Risk, Uncertainty, and Asset Prices

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Risk, Uncertainty, and Asset Prices Geert Bekaert, Eric Engstrom, and Yuhang Xing 2005-40 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Risk, Uncertainty and Asset Prices Geert Bekaert∗ Columbia University and NBER Eric Engstrom Federal Reserve Board of Governors Yuhang Xing Rice University This Draft: 20 February 2008 JEL Classifications G12, G15, E44 Keyphrases Equity Premium, Economic Uncertainty, Stochastic Risk Aversion, Time Variation in Risk and Return, Excess Volatility, External Habit, Term Structure, Heteroskedasticity Abstract: Weidentifytherelativeimportanceofchangesintheconditionalvarianceoffundamentals(which wecall“uncertainty”)andchangesinriskaversioninthedeterminationofthetermstructure,equity prices and risk premiums. Theoretically, we introduce persistent time-varying uncertainty about the fundamentals in an external habit model. The model matches the dynamics of dividend and consumptiongrowth, includingtheirvolatilitydynamicsandmanysalientassetmarketphenomena. While the variationin price-dividend ratios and the equity risk premium is primarily driven by risk aversion, uncertainty plays a large role in the term structure and is the driver of counter-cyclical volatility of asset returns. ∗Corresponding author: Columbia Business School, 802 Uris Hall, 3022 Broadway, New York New York, 10027; ph: (212)-854-9156;fx:(212)-662-8474;gb241@columbia.edu. WethankLarsHansen,BobHodrick,CharlieHimmelberg, Kobi Boudoukh, StijnVan Nieuwerburgh, Tano Santos, PietroVeronesi, Francisco Gomes and participants at presentationsattheFederalReserveBoard;theWFA,Portland;UniversityofLeuven,Belgium;CaesareaCenter3rd conference,Herzliya,Israel;BrazilFinanceSocietyMeetings,Vittoria,Brazil;AustralasianmeetingoftheEconometricSociety, Brisbane; Federal Reserve Bank of New Yorkfor helpful comments. The views expressed in this article arethoseoftheauthors andnotnecessarilyoftheFederalReserveSystem.

1 Introduction Without variation in discount rates, it is difficult to explain the behavior of aggregate stock prices within the confines of rational pricing models. In standard models, there are two main sources of fluctuations in asset prices and risk premiums: changes in the conditional variance of fundamentals (eitherconsumptiongrowthordividendgrowth)andchangesinriskaversionorriskpreferences. An old literature [Poterba and Summers (1986), Barsky (1989), Abel (1988), Kandel and Stambaugh (1990)]andrecentworkbyBansalandYaron(2004)andBansalandLundblad(2002)focusprimarily on the effect of changes in economic uncertainty on stock prices and risk premiums. However, the workofCampbellandCochrane(1999)(CChenceforth)hasmadechangesinriskaversionthemain focus of current research. They show that a model with counter-cyclical risk aversion can account for a large equity premium, substantial variation in equity returns and price-dividend ratios and significant long-horizonpredictability of returns. Inthisarticle,wetrytoidentifytherelativeimportanceofchangesintheconditionalvarianceof fundamentals(whichwecall“uncertainty”)andchangesinriskaversion1. Webuildontheexternal habit model formulated in Bekaert, Engstrom and Grenadier (2004) which features stochastic risk aversion, and introduces persistent time-varying uncertainty in the fundamentals. We explore the effects of both on price dividend ratios, equity risk premiums, the conditional variability of equity returns and the term structure, both theoretically and empirically. To differentiate time-varying uncertainty from stochastic risk aversion empirically, we use information on higher moments in dividendandconsumptiongrowthandtheconditionalrelationbetweentheirvolatilityandanumber of instruments. The model is consistent with the empirical volatility dynamics of dividend and consumption growth,matches the largeequity premium and low risk free rate observedin the data and produces realisticvolatilitiesofequityreturns,price-dividendratiosandinterestrates. Wefindthatvariation intheequitypremiumisdrivenbybothriskaversionanduncertaintywithriskaversiondominating. However,variationinassetprices(consolpricesandprice-dividendratios)isprimarilyduetochanges inrisk aversion. Theseresults arisebecause riskaversionacts primarilyasa levelfactorin the term structure while uncertainty affects both the level and the slope of the real term structure and also 1Hence, the term uncertainty is used in a different meaning than in the growing literature on Knightian uncertainty,seeforinstanceEpsteinandSchneider (2007). However,economicuncertaintyisthestandardtermtodenote heteroskedasticity in the fundamentals in both the asset pricing and macroeconomic literature. It is also consistent withasmallliteratureininternationalfinancewhichhasfocusedontheeffectofchangesinuncertaintyonexchange ratesandcurrencyriskpremiums,seeHodrick(1989, 1990)andBekaert(1996). TheHodrick(1989) paperprovided the obvious inspiration for the title to this paper. While “risk” is short for “risk aversion” in the title, we avoid confusionthroughout thepapercontrastingeconomicuncertainty (amountofrisk)andriskaversion(priceofrisk). 1

