Appendix to Bond Risk Premia

Transcription

1 Appendix o Bond Risk Premia John H. Cochrane and Monika Piazzesi December Revised Sep ; ypos fixedinequaions(d.17)-(d.20) A Addiional Resuls A.1 Unresriced forecass Table A1 repors he poin esimaes of he unresriced regression (1) ogeher wih sandard errors and es saisics. The coefficien esimaes in Table A1 are ploed in he op panel of Figure 1, and he R 2 and some of he saisics from Table A1 are repored in he righ side of Table 1B. The able shows ha he individual-bond regressions have he same high significance as he regression of average (across mauriy) excess reurns on forward raes documened in Table 1A. A.2 Fama-Bliss Table A2 presens a full se of esimaes and saisics for he Fama-Bliss regressions summarized in Table 2. The main addiions are he confidence inervals for R 2 and he small-sample disribuions. The R 2 confidence inervals show ha he Fama-Bliss regressions do achieve an R 2 ha is unlikely under he expecaions hypohesis. However, he Fama-Bliss R 2 is jus above ha confidence inerval where he 0.35 R 2 were much furher above he confidence inervals in Table A1. Again, he muliple regression provides sronger evidence agains he expecaions hypohesis, accouning for he larger number of righ hand variables, even accouning for small-sample disribuions. The small-sample sandard errors are larger, and χ 2 saisics smaller, han he largesample counerpars. The paern is abou he same as for he muliple regressions in Table 1. However, he small-sample saisics sill rejec. The rejecion in Table 1 is sronger wih small-sample saisics as well. A.3 ComparisonwihFama-Bliss If he reurn-forecasing facor really is an improvemen over oher forecass, i should drive ou oher variables, and he Fama-Bliss spread in paricular. Table A3 presens muliple regressions o address his quesion. In he presence of he Fama-Bliss forward spread, he coefficiens and significance of he reurn-forecasing facor from Table A2 are unchanged in 1

2 Table A3. The R 2 is also unaffeced, meaning ha he addiion of he Fama-Bliss forward spread does no help o forecas bond reurns. In he presence of he reurn-forecasing facor, however, he Fama-Bliss slope disappears. Clearly, he reurn-forecasing facor subsumes all he predicabiliy of bond reurns capured by he Fama-Bliss forward spread. Table A1. Regressions of 1-year excess reurns on all forward raes Mauriy n cons. y (1) f (2) f (3) f (4) f (5) R 2 Level R 2 χ 2 (5) Large T (0.69) (0.17) (0.40) (0.29) (0.22) (0.17) h0.00i Small T (0.86) (0.30) (0.50) (0.40) (0.30) (0.28) [0.19, 0.53] 32.9 EH [0.00, 0.17] h0.00i Large T (1.27) (0.30) (0.67) (0.47) (0.41) (0.30) h0.00i Small T (1.53) (0.53) (0.88) (0.71) (0.53) (0.50) [0.21, 0.55] 38.6 EH [0.00, 0.17] h0.00i Large T (1.73) (0.44) (0.87) (0.59) (0.55) (0.40) h0.00i Small T (2.03) (0.71) (1.18) (0.94) (0.71) (0.68) [0.24, 0.57] 46.0 EH [0.00, 0.17] h0.00i Large T (2.16) (0.55) (1.03) (0.67) (0.65) (0.49) h0.00i Small T (2.49) (0.88) (1.46) (1.16) (0.88) (0.85) [0.21, 0.55] 39.2 EH [0.00, 0.17] h0.00i Noes: The regression equaion is rx (n) +1 = β (n) 0 + β (n) 1 y (1) + β (n) 2 f (2) β (n) 5 f (5) + ε (n) +1. R 2 repors adjused R 2. Level R 2 repors he R 2 from a regression using he level excess reurn on he lef hand side, e r(n) +1 e y (1). Sandard errors are in parenheses (). Large T sandard errors use he 12 lag Hansen-Hodrick GMM correcion for overlap and heeroskedasiciy. Small T sandard errors are based on 50,000 boosrapped samples from an unconsrained 12 lag yield VAR. Square brackes [] are 95 percen boosrap confidence inervals for R 2. EH imposes he expecaions hypohesis on he boosrap: We run a 12 lag auoregression for he 1-year rae and calculae oher yields as expeced values of he 1-year rae. χ 2 (5) is he Wald saisic ha ess wheher he slope coefficiens are joinly zero. The 5 percen and 1 percen criical value for χ 2 (5) are 11.1 and All χ 2 saisics are compued wih 18 Newey-Wes lags. Small T Wald saisics are compued from he covariance marix of parameer esimaes across he boosrapped samples. Poined brackes <> repor probabiliy values. Daa source CRSP, sample 1964:1-2003:12. 2

3 TableA2.Fama-Blissexcessreurnregressions Mauriy n α β R 2 χ 2 (1) p-value Large T (0.30) (0.26) 18.4 h0.00i Small T (0.16) (0.33) [0.01, 0.33] 9.1 h0.00i EH [0.00, 0.12] h0.01i Large T (0.54) (0.35) 19.2 h0.00i Small T (0.32) (0.41) [0.01, 0.34] 10.8 h0.00i EH [0.00, 0.14] h0.01i Large T (0.75) (0.45) 16.4 h0.00i Small T (0.48) (0.48) [0.01, 0.34] 11.2 h0.00i EH [0.00, 0.14] h0.01i Large T (1.04) (0.58) 5.7 h0.02i Small T (0.64) (0.64) [0.00, 0.24] 4.0 h0.04i EH [0.00, 0.14] h0.13i Noes: The regressions are rx (n) +1 = α + β ³ f (n) y (1) + ε +1. (n) Sandard errors are in parenheses, boosrap 95 percen confidence inervals in square brackes [] and probabiliy values in angled brackes <>. The 5 percen and 1 percen criical values for a χ 2 (1) are 3.8 and 6.6. See noes o Table A1 for deails. Table A3. Cones beween γ > f and Fama-Bliss n a n σ (a n ) b n σ(b n ) c n σ(c n ) R (0.25) 0.46 (0.04) 0.05 (0.21) (0.48) 0.87 (0.12) 0.05 (0.41) (0.62) 1.22 (0.16) 0.05 (0.46) (0.71) 1.43 (0.15) 0.15 (0.35) 0.35 Noes: Muliple regression of excess holding period reurns on he reurn-forecasing facor and Fama-Bliss slope. The regression is rx (n) ³ +1 = a n + b n γ > f + cn f (n) y (1) + ε (n) +1. Sandard errors are in parenheses. See noes o Table A1 for deails. 3

