Cross-sectional forecasts of the equity premium $

Journal of Financial Economics 81 (2006) 101 141 www.elsevier.com/locae/jfec Cross-secional forecass of he equiy premium $ Chrisopher Polk a,, Samuel Thompson b, Tuomo Vuoleenaho c,d a Kellogg School of Managemen, Norhwesern Universiy, Evanson, IL 60208, USA b Deparmen of Economics, Harvard Universiy, Cambridge, MA 02138, USA c Arrowsree Capial L.P., Cambridge, MA 02138, USA d Naional Bureau of Economic Research, Cambridge, MA 02138, USA Received 15 April 2004; received in revised form 14 December 2004; acceped 16 March 2005 Available online 18 January 2006 Absrac If invesors are myopic mean-variance opimizers, a sock s expeced reurn is linearly relaed o is bea in he cross-secion. The slope of he relaion is he cross-secional price of risk, which should equal he expeced equiy premium. We use his simple observaion o forecas he equiy-premium ime series wih he cross-secional price of risk. We also inroduce novel saisical mehods for esing sock-reurn predicabiliy based on endogenous variables whose shocks are poenially correlaed wih reurn shocks. Our empirical ess show ha he cross-secional price of risk (1) is srongly correlaed wih he marke s yield measures and (2) predics equiy-premium realizaions, especially in he firs half of our 1927 2002 sample. r 2005 Published by Elsevier B.V. JEL classificaion: C12; C15; C32; C53; G12; G14 Keywords: Equiy premium; CAPM; Predicing reurns; Condiional inference; Neural neworks $ We would like o hank Nick Barberis, John Campbell, Xiaohong Chen, Randy Cohen, Ken Daniel, Ken French, Ravi Jagannahan, Mai Keloharju, Jussi Keppo, Jonahan Lewellen, Sefan Nagel, Vesa Puonen, Jeremy Sein, and seminar paricipans a he American Finance Associaion 2005 meeing, Boson Universiy Economics Deparmen, Brown Economics Deparmen, 2004 CIRANO conference, Darmouh Tuck School, Harvard Economics Deparmen, Kellogg School of Managemen, MIT Sloan School, Universiy of Michigan Indusrial and Operaions Engineering Deparmen, NYU Economics Deparmen, NYU Sern School, and Universiy of Rocheser Economics Deparmen for useful commens. We are graeful o Ken French and Rober Shiller for providing us wih some of he daa used in his sudy. All errors and omissions remain our responsibiliy. Corresponding auhor. E-mail address: c-polk@kellogg.norhwesern.edu (C. Polk). 0304-405X/$ - see fron maer r 2005 Published by Elsevier B.V. doi:10.1016/j.jfineco.2005.03.013

102 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 1. Inroducion The capial asse pricing model (CAPM) predics ha risky socks should have lower prices and higher expeced reurns han less risky socks (Sharpe, 1964; Linner, 1965; Black, 1972). The CAPM furher specifies he bea (he regression coefficien of a sock s reurn on he marke porfolio s reurn) as he relevan measure of risk. According o he Sharpe-Linner CAPM, he expeced-reurn premium per one uni of bea is he expeced equiy premium, or he expeced reurn on he value-weigh marke porfolio of risky asses less he risk-free rae. We use his CAPM logic o consruc equiy-premium forecass. We compue a number of cross-secional associaion measures beween socks expeced-reurn proxies (including he book-o-marke equiy raio, earnings yield, ec.) and socks esimaed beas. Low values of he cross-secional associaion measures should on average be followed by low realized equiy premia and high values by high realized equiy premia. Should his no be he case, here would be an incenive for a myopic mean-variance invesor o dynamically allocae his or her porfolio beween high-bea and low-bea socks. Given ha no all invesors can overweigh eiher high-bea or low-bea socks in equilibrium, prices mus adjus such ha he cross-secional price of risk and he expeced equiy premium are consisen. Our cross-secional bea-premium variables are empirically successful, as eviden from he following wo resuls. Firs, he variables are highly negaively correlaed wih he price level of he sock marke. Because a high equiy premium almos necessarily manifess iself wih a low price for he marke, negaive correlaion beween our variables and he Sandard and Poor s (S&P) 500 s valuaion muliples is reassuring. In paricular, our crosssecional measures have a correlaion as high as 0.8 wih he Fed model s ex ane equiypremium forecas (defined by us as he smoohed earnings yield minus he long-erm Treasury bond yield). Second, our cross-secional bea-premium measures forecas he equiy premium. In he US daa over he 1927:5-2002:12 period, mos of our cross-secional bea premium variables are saisically significan predicors a a beer han 1% level of significance, wih he predicive abiliy sronges in he pre-1965 subsample. These predicive resuls are also robus o a number of alernaive mehods of consrucing he cross-secional beapremium measure. We obain similar predicive resuls in an inernaional sample. Because of daa consrains (we only have porfolio-level daa for our inernaional sample), we define our cross-secional risk premium measure as he difference in he local-marke bea beween value and growh porfolios. If he expeced equiy premium is high (and he CAPM holds), a sor on valuaion measures will sor a disproporionae number of high-bea socks ino he value porfolio and low-bea socks ino he growh porfolio. Thus a high bea of a value minus growh porfolio should forecas a high equiy premium, holding everyhing else consan. In a panel of 22 counries, he pas local-marke bea of value minus growh is a saisically significan predicor of he fuure local-marke equiy premium, consisen wih our alernaive hypohesis. In muliple regressions forecasing he equiy premium, he cross-secional bea premium beas he erm yield spread (for all measures), bu he horse race beween he marke s smoohed price-earnings raio and he cross-secional bea premium is a draw. This is no inconsisen wih he heory. Campbell and Shiller (1988a, b) show ha if growh in a

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 103 cash-flow measure is nearly unpredicable, he raio of price o he cash-flow measure is mechanically relaed o he long-run expeced sock reurn, regardless of he economic forces deermining prices and expeced reurns. Because our variables are based on an economic heory and a cross-secional approach ha is no mechanically linked o he marke s expeced reurn, he fac ha he wo differen ypes of variables rack a common predicable componen in he equiy premium is no surprising if he logic underlying our variables is correc. In he pos-1965 subsample, he predicive abiliy of our cross-secional bea-premium measures is less srong han in he pre-1965 subsample. This is perhaps no surprising, given ha we generae our cross-secional forecass using a model, he CAPM, ha fails o empirically describe he cross-secion of average reurns in more recen subsamples (Fama and French, 1992, and ohers). An opimis, seeing our resuls, would poin ou ha 95% confidence inerval always covers a posiive value for he forecasing coefficien in all of he subsample pariionings. A pessimis would couner ha our cross-secional measure is no saisically useful in predicing he equiy-premium in he second half of he sample. The marke s smoohed earnings yield and our cross-secional bea-premium measures are much less correlaed in he second subsample han in he firs subsample, srongly diverging in he early 1980s. If he marke s smoohed earnings yield is a good predicor of he marke s excess reurn and he cross-secional bea premium a good predicor of he reurn of high-bea socks relaive o ha of low-bea socks, he divergence of he wo ypes of equiy-premium measures implies a rading opporuniy. Consisen wih his hypohesis, we show saisically significan forecasabiliy of he reurns on a hedged marke porfolio, consruced by buying he marke porfolio and bea hedging i by selling high-bea and buying low-bea socks. According o our poin esimaes, he annualized condiional Sharpe raio on his zero-bea zero-invesmen porfolio was close o one in early 1982. We also ackle a saisical quesion ha is imporan o financial economerics. In many ime-series ess of reurn predicabiliy, he forecasing variable is persisen wih shocks ha are correlaed wih reurn shocks. I is well known ha in his case he small-sample p- values obained from he usual suden- es can be misleading (Sambaugh, 1999; Hodrick, 1992, and ohers). Even in he Gaussian case, complex Mone-Carlo simulaions such as hose performed by Nelson and Kim (1993) and Ang and Bekaer (2001) have been he main mehod of reliable inference for such problems. We describe a mehod for compuing he small-sample p-values for he Gaussian error disribuions in he presence of a persisen and correlaed forecasing variable. Our mehod is an implemenaion of he Jansson and Moreira (2003) idea of condiioning he criical value of he es on a sufficien saisic of he daa. Specifically, we map he sufficien saisics of he daa o he criical value for he usual ordinary leas squares (OLS) -saisic using a neural nework (essenially a fancy look-up able). Our Mone Carlo experimens show ha his condiional criical value funcion produces a correcly sized es (i.e., he error is less han he Mone Carlo compuaional accuracy) wheher or no he daa series follows a uni roo process. The organizaion of he paper is as follows. In Secion 2, we recap he CAPM and he link beween he cross-secional bea premium and he expeced equiy premium. In Secion 3, we describe he consrucion of our cross-secional bea-premium measures. Secion 4 describes he saisical mehod. In Secion 5, we presen and inerpre our empirical resuls. Secion 6 concludes.

104 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 2. CAPM can link he ime series and cross-secion According o he Sharpe-Linner CAPM, he expeced-reurn premium per one uni of bea is he expeced equiy premium, or he expeced reurn on he value-weigh marke porfolio of risky asses less he risk-free rae: E 1 ðr i; Þ R rf ; 1 ¼ b i; 1 ½E 1 ðr M; Þ R rf ; 1 Š. (1) In Eq. (1), R i; is he simple reurn on asse i during he period. R rf ; 1 is he risk-free rae during he period known a he end of period 1. R M; is he simple reurn on he valueweigh marke porfolio of risky asses. b i; 1, or bea of sock i, is he condiional regression coefficien of R i; on R M;, known a ime 1. E 1 ðr M; Þ R rf ; 1 is he expeced marke premium. In our empirical implemenaion, we use he Cener for Research in Securiies Prices (CRSP) value-weigh porfolio of socks as our proxy for he marke porfolio. 1 Inuiively, a high expeced reurn on sock i (caused by eiher a high bea of sock i or a high equiy premium or boh) should ranslae ino a low price for he sock. Consisen wih his inuiion, Gordon (1962) proposes a sock-valuaion model ha can be invered o yield an ex ane risk-premium forecas: D i P i R rf þ Eðg i Þ¼EðR i Þ R rf (2) Eq. (2) saes ha he expeced reurn on he sock equals he dividend yield ðd i =P i Þ minus he ineres rae plus he expeced dividend growh Eðg i Þ. Reorganizing Eq. (2), subsiuing he Sharpe-Linner CAPM s predicion for expeced reurn, and assuming ha beas and he risk-free rae are consan yields D i; b P i E 1 ½R M; R rf Š Eðg i R rf Þ. (3) i; 1 In he reasonable cases in which he expeced equiy premium is posiive, he dividend yield on sock i can be high for hree reasons. Firs, he sock could have a high bea. Second, he premium per a uni of bea, ha is, he expeced equiy premium, could be high. Third, and finally, he dividends of he sock could be expeced o grow slowly in he fuure. Eq. (3) leads o a naural cross-secional measure of he equiy premium. Simply regress he cross-secion of dividend yields on beas and expeced dividend growh, D i; l 0; 1 þ l 1; 1 b P i þ l 2; 1 Eðg i Þ. (4) i; 1 If expeced excess reurns on he marke are consan, l 1; 1 recovers he expeced excess marke reurn. The cenral idea in our paper is o measure l 1; 1 for each period using 1 Roll (1977) argues ha his proxy is oo narrow, because i excludes many asses such as human capial, real esae, and corporae deb. Alhough Sambaugh (1982) shows some evidence ha inference abou he CAPM is insensiive o exclusion of less risky asses, a reader who is concerned abou he omission of asses from our marke proxy can choose o inerpre our subsequen resuls wihin he arbirage pricing heory (APT) framework of Ross (1976).

purely cross-secional daa, and hen use ha measuremen o forecas he nex period s equiy premium. 2 The CAPM does a poor job describing he cross-secion of sock reurns in he pos-1963 sample. However, ha failure does no necessarily invalidae our approach. Firs, Kohari e al. (1995) and Ang and Chen (2004) find a posiive univariae relaion beween average reurns and CAPM beas. Given ha boh sudies use a long sample as we do, heir evidence indicaes ha he CAPM is an adequae model for our purposes, a leas for our full-period ess. If he CAPM is a poor model in he second subsample, hen i is reasonable o expec our predicor o also perform poorly in he pos-1963 sample. Second, Cohen e al. (2003, 2005b) show ha alhough he CAPM perhaps does a poor job describing cross-secional variaion in average reurns on dynamic porfolios, ha model does a reasonable job describing he cross-secion of sock prices, which is essenially our lef-hand-side variable in Eq. (4). Third, for our mehod o work, we do no need he CAPM o be a perfec model. All we need is ha a higher expeced equiy premium (relaive o is ime-series mean) resuls in a more posiive relaion beween various pricelevel yield measures and CAPM beas. Fourh, and mos imporan, we do no simply assume ha l 1; 1 is he equiy premium bu es is predicive abiliy in our subsequen ime-series ess. Our mehodology can be easily exended o muli-facor models ha also include a marke facor, such as he Meron (1973) ineremporal capial asse pricing model (ICAPM) and many arbirage pricing heory (APT) specificaions of Ross (1976). For such models, one can regress he expeced-reurn proxies on muli-facor beas (including he loading on he marke in a muliple regression). The parial regression coefficien on he marke-facor loading is again relaed o he expeced excess reurn on he marke. Neiher he heory we rely on (he CAPM) nor our empirical ess provide insigh ino why he expeced equiy premium and cross-secional bea premium vary over ime. The hypohesis we es is wheher he pricing of risk is consisen enough beween he crosssecion and ime series o yield a useful variable for forecasing he equiy premium. Wheher he expeced equiy premium is he resul of ime-varying risk aversion (Campbell and Cochrane, 1999), invesor senimen (Shiller, 1981, 2000), invesor confusion abou expeced real cash-flow growh (Modigliani and Cohn, 1979; Rier and Warr, 2002; Cohen e al., 2005a), or some unmodeled hedging demand beyond our myopic framework (Meron, 1973; Fama, 1998) remains an unanswered quesion. 3. Daa and consrucion of variables C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 105 We consruc a number of alernaive proxies for he cross-secional risk premium. In consrucion of all hese cross-secional risk-premium measures, we avoid any look-ahead bias so ha all of our proxies are valid variables in regressions forecasing he equiy premium. 2 The Gordon model has he limiaion ha expeced reurns and expeced growh mus be consan, and hus using he Gordon model o infer ime-varying expeced reurns is in principle inernally inconsisen. Inerpreaing Eq. (4) in he conex of he Campbell and Shiller (1988a, b) log-linear dividend discoun model ha allows for ime-varying expeced reurns alleviaes his concern. If one repeas he above seps using he Campbell Shiller model and assumes ha he expeced one-period equiy premium E 1 ½R M; R rf Š follows a firs-order auoregressive process, he expeced one-period equiy premium is hen linearly relaed o he muliple regression coefficien l 1; 1.