governs the riskiness of the equity cash flow stream. Consequently, our work provides a new perspective on recent advances in asset pricing modelling. We confirm the importance of economic uncertainty as stressed by Bansal and Yaron (2004) and Kandel and Stambaugh (1990) but show thatchangesinriskaversionarecriticaltoo. However,themainchannelthroughwhichriskaversion affects asset prices in our model is the term structure, a channel shut off in the original CC paper butstressedbytheolderpartialequilibriumworkofBarsky(1989). Wemoregenerallydemonstrate thatinformationinthetermstructurehasimportantimplicationsfortheidentificationofstructural parameters. The remainder of the article is organized as follows. The second section sets out the theoretical modelandmotivatestheuseofourstatevariablestomodeltime-varyinguncertaintyofbothdividend and consumption growth. In the third section, we derive closed-from solutions for price-dividend ratios and real and nominal bond prices as a function of the state variables and model parameters. In the fourth section, we set out our empirical strategy. We use the General Method of Moments (Hansen(1982),GMMhenceforth)toestimatetheparametersofthemodel. Thefifthsectionreports parameter estimates and discusses how well the model fits salient features of the data. The sixth sectionreportsvariousvariancedecompositionsanddissectshowuncertaintyandriskaversionaffect assetprices. The seventhsectionexamines the robustnessofourresultsto the use ofpost-wardata andclarifiesthelinkanddifferencesbetweenourmodelandthoseofAbel(1988),Wu(2001),Bansal and Yaron (2004) and CC. Section 8 concludes. 2 Theoretical Model 2.1 Fundamentals and Uncertainty Tomodelfundamentalsanduncertainty,westartbymodellingdividendgrowthasanAR(1)process with stochastic volatility: ∆d t =µ d +ρ du u t−1 +√v t−1 σ dd εd t +σ dv εv t (1) v t =µ v +ρ vv v t−1 +σ vv √v t (cid:0) −1 εv t (cid:1) where d = log(D ) denotes log dividends, u is the demeaned and detrended log consumptiont t t dividend ratio (described further below) and v represents “uncertainty,” which is proportional to t 2

the conditional volatility of the dividend growth process. All innovations in the model, including εd and εv follow independent N(0,1) distributions. Consequently, covariances must be explicitly t t parameterized. With this specification, the conditional mean of dividend growth varies potentially withpastvaluesoftheconsumption-dividendratio,whichisexpectedtobeapersistentbutstationary process. Uncertainty itself follows a square-rootprocess andmay be arbitrarilycorrelatedwith dividend growth through the σ parameter.2 For identification purposes, we fix its unconditional dv mean at unity. While consumption and dividends coincide in the original Lucas (1978) framework and many subsequentstudies, recentpapers haveemphasizedthe importance of recognizingthatconsumption is financed by sources of income outside of the aggregate equity dividend stream [see for example Santos and Veronesi (2006)]. We model consumption as stochastically cointegrated with dividends, inafashionsimilartoBansal,DittmarandLundblad(2005),sothattheconsumptiondividendratio, u , becomes a relevant state variable. While there is a debate on whether the cointegrating vector t should be (1,-1) (see Hansen, Heaton and Li (2005)), we follow Bekaert, Engstrom and Grenadier (2004)who find the consumption-dividendratio to be stationary. We model u symmetrically with t dividend growth, u t =µ u +ρuuu t−1 +σ ud (∆d t − E t−1 [∆d t ])+σuu√v t−1 εu t . (2) By definition, consumption growth, ∆c , is t ∆c =δ+∆d +∆u t t t =(δ+µ u +µ d )+(ρ du +ρ uu − 1)u t−1 +(1+σ ud )√v t−1 σ dd εd t +σ dv εv t +σuu√v t−1 εu t . (3) (cid:0) (cid:1) Note that δ and µ cannot be jointly identified. We proceed by setting the unconditional mean u of u to zero and then identify δ as the difference in means of consumption and dividend growth.3 t Consequently, the consumption growth specification accommodates arbitrary correlation between dividend and consumption growth, with heteroskedasticity driven by v . The conditional means t of both consumption and dividend growth depend on the consumption-dividend ratio, which is an 2In discrete time, the square root process does not guarantee that vt is bounded below by zero. However, by imposing a lower bound on uv, the process rarely goes below zero. In any case, we use max[vt,0] under the square rootsigninanysimulation. Inderivingpricingsolutions,weignorethemassbelowzerowhichhasanegligibleeffect ontheresults. 3Thepresenceofδmeansthatut shouldbeinterpretedasthedemeanedanddetrendedlogconsumption-dividend ratio. 3