4 A.4 Forecasing he shor rae The reurn-forecasing facor also predics changes in shor-erm ineres raes. Shor rae forecass and excess reurn forecass are mechanically linked, as emphasized by Fama and Bliss (1987), bu seeing he same phenomenon as a shor rae forecas provides a useful complemenary inuiion and suggess addiional implicaions. Here, he expecaions hypohesis predics a coefficien of 1.0 if he forward rae is one percenage poin higher han he shor rae, we should see he shor rae rise one percenage poin on average. Table A4. Forecasing shor rae changes cons. f (2) y (1) y (1) f (2) f (3) f (4) f (5) R 2 χ 2 Fama-Bliss Large T (0.30) (0.26) h0.98i Small T (0.16) (0.33) [0.00, 0.33] 0.0 EH [0.05, 0.29] h1.00i Unconsrained Large T (0.69) (0.17) (0.40) (0.29) (0.22) (0.17) h0.00i Small T (0.86) (0.30) (0.50) (0.40) (0.30) (0.28) [0.15, 0.40] 25.0 EH [0.07, 0.32] h0.01i Noes: The Fama-Bliss regression is ³ y (1) +1 y (1) = β 0 + β 1 f (2) y (1) + ε +1. The unconsrained regression equaion is y (1) +1 y (1) = β 0 + β 1 y (1) + β 2 f (2) β 5 f (5) + ε +1. χ 2 ess wheher all slope coefficiens are joinly zero (5 degrees of freedom unconsrained, one degree of freedom for Fama-Bliss). Sandard errors are in parenheses, boosrap 95 percen confidence inervals in square brackes [] and percen probabiliy values in angled brackes <>. See noes o Table A1 for deails. The Fama-Bliss regression in Table A4 shows insead ha he wo year forward spread has no power o forecas a one year change in he one year rae. Thus, he Fama and Bliss s (1987) reurn forecass correspond o nearly random-walk behavior in yields. To find greaer forecasabiliy of bond excess reurns,our reurn-forecasingfacormusand does forecas changes in 1-year yields; a posiive expeced reurn forecass implies ha bond prices will rise. Indeed, in he unconsrained panel of Table A4, all forward raes ogeher 4

5 have subsanial power o predic one-year changes in he shor rae. The R 2 for shor rae changes rises o 19 percen, and he χ 2 es srongly rejecs he null ha he parameers are joinly zero. To undersand his phenomenon, noe ha we can always break he excess reurn ino a one year yield change and a forward-spo spread, 7 ³ ³ ³ (A.1) E rx (2) +1 = E y (1) +1 y (1) + f (2) y (1). Inuiively, you make money eiher by capial gains, or by higher iniial yields. Under he expecaions hypohesis, expeced excess reurns are consan, so any movemen in he forwardspreadmusbemachedbymovemensin he expeced 1-year yield change. If he forward rae is higher han he spo rae, i mus mean ha invesors mus expec a rise in 1-year raes (a decline in long-erm bond prices) o keep expeced reurns he same across mauriies. In Fama and Bliss s regressions, he expeced yield change erm is consan, so changes in expeced reurns move one-for one wih he forward spread. In our regressions, expeced reurns move more han changes in he forward spread. The only way o generae such changes is if he 1-year rae becomes forecasable as well, generaing expeced capial gains and losses for long-erm bond holders. Equaion (A.1) also means ha he regression coefficiens which forecas he 1-year rae change ³ in Table A4 are exacly equal o our reurn-forecasing facor b 2 γ > f, which forecass E rx (2) +1, minus a coefficien of 1 on he 2-year forward spread. The facor ha forecass excess reurns is also he sae variable ha forecass he shor rae. A.5 Addiional Lags Table 5 repors our esimaes of γ and α in he simple model for addiional lags, i rx +1 = γ hα > 0 f + α 1 f α k f 12 k + ε (n) Table A5 complees he model by showing how individual bond reurns load on he common reurn-forecasing variable, i.e. esimaes of b n in i rx (n) +1 = b n γ hα > 0 f + α 1 f α k f 12 k + ε (n) In Panel A, he b n rise wih mauriy. The b esimaes using addiional lags are almos exacly he same as hose using only f, as claimed in he paper. InPanelB,weseehaheR 2 for individual regressions mirror he R 2 for he forecass of bond average (across mauriy) reurns rx. We also see ha he R 2 from he resriced regressions are almos as high as hose of he unresriced regressions, indicaing ha he resricions do lile harm o he model s abiliy o fi he daa. This finding is especially 5

6 cogen in his case, as he unresriced regressions allow arbirary coefficiens across ime (lags) as well as mauriy. For example, wih 3 addiional lags, he unresriced regressions use 4 bonds 4lags (5 forward raes + 1 consan) = 96 parameers, while he resriced regressions use 4 b +6 γ +4α = 14 parameers. In sum, Table A5 subsaniaes he claim in he paper ha he single-facor model works jus as well wih addiional lags as i does using only ime righ hand variables. Table A5. Reurn forecass wih addiional lags A. b n esimaes and sandard errors Esimaes Sandard errors Lags Lags b (0.06) (0.04) (0.04) (0.04) b (0.11) (0.09) (0.08) (0.08) b (0.17) (0.13) (0.12) (0.11) b (0.21) (0.16) (0.15) (0.14) B. R 2 Resriced Unresriced Lags Lags rx (2) rx (3) rx (4) rx (5) rx Noes: Reurn forecass wih addiional lags, using he resriced model i rx (n) +1 = b n γ hα > 0 f + α 1 f 1 + α 2 f α k f 12 k + ε (n) Esimaes of γ and α are presened in Table 5. A.6 Eigenvalue facor models for yields We form he yield curve facors x used in Table 4 and Figure 2 from an eigenvalue decomposiion of he covariance marix of yields var(y) =QΛQ > wih Q > Q = I, Λ diagonal. Decomposiions based on yield changes, reurns, or excess reurns are nearly idenical. Then we can wrie yields in erms of facors as y = Qx ; cov(x,x > )=Λ. Here, he columns of Q give how much each facor x i moves all yields y. We can also wrie x = Q > y. Here, he columns of Q ell you how o recover facors from yields. The op righ panel of Figure 2 6

7 plos he firs hree columns of Q. We label he facors level, slope, curvaure, 4-5 and W based on he shape of hese loadings. We do no plo he las wo small facors for clariy, and because being so small hey are poorly idenified separaely. W is W shaped, loading mos srongly on he 3 year yield. 4-5 is mosly a 4-5 year yield spread. We compue he fracion of yield variance due o he kh facor as Λ kk /Q k Λ kk. To calculae he fracion of yield variance due o he reurn-forecasing facor, we firs run a regression y (n) = a + bγ > f + ε, and hen we calculae race cov(bγ > f) /race[cov(y)]. A.7 Eigenvalue facor model for expeced excess reurns We discuss in secion B. an eigenvalue facor decomposiion of he unconsrained expeced excess reurn covariance marix. We sar wih QΛQ > = cov E (rx +1 ),E (rx +1 ) > = βcov(f,f > )β >. Now we can wrie he unconsrained regression (A.2) E (rx +1 )=βf = QΓ > f. wih Γ > = Q > β. Equivalenly, we can find he facors Γ by regressing porfolios of expeced reurns on forward raes, Q > rx +1 = Γ > f + Q > ε +1. Table A6 presens he resuls. The firs column of Q in Panel A ells us how he firs expeced-reurn facor moves expeced reurns of bonds of differen mauriies. The coefficiens rise smoohly from 0.21 o This column is he equivalen of he b coefficiens in our single-facor model E (rx +1 )=bγ >. The corresponding firs row of Γ > in Panel B is very nearly our en-shaped funcion of forward raes. Expressed as a funcion of yields i displays almos exacly he paern of he op lef panel in Figure 2: a rising funcion of yields wih a srong 4-5 spread. The remaining columns of Q and rows of Γ > show he srucure summarized by simple porfolio regressions in he ex. When a porfolio loads srongly on one bond in Panel A, ha bond s yield is imporan for forecasing ha porfolio in Panel C. The remaining facors seem o be linear combinaions (organized by variance) of he paern shown in Table 7. Individual bond pricing errors in yields seem o be reversed. The boom wo rows of Panel A give he variance decomposiion. The firs facor capures almos all of he variaion in expeced excess reurns. Is sandard deviaion a 5.16 percenage poins dominaes he 0.26, 0.16 and 0.20 percenage poin sandard deviaions of he oher facors. Squared, o express he resul as fracions of variance, he firs facor accouns for 99.5 percen of he variance of expeced reurns. 7

