Journal of Empirical Finance

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1 Journal of Emprcal Fnance 17 (2010) Conens lss avalable a ScenceDrec Journal of Emprcal Fnance journal omepage: Sock and bond reurns w Moody Invesors Geer Bekaer a,b,, Erc Engsrom c, Seven R. Grenader b,d a Columba Unversy, Uned Saes b NBER, Uned Saes c Federal Reserve Board of Governors, Uned Saes d Sanford Unversy, Uned Saes arcle nfo absrac Arcle sory: Receved 14 Aprl 2010 Receved n revsed form 19 July 2010 Acceped 11 Augus 2010 Avalable onlne 17 Augus 2010 JEL classfcaon: G12 G15 E44 We presen a racable, lnear model for e smulaneous prcng of sock and bond reurns a ncorporaes socasc rsk averson. In s model, analyc soluons for endogenous sock and bond prces and reurns are readly calculaed. Afer esmang e parameers of e model by e general meod of momens, we nvesgae a seres of classc puzzles of e emprcal asse prcng leraure. In parcular, our model s sown o jonly accommodae e mean and volaly of equy and long erm bond rsk prema as well as salen feaures of e nomnal sor rae, e dvdend yeld, and e erm spread. Also, e model maces e evdence for predcably of excess sock and bond reurns. However, e sock bond reurn correlaon mpled by e model s somewa ger an a n e daa Elsever B.V. All rgs reserved. Keywords: Equy premum Excess volaly Sock bond reurn correlaon Reurn predcably Counercyclcal rsk averson Hab perssence 1. Inroducon Campbell and Cocrane (1999) denfy slow counercyclcal rsk premums as e key o explanng a wde varey of dynamc asse prcng penomena wn e conex of a consumpon-based asse-prcng model. Tey generae suc rsk premums by addng a slow movng exernal ab o e sandard power uly framework. Essenally, as we clarfy n e followng dscusson, er model generaes couner-cyclcal rsk averson. Ts dea as surfaced elsewere as well. Sarpe (1990) and praconers suc as Persaud (see, for nsance, Kumar and Persaud, 2002) developed models of me-varyng rsk appees o make sense of dramac sock marke movemens. Te frs conrbuon of s arcle s o presen a very racable, lnear model a ncorporaes socasc rsk averson. Because of e model's racably, becomes parcularly smple o address a wder se of emprcal puzzles an ose consdered by Campbell and Cocrane. Campbell and Cocrane mac salen feaures of equy reurns, ncludng e equy premum, excess reurn varably and e varably of e prce dvdend rao. Tey do so n a model were e rsk free rae s consan. Insead, we embed a fully socasc erm srucure no our model, and nvesgae weer e model can f salen feaures of bond and sock reurns smulaneously. Suc over-denfcaon s mporan, because prevous models a mac equy reurn momens Correspondng auor. Columba Busness Scool, 802 Urs Hall, 3022 Broadway, New York, NY, 10027, Uned Saes. Tel.: ; fax: E-mal address: [email protected] (G. Bekaer) /$ see fron maer 2010 Elsever B.V. All rgs reserved. do: /j.jempfn

2 868 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) ofen do so by ncreasng e varably of margnal raes of subsuon o e pon a a sasfacory f w bond marke daa and rsk free raes s no longer possble. Usng e general meod of momens (GMM), we fnd a our model can raer successfully f many feaures of bond and sock reurn daa ogeer w mporan properes of e fundamenals, ncludng a low correlaon beween fundamenals and reurns. Te NBER Workng Paper verson of Campbell and Cocrane also consdered a specfcaon w a socasc neres rae. Wle a model maced some salen feaures of neres rae daa, beng a one-facor model, necessarly could no provde a fully sasfacory f of erm srucure daa. Moreover, e one sock naure of e model mposes oo srong of a lnk beween bond and sock reurns, an ssue no examned n Campbell and Cocrane. Wacer (2006) and Burasc and Jlsov (2007) do provde exensons of e Campbell Cocrane framework bu focus almos enrely on erm srucure puzzles. In our model, socasc rsk averson s no perfecly negavely correlaed w consumpon grow as n Campbell and Cocrane, bu e perfec correlaon case represens a esable resrcon of our model. Once we model bond and sock reurns jonly, a seres of classc emprcal puzzles becomes esable. Frs, Sller and Belra (1992) pon ou a presen value models w a consan rsk premum mply a neglgble correlaon beween sock and bond reurns n conras o e moderae posve correlaon n e daa. We expand on e presen value approac by allowng for an endogenously deermned socasc rsk premum. Second, Fama and Frenc (1989) and Kem and Sambaug (1986) fnd common predcable componens n bond and equy reurns. Afer esmang e parameers of e model o mac e salen feaures of bond and sock reurns alluded before, we es ow well e model fares w respec o ese puzzles. Our model generaes a bond sock reurn correlaon a s somewa oo g relave o e daa bu maces e predcably evdence. Trd, o conver from model oupu o e daa, we use nflaon as a sae varable, bu ensure a nflaon s neural: a s e Fser ypoess olds n our economy. Ts s mporan n nerpreng our emprcal resuls on e jon properes of bond and sock reurns. More realsc modelng of e nflaon process s a prme canddae for resolvng e remanng falures of e model. Our model also fs no a long seres of recen aemps o break e g lnk beween consumpon grow and e prcng kernel a s e man reason for e falure of e sandard consumpon-based asse prcng models. Sanos and Verones (2006) add e consumpon/labor ncome rao as a second facor o e kernel, We (2004) adds lesure servces o e prcng kernel and models uman capal formaon, Pazzes, Scneder and Tuzel (2007) and Lusg and Van Neuwerburg (2005) model e ousng marke o ncrease e dmensonaly of e prcng kernel. Te remander of e arcle s organzed as follows. Secon 1 presens e model. Secon 2 derves closed-form expressons for bond prces and equy reurns. Secon 3 oulnes our esmaon procedure wereas Secon 4 analyzes e esmaon resuls, and e mplcaons of e model a e esmaed parameers. Secon 5 ess ow e model fares w respec o e neracon of bond and sock reurns. In e conclusons, we summarze e mplcaons of our work for fuure researc and we dscuss some recen papers a ave also consdered e jon modelng of bond and sock reurns. 2. Te Moody Invesor economy 2.1. Preferences Consder a complee markes economy as n Lucas (1978), bu modfy e preferences of e represenave agen o ave e form: E 0 " # ðc H Þ 1 γ 1 ; ð1þ 1 γ β =0 were C s aggregae consumpon and H s an exogenous exernal ab sock w C NH. One movaon for an exernal ab sock s e framework of Abel (1990, 1999) wo specfes preferences were H represens pas or curren aggregae consumpon, wc a small ndvdual nvesor akes as gven, and en evaluaes s own uly relave o a bencmark. Ta s, uly as a keepng up w e Joneses feaure. In Campbell and Cocrane (1999), H s aken as an exogenously modelled subssence or ab level. 1 Hence, e local coeffcen of relave rsk averson equals γ, C H were C H s defned as e surplus rao. 2 As e surplus rao goes o zero, e consumer's rsk averson goes o nfny. In our C model, we vew e nverse of e surplus rao as a preference sock, wc we denoe by Q. Tus, Q = C. Rsk averson s C H now caracerzed by γ Q, and Q N1. As Q canges over me, e represenave consumer/nvesor's moodness canges. C 1 In Heaon (1995) and Bekaer (1996), e ab sock depends on pas consumpon and also poenally reflecs e lasng servces of durable goods. 2 Of course, s s no acual rsk averson defned over weal wc depends on e value funcon. Te Appendx o Campbell and Cocrane (1995) examnes e relaon beween local curvaure and acual rsk averson, wc depends on e sensvy of consumpon o weal. In er model, acual rsk averson s smply a scalar mulple of local curvaure. In e presen arcle, we only refer o e local curvaure concep, and slgly abuse ermnology n callng rsk averson.

3 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Te margnal rae of subsuon n s model deermnes e real prcng kernel, wc we denoe by M. Takng e rao of margnal ules of me +1 and, we oban: M +1 = β C γ +1=C Q +1 =Q γ ð2þ = β exp γδc +1 + γ q +1 q ; were q =ln(q ) and Δc = lnðc Þ lnðc 1 Þ. Ts model may beer explan e predcably evdence an e sandard model w power uly because can generae couner-cyclcal expeced reurns and prces of rsk. To see s, frs noe a e coeffcen of varaon of e prcng kernel equals e maxmum Sarpe rao aanable w e avalable asses (see Hansen and Jagannaan, 1991). As Campbell and Cocrane (1999) also noe, w a log-normal kernel: σ M +1 = E M +1 qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff exp Var m +1 1 : ð3þ were m =ln(m ). Hence, e maxmum Sarpe rao caracerzng e asses n e economy s an ncreasng funcon of e condonal volaly of e prcng kernel. If we can consruc an economy n wc e condonal varably of e kernel vares roug me and s ger wen Q s g (a s, wen consumpon as decreased closer o e ab level), en we ave nroduced e requred counercyclcal varaon no e prce of rsk. Wereas Campbell and Cocrane (1999) ave only one source of uncerany, namely, consumpon grow, wc s modeled as an..d. process, we embed e Moody Invesor economy n e affne asse prcng framework. Te process for q ln(q )sncludedas an elemen of e sae vecor. Aloug e neremporal margnal rae of subsuon deermnes e form of e real prcng kernel roug Eq. (2), we sll ave a coce on ow o model Δc and q.snceq N1, we model q accordng o e specfcaon, pffffffff q +1 = μ q + ρ qq q + σ qq q 1 λ 2 1 = 2ε q +1 + λεc +1 ; ð4þ were μ q, ρ q and σ q and λ are parameers. 3 Here, ε q s a sandard normal nnovaon process specfcoq and ε c s a smlar process, represenng e sole source of condonal uncerany n e consumpon grow process. Bo are dsrbued as N(0, 1) and are ndependen. We wll sorly see a λ [ 1,1] s e condonal correlaon beween consumpon grow and q. Wen λ= 1, q and e consumpon grow wll be perfecly negavely correlaed wc s conssen w e ab perssence formulaon of Campbell and Cocrane (1999). Te fac a we model q as a square roo process makes e condonal varance of e prcng kernel depend posvely on e level of Q Fundamenals processes Wen akng a Lucas-ype economy o e daa, e deny of e represenave agen and e represenaon of e endowmen or consumpon process become crcal. Because we prce eques n s arcle, dvdend grow mus be a sae varable. Secon deals e modelng of consumpon and dvdend grow. To lnk a real consumpon model o e nomnal daa, we mus make assumpons abou e nflaon process, wc we descrbe n Secon Consumpon and dvdends In e orgnal Lucas (1978) model, a dvdend producng ree fnances all consumpon. Realscally, consumpon s fnanced by many sources of ncome (especally labor ncome) no represened n aggregae dvdends. 4 We erefore represen dvdends as consumpon dvded by e consumpon dvdend rao CD. Because dvdends and consumpon are nonsaonary we model consumpon grow and e consumpon dvdend rao, CD. Te man economerc ssue s weer CD s saonary or, more generally, weer consumpon and dvdends are conegraed. Bansal, Dmar and Lundblad (2004) recenly argue a dvdends and consumpon are conegraed, bu w a conegrang vecor a dffers from [1, 1], wereas Bansal and Yaron (2004) assume wo un roos. Table 1 repors some caracerscs of e consumpon dvdend rao usng oal nondurables consumpon and servces as e consumpon measure n addon o un roo ess for CD. For compleeness, we repor e same sascs for consumpon and dvdends as well. Te auocorrelaons of log consumpon and log dvdends clearly reveal non-saonary beavor, bu e frs auocorrelaon of e annual consumpon dvdend rao s Wen we es for a un roo n a specfcaon allowng for a me rend and addonal auocorrelaon n e regresson, we rejec e null ypoess of a un roo n e consumpon dvdend rao wle falng o rejec for consumpon and dvdends. Te evdence rffffffffffffffffffffffffffffffffffffffffffff 3 σ q s parameerzed as 1 1μ q 1 ρ 2 f qq were q = 1 ρ qq q. I s easly sown a f s e rao of e uncondonal mean o e uncondonal sandard devaon of q. By boundng f below a uny, we ensure a q s usually posve (under our subsequen esmaes, q s posve n more an 95% of smulaed draws). 4 In e NBER verson of s arcle, we provde a more formal movaon for our se-up n e conex of a mulple dvdend economy. Menzly e al. (2004) formulae a connuous-me economy exendng e Campbell Cocrane framework o mulple dvdend processes.