106 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 The firs se of proxies, l SRC, l REG, and l MSCI, is based on various ordinal associaion measures beween a sock s or porfolio s bea and is valuaion raios. These ordinal measures have he advanage of no only being robus o ouliers in he underlying daa bu also of never generaing exreme values hemselves. This robusness comes a a cos, however, because he ordinal measures have he disadvanage of hrowing away some of he informaion in he magniude of he cross-secional spread in valuaion muliples. The second se of cross-secional risk-premium proxies, l DP, l DPG, l BM, and l BMG,is measured on a raio scale and hus relaes more closely o Eq. (4). To alleviae he oulier problem associaed wih firm-level regressions, hese raios are compued from crosssecions of value-weigh porfolios sored on valuaion muliples. The hird ype of proxy ha we use, l ER, is perhaps mos direcly conneced o he CAPM marke premium bu perhaps he leas robus o errors in daa. This proxy preesimaes he funcion ha maps various firm characerisics ino expeced reurns and hen regresses he curren fied values on beas, recovering he marke premium implied by he firm-level reurn forecass. 3.1. l SRC measure of he cross-secional price of risk We consruc our firs measure of he cross-secional price of risk, l SRC, in hree seps. Firs, we compue a number of valuaion raios for all socks. Selecing appropriae proxies for a firm s valuaion muliple is he main challenge of our empirical implemenaion. Because dividend policy is largely arbirary a he firm level, i would be ill-advised o use firm-level dividend yield direcly as he only variable on he lef-hand side of regression Eq. (4). Insead, we use a robus composie measure of muliple differen valuaion measures. An addiional complicaion in consrucion of he lef-hand-side variable is ha here are likely srucural breaks in he daa series, semming from changes in dividend policy, accouning rules, and sample composiion. To avoid hese pifalls, we use an ordinal composie measure of he valuaion muliple by ransforming he valuaion raios ino a composie rank, wih a higher rank denoing higher expeced reurn. We calculae four raw firm-level accouning raios, dividend-o-price raio ðd=pþ, booko-marke equiy (BE=ME, he raio of he book value of common equiy o is marke value), earnings o price ðe=pþ, and cash flow o price ðc=pþ. The raw cross-secional daa come from he merger of hree daabases. The firs of hese, he CRSP monhly sock file, provides monhly prices; shares ousanding; dividends; and reurns for NYSE, Amex, and Nasdaq socks. The second daabase, he Compusa annual research file, conains he relevan accouning informaion for mos publicly raded US socks. The Compusa accouning informaion is supplemened by he hird daabase, Moody s book equiy informaion for indusrial firms as colleced by Davis e al. (2000). Deailed daa definiions are as follows. We measure D as he oal dividends paid by he firm from June year 1 o May year. We define BE as sockholders equiy, plus balance shee deferred axes (Compusa daa iem 74) and invesmen ax credi (daa iem 208, se o zero if unavailable), plus pos-reiremen benefi liabiliies (daa iem 330, se o zero if unavailable), minus he book value of preferred sock. Depending on availabiliy of preferred sock daa, we use redempion (daa iem 56), liquidaion (daa iem 10), or par value (daa iem 130), in ha order, for he book value of preferred sock. We calculae sockholders equiy used in he above formula as follows. We prefer he sockholders equiy number repored by Moody s or Compusa (daa iem 216). If neiher one is

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 107 available, we measure sockholders equiy as he book value of common equiy (daa iem 60), plus he book value of preferred sock. (The preferred sock is added a his sage, because i is laer subraced in he book equiy formula.) If common equiy is no available, we compue sockholders equiy as he book value of asses (daa iem 6) minus oal liabiliies (daa iem 181), all from Compusa. We calculae E as he hree-year moving average of income before exraordinary iems (daa iem 18). Our measure of C is he hree-year moving average of income before exraordinary iems plus depreciaion and amorizaion (daa iem 14). In boh he calculaion of E and C, we require daa o be available for he las hree consecuive years. We mach D along wih he BE, E, and C for all fiscal year ends in calendar year 1 (1926 2001) wih he firm s marke equiy a he end of May year o compue D=P, BE=ME, E=P, and C=P. Nex, we ransform hese accouning raios ino a single annual ordinal composie measure of firm-level valuaion. Specifically, each year we independenly ransform each raio ino a percenile rank, defined as he rank divided by he number of firms for which he daa are available. Afer compuing hese four relaive percenile rankings, we average he available (up o four) accouning-raio percenile ranks for each firm. This average is hen re-ranked across firms (o spread he measure for each cross-secion over he inerval from zero o one), resuling in our expeced reurn measure, VALRANK i;. High values of VALRANK correspond o low prices and, according o he logic of Graham and Dodd (1934) and he empirical findings of Ball (1978), Banz (1981), Basu (1977, 1983), Fama and French (1992), Lakonishok e al. (1994), Reinganum (1981), and Rosenberg e al. (1985), also o high expeced subsequen reurns. Second, we measure beas for individual socks. Our monhly measure of risk is esimaed marke bea, b i;. We esimae he beas using a leas one and up o hree years of monhly reurns in an OLS regression on a consan and he conemporaneous reurn on he value-weigh NYSE-Amex-Nasdaq porfolio. We skip hose monhs in which a firm is missing reurns. However, we require all observaions o occur wihin a four-year window. As we someimes esimae bea using only 12 reurns, we censor each firm s individual monhly reurn o he range ð 50%; 100%Þ o limi he influence of exreme firm-specific ouliers. In conras o he value measures, we updae our bea esimae monhly. Our resuls are insensiive o small variaions in he bea-esimaion mehod. Third, we compue he associaion beween valuaion rank and bea, and we use his associaion measure as our measure of he cross-secional bea premium. Our firs proxy is he Spearman rank correlaion coefficien, l SRC, a ime beween VALRANK i; and b i;. The resuling monhly series for he proxies begins in May 1927 and ends in December 2002. The l SRC proxy has he following advanages mosly resuling from simpliciy and robusness. Firs, averaging he ranks on available muliples convenienly deals wih missing daa for one or more of our valuaion muliples. Second, he use of ranks eliminaes any hardwired link beween he level of he marke s valuaion and he magniude of he cross-secional spread in valuaion levels. Third, ranks are a ransformaion of he underlying muliples ha is exremely robus o ouliers. This proxy also has he following disadvanages. Firs, in compuing l SRC we do no conrol for expeced growh and profiabiliy ha could be cross-secionally relaed o beas, causing an omied-variables bias in he esimaes. This omied-variable bias can be significan, if expeced growh and profiabiliy are correlaed wih beas. Second, if he

108 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 independen variaion in expeced firm-level growh (and profiabiliy) explains a small fracion of he cross-secional spread in valuaion muliples, he ordinal naure of l SRC could cause us o hrow away some significan informaion relaed o expansions and conracions of he cross-secional spread in beas and valuaion muliples. Appendix A shows via a calibraion exercise ha he laer disadvanage is unlikely o be a significan concern in our sample. 3.2. l REG measure of he cross-secional price of risk Our second measure, l REG, modifies l SRC o conrol for growh opporuniies. To conrol for growh opporuniies, we need proxies for expeced fuure growh Eq. (4) o serve as conrol variables in our empirical implemenaion. A exbook reamen of he Gordon growh model shows ha wo variables, reurn on equiy and dividend payou raio, drive a firm s long-erm growh. Thus, we use as our primary profiabiliy conrols hose seleced by Fama and French (1999) o predic firm level profiabiliy, excluding variables ha have an obvious mechanical link o our valuaion measures. Our firs profiabiliy conrol is D=BE, he raio of dividends in year o year 1 book equiy, for hose firms wih posiive book equiy. Fama and French moivae his variable by he hypohesis ha firms arge dividends o he permanen componen of earnings (Linner, 1956; Miller and Modigliani, 1961, and ohers). We censor each firm s D=BE raio o he range (0,0.15) o limi he influence of near-zero book equiy firms. Following Fama and French (1999), our second profiabiliy conrol is a non-dividend-paying dummy, DD, ha is zero for dividend payers and one for hose firms no paying dividends. Including DD in he regression in addiion o D=BE helps capure any nonlineariy beween expeced profiabiliy and dividends. As Fama and French (1999) show subsanial mean reversion in profiabiliy, our hird and fourh profiabiliy conrols are pas long-erm profiabiliy and ransiory profiabiliy. We calculae long-erm profiabiliy as he hree-year average clean-surplus profiabiliy, ROE ðbe BE 3 þ D 2 þ D 1 þ D Þ=ð3 BE 3 Þ. We define ransiory profiabiliy as ROE ROE, where ROE is curren profiabiliy and is equal o ðbe BE 1 þ D Þ=ðBE 1 Þ. Our fifh profiabiliy conrol is a loss dummy. Firms losing money ypically coninue o do poorly in he fuure. We moivae our final profiabiliy conrol from he exensive indusrial organizaion lieraure on produc marke compeiion. This proxy is he Herfindahl index of equiy marke capializaions for he op five firms in he wo-digi sandard indusrial classificaion (SIC) code indusry. Low concenraion wihin indusry should signal inense compeiion and hus lower profiabiliy. Because he selecion of growh proxies is a judgmen call, i is forunae ha our main subsequen resuls are insensiive o he inclusion or exclusion of hese expeced-growh measures. l REG is he cross-secional regression coefficien, l REG of VALRANK i; on b i; and growh/profiabiliy conrols, esimaed wih OLS: VALRANK i; ¼ l 0; þ l REG bb i; þ X6 l g GROWTHRANKg i; þ e i; (5) g¼1 GROWTHRANK g i; is he corresponding percenile rank for six firm-level profiabiliy conrols. Given ha Cohen e al. (2003, 2005b) show ha he majoriy of he crosssecional variaion in valuaion raios across firms is he resul of differences in expeced

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 109 fuure profiabiliy, no differences in fuure expeced reurns, hese conrols have he poenial o improve our measuremen of he cross-secional bea premium significanly. 3.3. l MSCI measure of he cross-secional price of risk We also measure he cross-secional price of risk for an inernaional sample of 22 counries using an ordinal measure. Because we do no have securiy-level daa for our inernaional sample, only porfolio reurns, we work wih value and growh porfolios consruced by Kenneh French and available on his websie. We ake he op 30% and boom 30% porfolios sored on four Morgan Sanley Capial Inernaional (MSCI) value measures: D=P, BE=ME, E=P, and C=P. We hen esimae he beas for hese porfolios using a hree-year rolling window and define he predicor variable l MSCI as he average bea of he four value porfolios minus he average bea of he four growh porfolios. The subsequen inernaional resuls are insensiive o changing he bea-esimaion window o four or five years (longer windows improve he resuls) and o selecing a subse of value measures for consrucing l MSCI. 3.4. l DP and l DPG measures of he cross-secional price of risk We also consruc cross-secional risk premium measures ha use valuaion muliples on a raio scale. The firs wo such measures, l DP and l DPG, are implemened using five valueweigh dividend-yield sored porfolios. We sor socks ino five porfolios on he end-of- May dividend yield. Then, for each porfolio we measure value-weigh average dividend yield (compued as aggregae dividends over aggregae marke value) and he value-weigh average pas esimaed bea using he rolling beas updaed each monh. We hen regress hese five porfolio-level dividend yields in levels on he porfolios beas and denoe he regression coefficien by l DP. l DPG modifies l DP by conrolling for pas dividend growh. In addiion o he dividend yield, we compue he value-weigh one-year dividend growh for he porfolios. l DPG is he muliple regression coefficien of he porfolios dividend yields on heir beas, conrolling for one-year pas dividend growh raes. 3.5. l BM and l BMG measures of he cross-secional price of risk We consruc book-o-marke based proxies l BM and l BMG analogously o l DP and l DPG. We sor socks ino five porfolios based on end-of-may BE=ME. Then, for each porfolio we measure value-weigh average BE=ME (compued as aggregae book value of equiy over aggregae marke value) and he value-weigh average pas esimaed bea using he rolling beas updaed each monh. We hen regress hese five porfolio-level book-o-marke raios in levels on he porfolios beas, and denoe he regression coefficien by l BM. l BMG is he muliple regression coefficien of he porfolios BE=MEs on heir beas, conrolling for one-year pas value-weigh ROEs. 3.6. l ER measure of he cross-secional price of risk In conras o our oher measures of cross-secional risk premium ha relae price levels o beas, we measure he cross-secional price of risk based on how well beas explain

110 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 esimaes of one-period expeced reurns. We exrac his measure using a wo-sage approach. Our firs sage is as follows. Each monh, using a rolling en-year panel of daa over he period 120 o 1, we regress cross-secionally demeaned firm-level reurns on lagged cross-secionally demeaned characerisics: VALRANK; b b; he raw valuaion muliples D=P, BE=ME, E=P, and C=P; and he raw profiabiliy conrols used in consrucion of l REG. 3 In his regression we replace missing values wih cross-secional means and drop E=P and C=P from he specificaion in subperiods in which daa for hose measures are no available for any firm. The resuling coefficien esimaes in conjuncion wih he ime observaions on he associaed characerisics produce forecass of firmlevel expeced reurns a ime. In our second sage, we regress hese forecass on our bea esimaes as of ime. We repea his process each monh, generaing our l ER series as he coefficiens of hese cross-secional regressions. 3.7. Oher variables We use wo measures of he realized equiy premium. The firs measure is he excess reurn on he value-weigh marke porfolio ðr e MÞ, compued as he difference beween he simple reurn on he CRSP value-weigh sock index ðr M Þ and he simple risk-free rae. The risk-free rae daa are consruced by CRSP from Treasury bills wih approximaely hree monhs o mauriy. The second measure ðr e mþ is he excess reurn on he CRSP equal-weigh sock index. For he inernaional sample, we use an equiy-premium series consruced from MSCI s sock marke daa and an ineres rae series from Global Financial Daa. We also consruc variables ha should logically predic he marke reurn if he expeced equiy premium is ime varying. Previous research shows ha scaled price variables and erm-srucure variables forecas marke reurns. We pick he smoohed earnings yield and erm yield spreads as examples of such variables and compare heir predicive abiliy agains ha of our variables. The log earnings price raio ðepþ is from Shiller (2000), consruced as a en-year railing moving average of aggregae earnings of companies in he S&P 500 index divided by he price of he S&P 500 index. Following Graham and Dodd (1934), Campbell and Shiller (1988a, b, 1998) advocae averaging earnings over several years o avoid emporary spikes in he price-earnings raio caused by cyclical declines in earnings. We follow he Campbell and Vuoleenaho (2003) mehod of consrucing he earnings series o avoid any forwardlooking inerpolaion of earnings. This ensures ha all componens of he ime earningsprice raio are conemporaneously observable by ime. The raio is log ransformed. The erm yield spread ðtyþ is provided by Global Financial Daa and is compued as he yield difference beween 10-year consan-mauriy axable bonds and shor-erm axable noes, in percenage poins. The moivaion of he erm yield spread as a forecasing variable, suggesed by Keim and Sambaugh (1986) and Campbell (1987), is he following: TY predics excess reurns on long-erm bonds. As socks are also long-erm asses, i should also forecas excess sock reurns, if he expeced reurns of long-erm asses move ogeher. 3 The variables are cross-secionally demeaned, because only cross-secional variaion in expeced sock reurns maers for our premium esimaes and because demeaning reduces he noise and adds o he precision of he esimaes.