8 Table A6. Facor decomposiion for expeced excess reurns A. Q marix of loadings Facor Mauriy σ(facor) Percen of var B. Γ > marix; forecasing he porfolios Facor y (1) f (2) f (3) f (4) f (5) C. Γ > marix wih yields Facor y (1) y (2) y (3) y (4) y (5) Noes: We sar wih he unconsrained forecasing regressions rx (n) +1 = β (n) f + ε (n) +1. Then, we perform an eigenvalue decomposiion of he covariance marix of expeced excess reurns, QΛQ > = cov E (rx +1 ),E (rx +1 ) > = βcov(f,f > )β >. Panel A gives he Q marix. The las wo rows of panel A give Λ i and Λ i / P Λ i respecively. Panels B and C give regression coefficiens in forecasing regressions Q > rx +1 = Γ > f + Q > ε +1 Q > rx +1 = Γ > y + Q > ε +1. 8

9 A.8 Wha measuremen error can and canno do To undersand he paern of Figure 4, wrie he lef hand variable as rx (n) +1 = p (n 1) +1 p (n) + p (1) = p (n 1) +1 + ny (n) y (1). Now, consider a regression of his reurn on o ime- variables. Clearly, measuremen error in prices, forward raes or yields anyhing ha inroduces spurious variaion in ime- variables will induce a coefficien of 1 on he one year yield and +n on he n year yield, as shown in he boom panel of Figure 4. Similarly, if we wrie he lef hand variable in erms of forward raes as rx (n) +1 = p (n 1) +1 p (n) + p (1) h p (n) = p (n 1) p (n 1) i + h p (n 1) i h i + p (n 2) + + p (2) + p (1) p (1) + p (1) = p (n 1) y (1) +1 f (1 2) +1 f (2 3) + +1 f (n 1 n) we see he sep-funcion paern shown in he op panel of Figure 4. The crucial requiremen for his paern o emerge as a resul of measuremen error is ha he measuremen error a ime mus be uncorrelaed wih in p (n 1) +1 on he lef hand side. If measuremen error a ime is correlaed wih he measured variable a ime +1,hen oher ime- variables may seem o forecas reurns, or he 1 and n year yield may forecas i wih differen paerns. Of course, he usual specificaion of i.i.d. measuremen error is more han adequae for his conclusion. Also, measuremen error mus be uncorrelaed wih he rue righ hand variables, as we usually specify. Measuremen errors correlaed across mauriy a a given ime will no change his paern. Muliple regressions orhogonalize righ hand variables. B Robusness checks We invesigae a number of desirable robusness checks. We show ha he resuls obain in he McCulloch-Kwan daa se. We show ha he resuls are sable across subsamples. In paricular, he resuls are sronger in he low-inflaion 1990s han hey are in he highinflaion 1970s. This finding comforingly suggess a premium for real raher han nominal ineres rae risk. We show ha he resuls obain wih real-ime forecass, raher han using he full sample o esimae regression coefficiens. We consruc rading rule profis, examine heir behavior, and examine real-ime rading rules as well. The rading rules improve subsanially on hose using Fama-Bliss slope forecass. 9

10 B.1 Oher daa The Fama-Bliss daa are inerpolaed zero-coupon yields. In Table A7, we run he regressions wih McCulloch-Kwon daa, which use a differen inerpolaion scheme o derive zero-coupon yields from Treasury bond daa. Table A7 also compares he R 2 and γ esimaes using McCulloch-Kwon and Fama-Bliss daa over he McCulloch-Kwon sample (1964:1-1991:2). Clearly, he en-shape of γ esimaes and R 2 are very similar across he wo daases. Table A7. Comparison wih McCulloch-Kwon daa A. R 2 All f γ > f f (n) y (1) n M-K F-B M-K F-B M-K F-B B. Coefficiens Daase γ 0 γ 1 γ 2 γ 3 γ 4 γ 5 McCulloch-Kwon Fama-Bliss Noes: The daa are McCulloch-Kwon and Fama-Bliss zero-coupon yields saring 1964:1 unil he end of he McCulloch-Kwon daase, 1991:12. The upper panel shows R 2 from he regressions corresponding o Table 1. The regressions run excess log reurns rx (n) +1 on he regressors indicaed on op of he able: all forwards f, he reurn-forecasing facor γ > f, and he forward spread f (n) y (1). The lower panel shows he esimaed γ coefficiens in he regression of average reurns on forward raes rx +1 = γ > f + ε +1.McCulloch-Kwondaaaredownloaded from hp:// B.2 Subsamples Table A8 repors a breakdown by subsamples of he regression of average (across mauriy) excess reurns rx +1 on forward raes. The firs se of columns run he average reurn on he forward raes separaely. The second se of columns runs he average reurn on he reurnforecasing facor γ > f where γ is esimaed from he full sample. The laer regression moderaes he endency o find spurious forecasabiliy wih five righ hand variables in shor ime periods. The firs row of Table A8 reminds us of he full sample resul he prey en-shaped coefficiens and he 0.35 R 2. Of course, if you run a regression on is own fied value you 10

