ONLINE APPENDIX. May 3, This online appendix presents additional results with regard to the performance of the model

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1 Asse pricing wih beliefs-dependen risk aversion and learning ONLINE APPENDIX May 3, 2017 This online appendix presens addiional resuls wih regard o he performance of he model and discusses he model specificaion ess used in he paper. Secion 1 provides basic evidence of performance based on ROC curves. Secion 2 illusraes differences beween hree specificaions of ineres, namely consan relaive risk aversion (CRRA), consan aversion o sae uncerainy (CASU) and beliefs-dependen risk aversion (BDRA), in heir abiliy o reproduce empirical properies of he equiy premium and he yield curve. Secion 3 examines he robusness of parameer esimaes wih respec o momen condiions employed in he esimaion procedure. Secion 4 carries ou specificaion ess for he hree alernaives under consideraion as well as for a unified specificaion conaining all alernaives as special cases. 1 Model as predicor of recessions: ROC curve The firs elemen suggesing a good fi of he model wih BDRA o momens ha were no argeed in he GMM esimaion is he evoluion of he condiional probabiliy of he bad sae (p 2 ). The rajecory of his condiional probabiliy depends on he esimaes of he ransiion marix Λ, he covariance marix and he drifs of he hree sae variables, namely consumpion, dividend and unemploymen rae. Figure 1 suggess ha p 2 correcly idenifies he recession periods measured by he NBER. To formally gauge he abiliy of he model o idenify he recession phases, we use a sandard ool for diagnosic es called he Relaive Operaing Characerisic (ROC) curve. 1 The ROC curve 1 ROC curves are widely used o evaluae diagnosic decision making in medicine, radiology, signal deecion, daa mining and psychology.

2 measures he sensiiviy wih respec o a diagnosic hreshold, of he True Posiive Rae (TPR), which is he rae of correc diagnosic, versus he False Posiive Rae (FPR), which is he rae of incorrec diagnosic. I plos power (one minus he probabiliy of a ype II error) agains he probabiliy of a ype I error. The ype I error is o accep H 1 when H 0 is rue. The ype II error is o accep H 0 when H 1 is rue. In our conex, he wo raes perain o recession diagnosic and are calculaed as follows. Fix a hreshold level c P r0, 1s. The TPR is he number of quarers in which p 2 is above he hreshold c and he economy is in recession divided by he oal number of quarers in which he economy is in recession (rae of correc recession diagnosic). The FPR is he number of quarers in which he recession probabiliy is above he hreshold c and he economy is no in recession divided by he oal number of quarers in which he economy is no in recession (rae of incorrec recession diagnosic). We perform hese calculaions for all values of c P r0, 1s and plo he resuling ROC curve in Figure 1. This wo-dimensional figure shows he relaion beween he TPR (y-axis) and he FPR (x-axis) for all levels of c P r0, 1s. The ROC curve is seen o diverge significanly from he 45 degree line in he norh-wes direcion. This divergence indicaes ha he model is a good recession diagnosic ool for all hreshold levels c. 2 Model comparison: equiy premium and yields This secion provides furher evidence of he incremenal performance of he model wih beliefsdependen risk aversion (BDRA) in comparison o (i) he model wih consan relaive risk aversion and consan aversion o sae uncerainy (CASU) and (ii) he model wih consan relaive risk aversion (CRRA). Figure 2 shows he rajecories of he equiy premium for he hree specificaions examined. In he model wih BDRA, he equiy premium displays he counercyclical behavior found in he daa (see Campbell and Cochrane (1999) and Melino and Yang (2003)). Wih CASU, i peaks during booms and also akes negaive values. Wih CRRA, he equiy premium remains nearly fla and close o zero. Among he hree specificaion, BDRA is bes a capuring empirical properies of he equiy premium. Figure 3 shows he ime series of he 10-year yield, calculaed as described in Secion 4.1.4, and he corresponding rajecories for he hree model specificaions. The models 2

3 True Posiive Rae False Posiive Rae Figure 1: Relaive Operaing Characerisic (ROC) curve for recession diagnosic. The model implied condiional probabiliy of he bad sae is used as a diagnosic for a recession quarer as idenified by he NBER. 3

4 Risk Premium and Recessions Figure 2: The plos compares he risk premium rajecories for he BDRA (red), CRRA (magena) and CASU (green) model. Parameers esimaes are given in Table 6 for CRRA and Table 7 for CASU. BDRA parameers are given in he paper in Table 2. Condiional probabiliies are updaed using innovaions from Consumpion, Dividend, and Unemploymen ime series. 4