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 111 In our informal illusraions, we also use he dividend-price raio, compued as he raio of railing 12-monh dividends and he price for he S&P 500 index. We also use he simple (no log) smoohed earnings yield, which is defined simply as expðepþ. In he Gordon (1962) model compuaions, any ineres rae adjusmens are performed using he same en-year consan-mauriy axable bond yield ðy 10Þ as is used in he compuaion of he erm yield spread. 4. Condiional ess for predicive regressions This secion describes he saisical mehodology for compuing he correc smallsample criical values of he usual -saisic in hose siuaions in which he forecasing variable is persisen and shocks o he forecasing variable are poenially correlaed wih shocks o he variable being forecas. 4.1. Inference in univariae regressions Consider he one-period predicion model y ¼ m 1 þ yx 1 þ u ; x ¼ m 2 þ rx 1 þ v, and ð6þ wih Eu ¼ Ev ¼ 0, Eu 2 ¼ s2 u, Ev2 ¼ s2 v, and Corrðu ; v Þ¼g. In a pracical example inroduced by Sambaugh (1999), y is he excess sock reurn on a sock marke index and x is he index dividend yield. Because dividends are smooh and reurns cumulae o price, we have srong a priori reasons o expec he correlaion g o be negaive. We wish o es he null hypohesis y ¼ 0, indicaing ha x does no predic y, or in he Sambaugh (1999) example ha he dividend yield does no predic sock reurns. The usual -saisic for his hypohesis is b ¼ bs 1 u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xðx 1 xþ 2 b y, (7) where b y is he leas squares esimae of y and bs 2 u is an esimaor of s2 u. Classical asympoic heory saes ha in a large sample he -saisic is approximaely disribued sandard normal. However, his is a poor approximaion of he rue sampling disribuion of b in small samples. For example, Sambaugh (1999) shows ha when x is he dividend yield and y is he marke excess reurn, he null disribuion of b is cenered a a posiive number, leading o over-rejecion of a rue null hypohesis. To ge he size of he es righ, we wan a criical value q equal o he 95% quanile of he null disribuion of b. When he errors are normal, he exac null disribuion of b depends on he parameer r. Thus here exiss a funcion kð rþ so ha under he null, Pr½ b4kð rþš ¼ 0:05. One can calculae kð rþ by he boosrap or using mehods described by Imhof (1961). We canno direcly use kð rþ as a criical value because we do no know r, and evaluaing kð rþ a he leas squares esimae br leads o size disorions. Recenly, Jansson and Moreira (2003) have proposed a soluion o his problem. Suppose ha he covariance parameers s 2 u, s2 v and g are known. Under he null ha y ¼ 0,

112 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 he saisics S ¼ ( P ðx 1 xþðx s v gy =s u Þ P ðx 1 xþ 2 ; X ) ðx 1 xþ 2 ; x; y; x 1 ; y 1 (8) are sufficien saisics for he parameer r, where x ¼ðT 1Þ 1 P T ¼2 x 1 and y ¼ ðt 1Þ 1 P T ¼2 y. The definiion of a sufficien saisic is as follows: A saisic S is sufficien for a parameer r if he condiional disribuion of he daa given S is independen of r. While he uncondiional disribuion of b depends on he unknown r, he condiional disribuion does no. The idea in heir mehod is o se he criical value o a quanile of he condiional disribuion. Le qðs; aþ denoe he a-quanile of he condiional null disribuion of b given S ¼ s: Pr½bpqðs; aþjs ¼ s; y ¼ 0Š ¼a. (9) When he covariance parameers are known, a es ha rejecs he null when b4qðs; aþ has he correc null rejecion probabiliy in any sample size and for any value of r. Jansson and Moreira (2003) do no provide a closed form expression for he condiional disribuion of given he sufficien saisics. Our conribuion is o devise a compuaionally feasible implemenaion of heir procedure. We approximae he criical funcion q wih q nn, a neural nework: qðs; aþ q nn a ðx; c; b b xþ, q nn a ðx; c; xþ signðbgþmðxþþsðxþf 1 ðaþ. ð10þ F 1 ðaþ is he quanile funcion for a sandard normal variable, so Pr½Nð0; 1Þp F 1 ðaþš ¼ a. The mean and variance are neural neworks in he sufficien saisics:! mðxþ ¼x m 0 þ X4 x m j gðc0 j ex Þ and sðxþ ¼exp x s 0 þ X4 x s j gðc0 j ex Þ, j¼1 j¼1 X ¼ 0; Tðbr R 1Þ=50; T X 0. 2 ðx 1 xþ 2 =bs 2 v ; log jbgj; T=100 ð11þ c j is a five-dimensional parameer vecor. The haed variables are he usual leas-squares esimaors of he covariance parameers. signðbgþ is þ1 ifbg is posiive, 1 oherwise. br R is he consrained maximum likelihood esimae for r, given ha he null is rue and he covariance parameers are known: br R ¼ P ðx 1 xþðx bs v bgy =bs u Þ P ðx 1 xþ 2. (12) g is called he acivaion funcion. We use he anh acivaion funcion gðxþ ¼anhðxÞ ¼ ex e x e x. (13) þ e x

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 113 We choose c and x o closely approximae he criical funcion q. The parameer values used are x m ¼ð 1:7383 4:5693 2:7826 0:0007 3:9894Þ 0, x s ¼ð0:1746 0:4631 0:1955 0:0210 0:4641Þ 0, c 1 ¼ð1:8702 2:1040 3:4355 0:5738 0:0119Þ 0, c 2 ¼ð3:7744 2:5565 1:9475 0:8120 0:0262Þ 0, c 3 ¼ð49:9034 2:9268 52:7576 5:0194 4:4890Þ 0, c 4 ¼ð 1:6534 1:0395 2:8437 0:2264 0:0084Þ 0. ð14þ We provide an algorihm for choosing he parameers in Appendix B. Fiing he neural nework is a compuaionally demanding ask, bu we should emphasize ha he applied researcher does no need o fi he ne. An applied researcher can use our parameer values o easily calculae he exac small-sample criical value for any quanile a and any sample size T, under he assumpions of daa-generaing process Eq. (6) and i.i.d. Gaussian errors. q nn can approximae he criical funcion q o arbirary accuracy. q nn implies ha he -saisic has a condiional normal disribuion wih mean and sandard deviaion given by neural neworks. This is a special case of he mixure of expers ne (see Bishop, 1995, pp. 212 222), which approximaes a condiional disribuion wih mixures of normal disribuions whose parameers are neural nes in he condiioning variables. The mixure of expers ne is a universal approximaor: Given enough acivaion funcions and enough mixure disribuions, he ne can approximae any condiional disribuion o arbirary accuracy (see Chen and Whie, 1999). We fi our simple ne Eq. (10) wih a single mixure disribuion and also fi larger nes wih more mixure disribuions. While he larger models are a bi more accurae, for pracical purposes he simple ne is accurae enough. Furhermore, he ne wih a single disribuion leads o convenien expressions boh for he condiional quanile q and he p-value of he es. For esing he null ha y ¼ 0 agains he one-sided alernaive y40, he p-value is pvalðb Þ1 F b signðbgþmðxþ. (15) sðxþ The vecor X differs from he sufficien saisics in several ways for compuaional convenience. X ransforms some of he sufficien saisics and omis he saisics x, y, x 1, and y 1. X also uses parameer esimaes bs u, bg, and bs v in place of he known covariance parameers. Forunaely, he omied saisics x, y, x 1 and y 1 are no paricularly informaive abou he nuisance parameer r. Size disorions caused by omiing hese saisics are very small. The Jansson Moreira heory delivers an exac es when he covariance parameers are known. In pracice one mus use parameer esimaes. We design he neural ne raining algorihm o correc for esimaion error in he covariance parameers. This is no a compleely clean applicaion of he saisical heory. I could be he case ha no exac es exiss in his model. Again, however, any size disorions caused by unknown covariance

114 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 0.15 0.1 0.05 0 0 0.5 1-0.5 0 0.5 0.15 0.15 0.1 0.1 0.05 0.05 0 0 0.5 1-0.5 0 0.5 0 0.5 1-0.5 0 0.5 Fig. 1. Size of he es in a Mone Carlo experimen. Consider he daa-generaing process: y ¼ m 1 þ yx 1 þ u ; x ¼ m 2 þ rx 1 þ v, wih i.i.d. normal error vecors. We are ineresed in esing he hypohesis y ¼ 0 agains y40. On he op are empirical rejecion frequencies from using he usual criical value of 1:65 for a one-ailed es. On he boom lef are resuls for he boosrap, and on he boom righ he condiional criical funcion q nn a is used. The grid values range over Corrðu; vþ 2f :9; :8;...;:8;:9g and r 2f0;:025;...;:975; 1g. For each grid poin in he op and boom righ picures here are 40 housand Mone Carlo rials of T ¼ 120 observaions. The boosrap is calculaed as follows. For each grid poin we simulae 30 housand sample daa ses, and for each simulaed sample we boosrap one housand new daa ses from he model wih normal errors, seing y ¼ 0 and he oher parameers o heir leas squares esimaes. We se he boosrapped criical value equal o he 95h percenile of boosrapped -saisics. parameers are small. Furhermore, esimaion error for he covariance parameers is asympoically negligible, wheher he x process is saionary or no. 4 A Mone Carlo experimen demonsraes he accuracy of our approximaion. Fig. 1 repors empirical rejecion frequencies over a range of values for r and Corrðu; vþ. For each ðr; Corrðu; vþþ pair, we simulae many samples of 120 observaions each and perform a -es of he null y ¼ 0 agains he alernaive y40. Nominal es size is 5%. The plo a he op repors resuls for he classical criical value of 1:65. The plo a he boom lef repors resuls from using he boosrap, and he boom righ gives resuls for he condiional criical funcion q nn a. The boosrap algorihm is described in he noes o he Fig. 1. When r is close o one, he classical criical value under-rejecs for posiive Corrðu; vþ and over-rejecs for negaive Corrðu; vþ. When r ¼ 1 and he correlaion is 0:9 he usual criical value rejecs a rue null abou 38% of he ime. The boosrap improves on his 4 While classical asympoic heory requires saionariy, he condiional esing heory is no sensiive o uni roo problems. See he argumen by Jansson and Moreira (2003) for deails.

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 115 resul bu remains flawed: For r ¼ 1 and g ¼ 0:9, he rejecion frequency of a nominal 5% size es is 17%. Our condiional criical funcion leads o accurae rejecion frequencies over he enire range of r and Corrðu; vþ values, wih rejecion raes ranging from 4:46% o 5:66%. X and he -saisic are exacly invarian o s 2 u and s2 v, so hese resuls hold for any variance parameers. The above experimens show ha he size of he es is correc. The es is also powerful. In Appendix B, we discuss he opimaliy properies of he es. Our conclusion from hese power consideraions is ha i is difficul o devise a es wih a significan power advanage relaive o our condiional es, a leas as long as we have no addiional informaion abou he parameers (such as r ¼ 1). The condiional esing procedures described above assume homoskedasic normal errors. Our condiional esing procedures can be modified o be more robus o heeroskedasiciy. Under he so-called local-o-uniy asympoic limi used by Campbell and Yogo (2006) and Torous e al. (2005), heeroskedasiciy does no aler he large sample disribuion of he -saisic. The local-o-uniy limi akes r ¼ 1 þ c=t for fixed c and increasing T, i.e., i akes r o be in a shrinking neighborhood of uniy. This is in conras o he radiional asympoics, which fix r as T becomes large. Under he radiional (fixed r) asympoic limi, heeroskedasiciy changes he null disribuion of he -saisic. However, under he local-o-uniy limi, heeroskedasiciy is asympoically irrelevan. In inerpreing he asympoic local-o-uniy resuls, one should noe ha i is a large sample resul ha holds only when r is very close o one. In a small sample, or when r is small enough so ha radiional asympoics work, heeroskedasiciy maers. In he empirical secions of he paper, we also carry ou condiional inference based on -saisics compued wih Eicker-Huber-Whie (Whie, 1980) sandard errors. We calibrae separae criical-value funcions analogous o Eq. (10) for his es saisic. This calibraion process is analgous o he calibraion process for he es using he usual OLS -saisic, and hus we omi he deails here o conserve space. Alhough he combinaion of Eicker-Huber-Whie sandard errors and condiional inference appears sensible, his es comes wih a cavea: The condiional disribuion of he Eicker-Huber-Whie -saisic has no been sudied, and i is no known wheher he condiional Eicker-Huber-Whie -saisic is robus o heeroskedasiciy. However, while we have no proven any formal analyical resuls, unrepored Mone Carlo experimens sugges ha he Eicker-Huber-Whie -saisic is much more robus o heeroskedasiciy in small samples han he uncorreced -saisic. Also, we do know ha under homoskedasiciy he size of his modified es is correc. 4.2. Inference in mulivariae regressions This secion exends he Janssen Moreira mehodology o a simple vecor auoregression. Consider he bivariae regression y ¼ m 1 þ y 0 x 1 þ u ; x ¼ l 2 þ Kx 1 þ V, and where x, l 2 and y are wo-dimensional column vecors, K is a 2 2 marix, and V is a wo-dimensional vecor of mean zero errors. For example, we could ake he elemens of x ð16þ

116 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 o be he index dividend yield and price earnings raio, in which case he coefficien vecor y deermines he predicive conen of each variable conrolling for he oher. We wish o es he null hypohesis ha he firs elemen of y is zero. The usual approach is o run a mulivariae regression and rejec he null for large values of he -saisic b ¼ b y 1 p ffiffiffiffiffiffiffi. (17) O 11 b y1 is he ordinary leas squares esimae of y 1, he firs elemen of y, and O 11 is an esimae of he variance of b y 1, he (1; 1) elemen of O ¼ bs 2 u ðp x 1 x 0 1 Þ 1. Classical asympoic heory approximaes he null disribuion of b wih a sandard normal variable. I is well known ha his could be a poor approximaion when he elemens of x are highly serially correlaed. In many cases of ineres, classical heory leads o over-rejecion of a rue null hypohesis. In principle i is easy o exend he Janssen Moreira mehodology o his model. Suppose ha he errors ðu ; V 0 Þ0 are i.i.d. mean zero normal variables wih known covariance marix S ¼ E½ðu ; V 0 Þ0 ðu ; V 0 ÞŠ. The null disribuion of b depends on he unknown marix K. However, he condiional null disribuion of b given sufficien saisics for K does no depend on unknown parameers. To consruc he sufficien saisics, define he ransformed variables ðey ; ex 0 Þ0 ¼ S 1=2 ðy ; x 0 Þ0, where S 1=2 is he lower diagonal choleski decomposiion of S and saisfies S 1=2 ðs 1=2 Þ 0 ¼ S. The sufficien saisics for K are n S ¼ ek; X o ðx 1 xþðx 1 xþ 0 ; x; y; x 1 ; y 1, (18) where x ¼ðT 1Þ 1 P T ¼2 x 1, y ¼ðT 1Þ 1 P T ¼2 y,and ek is he 2 2 marix of leas squares esimaes from regressing ex on x 1 and a consan, and premuliplying he resul by S 1=2. The -es will have correc size for any sample size if we rejec he null when b is bigger han he 1 a quanile of he condiional null disribuion of b given S. Compuing he quaniles of he condiional null disribuion for a mulivariae sysem is a dauning compuaional problem. In he univariae model Eq. (6) wih jus one regressor, he -saisic has a null disribuion ha depends on he wo parameers r and g. Our neural ne approximaion q nn a learns he condiional quanile funcion by searching over a grid of r and g values. In he wo dimensional case, i is compuaionally feasible o search over all grid poins ha are close o empirically relevan cases. In he mulivariae seing he null disribuion depends on he four elemens of K as well as he correlaion erms in S. I does no appear o be compuaionally feasible for our neural ne o learn all possible cases of his high dimensional parameer space. We experimened wih differen algorihms for fiing he neural ne bu were unable o achieve he accuracy aained for he univariae model. To carry ou condiional inference in he mulivariae seing, we propose a modified version of he usual parameric boosrap. If we could simulae from he condiional disribuion of b given S, we could use he empirical quanile of he simulaed b draws as he criical value. While we canno direcly simulae from he disribuion of b given S, iis sraighforward o simulae from heir join disribuion: For fixed parameer values simulae daa ses from he model and compue b and S. We simulae from he condiional null of b given S using a neares neighbor esimaor. We simulae B draws of b and S, and we consruc a sample of N condiional draws by choosing he b saisics corresponding o