11 ge a coefficien of 1.0 and he same R 2, as shown in he wo righ hand columns of he firs row. The second se of rows examine he period before, during, and afer he 1979:8-1982:10 episode, when he Fed changed operaing procedures, ineres raes were very volaile, and inflaion declined and sabilized. The broad paern of coefficiens is he same before and afer. The 0.78 R 2 looks dramaic during he experimen, bu his period really only has hree daa poins and 5 righ hand variables. When we consrain he paern of he coefficiens in he righ hand pair of columns, he R 2 ishesameasheearlierperiod. The las se of rows break down he regression by decades. Again, he paern of coefficiens is sable. The R 2 is wors in he 70s, a decade dominaed by inflaion, bu he R 2 rises o a dramaic 0.71 in he 1990s, and sill 0.43 when we consrain he coefficiens γ o heir full-sample values. This fac suggess ha he forecas power derives from changes in he real raher han nominal erm srucure. Table A8. Subsample analysis Regression on all forward raes γ > f only γ 0 γ 1 γ 2 γ 3 γ 4 γ 5 R 2 γ > f R : : : : : : : : : : : : : : : : : : Noes: Subsample analysis of average reurn-forecasing regressions. For each subsample, he firs se of columns presen he regression rx +1 = γ > f + ε +1. The second se of columns repor he coefficien esimae b and R 2 from rx +1 = b γ > f + ε+1 using he γ parameer from he full sample regression. Sample: B.3 Real ime forecass and rading rule profis How well can one forecas bond excess reurns using real-ime daa? Of course, he convenional raional-expecaions answer is ha invesors have hisorical informaion, and have 11

12 evolved rules of humb ha summarize far longer ime series han our daa se, so here is no error in using full-sample forecass. Sill, i is an ineresing robusness exercise o see how well he forecas performs based only on daa from 1964 up o he ime he forecas mus be made. Figure 8 compares real-ime and full-sample forecass. They are quie similar. Even hough he real-ime regression sars he 1970s wih only 6 years of daa, i already capures he paern of bond expeced reurns. The forecass are similar, bu are hey similarly successful? Figures 9 and 10 compare hem wih a simple calculaion. We calculae rading rule profis as rx +1 E (rx +1 )=rx +1 γ > (α 0 f + α 1 f 1 + a 2 f 2 ). This rule uses he forecas E (rx +1 ) o recommend he size of a posiion which is subjec o he ex-pos reurn rx +1. In Figure 10, we cumulae he profis so ha he differen cases can be more easily compared. 8 For he Fama-Bliss regressions, we calculae he expeced excess reurn of each bond from is mached forward spread, and hen we find he average expeced excess reurn across mauriies in order o compue E (rx +1 ). The full-sample vs. real-ime rading rule profis in Figure 9 are quie similar. In boh Figure 9 and he cumulaed profis of Figure 10 all of he rading rules produce surprisingly few losses. The lines eiher rise or say fla. The rading rules lie around waiing for occasional opporuniies. Mos of he ime, he forward curve is no really rising a lo, nor en shaped, so boh rules see a small E (rx +1 ). The rading rules hus recommend small posiions, leading o small gains and losses. On infrequen occasions, he forward curve is eiher rising or en-shaped, so he rading rules recommend large posiions E (rx +1 ), and hese posiions seem o work ou. The real-ime forecas looks quie good in Figure 9, bu he cumulaive difference amouns o abou half of he full-sample profis. However, he real-ime rading rule does work, and i even works beer han even he full sample Fama-Bliss forecas. We conclude ha he overall paern remains in real-ime daa. I does no seem o be he case ha he forecas power, or he improvemen over he Fama-Bliss forecass and relaed slope forecass, requires he use of ex-pos daa. Of course, real rading rules should be based on arihmeic reurns, and hey should follow an explici porfolio maximizaion problem. They also mus incorporae esimaes of he condiional variance of reurns. Bond reurns are heeroskedasic, so one needs o embark on a similar-sized projec o undersand condiional second momens and relae hem o he condiional firs momens we have invesigaed here. 12

13 10 5 Full sample 0 5,-5 Real ime Figure 8: Full-sample and real-ime forecass of average (across mauriy) excess bond reurns. In boh cases, he forecas is made wih he regression rx +1 = γ > f + ε +1. The real-ime graph re-esimaes he regression a each from he beginning of he sample o Full Sample Real Time Figure 9: Trading rule profis, using full-sampe and real-ime esimaes of he reurnforecasing facor. 13

14 CP full sample CP real ime 1000 FB full sample 500 FB real ime Figure 10: Cumulaive profis from rading rules using full sample and real ime informaion. Each line plos he cumulaive value of rx +1 E (rx +1 ). E (rx +1 ) are formed from he full sample or real ime daa from as marked. The CP lines use he forecas rx +1 = γ > (α 0 f + α 1 f α 2 f 2 ). The FB (Fama-Bliss) lines forecas each 12 excess reurn from he corresponding forward spread, and hen average he forecass across mauriies. 14

15 C Calculaions for he regressions C.1 GMM esimaes and ess The unresriced regression is rx +1 = βf + ε +1, The momen condiions of he unresriced model are (C.3) g T (β) =E(ε +1 f )=0. The resriced model is β = bγ >, wih he normalizaion b > 1 4 =4. We focus on a 2-sep OLS esimae of he resriced model firs esimae average (across mauriies) reurns on f, hen run each reurn on ˆγ > f: (C.4) (C.5) rx +1 = γ > f + ε +1, rx +1 = b ˆγ > f + ε+1. The esimaes saisfy 1 > 4 b = 4 auomaically. To provide sandard errors for he wo-sep esimae in Table 1, we use he momens corresponding o he wo OLS regressions (C.4) and (C.5), E ( ε+1 (b, γ) f g T (b, γ) = ) E ε +1 (b, γ) γ > =0. f Since he esimae is exacly idenified from hese momens (a = I) Hansen s (1982) Theorem 3.1 gives he sandard error, µ var = ˆγˆb 1 T d 1 S d 1> where d = g T [γ > b > ] = E f rx+1 f > γ [γ > b > ] E rx +1 bγ > f f > γ ) E f = f > E(rx +1 f > ) 2bγ > E f f > γ > E f f >. γ I4 Since he upper righ block is zero, he upper lef block of d 1 is E f f > 1. Therefore, he variance of γ is no affeced by he b esimae, and is equal o he usual GMM formula for a regression sandard error, var(ˆγ) =E(ff > ) 1 S(1 : 6, 1:6)E(ff > ) 1 /T. The variance of ˆb is affeced by he generaed regressor γ, viaheoff diagonal erm in d 1.This is an ineresing case in which he GMM sandard errors ha correc for he generaed regressor are smaller han OLS sandard errors ha ignore he fac. OLS has no way of knowing ha P n b n =1, while he GMM sandard errors know his fac. OLS sandard errors hus find a common 15

