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1 Journal of Moneary Economics 76 (205) Conens liss available a ScienceDirec Journal of Moneary Economics journal homepage: Asse pricing in producion economies wih exrapolaive expecaions David Hirshleifer a, Jun Li b, Jianfeng Yu c,d,n a Universiy of California, Irvine, Unied Saes b Universiy of Texas a Dallas, Unied Saes c Universiy of Minnesoa, 32 9h Avenue Souh, Suie 3-22, Carlson School of Managemen, Minneapolis, MN 55455, Unied Saes d The School of Managemen and Economics, CUHK Business School, The Chinese Universiy of Hong Kong, Shenzhen, China aricle info Aricle hisory: Received 20 November 203 Received in revised form 26 Augus 205 Acceped 27 Augus 205 Available online 8 Sepember 205 Keywords: Exrapolaion Producion-based model Long-run risk Recursive preferences absrac Inroducing exrapolaive bias ino a sandard producion-based model wih recursive preferences reconciles salien sylized facs abou business cycles (low consumpion volailiy, high invesmen volailiy relaive o oupu) and financial markes (high equiy premium, volaile sock reurns, low and smooh risk-free rae) wih plausible levels of risk aversion and ineremporal elasiciy of subsiuion. Furhermore, he model capures reurn predicabiliy based upon dividend yield, Q, and invesmen. Inuiively, exrapolaive bias increases he variaion in he wealh consumpion raio, which is heavily priced under recursive preferences; adjusmen coss decrease he covariance beween marginal uiliy and asse reurns. Empirical suppor for key implicaions of he model is also provided. & 205 Elsevier B.V. All righs reserved.. Inroducion During he millennial high ech boom, he U.S. economy grew rapidly, and expecaions among many invesors abou fuure growh were higher han subsequen realizaions. In conras, afer he credi crisis of 2008, growh has been low and pessimisic expecaions for fuure growh have been prevalen. This raises he quesions of wheher here is a general endency for individuals o overexrapolae recen economic growh, and if so wha effec his has on consumpion and asse pricing. Evidence from boh psychology and finance indicaes ha exrapolaive bias is pervasive in human judgemen and decisions (see, e.g., Gilovich e al., 985; Hirshleifer, 200; Barberis and Thaler, 2003; Fuser e al., 200). In laboraory experimens, Tversky and Kahneman (974) provide evidence consisen wih individuals following he represenaiveness heurisic, wherein observaions are perceived as being more indicaive (represenaive) of populaion disribuions han hey really are. This resuls in he so-called law of small numbers, a belief updaing process whereby individuals overweigh small numbers of observaions. In an invesmen seing, his would imply ha when invesors see a firm realizing high earnings growh, for example, hey may classify i as a growh firm and discoun inadequaely for he regression phenomenon. n Corresponding auhor a: Universiy of Minnesoa, 32 9h Avenue Souh, Suie 3-22, Carlson School of Managemen, Universiy of Minnesoa, Minneapolis, MN 55455, Unied Saes. Tel.: þ ; fax: þ address: jianfeng@umn.edu (J. Yu). hp://dx.doi.org/0.06/j.jmoneco /& 205 Elsevier B.V. All righs reserved.

2 88 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) Empirically, several field and experimenal sudies find ha he rading of individual and professional invesors seems o reflec exrapolaion of pas performance. Theoreical models and discussions have also emphasized how exrapolaion can affec capial marke behavior. In he model of Barberis e al. (998), he represenaiveness heurisic causes overreacion anomalies in he sock marke. Fuser e al. (200) sugges ha exrapolaion is imporan for undersanding macroeconomic flucuaions. Indeed, Barberis (20) proposes ha overexrapolaion may explain he 2008 credi crisis. As is well known, producion-based asse pricing models face an even greaer challenge han endowmen-based models in explaining consumpion and asse reurn behavior, as such models allow greaer scope for endogenous consumpion and dividend smoohing (see, e.g., Rouwenhors, 995; Jermann, 998; Leau and Uhlig, 2000; Boldrin e al., 200). 2 I has been recognized for some ime ha relaxing he assumpion of perfec raionaliy migh help explain macroeconomic and financial empirical puzzles. 3 Recenly, Fuser e al. (200, 20) argue ha quasi-raional models deserve greaer aenion. In his spiri, our paper inroduces exrapolaive expecaions ino a sandard dynamic sochasic general equilibrium (DSGE) model feauring recursive preferences o sudy he implicaions of he model for asse prices, consumpion, invesmen, and oupu. Specifically, he model assumes ha he rue average produciviy growh is unobservable, and ha he represenaive individual has o esimae i from hisorical daa. The individual uses a smoohed average of pas realized echnology growh o esimae fuure echnology growh, wih greaer weigh on he mos recen growh realizaion. Inroducing exrapolaive expecaions grealy improves upon radiional raional models in maching key sylized facs abou boh asse prices and macroeconomic quaniies. Specifically, our model produces large and volaile excess sock marke reurns and low and smooh risk-free raes, wih a relaive risk aversion (RRA) of four and a preference ineremporal elasiciy of subsiuion (IES) of wo. Moreover, he model can replicae he predicabiliy of excess marke reurns by he price dividend raio, Tobin's Q, and invesmen raes, consisen wih known evidence. Imporanly, exrapolaive expecaions also improves he model's abiliy o mach he relaive volailiies of invesmen growh. The inuiion for he high equiy premium in our model is sraighforward. Firs, exrapolaion of pas growh rends cause excess volailiy in produciviy growh expecaions. This increases he volailiy of invesmen, and since variaions in invesmen are driven by excessive volailiy in perceived produciviy, his resuls in excessive perceived volailiy of he consumpion growh rae. Second, owing o recursive preferences, his variaion in he perceived consumpion growh rae (i.e., he long-run risk) is heavily priced. Finally, wih reasonably high adjusmen coss, he limied flexibiliy of invesmen ends o direc he payoff from high realized produciviy growh more owards dividends and consumpion raher han invesmen. This reduces marginal uiliy of consumpion in he even of a favorable produciviy shock. Since he posiive produciviy shock is associaed wih a