5 year Yield - daa year Yield - Regime dependen risk aversion year Yield - Consan aversion o sae uncerainy year Yield - Regime independen risk aversion Figure 3: The plos compares he 10-year real yield rajecory for he daa (blue), BDRA (red), CASU (green) and CRRA (magena). Parameers esimaes are given in Table 2 in he paper (BDRA), Table 6 (CRRA) and Table 7 (CASU). Condiional probabiliies are updaed using innovaions from Consumpion, Dividend, and Unemploymen ime series. The 10-year yield daa series is compued using he mehodology described in Secion 4.1 in he paper. wih BDRA and CASU produce large flucuaions and fail o capure empirical properies of he yield. The large ampliude of he yield is due o he larger variabiliy of he marginal uiliy of consumpion in hese seings, which is essenial for capuring he dynamic behavior of he volailiy and he equiy premium. The model wih CRRA fares bes, in he sense ha i produces a more sable yield, precisely because i canno generae enough variabiliy o explain he behavior of he oher wo componens of reurns. Figure 4 shows he ime series of he volailiy for he models wih BDRA and CASU along wih he realized rolling volailiy of he sock index. The BDRA specificaion capures he empirical dynamics well, wih sharp increases during recessions, whereas he CASU specificaion display large increases in volailiy also during expansion periods, in paricular during he 90s and in he 5

6 afermah of he global financial crisis. We run simple regressions o es he relaionship beween he realized rolling volailiy and he model implied volailiies. We find ha only he BDRA specificaion produces a posiive and saisically significan slope coefficien. The esimaion resuls are provided in Table 1 Table 1: This able provides regression esimaes of he realized rolling volailiy agains model implied volailiy for he hree differen model specificaions. α -sa β -sa adj. R-squared BDRA CASU CRRA Figure 6 shows he yield curve for he CASU model. In conras o BDRA (Figure 5), he real yield curve lies ouside he 95% confidence inerval around he average real yield curve esimae. In conras o he daa and he BDRA specificaion, he real yield curve is slighly downward sloping. Shor erm yields are oo large and significanly differen from he daa. Figure 7 shows he yield curve for he CRRA model. The yield curve in his specificaion is fla, and for shor mauriies close o he boundary of he confidence inerval. The BDRA model, in conras, is able o capure he level and he slope of he average yield curve. 6

7 Volailiy and Recessions - BDRA Volailiy and Recessions - CASU Figure 4: The plos compares he volailiy rajecories for he daa (blue), BDRA (red) and CASU sae uncerainy (green). Parameer esimaes are given in Table 2 in he paper (BDRA) and Table 7 (CASU). Condiional probabiliies are updaed using innovaions from Consumpion, Dividend, and Unemploymen ime series. The 10-year yield daa series is compued using he mehodology described in Secion 4.1 in he paper. 7

8 Real yield curve : Daa (blue), Model (green) (red) Figure 5: The plo shows he average yield curve in he model over he period 01/1999 o 01/2014 (red), over he period 01/1957 o 01/2014 (green) and in he daa (blue) over he period 01/1999 o 01/2014. y-axis is in percenage erm. Dashed lines are 95% confidence bounds. Yield daa cover mauriies from 2 o 20 years. Risk aversion parameers are R , R and R Condiional probabiliies are updaed using innovaions from Consumpion, Dividend, and Unemploymen ime series. 8

9 6 Real yield curve : Jan Jan vs model yield curve (full sample in blue, in green) Figure 6: The plo shows he real yield curve in he CASU model compared o he average yield curve and 95% confidence bounds. The plo shows he average yield curve in he daa (red) wih 95% confidence inervals (red-doed) and he model implied yield curve (full sample (blue), laer sample (green diamonds)). Parameer esimaes are given in Table 7. Condiional probabiliies are updaed using innovaions from Consumpion, Dividend, and Unemploymen ime series. The 10-year yield daa series is compued using he mehodology described in Secion 4.1 in he paper. 9

10 Real yield curve : Jan Jan vs model yield curve (full sample in blue, in green) Figure 7: The plo shows he real yield curve in he CRRA model compared o he average yield curve and 95% confidence bounds. The plo shows he average yield curve in he daa (red) wih 95% confidence inervals (red-doed) and he model implied yield curve (full sample (blue), laer sample (green diamonds)). Parameer esimaes are given in Table 6. Condiional probabiliies are updaed using innovaions from Consumpion, Dividend, and Unemploymen ime series. The 10-year yield daa series is compued using he mehodology described in Secion 4.1 in he paper. 3 Robusness: alernaive momen condiions The risk aversion parameers R 1, R 3 are esimaed using he correlaions (conemporaneous and lagged one quarer) beween log simple reurns and changes in he log-pdr (M1). This secion shows ha hese risk aversion parameers esimaes are robus wih regard o alernaive ses of momen condiions given by firs and second order reurn auocorrelaions (M2) and by correlaions beween log simple reurns and one and wo quarer lagged changes log-pdr (M3). The risk aversion parameers esimaes ouside he minimal regime, pr 2, R 3 q, are (1.9251, ) for mo- 10