4 870 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Table 1 Consumpon dvdend rao caracerscs. Auocorr. ADF es p-values c (0.006) d (0.023) cd (0.077) Consan mean Lnear rend Sample unvarae auocorrelaon and augmened Dckey Fuller (1984) un roo ess for log consumpon, log dvdends, and e log consumpon dvdend rao. In e lef column, unvarae sample auocorrelaons are repored w GMM sandard errors n pareneses (1 Newey Wes lag). For e ADF ess, we repor p-values for e null ypoess of e presence of a un roo. Tese resuls are based on an OLS regresson of e form. y = α + δ + ζδy 1 + ρy 1 + u ð42þ and e prevously descrbed p-values are for e ypoess a ρ=1. For e column labeled consan mean, p-values are conduced for regressons w e δ erm omed. All seres excludng consumpon were obaned from Ibboson Assocaes, for , (74 years). Consumpon daa were obaned from e Bureau of Economc Analyss NIPA ables. Consumpon daa for e frs ree years of e sample ( ) are unavalable from e BEA. Aggregae consumpon grow was obaned from e webse of Rober Sller, and used for bo nondurables and servces consumpon seres for s perod. One observaon s los due o e esmaon of models requrng lags. See ex and Appendx A for addonal daa consrucon ssues. for consumpon and dvdends sows agan a s dffcul o dfferenae un roos from rends for non-saonary seres w a fne sample. As a resul we assume dvdends and consumpon are conegraed w [1, 1] as e conegrang vecor, and n our acual specfcaon, we do allow for a me rend o capure e dfferen means of consumpon and dvdend grow. We use aggregae nondurables and servces consumpon as e consumpon measure. Because many agens n e economy do no old socks a all, we cecked e robusness of e model o an alernave measure of consumpon a aemps o approxmae e consumpon of e sockolder. Mankw and Zeldes (1991) and A-Saala, Parker and Yogo (2004) ave poned ou a aggregae consumpon may no be represenave of e consumpon of sock olders. In parcular, we le e sockolder consumpon be a weged average of luxury consumpon and oer consumpon w e wegng equal o e sock marke parcpaon rae based on Amerks and Zeldes (2004). However, our model does no perform noceably beer w s consumpon measure and we do no repor ese resuls o conserve space. Our socasc model for consumpon grow and e consumpon dvdend rao becomes pffffffff Δ c +1 = μ c + ρ cc Δc + ρ cu u + σ cc q c ε +1 ð5þ cd +1 = ζ + δ + u +1 were μ c, ρ cc, ρ cu, and σ cc are parameers governng consumpon grow, Δc. Economcally, e model us mples a e agen s more rsk averse n mes of larger macroeconomc uncerany. Te model for u +1, e socasc componen of e consumpon dvdend rao, s symmerc w e model for consumpon grow: pffffffff c u +1 = μ u + ρ uu u + ρ uc Δc + σ uc q ε +1 + σ uu εu +1 : ð6þ Ts specfcaon mples a consumpon grow s an ARMA(2,1) process. Bansal and Yaron (2004) and Wacer (2006) also nroduce, effecvely, an MA componen n e dvdend and/or consumpon process. Noe a we ave allowed for eeroskedascy n e consumpon process as e condonal volaly of Δc +1 s proporonal o q. From a modellng perspecve, e square roo assumpon s mporan o arrve a closed-form soluons for asse prces as e varance of e error erm now smply depends on e expeced value of q. Forunaely, ere s subsanal evdence for suc eeroskedascy even n annual real consumpon grow, wc as (unrepored) uncondonal excess kuross of 7.00 n our sample. Durng esmaon, we are careful o ceck a our model mpled consumpon grow kuross does no exceed a n e daa. Te consan ζ s wou consequences once e model s pu n saonary forma, bu e rend erm, δ, accommodaes dfferen means for consumpon grow and dvdend grow. Specfcally, Δd +1 = Δc +1 δ Δu +1 ð7þ Te condonal covarance beween consumpon grow and preference socks can now be more explcly examned. In parcular, s covarance equals: Cov Δc +1 ; q +1 = σqq σ cc λq ð8þ

5 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) so a e covarance s mos negave wen λ= 1, a resrcon of perfecly couner-cyclcal rsk averson under wc our model mos closely approaces a of Campbell and Cocrane (1999). Anoer ssue a arses n modelng consumpon and socasc rsk averson dynamcs s weer e model preserves e noon of ab perssence. For s o be e case, e ab sock sould move posvely w (pas) consumpon. Campbell and Cocrane (1999) parameerze e process for e surplus rao suc a e dervave of e log of e ab sock s always posve w respec o log consumpon. Te ab sock n our model sasfes, H =v C, were v =1 1 s n (0,1) and s ncreasng n Q Q. Ta s, wen rsk averson s g, e ab sock moves closer o e consumpon level. I s now easy o see a Campbell and Cocrane's condon requres σqq λ N 1 Q σ for all cc, wc s no necessarly sasfed by our model Inflaon One callenge w confronng consumpon-based models w e daa s a e model conceps ave o be ranslaed no nomnal erms. Aloug nflaon could play an mporan role n e relaon beween bond and sock reurns, we wan o assess ow well we can mac e salen feaures of e daa wou relyng on nrcae nflaon dynamcs and rsk premums. Terefore, we append e model w a smple nflaon process: π +1 = μ π + ρ π π + σ π ε π +1 ð9þ Furermore, we assume a e nflaon sock s ndependen of all oer socks, n parcular socks o e real prcng kernel (or neremporal margnal rae of subsuon). Tese assumpons mpose a e Fser Hypoess olds n our economy. Te prcng of nomnal asses en occurs w a nomnal prcng kernel, ˆm +1 a s a smple ransformaon of e real prcng kernel, m +1. ˆm +1 = m +1 π +1 ð10þ 2.3. Te full model We are now ready o presen e full model. Te logarm of e prcng kernel or socasc dscoun facor n s economy follows from e preference specfcaon and s gven by: m +1 = lnðβþ γδc +1 + γδq +1 ð11þ Because of e logarmc specfcaon, e acual prcng kernel, M +1, s a posve socasc process a ensures a all asses are prced suc a 1=E M +1 1+R ;+1 ð12þ were R, +1 s e percenage real reurn on asse over e perod from o me (+1), and E denoes e expecaon condonal on e nformaon a me. Because M s srcly posve, our economy s arbrage-free (see Harrson and Kreps (1979)). Te model s compleed by e specfcaons, prevously nroduced, of e fundamenals processes, wc we collec ere: pffffffff q +1 = μ q + ρ qq q + σ qq q 1 λ 2 1 = 2ε q +1 + λεc +1 pffffffff Δc +1 = μ c + ρ cc Δc + ρ cu u + σ cc q c ε +1 pffffffff u +1 = μ u + ρ uu u + ρ uc Δc + σ uc q ε c +1+ σ uu ε u +1 ð13þ Δd +1 = Δc +1 δ Δu +1 π +1 = μ π + ρ π π + σ π ε π +1 Te real kernel process, m +1, s eeroskedasc, w s condonal varance proporonal o q. In parcular, Var m +1 = γ 2 q σ 2 cc + σqq 2σ 2 cc σ qq λ Consequenly, ncreases n q wll ncrease e Sarpe Rao of all asses n e economy, and e effec wll be greaer e more negave s λ. Ifq and Δc are negavely correlaed, e Sarpe rao wll ncrease durng economc downurns (decreases n Δc ). Noe a Campbell and Cocrane essenally maxmze e volaly of e prcng kernel by seng λ= 1.

6 872 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Bond and sock prcng n e Moody Invesor economy 3.1. A general prcng model We collec e sae varables n e vecor Y =[q,δc,u,π ]. As sown n Appendx B, e dynamcs of Y descrbed n Eq. (13) represen a smple, frs-order vecor auoregressve process: Y = μ + AY 1 + ðσ F F 1 + Σ H Þε F = ð ϕ + ΦY Þ I; ð14þ were Y s e sae vecor of leng k, μ and ϕ are parameer vecors also of leng k and A, Σ F, Σ H and Φ are parameer marces of sze (k k). ε N(0,I), I s e deny marx of dmenson k, denoes e non-negavy operaor for a vecor, 5 and denoes e Hadamard Produc. 6 Also, le e real prcng kernel be represened by: m +1 = μ m + Γ my + Σ mff + Σ mh ε +1 were μ m s a scalar and Γ m, Σ mf, and Σ mh are k-vecors of parameers. We requre e followng resrcons: Σ F F Σ H =0 Σ mff Σ mh =0 Σ H F Σ mf =0 Σ F F Σ mh =0 ϕ + ΦY 0 ð15þ Te man purpose of ese resrcons s o exclude ceran mxures of square-roo and Vascek processes n e sae varables and prcng kernel a lead o an nracable soluon for some asses. We can now combne e specfcaon for Y and m +1 o prce fnancal asses. Te deals of e dervaons are presened n Appendx B. I s mporan o noe a, due o e dscree-me naure of e model, ese soluons are only approxmae n e even a e las resrcon n Eq. (15) s volaed. If ese varables are forced o reflec a zero, our use of e condonal lognormaly feaures of e sae varables becomes ncorrec. I s for exacly s reason a n e specfcaon of q n Eq. (4), we model f drecly and bound from below us nsurng a suc nsances are suffcenly rare. Le us begn by dervng e prcng of e nomnal erm srucure of neres raes. Le e me prce for a defaul-free zerocoupon bond w maury n be denoed by P n,. Usng e nomnal prcng kernel, e value of P n, mus sasfy: P n; = E exp ˆm +1 Pn 1;+1 ; ð16þ were ˆm +1 = m +1 π +1 s e log of e nomnal prcng kernel as argued before. Le p n, =ln(p n, ). Te n-perod bond yeld s denoed by y n,, were y n, = p n, /n. Te soluon o e value of p n, s presened n e followng proposon, e proof of wc appears n Appendx B. Proposon 1. Te log of e me prce of a zero coupon bond w maury n, p n, can be wren as: p n; = a 0 n + a ny ð17þ were e scalar a 0 n and (k 1) vecor a n are defned recursvely by e equaons, a 0 n = a 0 n 1 + ða n 1 e π Þ μ + μ m + 1 ϕ Σ Fða 2 n 1 e π Þ Σ Fða n 1 e π Þ + 1 ð 2 a n 1 e π Þ Σ H Σ Hða n 1 e π Þ + 1 ð 2 Σ mf Σ mf Þ ϕ ϕ + Σ H Σ mh Φ Σ Fða n 1 e π Þ Σ Fða n 1 e π Þ Σ mhσ mh σ2 m + ða n 1 e π Þ Σ mf Σ F a n = ða n 1 e π Þ A + Γ m ð 2 Σ mf Σ mf Þ Φ + a n 1 e π ð Þ Σ mf Σ F Φ ð18þ were e π s a vecor suc a π =e π Y and a 0 0 =0 and a 0 = e π. 5 p Specfcally, f v s a k-vecor, en v = ffffffff w were w =max(v,0) for =1,,k. 6 Te Hadamard Produc operaor denoes elemen-by-elemen mulplcaon. We defne formally n Appendx B. A useful mplcaon of e Hadamard Produc s a f ϕ +ΦY 0, for all elemens, en F F =(ϕ +ΦY ) I.