high asse reurn, higher adjusmen coss end o decrease he covariance beween reurn and marginal uiliy of consumpion, hereby increasing he equiy risk premium. Thus, aken ogeher, he combinaion of exrapolaion bias, adjusmen coss, and recursive preferences can generae a large equiy premium. All hree of adjusmen coss, exrapolaion and recursive preferences conribue o his oucome. Wihou capial adjusmen coss, consumpion is excessively smoohed, because afer high growh realizaions, firms would heavily reinves cash flows, reducing he payous and consumpion or making hem counercyclical; his would reduce equiy risk. Wihou exrapolaion bias, he perceived volailiy of invesmen and consumpion growh would be oo small o explain a high equiy premium. Finally, wihou recursive preferences, he ime-variaion in perceived consumpion growh is no priced, and hus he equiy premium is small. The inuiion for condiional reurn predicabiliy derives from exrapolaive expecaions and overreacion. By a sandard argumen, overreacion resuls in higher volailiy in he sock marke and he wealh consumpion raio, and predicabiliy of sock reurns by valuaion raios and invesmen raes. As observed by Barlevy (2004) and Lansing (202), a raional model wih capial adjusmen coss faces difficuly in generaing sufficien invesmen volailiy. Owing o capial adjusmen coss, invesmen growh in he raional model exhibis abou he same volailiy as oupu growh, whereas invesmen growh in he daa is abou wo imes more volaile han oupu growh. In our model, he excess volailiy of perceived produciviy growh causes excess volailiy of invesmen, hereby improving he fi wih invesmen volailiy in he daa. Lusig e al. (203) documen ha boh he wealh consumpion raio and he reurn on he consumpion claim are volaile, a challenge for radiional leading asse-pricing models. 4 Our paper shows ha exrapolaion can help produce high See, e.g., Smih e al. (988), Benarzi (200), Haruvy e al. (2007), Greenwood and Nagel (2009), and Choi e al. (200). In survey evidence, Case and Shiller (988) repor higher expecaions of fuure housing price growh in ciies ha have experienced pas recen price growh. Using boh survey and experimenal daa, de Bond (993) finds ha he forecass of individual invesors saisfy a simple rend-following mechanism. Vissing-Jorgensen (2003) provides survey evidence ha invesors who have experienced high porfolio reurns in he pas expec higher reurns in he fuure. Furhermore, Ederingon and Golubeva (200) find ha muual fund invesors reallocae oward sock funds afer sock price increases, and ino bond funds afer bond price increases. 2 Several endowmen-based asse-pricing models can successfully mach he firs wo momens of he excess sock marke reurn and he risk-free rae (e.g., Campbell and Cochrane, 999; Bansal and Yaron, 2004; Barro, 2006). However, he reconciliaion of asse markes wih aggregae quaniies has proved o be a challenge for DSGE models. 3 Early noable sudies include de Long e al. (990a,b) and Barsky and de Long (993). 4 In a recen paper, Ai (200) proposes a learning model in producion economy which can accoun for he dynamics of he wealh consumpion raio. However, Ai (200) does no address he condiional momens of he sock reurns or he quaniies.

3 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) volailiy for boh he wealh consumpion raio and he reurn on he aggregae wealh, again owing o excessive variaion in expecaions abou echnological growh. Two key refuable implicaions of he model are also invesigaed. Firs, owing o overexrapolaion, invesor expecaions of fuure produciviy growh is prediced o forecas fuure reurns negaively. Second, invesmen raes are prediced o be posiively conemporaneously correlaed wih perceived produciviy growh. Using several proxies for perceived produciviy growh, we provide empirical evidence supporing hese implicaions. Several previous sudies examine he effecs of exrapolaive expecaions. In a parial equilibrium model, Barsky and de Long (993) show ha exrapolaion in esimaing expeced dividend growh conribues o volailiy in price dividend raios. Cecchei e al. (2000), Choi (2006), Lansing (2006), and Bansal and Shaliasovich (200) sudy he effecs of recency bias in an exchange economy. 5 They show ha exrapolaive bias can help explain a high equiy premium and high sock marke volailiy. Fuser e al. (200) sudy he implicaions for macroeconomic flucuaions of naural expecaions (a weighed average of raional and exrapolaive expecaions) in an endowmen economy wih consan relaive risk aversion (CRRA) preferences. Barberis e al. (205) sudy a consumpion-based asse pricing model and find ha he sock price exrapolaive bias capures many feaures of acual prices and reurns, as well as he survey evidence on invesor expecaions. By embedding he his ime is differen learning bias (a bias akin o exrapolaion) and overlapping generaions ino oherwise sandard macro-finance models, Collin-Dufresne e al. (205b) show ha his bias can be a key deerminan of he join dynamics of macro-aggregaes and asse prices. Empirically, de Bond and Thaler (985), Poerba and Summers (988), Lakonishok e al. (994), and La Pora e al. (997) provide evidence of overreacions, which suggess ha exrapolaion can help explain sylized facs abou predicabiliy of aggregae marke reurns and he cross-secion of sock reurns (bu see also Daniel and Timan, 2006). Our paper differs from all of hese sudies in ha our paper sudies he effecs of exrapolaion on boh quaniies and prices simulaneously. Our producion-based model has implicaions for boh price-relaed variables such as he equiy premium and ineres raes, and for quaniy variables such as consumpion growh, oupu growh, and invesmens. The model can herefore be subjec o sringen calibraion o see wheher i can accuraely reflec he real-world daa on boh macroeconomic quaniies and capial marke prices simulaneously. As menioned earlier, i is more challenging o produce a large equiy premium when consumpion can be endogenously smoohed by firms invesmen decisions as in our model. Our approach builds on a growing lieraure on long-run risk, especially as applied o producion economies. Bansal and Yaron (2004) demonsrae ha in an endowmen economy wih long-run risk in consumpion and recursive preferences, consumpion and asse-price properies can be reconciled wih moderae risk aversion and an IES greaer han one. Our paper differs in examining a producion economy, so ha aggregae