16 componen o he b sandard errors, while GMM knows ha common movemen in he b is soaked up in he γ esimaes. We invesigaed a number of ways of compuing S marices. The regression is rx +1 = γ > f + ε +1. The Hansen-Hodrick ( HH ) saisics are based on " kx # cov(ˆγ) =E(f f > ) 1 E(f f j > ε +1 ε +1 j ) E(f f > ) 1. The Newey-Wes ( NW ) saisics use cov(ˆγ) =E(f f > ) 1 " kx j= k j= k # k j E(f f j > ε +1 ε +1 j ) E(f f > ) 1. k The Simplified HH saisics assume E(f f j > ε +1 ε +1 j )=E(f f j)e( ε > +1 ε +1 j )and E( ε +1 ε +1 j )= k j E( ε 2 k +1), hence " kx # cov(ˆγ) =E(f f > ) 1 k j E(f f > k j) E(f f > ) 1 E( ε 2 +1). No overlap saisics use j= k cov(ˆγ) =E(f f > ) 1 E(f f > ε 2 +1)E(f f > ) 1 averaged over 12 iniial monhs. To es he (inefficien) wo sep esimae in Table 6, we apply Hansen s Lemma 4.1 he counerpar o he J T es ha handles inefficien as well as efficien esimaes. To do his, we mus firs express he resriced esimae as a GMM esimae based on he unresriced momen condiions (C.3). The wo sep OLS esimae of he resriced model ses o zero a linear combinaion of he unresriced momens: (C.6) a T g T =0, where a T = I 6 I 6 I 6 I 6 γ > γ > γ > = 1 > 4 I 6 I 3 γ > The firs row of ideniy marices in a T sums across reurn mauriies o do he regression of average reurns on all forward raes. The las hree rows sum across forward raes a a given reurn mauriy o do he regression of each reurn on γ > f. An addiional row of a T of he form γ > o esimae he las elemen of b would be redundan he b 4 regression is implied by he firs hree regressions. The esimae is he same wheher one 16

17 runs ha regression or jus esimaes b 4 =1 b 1 b 2 b 3. We follow he laer convenion since he GMM disribuion heory is wrien for full rank a marices. I is iniially roubling o see a parameer in he a marix. SinceweuseheOLSγ esimae in he second sage regression, however, we can inerpre γ in a T as is OLS esimae, γ = E T (ff > ) 1 E T (rx f). Then a T is a random marix ha converges o a marix a as i should in he GMM disribuion heory. (I.e. we do no choose he γ in a T o se a T (γ)g T (γ,b) =0.) We need he d marix, g T d b > γ. > Recalling b 4 =4 b 1 b 2 b 3, he resul is 1 d = E(ff> )γ b E(ff > ). Now, we can invoke Hansen s Lemma 4.1, and wrie he covariance marix of he momens under he resriced esimae, The es saisic is cov(g T )= 1 T (I d(ad) 1 a)s(i d(ad) 1 a) >. g > T cov(g T ) + g T χ There are 4 6 = 24 momens and 6(γ) +3(b) parameers, so here are 15 degrees of freedom. The cov(g T ) marix is singular, so he + operaor represens pseudoinversion. Onecanuseaneigenvaluedecomposiionforcov(g T ) and hen reain only he larges 15 eigenvalues, i.e. wrie cov(g T )=QΛQ > where Q is orhogonal and Λ is diagonal and hen cov(g T ) + = QΛ + Q > where Λ + invers he 15 larges diagonal elemens of Λ and ses he remainder o zero. We use he malab pinv command. To conduc he corresponding Wald es in Table 6, we firs find he GMM covariance marix of he unresriced parameers. We form a vecor of hose parameers, vec(β), and hen cov(vec(β)) = 1 T d 1 S (d) 1>, d = g T (vec(β) > ) = I 4 E(ff > ). To consruc sandard errors and es saisics for he resriced model wih lags in Table 5, we proceed similarly. The esimaed parameers are α and γ. Themomensg T se o zero 17

18 he wo regressions on which we ierae, 0=E(γ > f ε +1 ) 0=E(γ > f i ε +1 ) h i 0=E (α 0 f (1) + α 1 f (1) 1 +..) ε +1 h i 0=E (α 0 f (2) + α 1 f (2) 1 +..) ε The sandard error formula is as usual cov( α > γ > )=(d 0 S 1 d) 1 /T.Thed marix is d = dg T d [α > γ > ]. Noe ha he firs se of momens do no involve α and he second se of momens do no involve γ Thus, he d marix is block-diagonal. This means ha α sandard errors are no affeced by γ esimaion and vice versa. Therefore, we can use regular GMM sandard errors from each regression wihou modificaion. C.2 Simulaions for small-sample disribuions We simulae yields based on hree differen daa-generaing processes. Each of hese processes is designed o address a specific concern abou he saisical properies of our es saisics. As he generic boosrap for Wald saisics, we use a vecor auoregression wih 12 lags for he vecor of yields y = A 0 + A 1 y 1/ A 12 y 1 + ε. VARs based on fewer lags (such as one or wo) are unable o replicae he one year horizon forecasabiliy of bond reurns or of he shor rae documened in Table 5. Second o address uni roo fears, we use a VAR wih 12 lags ha imposes a common rend: y (1) 1/12 y(5) 1/12 y = A 0 + B where y = y y 1/12. y (2) 1/12 y(5) 1/12 y (3) 1/12 y(5) 1/12 y (4) 1/12 y(5) 1/ X i=1 A i y i/12 + ε, Third, we impose he expecaions hypohesis by saring wih an AR(12) for he shor 18

19 rae y (1) = a 0 + a 1 y (1) 1/ a 12y (1) 1 + ε. We hen compue long yields as à nx! = 1 n E,n=2,...,5. y (n) i=1 y (n) +i 1 To compue he expeced value in his las expression, we expand he sae space o rewrie he dynamics of he shor rae as a vecor auoregression wih 1 lag. The 12-dimensional h > vecor x = y (1) y (1) 1/11... y(1) 11/12i follows x = B 0 + B 1 x 1/12 + Σu for B 0 = µ a0 0 µ µ a1 a,b 1 = , Σ =. I The vecor e 1 =[ ]pickshefirs elemen in x, which gives y (1) = e 1 x. Longer yields can hen be easily compued recursively as Ãà y (n) = n 1 n y(n 1) + 1 n 1!! X n e 1 B1 i B 0 + B n1 x,n=2,...,5, wih he undersanding ha B i 1 for i =0ishe12 12 ideniy marix. i=0 D Affine model The main poin of his secion is o show ha one can consruc an affine erm-srucure model ha capures our reurn-forecasing regressions exacly. To ha end, we sar by defining how an affine model works. Then we show how o consruc an affine model how o pick marke prices of risk so ha a VAR of our five bond prices p +1 = μ + φp + v +1 is an affinemodel.theissuehereishowomakehemodel self-consisen, howomake sure ha he prices ha come ou ofhemodelarehesameashepriceshagoin o he model as sae variables. Reurn regressions carry exacly he same informaion as a price VAR of course, so in a formal sense, finding marke prices of risk o jusify p +1 = μ + φp + v +1 as an affine model is enough o make our poin. However, i is ineresing o connec he affine model direcly o reurn regressions, exhibiing he ransformaion from reurn regressions rx +1 = a+βf +ε +1 o he price VAR and providing reurn-based inuiion for he choice of he marke prices of risk. We show he marke prices of risk generae a self-consisen model if and only if hey saisfy he one-period expeced reurn equaion (7). Our choice (8) is consruced o capure 19