7 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Noce a e log prces of all zero-coupon bonds (as well as er yelds) ake e form of affne funcons of e sae varables. Gven e srucure of Y, e erm srucure wll represen a dscree-me muldmensonal mxure of e Vascek and CIR models. 7 Te process for e one-perod sor rae process, r y 1,, s erefore smply (a 0 1 +a 1 Y ). b b Le R n, +1 and r n, +1 denoe e nomnal smple ne reurn and log reurn, respecvely, on an n-perod zero coupon bond beween daes and +1. Terefore: R b n; +1 = exp a 0 n 1 a 0 n + a n 1 Y +1 a ny 1; r b n;+1 = a 0 n 1 a 0 n + a n 1 Y +1 a ny : ð19þ We now use e prcng model o value equy. Le V denoe e real value of equy, wc s a clam on e sream of real dvdends, D. Usng e real prcng kernel, V mus sasfy e equaon: V = E exp m +1 D+1 + V +1 : ð20þ Usng recursve subsuon, e prce dvdend rao (wc s e same n real or nomnal erms), PD, can be wren as: ( " PD = V n # ) = E D exp m + j + Δd + j ; ð21þ j =1 were we mpose e ransversaly condon, lm n E " # exp m +j V+n =0: n j =1 In e followng proposon, we demonsrae a e equy prce dvdend rao can be wren as e (nfne) sum of exponenals of an affne funcon of e sae varables. Te proof appears n Appendx B. Proposon 2. Te equy prce dvdend rao, PD, can be wren as: PD = exp b 0 n + b ny ð22þ were e scalar b n 0 and (k 1) vecor b n are defned recursvely by e equaons, b 0 n = b 0 n 1 + ðb n 1 + e d1 Þ μ + μ m + 1 ϕ Σ Fðb 2 n 1 + e d1 Þ Σ Fðb n 1 + e d1 Þ + 1 ð 2 b n 1 + e d1 Þ Σ H Σ Hðb n 1 + e d1 Þ + 1 ð 2 Σ mf Σ mf Þ ϕ ϕ + Σ H Σ mh Φ Σ Fðb n 1 + e d1 Þ Σ Fðb n 1 + e d1 Þ Σ mhσ mh σ2 m + ðb n 1 + e d1 Þ Σ mf Σ F b n = e d2 + ðb n 1 + e d1 Þ A + Γ m ð 2 Σ mf Σ mf Þ Φ + b n 1 + e d1 ð Þ Σ mf Σ F Φ ð23þ were e d1 and e d2 are selecon vecors suc a Δd = e d1 Y + e d2 Y 1. s s Le R +1 and r +1 denoe e nomnal smple ne reurn and log reurn, respecvely, on equy beween daes and +1. Terefore: 0 R e +1 = exp π +1 + Δd exp b 0 n + b ny exp A 1 ð24þ b0 n + b ny 0 r e exp b 0 n + b 1 ny = π +1 + Δd +1 + ln@ exp A: b0 n + b ny Te only nuon mmedaely apparen from comparng Eqs. (18) and (23) s a e coeffcen recursons look dencal excep for e presence of e vecor e π n e bond equaons and e d1 and e d2 n e equy equaons. Because e π selecs nflaon from e sae varables, s presence accouns for e nomnal value of e bond's cas flows w nflaon depressng e bond 7 For an analyss of connuous me affne erm srucure models, see Da and Sngleon (2000).

8 874 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) prce. Because e d1 and e d2 selec dvdend grow from e sae varables, er presence reflecs e fac a equy s essenally a consol w real, socasc coupons Te rsk free rae and e erm srucure To oban some nuon abou e erm srucure, we sar by calculang e log of e nverse of e condonal expecaon of e (gross) prcng kernel, fndng, r real = lnðβþ + γ μ c μ q + γρ cc Δ c + γρ cu u + γ 1 ρ q 1 2 γ2 σ 2 cc + σ2 qq 2σ cc σ qq λ q : ð25þ Hence, e real neres rae follows a ree-facor model w wo observed facors (consumpon grow and e consumpon dvdend rao) and one unobserved facor a preference sock. I s useful o compare s o a sandard verson of e Lucas economy wn wc Mera and Presco (1985) documened e so-called low rsk-free rae puzzle. Te real rsk free rae n e sandard Mera Presco economy s gven by r real;m P 1 = lnðβþ + γe Δc +1 2 γ2 V Δc +1 : ð26þ Te frs erm represens e mpac of e dscoun facor. Te second erm represens a consumpon-smoong effec. Snce n a growng economy agens w concave uly (γn0) ws o smoo er consumpon sream, ey would lke o borrow and consume now. Ts desre s greaer, e larger s γ. Tus, snce s ypcally necessary n Mera Presco economes o allow for large γ o generae a g equy premum, ere wll also be a resulng real rae a s ger an emprcally observed. Te rd erm s e sandard precauonary savngs effec. Uncerany nduces agens o save, erefore depressng neres raes and mgang e consumpon-smoong effec. Because aggregae consumpon grow exbs que low volaly, e laer erm s ypcally of second-order mporance. Te real rae n e Moody Invesor economy, r real, equals e real rae n e Mera Presco economy, plus wo addonal erms: r real = r real;m P + γ 1 ρ q q μ q 1 2 γ2 σqq 2σ 2 cc σ qq λ q ð27þ Te frs of e exra erms represens an addonal consumpon-smoong effec. In s economy, rsk averson s also affeced by q, and no only γ. Wen q s above s uncondonal mean, μ q /(1 ρ qq ), e consumpon-smoong effec s exacerbaed. Te second of e exra erms represens an addonal precauonary savngs effec. Te uncerany n socasc rsk averson as o be edged as well, depressng neres raes. Taken ogeer, ese addonal erms provde suffcen cannels for s economy o mgae, n eory, e rsk-free rae puzzle. In e daa, we measure nomnal neres raes. Te nomnal rsk free neres rae n s economy smply follows from, exp r f = E exp m +1 π +1 : ð28þ Because of e assumpons regardng e nflaon process, e model yelds an approxmae verson of e Fser equaon, were e approxmaon becomes more exac e lower e nflaon volaly 8 : r f = r real + μ π + ρ π π 1 2 σ2 π: ð29þ Te nomnal sor rae s equal o e sum of e real sor rae and expeced nflaon, mnus a consan erm (σ π 2 /2) due o Jensen's nequaly. Because of e neuraly of nflaon, e model mus generae an upward slopng erm srucure, a salen feaure of erm srucure daa, roug e real erm srucure. To oban some smple nuon abou e deermnans of e erm spread, we nvesgae a wo perod real bond. For s bond, e erm spread can be wren as: r f ;2 r f = 1 2 E r f +1 r f Cov m +1 ; r f Var r f +1 ð30þ Te erm n e mddle deermnes e erm premum, ogeer w e rd erm, wc s a Jensen's nequaly erm. Te full model mples a que complex expresson for e uncondonal erm premum a canno be sgned. Under some smplfyng 8 Te expeced gross ex-pos real reurn on a nomnal one-perod conrac, E [exp(r f π +1 )] wll be exacly equal o e gross ex-ane real rae, exp(r real ).

9 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) assumpons, we can develop some nuon. Frs, we proceed under e assumpon a e Jensen's nequaly erm s second order and can be gnored. Hence, we focus on e mddle erm. In general, we can wre: Cov m +1 ; r f +1 = v 0 + v 1 q ð31þ Te me-varaon n e erm premum s enrely drven by socasc rsk averson. Furer assume a ere s lle movemen n e condonal mean of consumpon grow (ρ cc =ρ cu =0). In s case, v 0 =0 and v 1 = γθ σqq σ 2 qq σ cc λ ð32þ 1 w θ = γ 1 ρ q 2 γ2 σcc 2 + σ2 qq 2σ cσ q λ. Assumng λ s negave, e nerpreaon s sragforward. Te parameer θ measures weer e precauonary savngs or consumpon smoong effec domnaes n e deermnaon of neres raes. Wacer (2006) also generalzes e Campbell Cocrane seng o a wo-facor model w one parameer governng e domnance of eer one of ese effecs. If θn0, e consumpon smoong effec domnaes and ncreases n q ncrease sor raes. We see a s wll also ncrease e erm premum and gve rse uncondonally o an upward slopng yeld curve: bonds are rsky n suc a world. In conras, wen θb0, e precauonary savngs effec domnaes. Increases n q now lower sor raes, drvng up e prces of bonds. Consequenly, bonds are good edges agans movemens n q and do no requre a posve rsk premum Equy prcng In order o develop some nuon on e sock prcng equaon n Eq. (23), we spl up e b n vecor no s four componens. Frs, e componen correspondng o nflaon, denoed b π n, s zero because nflaon s neural n our model. Second, e coeffcen mulplyng curren consumpon grow s gven by b c n = b c n 1 +1 ρcc + b u n 1 1 ρuc γρ cc ð33þ Consumpon grow affecs eques roug cas-flow and dscoun rae cannels. In our model, dvdend grow equals consumpon grow mnus e cange n e consumpon dvdend rao. Because dvdends are e cas flows of e equy sares, an ncrease n expeced dvdends sould rase e prce dvdend rao. Consumpon grow poenally forecass dvdend grow roug wo cannels fuure consumpon grow and e fuure consumpon dvdend rao. Ts s refleced c u n e erms, (b n 1 +1)ρ cc and (b n 1 1)ρ uc. Addonally, consumpon grow may forecas self and because s an elemen of e prcng kernel, s nduces a dscoun rae effec. For example, f consumpon grow s posvely auocorrelaed, an ncrease n consumpon lowers expeced fuure margnal uly. Te resulng ncreased dscoun rae depresses e prce dvdend rao. Ts effec s represened by e erm, γρ cc. In a sandard Lucas-ype model were consumpon equals dvdends and consumpon grow s e only sae varable, ese are e only wo effecs affecng sock prces. Because ey end o be counervalng effecs, s dffcul o generae muc varably n prce dvdend raos n suc a model. Te consumpon dvdend rao effec on equy valuaon s smlar. Te effec of e consumpon dvdend rao, b u n s gven by, b u n = b c n 1 +1 ρcu + b u n 1 1 ρuu +1 γρ cu ð34þ Te frs ree erms represen e effecs of e consumpon dvdend rao forecasng dvdends cas flow effecs and e four erm arses because e consumpon dvdend rao may forecas consumpon grow, leadng o a dscoun rae effec. Fnally, e prce dvdend rao s affeced by canges n rsk averson, q. Te effec of q on e prce dvdend rao s very complex: b q n = b q n 1 ρ qq + γ ρ qq γ2 σ qq λ σ cc + 2 γ2 σ 2 qq 1 λ σ cc bc n σuc b u n σqq λb q n 1 2 σ 2 qq bq 2 n 1 1 λ + γ σ qq λ σ cc b c n 1 +1 σcc + b u n 1 1 σuc + b q n 1 σ qqλ + γ σ qq b q 2 2 n 1 1 λ ð35þ I s empng o nk a ncreases n rsk averson unambguously depress prce dvdend raos, bu s s no necessarly rue because q affecs e prce dvdend raos roug many cannels. Te frs erm on e rg and sde of Eq. (35) arses only due o perssence n q. Te second lne of Eq. (35) summarzes e effec of q on real neres raes. Te frs of ese

10 876 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) erms capures e nuon a f rsk averson (low surplus consumpon) s g oday, s expeced o be lower n e fuure. Ts nduces a move for nvesors o borrow agans fuure beer mes, so neres raes mus ncrease n equlbrum o dscourage s borrowng, nducng a fall n e prces of long lved asses. Te second and rd erms on e second lne of Eq. (35) are precauonary savngs effecs. Hg q mples g uncerany, wc serves o lower raes and rase prces of long lved asses. Te rd lne of Eq. (35) s comprsed of Jensen's nequaly erms, n effec reflecng an addonal precauonary savngs effec for asses w rsky cas flows. Hg q rases e volaly of e dvdend sream, and n a log-normal framework, s ncreases valuaons. Te four lne of Eq. (35) s e mos neresng because capures e effec of e rskness of e dvdend sream on valuaons, or more precsely, e effec of q on a rskness. To clean up e algebra, le us consder e drec mpac of q (a s, excludng e b n 1 erms). Ten e las lne of Eq. (35) reduces o γ σ qq λ σ cc ðσ cc σ uc Þ ð36þ Assumng a λb0, e second erm s negave. Now, f dvdend grow s procyclcal, covaryng posvely w consumpon grow, en (σ cc σ uc )N0 and e overall expresson n Eq. (36) s negave. Hence, n mes of g rsk averson and g marke volaly (g q ), equy valuaons fall. Ts effec makes equy rsky and nduces a posve rsk premum. Any negave covarance beween e kernel and sock reurns can conrbue o a posve equy premum, ncludng a posve correlaon beween consumpon grow and dvdend grow. 4. Esmaon and esng procedure 4.1. Esmaon sraegy Our economy as four sae varables, wc we collec n e vecor Y. Excep for q, we can measure ese varables from e daa wou error, w u beng exraced from consumpon dvdend rao daa. We are neresed n e mplcaons of e model for fve endogenous varables: e sor rae, r f, e erm spread, spd, e dvdend yeld, dp, 9 e log excess equy reurn, r ex, and e log excess bond reurn, r bx. For all ese varables we use raer sandard daa, comparable o wa s used n e classc sudes of Campbell and Sller (1988) and Sller and Belra (1992). Terefore, we descrbe e exracon of ese varables ou of e daa and e daa sources n a daa appendx (Appendx A). We collec all e measurable varables of neres,. e ree observable sae varables and e fve endogenous varables n e vecor Z. Ta s, Z = Δc ; u ; π ; r f ; spd ; dp ; r ex ; r bx Also, we le Ψ denoe e srucural parameers of e model: Ψ = μ c ; μ π ; μ u ; μ q ; δ; ρ cc ; ρ uc ; ρ cu ; ρ uu ; ρ ππ ; ρ qq ; σ cc ; σ uc ; σ uu ; σ ππ ; σ qq ; λ; β; γ ð37þ Trougou e esmaon, we requre a Ψ sasfes e condons of Eq. (15). Tere are a oal of 19 parameers. If we resrc ourselves o e erm srucure, e fac a e relaon beween endogenous varables and sae varables s affne grealy smplfes e esmaon of e parameers. As s apparen from Eqs. (22) and (24), e relaonsp beween e dvdend yeld and excess equy reurns and e sae varables s non-lnear. In e Appendx C, we lnearze s relaonsp and sow a e approxmaon s very accurae. Noe a s approac s very dfferen from e popular Campbell and Sller (1988) and Campbell (1990) lnearzaon meod, wc lnearzes e reurn expresson self before akng e lnearzed reurn equaon roug a presen value model. We frs fnd e correc soluon for e prce dvdend rao and lnearze e resulng expresson. Appendx C demonsraes a e dfferences beween e analyc and approxmae momens do no affec our resuls. Condonal on e lnearzaon, e followng propery of Z obans, Z = μ z + Γ z Y 1 + ΣFF z 1 + Σ z H ε ð38þ were e coeffcens superscrped w z are nonlnear funcons of e model parameers, Ψ. We esmae e model n a wosep GMM procedure ulzng seleced condonal momens and exracng e laen sae vecor usng e lnear Kalman fler. We frs descrbe e flerng process and en e calculaon of e condonal GMM resduals and objecve funcon. Te nex subsecon descrbes e specfc momens and GMM wegng marx employed. To fler e sae vecor, we represen e model n sae-space form usng Eq. (14) as e sae equaon and appendng Eq. (38) w measuremen error for e observaon equaon, Z = μ z + Γ z Y 1 + ΣFF z 1 + Σ z H ε + Dv ð39þ 9 Ts dvdend yeld s no compued as e log of e dvdend prce rao as n Campbell and Cocrane (1999), bu as e log of 1 plus e annual percenage dvdend yeld.