consumpion is endogenous, and in using a lower risk aversion coefficien (which is arguably more realisic) in our calibraion. Tallarini (2000) works wih a represenaive agen in a producion economy wih recursive preferences, bu his model focuses on he case of a fixed IES and no capial adjusmen coss. He shows ha even wih high risk aversion, his model has implicaions for macroeconomic quaniies comparable o hose obained by Kydland and Presco (982). The producion economy of Tallarini (2000) can generae a high Sharpe raio wih an high risk aversion. His model, however, generaes a low equiy premium. The mos closely relaed paper o his one is Kalenbrunner and Lochsoer (200) (KL (200) hereafer), who show ha long-run consumpion risk can be endogenously generaed even if he echnology is i.i.d.. Our model exends ha of KL (200) by inroducing exrapolaion; heir model is he special case of ours in which here is no exrapolaive bias. KL (200) feaures an IES larger han one and can produce a high Sharpe raio wih relaively low risk aversion. However, he volailiy of equiy reurns is sill very low, and hence his approach does no replicae he high equiy premium found in he daa. Anoher relaed sudy is Croce (204), who sudies a model feauring long-run produciviy risk direcly. As compared wih Croce (204), exrapolaion bias in our model reduces he predicabiliy in realized consumpion growh o a more realisic level, produces a larger equiy premium wih lower risk aversion, and generaes asse reurn predicabiliy as well. An earlier lieraure sudies asse prices in a producion economy wih habi preferences, including influenial papers by Jermann (998), Leau and Uhlig (2000), and Boldrin e al. (200). Mehodologically, our paper is closely relaed o Jermann (998), who finds ha he combinaion of capial adjusmen coss and habi preferences can generae a low risk-free rae, a high equiy premium, high volailiy of excess reurns, and high relaive invesmen and low consumpion volailiy. More recenly, Campanale e al. (200) show ha a producion economy wih convex capial adjusmen coss and disappoinmen aversion can produce a high equiy premium as well. These models ypically feaure a very low IES, and hence imply excessively high volailiy for he risk-free rae. This ends o resul in an abnormally large erm premium. Our model produces a low volailiy for he risk-free rae and high volailiy for he equiy reurns simulaneously. 6 5 Recency bias in Bansal and Shaliasovich (200) endowmen-based model is somewha akin o exrapolaion. In heir seing, invesors receive exogenous signals abou consumpion growh raher han simply observing pas realizaions, and overweigh recen signals in forming heir forecass. In conras, in our model invesors updae beliefs afer observing pas produciviy realizaions. Their quaniaive analysis is based upon a very srong recency bias (50% weigh on he laes signal), whereas in our analysis he exrapolaion bias pus 2 5% weigh on he laes observaion. Also, heir focus is on confidence risk (second momens) raher han he effecs of learning per se. 6 There is also a large lieraure examining he role of Bayesian learning in asse markes (e.g., Timmerman, 993, 996; Veronesi, 999; Brennan and Xia, 200; Brand e al., 2004).

4 90 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) In sum, our paper shows ha incorporaing exrapolaive bias ino a sandard producion-based equilibrium asse pricing model subsanially improves he model's abiliy o mach boh macroeconomic quaniies and asse prices. 2. A producion-based model wih exrapolaion This secion presens a producion-based equilibrium model wih exrapolaive expecaions o examine he join dynamics of consumpion, invesmen, oupu, and asse prices. For simpliciy, he model assumes a represenaive agen economy. In he special case where informaion is complee and here is no exrapolaive bias, our model is he same as he permanen shock model of KL (200). We hus follow heir noaions in seing up he model. 2.. Household's preferences The erms invesor, individual, and household are inerchangeably used o refer o he represenaive household. Following he long-run risk lieraure, he represenaive household's preferences over he uncerain consumpion sream C are described by he Epsein Zin Weil recursive uiliy funcion (e.g., Epsein and Zin, 989; Weil, 989) / / / ^ θ θ ( γ ) ( γ) θ γ V = C + EV +, β β( ) where ^ E ( ) is he expecaion under he individual's subjecive belief condiional on informaion available up o ime, he parameer 0 < β < is he ime discoun facor, γ 0 is he risk-aversion parameer, ψ 0 is he ineremporal elasiciy of subsiuion (IES) preference parameer, and θ γ =. ψ The sign of θ is deermined by he values of risk aversion and IES. When he risk aversion parameer exceeds he reciprocal of IES, he individual prefers early resoluion of he uncerainy of consumpion pah. Hence, hese preferences allow for a preference over he iming of he resoluion of uncerainy. The Euler equaion describing he represenaive individual's opimizaion holds under he individual's belief, which, owing o exrapolaive bias, generically does no mach he rue probabiliy disribuion. Thus, he pricing kernel is (e.g., Bansal and Yaron, 2004) θ m+ log( M+ ) = θ log β g+ + ( θ ) ra, +, ψ where r a, + is he logarihm of he gross reurn on an asse ha delivers aggregae consumpion as is dividends each period. For any coninuous reurn r+ = log( R+ ), including he one on he consumpion claim ^ E [ exp ( m + r )] =. ( 3 ) + + The expecaion operaor ^ E ( ) applies o he individual's biased subjecive belief; his is he key difference from a raional expecaions model. ( ) ( 2) 2.2. Produciviy, capial accumulaion, and belief updae There is a represenaive firm owned by he represenaive household, and he oupu, Y, is produced by a consan reurn-o-scale neoclassical producion funcion ( ) α α Y = A L K, ( 4) where L is he normalized labor supply, 7 A is he producion-enhancing echnology, and he capial level, K, evolves as K I =( δ ) K + ϕ K K, ( 5 ) + K where I is he invesmen in period, δ K is he rae of depreciaion of he capial, and ϕ( ) is a concave funcion from Jermann (998) ha allows for convex capial adjusmen coss 7 In oher words, he model assumes an exogenous wage process such ha i is opimal for he firm o always hire a full capaciy (L ¼). In his case, one can show ha he operaing profi funcion of he represenaive firm is linearly homogenous in capial (see KL, 200).

