v ) y = + + jk j k j k * ( B j 'var( x ) Bk δ jk var( ej) )( var( e) B'var( v) B) The block off-dagonal marx W h ( h) represens he varance-covarance marx beween he orhogonaly condons for asses and h. Ths marx s nonzero because an nsrumen used for asse may also be used for asse h. In addon, asse s also an nsrumen for asse h and vce versa. Thus he orhogonaly condons assocaed wh dfferen equaons. are correlaed. The ( jk, ) h elemen of hs marx, h w jk, equals E( ujuhk ) = E yj ( e v ' B )( eh Bh ' v ) y k, where j and k h by defnon of IV. We derve he formulae for w, he (, ) h jk jk h elemen of he block off-dagonal marx W h, n each of he four possble cases n Appendx C. We now have he whole weghng marx W. GMM s esmaed by mnmzng mn ub ( )' W ub ( ) B. The esmae of bea s: where ( ) B = (( I X)' ZW Z'( I X)) ( I X)' ZW Z' Y, (2) Y = Y ',, Y N ' '. Z s a block dagonal marx, wh Z = dag( Z, Z2,..., Z N ), where Z [, ι Y ]. Jon GMM wll no be used n hs paper because he number of momen condons, N(N-), s oo large. Bu f N s small, jon GMM wll be useful. In he nex smulaon secon, 5

18 when comparng OLIVE wh oher esmaors, we only consder he wo-sep equaon-byequaon GMM esmaor (2GMM), n addon o he IV esmaors dscussed n Secon Smulaon Sudy 4.. Mone Carlo Desgn We conduc a Mone Carlo smulaon sudy o compare he performance of our smple OLIVE esmaor wh oher esmaors. The daa generang process (DGP) for our smulaon sudy s as follows. We assume no nercep,.e., arbrage prcng heory (APT) or capal asse prcng model (CAPM) holds, as n Connor and Korajczyk (993) and Jones (200). Alhough he esmaon framework s general for any facor model, we mplemen our smulaon wh a sock marke applcaon n mnd. The DGP below s very smlar o he one n Connor and Korajczyk (993). We frs generae a secury wh a rue bea of one, whch s o be esmaed. Then we generae K = N- nsrumens usng he followng: x = x * + v, =,..., T y = β ' x * + e = β ' x + ( β ' v + e ) = β ' x + ε, =,..., N x MVN πσ I 2 J * (, x ) v MVN σ I J 2 J (0, v ) β MVN ι σ J 2 J (, β I ) e MVN σ I N 2 N (0, e ) y = ( y, y,..., y, y,..., y ), 2, +, N We use K = (, 2, 3, 5, 0, 30, 45, 55, 90, 50, 300, 600), T = 60, π = 0., σ x = 0., σ β = and 000 replcaons. Whou loss of generaly, we assume J, he number of explanaory varables o be, whch makes he model specfcaon equvalen o he CAPM for he excess reurn. We allow x and β o be normally generaed. One advanage of OLIVE s ha when K s larger han T, sll works whle mos oher IV esmaors do no. 6

19 Two mporan parameers for he performance of he esmaors are he sandard devaon of he error n reurns, σ e, and he sandard devaon of he measuremen error, We allow hese wo parameers o change from low (0.0), medum (0.), o hgh (),.e., σ e (0.0, 0., ) and σ v (0.0, 0., ). When σ e ncreases from 0.0 o, he nsrumens becomes weaker. When σ v ncreases, he magnude of measuremen error ncreases. Table presens smulaon resuls when boh σ v and σ v are se equal o 0., whch s he medum measuremen error and medum nsrumens case. σ v Mone Carlo Resuls In Tables, a varey of summary sascs s compued for each esmaor. When K s se from o 55 (K<T), all esmaors are compued. When K>T, only OLS, OLIVE, and he wo-sep equaon-by-equaon GMM esmaor (2GMM) are compued because oher IV esmaors become nfeasble. Followng Donald and Newey (200), we compue he medan and mean bas and he medan and mean absolue devaon (AD), for each esmaor from he rue value of β generaed. We examne dsperson of each esmaor usng boh he ner-quarle range (IQR) and he dfference beween he 0. and 0.9 decles (Dec. Rge) n he dsrbuon of each esmaor. Throughou, OLS offers he smalles dsperson n erms of boh IQR and Dec. Rge. Ths fndng s conssen wh Hahn, Hausman, and Kuersener (2004). We also repor he coverage rae of a nomnal 95% confdence nerval (Cov. Rae). When here s only one nsrumen, 2SLS, LIML, OLIVE, and 2GMM are all equvalen. Throughou, boh OLS and 2GMM seem o be based downwards. As Newey and Smh (2004) pon ou, he asympoc bas of GMM ofen grows wh he number of momen resrcons. Our smulaon resuls show ha he performance of he wo-sep GMM esmaor becomes worse as Smulaon resuls under oher parameer values are generally smlar o hose n Table. These supplemenal ables are no ncluded o save space and are readly avalable upon reques. 7