11 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) were v s an ndependen sandard normal measuremen error nnovaon, and D s a dagonal marx w e sandard devaon of e measuremen errors along e dagonal. I s necessary o nroduce measuremen error because e dmensonaly of e observaon equaon s greaer an a of e sae equaon (a s, e model as a socasc sngulary). To avod esmang e measuremen error varances and o keep em small, we smply fx e dagonal elemens of D suc a e varance of e measuremen error s equal o one percen of e uncondonal sample varance for eac varable. Togeer, e sae and measuremen equaons may be used o exrac e sae vecor n e usual fason usng e sandard lnear Kalman fler (see Harvey, 1989). Gven condonal (flered) esmaes for Y, denoed, Ŷ, s sragforward o calculae condonal momens of Z +1 usng Eq. (39), E Z +1 = μ z + Γ z Ŷ VAR Z +1 = Σ z ˆF F + Σ z H Σ z ˆF F + Σ z H + DD ð40þ were ˆF s defned analogously o Eq. (14). Resduals are defned for eac varable as v z =Z E 1 [Z ] Momen condons, sarng values and wegng marx We use a oal of 30 momen condons o esmae e model parameers. Tey can be ordered no several groups. v z ½1Š for Z = Δd ; Δc ; π ; r f ; dp ; spd ; r ex ; r bx ð8þ v z Z 1 for Z = Δd ; Δc ; π ; r f ; u ð5þ v z 2 ½1Š for Z = Δd ; Δc ; π ; r f ; dp ; spd ; r ex ; r bx ; u ð9þ v z 1 v z 2 for Z 1 = ½Δd ; Δc Š; Z 2 = r ex ; r bx ð4þ v Δc v Δd ; v Δc v u ; v Δc u 1 ; v u Δc 1 ð4þ ð41þ Te frs lne of Eq. (41) essenally capures e uncondonal mean of e endogenous varables. In s group only e mean of e spread and e excess bond reurn are momens a could no be nvesgaed n e orgnal Campbell Cocrane framework. We also explcly requre e model o mac e mean equy premum. Te second group uses lags of e endogenous varables as nsrumens o capure condonal mean dynamcs for e fundamenal seres and e sor rae. Ts explcly requres e model o address predcably (or lack ereof) of consumpon and dvdend grow. Te rd se of momens s ncluded so a e model maces e volaly of e endogenous varables. Ts ncludes e volaly of bo e dvdend yeld and excess equy reurns, so a e esmaon ncorporaes e excess volaly puzzle and adds o a e volaly of e erm spread and bond reurns. Inuvely, s may be a ard rade-off (see, for nsance Bekaer (1996)). To mac e volaly of equy reurns and prce dvdend raos, volale neremporal margnal raes of subsuon are necessary, bu neres raes are relavely smoo and bond reurns are muc less varable an equy reurns n e daa. Ineres raes are funcons of expeced margnal raes of subsuon and er varably mus no be excessvely g o yeld realsc predcons. Te four group capures e covarance beween fundamenals and reurns. Tese momens confron e model drecly w e Cocrane and Hansen (1992) puzzle. Fnally, e ff se s ncluded so a e model may mac e condonal dynamcs beween consumpon and dvdends. Because n Campbell and Cocrane (1999) consumpon and dvdends concde, macng e condonal dynamcs of ese wo varables s an mporan deparure and exenson. Ts se of GMM resduals forms e bass of our model esmaon. To opmally weg ese orogonaly condons and provde e mnmzaon roune w good sarng values, we employ a prelmnary esmaon wc yelds a conssen esmae of Ψ, denoed Ψ 1. Te prelmnary esmaon uses only uncenered, uncondonal momens of Z and does no requre flerng of e laen sae varables or a parameer dependen wegng marx. Deals of e frs sage esmaon are relegaed o Appendx D. Gven Ψ 1, e resduals n Eq. (41) are calculaed, and er jon specral densy a frequency zero s calculaed usng e Newey and Wes (1987) procedure. Te nverse of s marx s used as e opmal GMM wegng marx n e man esmaon sage. Noe a ere are 13 over-denfyng resrcons and a we can use e sandard J-es o assess e f of e model Tess of addonal momens If e model can f e base momens, would be a raer successful sock and bond prcng model. Nevereless, we wan o use our framework o fully explore e mplcaons of a model w socasc rsk averson for e jon dynamcs of bond and sock reurns, parally also o gude fuure researc. In Secon 6, we consder a se of addonal momen resrcons a we

12 878 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) would lke o es. In parcular, we are neresed n ow well e model fs e bond sock reurn correlaon and reurn predcably. To es conformy of e esmaed model w momens no explcly f n e esmaon sage, we consruc a GMM-based es sasc a akes no accoun e samplng error n esmang e parameers, Ψ. Appendx C descrbes e exac compuaon. 5. Esmaon resuls Ts secon examnes resuls from model esmaon and mplcaons for observable varables under e model Parameers Table 2 repors e parameer esmaes for e model. Te frs column repors mean parameers. Te negave esmae for δ ensures a average consumpon grow s lower an average dvdend grow, as s rue n e daa. Imporanly, neer μ q nor μ u are esmaed, bu fxed a uny and zero respecvely. Ts s necessary for denfcaon of e model and reduces e number of esmaed parameers o 17. Because rsk averson under s model s proporonal o exp(q ), e uncondonal mean and volaly of q are dffcul o jonly denfy under e lognormal specfcaon of e model. Resrcng μ q o be uny does no sgnfcanly reduce e flexbly of e model. Te second column repors feedback coeffcens. Consumpon grow sows modes seral perssence as s rue n e daa. Te consumpon dvdend rao s que perssen, and ere s some evdence of sgnfcan feedback beween (pas) consumpon grow and e fuure consumpon dvdend rao. Bo nflaon and socasc rsk averson, q, are very perssen processes. Table 2 Esmaon of e Moody Invesor model. Means Feedback Volales Preferences E[Δc] (0.0017) E[π] (0.0076) δ (0.0046) J-sa(13) (pval) (0.0291) Te esmaed model s defned by ρ cc (0.0926) ρ cu (0.0052) ρ uc (0.9969) ρ uu (0.0653) ρ ππ (0.0235) ρ qq (0.0290) pffffffff q +1 = μ q + ρ qq q + σ qq q 1 λ 2 1 = 2ε q +1 + λεc +1 pffffffff Δc +1 = μ c + ρ cc Δc + ρ cu u + σ cc q ε c +1 pffffffff u +1 = μ u + ρ uu u + ρ uc Δc + σ uc q ε c +1 + σ uu ε u +1 σ cc (0.0018) σ uc (0.0136) σ uu (0.0122) σ ππ (0.0026) σ qq (0.1525) λ (0.1182) ln(β) (0.0119) γ (0.4093) Δd +1 = Δc +1 δ Δu +1 π +1 = μ π + ρ π π + σ π ε π +1 m +1 = lnðβþ γδc +1 + γδq +1 Te momens f are (30 oal) v z ðz E 1 ½Z ŠÞ v z ½1Š forz = Δd ; Δc ; π ; r f ; dp ; spd ; r ex ; r bx ð8þ v z Z 1 forz = Δd ; Δc ; π ; r f ; u ð5þ v z 2 ½1Š forz = Δd ; Δc ; π ; r f ; dp ; spd ; rex ; r bx ; u v z 1 v z 2 for Z 1 = ½Δd ; Δc Š; Z 2 = r ex ; r bx ð4þ v Δc v Δd ; v Δc v u ; v Δc u 1 ð9þ ; v u ; Δc 1 ð4þ GMM sandard errors are n pareneses. See es for a dscusson of e esmaon procedure. Noe a e uncondonal means of u and q are fxed a zero and one respecvely. Tere are a oal of 17 esmaed parameers. Daa are annual from (74 years). See e daa appendx for addonal daa consrucon noes.

13 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Te volaly parameers are repored n e rd column. Consumpon grow s negavely correlaed w e consumpon dvdend rao, bu e coeffcen s only sgnfcanly dfferen from zero a abou e 10% level. Te condonal correlaon beween consumpon grow and q s 0.19, and s value s sascally dfferen from zero a abou e 10% level. Fnally, we repor e dscoun facor β and e curvaure parameer of e uly funcon, γ. Because rsk averson s equal o γq, and e economy s growng, ese coeffcens are dffcul o nerpre by emselves. We also repor e es of e over-denfyng resrcons. Tere are 17 parameers and 30 momen condons, makng e J- es a χ 2 (13) under e null. Te es fals o rejec a e 1% level of sgnfcance, bu rejecs a e 5% level. Te fac a all - sascs are over 1.00 suggess a e daa conan enoug nformaon o denfy e parameers Impled momens Here, we assess wc momens e model fs well and wc momens fals o f perfecly. Table 3 sows a large array of frs and second momens regardng fundamenals (dvdend grow, consumpon grow and nflaon), and endogenous varables (e rsk free rae, e dvdend yeld, e erm spread, excess equy reurns and excess bond reurns). We sow e means, volales, frs-order auocorrelaon and e full correlaon marx. Numbers n pareneses are GMM based sandard errors for e sample momens. Numbers n brackes are populaon momens for e model (usng e log-lnear approxmaon for e prce dvdend rao descrbed prevously for dp and r ex ). In our dscusson, we nformally compare sample w populaon momens usng e daa sandard errors as a gude o assess goodness of f. Ts of course gnores e samplng uncerany n e parameer esmaes Te equy premum and rsk free rae puzzles Table 3 ndcaes a our model mples an excess reurn premum of 5.2% on equy, wc maces e daa momen of 5.9% que well. Te man mecansm nducng a g equy premum s e posve covarance beween dvdend yelds and rsk averson (wc we dscuss furer laer). Sandard power uly models ypcally do so a e cos of exorbanly g-rsk free Table 3 Impled momens for Moody Invesor model. Δd Δc π r f dp spd r ex Mean [0.037] [0.032] [0.039] [0.057] [0.033] [0.004] [0.052] [0.011] (0.016) (0.003) (0.006) (0.005) (0.002) (0.002) (0.024) (0.009) Sd.dev. [0.084] [0.016] [0.036] [0.033] [0.012] [0.014] [0.172] [0.096] (0.022) (0.005) (0.007) (0.004) (0.002) (0.001) (0.022) (0.007) Auo. corr. [0.196] [0.413] [0.881] [0.868] [0.879] [0.826] [ 0.057] [ 0.001] (0.164) (0.185) (0.160) (0.105) (0.286) (0.106) (0.162) (0.164) Correlaons r bx Δd Δc π r f dp spd r ex r bx Δc [0.53] 0.59 (0.14) π [0.00] [0.00] (0.21) (0.19) f r [0.10] [0.17] [0.96] (0.14) (0.14) (0.15) dp [0.03] [ 0.02] [0.00] [0.22] (0.14) (0.15) (0.23) (0.11) spd [ 0.20] [ 0.37] [ 0.91] [ 0.90] [0.16] (0.14) (0.13) (0.14) (0.10) (0.11) ex r [0.19] [0.30] [0.04] [0.05] [ 0.21] [ 0.18] (0.13) (0.17) (0.22) (0.10) (0.14) (0.09) bx r [ 0.06] [ 0.07] [ 0.43] [ 0.44] [ 0.04] [0.41] [0.28] (0.08) (0.10) (0.11) (0.13) (0.10) (0.07) (0.09) Te numbers n square brackes are smulaed momens of e Moody Invesor model. Usng e pon esmaes from Table 2, e sysem was smulaed for 100,000 perods. Dvdend yeld and excess equy reurn smulaed momens are based upon e log-lnear approxmaon descrbed n e ex. Te second number n eac enry s e sample momen based on e annual daase ( ) and e rd number n pareneses s a GMM sandard error for e sample momen (one Newey Wes lag). Asersks denoe sample momens more an wo sandard errors away from e model mpled value. See daa appendx for addonal daa consrucon noes.