5 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) I a2 I ϕ = a + K K ξ ( / ξ), ξ > 0. ( 6) The adjusmen cos is parameerized inversely by ξ. The consans a and a 2 are se such ha here are no adjusmen coss in he nonsochasic seady sae (see, also Boldrin e al., 200; KL, 200). The adjusmen cos allows he shadow price of insalled capial o diverge from he price of an addiional uni of capial, and hence i permis variaion in Tobin's Q. The aggregae resource consrain is Y = C + I, where C is he aggregae consumpion. Labor is paid a is marginal produc. Thus, wages, ω, and firm dividend payous, D, saisfy ω =( α) Y, and D = αy I, respecively. Leing he oal facor produciviy (TFP) growh rae be denoed by g A, A = log, A he dynamics of he daa-generaing process for he produciviy growh is assumed o saisfy g = μ + σ ϵ, ( 7) A, + A A A, + where ϵ A, + is i.i.d. sandard normal. I is assumed ha s A is known o he represenaive individual, bu ha he rue growh rae in produciviy μ A is no observable. In pracice, i is much easier o esimae he variance han he mean (see, e.g., Meron, 980). The individual is subjec o exrapolaive bias and updaes his perceived growh rae a ime for period +, μ^, as μ^ =( ρ ρ ) μ + ρμ^ + ρ g, ( 8) A, where ρ reflecs he degree of overexrapolaion of he mos recen growh shock, ρ reflecs he persisence of exrapolaions made from pas growh shocks, and μ is he long-run mean in he individual's belief. In our calibraion, μ is equal μ A, he rue expeced rae of produciviy growh. So by Eq. (7), he individual believes ha produciviy growh follows g = μ^ + σ ϵ^, ( 9) A, + A A, + where he individual perceives ϵ^ A, + o be i.i.d. sandard normal. By Eq. (8), he individual akes an average of recen pas produciviy growh wih geomerically declining weighs and projecs ha growh rae forward o forecas he fuure. The seepness of he decline (measured inversely by ρ) can be viewed as he degree of myopia in updaing. If ρ is small and ρ is large, he individual exrapolaes placing a heavy weigh on recen realizaions of echnological growh raes. On he oher hand, when ρ is close o one, he individual places heavy weigh on disan pas growh raes. In a sense, when ρ is small and ρ is large, he individual is boh exrapolaive and myopic. Finally, in he special case where ρ = ρ, he above seing is similar o ha of Barsky and de Long (993). Moreover, in his case, he exrapolaive learning maches he consan-gain learning rule, which is popular in adapive learning lieraure (see Sargen, 993). The learning scheme in (8) is quie inuiive as a way of capuring overexrapolaion. A possible moivaion derives from srucural breaks in produciviy growh. If individuals believe ha a srucural break migh have occurred, hey pu less weighs on disan pas observaions and more weighs on recen observaions. Anoher possible moivaion for he updaing rule in (8) is ime-varying expeced produciviy growh. Indeed, Barsky and de Long (993) presen an example in which he rue long-run growh rae is a random walk, and he opimal esimaion of he expeced growh rae exacly follows Eq. (8) wih ρ = ρ. They use his specificaion o sudy excess sock marke volailiy in a parial equilibrium framework. In our general equilibrium framework, o ensure saionariy and he finieness of asse prices, ρ + ρ is always fixed a < in our calibraion. 8 Therefore, in he calibraions wih overexrapolaion, he degree of overexrapolaion is measured inversely by a single parameer, ρ. Lasly, in principle, based on produciviy realizaions invesors could evenually learn o correc heir misaken exrapolaive belief as given in Eq. (8). However, as is sandard in behavioral models, invesors are assumed o be subjec o psychological bias, and are no able o compleely learn heir way ou of heir bias. Psychological lieraure suggess ha such imperfec learning can be a consequence of inheren cogniive consrains, or of overconfidence. Also, i could be argued ha invesors are even less raional han assumed in our model, and in paricular do no fully undersand he srucure of he economy in which hey paricipae. This is probably he case, bu for reasons of parsimony we believe i is useful, a leas as a firs sep, o ry o undersand he consequences of a single deviaion from raionaliy before sudying more complicaed combinaions of psychological effecs. 8 If ρ + ρ =, he perceived growh rae is a random walk as shown in (0). Under some preferences, i is possible ha he value funcion is infinie. To rule ou his possibiliy, ρ + ρ is se o be less han one.