20 he number of nsrumens grows. As he number of nsrumens becomes very large (e.g., when K = 50, 300, and 600), 2GMM has even worse performance han OLS. As expeced, LIML performs well n erms of medan Bas when s feasble (when K = 3, 0, 30, and 55). In erms of mean Bas, FULLER usually performs well (when K = 2, 5, 0, and 30). In general, OLIVE does que well n erms of bas. I s comparable o hese unbased esmaors and somemes he bas of OLIVE s even smaller (for example, when K = 2, 5, 45, and 55 for medan bas, and when K = 3, 45, and 55 for mean bas). As he number of nsrumens ncrease, he advanage of OLIVE n erms of absolue devaon becomes more sgnfcan. When K equals 0 and larger, OLIVE has he smalles medan and mean absolue devaons. Moreover, when K s larger han 0, OLIVE also has he smalles mean squared error. When he number of nsrumens s larger han he number of me perods (K>T), nsrumenal varable esmaors such as 2SLS, LIML, B2SLS, and FULLER all become nfeasble. Among he hree esmaors ha are sll feasble, OLIVE performs sgnfcanly beer han boh OLS and 2GMM n erms of medan and mean bas, medan and mean absolue devaon, and mean squared error. Overall, when he number of nsrumens ncreases, he advanage of OLIVE becomes more and more sgnfcan (hs s also rue n he supplemenal ables). The performance of OLIVE mproves almos monooncally as he number of nsrumens ncreases (levels off when K becomes very large). On he oher hand, oher IV esmaors usually peak a a ceran number of nsrumens hen deerorae as he number of nsrumens furher ncrease. Ths demonsraes anoher advanage of OLIVE: one can smply use all vald nsrumens a hand whou havng o selec nsrumens or deermne he opmal number of nsrumens. 8

21 5. Emprcal Applcaon: Leau and Ludvgson (200b) 5.. Background One of he mos successful mulfacor models for explanng he cross-secon of sock reurns s he Fama-French hree-facor model. Fama and French (993) argue ha he new facors hey denfy, small-mnus-bg (SMB) and hgh-mnus-low (HML), proxy for unobserved common rsk facors. However, boh SMB and HML are based on reurns on sock porfolos sored by frm characerscs, and s no clear wha underlyng economc rsk facors hey proxy for. On he oher hand, even hough macroeconomc facors are heorecally easy o movae and nuvely appealng, hey have had lle success n explanng he cross-secon of sock reurns. 2 Leau and Ludvgson (200b) specfy a macroeconomc model ha does almos as well as he Fama-French hree-facor model n explanng he 25 Fama-French porfolo reurns. They explore he ably of condonal versons of he CAPM and he Consumpon CAPM (CCAPM) o explan he cross-secon of average sock reurns. They express a condonal lnear facor model as an uncondonal mulfacor model n whch addonal facors are consruced by scalng he orgnal facors. Ths mehodology bulds on he work n Cochrane (996), Campbell and Cochrane (999), and Ferson and Harvey (999). The choce of he condonng (scalng) varable n Leau and Ludvgson (200b) s unque: cay - a conegrang resdual beween log consumpon c, log asse wealh a, and log labor ncome y. Leau and Ludvgson (200a) fnds ha cay has srong forecasng power for excess reurns on aggregae sock marke ndces. Leau and Ludvgson (200b) argue ha cay may have mporan advanages as a scalng varable n cross-seconal asse prcng ess because summarzes nvesor expecaons abou he enre marke porfolo. We conjecure ha, as wh mos facors consruced usng macroeconomc daa, cay may conan measuremen error. If so, our OLIVE mehod should mprove he fndngs n Leau and 2 See Ludvgson and Ng (2007) for a dscusson of curren emprcal leraure on he rsk-reurn relaon. 9