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 117 he N draws of S ha are closes o he sufficien saisics observed in he daa. We call his procedure he condiional boosrap. We also carry ou heeroskedasiciy-robus condiional inference using he same condiional-boosrap procedure based on -saisics compued wih Eicker-Huber-Whie (Whie, 1980) sandard errors. Deails of hese procedures are given in Appendix B. 5. Empirical resuls Our empirical resuls can be summarized wih wo findings. Firs, he cross-secional price of risk is highly negaively correlaed wih he marke price level and highly posiively correlaed wih popular ex ane equiy-premium measures derived from he Gordon (1962) growh model, such as he smoohed earnings yield minus he long-erm Treasury bond yield. Second, he cross-secional bea-premium forecass fuure excess-reurn realizaions on he CRSP value-weigh index. For he 1927:5-2002:12 period, he cross-secional bea premium is saisically significan a a level beer han 1%, wih mos of he predicive abiliy coming from he pre-1965 subsample. We also deec predicabiliy in a largely independen inernaional sample, indicaing ha our resuls are no sample specific. 5.1. Correlaion wih ex ane equiy-premium measures As an informal illusraion, we graph he ime-series evoluion of popular ex ane equiypremium measures and our firs cross-secional measure, l SRC,inFig. 2. (We focus on l SRC in hese illusraions o save space, bu similar resuls can be obained for our oher crosssecional variables.) One popular ex ane measure is based on he comparison deemed he Fed model, in which he equiy risk premium equals he equiy yield (eiher dividend yield or smoohed earnings yield) minus he long-erm Treasury bond yield. This measure is ofen called he Fed model, because he Federal Reserve Board supposedly uses a similar model o judge he level of equiy prices. 5 The Fed model and is variaions provide an inuiive esimaor of he forward-looking equiy risk premium. The earnings-yield componen of he Fed model is easily moivaed wih he Gordon (1962) growh model. As for he ineres rae componen, here are wo argumens why he earnings yield should be augmened by subracing he ineres rae. Firs, if one is ineresed in he equiy premium insead of he oal equiy reurn, subracing he ineres rae from he earnings yield is naural. Second, many argue ha an environmen of low ineres raes is good for he economy and hus raises he expeced fuure earnings growh. Asness (2002) poins ou ha, while seeming plausible, hese argumens are flawed in he presence of significan and ime-varying inflaion. In he face of inflaion, cash flows for he sock marke should ac much like a coupon on a real bond, growing wih inflaion. Holding real growh consan, low inflaion should forecas low nominal earnings growh. 5 The Federal Reserve Board s Moneary Policy Repor o he Congress of July 1997 argues: Sill, he raio of prices in he S&P 500 o consensus esimaes of earnings over he coming welve monhs has risen furher from levels ha were already unusually high. Changes in his raio have ofen been inversely relaed o changes in longerm Treasury yields, bu his year s sock price gains were no mached by a significan ne decline in ineres raes. The Federal Reserve has no officially endorsed any sock-valuaion model.

118 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 4 3 Sandard deviaions 2 1 0-1 -2 1927 1937 1947 1957 1967 1977 1987 1997 Year Fig. 2. Time-series evoluion of he ex ane equiy-premium forecass. This figure plos he ime-series of hree equiy-premium measures: (1) l SRC, he cross-secional Spearman rank correlaion beween valuaion levels and esimaed beas, marked wih a hick solid line; (2) expðepþ, he raio of a en-year moving average of earnings o price for Sandard and Poor s (S&P) 500, marked wih a dash-doed line; and (3) expðepþ Y10, he raio of a en-year moving average of earnings o price for S&P 500 minus he long-erm governmen bond yield, marked wih riangles. All variables are demeaned and normalized by heir sample sandard deviaions. The sample period is 1927:5-2002:12. In a sense, socks should be a long-erm hedge agains inflaion. (Modigliani and Cohn, 1979; Rier and Warr, 2002, argue ha he expeced real earnings growh of levered firms increases wih inflaion.) Thus, in he presence of ime-varying inflaion, he Fed model of equiy premium should be modified o subrac he real (insead of nominal) bond yield, for which daa unforunaely do no exis for he majoriy of our sample period. An alernaive o he implici consan-inflaion assumpion in he Fed model is o assume ha he real ineres rae is consan. If he real ineres rae is consan and earnings grow a he rae of inflaion (plus perhaps a consan), he earnings yield is a good measure of he forward-looking expeced real reurn on equiies. Under his assumpion, he earnings yield is also a good measure of he forward-looking equiy premium. Fig. 2 also plos he smoohed earnings yield wihou he ineres rae adjusmen. The hree variables in Fig. 2 are demeaned and normalized by he sample sandard deviaion. Our sample period begins only wo years before he sock marke crash of 1929. This even is clearly visible from he graph in which all hree measures of he equiy premium shoo up by an exraordinary five sample sandard deviaions from 1929 o 1932. Anoher sriking episode is he 1983 1999 bull marke, during which he smoohed earnings yield decreased by four sample sandard deviaions. However, in 1983 boh he

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 119 Table 1 Explaining he cross-secional risk premium wih he Fed model s equiy premium forecas and smoohed earnings yield Variable Consan expðep Þ expðep Þ Y10 Adjsued R 2 l SRC l SRC l SRC ¼ 0.1339 0.0628 4.6292 71.87% ( 10.54) (0.30) (36.54) ¼ 0.3996 5.561 30.45% ( 24.39) (19.95) ¼ 0.1303 4.6536 71.90% ( 34.02) (48.19) The able shows he ordinary leas squares (OLS) regression of cross-secional risk-premium measure, l SRC,on expðepþ and expðepþ Y10. l SRC is he Spearman rank correlaion beween valuaion rank and esimaed bea. Higher han average values of l SRC imply ha high-bea socks have lower han average prices and higher han average expeced reurns, relaive o low-bea socks. ep is he log raio of Sandard and Poor s (S&P) 500 s 10- year moving average of earnings o S&P 500 s price. Y10 is he nominal yield on 10-year consan-mauriy axable bonds in fracions. The OLS -saisics (which do no ake ino accoun he persisence of he variables and regression errors) are in parenheses, and R 2 is adjused for he degrees of freedom. The regression is esimaed from he full sample period 1927:5-2002:12 wih 908 monhly observaions. smoohed earnings yield less he bond yield (i.e., he Fed model) and our cross-secional bea-premium variable are already low and hus diverged from he earnings yield. I is eviden from he figure ha our cross-secional risk premium racks he Fed model s equiy-premium forecas wih an incredible regulariy. This relaion is also shown in Table 1, in which we regress he cross-secional premium l SRC on expðepþ and expðepþ Y10. Essenially, he regression fis exremely well wih an R 2 of 72%, and he explanaory power is enirely due o he Fed model (expðepþ Y10). (The OLS -saisics in he able do no ake ino accoun he persisence of he variables and errors and are hus unreliable.) Our conclusion from Table 1 and Fig. 2 is ha he marke prices he crosssecional bea premium o be consisen wih he equiy premium implied by he Fed model. There is poenially a somewha mechanical link beween he marke s earnings yield and our cross-secional measure. Our l measures are cross-secional regression coefficiens of earnings yields (and oher such muliples) on beas. If he marke has recenly experienced high pas reurns, high-bea socks should have also experienced high pas reurns relaive o low-bea socks. The high reurn on high-bea socks implies a lower yield on hose socks, if earnings do no adjus immediaely. Therefore, high reurns on he marke cause low values of our cross-secional bea premium, which migh explain he srong link beween he marke s valuaion muliples and our cross-secional measures. Unrepored experimens confirm ha our resuls are no driven by his link. 6 6 We firs regressed l SRC on five annual lags of he annual compound reurn on he CRSP value-weigh index. The coefficiens in his regression are negaive, bu he R 2 is low a 12%. Then, we ook he residuals of his regression and compared hem wih he earnings yield and Fed model s forecas. Even afer filering ou he impac of pas marke reurns, he residuals of l SRC plo almos exacly on op of he Fed model s forecas, wih a correlaion of approximaely 0.8. Furhermore, using he residuals of l SRC in place of l SRC in he subsequen predicabiliy ess does no aler our conclusions. Thus, we conclude ha our resuls are no driven by a mechanical link beween he marke s pas reurns and our cross-secional measures.

120 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 Fig. 2 also cass ligh on he Franzoni (2002), Adrian and Franzoni (2002), and Campbell and Vuoleenaho (2003) resul ha he beas of value socks have declined relaive o beas of growh socks during our sample period. This rend has a naural explanaion if he CAPM is approximaely rue and he expeced equiy premium has declined, as suggesed by Fama and French (2002), Campbell and Shiller (1998), and ohers. Value socks are by definiion socks wih low prices relaive o heir abiliy o generae cash flow. On he one hand, if he marke premium is large, i is naural ha many high-bea socks have low prices, and hus end up in he value porfolio. On he oher hand, if he marke premium is near zero, here is no obvious reason o expec high-bea socks o have much lower prices han low-bea socks. If anyhing, if growh opions are expeced o have high CAPM beas, hen growh socks should have slighly higher beas. Thus, he downward rend in he marke premium we deec provides a naural explanaion o he seemingly puzzling behavior of value and growh socks beas idenified by Franzoni (2002) and ohers. 5.2. Univariae ess of predicive abiliy in he US sample While he above illusraions show ha he cross-secional price of risk is highly correlaed wih reasonable ex ane measures of he equiy premium, i remains for us o show ha our variable forecass equiy-premium realizaions. We use he new saisical ess inroduced in Secion 4 o conclusively rejec he hypohesis ha he equiy premium is unforcasable based on our variables. Table 2 shows descripive saisics for he variables used in our formal predicabiliy ess. To save space we repor only he descripive saisics for one cross-secional riskpremium measure, l SRC. A high cross-secional bea premium suggess ha a ha ime high-bea socks were cheap and low-bea socks expensive. The correlaion marix in Table 2 shows clearly ha he variaion in he cross-secional measure, l SRC, appears posiively correlaed wih he log earnings yield, high overall sock prices coinciding wih low cross-secional bea premium. The erm yield spread ðtyþ is a variable ha is known o rack he business cycle, as discussed by Fama and French (1989). The erm yield spread is very volaile during he Grea Depression and again in he 1970s. I also racks l SRC, wih a correlaion of 0.31 over he full sample. Table 3 presens he univariae predicion resuls for he excess CRSP value-weigh index reurn, and Table 4 for he excess CRSP equal-weigh index reurn. The firs panel of each able forecass he equiy premium wih he cross-secional risk-premium measure l SRC. 7 The second panel uses he log smoohed earnings yield (ep) and he hird panel he erm yield spread (TY) as he forecasing variable. The fourh panel shows regressions using alernaive cross-secional risk-premium measures. While he firs hree panels also show subperiod esimaes, he fourh panel omis he subperiod resuls o save space. The regressions of value-weigh equiy premium in Table 3 reveal ha our crosssecional risk-premium measures do forecas fuure marke reurns. For all measures 7 A firs, i could appear ha our saisical ess are influenced by he so-called generaed regressor problem. However, because our proxy variables for he expeced marke premium is a funcion only of informaion available a 1, our predicabiliy ess do no over-rejec. While b y is a biased esimae of he coefficien on he rue, unknown marke expecaion of he equiy premium, i is a consisen esimaor of he coefficien on he proxy variable x 1. Furher deails are available from he auhors on reques.

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 121 Table 2 Descripive saisics of he vecor auoregression sae variables Variable Mean Median S.D. Min Max R e M; 0.0062 0.0095 0.0556 0.2901 0.3817 R e m; 0.0097 0.0114 0.0758 0.3121 0.6548 0.0947 0.1669 0.2137 0.5272 0.5946 l SRC ep 2.8769 2.8693 0.3732 3.8906 1.4996 TY 0.6232 0.5500 0.6602 1.3500 2.7200 Correlaions R e M; R e m; l SRC ep TY R e M; 1 0.9052 0.1078 0.0305 0.0474 R e m; 0.9052 1 0.1333 0.0658 0.0798 l SRC 0.1078 0.1333 1 0.5278 0.3120 ep 0.0305 0.0658 0.5278 1 0.2223 TY 0.0474 0.0798 0.3120 0.2223 1 R e M; 1 0.1048 0.2052 0.0825 0.0475 0.0428 R e m; 1 0.1070 0.2059 0.1075 0.0010 0.0726 l SRC 1 0.0930 0.1321 0.9748 0.5196 0.3011 ep 1 0.1140 0.1509 0.5359 0.9923 0.2279 TY 1 0.0469 0.0812 0.3219 0.2188 0.9131 The able shows he descripive saisics esimaed from he full sample period 1927:5-2002:12 wih 908 monhly observaions. R e M is he excess simple reurn on he Cener for Research in Securiies Prices (CRSP) value-weigh index. R e m is he excess simple reurn on he CRSP equal-weigh index. lsrc is he Spearman rank correlaion beween valuaion rank and esimaed bea. Higher han average values of l SRC imply ha high-bea socks have lower han average prices and higher han average expeced reurns, relaive o low-bea socks. ep is he log raio of Sandard and Poor s (S&P) 500 s en-year moving average of earnings o S&P 500 s price. TY is he erm yield spread in percenage poins, measured as he yield difference beween 10-year consan-mauriy axable bonds and shor-erm axable noes. S.D. denoes sandard deviaion. excep l DPG, we can rejec he null hypohesis of a zero coefficien in favor of a posiive coefficien a beer han a 1% level of significance in full-sample ess assuming homoskedasiciy. Using he version of condiional es ha is robus o heeroskedasiciy, he p-values are slighly higher and he evidence slighly less uniform: The measures based on firm-level daa coninue o be significan a beer han 1%, bu he measures based on porfolio-level daa are no longer significan. Comparing he small-sample p-values o he usual criical values for hese -saisics, i is clear ha he usual ess would perform adequaely in hese cases. This is no surprising, given ha he correlaion beween equiypremium shocks and our cross-secional forecasing-variable shocks is small in absolue value. The subperiod resuls for l SRC show ha he predicabiliy is sronger in he firs half of he sample han in he second half. The coefficien on l SRC drops from 0.0368 for he 1927:5-1965:2 period o 0.0088 for he 1965:2-2002:12 period. A similar drop is observed for he oher cross-secional measures, excep for l ER, which performs well in all subsamples (resuls unrepored). However, he 95% confidence inervals sugges ha one should no read oo much ino hese subperiod esimaes. The poin esimae for he firs subperiod is conained wihin he confidence inerval of he second subperiod and he poin esimae of he second subperiod wihin he confidence inerval of he firs subperiod. Furhermore, for every subperiod we examine, a posiive coefficien is conained wihin he