6 92 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) Model soluion Solving he model numerically is sraighforward. Since he quaniies in he economy are coinegraed wih he aggregae produciviy and he problem is homogeneous in A, variables are firs scaled by he aggregae produciviy. Then he value funcion is solved wih he usual value funcion ieraion. Readers can refer o KL (200) for deails on he numerical soluion. The only difference from he sandard raional model is ha, under he percepion of he individual, by (9) he dynamics of he sae variable, μ^, are μ^ =( ρ ρμ ) + ρμ^ + ρ g =( ρ ρμ ) + ( ρ+ ρμ ) ^ + ρσ ϵ^. ( 0) + A A, + A A A, + Thus, ρ + ρ deermines he persisence of he perceived echnological growh rae under he individual's own belief. Thus, he perceived growh is quie persisen. This propery is no unique o our exrapolaive learning scheme. In an i.i.d growh economy wih Bayesian learning on he rue mean growh rae, Collin-Dufresne e al. (205a) show ha in he agen's filraion, he mean expeced consumpion growh rae is ime-varying wih a uni roo. Once he value funcion is solved numerically, variables of ineres can be obained. For example, from Epsein and Zin (989), he log wealh consumpion raio is W V wc log log log = +. C β ψ C ( ) Following a sandard argumen of Cochrane (99), he reurn on invesmen is R α A+ δk ϕ I / K = ϕ ( I/ K ) α + + K+ ϕ ( I+ / K+ ) I, + ( + + ) I K + +. ( 2) The log reurn on invesmen is herefore ri, = log( RI, ). Noice ha he reurn on invesmen is he same as he reurn on he equiy claim. 9 Finally, i follows from Epsein and Zin (989), he risk-free rae can be calculaed numerically as r f, ^ C + = log E β C / ψ ^ E ( ) V K, μ^ + + +, A+ γ ( V ( K, μ^ + + +, A + ) ) / ( γ) ( / ψ) γ In calibraion, resuls on levered equiy marke reurns, r E,, are repored. Following Boldrin e al. (995) and Croce (204), our model inroduces consan financial leverage, and he levered excess reurn is defined as re, + rf, ( ri, + rf, )( + B/ E), where B/ E is he average deb equiy raio. B/ E is se o 2/3 since he acual deb o equiy raio is around 2/3 (see, e.g., Benninga and Proopapadakis, 990). Alernaive ways o inroduce leverage are also discussed in Secion The basic idea The abiliy of he model o reconcile a high equiy premium, a low risk-free rae, and a smooh consumpion growh wih low risk aversion comes from he way ha exrapolaive bias ineracs wih recursive preferences. Exrapolaion bias causes excessive variaion in perceived produciviy growh, which in urn induces volaile flucuaions in invesors expecaions of consumpion growh. Wih recursive preferences, flucuaions in expeced consumpion growh are priced. Following Epsein and Zin (989), he pricing kernel can be rewrien as γ ψ m E ^ ( m) γϵ^ c, ϵ^ wc,, ψ ( 4) where ϵ^ c, is he shor-run shock in consumpion growh, and ϵ^ wc, is he shock in he log wealh consumpion raio. These shocks are derived under he subjecive belief since i is he invesor percepion ha deermines asse prices. Under loglinear approximaion in he spiri of Campbell e al. (2003) and Bansal and Yaron (2004), shocks o he wealh consumpion raio can be approximaed by j ϵ^ wc, ( E E ) κ ( / ψ)δc+ j, W/ C j= where κ = and W/ C is he uncondiional mean of he wealh consumpion raio. Thus, long-run risk comes from W/ C shocks o he wealh consumpion raio, or shocks o expeced fuure consumpion growh. To generae a highly volaile. ( 3) ( 5) 9 As in KL (200), one can show ha he condiions in Resoy and Rockinger (994) are saisfied, and hus he invesmen reurn and he sock reurn are he same.

7 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) Perceived (log) TFP level by he agen Subjecive Objecive No bias Perceived TFP growh by he agen Quarer Quarer (log) Consumpion level Perceived consumpion growh by he agen Quarer Quarer Fig.. Impulse response funcions. This figure plos he impulse response funcions for he perceived (log) TFP level and perceived TFP growh by he agen, (log) consumpion level, and perceived consumpion growh by he agen o a posiive one-sandard-deviaion TFP growh shock from he model under he subjecive and objecive measures wih and wihou exrapolaive bias. The impulse response funcions are esimaed from he log-linearizaion as in KL (200) under he benchmark parameerizaion. The process of each variable is expressed using lag operaors, and he coefficien of he order T in he Taylor expansion around zero represens he impulse response in quarer T. The op lef panel plos he response funcion for he agen's percepion of he (log) TFP level. The op righ panel plos he response funcion for he perceived TFP growh by he agen. The boom lef panel presens he response funcion for he (log) consumpion level, and he boom righ panel presens he response funcion for he perceived consumpion growh by he agen. The solid and dashed lines are for resul under he subjecive and objecive measures, respecively, for he specificaion where here is exrapolaive bias. The doed line is for he specificaion where here is no exrapolaive bias. pricing kernel, which is a prerequisie for maching evidence of a high equiy premium, our model needs a persisen and volaile expeced consumpion growh under he subjecive belief. In he supplemenary maerial, 0 we follow he approximaion argumen in KL (200) and provide an approximae analyical soluion for he model. The key findings from he log-linearizaion approximaion are summarized as follows. In conras wih an ARMA(, ) process for consumpion growh and an AR ( ) process for he expeced consumpion growh in KL (200), when here is exrapolaive bias he consumpion growh is approximaely an ARMA( 2, 2) and he perceived expeced consumpion growh follows an ARMA( 2, ) process under boh he subjecive and he objecive measures. In paricular, i is shown ha boh endogenous capial accumulaion (jus as in KL, 200) and invesor's mispercepion on he persisence of he expeced TFP growh (i.e., Eq. (0)) conribue o he perceived persisence of consumpion growh. Alhough he produciviy shocks are i.i.d. in our model, a persisen and predicable componen in consumpion growh 0 The supplemenary maerial is available a hp://