11 men condiions M1, (1.7895, ) for M2, and (1.9396,1.775) for M3. Parameer values change slighly bu he Ş -shape of risk aversion parameers across regimes is preserved. This finding esablishes ha he esimaes of parameers in Θ 3 are robus wih regard o momen selecion. 4 Alernaive models: specificaion ess Equilibrium formulas for CASU and CRRA are provided in Secions 4.1 and 4.2. Salien differences beween he various model specificaions are discussed in Secion 4.3. A unified framework conaining CASU, CRRA and BDRA as subcases is developed in Secion 4.4. Model specificaion ess are carried ou in Secion CASU (Veronesi (2004)) Le m 1 exp ` ϱµ C 1,, exp ` ϱµ C K where ϱ denoes he consan aversion o sae uncerainy parameer. Wih CASU, marginal uiliy is u c p, c, p q exp pϱg n βq C R m 1 p. The sae price densiy (SPD) is, ξ u c p, c, p q {u c p0, c 0, p 0 q MPR Marke prices of risk are obained from he covariaion of he SPD wih he innovaions `ν C, ν G, ν Y. This gives, θ C Rσ C m 1 diag µ C k pµ C σ C ı p m 1 p (1) ı m 1 µ G diag k pµ G p θ G σ G m 1 p (2) ı m 1 µ Y diag k pµ Y p θ Y σ Y m 1. (3) p Price-dividend raio The price-dividend raio (PDR) is, S D şt ı m 1 E e βv Cv κ R G v p v dv m 1 e β C κ R G p. 11

12 ˆˆpσC q 2 Leing Φ 2 pκ Rq pκ R 1q β I K ` Λ 1 ` diag pκ Rq µ C k µg k and noicing ha, d e βv C κ R v G v p v Φe βv Cv κ R G v p v dv ` dm v where M v is a maringale, i follows ha E e β v Cv κ R G v p v exp ` Φ pv q e β C κ R G p. As long as he real par of he larges eigenvalue of Φ is negaive, we have, ż 8 E e βv Cv κ R G v p v ı dv Φ 1 e β C κ R G p and he PDR becomes, S D m1 Φ 1 p m 1 p. (4) Sock price volailiy Given (4) for he PDR, he sock volailiy coefficiens are obained from he covariaions of he log-pdr wih he innovaions in `µ C, µ G, µ Y as follows, ı ı m 1Φ 1 µ C diag k pµ C p σ SC κσ C σ ` C m 1 µ C diag k pµ C p σ m 1Φ 1 C p m 1 p κσ C ` m 1 ˆ S D 1 Φ 1 I K diag µ C k pµ C σ C ı p ı ı m 1Φ 1 µ G diag k pµ G p σ SG σ G σ ` G m 1 µ G diag k pµ C p σ m 1Φ 1 G p m 1 p σ G ` m 1 ˆ S D 1 Φ 1 I K diag m 1 p (5) µ G k pµ G σ G ı p m 1 p (6) σ SY ı ı m 1Φ 1 µ Y diag k pµ Y p σ Y m 1 µ Y diag k pµ Y p σ m 1Φ 1 Y p m 1 p m 1 ˆ S D 1 Φ 1 I K diag µ Y k pµ Y σ Y ı p m 1 p. (7) 12

13 4.1.4 Bond prices The bond price is, B `τ ı m 1 E C R T p T exp p β pt qq m 1C R. p ˆpσC Leing Φ B q 2 2 R p1 ` Rq β I K ` Λ 1 Rdiag µ C k and noicing ha, d e βv C R v p v Φ B e βv C R v p v dv ` dm B v for some maringale M B, i follows ha E e βt C R T p T ı exp Φ B pt q e βc R p and m 1 exp Φ B τ p B `τ m 1 p. (8) Bond yields and shor rae Given he bond prices (8), he bond yields and shor rae are, Y `τ 1 m1 exp Φ B pt q T log m 1 p p r lim ` BT log B `τ m 1 Φ B p ÓT m 1. p Bond and shor rae volailiies Bond volailiy σ B,α p, τq Using he expression for he bond price wih α P C, G, Y u leads o, ı m 1 exp Φ B µ α pt q diag k pµ α σ p α m 1 exp pφ B pt qq p m 1 `B`τ 1 exp Φ B pt q m 1 diag µ α k pµ α σ α ı p m 1 p ı µ α I K diag k pµ α σ p α m 1. p 13