122 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 Table 3 Univariae predicors of he excess value-weigh marke reurn ðr e M Þ Specificaion b y -saisic p p W 95% CI br bg bs 1 bs 2 Predicion by he cross-secional bea premium, x ¼ l SRC 1927:5-2002:12 0.0242 2.811 o0:01 0.030 [0.008, 0.041] 0.975 0.0773 0.0553 0.0477 1927:5-1965:2 0.0368 2.450 o0:01 0.043 [0.008, 0.066] 0.960 0.0152 0.0633 0.0562 1965:2-2002:12 0.0088 0.413 0.309 0.297 [ 0.031, 0.054] 0.931 0.278 0.0460 0.0368 1927:5-1946:3 0.0663 1.967 0.030 0.091 [ 0.001, 0.131] 0.934 0.0413 0.0823 0.0585 1946:3-1965:2 0.0395 3.113 o0:01 o0:01 [0.016, 0.065] 0.957 0.080 0.0348 0.0534 1965:2-1984:1 0.0147 0.6027 0.240 0.249 [ 0.03, 0.065] 0.942 0.188 0.0458 0.0429 1984:1-2002:12 0.0190 0.4181 40:5 40:5 [ 0.099, 0.089] 0.885 0.416 0.0462 0.0292 Predicion by log smoohed earnings/price, x ¼ ep 1927:5-2002:12 0.0170 3.454 0.014 0.215 [0.003, 0.024] 0.993 0.669 0.0552 0.0464 1927:5-1965:2 0.0317 3.282 0.018 0.349 [0.003, 0.046] 0.987 0.671 0.0630 0.0549 1965:2-2002:12 0.00756 1.319 40:5 40:5 [ 0.009, 0.011] 0.996 0.668 0.0459 0.0359 1927:5-1946:3 0.0410 2.670 0.096 0.374 [ 0.006, 0.061] 0.981 0.659 0.0817 0.0707 1946:3-1965:2 0.0294 2.344 0.168 0.204 [ 0.009, 0.043] 0.994 0.727 0.0351 0.0322 1965:2-1984:1 0.0204 1.817 0.291 0.380 [ 0.012, 0.028] 0.987 0.662 0.0455 0.0362 1984:1-2002:12 0.0105 1.251 40:5 0.483 [ 0.012, 0.013] 0.990 0.668 0.0460 0.0352 Predicion by erm yield spread, x ¼ TY 1927:5-2002:12 0.00396 1.413 0.075 0.095 [ 0.001, 0.009] 0.917 0.0111 0.0555 0.269 1927:5-1965:2 0.00489 1.015 0.178 0.214 [ 0.005, 0.013] 0.968 0.156 0.0636 0.151 1965:2-2002:12 0.00270 0.862 0.150 0.179 [ 0.003, 0.008] 0.871 0.111 0.0460 0.346 1927:5-1946:3 0.00497 0.711 0.316 0.300 [ 0.010, 0.017] 0.969-0.174 0.0829 0.184 1946:3-1965:2 0.0201 1.978 0.030 0.030 [0.000, 0.039] 0.886 0.0707 0.0352 0.108 1965:2-1984:1 0.00868 1.677 0.043 0.045 [ 0.001, 0.019] 0.765 0.218 0.0455 0.378 1984:1-2002:12 0.00221 0.521 40:5 40:5 [ 0.010, 0.005] 0.918 0.00463 0.0462 0.301 Full sample predicive resuls for alernaive cross-secional measures x ¼ l REG 0.0908 3.605 o0:01 o0:01 [0.042, 0.141] 0.937 0.0644 0.0552 0.0255 x ¼ l DP 0.03539 2.53 o0:01 0.138 [0.007, 0.062] 0.926 0.167 0.0554 0.0498 x ¼ l DPG 0.02419 1.75 0.044 0.204 [ 0.003, 0.051] 0.917 0.107 0.0555 0.0531 x ¼ l BM 0.001121 2.63 o0:01 0.156 [0.0003, 0.0019] 0.942 0.236 0.0554 1.440 x ¼ l BMG 0.001449 2.98 o0:01 0.146 [0.0005, 0.0023] 0.919 0.222 0.0553 1.500 x ¼ l ER 2.175 3.15 o0:01 o0:01 [0.80, 3.53] 0.979 0.0331 0.0459 0.0005 This able shows resuls from he model R e M; ¼ m 1 þ yx 1 þ u ; x ¼ m 2 þ rx 1 þ v, wih E u ¼ s 2 1,Ev ¼ s 2 2, and Corrðu ; v Þ¼g. The p-value ess he null y ¼ 0 agains he one-sided alernaive y40, p denoing he p-value compued using he regular -saisic and p W using heeroskedasiciy-robus Whie -saisic. The confidence inerval is a wo-sided inerval for y compued assuming homoskedasiciy. The haed variables are unresriced ordinary leas squares esimaes. 95% CI is he 95% confidence inerval. 95% confidence inervals. Again, hese conclusions are no alered even if we focus our aenion on he heeroskedasiciy robus version of he condiional es. Of he wo exan insrumens we sudy, he log smoohed earnings yield is he sronger forecaser of he equiy premium, while he erm yield spread has only weak predicive abiliy. Consisen wih economic logic, he coefficien on ep is posiive for all subsamples, and he -saisic esing he null of no predicabiliy is 3.45 for he full sample. Our new

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 123 Table 4 Univariae predicors of he excess equal-weigh marke reurn ðr e m Þ Specificaion b y -sa p p W 95% CI br bg bs 1 bs 2 Predicion by he cross-secional bea premium, x ¼ l SRC 1927:5-2002:12 0.0469 4.012 o0:01 o0:01 [0.025, 0.070] 0.975 0.0202 0.0752 0.0477 1927:5-1965:2 0.0786 3.755 o0:01 o0:01 [0.037, 0.119] 0.960 0.0613 0.0882 0.0562 1965:2-2002:12 0.0345 1.260 0.100 0.098 [ 0.017, 0.092] 0.931 0.226 0.0589 0.0368 1927:5-1946:3 0.147 3.041 o0:01 0.030 [0.048, 0.238] 0.934 0.0960 0.118 0.0585 1946:3-1965:2 0.0466 3.256 o0:01 o0:01 [0.020, 0.075] 0.957 0.0603 0.0392 0.0534 1965:2-1984:1 0.0457 1.357 0.076 0.082 [ 0.017, 0.115] 0.942 0.139 0.0633 0.0429 1984:1-2002:12 0.00571 0.107 0.405 0.393 [ 0.089, 0.132] 0.885 0.379 0.0542 0.0292 Predicion by log smoohed earnings/price, x ¼ ep 1927:5-2002:12 0.0307 4.594 o0:01 0.234 [0.011, 0.040] 0.993 0.683 0.0750 0.0464 1927:5-1965:2 0.0662 4.943 o0:01 0.406 [0.026, 0.086] 0.987 0.683 0.0872 0.0549 1965:2-2002:12 0.0104 1.420 0.470 40:5 [ 0.011, 0.015] 0.996 0.689 0.0589 0.0359 1927:5-1946:3 0.0839 3.833 0.014 0.284 [0.016, 0.112] 0.981 0.683 0.117 0.0707 1946:3-1965:2 0.0285 2.004 0.250 0.299 [ 0.015, 0.045] 0.993 0.707 0.0398 0.0322 1965:2-1984:1 0.0290 1.866 0.300 0.381 [ 0.017, 0.039] 0.987 0.705 0.0631 0.0362 1984:1-2002:12 0.00131 0.132 40:5 40:5 [ 0.026, 0.004] 0.990 0.684 0.0542 0.0352 Predicion by erm yield spread, x ¼ TY 1927:5-2002:12 0.00935 2.452 o0:01 o0:01 [0.002, 0.016] 0.915 0.0140 0.0756 0.269 1927:5-1965:2 0.0106 1.572 0.074 0.089 [ 0.003, 0.023] 0.968 0.151 0.0893 0.151 1965:2-2002:12 0.00774 1.933 0.026 0.020 [0.00, 0.015] 0.871 0.124 0.0588 0.346 1927:5-1946:3 0.00867 0.857 0.243 0.220 [ 0.013, 0.026] 0.969 0.171 0.120 0.186 1946:3-1965:2 0.0208 1.808 0.043 0.029 [ 0.002, 0.043] 0.886 0.0699 0.0398 0.108 1965:2-1984:1 0.0172 2.410 o0:01 o0:01 [0.004, 0.032] 0.765 0.184 0.0628 0.378 1984:1-2002:12 0.00420 0.844 0.138 0.192 [ 0.005, 0.013] 0.918 0.0809 0.0541 0.301 Full sample predicive resuls for alernaive cross-secional measures x ¼ l REG 0.165 4.811 o0:01 o0:01 [0.098, 0.232] 0.937 0.0238 0.0749 0.0255 x ¼ l DP 0.07916 4.17 o0:01 0.044 [0.041, 0.115] 0.926 0.206 0.0751 0.0498 x ¼ l DPG 0.06813 3.63 o0:01 0.051 [0.031, 0.104] 0.917 0.136 0.0753 0.0531 x ¼ l BM 0.002609 4.51 o0:01 0.043 [0.0015, 0.0037] 0.942 0.245 0.075 1.440 x ¼ l BMG 0.003129 4.76 o0:01 0.052 [0.0018, 0.0043] 0.919 0.242 0.0749 1.500 x ¼ l ER 3.371 3.74 o0:01 o0:01 [1.55, 5.12] 0.979 0.0697 0.0599 0.0005 This able shows resuls from he model R e m; ¼ m 1 þ yx 1 þ u ; x ¼ m 2 þ rx 1 þ v, wih E u ¼ s 2 1,Ev ¼ s 2 2, and Corrðu ; v Þ¼g. The p-value ess he null y ¼ 0 agains he one-sided alernaive y40, p denoing he p-value compued using he regular -saisic and p W using heeroskedasiciy-robus Whie -saisic. The confidence inerval is a wo-sided inerval for y compued assuming homoskedasiciy. The haed variables are unresriced ordinary leas squares esimaes. 95% CI is he 95% confidence inerval. saisical mehodology maps his -saisic o a one-sided p-value of 1.4% under he homoskedasiciy assumpion. While he -saisic on ep is higher han on our firs crosssecional measure l SRC (2.81 versus 3.45), he p-value for ep is higher han he p-value for l SRC. This is he moivaion for our economeric work in Secion 4; he earnings yield is very persisen, and is shocks are srongly negaively correlaed wih equiy-premium shocks, making sandard saisical inference misleading.

124 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 Using he heeroskedasiciy-robus version of our condiional es, which is based on he Eicker-Huber-Whie -saisics, grealy weakens he case for predicabiliy based on ep. When predicing he value-weigh CRSP excess reurn over he enire sample, ep is no a saisically significan predicor, wih a one-sided p-value of 21.5%. The p-value for he erm yield spread is less affeced by he heeroskedasiciy adjusmen. However, i is also only marginally saisically significan predicor as is p-value is 9.5%. As wih mos reurn-predicion exercises, equal-weigh index resuls are a more exreme version of hose for he value-weigh index. Table 4 shows ha our main cross-secional measure, l SRC, forecass monhly excess equal-weigh reurns wih a -saisic of 4.01. Similarly high -saisics are obained for he earnings yield (4.59) and alernaive crosssecional measures (ranging from 3.63 o 4.81), while he erm yield spread s is slighly lower (2.45). All OLS -saisics imply rejecion of he null a a beer han 1% level, even afer accouning for he problems caused by persisen and correlaed regressors. However, as above, heeroskedasiciy adjusmen has a dramaic impac on he saisical evidence concerning ep (bu no for oher predicors). While under he homoskedasiciy assumpion ep is significan a beer han 1% level, he heeroskedasiciy-robus p-value is 23%. Anoher way of addressing he issue of heeroskedasiciy is o noe ha sock reurns were very volaile during he Grea Depression. A simple check for he imporance of heeroskedasiciy is o omi his volaile period from esimaion. When we esimae he model and p-values from he 1946 2002 sample, l SRC remains saisically significan a a beer han 1% level, while he log earnings yield is no longer significan, even a he 10% level. 5.3. Univariae ess of predicive abiliy in he inernaional sample We also examine he predicive abiliy of cross-secional risk-premium measures in an inernaional sample and obain similar predicive resuls as in he US sample. Because of daa consrains (we only have porfolio-level daa for our inernaional sample), we define our cross-secional risk premium measure as he difference in he local-marke bea beween value and growh porfolios. We work wih value and growh porfolios consruced by Kenneh French and available on hisweb sie, focusing on he op 30% and boom 30% porfolios sored on four MSCI value measures: D=P, BE=ME, E=P, and C=P. We hen esimae he beas for hese porfolios using a 36-monh rolling window and define he predicor variable l MSCI as he average bea of he four value porfolios minus he average bea of he four growh porfolios. If he CAPM holds, he bea difference beween wo dynamic rading sraegies, a lowmuliple value porfolio and a high-muliple growh porfolio, is a naural measure of he expeced equiy premium. The underlying logic is perhaps easies o explain in a simple case in which individual socks growh opporuniies and beas are consan for each sock and cross-secionally uncorrelaed across socks. During years when he expeced equiy premium is high, he high-bea socks have low prices (relaive o curren cash-flow generaing abiliy) and are hus mosly sored ino he value porfolio. Symmerically, lowbea socks have relaively high prices and hose socks mosly end up in he growh or high-muliple porfolio. Consequenly, a high expeced equiy premium causes he value porfolio s bea o be much higher han ha of he growh porfolio. In conras, during years when he expeced equiy premium is low, muliples are deermined primarily by

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 125 growh opporuniies. The high bea and low-bea socks have approximaely he same muliples and are hus approximaely equally likely o end up in eiher he low-muliple value porfolio or he high-muliple growh porfolio. Thus during years when he expeced equiy premium is low, he bea difference beween value and growh porfolio should be small. This simple logic allows us o consruc a cross-secional risk-premium proxy wihou securiy-level daa. We find ha he pas local-marke bea of value minus growh is generally a saisically significan predicor of he fuure local-marke equiy premium. In he individual counry regressions of Table 5, 17 ou of 22 counries have he correc sign in he associaed localmarke equiy premium predicion regression, wih nine ou of 22 esimaes saisically significan a he 10% level. Moreover, he five negaive esimaes are no measured Table 5 Predicing he equiy premium, counry-by-counry regressions Counry Time period N b y OLS Whie br bg Ausralia 1975:1-2001:12 324 0.0237 1.70 (0.111) 1.60 (0.132) 0.989 0.229 Ausria 1987:1-2001:12 180 0.0251 0.61 (0.584) 0.48 (0.540) 0.935 0.241 Belgium 1975:1-2001:12 324 0.0234 1.36 (0.064) 1.43 (0.056) 0.972 0.196 Denmark 1989:1-2001:12 156 0.0149 0.78 (0.240) 0.78 (0.238) 1.02 0.027 Finland 1988:1-2001:12 168 0.0150 0.81 (0.710) 0.78 (0.701) 1.00 0.151 France 1975:1-2001:12 324 0.0444 2.08 (0.028) 2.08 (0.028) 1.00 0.033 Germany 1975:1-2001:12 324 0.0226 1.32 (0.078) 1.25 (0.089) 0.985 0.018 Hong Kong 1975:1-2001:12 324 0.0200 0.50 (0.278) 0.49 (0.282) 0.977 0.11 Ireland 1991:1-2001:12 132 0.0100 0.39 (0.374) 0.36 (0.386) 0.911 0.061 Ialy 1975:1-2001:12 324 0.0268 0.92 (0.233) 0.92 (0.234) 1.01 0.081 Japan 1975:1-2001:12 324 0.0172 1.40 (0.095) 1.66 (0.058) 0.992 0.037 Malaysia 1994:1-2001:10 94 0.0418 0.81 (0.089) 0.76 (0.096) 0.918 0.306 Mexico 1982:1-1987:12 72 0.3490 1.41 (0.044) 1.50 (0.036) 0.844 0.311 Neherland 1975:1-2001:12 324 0.0061 0.37 (0.596) 0.30 (0.573) 0.984 0.105 New Zealand 1988:1-2001:12 168 0.0456 1.95 (0.018) 1.98 (0.017) 0.959 0.023 Norway 1975:1-2001:12 324 0.0053 0.54 (0.708) 0.50 (0.697) 0.994 0.024 Singapore 1975:1-2001:12 324 0.0159 0.76 (0.201) 0.66 (0.229) 0.977 0.094 Spain 1975:1-2001:12 324 0.0366 2.76 (0.004) 2.79 (0.004) 0.986 0.051 Sweden 1975:1-2001:12 324 0.0177 1.57 (0.046) 1.37 (0.068) 1.01 0.019 Swizerland 1975:1-2001:12 324 0.0025 0.15 (0.552) 0.15 (0.552) 0.974 0.030 Unied Kingdom 1975:1-2001:12 324 0.0115 0.53 (0.341) 0.44 (0.372) 0.971 0.153 Unied Saes 1926:7-2002:12 918 0.0166 2.41 (0.008) 2.02 (0.019) 0.993 0.089 This able shows resuls from he model R e M;;i ¼ m 1;i þ y i x 1;i þ u ;i ; x ;i ¼ m 2;i þ r i x 1;i þ v ;i, wih Corrðu ; v Þ¼g. x ;i ¼ l MSCI ;i for counry i in year. l MSCI ;i is consruced by aking he op 30% and boom 30% porfolios sored on four Morgan Sanley Capial Inernaional value measures: D=P, BE=ME, E=P, and C=P. We hen esimae he beas for hese porfolios using a hree-year rolling window and define he predicor variable l MSCI as he average bea of he four value porfolios minus he average bea of he four growh porfolios. The dependen variable in he regressions is he local-marke equiy premium, for which he sock marke reurns are from Kenneh French s files and he local risk-free reurns are from Global Financial Daa. The regressions are esimaed using counry-by-counry ordinary leas squares regressions. The OLS is he homoskedasic -saisic for esing he null ha y ¼ 0. Whie is he -saisic robus o heeroskedasiciy. The p-values in parenheses are based on he condiional criical funcions and es he null y ¼ 0 agains he one-sided alernaive y40. N is he number of observaions. The haed variables are unresriced ordinary leas squares esimaes.