8 94 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) (i.e., long-run risk) is endogenously generaed as a consequence of consumpion smoohing, he same as in KL (200). More imporanly, owing o exrapolaion-induced overreacion, he volailiy of he perceived expeced consumpion growh ends o increase as he exrapolaion bias becomes more severe (i.e., as ρ increases). To illusrae his mechanism more clearly, he impulse response funcions for (log) perceived TFP level, perceived TFP growh, (log) consumpion, and perceived consumpion growh are ploed under boh subjecive and objecive measures wih and wihou exrapolaive bias in Fig.. Fig. displays he impulse response funcions o a posiive one-sandarddeviaion shock o TFP growh. Firs consider he impulse response funcions when he represenaive agen is subjec o exrapolaive bias. Because ρ + ρ is close o one in our benchmark calibraion, he expeced TFP growh is very persisen under he subjecive belief (he op wo panels). This is in conras wih a less persisen TFP growh rae under he objecive measure, whose dynamics are he weighed average of he perceived and realized TFP growh (i.i.d.). The endogenous consumpion responses are ploed in he boom wo panels. Due o he difference in persisence and volailiy in he perceived TFP growh under hese wo measures, he expeced consumpion growh is also more volaile and more persisen under he subjecive measure han under he objecive measure. Since asse prices are deermined by he dynamics under he subjecive measure, our model is able o produce a volaile pricing kernel. Our nex sep is o compare he cases wih and wihou exrapolaive bias. Because he underlying process for he TFP is random walk, here is no predicable componen in TFP when he individual is raional. A posiive realized TFP shock increases he TFP level permanenly, bu does no change he expeced TFP growh. More ineresingly, when here is exrapolaion bias, he consumpion response o TFP shocks is weaker han in he case of no exrapolaion bias as in KL (200). This is because higher perceived TFP growh induces more invesmen now and less curren consumpion. However, he perceived fuure consumpion growh by he agen would be higher wih exrapolaion bias han wihou exrapolaion bias. Therefore, he inroducion of exrapolaive bias subsanially increases he amoun of long-run consumpion risks under he subjecive measure. Again, since asse prices are deermined by he dynamics under he subjecive measure, he pricing kernel in our exrapolaion model is more volaile han ha in a sandard long-run risk model (KL, 200). A he same ime, he smaller consumpion response wih exrapolaive bias allows us o increase he capial adjusmen cos, which helps raise he volailiy of sock reurns or he quaniy of risk in he economy. In addiion, by comparing he impulse responses of consumpion under he objecive measure wih and wihou he exrapolaive bias, i is seen ha he consumpion dynamics under hese wo siuaions are similar and he expeced consumpion growh under hose wo cases is less volaile and less persisen han ha under he subjecive measure wih exrapolaion bias. Since he realized consumpion is generaed by he dynamics under he objecive measure, i is smooh relaive o he oupu. In sum, hese feaures combined explain why our model generaes a large and volaile equiy premium while keeping a smooh consumpion process relaive o he oupu. The supplemenary maerial provides more deailed analysis regarding he dynamics of consumpion growh and reurns under boh he subjecive and objecive beliefs using he log-linear approximaion. I also provides a more deailed approximaion analysis of how exrapolaion bias raises he volailiy of he pricing kernel. Here we now proceed o numerical analysis of model calibraion. 3. Calibraion This secion examines differen versions of he model o explore he imporance of he differen model assumpions for explaining sylized facs abou asse pricing while replicaing salien business cycle evidence abou oupu, consumpion and invesmen volailiy. In addiion, his secion also examines he condiional performance of he model such as he Table Parameer choices for benchmark calibraion. This able repors he parameer choices for he benchmark calibraion. The model is simulaed in quarerly frequency. These parameers are in quarerly frequency as well. The firs five parameers are fixed for all of our calibraion, whereas he las five parameers vary across calibraions. Saisics Variable B.I Fixed parameer: Mean echnology growh (%) μ A 0.4 Volailiy of he innovaion in echnology growh (%) s A 4. Share of capial α 0.36 Depreciaion (%) δ K 0.02 Leverage BE / 2/3 Varying parameer: Exrapolaion parameer ρ 0.98 Risk aversion γ 4.00 IES ψ 2.00 Capial adjusmen cos ξ.50 Subjecive discoun facor β 0.99

9 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) reurn predicabiliy by price dividend raio, invesmen, and aggregae Q. As is sandard in he real business cycle lieraure, he model is calibraed a a quarerly frequency. Since he model is in real and per capia form, all calibraion is done wih real, per capia empirical counerpars. 3.. Parameer choices Table repors he parameer values used for our benchmark calibraion. Mos of he parameer values are borrowed from he real business cycle lieraure. Following Boldrin e al. (200), he capial share (α) is se o a value of 0.36, he quarerly depreciaion rae (δ K ) is se a 0.02, and he quarerly average log produciviy growh rae (μ A )isfixed a 0.4%. This se of parameers is chosen o mach he long-run growh rae of he economy; hese parameers do no subsanially affec model dynamics. Finally, he volailiy of produciviy growh is fixed a σ A = 0.04, he same value as in KL (200), o mach observed oupu volailiy since 929. As our model is an exogenous growh model, all endogenous variables in he long run grow a he same rae as produciviy. Mehra and Presco (985) sugges ha he convenional range of risk aversion should be less han 0. Our benchmark calibraion herefore chooses γ = 4. This risk aversion coefficien is lower han ha ypically employed in he exising asse pricing lieraure. Following Ai (200) and Croce (204), he IES ( ψ ) is fixed a 2, which is consisen wih esimaes of Aanasio and Vissing-Jorgensen (2003), Bansal e al. (2007), Bansal e al. (2007), and van Binsbergen e al. (202). For example, he esimaed IES ranges from.73 o 2.09 in van Binsbergen e al. (202). Empirical sudies do no offer precise guidance for calibraing he pure ime discoun facor ( β), capial adjusmen coss ( ξ), and exrapolaive bias ( ρ). Given he cenral role played by hese parameers for business cycles and asse reurns, reasonable ranges for hem are examined o verify wheher he model can mach he empirical momens. For example, he ime discoun parameer β is chosen o keep he level of ineres raes low. For he capial adjusmen cos parameer ξ, KL (200) choose ξ = 8 and 0.7 for wo baseline models, while Jermann (998) and Boldrin e al. (200) choose ξ = As will be shown below, reducing he value of ξ (i.e., raising he adjusmen coss) ends o improve he performance of our model. An inermediae value of.5 is chosen for our benchmark calibraion. In addiion o he