14 Shor rae volailiy Using he expression for he shor rae for α P C, G, Y u leads o, ı m 1 Φ B µ α diag k `pµ α σ r,α σ p α m 1 p ı µ α m1 diag k pµ α σ p ` r α m 1 p ı m 1 Φ B µ α r I K diag k pµ α σ p α m 1. p 4.2 CRRA The equilibrium coefficiens in he CRRA model are obained by seing ϱ 0, herefore m K, in he CASU model. Table 2 summarizes he equilibrium quaniies for all models considered. 4.3 Model comparison CASU The funcional form of equilibrium in he CASU model is similar o ha in he BDRA model as `Z, Υ, A pτq, H, H τ,b, H r are replaced by Z, Υ, A pτq, H, H τ,b, H r. BDRA generaes addiional erms in he sock and bond volailiy componens ha are ied o consumpion, namely Z 1 diag rr k s H p σ C and Z 1 H τ,b p σ C. These erms emerge as Z depends on boh C and p, whereas wih CASU, Z only depends on p. These addiional volailiy erms vanish asympoically as consumpion becomes large. If arg min j µ C j arg min j R j and pϱ, C q Ñp8, 8q, hen he asympoic equilibrium coefficiens wih CASU and BDRA are idenical CRRA Wih CRRA, learning only affecs he drif of he sae dynamics. MPRs are θ C Rσ C, θ G θ Y 0. The risk premium is herefore he same as in he i.i.d. seing of Mehra and Presco (1982). Similarly, he shor rae becomes r β ` Rpµ C 1 2 R p1 ` Rq `σ C 2 and only differs from he ineres rae in he i.i.d. model of Mehra and Presco by he sochasic consumpion growh rae pµ C. Empirically, as shown in he original aricles on his opic, he i.i.d. model produces a poor fi for he equiy premium and he risk free rae for moderae levels of risk aversion. Marke volailiy and he equiy premium are boh small. Furhermore, a correlaion puzzle emerges as 14

15 Table 2: This able shows he equilibrium MPRs θ α, PDRs S {D, sock volailiies σ Sα prices B `τ, yields Y `τ BDRA, CASU and CRRA model specificaions., shor raes r, bond and shor rae volailiies σ Bα p, τq and σ rα, bond for he BDRA CASU CRRA MPRs θ C Zdiag 1 rr k s p σ C Zσ 1 p,c p Rσ C Z 1 σ p,c p Rσ C 1 1 Kσ p,c p Rσ C θ G Zσ 1 p,g p Z 1 σ p,g p 1 1 Kσ p,g p 0 θ Y Zσ 1 p,c p Z 1 σ p,y p 1 1 Kσ p,y p 0 PDR S D ZΥp 1 Z 1 Υp 1 1 KΥp Sock volailiy σ SC σ SG σ SY ρ DC σ D ` ZH 1 σ p,c p ρ DC σ D ` Z 1 H σ p,c p ρ DC σ C ` 1 1 KH σ p,c p b Zdiag 1 rr k s H p σ C b b σ D 1 pρ DC q 2 ` ZH 1 σ p,g p σ D 1 pρ DC q 2 ` Z 1 H σ p,g p σ D 1 pρ DC q 2 ` 1 1 KH σ p,g ZH 1 σ p,y p Z 1 H σ p,y p 1 1 KH σ p,y p Bond price, yields, and shor rae B `τ ZA 1 pt q p Z 1 A pt q p 1 1 KA pt q p Y `τ logpz1 ApT qp q logpz1 ApT qp q logp11 K ApT qp q T T T r Z 1 A 9 p0q p Z 1 9 A p0q p KA p0q p β ` Rpµ C 1 C 2 R p1 ` Rq `σ 2 Bond and shor rae volailiy σ BC p, τq σ Bα p, τq σ rα σ p,α m 1 Z 1 H τ,b p Z 1 H τ,b σ p,c p 1 1 KH τ,b σ p,c Zdiag 1 rr k s H τ,b T pσc µ Y diag k pµ Y σ p,c p ZH 1 τ,b σ p,α p Z 1 H τ,b σ p,α p 1 1 KH τ,b σ p,α p ZH 1 r σ p,α p Z 1 H r σ p,α p 1 1 KH r σ p,α p RCOV `µc ps q, µ α ps q σ Y exp ϱµ C 1 Υ Φ 1 ; Φ j ; Z m m 1 ; Z M p Mp 1,, exp `σc 2 Υ ij e 1 iφ 1 i e j; Φ i H ˆ S D 1 Υ I K; H ϱµ C K ı ; M 1 C R 1 ı,..., C R k pκ Rq pκ R 1q β 2 `σc 2 pκ R iq pκ R i 1q β 2 ˆ S D A pt q exp Φ B pt q ; Φ B A ij pt q e 1 i exp Φ B i 9A ij p0q e 1 iφ B i e j; H τ,b 1 Υ I K; `σc 2 R p1 ` Rq β 2 `σc 2 pt q e j; Φ B i R i p1 ` R iq β 2 9 A p0q Φ B `τ 1 `B A pτq IK; H τ,b H r 9 A p0q r I K; H r 9 A p0q r I K `τ 1 `B A pτq IK, ı I K ` Λ 1 ` diag pκ Rq µ C k µ G k ı I K ` Λ 1 ` diag pκ R iq µ C k µ G k ı I K ` Λ 1 Rdiag µ C k ı I K ` Λ 1 diag R iµ C k p 15