20 he one-period expeced reurn equaion, so we now know ha i will form a self-consisen affine model. There are many choices of marke price of risk ha saisfy he one-period expeced reurn equaion. We show ha he choice (8) minimizes variance of he discoun facor, and is hus analogous o he Hansen-Jagannahan (1991) discoun facor and shares is many appealing properies. D.1 Model seup Firs, we se up he exponenial-normal class of affine models ha we specialize o accoun for reurn predicabiliy. A vecor of sae variables follows (D.7) X +1 = μ + φx + v +1 ; v +1 N (0,V). The discoun facor is relaed o hese sae variables by M +1 =exp µ δ 0 δ >1 X 12 (D.8) λ> Vλ λ > v +1 λ = λ 0 + λ 1 X. These wo assumpions are no more han a specificaion of he ime-series process for he nominal discoun facor. We find log bond prices p (n) by solving he equaion p (n) =lne (M +1 M +n ). Proposiion. The log prices are affine funcions of he sae variables (D.9) p (n) = A n + B > n X. The coefficiens A n and B n can be compued recursively: (D.10) (D.11) A 0 =0; B 0 =0 B n+1 > = δ 1 > + B n > φ A n+1 = δ 0 + A n + B > n μ B> n VB n where μ and φ are defined as (D.12) (D.13) φ φ Vλ 1 μ μ Vλ 0. Proof. We guess he form (D.9) and show ha he coefficiens mus obey (D.10)- (D.11). Of course, p (0) =0,soA 0 =0andB 0 = 0. For a one-period bond, we have p (1) =lne (M +1 )= δ 0 δ 1 > X, 20

21 which saisfies he firs (n = 1) version of (D.10)-(D.11). We can herefore wrie he one-period yield as y (1) = δ 0 + δ 1 > X. The price a ime of a n + 1 period mauriy bond saisfies P n+1 = E M+1 P n +1. Thus, we mus have exp A n+1 + B n+1x > = E exp µ δ 0 δ >1 X 12 λ> Vλ λ > v +1 + A n + B >n X +1 =exp µ δ 0 δ >1 X 12 (D.14) λ> Vλ + A n E exp λ > v +1 + B n > X +1. We can simplify he second erm in (D.14): E exp λ > v +1 + B n > X +1 = E exp λ > v +1 + B n > μ + B n > φx + B n > v +1 = E exp ( λ > + B n > )v +1 + B n > μ + B n > φx =exp B n > μ + B n > 1 φx exp 2 ( λ> + B n > )V ( λ + B n ) =exp B n > μ + B n > 1 φx exp 2 (λ> Vλ 2B n > Vλ + B n > VB n ). Now, coninuing from (D.14): A n+1 + B n+1x > = µ δ 0 δ >1 X 12 λ> Vλ + A n + B n > μ + B n > φx + µ λ> Vλ B n > Vλ B> n VB n = δ 0 δ > 1 X + A n + B > n μ + B > n φx B > n Vλ B> n VB n = δ 0 + A n + B n > μ B n > Vλ B> n VB n δ 1 > X + B n > φx B n > Vλ 1 X = µ δ 0 + A n + B >n μ B >n Vλ B>n VB n + δ 1 > + B n > φ B n > Vλ 1 X. Maching coefficiens, we obain B > n+1 = δ > 1 + B > n φ B > n Vλ 1 A n+1 = δ 0 + A n + B > n μ B > n Vλ B> n VB n. Simplifying hese expressions wih (D.12) and (D.13), we obain (D.10)-(D.11). 21

22 Commens Ieraing (D.10)-(D.11), we can also express he coefficiens A n, B n in p (n) explicily as = A n + B > n X (D.15) (D.16) A 0 =0; B 0 =0 B n+1 > = δ 1 > + B n > φ A n+1 = δ 0 + A n + B > n μ B> n VB n (D.17) (D.18) B > n = δ > 1 Xn 1 φ j = δ 1 > (I φ n )(I φ ) 1 j=0 Xn 1 A n = nδ 0 + µb >j μ + 12 B>j VB j. j=0 Given prices, we can easily find formulas for yields, forward raes, ec. as linear funcions of he sae variable X. Yields are jus Forward raes are y (n) = A n n B> n n X. f (n 1 n) = p (n 1) p (n) =(A n 1 A n )+(B n 1 > B n > )X = A f n + B f> n X. We can find A f and B f from our previous formulas for A and B. From (D.17) and (D.18), (D.19) (D.20) B f> n = δ > 1 φ n 1 A f n = δ 0 B > n 1μ 1 2 B> n 1VB n 1. In a risk neural economy, λ 0 = λ 1 = 0. Thus, looking a (D.12) and (D.13), we would have he same bond pricing formulas (D.10)-(D.11) in a risk-neural economy wih φ = φ and μ = μ. Equaions (D.17) and (D.18) are recognizable as risk-neural formulas wih risk-neural probabiliies φ and μ. The forward rae formula (D.19) is even simpler. I says direcly ha he forward rae is equal o he expeced value of he fuure spo rae and a Jensen s inequaliy erm, under he risk-neural measure (where φ is he auocorrelaion marix). Noe in (D.12) and (D.13) ha λ 0 conribues only o he difference beween μ and μ,and 22

23 hus conribues only o he consan erm A n in bond prices and yields. A homoskedasic discoun facor can only give a consan risk premium. The marix λ 1 conribues only o he difference beween φ and φ, and only his parameer can affec he loading B n of bond prices on sae variables o give a ime-varying risk premium. Equivalenly, a ime-varying risk premium needs condiional heeroskedasiciy in he discoun facor (D.8), and his is provided by he λ v +1 erm of (D.8) and he variaion in λ provided by λ 1 6=0. D.2 One period reurns Here we derive he one-period expeced reurn relaion. Proposiion. One-period reurns in he affine model follow (D.21) E [rx +1 ]+ 1 2 σ2 (rx +1 )=cov (rx +1,v +1)λ >. This equaion shows ha he loadings λ which relae he discoun facor o shocks in (D.8) are also he marke prices of risk ha relae expeced excess reurns o he covariance of reurns wih shocks. (This equaion is similar o equaion (7) in he paper. Here we use shock o prices, while he paper uses shocks o ex-pos reurns, bu he wo shocks are idenical since rx (n) +1 = p (n 1) +1 p (n) + p (1).) h i Proof. To show Equaion (D.21), sar wih he pricing equaion 1 = E M +1 R (n) +1 which holds for he gross reurn R (n) +1 on any n-period bond. Then, we can wrie h i i 1=E M +1 R (n) +1 = E he m +1+r (n) +1 0=E [m +1 ]+E h r (n) +1 i σ2 (m +1 )+ 1 2 σ2 (r (n) +1)+cov (r (n) +1,m +1 ) wih m =lnm and r (n) =lnr (n). Subracing he same expression for he oneyear yield, 0=E [m +1 ]+y (1) σ2 (m +1 ), and wih he 4 1vecorrx +1 = r +1 y (1),wehave E [rx +1 ]+ 1 2 σ2 (rx +1 )= cov (rx +1,m +1 ), where σ 2 (rx +1 )denoeshe4 1 vecor of variances. Now, we subsiue in for m +1 from (D.8) o give (D.21). 23