22 Ludvgson (200b). Indeed, our emprcal resuls sugges he presence of large measuremen error n cay and oher macroeconomc facors, bu no n reurn-based facors, such as he Fama- French facors Daa and Mehodology Our sample s formed usng daa from he hrd quarer of 963 o he hrd quarer of 998. We choose he same me perod as Leau and Ludvgson (200b), so ha our resuls are drecly comparable. As n Leau and Ludvgson (200b), he reurns daa are for he 25 Fama- French (992, 993) porfolos. These daa are value-weghed reurns for he nersecons of fve sze porfolos and fve book-o-marke equy (BE/ME) porfolos on NYSE, AMEX and NASDAQ socks n CRSP and COMPUSTAT. We conver he monhly porfolo reurns o quarerly daa. The Fama-French facors, SMB and HML, are consruced he same way as n Fama and French (993). R vw s he value-weghed CRSP ndex reurn. The condonng varable, cay, s consruced as n Leau and Ludvgson (200a,b). We use he measure of labor ncome growh, y, advocaed by Jagannahan and Wang (996). Labor ncome growh s measured as he growh n oal personal, per capa ncome less dvdend paymens from he Naonal Income and Produc Accouns publshed by he Bureau of Economc Analyss. Labor ncome s lagged one monh o capure lags n he offcal repors of aggregae ncome. Our mehodology can be vewed as a modfed verson of Fama and MacBeh s (973) wo-pass mehod. Leau and Ludvgson (200b) dscuss dfferen mehods avalable, and argue ha he Fama-MacBeh procedure has mporan advanages for her applcaon. In he frspass, he me-seres beas are compued n one mulple regresson of he porfolo reurns on he facors. In addon o esmang beas by runnng me-seres OLS regressons lke n Leau and Ludvgson (200b), we also use OLIVE o esmae beas. For a gven porfolo (R ), reurns on he oher porfolos serve as nsrumens (R - ). As shown by our smulaon resuls, f facors 20

23 conan measuremen error, beas esmaed usng OLIVE are much more precse han beas esmaed usng OLS (and more precse han oher IV mehods). In he second-pass, cross-seconal OLS regressons usng 25 Fama-French porfolo reurns are run on beas esmaed usng eher OLS or OLIVE n he frs-pass o draw comparsons: ER (, + ) = ER ( 0, ) + β' λ. (22) 5.3. Emprcal Resuls Tables 2 and 3 repor he Fama-MacBeh cross-seconal regresson (second-pass) coeffcens, λ, wh wo -sascs n parenheses for each coeffcen esmae. The op -sasc uses uncorreced Fama-MacBeh sandard errors, and he boom -sasc uses he Shanken (992) correcon. The cross-seconal R 2 s also repored. Table 2 (Table 3) corresponds o Table (Table 3) n Leau and Ludvgson (200b), wh he same row numbers represenng he same models. For each row, he OLS resuls are replcaons of Leau and Ludvgson (200b). Afer numerous correspondences wh he auhors (we are graeful for her mely responses), we are able o oban very smlar resuls, hough no compleely dencal. The OLIVE resuls are based on our OLIVE bea esmaes n he frs-pass. Secons 5.3. and dscuss resuls n Table 2, whle secon dscusses resuls n Table Uncondonal Models Followng Leau and Ludvgson (200b), we begn by presenng resuls from hree uncondonal models. Row of Table 2 presens resuls from he sac CAPM, wh he CRSP value-weghed reurn, R vw, used as a proxy for he unobservable marke reurn. Ths model mples he followng cross-seconal specfcaon: 2