16 only one risk facor is priced Seady sae Seady sae values in he model wih BDRA, if here is a single minimal risk aversion regime, are obained by aking Z 8 e 1 1 {p 18 and p p 8. In he model wih CASU, hey are found by seing Z equal o Z 8 m{m 1 p 8. In he model wih CRRA, hey are obained by aking p p 8. Noe ha in he seady sae, he addiional volailiy coefficiens of he BDRA model, i.e., Zdiag 1 rr k s H τ,b p σ C and Zdiag 1 rr k s H p σ C, vanish. These coefficiens conribue o he shor run dynamics of he model. 4.4 Unified model To perform specificaion ess, he models wih CASU and BDRA are embedded in a unified framework where he marginal uiliy of he represenaive agen is ř K formulas are he same as wih BDRA excep ha Z is replaced by, k 1 e βe ϱµc k C R k p k. Equilibrium rz k exp ` ϱµ C k ř K k 1 exp ` ϱµ C k C R k C R k p k. In he seady sae, when C Ñ 8, Z r Z Ñ 0. Hence, he seady sae equilibrium quaniies in he unified model are he same as in he model wihou sae uncerainy aversion or beliefsdependen risk aversion. As a consequence, a specificaion es for BDRA versus CASU mus rely on dynamic quaniies ha pin down he parameers in Θ 3. Furhermore, if he minimal risk aversion regime coincides wih he minimal consumpion growh regime, i.e., if arg min k R k arg min k µ C k, hen he seady sae momen condiions wih CASU converge o hose wih BDRA as he sae uncerainy aversion parameer ϱ Ñ 8. Therefore, if he rue model has BDRA, inference based on saic momen condiions for CASU will rely on parameer esimaes on he boundary of he parameer space. Sandard specificaion ess are hen invalid. For his reason, dynamic momen condiions mus be used. The corresponding specificaion ess are discussed nex. 16

17 4.5 Specificaion ess Boh nesed and non-nesed specificaion ess are performed. The various nesed model specificaion ess performed nex are based on he D-saisic, D T T pjt rj T where J p T (resp. JT r ) is he unresriced (resp. resriced) GMM objecive funcion. This saisic has been pioneered for insrumenal variable esimaion by Gallan and Jorgenson (1979) and sudied for mehod of momen esimaors by Wes and Whiney (1987). Similar o a likelihood raio es, he saisic corresponds o an appropriaely scaled difference beween unresriced and resriced model esimaes. Asympoically, for large T, he saisic is χ 2 r-disribued where r is he number of consrains. 2 Non-nesed specificaion ess are based on he N-saisic, N T T 1{2 pj model1 T pj model2 T {pσ J where J p T model1, J p T model2 are he GMM-objecive funcions and pσ J is an esimaor of he asympoic ı variance of he numeraor of N, σj 2 lim T Ñ8 V AR T 1{2 pj model1 T pj model2 T. This es saisic for non-nesed hypohesis esing was pioneered by Rivers and Vuong (2002). Under he null hypohesis ha he wo models, i.e., he wo ses of momen resricions, provide an equivalen fi, he N-saisic is asympoically sandard normal if boh models are misspecified and, as demonsraed by Hall and Pelleier (2011), i has a non-sandard disribuion ha depends on nuisance parameers if boh models are (locally) correcly specified. 3 Rivers and Vuong assume ha σ 2 J 0. Hall and Pelleier (2011) show ha his is only he case if boh models are misspecified. The N-es rejecs he null hypohesis agains he alernaive H 1a ha model 2 provides a beer fi if he N-saisic is larger han he 1 α% quanile of he asympoic limi disribuion. I rejecs he null hypohesis agains he alernaive H 1b ha model 1 dominaes if he es saisic is smaller han he α% quanile of he asympoic limi disribuion. To calculae he N-saisic, he sandard deviaion in he denominaor is obained from a saionary boosrap esimaor based on Romano and Poliis (2004). As he es based on he asympoic disribuion has poenially limied power in finie samples and as he disribuion iself depends on auxiliary assumpions abou model misspecificaion, he N-es is also implemened using empirical quaniles and p-values from he empirical cumulaive disribuion funcion calculaed using he saionary boosrap. 2 As shown by Wes and Whiney (1987), he D saisic used for he es is numerically equivalen o he Wald or LM/score saisics in he jus idenified case. 3 They show ha if he model is (locally) correcly specified, σ J 0, and herefore, he limi disribuion of he he N-saisic is no sandard normal bu given by he raio of a quadraic form involving sandard normal random variables in he numeraor and he square roo of a differen quadraic form involving he same normal random variables in he denominaor. 17