126 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 precisely. Finally, he parameer esimaes for all he counries are similar o hose obained for he Unied Saes (wih he excepion of Mexico wih is exremely shor sample). In addiion o he counry-by-counry regressions, we pool he daa and hus consrain he regression coefficiens o be equal across counries. Forunaely, our pooled regression specificaion does no suffer significanly from he usual problems associaed wih equiypremium predicion regressions. This is because of wo reasons. Firs, he shocks o he predicor variable are largely uncorrelaed wih he reurn shocks. In fac, he correlaion poin esimaes are close o 0.05, suggesing ha he usual asympoic es is slighly conservaive. Second, even if he shocks for a given counry were negaively correlaed, he cross-secional dimension in he daa se lowers he pooled correlaion beween he predicor variable and pas reurn shocks. However, he usual OLS sandard errors (and hypohesis ess based on hem) suffer from anoher problem. The OLS sandard errors ignore he poenial cross-correlaion beween he residuals. To deal wih his problem, we compue sandard errors ha cluser by cross-secion. Our Mone Carlo experimens show ha for our parameer values, clusered sandard errors provide a slighly conservaive hypohesis es. Table 6 Predicing he equiy premium, pooled inernaional regressions No FE Counry FE Counry, ime FE All Excluding US All Excluding US All Excluding US b y 0.0102 0.0090 0.0132 0.0123 0.00961 0.00756 Homoskedasic 3.21 2.53 3.76 3.09 3.32 2.34 p Homoskedasic (0.0007) (0.0057) (0.0001) (0.0010) (0.0004) (0.0010) Heeroskedasic 2.69 2.12 3.31 2.73 2.97 2.14 p Heeroskedasic (0.0036) (0.0171) (0.0005) (0.0032) (0.0015) (0.016) p Clusering by year 2.08 1.65 2.31 1.89 2.57 1.78 p Clusering by year (0.0189) (0.0494) (0.0105) (0.0295) (0.0051) (0.0376) br 0.992 0.992 0.990 0.990 0.988 0.987 bg 0.0519 0.0469 0.0545 0.0497 0.0578 0.0483 This able shows resuls from he model R e M;;i ¼ m 1;;i þ yx 1;i þ u ;i ; x ;i ¼ m 2;;i þ rx 1;i þ v ;i, wih Corrðu ;i ; v ;i Þ¼g. x ;i ¼ l MSCI ;i for counry i in year. l MSCI ;i is consruced by aking he op 30% and boom 30% porfolios sored on four Morgan Sanley Capial Inernaional value measures: D=P, BE=ME, E=P, and C=P. We hen esimae he beas for hese porfolios using a hree-year rolling window and define he predicor variable l MSCI as he average bea of he four value porfolios minus he average bea of he four growh porfolios. The dependen variable in he regressions is he local-marke equiy premium, for which he sock marke reurns are from Kenneh French s files and he local risk-free reurns are from Global Financial Daa. FE denoes fixed effecs, meaning we esimae differen inerceps m 1;;i and m 2;;i for each counry or each counry and ime poin. No FE indicaes ha we esimae a common inercep for all counries and ime poins. homoskedasic and heeroskedasic indicae he usual ordinary leas squares (OLS) -saisic and he heeroskedasiciy-robus Whie -saisic. clusering by year indicaes ha we calculae sandard errors robus o correlaions beween firms as well as heeroskedasiciy, bu assume independence over ime. The analogously defined p-values in parenheses es he null y ¼ 0 agains he one-sided alernaive y40. p-values are based on he usual sandard normal approximaion o he null disribuion of a -saisic. The haed variables are unresriced OLS esimaes.

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 127 Table 6 shows ha we can rejec he null hypohesis of no predicabiliy in favor of he alernaive ha he beas of he counry-specific value minus growh porfolios are posiively relaed o he counry-specific expeced equiy premiums. This conclusion is robus o inclusion or exclusion of he US daa and inclusion or exclusion of counry fixed effecs in he pooled regression. All p-values are under 5%. Thus we conclude ha our simple proxy, l MSCI, predics equiy premium realizaions in a sample largely independen of our main US sample, as well as in he US sample. 5.4. Mulivariae predicabiliy ess The above ess demonsrae ha our new cross-secional variables can forecas he equiy premium. In his secion, we perform mulivariae ess o see wheher he predicive informaion in our new variables subsumes or is subsumed by ha in he earnings yield and erm yield spread. We show he resuls from hese horse races for he value-weigh index in Table 7. Unrepored resuls for he equal-weigh index are similar bu saisically sronger. The horse race beween l SRC and ep is a draw, a leas under he homoskedasiciy assumpion. In regressions forecasing he value-weigh reurn over he full period, we fail o rejec a he 5% level of significance he hypohesis ha l SRC has no predicive abiliy independen of ep (p-value 15.8%). Likewise, we canno rejec he hypohesis ha ep has no predicive abiliy conrolling for l SRC (p-value 10.8%). Because hese p-values are relaively close o 10% for boh he earnings yield and our cross-secional measures, we are cauious abou drawing clear conclusions abou he independen predicive abiliy of hese variables. Allowing for heeroskedasiciy changes his conclusion, however. Using he heeroskedasiciy-robus es, he p-values are always much larger for ep. Though he horse race beween l SRC and ep is a draw under he homoskedasiciy assumpion, many of our alernaive cross-secional measures win heir respecive races wih ep. When l REG, l BMG, and l ER are raced agains ep, he above conclusions change. We now fail o rejec he hypohesis ha ep has no independen predicive power ( p-values ranging from 7.8% o 28.4%) bu do rejec he hypohesis ha l REG, l BMG, and l ER have no independen predicive power ( p-values ranging from 1.5% o 5.0%). These conclusions change slighly in unrepored forecass of he fuure excess reurn on an equal-weigh porfolio of socks. For all combinaions, boh he cross-secional risk premium and he marke s earnings yield are saisically significan. Our resul ha equalweigh reurns are more predicable is consisen wih resuls in he previous lieraure. The erm yield spread is unimpressive in muliple regressions. All oher variables bea he erm yield spread, and TY is insignifican even in mos regressions ha forecas he equal-weigh equiy premium. 5.5. Implicaions of premia divergence in he 1980s Across specificaions, our cross-secional bea-premium variables show heir poores performance as predicors of he equiy premium in he second subsample, especially in he 1980s. Curiously, as Fig. 2 shows, he second subsample also exhibis occasionally large divergences beween he marke s smoohed earnings yield and he cross-secional bea premium. For example, in 1982 boh our cross-secional measures and he Fed model

128 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 Table 7 Mulivariae predicors of he excess value-weigh marke reurn ðr e M Þ Specificaion b y1 1 p 1 p W 1 b y2 2 p 2 p W 2 Fp Fp W Predicion equaion: R e M; ¼ y 0 þ y 1 l SRC 1 þ y 2ep 1 þ u 1927:5-2002:12 0.012 1.17 0.158 0.191 0.013 2.32 0.108 0.294 0.020 0.409 1927:5-1965:2 0.001 0.045 0.532 0.533 0.031 2.17 0.091 0.366 0.056 0.596 1965:2-2002:12 0.011 0.527 0.285 0.295 0.008 1.36 0.447 0.516 0.592 0.696 1927:5-1946:3 0.002 0.030 0.573 0.571 0.042 1.79 0.176 0.342 0.139 0.585 1946:3-1965:2 0.037 2.03 0.038 0.039 0.004 0.225 0.798 0.784 0.086 0.082 1965:2-1984:1 0.027 1.08 0.183 0.209 0.023 2.03 0.172 0.209 0.210 0.283 1984:1-2002:12 0.034 0.738 0.778 0.756 0.012 1.39 0.341 0.371 0.527 0.566 Predicion equaion: R e M; ¼ y 0 þ y 1 l SRC 1 þ y 2TY 1 þ u 1927:5-2002:12 0.023 2.49 0.006 0.051 0.002 0.564 0.289 0.307 0.017 0.126 1927:5-1965:2 0.038 2.23 0.017 0.095 0.001 0.132 0.604 0.582 0.051 0.208 1965:2-2002:12 0.005 0.233 0.357 0.360 0.003 0.791 0.184 0.181 0.670 0.666 1927:5-1946:3 0.069 1.84 0.043 0.162 0.001 0.158 0.634 0.614 0.160 0.337 1946:3-1965:2 0.035 2.55 0.006 0.014 0.010 0.912 0.201 0.233 0.005 0.003 1965:2-1984:1 0.010 0.398 0.280 0.283 0.008 1.61 0.038 0.047 0.229 0.223 1984:1-2002:12 0.012 0.257 0.524 0.514 0.002 0.403 0.633 0.649 0.849 0.849 Full sample resuls for alernaive cross-secional risk premium measures Predicion equaion: R e M; ¼ y 0 þ y 1 x 1 þ y 2 ep 1 þ u x ¼ l REG x ¼ l DP x ¼ l DPG x ¼ l BM x ¼ l BMG x ¼ l ER 0.066 2.36 0.015 0.030 0.012 2.12 0.151 0.350 0.003 0.283 0.0200 1.32 0.107 0.265 0.0143 2.69 0.056 0.175 0.005 0.336 0.0076 0.516 0.321 0.409 0.0160 3.02 0.027 0.140 0.010 0.261 0.0005 1.03 0.174 0.330 0.0140 2.46 0.062 0.101 0.007 0.290 0.0009 1.71 0.050 0.257 0.0131 2.44 0.078 0.122 0.003 0.294 1.766 2.40 0.019 0.029 0.0081 1.57 0.284 0.393 0.011 0.057 Full sample resuls for alernaive cross-secional risk premium measures Predicion equaion: R e M; ¼ y 0 þ y 1 x 1 þ y 2 TY 1 þ u x ¼ l REG 0.088 3.35 0.000 0.016 0.001 0.474 0.316 0.333 0.001 0.046 x ¼ l DP 0.0335 2.38 0.013 0.172 0.0032 1.13 0.136 0.158 0.017 0.248 x ¼ l DPG 0.0215 1.53 0.066 0.225 0.0032 1.13 0.129 0.153 0.121 0.343 x ¼ l BM 0.0010 2.39 0.012 0.226 0.0026 0.910 0.184 0.223 0.028 0.275 x ¼ l BMG 0.0014 2.82 0.003 0.211 0.0029 1.03 0.159 0.174 0.006 0.272 x ¼ l ER 2.154 3.08 0.001 0.001 0.0005 0.172 0.400 0.411 0.006 0.014 b yi is he ordinary leas squares esimae of y i, wih y i ¼ðy 1 y 2 Þ 0 in he model R e M; ¼ m 1 þ y 0 x 1 þ u ; x ¼ l 2 þ Kx 1 þ V. is he usual -saisic for esing he null y i ¼ 0. The able also repors he p-values for esing he null y i ¼ 0 agains y i 40, p denoing he p-value compued using he regular -saisic () and p W using heeroskedasiciy robus Whie -saisic. Fpand Fp W denoe he p-values for he F-es of he null y 1 ¼ y 2 ¼ 0 wih and wihou imposing he homoskedasiciy assumpion. All p-values are compued using he condiional boosrap described in Appendix B. forecas a low equiy premium, while he smoohed earnings yield forecass a high equiy premium. If ep is a good predicor of marke s excess reurn and l SRC of he reurn of high-bea socks relaive o ha of low-bea socks, he divergence implies a rading opporuniy. In

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 129 1982, an invesor could have bough he marke porfolio of socks (which had a high expeced reurn) and hen hedged his invesmen by a zero-invesmen porfolio long lowbea socks and shor high-bea socks (which had a low expeced reurn). A his ime, his hedged marke porfolio should have had a high expeced reurn relaive o boh is sysemaic and unsysemaic risk. We es his hypohesis by consrucing a zero-invesmen porfolio consising of 1.21 imes he CRSP value-weigh excess reurn, minus he reurn difference beween he highes-bea (10) and lowes-bea (1) deciles. The bea-decile porfolios are formed on pas esimaed beas, value weighed, and rebalanced monhly. We picked he coefficien 1.21 o give he porfolio an approximaely zero in-sample uncondiional bea, bu our subsequen resuls are robus o using more elaborae and complex porfolio consrucions schemes. The excess reurn on his bea-hedged marke porfolio is denoed by R e arb. Table 8 confirms his implicaion of premia difference. When we forecas he beahedged marke reurn wih l SRC and ep, he former has a negaive coefficien and he laer a posiive coefficien (alhough ep s -saisic is only 1.13). Alhough we do no have a clear prior abou he uncondiional mean of R e arb, a naural alernaive hypohesis is ha he coefficien on l SRC should be negaive while he coefficien on ep should be posiive. R e arb is a combinaion of wo bes: (1) buying he marke on margin and (2) hedging his equiy-premium be by shoring high-bea socks and invesing he proceeds in low-bea socks. Firs, holding he cross-secional bea premium among socks consan, a higher Table 8 Mulivariae predicors of he hedged value-weigh marke reurn ðr e arb Þ Specificaion b y1 1 p 1 p W 1 b y2 1 p 1 p W 1 Fp Fp W Predicion equaion: R e arb; ¼ y 0 þ y 1 l SRC 1 þ y 2ep 1 þ u 1927:6-2002:12 0.030 2.87 0.002 0.001 0.007 1.13 0.140 0.249 0.013 0.013 1927:6-1965:2 0.018 0.965 0.161 0.168 0.014 1.10 0.877 0.763 0.018 0.014 1965:2-2002:12 0.063 2.29 0.016 0.016 0.010 1.34 0.108 0.196 0.022 0.007 1927:6-1946:3 0.004 0.087 0.522 0.514 0.018 0.907 0.821 0.764 0.440 0.728 1946:3-1965:2 0.016 1.05 0.150 0.139 0.009 0.600 0.486 0.468 0.605 0.608 1965:2-1984:1 0.070 2.75 0.006 0.009 0.007 0.599 0.300 0.308 0.012 0.011 1984:1-2002:12 0.083 1.19 0.121 0.152 0.025 1.89 0.028 0.042 0.126 0.108 Full sample resuls for alernaive cross-secional risk premium measures Predicion equaion: R e arb; ¼ y 0 þ y 1 x 1 þ y 2 ep 1 þ u x ¼ l REG 0.062 2.32 0.011 0.015 0.001 0.467 0.305 0.384 0.062 0.085 x ¼ l DP 0.072 4.66 0.000 0.000 0.0073 1.34 0.099 0.189 0.000 0.005 x ¼ l DPG 0.066 4.36 0.000 0.000 0.0064 1.19 0.130 0.225 0.000 0.006 x ¼ l BM 0.0029 5.80 0.000 0.000 0.0146 2.53 0.007 0.033 0.000 0.001 x ¼ l BMG 0.0029 5.31 0.000 0.001 0.0098 1.79 0.041 0.097 0.000 0.004 x ¼ l ER 2.21 2.61 0.005 0.011 0.0114 1.94 0.036 0.104 0.019 0.063 b yi is he ordinary leas squares esimae of y i, wih y i ¼ðy 1 y 2 Þ 0 in he model R e arb; ¼ m 1 þ y 0 x 1 þ u ; x ¼ l 2 þ Kx 1 þ V. R e arb is he reurn on a zero-invesmen porfolio consising of 1.21 imes he value-weigh excess marke reurn, minus he reurn difference beween he highes-bea (10) and lowes-bea (1) deciles. The able repors he p-values for esing he null y i ¼ 0 agains y i 40, p denoing he p-value compued using he regular - saisic () and p W using heeroskedasiciy-robus Whie -saisic. Fp and Fp W denoe he p-values for he F-es of he null y 1 ¼ y 2 ¼ 0 wih and wihou imposing he homoskedasiciy assumpion. All p-values are compued using he condiional boosrap described in Appendix B.