24 ER (, + ) = ER ( 0, ) + βvwλvw. (23) The OLS resuls n Row hghlgh he falure of he sac CAPM, as documened by prevous sudes (see, e.g., Fama and French (992)). Only % of he cross-seconal varaon n average reurns can be explaned by he bea for he marke reurn. The esmaed value of λ vw s sascally nsgnfcan and has he wrong sgn (negave nsead of posve) accordng he CAPM heory. The consan erm, whch s an esmae of he zero-bea rae, s oo hgh (4.8% per quarer). Esmang beas usng OLIVE nsead of OLS provdes lle mprovemen n erms of cross-seconal explanaory power: he R 2 s sll %. However, he sgn of he esmaed value of λ vw changes from negave o posve, hough sll sascally nsgnfcan, and he esmaed zero-bea rae decreases from 4.8% o 3.48% per quarer. We expec he advanage of OLIVE esmaon o be small here, snce R vw s a reurn-based facor lkely wh lle measuremen error. Row 2 of Table 2 presens resuls for he human capal CAPM, whch adds he bea for labor ncome growh, y, no he sac CAPM (Jagannahan and Wang (996)): ER ( ) = ER ( ) + β λ + β λ. (24), + 0, vw vw y y The human capal CAPM performs much beer han he sac CAPM, explanng 58% of he cross-seconal varaon n reurns. Labor ncome growh s a macroeconomc facor, whch probably conans measuremen error. When OLIVE s used o esmae beas, he R 2 jumps from 58% o 78%. However, for boh OLS and OLIVE resuls, he esmaed value of λ vw has he wrong sgn and he esmaed zero-bea rae s oo hgh. Row 3 of Table 2 presens resuls for he Fama-French hree-facor model: ER (, + ) = ER ( 0, ) + βvwλvw+ βsmbλsmb+ βhmlλhml. (25) Ths specfcaon performs exremely well wh OLS esmaed beas: he R 2 becomes 8%; he esmaed value of λ vw has he correc posve sgn; and he esmaed zero-bea rae s reasonable (.76% per quarer). The Fama-French facors should conan lle measuremen error, snce hey 22

25 are consruced from sock reurns. As one would expec, usng OLIVE esmaed beas yelds almos dencal coeffcen esmaes. The R 2 only margnally mproves o 83% Condonal/Scaled Facor Models Row 4 of Table 2 repors resuls from he scaled, condonal CAPM wh one fundamenal facor, he marke reurn, and a sngle scalng varable, cay : ER (, + ) = ER ( 0, ) + βcayλcay + βvwλvw + βvwcayλvwcay. (26) Under hs specfcaon, usng OLIVE nsead of OLS o esmae beas dramacally mproves he cross-seconal explanaory power from 3% o 80%, whch s smlar o he performance of he Fama-French hree-facor model. Ths s conssen wh our conjecure ha snce cay s consruced usng macroeconomc daa, conans large measuremen error. Usng OLIVE also changes he sgn of he esmaed value of λ vw from negave o posve, hough he esmaed coeffcens are close o zero for boh OLS and OLIVE. Usng OLIVE also reduces he esmaed zero-bea rae from 3.69% o 3.09% per quarer, hough hey are sll oo hgh. Rows 5 and 5 are varaons of Row 4. Gven he fndng ha he esmaed value of λ cay s no sascally dfferen from zero n Row 4, Row 5 oms β cay as an explanaory varable n he second-pass cross-seconal regressons, bu sll ncludes cay n he frs-pass me-seres regressons. Row 5 furher excludes cay n he frs-pass me-seres regressons (resuls from hs specfcaon are no repored n Leau and Ludvgson (200b)). Resuls n Rows 5 and 5 are very smlar o hose n Row 4, suggesng ha he me-varyng componen of he nercep s no an mporan deermnan of cross-seconal reurns. The mpac of usng OLIVE o esmae beas s also very smlar: he cross-seconal R 2 jumps from abou 30% o abou 80%. Row 6 of Table 2 repors resuls from he scaled, condonal verson of he human capal CAPM: 23