18 To consider model specificaion ess based on a unified model srucure has he advanage ha, in conras o specificaion ess of non-nesed models, i does no require addiional generic assumpions on he asympoic variance σj 2, which poenially imply non-sandard asympoic disribuions of he N-saisic. Furhermore, he saisic for ess of nesed models is no sensiive o he weighing of momen condiions. In non-nesed specificaion ess, he es saisic depends on he weighing of momen condiions boh hrough he numeraor and he denominaor. In conras o correcly specified models, he choice of he opimal weighing scheme canno rely on asympoic efficiency consideraions if he models are misspecified. As emphasized by Hall and Pelleier (2011) an opimal choice of insrumens and herefore an opimal weighing of momen condiions for misspecified models has ye o be developed. The fac ha he asympoic disribuion of he non-nesed es relies on auxiliary assumpions abou misspecificaion renders inference difficul. Ideally, using overidenifying resricions, a saisical es should decide wheher σ 2 J is zero or no, i.e. wheher models are misspecified. Hall and Pelleier (2011) show ha such a es is no feasible as i depends on he unknown bias of he misspecified model. The non-nesed specificaion es resuls presened in his paper have been obained under non-adaped choices of weighing schemes, i.e., he weighing is no model-dependen. To use a fixed weighing for boh CASU and BDRA alleviaes some of he concerns expressed in Hall and Pelleier (2011). To esimae he unified model and perform he various specificaion ess, he correlaion beween he shor and long erm yields is used as an addiional saionary momen condiion. Similarly, he dynamic momen condiions used are he auocorrelaion of simple reurns for one and wo quarers, he cross-correlaion beween simple sock reurns and shor erm yields lagged by one quarer, as well as he cross-correlaion beween simple sock reurns and shor and long erm yields lagged by wo quarers. The following nesed model specificaion ess are performed H 0 : R k R, β k β and ϱ 0 versus H 1 : R k R, β k β and ϱ 0 or ϱ 0 (CRRA (A1)) as well as H 0 : ϱ 0, β k β and R n R some k versus H 1 : ϱ 0, β k β and R k R some k (CASU (A2)). Noe ha under he alernaive hypohesis, as a minimal regime exiss, he saionary momen consrains are he same wheher or nor he sae uncerainy parameer ϱ 0. I is herefore appropriae o base he es of CRRA agains BDRA jus on saionary momen condiions. Compared o a es based on all momen condiions, his es is conservaive, i.e., if he null hypohesis is rejeced, i 18

19 will also be rejeced in a es ha includes dynamic momen condiions. Similarly, o es wheher he sae uncerainy parameer is null mus be based exclusively on dynamic momen condiions. This follows because he saionary momen condiions do no depend on he aversion o sae uncerainy parameer in he presence of a minimal risk aversion regime. As a resul, he D- saisic calculaed for he es H 0 : ϱ 0 versus H 1 : ϱ 0 only depends on he dynamic momen condiions. 4 Table 3 summarizes he es resuls. Parameer esimaes for he consrained model are given in Table 6 for CRRA and Table 7 for CASU. The null hypohesis of consan risk aversion is rejeced (A1, B1) whereas he null hypohesis of no sae uncerainy aversion canno be rejeced (A2). These wo ess provide evidence ha he model wih beliefs-dependen risk aversion, i.e., he BDRA model is he bes model. Two addiional ess provide furher evidence for his claim. Wihin he BDRA model specificaion (ϱ 0), he null hypohesis H 0 : β k β for all k and R k R some k vs. H 1 : β k β and R k R some k canno be rejeced (B3). There is no evidence for regime-dependen subjecive discoun rae parameers. Similarly, he null hypohesis H 0 : R 1 R 3 vs. H 1 : R 1 R 3 can be rejeced (B2). Resricing he number of risk aversion regimes and forcing a symmeric Ş -shaped srucure of risk aversion parameers does no improve he model fi. Non-nesed specificaion ess and AIC and BIC model selecion crieria srongly favor he BDRA model over CASU. The null hypohesis ha BDRA and CASU provide equivalen fis of he momen condiions is srongly rejeced agains he hypohesis ha BDRA provides a beer fi. Boh model selecion crieria srongly favor BDRA even hough BDRA has one more parameer. Non-nesed specificaion ess and model selecion crieria confirm he resuls of nesed specificaion ess. To summarize, all he specificaion ess sugges ha he bes model is he BDRA model wih hree risk aversion parameers, no sae uncerainy aversion and consan subjecive discoun raes. 4 If only saionary momen condiions are used and he BDRA model is he correc model, he esimaed sae uncerainy parameer mus be infinie. Esimaion resuls obained from saionary momen condiions alone confirm his, providing furher evidence agains he CASU model. 19