benchmark calibraion, a deailed sensiiviy analysis is performed o provide us wih insighs abou he model mechanisms a work. Since he more innovaive feaure of he model, overexrapolaion, is refleced inversely in he parameer ρ, resuls for differen values of he exrapolaion parameer are repored wih benchmark value of ρ = Firs, bearing in mind ha exrapolaive learning here is basically he same as he consan-gain learning in macro lieraure, a value of ρ = 0.98 is consisen wih exising sudies. Orphanides and Williams (2005), for example, choose ρ = 0.98 o mach he inflaion forecass from he Survey of Professional Forecasers (SPF). Milani (2007) esimaes a DSGE model on several macroeconomic ime series, and find ha ρ = fis he daa he bes. More recenly, using micro-level forecas daa, Malmendier and Nagel (205) also confirm ha ρ 0.98 is a good approximaion for aggregae-level exrapolaion parameer. An alernaive way o gauge he exrapolaion parameer is o use survey daa on produciviy or GDP growh. The volailiy of produciviy growh is 0.22% based on he SPF 0-year produciviy median forecas. However, his long-erm survey sars wih 992, a well-known sable period (grea moderaion) compared o he full sample, despie he recen financial crisis. For example, he volailiy of realized GDP growh from 992 o 200 is only abou 37% of he volailiy in he full sample from 929 o 200. Taking his effec ino accoun, he volailiy of perceived produciviy growh in he full sample should be around 0.22%/0.37 = 0.60%. Wih ρ = 0.98, he implied volailiy of he perceived produciviy growh is 0.4%, which is smaller han 0.60%. Anoher imporan cavea is ha survey informaion may be unreliable since he respondens do no have a srong sake in undersanding he survey quesions. Indeed, Abel e al. (2008) find ha inflaion expecaions of marke paricipans are abou wo imes more volaile han hose of professional forecasers. Thus, even if he perceived long-run growh rae is smooh based on survey evidence, he rue invesor percepion could be quie volaile, implying a larger rue exrapolaion bias. Given hese concerns, ρ is se o 0.95 in an alernaive calibraion, which implies volailiy of 0.66% for perceived produciviy growh, slighly higher han he 0.60% benchmark. To furher discipline our parameer choice, we also verify ha he model-implied volailiy of perceived consumpion growh is comparable o ha in a sandard Bansal and Yaron (2004) seing. As will be shown laer, even wih ρ = 0.95 he model-implied volailiy of he perceived consumpion growh rae is indeed similar in magniude o he volailiy of expeced growh in a sandard long-run risk model. Alhough he values for μ A, s A, α, and δ K are fixed, ρ, ψ, γ, β, and ξ are allowed o vary across differen calibraions o mach he key momens in he daa. As menioned earlier, ρ + ρ is always fixed a excep he cases in which here is no exrapolaive bias ( ρ = and ρ = 0). In conras, using aggregae daa, early sudies, such as Hall (988) and Campbell e al. (989), ypically found he IES o be much less han. However, Vissing-Jorgensen (2002) and Guvenen (2006) poin ou ha here is a downward bias in he IES esimaion using aggregae daa. Once heerogeneiy especially limied asse marke paricipaion is aken ino accoun, he esimaed IES is much larger.

10 96 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) Table 2 Momens for quaniies and prices. This able repors he resuls for six calibraions. Calibraions B.I-V are based on pos-929 sample, where he volailiy of he innovaion in echnology growh σa=4.. Calibraion B.I is he benchmark calibraion. Calibraions II considers a higher exrapolaive bias (i.e., lower ρ). Calibraions III and IV consider a similar specificaion as he permanen TFP shocks case in Kalenbrunner and Lochsoer (200) wih low and high capial adjusmen cos (i.e., higher and lower ξ), respecively. Based on he benchmark specificaion, Specificaion V considers a lower adjusmen cos (i.e., higher ξ). Calibraion VI is based on he pos-war II sample, where s A is se o.93%. All he numbers are annualized. The mean and he volailiy are repored in percenages. Δc denoes consumpion growh; Δy denoes oupu growh; Δi denoes invesmen growh; IK denoes invesmen over capial; x denoes he perceived consumpion growh by he agen in he model; R f, denoes he risk-free rae; RE, Rf, denoes he levered equiy excess reurns; w c denoes he log wealh consumpion raio; r wc denoes he log reurn on he aggregae consumpion claim. Parameer Daa B.I II III IV V VI Risk aversion (γ) NA IES (ψ) NA Time discoun (β) NA Adjusmen cos (ξ) NA Exrapolaion (ρ) NA TFP volailiy (s A ) (%) NA σ (Δ c ) σ (Δ c )/ σ (Δ y ) σ (Δ i )/ σ (Δ y ) Adj. cos/oupu (%) NA E ( R f, ) σ ( R f, ) E ( RE, Rf, ) σ ( RE, Rf, ) Sharpe raio E (0-year erm premium) s (0-year erm premium) NA σ ( w c) σ ( r wc ) E ( w c) E ( r wc ) σ ( x ) NA σ( x) / σ(δ c) NA AC ( x ) NA Uncondiional momens The model is simulaed for 400,000 quarers of arificial daa o esimae populaion values for a variey of saisics. Small sample properies of he model are also examined by simulaing 400 quarers of arificial daa each ime and repeaing he procedure 000 imes. The main resuls are found in Table 2, which includes he summary saisics of boh quaniies and asse prices from he six differen parameerizaions. In general, consumpion growh is smooher han oupu growh, while invesmen growh is more volaile han oupu growh, consisen wih he daa. The momens for consumpion, GDP, invesmen, risk-free rae and sock marke reurns in he daa column are aken direcly from Bansal and Yaron (2004) and KL (200), boh of which are based on a daa sample from 929 o The benchmark calibraion In our benchmark calibraion (i.e. Model I in Table 2), he oupu volailiy and he mean growh raes of he economy are pinned down by he echnology parameers and are chosen o mach he daa. We herefore omi hem and only repor he volailiy of he variables of ineres. Wih an adjusmen cos of ξ =.5 and an exrapolaion parameer of ρ = 0.98, he model generaes smooh consumpion and volaile invesmen relaive o oupu ha is in a magniude similar o, bu sill smaller han, ha in he daa as repored in KL (200). Moreover, in he second calibraion wih sronger exrapolaion bias, he quaniies are even closer o hose in he daa. The benchmark model I maches he momens of asse prices well. Wih a risk aversion coefficien of 4 and an IES of 2, he model produces a sizable equiy premium of 5.75%, compared wih 6.33% in he hisorical daa. The volailiy of he excess reurn is 0.42%, a reasonable number given he relaively small risk aversion coefficien. Since he model roughly maches he macroeconomic quaniies, his is a significan success for a producion-based model. For example, KL (200) produce a volailiy of 0.66% for he excess reurn on he (unlevered) equiy claim in one of heir benchmark calibraions. In addiion, his volailiy can be furher increased if one chooses ρ = 0.95 as in our model II. Inuiively, invesor overexrapolaion causes a percepion of high volailiy of expeced consumpion growh; ogeher wih recursive preference his can resul in high risk premia wihou high consumpion volailiy. The benchmark calibraion also produces smooh ineres raes, even smooher han hose in he daa. This is a sandard feaure in models wih Epsein Zin preferences and IES greaer han one. On he oher hand, sandard habi models in