26 ER ( ) = ER ( ) + β λ + β λ + β λ + β λ + β λ. (27), + 0, cay cay vw vw y y vwcay vwcay ycay ycay We focus our dscussons on hs complee specfcaon. Usng OLIVE nsead of OLS n he frs-pass o esmae beas mproves he second-pass cross-seconal R 2 from 77% o 83% (smlar o he performance of he Fama-French hree-facor model). More mporanly, our resuls here help o resolve wo puzzlng fndngs by Leau and Ludvgson (200b) and Jagannahan and Wang (996). The second-pass rsk prema and zerobea rae esmaes wll be nconssen f facors are measured wh error and OLS s used o esmae beas n he frs-pass. On he oher hand, he rsk prema and zero-bea rae can be conssenly esmaed n he second-pass f OLIVE s used n he frs-pass o esmae beas, because OLIVE bea esmaes are conssen even when facors conan measuremen error. Leau and Ludvgson (200b) use OLS o esmae beas and fnd ha wo feaures of her cross-seconal resuls n her Table bear nong. 3 Frs, hey sae ha [a] problem wh hs model, however, s ha here s a negave average rsk prce on he bea for he valueweghed reurn. They also noe ha Jagannahan and Wang (996) repor a smlar fndng for he sgns of he rsk prces on he marke and human capal beas. Indeed, n our OLS resuls n Row 6 of Table 2, he esmaed value of λ vw (coeffcen on he marke reurn bea) s -2.00, and he esmaed value of λ ycay (coeffcen on he scaled human capal bea) s -0.7, boh negave whch s nconssen wh he heory. However, when we use OLIVE o esmae beas n he frs-pass, he esmaed value of λ vw becomes posve (.33), and he esmaed value of becomes close o zero ( ), more conssen wh he heory. λ ycay Second, Leau and Ludvgson (200b) sae ha he esmaed value of he average zero-bea rae s large. The average zero-bea rae should be beween he average rskless borrowng and lendng raes, and he esmaed value s mplausbly hgh for he average nvesor. Alhough he (C)CAPM can explan a subsanal fracon of he cross-seconal varaon n hese 3 See pages of Leau and Ludvgson (200b). 24

27 25 porfolo reurns, hs resul suggess ha he scaled models do a poor job of smulaneously prcng he hypohecal zero-bea porfolo. Ths fndng s no uncommon n sudes ha use macro varables as facors. For example, he esmaed values for he zero-bea rae we fnd here have he same order of magnude as ha found n Jagannahan and Wang (996). The auhors noe ha [] s possble ha he greaer samplng error we fnd n he esmaed beas of he scaled models wh macro facors s conrbung o an upward bas n he zero-bea esmaes of hose models relave o he esmaes for models wh only fnancal facors. They also noe ha [s]uch argumens for large zero-bea esmaes have a long radon n he cross-seconal asse prcng leraure (e.g., Black e al. 972; Mller and Scholes 972). However, he auhors conclude ha [p]rocedures for dscrmnang he samplng error explanaon for hese large esmaes of he zero-bea rae from ohers are no obvous, and s developmen s lef o fuure research. Our resuls sugges ha measuremen error n facors s he cause of hs problem. Samplng error s a second-order ssue; becomes neglgble as he sample sze T becomes large. Unlke samplng error, he measuremen error problem does no dmnsh as he sample sze T becomes large. When macroeconomc facors wh measuremen error are ncluded n he model, OLIVE can provde more precse bea esmaes n he frs-pass, whch lead o more precse esmaes of he zero-bea rae n he second-pass. In Row 6 of Table 2, he esmaed zero-bea rae based on OLS esmaed beas s oo hgh a 5.9% per quarer. However, when we use OLIVE o esmae beas, he esmaed zero-bea rae drops dramacally o a reasonable.9% per quarer. Rows 7 and 7 are varaons of Row 6. Row 7 oms β cay as an explanaory varable n he second-pass cross-seconal regressons, bu sll ncludes cay n he frs-pass me-seres regressons. Row 7 furher excludes cay n he frs-pass me-seres regressons (resuls from hs specfcaon are no repored n Leau and Ludvgson (200b)). Resuls n Rows 7 and 7 are very smlar o hose n Row 6. The mpac of usng OLIVE nsead of OLS o esmae beas 25