14 880 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Table 4 Varance decomposon under e Moody Invesor model. r f Δc u π q Δc 1 u 1 π 1 q dp spd ex r bx r Under e Moody Invesor model, e varable n eac row can be expressed as a lnear combnaon of e curren and lagged sae vecor. Generally, for e row varables, x = μ + Γ Y c were Y c s e companon form of Y (a s Y c sacks Y, Y 1 ). Based on μ and Γ, e proporon of e varaon of eac row varable arbued o e k elemen of e sae vecor s calculaed as Γ vy c Γ ðkþ Γ vy ð c ÞΓ were Γ (k) s a column vecor w e k elemen equal o ose of Γ and zero elsewere. Essenally, e numeraor compues e covarance of Y w e sae varable. raes, a penomenon called e rsk free rae puzzle (Wel, 1989). Our neres rae process does ave a mean a s oo g by 1.6 percenage pons, bu we also generae an average excess bond reurn of 1.1%, wc s very close o e 1.0% daa mean Excess volaly Sock reurns are no excessvely volale from e perspecve of our model. Wle e sandard devaon of excess reurns n e daa s 19.7%, we generae excess reurn volaly of 17.2%. Wa makes s especally surprsng s a e model slgly undersoos e volaly of e fundamenals. Ta s, aloug ey are wn e wo sandard error bound around e sample esmae, e volales of dvdend grow and consumpon grow are bo lower an ey are n e daa. To nevereless generae subsanal equy reurn volaly, e neremporal margnal rae of subsuon mus be raer volale n our model, and a ofen as e mplcaon of makng bond reurns excessvely volale (see, for example, Bekaer, 1996). Ts also does no appen n e model, wc generaes an excess bond reurn volaly of 9.6% versus 8.1% n e daa. Sor rae volaly s acually wn 10 bass pons of e sample volaly. Table 4 elps nerpre ese resuls. I provdes varance decomposons for a number of endogenous varables n erms of curren and lagged realzaons of e four sae varables. Te sae varables are elemens of Y,defned prevously. Abou 5.5% of e varaon n excess sock reurns s explaned by consumpon grow and e consumpon dvdend rao. Te bulk of e varance of reurns (over 90%) s explaned by socasc rsk averson. In Campbell and Cocrane (1999), s proporon s 100% because consumpon and dvdend grow are modeled as..d. processes. Wereas e Campbell and Cocrane (1999) model feaured a non-socasc erm srucure, 10 we are able o generae muc varably n bond reurns smply usng a socasc nflaon process, wc accouns for 80% of e varaon. Te remander s prmarly due o socasc rsk averson, and only abou 10% s due o consumpon grow or e consumpon dvdend rao. Ts s conssen w e lack of a srong relaonsp beween bond reurns and ese varables n e daa. 11 Te excess volaly puzzle ofen refers o e nably of presen value models o generae varable prce dvdend raos or dvdend yelds (see Campbell and Sller, 1988; Cocrane, 1992). In models w consan excess dscoun raes, prce dvdend raos mus eer predc fuure dvdend grow or fuure neres raes and s unlkely a predcable dvdend grow or neres raes can fully accoun for e varaon of dvdend yelds (see Ang and Bekaer, 2007; Leau and Ludvgson, 2005 for recen arcles on s opc). Table 3 sows a our model maces e varance of dvdend yelds o wn 20 bass pons. Table 4 sows a e bulk of s varaon comes from socasc rsk averson and no from cas flows Term srucure dynamcs One of e man goals of s arcle s o develop an economy a maces salen feaures of equy reurns as n Campbell and Cocrane, wle nroducng a socasc bu racable erm srucure model. Table 3 repors ow well e model performs w respec o e sor rae and e erm spread. Te volales of bo are maced near perfecly. Te model also reproduces a perssen sor rae process, w an auocorrelaon coeffcen of 0.87 (versus 0.90 n e daa). Addonally, e model mpled erm spread s a b more perssen, a 0.83, an e daa value of Te erm srucure model n s economy s affne and varaon n yelds s drven by four facors: consumpon grow, e consumpon dvdend rao, nflaon and socasc rsk averson. Table 4 sows ow muc of e varaon of e sor rae and 10 An earler unpublsed verson of Campbell and Cocrane (1999) relaxes s condon. 11 In s daa sample, a regresson of excess bond reurns on conemporaneous and lagged consumpon grow and e consumpon dvdend rao yelds an r-squared sasc of A regresson run on smulaed daa under e model repored n Table 2 yelds an r-squared of 0.01.

15 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Table 5 Properes of q and rsk averson under e Moody Invesor model. RA percenle 1% 5% 10% 25% 50% 75% 90% 95% 99% q correlaon Δd Δc π r f dp spd r ex Ts able presens smulaed rsk averson momens under e Moody Invesor model and correlaons w q and observable varables. Te sysem was smulaed for 100,000 perods usng e daa generang process and pon esmaes from Table 2. Rsk averson s calculaed as RA = γexpðq Þ r bx e erm spread eac of ese facors explans. Ineresngly, nflaon socks drve abou 92% of e oal varaon of e sor rae, bu only 83% of e varaon n e erm spread. Ts s of course no surprsng snce e frs-order effec of expeced nflaon socks s o ncrease neres raes along e enre yeld curve. Because ere s no nflaon rsk premum n s model, e spread acually reacs negavely o a posve nflaon sock, as nflaon s a mean reverng process. Wereas consumpon grow explans 3% of e varaon n sor raes, drves 13% of e varaon n e erm spread. Ts s naural as consumpon grow s less perssen an e man drver of e sor rae, nflaon, makng s relave weg for erm spreads (wc depend on expeced canges n neres raes) larger (see Eq. (25)). Table 3 sows a n our economy negave consumpon or dvdend grow socks (recessons) are assocaed w lower nomnal sor raes and ger spreads. Suc pro-cyclcal neres raes and couner-cyclcal spreads are conssen w convenonal wsdom abou neres raes, bu e effecs as measured relave o annual dvdend and consumpon grow are no very srong n e curren daa sample. 12 For example, e uncondonal correlaon beween dvdend grow and neres raes s only slgly negave ( 0.01) and ndsngusable from zero n e sample daa, and e correlaon beween aggregae consumpon grow and neres raes s slgly posve (0.01). In e model, bo of ese correlaons are posve. Te model generaed correlaon beween e sor rae and dvdend grow s 0.10, well wn one sandard devaon of e sample sasc, and e model generaed correlaon beween e sor rae and consumpon grow s 0.17, jus more an one sandard devaon above e sample esmae. Te correlaon beween e erm spread and bo consumpon and dvdend grow s negave as n e daa, bu e magnudes are a b oo large. Were e model as some rouble s n fng e correlaon beween nflaon and e erm srucure. Nevereless, e varaon of e erm spread accouned for by nflaon s almos dencal o varaon esmaed by Ang e al. (2008), wereas e conrbuon of nflaon o e sor rae varance seems slgly oo large. To elp us nerpre ese fndngs, we deermne e coeffcens n e affne relaon of e (nomnal) sor rae w e facors. We fnd a e sor rae reacs posvely o ncreases n rsk averson ndcang a w respec o e q-sock e consumpon smoong effec domnaes (see Eq. (30)). Addonally, e sor rae reacs posvely o consumpon grow. Te erm spread reacs less srongly o preference socks and also reacs negavely o an ncrease n consumpon grow. Te fac a e consumpon smoong effec of q domnaes s precauonary savngs effec nduces a posve correlaon beween e kernel and neres raes, makng bonds rsky. Terefore e model generaes a posve erm spread and bond rsk premum Lnk beween fundamenals and asse reurns Equlbrum models ypcally mply a consumpon grow and sock reurns are gly correlaed. In our model, several cannels break e g lnk beween sock reurns and consumpon grow presen n sandard models. Frs, we model equy correcly as a clam o dvdends, no o consumpon. Ts elps reduce e correlaon beween sock reurns and consumpon grow, bu generaes anoer puzzle. For equy o earn a rsk premum, s reurns mus be correlaed w e prcng kernel, and dvdend grow and consumpon grow are reporedly no gly correlaed (see Campbell and Cocrane, 1999). Our second mecansm o break e g lnk beween consumpon grow and sock reurns comes n play ere as well: socasc rsk averson s e man drver of e varably of e prcng kernel. Table 3 sows a e f of our model w respec o e lnks beween fundamenals and asse reurns s penomenal. Frs, aggregae consumpon grow and dvdend grow ave a realsc 0.5 correlaon n our model. Second, we generae a correlaon beween dvdend and consumpon grow and excess equy reurns of 0.19 and 0.30 respecvely, wc s no sgnfcanly above e correlaon n e daa (wc s respecvely 0.09 for dvdend grow and 0.19 for consumpon grow). Trd, n e daa, bond reurns and dvdend and consumpon grow are negavely correlaed bu e correlaon s small. We generae small correlaons n e model, macng e sgn n bo cases and closely approxmaely e magnudes Tme-varyng rsk appees Socasc rsk averson n our model equals γq. Because s unobserved, we caracerze s properes roug smulaon n Table 5. Medan rsk averson equals 2.06 and s nerquarle range s [1.35, 4.08]. Rsk averson s posvely skewed and e 90 percenle observaon equals I exceeds 100 n less an 1% of draws. However, rsk averson s occasonally exremely g 12 Ang e al. (2008) fnd a ese noons beer apply o real raes and fnd evdence a real-raes are ndeed pro-cyclcal.

16 882 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) n s economy as e process for γq as a very long al. In fac, we verfed a e mean does no exs. Ts s analogous o a feaure of e model of Campbell and Cocrane (1999), wo sress a e dsrbuon of e surplus rao rased o e power of γ s non-saonary. Te boom panel of Table 5 repors e correlaon of q w all of our endogenous and exogenous varables. As expeced, e varable s counercyclcal, sowng a negave correlaon w bo dvdend and consumpon grow, bu somewa weakly so. Wen rsk averson s g, dvdend yelds ncrease (a s, prce dvdend raos decrease) makng e dvdend yeld-rsk averson correlaon posve. From Table 4, we already know a q s e sole drver of me-varaon n rsk and expeced reurn n s model, drvng up expeced reurns on bo socks and bonds n mes of g-rsk averson. Terefore, perods of g rsk averson are caracerzed by negave realzaons of unexpeced reurns as well as ncreased posve expeced reurns. Te ne uncondonal correlaon beween rsk averson and reurns s ndeermnae and ends o be small. Te op wo panels of Fg. 1 plo e condonally flered values for e laen varable, ˆq, and local rsk averson, ˆRA = ˆγ exp ˆq. Te model denfes e ges rsk averson followng e Grea Depresson n e 1930s, w values brefly exceedng 50. Wle rsk averson generally decreased aferwards, remaned relavely g roug e 1950s. Rsk averson was low durng mos of e 1960s and 1970s, bu ramped up n e early 1980s. Te sock marke boom of e 1980s and 1990s was accompaned by a sgnfcan declne n rsk averson. 6. Te jon dynamcs of bond and sock reurns Te economy we ave creaed so far manages o mac more salen feaures of e daa an e orgnal Campbell and Cocrane (1999) arcle n an essenally lnear framework. Nevereless, e man goal of s arcle s o asceran ow many of e salen feaures of e jon dynamcs of bond and sock reurns can capure. In s secon, we frs analyze e q RA Corr[rex,rbx] Fg. 1. Flered condonal momens. Ts fgure plos varous flered seres under e Moody Invesor model and e pon esmaes from Table 2. Eac of e ploed seres s a funcon of q alone, condonal on e model parameers. Te frs wo frames plo e flered values for e laen sae varable q and rsk averson, RA =γexp(q ). Te remanng frame plos e model mpled condonal correlaon of excess sock and bonds reurns (blue) and 15-year-rollngwndow realzed correlaons beween sock and bond reurns (red, crcles).