24 D.3 A self-consisen model Equaion (D.9) shows ha log bond prices are linear funcions of he sae variables X. The nex sep is obviously o inver his relaionship and infer he sae variables from bond prices, so ha bond prices (or yields, forward raes, ec.) hemselves become he sae variables. In his way, affine models end up expressing each bond price as a funcion of a few facors (e.g. level, slope and curvaure) ha are hemselves linear combinaions of bond prices. Given his fac, i is emping o sar direcly wih bond prices as sae variables, i.e. o wrie X = p, and specify he dynamics given by (9) as he dynamics of bond prices direcly, (D.22) p +1 = μ + φp + v +1. (I is more convenien here o keep he consans separae and define he vecors p, f o conain only he prices and forward raes, unlike he case in he paper in which p, f include a one as he firs elemen.) I is no obvious, however, ha one can do his. The model wih log prices as sae variables may no be self-consisen; he prices ha come ou of he model may no be he same as he prices we sared wih; heir saus iniially is only ha of sae variables. The bond prices ha come ou of he model are, by (D.11) and (D.10), funcions of he marke prices of risk λ 0,λ 1 and risk-free rae specificaion δ 0 δ 1 as well as he dynamics μ and φ, so in fac he model will no be self-consisen for generic specificaions of {δ 0,δ 1,λ 0,λ 1 }.Bu here are choices of {δ 0,δ 1,λ 0,λ 1 } ha ensure self-consisency. We show by consrucion ha such choices exis; we characerize he choices, and along he way we show ha any marke prices of risk ha saisfy he one-period reurn equaion will generae a self-consisen model. In his sense, he marke prices of risk λ which we consruc o saisfy he reurnforecasing regressions do, when insered in his exponenial-gaussian model, produce an affine model consisen wih he reurn regressions. Raher han jus show ha he choice (8)works,wecharacerizeheseofmarkepriceshaworkandhowhechoice(8)isa paricular member of ha se. Definiion The affine model is self-consisen if he sae variables are he prices, ha is, if A n =0,B n = e n where e n is a vecor wih a one in he nh place and zeros elsewhere. Proposiion. The affine model is self-consisen if and only if δ 0 =0, δ 1 = e 1 and he marke prices of risk λ 0,λ 1 saisfy (D.23) (D.24) QV λ 1 = Qφ R QV λ 0 = Qμ Q diag(v ) 24

25 Here, Q ; R Q is a marix ha removes he las row, and R is a form of he firs four rows of φ ha generaes he expecaions hypohesis under he risk neural measure, as we show below.. Proof. The proof is sraighforward: we jus look a he formulas for A n and B n in (D.10) and (D.11) and abulae wha i akes o saisfy A n =0,B n = e n. For he one-period yield we need B 1 > = e > 1 and A 1 = 0. Looking a (D.10) and (D.11) we see ha his requiremen deermines he choice of δ, δ 0 =0; δ 1 = e 1. This resricion jus says o pick δ 0,δ 1 so ha he one-year bond price is he firs sae variable. We can ge here direcly from p (1) = δ 0 δ > 1 p. For he n =2, 3, 4, 5 period yield, he requiremen B > n = e > n in (D.10) and A n = 0 in (D.11) become (D.25) e > n = e > 1 + e > n 1φ 0=e > n μ e> n Ve n. In marix noaion, he self-consisency resricion B n = e n is saisfied if φ has he form (D.26) φ = a b c d e The las row is arbirary. We only use 5 prices as sae variables, so here is no requiremen ha e > 6 = e > 1 + e > 5 φ. The resricion A n = 0 in equaion (D.25) amouns o choosing consans in he marke prices of risk o offse 1/2 σ 2 erms. The resricion is saisfied if (D.27) μ n = 1 2 e> n Ve n. μ 5 is similarly unresriced. Since only he firs four rows of φ and μ are resriced, we can express he necessary resricions on λ 0,λ 1 inmarixformbyusingourq marix ha delees 25

26 he fifh row, Qφ = R Qμ = 1 2 Q diag(v ) Finally, using he definiions of φ and μ in (D.12) and (D.13) we have Q (φ Vλ 1 )=R Q (μ Vλ 0 )= 1 2 Q diag(v ) and hence we have (D.23) and (D.24). Commens The form of (D.26) and (D.27) have a nice inuiion. The one-period pricing equaion is, from (D.21), ³ E rx (n) ³ 2 σ2 rx (n) +1 = cov(rx (n) +1,v +1)λ >. Under risk neural dynamics, λ = 0. Now, from he definiion so we should see under risk neural dynamics E σ 2 ³ p (n 1) +1 ³ rx (n) +1 rx (n) +1 = p (n 1) +1 p (n) + p (1), = p (n) p (1) + 1 ³ 2 σ2 p (n 1) +1 ³ = σ 2 = σ 2 (v n 1 )=V ii. p (n 1) +1 The condiional mean in he firs line is exacly he form of (D.26), and he consan in he firs line is exacly he form of (D.27). In sum, he proposiion jus says if you ve picked marke prices of risk so ha he expecaions hypohesis holds in he risk-neural measure, you have a self-consisen affine model. ³ p (5) +1 sincewedonoobservep (6), and ha is why he This logic does no consrain E las row of φ is arbirary. Inuiively, he 5 prices only define 4 excess reurns and hence 4 marke prices of risk. From he definiions φ = φ Vλ 1 ; μ = μ Vλ 0 we can jus pick any φ and μ wih he required form (D.26) and (D.27), and hen we can consruc marke prices of risk by λ 1 = V 1 (φ φ ) λ 0 = V 1 (μ μ ) This is our firs proof ha i is possible o choose marke prices of risk o incorporae he 26

27 reurn regressions, since reurn regressions amoun o no more han a paricular choice of values for φ, μ. Since he las rows are no idenified, many choices of φ and μ will generae an affine model, however. Equivalenly, since we only observe four excess reurns, we only can idenify four marke prices of risk. By changing he arbirary fifh rows of φ and μ,weaffec how marke prices of risk spread over he 5 shocks, or, equivalenly, he marke price of risk of he fifh (orhogonalized) shock. The remaining quesion is how bes o resolve he arbirariness how bes o assign he fifh rows of φ and μ. A he same ime, we wan o produce a choice ha ies he marke prices of risk more closely o he acual reurn regressions han merely he saemen ha any reurn regression is equivalen o some φ, μ. D.4 Connecionoreurnregressions Our nex ask is o connec he condiions (D.23) and (D.24) o reurn regressions raher han o he parameers of he (equivalen) price VAR. One reason we need o do his is o verify ha he marke prices of risk defined in erms of reurn regressions (8) saisfy condiions (D.23) and (D.24) defined in erms of he price VAR. Denoe he reurn regression (D.28) rx +1 = α + βf + ε +1 ; E(ε +1 ε 0 +1) =Σ. Proposiion. The parameers μ, φ, V of he affine model (D.22) and he parameers α, β, Σ of he reurn forecasing regressions (D.28) are conneced by Here, F generaes forward raes from prices, f (n) α = Qμ β =(Qφ R)F 1 Σ = QV Q > = p (n 1) p (n) so f = Fp. Proof To connec reurn noaion o he price VAR, we sar wih he definiion ha connecs reurns and prices, rx (n) +1 = p (n 1) +1 p (n) + p (1), 27