20 Table 3: This able summarizes all he specificaion ess of he models and he model selecion crieria based on he sample period 01/ /2014. Boh nesed (panels (A) and (B)) and non-nesed (panel (C)) ess are performed (see online Appendix for more deails). GMM-AIC and GMM-BIC model selecion crieria are calculaed as described in Andrews (1999). I. Specificaion ess Tes-saisic (D T resp. N T ) criical value p-value (A) Nesed model specificaion ess unified model (D-es, size α 5%) (A1) Tes CRRA: p q H 0 : R k R, β k β and ϱ 0 vs. H 1 : R k R, β k β and ϱ 0 or ϱ 0 ą (A2) Tes CASU: H 0 : ϱ 0, β k β, R k R some k vs. H 1 : ϱ 0, β k β, R k R some k (A3) Tes bea: H 0 : β k β, ϱ 0, and R k R some k vs. H 1 : β k β, ϱ 0 and R k R some k (B) Nesed model specificaion es wihin BDRA (ϱ 0) (D-es, size α 5%) (B1) Tes CRRA: p q H 0 : R k R vs. H 1 : R k R ą (B2) Tes # R-regimes: H 0 : R 1 R 3 vs. H 1 : R 1 R (B3) Tes bea: H 0 : β k β all k, and R k R some k vs. H 1 : β k β and R k R some k (C) Non-nesed model specificaion ess (Rivers-Vuong es, size α 5%) (C1) Asympoic, non-nesed Tes CASU: p q H 0 : CASU BDRA vs. H 1a,b : BDRA ą ăcasu (C2) Boosrapped, non-nesed Tes CASU: H 0 : CASU BDRA vs. H 1a,b : BDRA ą ăcasu II. GMM-AIC and GMM-BIC model selecion crieria Model GMM-AIC GMM-BIC CASU BDRA ( ) This es only depends on saionary momen condiions and is herefore conservaive. ( ) This es is a conservaive es wihin he BDRA specificaion. ( ) This es assumes ha boh CASU and BDRA are misspecified such ha he asympoic disribuion is Gaussian. Table 4 compares saionary momens of various model specificaions. The CRRA model gen- 20

21 eraes low asse volailiy and low risk premium, consisen wih he predicions above. The CASU model provides reasonable uncondiional momens, bu he correlaions are off as well. In fac, as shown in Figure 8, he mos imporan volailiy componen in he CASU model is he consumpion componen, σ SC. This ranslaes ino a srong consumpion componen in he risk premium and, as a consequence, a correlaion beween consumpion and sock reurns ha is oo high. The same occurs for he correlaion beween he changes in log-pdr and consumpion growh. In he CASU model, momens are affeced by boh cash-flow and discoun facor risks, bu he discoun facor risk is heavily affeced by consumpion variaions. This is he case despie he presence of sae probabiliies in he sochasic discoun facor, because he consumpion componen dominaes from an empirical poin of view. The ension beween maching he equiy premium and he sock volailiy, and maching correlaions beween cash-flows and sock reurns also affecs he parameer esimaion in he CRRA model. To mach correlaions in he CRRA model, he risk aversion coefficien mus be low. This resuls in a risk premium ha is oo small. Alernaively, if risk aversion is high, he risk premium is larger, bu as his risk premium is generaed exclusively by consumpion risk, a srong correlaion beween reurns and cash flows emerges, i.e., here is a correlaion puzzle. 21

22 Variance decomposiion and variance (red) - sored - CASU Specificaion consumpion share dividend share informaion share sock variance Figure 8: The plo shows he sock reurn variance decomposiion along wih he model implied sock variance displayed by increasing level of variance, for he CASU specificaion. The hree componens correspond o consumpion source (blue), dividend source (green) and informaion source (yellow). In conras, in he BDRA specificaion, he mos imporan volailiy componen is σ SY, i.e., he informaion risk facor (see Figure 6 in he paper and Table 5 below). This solves he correlaion puzzles: he correlaion beween sock reurns and consumpion and beween sock reurns and dividends are boh low. In addiion, oher momens such as risk premia and volailiies mach he daa well. This is a direc consequence of he differences in model specificaions beween CASU and BDRA. Table 5 provides he decomposiions of volailiies in heir orhogonalized consiuens for each model. This decomposiion highlighs he imporance of he informaion facor for volailiy in he BDRA specificaion. In conras o CASU and CRRA, he BDRA model can generae a high risk premium hrough he discoun facor risk channel (informaion premium) and keep correlaions beween cash-flows (consumpion and dividends) and sock reurns low and as such resolve he correlaion and equiy premium puzzles. 22