17 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) comovemens of bond and sock reurns and en look a e predcably of bond and sock reurns. Before we do, noe a Table 3 reveals a e f of e model w respec o bond and sock marke reurns (las wo columns and las wo rows) s mpressve. Of 19 momens, only wo model-mpled momens are ousde of a wo-sandard error range around e sample momen Te bond sock reurns correlaon Sller and Belra (1992) sow a n a presen value framework w consan rsk premums, e correlaon beween bond and sock reurns s oo low relave o e correlaon n e daa. Nevereless, Table 3 sows a e correlaon beween bond and sock reurns durng e sample s only 0.15 w a relavely large sandard error. In our model, expeced excess bond and sock reurns bo depend negavely on socasc rsk averson and s common source of varaon nduces addonal correlaon beween bond and sock reurns. We generae a correlaon of 0.28, wc s less an wo sandard errors above e sample momen. Table 6, Panel A, presens a formal es o see ow well e model fs e condonal covarance beween sock and bond reurns, ncorporang bo samplng error and parameer uncerany (see Appendx C). Te es fals o rejec. A condonal es usng e sor rae, dvdend yeld and spread as nsrumens, also does no rejec e model's predcons. Te boom panel of Fg. 1 plos e model mpled condonal correlaon beween sock and bond reurns over e sample perod (sold blue) along w sample values of e correlaon over a backward-lookng rollng wndow of wd 15 years. Te low-frequency dynamcs of e wo seres are que smlar, fallng smulaneously n e lae 1960s and agan n e lae 1990s. Te reason we oversoo e correlaon on average as o do w e effec of q on asse prces. Wle equy prces decrease wen q ncreases, e effec on bond reurns s ambguous because e effec of q on neres raes s ambguous. Emprcally, we ave sown a q ncreases neres raes and ence lowers bond reurns. Terefore, q provdes a cannel for ger correlaon, bo beween expeced and unexpeced bond and sock reurns. Clearly, s s one dmenson a could be a useful yardsck for fuure models. Da (2003) argues a e bond marke requres a separae facor a does no affec sock prces. We beleve a mg be more fruful o nk abou poenal socasc componens n cas flows a are no relevan for bond prcng and a our smple dvdend grow model may ave mssed. Anoer fruful avenue for exendng e model s o nvesgae e dynamcs of nflaon more closely. Campbell and Ammer (1993) emprcally decompose bond and sock reurn movemens no varous componens and fnd nflaon socks o be an mporan source of negave correlaon beween bond and sock reurns. Table 6 Tess of addonal momens. Panel A: Sock bond covarance orogonaly ess Momen(s) p-val u rex u bex C 1 [r ex,r bx ] (0.34) u rex u bex C 1 r ex ; r bx z 1 (0.77) Panel B: Reurn condonal mean orogonaly ess Momen(s) p-val Momen(s) p-val Jon p-val u rex f r 1 (0.36) u rbx f r 1 (0.71) (0.65) u rex dp 1 (0.71) u rbx dp 1 (0.25) (0.47) u rex spd 1 (0.56) u rbx spd 1 (0.36) (0.65) u rex z 1 (0.25) u rbx z 1 (0.48) (0.77) Panel C: Reurn predcably unvarae slope coeffcens r ex bx r 1 f r 1 f dp 1 spd 1 (dp 1 r 1 ) [0.2365] [3.2598] [0.7057] [0.1889] (0.5992) (1.4016) (1.4603) (0.4587) [0.0927] [1.1975] [0.2332] [0.0634] (0.2496) (0.5828) (0.5846) (0.2107) Ts able repors e covarance and predcably performance of e Moody Invesor model. In bo panels, u x denoes (x E 1 [x ]) were x s an observable varable. Te condonal expecaon s a mpled by e model. z 1 = ½r f 1 ; dp 1; spd 1 Š In Panel A, we es momens correspondng o e uncondonal and condonal covarance beween sock and bond reurns. C 1 [r ex,r bx ] denoes e condonal covarance mpled by e model. In panel B, we es momens capurng e condonal mean of sock and bond excess reurns: e condonal rsk premums. Te columns labeled p-val repor GMM based orogonaly ess of e correspondng momen(s) condon. Te fnal column repors a jon es for e momens across e rows. Daa are annual from , (74 years). See e daa appendx for addonal daa consrucon noes. In Panel C, we presen slope coeffcens from smulaed unvarae reurn predcably regressons usng varous nsrumens under e model (op number n cell w square brackes) and e sample coeffcens n e daa (second and rd numbers are daa slope coeffcens w sandard errors n pareneses).

18 884 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Bond and sock reurn predcably Table 6, Panels B and C repor on e conssency of e model w e predcably of reurns n e daa. We run unvarae regressons of excess bond and sock reurns usng four nsrumens: e rsk free rae, e dvdend yeld, e yeld spread, and e excess dvdend yeld (e dvdend yeld mnus e neres rae). A long ls of arcles as demonsraed e predcve power of ese nsrumens for excess equy reurns. However, a more recen leraure cass doub on e predcve power of e dvdend yeld, wle confrmng srong predcably for equy reurns usng e neres rae or erm spread as a predcor, a leas n pos daa, see for example Ang and Bekaer (2007) and Campbell and Yogo (2006). Panel B demonsraes a n our annual daa se, e only sgnfcan predcor of equy reurns s e yeld spread. Te - sascs n e sor rae and excess dvdend yeld regresson are above 1.00 bu do no yeld a 5% rejecon. Wen we nvesgae bond reurns, we also fnd e yeld spread o be e only sgnfcan predcor. Ts predcably reflecs e well-known devaons from e Expecaons Hypoess (see Campbell and Sller, 1991; Bekaer e al. 2001). A ger yeld spread predcs g expeced excess reurns on bo socks and bonds. Te predcably coeffcens mpled by our model are repored n square brackes. Of course, excep for e yeld spread regresson, ese ess ave lle power and are no useful o nvesgae. Wa s neresng s o ceck weer e model ges e sgns rg. Te one mss ere s e negave sgn of e sor rae coeffcen n e reurn regressons. Ts puzzle, more prevalen w pos-treasury accord daa and known snce Fama and Scwer (1977), can poenally be resolved n our model, because e equy premum ncreases wen rsk averson (q ) ncreases, wereas e sor rae can ncrease or decrease w ger rsk averson dependng on weer e consumpon smoong or precauonary savngs effec domnaes. Wacer (2006) nvesgaes a wo-facor exenson of Campbell and Cocrane (1999) w exacly s purpose. Because a our esmaed values, an ncrease n q ncreases e sor rae, we generae a posve correlaon beween curren neres raes and e equy premum. A full nvesgaon of s puzzle requres a more serous nvesgaon of nflaon dynamcs, because e emprcal relaonsp may be due o e expeced nflaon componen n nomnal neres raes, raer an e real sor rae componen. W respec o e predcve power of e yeld spread, e model does reasonably well. I generaes subsanal posve predcably coeffcens for bo sock and bond reurns and also maces e fac a e coeffcen s larger for e equy an for e bond reurn regresson. In a recen paper, Burasc and Jlsov (2007) also fnd a an exernal ab model elps f devaons of e Expecaons Hypoess. In Panel C, we presen alernave ess compung e model mpled nnovaons o e reurn seres and esng weer ey are orogonal o observable nsrumens. Te p-values for ese ess sow a falure o rejec n eac and every case. Tese ess presen furer evdence a our model s conssen w e dynamcs of expeced reurns. One srong mplcaon of e model s a e predcable componens n e excess reurns of socks and bonds are perfecly correlaed because of e dependence on q. In e daa, s would be e case f e yeld spread was really e only rue predcor. To nvesgae ow realsc s mplcaon of e model s, we projec e excess reurns n e daa ono e neres rae, e yeld spread and e dvdend yeld and compue e correlaon of e wo fed values. We fnd s correlaon o be 0.81 w a sandard error of Ts suggess a e assumpon of perfec correlaon beween expeced excess reurns on bonds and socks s a raer accurae approxmaon of e ru. 7. Concluson In s arcle, we ave presened a prcng model for socks and bonds were poenally couner-cyclcal preference socks generae me-varaon n rsk premums. Te model can be nerpreed as a racable verson of e exernal ab model of Campbell and Cocrane (1999) accommodang a fully socasc erm srucure. Our fundamenals nclude bo consumpon (wc eners e uly funcon) and dvdends (wc s e relevan cas flow process), wc are assumed o be conegraed processes. A GMM esmaon reveals a e model s rejeced a e 5% level, bu sll fs a large number of salen feaures of e daa, ncludng e level and varably of neres raes, bond and sock reurns, erm spreads and dvdend yelds. Te model also maces e correlaon beween fundamenals (consumpon and dvdend grow) and asse reurns. We furer examne e f of e model w respec o bond and sock reurn dynamcs, fndng a produces a somewa oo g correlaon beween sock and bond reurns bu maces e fac a e erm spread sgnals g rsk premums on bo. Te model also does no generae a negave relaon beween e equy premum and sor raes, aloug could eorecally do so. Ts relaonsp deserves furer scruny n a model were e nflaon process ges more aenon. Our arcle s par of a growng leraure a explores e effecs of socasc rsk averson on asse prce dynamcs. A number of arcles ave sayed farly close o e Campbell and Cocrane framework and emprcally focused manly on e erm srucure and devaons of e Expecaons Hypoess. Tese arcles nclude Wacer (2006), Brand and Wang (2003), wo model rsk averson as a funcon of unexpeced nflaon, and Da (2003), wo consrucs a model nesng nernal and exernal ab. Oer auors ave explored alernave preference specfcaons were rsk averson vares roug me. Tese nclude e regmeswcng rsk averson model of Gordon and S Amour (2000, 2004) and e preference sock model of Leau and Wacer (2007) (wo focus on explanng e value premum). Leau and Wacer sress a s mporan a ere s no correlaon beween preference socks n er model and fundamenals, and a an exernal ab model mposng a perfecly negave correlaon would no work. Te sreng of our framework s a we reman ed o fundamenals bu relax e perfec negave

19 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) correlaon assumpon. Fnally, Bekaer e al. (2009) ry o dsenangle rsk averson and economc uncerany, wc ey ake as e eeroskedascy of fundamenals. Our researc reveals a fuure modelng effors mus searc for facors a drve a sronger wedge beween bond and sock prcng. Possble canddaes are more nrcae modelng of e nflaon process and a cas flow componen uncorrelaed w e dscoun rae. Acknowledgemens We ank Pero Verones, Jon Campbell, e parcpans a e NBER Summer Insue 2004, wo anonymous referees, and e edor, Franz Palm for e useful commens. All remanng errors are e sole responsbly of e auors. Ts work does no necessarly reflec e vews of e Federal Reserve Sysem or s saff. Appendx A. Daa appendx In s appendx, we ls all e varables used n e arcle and descrbe ow ey were compued from e orgnal daa sources. 1. r ex. To calculae excess equy reurns, we sar w e CRSP dsaggregaed monly sock fle, and defne monly US aggregae equy reurns as: RET m N prc ; 1 srou ; 1 = re ; MCAP 1 m ð43þ MCAP m N = prc ; srou ; were e unverse of socks ncludes ose lsed on e AMEX, NASDAQ or NYSE, re, s e monly oal reurn o equy for a frm, prc, s e closng monly prce of e sock, and srou, are e number of sares ousandng a e end of e mon for sock. We creae annual end-of-year observaons by summng ln(1+ret m ) over e course of eac year. Excess reurns are en defned as: r ex lnð1+ret Þ r f 1 ð44þ were e rsk free rae, r f,sdefned n wa follows. Noe a e lagged rsk free rae s appled o mac e perod over wc e wo reurns are earned (r f s daed wen eners e nformaon se). 2. r bx. Excess long bond reurns are defned as, r bx lnð1+ltbr Þ r f 1 ð45þ were LTBR s e annually measured long erm governmen bond oldng perod reurn from e Ibboson Assocaes SBBI yearbook. 3. Δd. Log real dvdend grow s defned as: Δ d lnðdiv DIV = 12 mon =1 Þ lnðdiv 1 Þ π N re ; rex ;! ð46þ prc ; 1 srou ; 1 were re,, rex,, prc, and srou, are oal reurn, oal reurn excludng dvdends, prce per sare and number of common sares ousandng for all ssues raded on e AMEX, NASDAQ and AMEX as repored n e CRSP monly sock fles. π, nflaon, s defned as follows. 4. spd. Te yeld spread s defned as: spd lnð1+ltby Þ r f ð47þ were LTBY s e annually measured long erm governmen bond yeld as repored by Ibboson Assocaes n e SBBI yearbook. 5. π. Log nflaon s defned as: π lnð1+infl Þ ð48þ