AR(1) process. Consequently, the reduced form model for both dividend and consumption growth isanARMA(1,1)whichcanaccommodateeitherthestandardnearlyuncorrelatedprocesseswidely assumed in the literature, or the Bansal and Yaron (2004) specification where consumption and dividend growth have a long-run predictable component. Bansal and Yaron (2004) do not link the long run component to the consumption-dividend ratio as they do not assume consumption and dividends are cointegrated. Our specification raises two important questions. First, is there heteroskedasticity in consumption and dividend growth data? Second, can this heteroskedasticity be captured using our single latentvariablespecification? Insection4,wemarshalaffirmativeevidenceregardingbothquestions. 2.2 Investor Preferences FollowingCC,considera complete marketseconomyas inLucas (1978),but modify the preferences of the representative agent to have the form: ∞ (C H )1−γ 1 E βt t − t − , (4) 0 1 γ " # t=0 − X where C is aggregate consumption and H is an exogenous “external habit stock” with C >H . t t t t One motivation for an “external” habit stock is the “keeping up with the Joneses” framework of Abel (1990,1999) where H represents past or current aggregateconsumption. Small individual t investors take H as given, and then evaluate their own utility relative to that benchmark.4 In t CC, H is taken as an exogenously modelled subsistence or habit level. In this situation, the local t coefficient of relative risk aversion can be shown to be γ Ct , where Ct−Ht is defined as the Ct−Ht Ct surplus ratio5. As the surplus ratio goes to zero, the consumer’s risk aver (cid:16) sionten (cid:17) ds toward infinity. Inourmodel,weview the inverseofthe surplusratioasapreferenceshock,whichwedenote byQ . t Thus, we have Q Ct , so that local risk aversion is now characterized by γQ , and Q > 1. t ≡ Ct−Ht t t As Q changes over time, the representative consumer investor’s “moodiness” changes, which led t Bekaert, Engstrom and Grenadier (2004) to label this a “moody investor economy.” The marginal rate of substitution in this model determines the real pricing kernel, which we 4For empirical analyses of habit formationmodels where habit depends on past consumption, see Heaton (1995) andBekaert(1996). 5Riskaversionistheelasticityofthevaluefunctionwithrespecttowealth,butthelocalcurvatureplaysamajor roleindeterminingitsvalue,seeCC. 4

denote by M . Taking the ratio of marginal utilities at time t+1 and t, we obtain: t (C /C ) −γ t+1 t M =β (5) t+1 (Q /Q ) −γ t+1 t =βexp[ γ∆c +γ(q q )], t+1 t+1 t − − where q =ln(Q ). t t We proceed by assuming that q follows an autoregressive square root process which is contemt poraneously correlated with fundamentals, but also possesses its own innovation, q t =µ q +ρ qq q t−1 +σ qc (∆c t − E t−1 [∆c t ])+σ qq √q t−1 εq t (6) As with v , q is a latent variable and can therefore be scaled arbitrarily without economic conset t quence; we therefore set its unconditional mean at unity. In our specification, Q is not forced to t be perfectly negatively correlatedwith consumption growth as in CC. In this sense, our preference shock specificationis closerin spirit to that ofBrandtand Wang (2003)who allow for Q to be cort related with other business-cycle factors, or Lettau and Watcher (2007), who also allow for shocks to preferences uncorrelated with fundamentals. Only if σ = 0 and σ < 0 does a Campbell qq qc Cochrane-likespecificationobtainwhere consumptiongrowthandrisk aversionshocks are perfectly negatively correlated. Consequently, we can test whether independent preference shocks are an important part of variation in risk aversionor whether its variation is dominated by shocks to fundamentals. Note that the covariance between q and consumption growth and the variance of q t t both depend on v and consequently may inherit its cyclical properties. t 2.3 Inflation When confronting consumption-based models with the data, real variables have to be translated into nominal terms. Furthermore, inflation may be important in realistically modeling the joint dynamics of equity returns, the short rate and the term spread. Therefore, we append the model with a simple inflation process, π t =µ π +ρ ππ π t−1 +κE t−1 [∆c t ]+σ π επ t (7) 5