11 D. Hirshleifer e al. / Journal of Moneary Economics 76 (205) producion economies end o produce an excessively volaile ineres rae. A usual side effec of highly volaile ineres raes is an excessively large erm premium (see, e.g., Jermann, 998; Abel, 999). Large bond erm premia are relaed o overly volaile ineres raes, as erm premia are compensaion for real ineres rae risk. In our model, owing o he high IES, he ineres rae is very smooh despie he exrapolaive expecaions, and hence he erm premium (defined as he difference beween he 0-year bond yield and 3-monh ineres rae) is also small, consisen wih he daa. 2 For he benchmark model, i produces a downward-sloping real yield curve. The shor-erm real rae is already low a.29%, while he long real raes are an addiional 0.70 percenage poin lower. This paern does no mach he posiive real erm premium of.79% for he U.S. TIPS daa beween January 998 and Ocober 2009, bu in erm of magniude, our erm premium is only slighly negaive. 3 The boom of Table 2 repors summary saisics for he perceived expeced consumpion growh by he individual, and he aggregae wealh porfolio. The volailiy of perceived expeced growh rae in he model is 0.79%. The raio of he volailiy of he perceived expeced growh o he volailiy of realized growh rae is slighly less han 2%. The perceived expeced growh rae is no direcly observable, bu is magniude can be compared wih exising sudies. In he one channel model of Bansal and Yaron (2004), for example, he volailiy raio of expeced growh and realized growh is 34.4% for quarerly frequency calibraion. Thus, he volailiy of he perceived growh rae in our calibraion does no appear oo large. Owing o he high variaion in he perceived expeced growh rae, our model produces an annual volailiy of he wealh consumpion raio of abou 2.40%, which is abou half of ha in he daa as repored in Lusig e al. (203). This is a success compared o he sandard long-run risk models; Ai (200), for example, shows ha he sandard long-run risk model only produces a volailiy of 4.70% (see Table I in Ai, 200). From Eq. (4), his high volailiy in he wealh consumpion raio is he main driving force underlying he high and volaile equiy reurns. Our model also generaes a high volailiy of he reurn on aggregae wealh, again consisen wih he daa. Finally, when ρ is reduced o 0.95 (as in model II), he model maches he momens in he daa even more closely. For example, consumpion growh is smooher, and invesmen growh and asse reurns are more volaile. Wih larger exrapolaion bias, he perceived produciviy growh becomes more volaile, and hence he volailiy of invesmen and perceived expeced consumpion growh is also larger. Relaive invesmen volailiy increases from.72 o 2.4, which is very close o ha in he daa. Moreover, due o high long-run consumpion risk, he equiy premium is sill very large despie a lower risk aversion in model II, and equiy reurns are highly volaile. The risk-free rae is also slighly more volaile owing o greaer volailiy of he perceived expeced consumpion growh rae. However, in boh calibraions, he mean reurn on he wealh porfolio is larger han ha in he daa, as calculaed by Lusig e al. (203). In he model, he consumpion claim is riskier han he firm's dividend payou claim since he payou is less procyclical han consumpion. Thus, he reurn on he consumpion claim is oo high relaive o he daa. Poenial resoluions for his issue are discussed in Secion The mechanism of he model To idenify he key mechanism behind he empirical success of he model, below we follow Jermann (998) by calibraing he model a differen parameer combinaions. This way helps idenify which ingrediens are key o replicaing differen aspecs of he daa. We sar wih a calibraion (model III) wih a very low capial adjusmen cos and no exrapolaion ( ρ =, ρ = 0, and μ^ = μa), and hen sequenially add he adjusmen cos and exrapolaion ino he model. The oucome of model III is consisen wih he sandard RBC models and KL (200). Consumpion is smooh, and invesmen is more volaile han oupu. These paerns are consisen wih he quaniy daa. However, he equiy premium is very low (less han % ), as is he volailiy of he sock reurn (less han 3%). Inuiively, owing o low adjusmen coss, he individual can easily smooh consumpion by adjusing he amoun of invesmen. This reduces consumpion risk and herefore equiy risk premium. Model IV increases he capial adjusmen cos. No surprisingly, his grealy reduces invesmen volailiy. Consisen wih he inuiion above, his in urn increases he volailiy of consumpion growh. So he model sacrifices good maching of his quaniy momen, bu does have he benefi of raising he equiy premium slighly, and generaing greaer reurns volailiy. This ype of finding is well known for sandard DSGE models (e.g., Jermann, 998). To produce a high equiy premium, he adjusmen cos canno be oo small; oherwise, consumpion is oo smooh, and hence he equiy premium is small. However, if he capial adjusmen cos is high, he invesmen volailiy is oo low compared wih he daa. Thus, i is hard o mach boh he quaniies and asse prices simulaneously. In wo imporan papers, Jermann (998) and Boldrin e al. (200) show ha inroducing habi preference can help o mach hese sylized facs. 2 Campbell e al. (2003) argue ha exrapolaive expecaions in general equilibrium end o lead o volaile ineres raes. In our benchmark models, owing o high adjusmen coss, he perceived expeced consumpion growh is no very volaile. This fac, ogeher wih a high IES value, resuls in a smooh risk-free rae. 3 The daa on he US real erm srucure is downloaded from Professor J. Huson McCulloch's Web sie (hp:// hml). For he inernaional evidence, Piazzesi and Schneider (2006) find a negaive real erm premium (he difference in real yield beween 0-year bond and 2.5-year bond) of 0.8% in he sample of December 995 March 2006 in UK. However, in he longer sample of January 985 June 205, he real erm premium in UK is slighly posiive a 0.33%.