28 s also very smlar: he cross-seconal R 2 ncreases; he sgn of he esmaed value of λ vw changes from negave o posve; and he esmaed zero-bea rae drops sgnfcanly o a reasonable magnude. To summarze, our resuls n Table 2 confrm he exsence of large measuremen error n macroeconomc facors, such as cay and labor ncome growh, and valdae he use of OLIVE o help mprove bea esmaon under hese crcumsances. In addon, o he exen ha our emprcal resuls help resolve some puzzlng fndngs n Leau and Ludvgson (200b) and Jagannahan and Wang (996), we also srenghen her resuls Consumpon CAPM Table 3 presens, for he consumpon CAPM, he same resuls presened n Table 2 for he sac CAPM and he human capal CAPM. The scaled mulfacor consumpon CAPM, wh cay as he sngle condonng varable akes he form: ER ( ) = ER ( ) + β λ + β λ + β λ, (28), + 0, cay cay c c ccay ccay where c denoes consumpon growh (log dfference n consumpon), as measured n Leau and Ludvgson (200a). As a comparson, Row of Table 3 repors resuls of he uncondonal consumpon CAPM. The performance of hs specfcaon s poor, explanng only 6% of he cross-seconal varaon n porfolo reurns. Usng OLIVE bea esmaes seems o have made he performance even worse. Row 2 of Table 3 presens he resuls of esmang he scaled specfcaon n equaon (28). The R 2 jumps o 70%, n sharp conras o he uncondonal resuls n Row. When OLIVE s used o esmae beas, he R 2 furher ncreases o 82%. For boh OLS and OLIVE resuls, he esmaed value of sgnfcan. λ ccay (scaled consumpon growh) s posve and sascally 26

29 Row 3 excludes β cay as an explanaory varable n he second-pass cross-seconal regressons, bu sll ncludes cay n he frs-pass me-seres regressons. Ths seems o have made very lle dfference, as he resuls n Row 3 are very smlar o hose n Row 2. Agan, when OLIVE esmaed beas are used, he R 2 ncreases from 69% o 8%. Row 3 furher excludes cay n he frs-pass me-seres regressons. As noed by Leau and Ludvgson (200b), he resuls here are somewha sensve o hs excluson. 4 The R 2 drops o 27% for OLS resuls and 34% for OLIVE resuls. These resuls sugges ha ncludng he scalng varable cay as a facor n he prcng kernel can be mporan even when he bea for hs facor s no prced n he cross-secon. Our resuls n Table 3 sugges ha usng OLIVE nsead of OLS o esmae beas n he condonal consumpon CAPM generally ncreases he cross-seconal varaon of porfolo reurns explaned by he model, as measured by he R 2. However, unlke n Table 2, he esmaed zero-bea raes reman hgh. 6. Concluson In hs paper, we pu forh a smple mehod for esmang beas (facor loadngs) when facors are measured wh error, whch we call OLIVE. OLIVE uses all avalable nsrumens a hand, and s nuve and easy o mplemen. OLIVE acheves beer performance n smulaons han OLS and oher nsrumenal varable esmaors such as 2SLS, B2SLS, LIML, and FULLER, when he number of nsrumens s large. OLIVE can be nerpreed as a GMM esmaor when seng he weghng marx equal o he deny marx and has beer fne sample properes han he effcen wo-sep GMM esmaor. OLIVE also has an mporan advanage over he Asympoc Prncple Componens (APC) because he sascal facors of he prncple 4 Resuls from hs specfcaon are no repored n Leau and Ludvgson (200b). See foonoe 25 on page 26 of her paper. 27

30 componens mehod lack clear economc nerpreaons, whle OLIVE drecly makes use of he observed economc facors. OLIVE has many poenal applcaons. For example, can be used n cross-counry sudes, where daa from oher counres can be used as nsrumens for he counry n queson. OLIVE s especally suable for esmang asse reurn beas when facors are measured wh error, snce hs s ofen a large N and small T seng. Inuvely, snce all asse reurns vary ogeher wh a common se of facors, one can use nformaon conaned n oher asse reurns o mprove he bea esmae for a gven asse. As an emprcal applcaon, we reexamne Leau and Ludvgson s (200b) es of he (C)CAPM usng OLIVE n addon o OLS o esmae beas. Leau and Ludvgson s facor cay has been found o have srong forecasng power for excess reurns on aggregae sock marke ndces, bu may conan measuremen error. We fnd ha n regressons where macroeconomc facors are ncluded, usng OLIVE nsead of OLS mproves he R 2 sgnfcanly. Perhaps more mporanly, our resuls from OLIVE esmaon help o resolve wo puzzlng fndngs by Leau and Ludvgson (200b) and Jagannahan and Wang (996): frs, he sgn of he average rsk premum on he bea for he marke reurn changes from negave o posve, whch s n accordance wh he heory; second, he esmaed value of average zero-bea rae s no longer oo hgh. These resuls sugges ha when macroeconomc facors wh measuremen error are ncluded n he model, OLIVE can provde more precse bea esmaes n he frs-pass, whch lead o more precse esmaes of he rsk prema and zero-bea rae n he second-pass. Overall, our resuls from hs emprcal applcaon valdae he use of OLIVE o help mprove bea esmaon when facors are measured wh error. Our emprcal resuls suppor and srenghen Leau and Ludvgson (200b) and Jagannahan and Wang (996) by resolvng some of her puzzlng fndngs. Our fndngs are also conssen wh he heme n Ferson, Sarkssan, and Smn (2007) ha he (C)CAPMs mgh work beer han prevously recognzed n he leraure. 28