20 886 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) were INFL s e annually measured gross rae of cange n e consumer prce ndex as repored by Ibboson Assocaes n e SBBI yearbook. 6. Consumpon. C. Toal real aggregae consumpon s calculaed as oal consan dollar non-durable plus servces consumpon as repored n e NIPA ables avalable from e webse of e US Bureau of Economc Analyss. As descrbed n Secon 2, we cecked e robusness of our resuls o e use of alernave consumpon measures a more closely approxmae e consumpon of sockolders. C LX denoes real luxury consumpon, defned as e sum of ree dsaggregaed consan-dollar NIPA consumpon seres: boas and arcraf, (C BA ), jewelry and waces, (C JW ) and foregn ravel, (C FT ). C WT, parcpaon weged consumpon, s defned as follows C WT = PART C AG C FT + C LX ð49þ Te seres sould more accuraely reflec e consumpon baske of sock marke parcpans. Te ger e sock marke parcpaon rae, PART, e more relevan s aggregae (non-luxury) consumpon. C FT s subraced from C AG o avod doublecounng (e oer elemens of luxury consumpon are classfed as durables, and us no ncluded n oal nondurable and servce consumpons, wc comprse C AG ). PART s e US sock marke parcpaon rae aken from daa provded by Seve Zeldes (see Amerks and Zeldes (2004)): e percen of US ouseolds w drec or ndrec ownersp of socks: Year Rae : : : : :570 ð50þ From ese daa, an nerpolaed parcpaon rae, PART was calculaed by e auors as ð1 + exp½ 62:0531 ð + 0:03118 YEAR ÞŠÞ 1, e resul of esmang a deermnsc rend lne roug e numbers n Eq. (50). Tofll n consumpon daa pror o NIPA coverage, (nclusve), we appled (n real erms) e grow rae of real consumpon repored a e webse of Rober Sller for ose years. Te real log grow raes of all consumpon seres are calculaed as: Δc = lnðc Þ lnðc 1 Þ π ð51þ Noe a e same nflaon seres, defned prevously, s appled o deflae all ree consumpon measures. 7. dp. Te dvdend yeld measure used n s paper s: dp ln 1+ DIV MCAP MCAP = MCAP m ;DEC ð52þ m were DIV s defned prevously and MCAP, DEC 8. r f. Te sor erm rsk free rae s defned as: corresponds o e December value of MCAP m for eac year. r f lnð1+stby Þ ð53þ were STBY s e sor erm governmen bond yeld repored by e S. Lous federal reserve sascal release webse (FRED). From s monly seres, we ook December values o creae annual end of year observaons. Noe a r f s daed wen eners e nformaon se, e end of e perod pror o a over wc e reurn s earned. For nsance, e rsk free rae earned from January 1979 o December 1979 s daed as (end-of-year) Appendx B. Te general prcng model Here we collec proofs of all e prcng proposons. For compleeness, we repor e general prcng model equaons. We begn by defnng e Hadamard Produc, denoed by. Te use of s operaor s solely for ease of noaonal complexy. Defnon. Suppose A=(a j ) and B=(b j ) are eac N N marces. Ten A B=C, were C=(c j )=(a j b j )sann N marx. Smlarly, suppose a=(a )sann-dmensonal column vecor and B=(b j )sann N marx. Ten a B=C, were C=(c j )= (a b j )sann N marx. Agan, suppose a=(a j )sann-dmensonal row vecor and B=(b j )sann N marx. Ten a B=C, were C=(c j )=(a j b j )sann N marx. Fnally, suppose a=(a ), and b=(b ) are N 1 vecors. Ten a b=c, were C=(c )= (a b )sann 1 vecor.

21 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) B.1. Defnon of e sysem Te sae vecor s descrbed by, Y = μ + AY 1 + ðσ F F 1 + Σ H Þε F = ðϕ + ΦY Þ I; ð54þ were Y s e sae vecor of leng k, μ and ϕ are parameer vecors also of leng k and A, Σ F, Σ H and Φ are parameer marces of sze (k k). ε s a k-vecor of zero mean..n. nnovaons. Te log of e real socasc dscoun facor s modeled as, m r +1 = μ m + Γ my + Σ mff + Σ mh ε +1 were μ m, s scalar and Γ m, Σ mf, and Σ mh are k-vecors of parameers. Te followng resrcons are requred: Σ F F Σ H =0 Σ mff Σ mh =0 Σ H F Σ mf =0 Σ F F Σ mh =0 ϕ + ΦY 0 ð55þ Tese resrcons are convenen for e calculaon of condonal expecaons of funcons of Y. B.2. Some useful lemmas Lemma 1. Te condonal expecaon of an exponenal affne funcon of e sae varables and e prcng kernel s gven by: E exp a + c 1 Y +1 + d Y + m r +1 = exp g 0 + g Y ð56þ Proof. By lognormaly, E exp a + c Y +1 + m r +1 = exp a + c E Y +1 + E m r V 2 c Y +1 + V m r +1 +2C c Y +1 ; m r +1 ð57þ We wll ake eac of e fve condonal expecaons prevously descrbed separaely. In e followng, (e Hadamard produc) denoes elemen-by-elemen mulplcaon. c E Y +1 = c ð μ + AY Þ ð58þ E m r +1 = μm + Γ m Y ð59þ VAR c Y +1 = c ð ΣF F + Σ H ÞðΣ F F + Σ H Þ c = c Σ F F F Σ F c + c Σ H Σ H c ð60þ = Σ F c ΣF c ðϕ + ΦY Þ + c Σ H Σ H c ð61þ were e second lne uses resrcons n Eq. (55) and e rd lne follows from properes of e operaor. Smlar calculaons lead o e resuls, VAR m r +1 = ð ΣmF Σ mf Þ ðϕ + ΦY Þ + Σ mh Σ mh COV c Y +1 ; m r +1 = c ΣmF Σ F ð ϕ + ΦY Þ + Σ H Σ mh ð62þ

22 888 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Subsung, 0 a + c ðμ + AY Þ + μ m + Γ m Y + 1 Σ 2 F c ΣF 1 c ðϕ + ΦY Þ E exp a + c Y +1 + m r +1 = exp + 1 c Σ 2 HΣ H c + 1 ð Σ 2 mf Σ mf Þ ðϕ + ΦY Þ + 1 Σ 2 mhσ mh + 1 σ 2 B 2 m A + c Σ mf Σ F ðϕ + ΦY Þ + Σ H Σ mh ð63þ Evdenly, g 0 = a + c μ + μ m Σ 1 F c ΣF c ϕ + 2 c Σ H Σ H c + 1 ð 2 Σ mf Σ mfþ ϕ Σ mhσ mh σ2 m + c Σ mf Σ F ϕ + ΣH Σ mh g = d + c A + Γ m Σ 1 F c ΣF c Φ + 2 Σ ð mf Σ mf Þ Φ + c Σ mf Σ F Φ ð64þ B.3. Represenaon of e nomnal rsk free rae I s well known a e gross rsk free rae s gven by e nverse of e condonal expecaon of e nomnal prcng kernel. Le e π denoe e k-vecor wc selecs log nflaon from e sae vecor. Ten, exp r f = E exp m r 1 +1 e π Y +1 ð65þ Applyng Lemma 1 w a=0 and c= e π and d=0, s mmedae a r f = a 0 1 a 1 Y ð66þ were a 0 and a 1 are gven by e Lemma 1. B.4. Represenaon of e full erm srucure Te proof o demonsrae e affne form for e erm srucure s accomplsed by nducon. Recall a e nomnal one perod rsk free rae s gven by, r f = a 0 1 a 1 Y ð67þ p 1; = a a 1 Y were a 0 1 and a 1 are gven n e prevous subsecon. Recall also e recursve relaon of dscoun bond prces n e socasc dscoun facor represenaon, P n; = E M+1P N n 1;+1 : ð68þ Suppose, for e purposes of nducon, a P n 1, can be expressed as, P n 1; = exp a 0 n 1 + a n 1 Y : ð69þ Ten, leadng s expresson by one perod and subsung no e recursve relaon, Eq. (68), we ave, P n; = E exp m r +1 e π Y +1 + a 0 n 1 + a n 1 Y +1 ð70þ = exp a 0 n + a n Y were e coeffcens are gven (recursvely) by Lemma 1. Upon subsuon, e followng recurson s revealed a 0 n = a0 n 1 + ð a n 1 e πþ μ + μ m Σ F ða n 1 e π Þ ΣF 1 ða n 1 e π Þ ϕ + ð 2 a n 1 e π Þ Σ H Σ H ða n 1 e π Þ + 1 ð 2 Σ mf Σ mf Þ ϕ Σ mh Σ mh σ2 m + ða n 1 e π Þ Σ mf Σ F ϕ + ΣH Σ mh an = ða n 1 e π ÞA + Γ m Σ F ða n 1 e π Þ ΣF 1 ða n 1 e π Þ Φ + 2 Σ ð mf Σ mf Þ Φ + ða n 1 e π Þ Σ mf Σ F Φ ð71þ

23 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) B.5. Represenaon of e equy prces To demonsrae e dependence of e prce dvdend rao on Y, we use a proof by nducon. Le e d1 and e d2 be e wo selecon vecors suc a Δd = e d1 Y + e d2 Y 1. Te prce dvdend rao s gven by P D = E exp n j =1! m r + j + Δd + j = E exp n m r + j + e d1 Y + j + e d2 j =1 Y + j 1! = q 0 n; were q 0 n; E exp n j =1 mr + j + e d1 Y + j + e d2 Y + j 1 are scalars. We wll prove a q 0 n, =exp(b 0 n +b n Y ) were b 0 n (scalar) and b n (k-vecors) are defned as follows. Consder q 0 1, : q 0 1; = E exp m r + j + e d1 Y + j + e d2 Y + j 1 ð72þ ð73þ By Lemma 1, q 0 1; = exp b b 1Y ð74þ were b and b 1 are gven by Lemma 1. Nex, suppose a q n 1, 0 =exp(b n 1 +b n 1 Y ). Ten rearrange q 0 n, as follows. q 0 n; = E exp n j =1! m r + j + e d1 Y + j + e d2y + j 1 ( = E exp m r +1 + e d1 Y +1 + e d2 Y!) n 1 E+1 exp m r + j +1 + e d1 Y + j +1 + e d2 Y + j j =1 n = E exp m r +1 + e d1y +1 +e d2 Y o q 0 n 1;+1 n o = E exp b 0 n 1 + m r +1 + ðe d + b n 1 Þ Y +1 + e d2 Y = exp b 0 n + b n Y ð75þ were b n 0 and b n are easly calculaed usng Lemma 1. Upon subsuon, e recursons are gven by, b 0 n = b 0 n 1 + ðb n 1 + e d1 Þ μ + μ m Σ F ðb n 1 + e d1 Þ ΣF 1 ðb n 1 + e d1 Þ ϕ + ð 2 b n 1 + e d1 Þ Σ H Σ H ðb n 1 + e d1 Þ ð76þ + 1 ð 2 Σ mf Σ mf Þ ϕ Σ mhσ mh σ2 m + ðb n 1 + e d1 Þ Σ mf Σ F ϕ + Σ H Σ mh b n = e d2 + ðb n 1 + e d1 Þ A + Γ m + 1 Σ Fð 2 b n 1 + e d1þ Σ Φ Fð b n 1 + e 1 d1þ + ð 2 Σ mf Σ mfþ Φ + ðb n 1 + e d1 Þ Σ mf Σ F Φ For e purposes of esmaon e coeffcen sequences are calculaed ou 200 years. If e resulng calculaed value for PD as no converged, en e sequences are exended anoer 100 years unl eer e PD value converges, or becomes greaer an 1000 n magnude. Appendx C. Te esmaed model Te sae varables and dynamcs for e esmaed model are gven by pffffffff q +1 = μ q + ρ qq q + σ qq q 1 λ 2 1 = 2ε q +1 + λεc +1 pffffffff c Δc +1 = μ c + ρ cc Δc + ρ cu u + σ cc q ε +1 pffffffff c u +1 = μ u + ρ uu u + ρ uc Δc + σ uc q ε +1 + σ uu ε u +1 ð77þ Δd +1 = Δc +1 δ Δu +1 π +1 = μ π + ρ π π + σ π ε π +1 m +1 = lnðβþ γδc +1 + γ q +1 q ð78þ