130 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 expeced equiy premium (as evidenced by a high ep) should ranslae ino a high expeced reurn on R e arb. Second, holding he expeced equiy premium consan, a higher crosssecional bea premium (manifes by a high l SRC ) should ranslae ino a low expeced reurn on R e arb. Thus, one-sided ess are appropriae. The variables are joinly significan for he full period as well as for boh subperiods. However, because l SRC and ep are so highly correlaed in he firs subsample, he idenificaion for he parial regression coefficiens mus come from he second sample. Consisen wih his conjecure, he nulls for boh variables are rejeced a a beer han 10% level in he second subsample, while he p-values are consderably higher in he firs subsample. Similar conclusions can be drawn from regressions ha use oher measures of cross-secional bea premium. These resuls on he predicabily of R e arb are relaively insensiive o using ess ha are more robus o heeroskedasiciy. While he volailiy of he realized equiy premium is sysemaically relaed o he earnings yield, he volailiy of he bea-hedged marke reurn is much less so. Therefore, he relaive insensiiviy of hese ess o heeroskedasiciy adjusmens makes sense. Even a cursory examinaion of he fied values suggess ha hese predicabiliy resuls are also economically significan. In he beginning of year 1982, he prediced value for R e arb is over 20% annualized in he regression ha uses lsrc and ep as forecasing variables. (For reference, his condiional mean is more han hree sandard errors from zero.) Because he uncondiional volailiy of R e arb is under 20% annualized (and various condiional volailiy esimaes even lower), he fied values imply a condiional annualized Sharpe raio of over one a he exreme poin of divergence. In summary, he evidence in Table 8 clearly shows ha divergence of l SRC and ep creaes a boh economically and saisically significan rading opporuniy for an invesor who can borrow a he Treasury bill rae. An alernaive bu equivalen way o describe our resuls is ha he zero-bea rae in he universe of socks deviaes predicably from he Treasury bill rae. 6. Conclusions This paper ells a coheren sory connecing he cross-secional properies of expeced reurns o he variaion of expeced reurns hrough ime. We use he simples risk model of modern porfolio heory, he Sharpe-Linner CAPM, o relae he cross-secional bea premium o he equiy premium. When he cross-secional bea premium is high, he Sharpe-Linner CAPM predics ha he equiy premium should also be expeced o be high. We consruc a class of cross-secional bea-premium variables by measuring he crosssecional associaion beween valuaion muliples (book-o-price, earnings yield, ec.) and esimaed beas. Consisen wih he Sharpe-Linner CAPM, our ime-series ess show ha he cross-secional bea premium is highly correlaed wih he marke s yield measures. Furhermore, he cross-secional variable forecass he equiy premium, boh on is own and in a muliple regression wih he smoohed earnings yield, alhough he high correlaion beween he wo variables makes he muliple-regression resuls less conclusive. Resuls obained from an inernaional sample suppor our main conclusions drawn from he US sample.

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 131 Because equiy-premium realizaions are very noisy, forecasing he equiy premium wih univariae mehods is a nearly impossible ask. Forunaely, simple economic logic makes predicions abou he equiy premium, such as high sock prices should imply a low equiy premium (Campbell and Shiller, 1988a, b; Fama and French, 1989), he equiy premium should usually be posiive because of risk aversion (Meron, 1980), and he crosssecional pricing of risk should be consisen wih he ime-series pricing of risk. We join ohers in arguing ha imposing such economically reasonable guidelines can be of grea pracical uiliy in formulaing reasonable equiy-premium forecass. Beyond simply forecasing he equiy premium, our resuls provide insigh ino he process by which he marke prices he cross-secion of equiies. According o our esimaes, he sock marke prices one uni of bea in he cross-secion wih a premium ha is highly correlaed wih he equiy premium derived from he Fed model, he earnings yield minus he long-erm bond yield. In our sample, he Fed model explains 72% of he ime-series variaion in our main cross-secional risk-price measure. Our claim is no ha one should use he CAPM and he Fed model for relaive valuaion of socks. We merely show ha he cross-secion prices are se approximaely as if he marke paricipans did so. We also provide a pracical soluion o a long-sanding inference problem in financial economerics. A volume of sudies has asked wheher he equiy premium can be prediced by financial variables such as he dividend or earnings yield (Rozeff, 1984; Keim and Sambaugh, 1986; Campbell and Shiller, 1988a, b; Fama and French, 1988, 1989; Hodrick, 1992). Alhough he usual asympoic p-values indicae saisically reliable predicabiliy, Sambaugh (1999) noes ha he small-sample inference is complicaed by wo issues. Firs, he predicor variable is ofen very persisen, and second, he shocks o he predicor variable are correlaed wih he unexpeced componen of he realized equiy premium. Togeher, hese wo issues can cause large small-sample size disorions in he usual ess. Consequenly, elaborae simulaion schemes (e.g., Ang and Bekaer, 2001) have been necessary for finding reasonably robus p-values even in he case of homoskedasic Gaussian errors. We use a novel mehod o solve for he exac small-sample p-values in he case of homoskedasic Gaussian errors. The mehod is based on he Jansson and Moreira (2003) idea of firs reducing he daa o a sufficien saisic and hen creaing he nonlinear mapping from he sufficien saisic o he correc criical value for he OLS -saisic. For a single forecasing variable and he now usual seup proposed by Sambaugh (1999) wih homoskedasic Gaussian errors, we provide he finance communiy wih a funcion ha enables an applied researcher o implemen a correcly sized es of predicabiliy in seconds. Appendix A. l SRC idenificaion requires cross-secional variaion in growh raes This appendix explains how idenifying expeced-equiy premium variaion wih he measure l SRC requires cross-secional variaion in expeced growh raes. Le D=P i be he dividend yield, g i he expeced growh rae (in excess of he risk-free rae), and b i he bea for sock i. Le k be he ypical expeced excess reurn on he marke and k e he deviaion of he expeced excess reurn on he marke from k. The Gordon model and he CAPM sae ha D=P i ¼ b i ðk þ k e Þ g i. We calibrae he model such ha when he equiy premium is a k, he cross-secional variance of g i is c imes ha of b i k. We make wo

132 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 furher simplifying assumpions ha (1) expeced growh raes and discoun raes are uncorrelaed and ha (2) boh variables are uniformly disribued. (Uniform disribuions are convenien, because in his case simple correlaions are equal o rank correlaions in large samples.) Simple algebra shows ha he (rank) correlaion (G) beween dividend yields and beas is equal o k þ k e G ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. (19) ðk þ k e Þ 2 þ ck 2 The change in his correlaion in response o a small change in k e a k e ¼ 0is qg ck qk ¼ e ke ¼0 ½k 2 þ ck 2. (20) 3=2 Š As long as c is posiive and growh raes vary across socks, he (rank) correlaion of b i wih D=P i will vary wih he equiy premium. Furhermore, as one can see from hese formulas, increasing he cross-secional variaion in expeced growh raes makes many of our ordinal equiy-premium measures more sensiive o changes in he equiy premium, a leas as long as beas and expeced growh raes are no correlaed in he cross-secion. We calibrae he above equaions using an esimae of c from Cohen e al. (2003, 2005b). Those auhors esimae ha approximaely 75% of he cross-secional variaion in valuaion muliples is caused by expeced growh raes and only 25% by discoun raes, giving a variance raio of c ¼ 75%=25% ¼ 3. The following exhibi illusraes how even a very modes level of cross-secional spread in growh raes allows changes in he (rank) correlaion beween bea and dividend yield o be srongly relaed o changes in he equiy premium. c 0 0.5 1 2 3 4 5 k ¼ 0.07 0.07 0.07 0.07 0.07 0.07 0.07 G 1.00 0.82 0.71 0.58 0.50 0.45 0.41 qg qk e 0.00 55.54 72.15 78.55 76.53 73.01 69.43 qg qk e =100 0.00 0.56 0.72 0.79 0.77 0.73 0.69 qg qk e =G 0.00 68.03 102.04 136.05 153.06 163.27 170.07 qg qk e =ð100gþ 0.00 0.68 1.02 1.36 1.53 1.63 1.70 Appendix B. Saisical appendix B.1. Algorihm for compuing q nn a In his secion, we describe he algorihm for compuing q nn a, he neural nework approximaion o he criical values defined in Eq. (10). We choose he parameers of he neural ne o minimize he sum of squared size disorions over a grid of

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 133 ðr; g; TÞ values: ð b c; b xþ¼argmin ðc;xþ X J j¼1 X N i¼1 XB 1 a j B b¼1 1 h ½b b;i 4q nn a j ðx b;i ; c; xþš! 2. (21) i ¼ 1;...; N indexes a grid of ðr i ; g i ; T i Þ riples. For each i, we simulae B daa ses from he null model, wih y ¼ 0, i.i.d. normal errors, m 1 ¼ m 2 ¼ 0, and s 2 u ¼ s2 v ¼ 1. b b;i is he - saisic for he b h simulaed sample generaed from ðr i ; g i ; T i Þ, and X b;i is he X vecor generaed from his sample. We can esimae he rejecion frequency based on he criical value q nn a by averaging over he simulaed draws: Pr½b4q nn a ðx; c; xþ; r XB 1 i; g i ; T i ŠB b¼1 1½b b;i 4q nn a ðx b;i; c; xþš. (22) 1ðxÞ is he indicaor funcion, equal o one when xx0 and zero oherwise. Thus our minimizaion problem is a simulaion-based way o minimize he sum of squared size disorions. Because he indicaor funcion is no differeniable, we replace i wih he differeniable funcion 1 h ðxþ ¼ð1 þ e x=h Þ 1. (23) As h goes o zero, 1 h converges poinwise o he indicaor funcion. Because our objecive funcion is differeniable in c and x, we can use efficien minimizaion mehods. The neural ne criical values used in his paper were compued seing B ¼ 10; 000 and h ¼ 0:01. Reparameerizing r ¼ 1 þ c=t, he grid poins were all possible combinaions of c 2ðT; 75; 50; 30; 20; 15; 12; 10; 8; 6; 4; 2; 0Þ=T, g 2ð0; 0:2; 0:4; 0:5; 0:6; 0:7; 0:8; 0:85; 0:86; 0:88; 0:90; 0:92; 0:94; 0:96; 0:98Þ, T 2ð60; 120; 240; 480; 840Þ, a 2ð0:01; 0:025; 0:05; 0:10; 0:50; 0:90; 0:95; 0:975; 0:99Þ. ð24þ We do no need o simulae over differen values of m 1, m 2, s 2 u or s2 v because boh b b;i and X b;i are exacly invarian o hese parameers. Even hough we simulaed only over negaive correlaions, q nn a is valid over he enire range, 1ogo1. To undersand why, firs consider he case in which g is negaive. Our rejecion rule is if bgo0 rejec when b4 mðxþþsðxþf 1 ð:95þ. (25) Nex consider he case in which g is posiive. We replace x by x, hus reversing he sign of he correlaion and making our approximae quanile funcion valid. This ransformaion also reverses he sign of y, so insead of esing he null y ¼ 0 agains he posiive alernaive y40, we es he null y ¼ 0 agains he negaive alernaive yo0. Insead of rejecing when he -saisic b (compued from x ) is greaer han he 95% quanile of he condiional null disribuion we rejec when he ransformed -saisic b (compued from x ) is less han he 5% quanile of he condiional null. Because X is invarian o replacing x wih x, we rejec when bo mðxþþsðxþf 1 ð:05þ. (26)

134 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 Because F 1 ðaþ ¼ F 1 ð1 aþ, he rejecion rule becomes if bg40; rejec when b4mðxþþsðxþf 1 ð:95þ. (27) This leads o our general rejecion rule b4signðbgþmðxþþsðxþf 1 ð0:95þ, valid wheher bg is posiive or negaive. The simulaed sample sizes range from 60 o 840 observaions. The quanile funcion perhaps is no accurae for samples smaller han 60 observaions, bu unrepored Mone Carlo simulaions indicae ha i is accurae for any sample size greaer han 60 observaions, including samples larger han 840 observaions. There are sound heoreical reasons o believe ha he funcion works for samples larger han 840 observaions. As T becomes large, asympoic approximaions become accurae. If x is saionary, he inpu vecor X converges o ð1; 0; 0; 1; gþ 0 and he criical funcion reurns he usual sandard normal approximaion. If r is modeled as a uni roo (or local o uniy) as in Jansson and Moreira (2003), heir asympoic approximaions imply ha he condiional criical funcion qðs; aþ converges o a funcion ha does no depend on T and delivers he correc es size in any large sample. So our criical funcion reurns asympoically sensible criical values wheher we have a uni roo or no. Minimizing objecive funcions in neural neworks is compuaionally demanding. The objecive funcion is no convex in he parameers and has many local minima. We used he following algorihm, which draws on suggesions in Bishop (1995, Chaper 7) and Masers (1993, Chaper 9). Afer generaing all he X and b values, we sandardize hem o have zero sample means and uni variances. Following Bishop (1995, p. 262), we randomly draw each elemen of c and x from an independen Normal(0; 1=5) disribuion. We hen ierae from he saring values for c and x using he Broyden-Flecher-Goldfarb-Shanno opimizaion algorihm. All compuaions were done using Ox 3.00, a programming language described in Doornik (2001). We repeaed his algorihm for many differen randomly drawn saring values for c and x. Some of he saring values led o soluions wih minimal size disorions; he rejecion frequencies were visually similar o hose in Fig. 1. Some saring values converged a parameers ha did no lead o accurae soluions. B.2. Consrucing confidence inervals for he univariae case Confidence inervals consis of all he nulls we fail o rejec. We consruc confidence inervals for y by invering a sequence of hypohesis ess. Le P ¼ðm 1 ; m 2 ; y; r; s 2 u ; s2 v ; gþ0 denoe he parameers of he model. A 100a% confidence se C for y has he propery ha i conains he rue parameer value wih probabiliy a leas a: inf P Pr½y 2 C; PŠXa for all P. (28) C is a random inerval, because i is a funcion of he daa, and Pr½y 2 C; PŠ denoes he probabiliy ha y is in C given he parameers P. Suppose ha, for each poin y in he parameer space, we carry ou he condiional es of size 1 a for he hypohesis y ¼ y. We define C as he se of all y ha we fail o rejec. C is a valid inerval because i conains