31 Appendces Appendx A. Proof of Proposon (see Secon 2.2). To smplfy noaon, we consder a more absrac seng. Le y = β ' x + ε = x ' β + ε, (A) where x and β are M vecors, Exε ( ) 0, and x = x* + v. Le z = β ' x * + e be nsrumens ( =,, N; =,, T). Here we assume here are N nsrumens (.e., N+ asses). For example, o esmae B n he noaon of Secon 2, we le β = B, and y = y, ε = e, and z = y +, for. Then or can be smplfed as T T T z y = z x ' β + z ε T T T, (A2) = = = y = x' β + ε, (A3) where T x = xz ', T = T z T = ε = ε, and y T z y T = =. The esmaor OLIVE s OLIVE ( ) β = x ' x x' y. Now T T Ex ( ε) = E ' 2 zx zsεs T = s= T T T = E z x z E z x z T T + Σ Σ T T ' ε ( ' sε s ). (A4) = s= = s = O 0 as T Therefore, OLIVE N N β β = xx' xε = = N N = xx' xε N = N = (A5) 29

32 N OLIVE N T ( β β) = xx' T xε N = N = N N T T = xx ' xz ' z N ε = N = T = T = = A N N DTξ T N =, (A6) where A N N = xx', N = D T = xz ', and ξ T T = T T z T = = ε. Noe ha ξ T and ξ jt are dependen hrough he common erm T = x * ε, see (A8) below. The nsrumens z are deermned by rue facor x * : z = β ' x * + e, (A7) herefore, T T ξ = β ' x * ε + e ε, (A8) T T = T = and T T D ξ = D β ' x * ε + D e ε. (A9) T T T T T = T = N N T N T DTξT = DT ' x * DT e N β N ε + ε = = T = N = T = We have. (A0) N T DT β ' x * N(0, ) N ε ΓΩ = T =, (A) N T where DTβ ', and x * ε N(0, ) N Γ Ω = T =. We also have = N N T /2 DT eε Op( N ) 0 = T =. (A2) 30

33 and E ( D ) To see (A2) s /2 Op( N ), we wre D = x z = x + v x + e = x x + O T T T T /2 ' ( * )( *' β ) * *' β ( ), T p T = T = T = = Gβ, where equaon (A2) can be rewren as T G = E x * x *' T = N T NT. Addng and subracng E ( D T ) N T N T, hen /2 DT E( DT ) eε + GN βeε = I + II. (A3) = = = = /2 Snce DT E( DT ) Op ( T ) =, erm I s domnaed by II. From /2 we have II Op ( N ) =. NT N T βeε= Op(), (A4) = = If N s fxed, (A2) s O () and s no neglgble. Ths erm wll conrbue o he p lmng dsrbuon; bu he T conssency and he asympoc normaly sll hold. Appendx B. Proof of Proposon 2 (see Secon 2.2). The proof of Proposon remans vald up o (A). We show (A2) s sll asympocally neglgble f T / N 0. I s suffcen o consder II n (A3). Le ( ε ) γ = E e, wh γ 0, equaon (A4) wll no longer hold. Bu can be rewren as N T N T N T βeε = β eε E( eε) + βγ. (A5) NT NT NT = = = = = = The frs erm on he rgh hand sde s O (). Assumng β M for all, he second N erm s bounded by M ( T/ N) γ = O( T / N) because = p N γ = O() by assumpon (9). /2 Thus (A5) s Op() + Op( T/ N). Ths mples ha, nong he exra erm N, II n (A3) /2 O ( / ) p N + Op T N, whch converges o zero f T / N 0. s equal o ( ) = 3