24 890 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) Ts model s clearly a specal case of e model descrbed n Appendx B. Te mpled sysem marces are, μ q ρ qq μ μ = 6 c 7 4 μ u 5 ; A = 0 ρ cc ρ cu ρ uc ρ uu 0 5 ; Σ H = σ cc μ π ρ π σ ππ 1 λ 2 1 = 2σqq λσ qq Σ F = 6 0 σ cc ; ϕ = : ; Φ = ð79þ rffffffffffffffffffffffffffffffffffffffffffffffffff were σ qq 1 μ f q 1 ρ 2 qq. Noe a n s formulaon, f q s e rao of e uncondonal mean o e uncondonal varance of q q. I s esmaed drecly. Te prcng kernel for e model can be wren as: m +1 = lnðβþ γδ c +1 + γ q +1 q 0 pffffffff μ c + ρ cc Δc + ρ cu u + σ cc q ε c = lnðβþ γb pffffffff μ q + ρ qq 1 q + σ qq q 1 λ 2 1 = 2ε q +1 + A λεc +1 ð80þ We can now read off e prcng kernel marces: μ m = lnðβþ γμ c + γμ q 2 1 λ 2 1 = 2σqq 3 Σ mf = γ σ cc + λσ 6 qq : 0 2 Γ m = γ6 4 ρ qq 1 ρ cc ρ cu Σ mh =0 ð81þ C.1. Alernae models Several oer models were explored durng s sudy. Frs, e alernae consumpon measures Δc lx and Δc w were red n place of aggregae consumpon. Te resuls for weged consumpon were nearly dencal o ose repored prevously. For luxury consumpon, e model faled o converge. Ts may be due o a lack of conegraon beween aggregae dvdends and e small componen of consumpon represened by e few luxury seres we denfed. Secondly, models weren consumpon grow and dvdend grow are no conegraed were consdered. Specfcally, aemps were made o esmae models of e form Δ d +1 = μ d + ρ d Δd + ρ dq q + σ d ε d pffffffff +1 + κ 1 q q ε +1 Δc +1 = μ c + ρ c Δc + ρ cq q + σ cd ε d +1 + σ c ε c pffffffff +1 + κ 2 q q ε +1 ð82þ π +1 = μ π + ρ π π + σ π ε π +1 q +1 = μ q + ρ q q + 1 rffffffffffffffffffffffffffffffffffffffffffffff μ f q 1 ρ 2 pffffffff q q m r +1 = lnðβþ γδc +1 + γδq +1 q ε +1 Esmaon was aemped for eac of e ree consumpon measures. However, none of ese models converged. Ts s almos ceranly due o e very dfferen prcng mplcaons of a non-saonary consumpon dvdend rao.

25 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) C.2. Log lnear approxmaon of equy prces In e esmaon, we use a lnear approxmaon o e prce dvdend rao. From Eq. (72), we see a e prce dvdend rao s gven by P D = q 0 n; = exp b 0 n + b n Y ð83þ and e coeffcen sequences, b 0 n and b n, are gven prevously. We seek o approxmae e log prce dvdend rao usng a frs order Taylor approxmaon of Y abou Y, e uncondonal mean of Y. Le q 0 n = exp b 0 n + b n Y ð84þ and noe a Y q 0 n; = q 0 n; = q 0 Y n; b n ð85þ Approxmang, pd ln q 0 1 n + q 0 n b n Y Y q0 n ð86þ = d 0 + d Y were d 0 and d are mplcly defned. Smlarly, gpd ln 1+ P ln 1+ D q 0 n q0 n q 0 n b n Y Y = 0 + Y ð87þ were 0 and are mplcly defned. Noe also a e dvdend yeld measure used n s sudy can be expressed as follows dp ln 1+ D = gpd P pd ð88þ so a s also lnear n e sae vecor under ese approxmaons. Also, log excess equy reurns can be represened follows. Usng e defnon of excess equy reurns, r x +1 = r f pd + gd +1 + π +1 + gpd +1 ð 0 d 0 Þ + e d + e π + Y+1 + e rf + d Y ð89þ = r 0 + r 1 Y +1 + r 2 Y were r 0, r 1 and r 2 are mplcly defned. C.3. Accuracy of e prce dvdend rao approxmaon To assess e accuracy of e log lnear approxmaon of e prce dvdend rao, e followng expermen was conduced. For e model and pon esmaes repored n Table 2, a smulaon was run for 10,000 perods. In eac perod, e exac prce dvdend rao and log dvdend yeld are calculaed n addon o er approxmae counerpars derved n e prevous subsecon. Te resulng seres for exac and approxmae dvdend yelds and excess sock reurns compare as follows: Appx dp Exac dp Appx r x Mean Sd. dev Correlaon Exac r x

26 892 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) C.4. Analyc momens of Y and Z Recall a e daa generang process for Y s gven by, Y = μ + AY 1 + ðσ F F 1 + Σ H Þε F = sqrðdagðϕ + ΦY ÞÞ ð90þ I s sragforward o sow a e uncenered frs, second, and frs auocovarance momens of Y are gven by, Y = ði k AÞ 1 μ vec Y Y = I k 2 A A 1 vec μμ + μy A + AY μ + Σ F F 2 Σ F + Σ H Σ H vec Y Y 1 = I k 2 A A 1 vec μμ + μy A + AY μ + A Σ F F 2 Σ F + Σ H Σ H ð91þ were overbars denoe uncondonal means and F 2 = dag ϕ + ΦY. Now consder e uncondonal momens of a n-vecor of observable varables Z wc obey e condon Z = μ w + Γ w Y 1 + Σ w F F 1 + Σ w H ε ð92þ were μ w s an n-vecor and Σ F w, Σ H w and Γ w are (n k) marces. I s sragforward o sow a e uncenered frs, second, and frs auocovarance momens of Z are gven by, Z = μ w + Γ w Y Z Z = μ w μ w + μ w Y Γ w + Γ w Y μ w + Γ w Y Y Γ w + Σ w F F 2 Σ w F + Σ w H Σ w H Z Z = μ w μ w + μ w Y Γ w + Γ w Y μ w + Γ w Y Y 1 Γw + Γ w Σ F F 2 Σw F + Σ H Σ w H ð93þ I remans o demonsrae a e observable seres used n esmaon obey Eq. (92). Ts s rvally rue for elemens of Z wc are also elemens of Y suc as Δd, Δc, π. Usng Eqs. (66), (89) and (86), s apparen a r f, dp and r x sasfy Eq. (92) as well. C.5. Tes of addonal momens To es conformy of e esmaed model w momens no explcly f n e esmaon sage, e a GMM based sasc s consruced a akes no accoun e samplng error n esmang e parameers, Ψ. Le g 2T (Ψ 0,X ) be e sample mean of e resrcons we ws o es. By e Mean Value Teorem, g 2T ˆΨ = a:s: g2t Ψ 0 ð Þ + D 2T ˆΨ Ψ 0 were D 2T = g 2TðΨÞ Ψ. Snce ˆΨ s esmaed from e frs se of orogonaly condons, a:s: ˆΨ Ψ 0 = ð A11 D 1T Þ 1 A 11 g 1T ðψ 0 Þ ð95þ w ð94þ D 1T = g 1TðΨ 0 Þ Ψ ð96þ A 11 = D 1T S 1 11 ð97þ Subsung, g 2T a:s: ˆΨ = LgT ðψ 0 Þ ð98þ were L = D 2T ða 11 D 1T Þ 1 A 11 ; I ð99þ g T ðψ 0 Þ = ½g 1T ðψ 0 Þ ; g 2T ðz ; Ψ 0 Þ Š ð100þ

27 G. Bekaer e al. / Journal of Emprcal Fnance 17 (2010) pffffff Snce T gt ðψ 0 Þ Nð0; SÞwere S s e specral densy a frequency zero of all e orogonaly condons, and S 11 s e op lef quadran of S, e sasc, Tg 2T ˆΨ LSL 1g2T ˆΨ ð101þ as a χ 2 (k) dsrbuon under e null, were k s e number of momens consdered n g 2T ˆΨ. Appendx D. Uncondonal parameer esmaon (frs sage) Collec all e measurable varables of neres, e ree observable sae varables and e fve endogenous varables n e vecor Z. Also, we le Ψ denoe e srucural parameers of e model: Ψ = μ c ; μ π ; μ u ; μ q ; δ; ρ cc ; ρ uc ; ρ cu ; ρ uu ; ρ ππ ; ρ qq ; σ cc ; σ uc ; σ uu ; σ ππ ; σ qq ; λ; β; γ ð102þ Applyng e log-lnear approxmaon of Appendx C.2, e followng propery of Z obans, Z = μ z + Γ z Y 1 + ΣFF z 1 + Σ z H ε ð103þ were e coeffcens superscrped w z are nonlnear funcons of e model parameers, Ψ. Because Y follows a lnear process w square-roo volaly dynamcs, uncondonal momens of Y are avalable analycally as funcons of e underlyng parameer vecor, Ψ. Le X(Z ) be a vecor valued funcon of Z. For e curren purpose, X( ) wll be comprsed of frs and second order monomals, uncondonal expecaons of wc are uncenered frs and second momens of Z. Usng Eq. (38), we can also derve e analyc soluons for uncenered momens of Z as funcons of Ψ. Specfcally, EX ½ ðz ÞŠ = f ðψþ ð104þ were f( ) s also a vecor valued funcon (appendces provde e exac formulae). Ts mmedaely suggess a smple GMM based esmaon sraegy. Te GMM momen condons are, g 1T ðz ; Ψ 0 Þ = 1 T T XZ ð Þ f ðψ 0 Þ: ð105þ =1 Moreover, e addve separably of daa and parameers n Eq. (105) suggess a fxed opmal GMM wegng marx free from any parcular parameer vecor and based on e daa alone. We denoe e daa used as X T = fx 1 ; X 2 ; X T g. Te opmal GMM wegng marx s e nverse of e specral densy a frequency zero of g 1T X T ; Ψ 0, wc we denoe as S 11 X T, because only e frs erm on e rg and sde of Eq. (105) conans any random varables (daa). Furer, o reduce e number of parameers mplcly esmaed n calculang e opmal GMM wegng marx wle sll accommodang g perssence n e orogonaly condons, we explo e srucure mpled by e model. In parcular, we compue e specral densy n wo seps. Frs, we consder e specral densy of Y c = Y P;, YP 1 were P Y s an observable proxy for e sae vecor (wc ncludes a laen varable, q ). In pracce, we use Y P = ½Δc ; u ; π ; rf Š w rf proxyng for q. Because Y P s que perssen, we use a sandard VAR(1) pre-wenng ecnque. Denoe e specral densy a frequency zero esmae of Y c as Ŝ 11 YT c. Second, we projec X T ono Y c. Le ˆB denoe e leas squares projecon coeffcens and ˆD e (dagonal) varance covarance marx of e resduals of s projecon. Ten, our esmae for Ŝ 11 ðz T Þ s Ŝ 11 X T = ˆB Ŝ 11 Y c T ˆB + ˆD ð106þ Te nverse of Ŝ 11 X T s e opmal wegng marx. To esmae e sysem, we mnmze e sandard GMM objecve funcon, J W T ; ˆΨ = g 1T ˆΨ Ŝ 11 X T 1 g1t ˆΨ ð107þ n a one sep opmal GMM procedure. Because s sysem s exremely non-lnear n e parameers, we ook precauonary measures o assure a a global mnmum as ndeed been found. Frs, over 100 sarng values for e parameer vecor are cosen a random from wn e parameer space. From eac of ese sarng values, we conduc prelmnary mnmzaons. We dscard e runs for wc esmaon fals o converge, for nsance, because e maxmum number of eraons s exceeded, bu rean converged parameer values as canddae esmaes. Nex, eac of ese canddae parameer esmaes s aken as a new sarng pon and mnmzaon s repeaed. Ts process s repeaed for several rounds unl a global mnmzer as been denfed as e parameer vecor yeldng e lowes value of e objecve funcon. In s process, e use of a fxed wegng marx s crcal. Indeed, n e presence of a parameer-dependen wegng marx, s searc process would no be well defned. Fnally, e parameer

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