he rue y wih probabiliy equal o a: C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 135 Pr½y 2 C; PŠ Pr½fail o rejec null y ¼ y when null is rue; PŠ ¼ 1 a for all P. ð29þ Thus we have an algorihm for consrucing confidence inervals. We (1) consruc a grid of J null hypoheses y 1 oy 2 o oy J, (2) es each null y ¼ y j versus he wo-sided alernaive yay j and (3) ake he confidence inerval o be all he y j s ha are no rejeced. 8 The condiional ess we have described so far are designed o es he null ha y is zero. To es he general null y ¼ y j, ransform he model so ha he null is again zero. Creae he variable ey ¼ y y j x 1, so he firs equaion becomes ey ¼ m 1 þ e yx 1 þ u, (30) wih e y ¼ y y j. Then compue a condiional es of he null e y ¼ 0. B.3. Condiional boosrap algorihm for he mulivariae model In his secion, we describe he condiional boosrap used o carry ou inference in he mulivariae model. (1) Compue bs, he unresriced regression esimae of S. Compue he ransformed vecor ðey ex 0 Þ0 ¼ bs 1=2 ðy x 0 Þ0, where bs 1=2 is he lower diagonal choleski decomposiion of bs and saisfies bs 1=2 ðbs 1=2 Þ 0 ¼ bs. Compue b y 2;R by regressing ey on he second elemen of ex 1 and a consan. Compue bk R by regressing ex on x 1 and a consan and premuliplying he resul by bs 1=2. b y 2;R and bk R are he maximum likelihood esimaors for y 2 and K when bs is he known covariance marix and he null y 1 ¼ 0 is imposed. Define he vecor X ¼ðvecð bk R Þ 0 seðx 1 Þ seðx 2 Þ d Corrðx 1 ; x 2 ÞÞ 0, (31) where x ¼ðx 1; ; x 2; Þ, ½seðx 1 ÞŠ 2 ¼ 1=ðT 1Þ P ðx 1; 1 x 1 Þ 2 is he esimaed variance of x 1, ½seðx 2 ÞŠ 2 is he esimaed variance of x 2,andCorrðx d 1 ; x 2 Þ is heir esimaed covariance. (2) Simulae B daa ses from he parameer values y 1 ¼ 0, b y 2;R, bk R,andbS. Le b denoe he -saisic for he bh simulaed daa se, and le X b denoe he X vecor for he bh sample. (3) Creae he variable d b ¼ max i jðx i X b;i Þ=s i j, where X i and X b;i are he ih elemens of X and X b,ands 2 i ¼ðB 1Þ 1 P b ðx b;i X i Þ 2, he sandard deviaion of X b;i. d b is a measure of he disance beween he sufficien saisics compued from he acual and he simulaed daa. (4) Le d ðbþ denoe he bh sored d value, sored in ascending order, so d ð1þ pd ð2þ p pd ðbþ. Le D denoe he se of b b where he corresponding X b is among he N ha are neares o he acual sufficien saisics: b b 2 D iff d ðbþ pd ðnþ. (32) 8 Throughou he paper a size 1 a es rejecs he null y ¼ y j in favor of yay j when b4qðs; ð1 þ aþ=2þ or boqðs; ð1 aþ=2þ.

136 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 (5) The se of draws D are reaed as draws from he condiional disribuion of b given S. We esimae he 100ah quanile of he condiional disribuion wih he 100ah empirical quanile of he sample of draws D. This boosrap procedure compues a nonparameric neares neighbor esimae of he condiional quanile of b given X. Chaudhuri (1991) shows ha as B and N increase o infiniy, wih N becoming large a a slower rae han B, he boosrapped quanile converges in probabiliy o he rue condiional quanile. However, because X is a high-dimensional vecor he curse of dimensionaliy requires B o be exraordinarily large, possibly in he billions. Thus if we ake he Chaudhuri (1991) heory lierally, i is no compuaionally feasible o precisely esimae he condiional quanile. However, he Mone Carlo resuls repored below sugges ha he condiional boosrap accomplishes he more modes goal of improving on he parameric boosrap. We simulae five housand samples of 120 observaions each, seing y 1 ¼ 0 and he res of he model parameers equal o he unresriced leas squares esimaes when y is R e M, he value-weighed CRSP excess reurn, and x conains he wo predicors l SRC and ep. For each simulaed sample we es he null y 1 ¼ 0 agains he one-sided alernaive y 1 40. We compue criical values using he parameric boosrap and he condiional boosrap. For he parameric boosrap we simulae 20 housand new daa ses from he model wih normal errors, seing y 1 ¼ 0 and he oher parameers o heir unresriced leas squares esimaes. The condiional boosrap is compued aking B ¼ 20; 000 and N ¼ 1; 000. The above experimen yields he following resuls. When y 1 is he coefficien on l SRC, he parameric boosrap rejecs he null 5:48% of he ime and he condiional boosrap rejecs 3:84% of he ime. When y 1 is he coefficien on ep he rejecion frequencies are 11.46% and 4.78%, respecively. We hen simulae from he model wih K ¼ I, o see how he boosraps perform when he predicors follow uni roos. When y 1 is he coefficien on l SRC, he parameric boosrap rejecs 6:36% of he ime and he condiional boosrap rejecs 3:32% of he ime. When y 1 is he coefficien on ep he rejecion frequencies are 15:64% and 7:80%. These Mone Carlo resuls suggess ha condiional inference yields a significan improvemen even in he compuaionally more challenging mulivariae problem. We choose he N and B used for Tables 7 and 8 as follows. For he p-values ha es he nulls y 1 ¼ 0, y 2 ¼ 0, and y 1 ¼ y 2 ¼ 0, we se N ¼ 10; 000 and B ¼ 200; 000. B.4. Discussion of es power The Mone Carlo experimens in Secion 4, as well as he heory underlying condiional inference, demonsrae ha he size of our condiional es is correc. The es is also powerful. Jansson and Moreira (2003) show ha he condiional es is mos powerful among he class of similar ess. Similar ess deliver correc size for all values of he nuisance parameers r, m 1, and m 2. Therefore, a es can only be more powerful han ours if i eiher under-rejecs or over-rejecs for some values of he nuisance parameers. One way o add power o a es is o make srong a priori assumpions abou he values of r, m 1, and m 2. For example, if one knows he value of r, hen i is sraighforward o consruc a es ha is more powerful han ours. If one s belief in r is incorrec, he es has power or size problems or boh. In paricular, our procedure has good power relaive o alernaive procedures proposed by Lewellen (2004), Torous e al. (2005), and Campbell and Yogo (2006). Lewellen (2004) provides a good example of a clever es ha improves power by making srong

C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 137 assumpions abou nuisance parameers. Lewellen (2004) derives an one-sided es (alernaive hypohesis is y40) assuming ha r ¼ 1 and go0. If we know for sure ha r equals one, hen Lewellen s es has a power advanage over our es. However, Lewellen (2004) shows ha he power of his es declines dramaically as r declines. To address he power issue, Lewellen (2004) also describes a Bonferroni procedure o improve power for r below one. (For some of our predicive variables ro1 and g40, hus Lewellen s ess would over-rejec he null.) Torous e al. (2005) use a Bonferroni procedure firs proposed by Cavanaugh e al. (1995). They form a confidence inerval for r, hen consruc he opimal es of y ¼ 0 a all values of r in he inerval. If none of he ess rejecs for any r in he inerval, hey do no rejec he null ha y ¼ 0. Their es is also designed o have high power if r is close o one. Campbell and Yogo (2006) make sronger assumpions abou he nuisance parameers han we do, and if heir assumpions are correc, heir es could herefore be more powerful. Campbell and Yogo (2006) and Torous e al. (2005) boh use Bonferroni ess ha rely on a firs-sage confidence inerval for r. Campbell and Yogo (2006) differ in ha hey form he inerval using a newer, more powerful es: he Dickey-Fuller generalized leas squares (DF-GLS) es of Ellio e al. (1996). The DF-GLS es is poenially more powerful han radiional mehods bu also makes sronger assumpions abou nuisance parameers. To beer undersand Campbell and Yogo s approach in pracice, suppose ha he inercep in he second equaion, m 2, is known o be zero and does no need o be esimaed. Incorporaing his prior informaion ino a es should improve he power of he es. In pracical applicaions m 2 is probably no exacly zero. However, suppose ha m 2 is by some meric small, and furher suppose ha r is close o one. Then, over ime, x varies so much ha i dominaes any small value of m 2 (in oher words, m 2 is small relaive o he sandard deviaion of x,) and he daa behave approximaely as if m 2 were zero. Therefore, one can consruc ess under he assumpion ha m 2 is zero, and in a large sample he assumpion does no lead o size problems. This approximaion is he basis for he DF-GLS es used by Campbell and Yogo. In summary, Campbell and Yogo are hus implicily making he assumpion ha r is close o one and m 2 is close o zero. Mone Carlo experimens confirm he above logic. The ess by Lewellen (2004), Torous e al. (2005), andcampbell and Yogo (2006) have beer power han our es if r is close o one and g is close o negaive one. As r decreases and g increases, our es becomes more powerful. In he simulaions, we focus on a long (1926 2002) monhly ime series of log dividend yields and excess reurns, available from Moohiro Yogo s websie. We use his ime series for he following reasons. Firs, his series is he mos persisen of he commonly used predicor variables. Second, shocks o dividend yield have much sronger negaive correlaion wih reurns han shocks o any oher of he commonly used predicor variables. Third, his series has a long ime span, and hus many of he asympoic approximaions are likely be accurae. Fourh, a series ha sars in 1926 has a relaively low inial value, which is relevan for some of he ess we examine. For our Mone Carlo experimens, we esimae he following model: y ¼ m 1 þ yx 1 þ u, x ¼ em 2 þ z ; and z ¼ rz 1 þ v, ð33þ

power power 138 C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 1 γ=-0.95, c=0 1 γ =-0.50, c =0 0.8 0.8 0.6 0.6 0.4 PTV CY L 1 0.2 L 2 TVY 0 0 5 10 15 20 b 1 0.8 0.6 γ = -0.95, c = -15 0.4 0.2 0 0 5 10 15 20 1 0.8 0.6 γ = -0.50, c = -15 0.4 0.2 0 0 5 10 15 20 0.4 0.2 0 0 5 10 15 20 Fig. 3. Power of alernaive ess. The plos compare small-sample rejecion probabiliies for various ess. The labels are Polk-Thompson-Vuoleenaho (PVT), Campbell-Yogo (CY), Lewellen s es ðl 1 Þ, Lewellen s Bonferroni es ðl 2 Þ, and Torous, Valkanov, and Yan s Bonferroni es (TVY). We provide resuls for various values of c ¼ Tðr 1Þ, b ¼ y=t, and g. There are en housand Mone Carlo simulaions. wih r ¼ 1 þ c=t and y ¼ b=t. This specificaion appears in he curren version of Campbell and Yogo s working paper. I is equivalen o he wo-equaion sysem in Eq. (6) ha we use in he res of he paper, wih m 2 ¼ em 2 ð1 rþ. The poin esimaes are m 1 ¼ 0:0319; em 2 ¼ 3:3621; x 0 ¼ 3:0015; z 0 ¼ 0:3606; and p s 1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Varðu Þ ¼ 0:0545; s 2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðv Þ ¼ 0:0565, g ¼ Corrðe ; v Þ¼ 0:9543. ð34þ The confidence inerval for c Tðr 1Þ, based on invering he Dickey-Fuller saisic, ranges approximaely from c ¼ 16 o c40. Fig. 3 compares he small-sample rejecion probabiliies of five alernaive ess: our es (PTV), he Campbell-Yogo es (CY), Lewellen s es ðl 1 Þ, Lewellen s Bonferroni es ðl 2 Þ, and he Bonferroni es of Torous, Valkanov, and Yan (TVY). The Mone Carlo resuls use he parameer esimaes and saring values given above, wih a few variaions. We se c ¼ 0, i.e., a uni roo, and c ¼ 15, boh of which are in he confidence inerval. The value c ¼ 15 calibraes o r ¼ 1 þ c=t ¼ 0:9836 for 913 observaions. We se he correlaion g ¼ 0:95, he correlaion in Campbell and Yogo s daa, and g ¼ 0:50, which is closer o he correlaions we see for many oher predicor series.

We calculae he CY es by following he insrucions in heir Appendix C, excep ha we do no implemen any of he correcions for serial correlaion in he errors e or v.to implemen he CY es we elecronically sore he criical-value ables from he mos recen version of heir working paper. The Lewellen es ðl 1 Þ is opimal for r ¼ 1. As Lewellen has shown, i has poor power when r is below one. Lewellen proposes bu did no empirically implemen a Bonferroni version of he es ðl 2 Þ o improve power for small r. Our implemenaion of he ðl 2 Þ es is parly based on suggesions and clarificaions by Jonahan Lewellen in privae correspondence. The p-value for Lewellen s Bonferroni es is p bonferroni ¼ min½2p; P þ DŠ, (35) where P ¼ min½p lewellen ; p sambaugh Š, (36) and D is he p-value for a uni roo es of r ¼ 1, based on he sampling disribuion of br. p lewellen is he p-value for he Lewellen es ðl 1 Þ,andp sambaugh is he p-value for he Sambaugh es. We calculae p sambaugh assuming ha he Sambaugh bias-correced -saisic is normally disribued. We also ran he simulaions using a boosrap procedure o calculae he p-values. The resuls are qualiaively similar. When c equals zero and he correlaion g is very close o negaive one, he compeing procedures (CY, L 1,L 2, TVY) have more power han our es. The compeing procedures also do no over-rejec in his case. This is no surprising, as he alernaive ess are developed wih his paricular siuaion in mind. Our es is superior in erms of power for smaller values of c and larger values of g. When c ¼ 15, many of he alernaive procedures have size disorions, causing hem o under-rejec he null and have low power. The CY and L 1 ess have paricularly low power for c ¼ 15 and g ¼ 0:50. In paricular, for he predicor variables and sample periods we es in our paper, c poin esimaes range approximaely from 1 o 75 and g poin esimaes from 0:7 oþ0:4. As he above experimens show, hese ranges include many values of g for which our es is superior. References C. Polk e al. / Journal of Financial Economics 81 (2006) 101 141 139 Adrian, T., Franzoni, F., 2002. Learning abou bea: an explanaion of he value premium. Unpublished working paper. Massachuses Insiue of Technology, Cambridge, MA. Ang, A., Bekaer, G., 2001. Sock reurn predicabiliy: is i here? Unpublished working paper. Columbia Universiy Graduae School of Business, New York. Ang, A., Chen, J., 2004. CAPM over he long-run: 1926 2001. Unpublished working paper. Universiy of Souhern California, Los Angeles, CA. Asness, C.S., 2002. Figh he Fed model: he relaionship beween sock marke yields, bond marke yields, and fuure reurns. Unpublished working paper. AQR Capial Managemen, LLC, Greenwich, CT. Ball, R., 1978. Anomalies in relaionships beween securiies yields and yield-surrogaes. Journal of Financial Economics 6, 103 126. Banz, R.W., 1981. The relaion beween reurn and marke value of common socks. Journal of Financial Economics 9, 3 18. Basu, S., 1977. Invesmen performance of common socks in relaion o heir price-earnings raios: a es of he efficien marke hypohesis. Journal of Finance 32, 663 682. Basu, S., 1983. The relaionship beween earnings yield, marke value, and reurn for NYSE common socks: furher evidence. Journal of Financial Economics 12, 129 156. Bishop, C.M., 1995. Neural Neworks for Paern Recogniion. Oxford Universiy Press, New York.

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