28 or rx (2) +1 rx (3) +1 rx (4) +1 rx (5) +1 = or, in marix noaion, p (1) +1 p (2) +1 p (3) +1 p (4) +1 p (5) +1 + rx +1 = Qp +1 Rp p (1) p (2) p (3) p (4) p (5) Thus, if we have an affine model p = μ+φp 1 +v i implies ha we can forecas excess reurns from prices p by (D.29) rx +1 = Qμ +(Qφ R)p + Qv +1. Noe ha he reurn shocks ε +1 are exacly he firsfourpriceshocksqv +1. The fied value of his regression is of course exacly equivalen o a regression wih forward raes f on he righ hand side, and our nex ask is o make ha ransformaion. From he definiion f (n) raes and prices wih f = Fp, yielding = p (n 1) p (n) rx +1 = Qμ +(Qφ R)F 1 f + Qv +1.,,wecanconnecforward Maching erms wih (D.28) we obain he represenaion of he proposiion. We also have he covariance marix of reurns wih price shocks, cov(rx +1,v > +1) =Qcov(v +1,v > +1) =QV. A his poin, i s worh proving he logic saed in he ex. Proposiion. If marke prices of risk saisfy he one period reurn equaion, he affine model wih log prices as sae variables is self-consisen. In equaions, if he λ saisfy (D.30) hen hey also saisfy (D.23) and (D.24). h i E rx (n) ³ 2 σ2 rx (n) +1 = cov(rx (n) +1,v +1)λ >, Proof. Using he form of he reurn regression rx +1 = Qμ +(Qφ R)p + Qv +1, he reurn pricing equaion (D.30) is Qμ +(Qφ R)p diag(qv Q> )=QV λ = QV (λ 0 + λ 1 p ). 28

29 Thus, maching he consan and he erms muliplying p,weobainexaclyhe condiions (D.23) and (D.24) again Qμ diag(qv Q> )=QV λ 0 Qφ R = QV λ 1. More inuiively, bu less explicily, we can always wrie a price as is payoff discouned by expeced reurns. For example, p (3) = p (3) p (2) +1 + p (2) +1 p (1) +2 + p (1) +2 h i h i = p (2) +1 p (3) y (1) y (1) p (1) +2 p (2) +1 y (1) +1 = y (1) y (1) +1 y (1) +2 rx (3) +1 rx (2) +2 y (1) +1 y (1) +2 so we can wrie p (3) h i = E y (1) y (1) +1 y (1) +2 rx (3) +1 rx (2) +2. Obviously, if a model ges he righ hand side correcly, i mus ge he lef hand side correcly. D.5 Minimum variance discoun facor Now, here are many choices of he marke price of risk ha saisfy our condiions; (D.23) and (D.24) only resric he firs four rows of Vλ 0 and Vλ 1. We find one paricular choice appealing, and i is he one given in he ex as Equaion (8). The choice is λ 1:4, = Σ µe 1 [rx +1 ]+ 1 (D.31) 2 σ2 (rx +1 ) µ = Σ 1 α + βf diag(σ) ; λ 5, =0 (Equaion (8) in he paper expresses a 4 1 vecor marke prices of risk, which muliply he four reurn shocks. Here, we wan a 5 1 vecor λ ha can muliply he 5 price shocks in he self-consisen affine model p +1 = μ + φp + v +1. The reurn shocks are equal o he firs four price shocks, so (D.31) and (8) are equivalen. Also, in his appendix we are using f o denoe a vecor ha does no include a consan, so α + βf in (D.31) corresponds o βf in (8.)) 29

30 Proposiion. The choice (D.31) is, in erms of he parameers {μ, φ, V } of a price VAR, λ 0 = Q > QV Q > 1 µqμ + 12 Qdiag (V ) λ 1 = Q > QV Q > 1 (Qφ R). The choices form a self-consisen affine model. (They are soluions o (D.23) and (D.24).) Proof. As wih Equaion (8), Equaion (D.31) saisfies he one period reurn equaion (D.30) by consrucion, so we know i forms a self-consisen affine model. The res is ranslaing reurn noaion o price noaion. The marix Q removes a row from a marix wih 5 rows, so he marix Q > adds a row of zeros o a marix wih 4 rows. Therefore, we can wrie (D.31) as λ = Q > Σ µe 1 [rx +1 ]+ 1 2 σ2 (rx +1 ). Subsiuing he regression in erms of he price-var, (D.29), and wih Σ = QV Q >, λ = Q > QV Q > µ 1 Qμ +(Qφ R)p diag QV Q >. Maching consan and price erms, and wih diag QV Q > = Qdiag(V ) λ 0 = Q > QV Q > 1 µqμ + 12 Qdiag (V ) λ 1 = Q > QV Q > 1 (Qφ R). These are easy soluions o (D.23) and (D.24); when we ake QV λ i, he erms in fron of he final parenheses disappear. This choice of marke price of risk is paricularly nice because i is he minimal choice. Jusifying asse pricing phenomena wih large marke prices of risk is always suspicious, so why no pick he smalles marke prices of risk ha will do he job? We follow Hansen and Jagannahan (1991) and define smalles in erms of variance. In levels, hese marke prices of risk give he smalles Sharpe raios. Our choice is he discoun facor wih smalles condiional variance ha can do he job. 30

31 Proposiion. The marke prices of risk (D.31) are he smalles marke prices of risk possible, in he sense ha hey produce he discoun facor wih smalles variance. Proof. For any lognormal M, we have σ 2 (M) =e 2E(m)+2σ2 (m) e 2E(m)+σ2 (m) = e 2E (m)+σ ³e 2(m) σ2 (m) 1. Thus, given he form (D.8) of he discoun facor, we have σ 2 (M) =e 2 (δ 0 +δ 1 > X ) ³ e λ> Vλ 1. Our objecive is min σ 2 (m). We will ake he one period pricing equaion as he consrain, knowing ha i implies a self-consisen model, h i E rx (n) ³ 2 σ2 rx (n) +1 = cov(rx (n) +1,v +1)λ > = Cλ, where C is a 4 5 marix. Thus, he problem is 1 min λ 2 λ> Vλ s.. α + βf Qdiag(V )=Cλ. The firs-order condiion o his problem is Vλ = C > ξ, where ξ is a 4 1 vecor of Lagrange mulipliers. We can now solve λ = V 1 C > ξ. Plugging his soluion back ino he consrain gives α + βf Qdiag(V )=CV 1 C > ξ, whichwecansolveforξ,sohawege λ = V 1 C > CV 1 C > 1 α + βf + 12 Qdiag(V ). Wih C = QV, we have he same marke prices of risk we derived above, λ = V 1 VQ > QV V 1 V > Q > 1 12 α + βf + Qdiag(V ) = Q > QV Q > 1 12 α + βf + Qdiag(V ). 31