23 Table 4: Model comparison: The following able compares various momens for he differen specificaions wih empirical esimaes and confidence inervals. BDRA CASU CRRA daa lower bound upper bound µ C µ D log PDR yield 10 years σ sock σ yield excess reurn ρ SC ρ SD ρ Y C ρ Y D µ unem ρ rc ρ rd σ logppdrq ρ logppdrq,c yield 3 monhs ρ r,y Table 5: Volailiy decomposiion: σ Sα {σ S where α P C, G, Y u. lower ier mid ier op ier BDRA cons (C) 12.7% 37.6% 21.4% div (G) 48.4% 11.2% 2.8% info (Y) 38.9% 51.2% 75.8% CASU cons (C) 3.0% 4.29% 50.25% div (G) 97.0% 95.59% 48.82% info (Y) 0.0% 0.12% 0.93% 23

24 Table 6: Esimaed parameers (sandard errors) CRRA model: GMM parameer esimaes wih sandard errors obained from saionary boosrap (Poliis and Romano (1994). Growh Regime Normal Low High Consumpion µ C 1 µ C 2 µ C ( ) ( ) ( ) Dividend µ D 1 µ D 2 µ D ( ) ( ) ( ) Growh Regime Normal Low High Unemploymen µ UE 1 µ UE 2 µ UE ( ) ( ) ( ) Preferences: risk aversion R ( ) Preferences: subjecive discoun rae ( ) Sandard Deviaions and Correlaions Consumpion Dividend Unemploymen Consumpion (0.0006) (0.0653) (0.0594) Dividend (0.0653) (0.0079) (0.0596) Unemploymen (0.0594) (0.0596) (0.0137) Infiniesimal Generaor Normal Low High Seady sae probabiliies Normal e ( ) (7.728e-07) Low ( ) - ( ) High e ( ) (3.0869e-06) - 24

25 Table 7: Esimaed parameers (sandard errors) CASU model: GMM parameer esimaes wih sandard errors obained from saionary boosrap (Poliis and Romano (1994). Growh Regime Normal Low High Consumpion µ C 1 µ C 2 µ C ( ) ( ) ( ) Dividend µ D 1 µ D 2 µ D ( ) ( ) ( ) Preferences: subjecive discoun rae ( ) Growh Regime Normal Low High Unemploymen µ UE 1 µ UE 2 µ UE ( ) ( ) ( ) Preferences: risk aversion R ( ) Preferences: sae uncerainy aversion (6.0630) Sandard Deviaions and Correlaions Consumpion Dividend Unemploymen Consumpion (0.0006) (0.0653) (0.0594) Dividend (0.0653) (0.0079) (0.0596) Unemploymen (0.0594) (0.0596) (0.0137) Infiniesimal Generaor Normal Low High Seady sae probabiliies Normal e ( ) (1.5733e-06) Low ( ) - ( ) High e ( ) (1.6672e-06) - 25

26 Table 8: Esimaed parameers (sandard errors) BDRA model: GMM parameer esimaes wih sandard errors obained from saionary boosrap (Poliis and Romano (1994)). The sample period is January December Growh Regime Normal Low High Consumpion µ C 1 µ C 2 µ C ( ) ( ) ( ) Dividend µ D 1 µ D 2 µ D ( ) ( ) ( ) Growh Regime Normal Low High Unemploymen µ UE 1 µ UE 2 µ UE ( ) ( ) ( ) Preferences: risk aversion R 1 R 2 R ( ) ( ) ( ) Preferences: subjecive discoun rae β ( ) Sandard Deviaions and Correlaions Consumpion Dividend Unemploymen Consumpion (0.0618) (0.2110) (0.2239) Dividend (0.2110) (0.0221) (0.1801) Unemploymen (0.2239) (0.1801) (0.0773) Infiniesimal Generaor Normal Low High Seady sae probabiliies Normal ( ) (1.7331e-06) Low ( ) - ( ) High ( ) (1.8853e-06) - 26

27 References Gallan, A.R., Jorgenson, D.W., Saisical inference for a sysem of simulaneous nonlinear implici equaions in he conex of insrumenal variable esimion, Journal of Economerics 11, Hall, A.R., Pelleier D., Nonnesed esing in models esimaed via generalized mehod of momens, Economeric Theory 27, Newey, W.K., Wes, K.D., Hypohesis esing wih efficien mehods of momen esimaion, Inernaional Economic Review 28, Rivers, D., Vuong, Q., Model selecion ess for nonlinear dynamic models, Economerics Journal 5,