The impact of expected “real” growth on inflation can be motivated by macroeconomic intuition, suchasthePhillipscurve(inwhichcaseweexpectκtobepositive). Becausethereisnocontemporaneous correlation between this inflation process and the real pricing kernel, the one-period short rate willnotinclude aninflationrisk premium. However,non-zerocorrelationsbetweenthe pricing kernel and inflation may arise at longer horizons due to the impact of E t−1 [∆c t ] on the conditional mean of inflation. Note that expected real consumption growth varies only with u ; hence, the t specification in Equation (7) is equivalent to one where ρ πu u t−1 replaces κE t−1 [∆c t ]. To price nominal assets, we define the nominal pricing kernel, m , which is a simple transfort+1 mation of the log real pricing kernel, m , t+1 b m =m π . (8) t+1 t+1 t+1 − b To summarize, our model has five state variables with dynamics described by the equations, ∆d t =µ d +ρ du u t−1 +√v t−1 σ dd εd t +σ dv εv t v t =µ v +ρ vv v t−1 +σ vv √v t (cid:0) −1 εv t (cid:1) u t =ρuuu t−1 +σ ud (∆d t − E t−1 [∆d t ])+σuu√v t−1 εu t q t =µ q +ρ qq q t−1 +σ qc (∆c t − E t−1 [∆c t ])+σ qq √q t−1 εq t π t =µ π +ρ ππ π t−1 +ρ πu u t−1 +σ ππ επ t (9) with ∆c =δ+∆d +∆u . t t t As discussed above, the unconditional means of v and q are set equal to unity so that µ and t t v µ are not free parameters. Finally, the real pricing kernel can be represented by the expression, q m =ln(β) γ(δ+∆u +∆d )+γ∆q (10) t+1 t+1 t+1 t+1 − We collect the 19 model parameters in the vector, “Ψ,” ′ µ d ,µ π ,ρ du ,ρ ππ ,ρ πu ,ρuu,ρ vv ,ρ qq ,... Ψ= . (11)   σ dd ,σ dv ,σ ππ ,σ ud ,σuu,σ vv ,σ qc ,σ qq ,δ,β,γ     6

3 Asset Pricing In this section, we present exact solutions for asset prices. Our model involves more state variables and parametersthan much of the existing literature, making it difficult to trace pricing effects back to any single parameter’svalue. Therefore we defer providing part of the economic intuition for the pricing equations to Section 6. There, we discuss the results and their economic interpretation in the context of the model simultaneously. The general pricing principle in this model follows the framework of Bekaert and Grenadier (2001). Assume an asset pays a real coupon stream K , τ = 1,2...T. We consider three assets: t+τ a real consol with K = 1, T = , a nominal consol with K = Π −1, T = , (where Π t+τ ∞ t+τ t,τ ∞ t,τ represents cumulative gross inflation from t to τ) and equity with K =D , T = . The case t+τ t+τ ∞ of equity is slightly more complex because dividends are non-stationary (see below). Then, the price-coupon ratio can be written as n=T n PC =E exp (m +∆k ) (12) t t  t+j t+j  n X =1 X j=1    By induction, it is straightforwardtoshow that  n=T PC = exp(A +C u +D π +E v +F q ) (13) t n n t n t n t n t n=1 X with X n =fX(A n−1 ,C n−1 ,D n−1 ,E n−1 ,F n−1 ,Ψ) forX [A,C,D,E,F]. Theexactformofthesefunctionsdependsontheparticularcouponstream. ∈ Note that ∆d t is not strictly a priced state variable as its conditional mean only depends on u t−1 . The Appendix provides a self-contained discussion of the pricing of real bonds (bonds that pay out 1 unit of the consumption good at a particular point in time), nominal bonds and finally equity. Here we provide a summary, with proposition numbers referring to the Appendix. 7

3.1 Term Structure The basic building block for pricing assets is the term structure of real zero coupon bonds. The well known recursive pricing relationship governing the term structure of these bond prices is Prz =E M Prz (14) n,t t t+1 n−1,t+1 (cid:2) (cid:3) where Prz is the price of a real zero coupon bond at time t with maturity at time t+n. The n,t followingpropositionsummarizesthesolutionforthesebondprices. Wesolvethemodelforaslightly generalized (but notation saving) case where q t = µ q +ρ qq q t−1 +√v t−1 σ qd εd t +σ qu εu t +σ qv εv t + √q t−1 σ qq εq t . Our current model obtains when (cid:0) (cid:1) σ =σ σ (1+σ ) qd qc dd ud σ =σ σ qu qc uu σ =σ σ (1+σ ). (15) qv qc dv ud Proposition 1 For the economy described by Equations (9) and (10), the prices of real, risk free, zero coupon bonds are given by Prz =exp(A +C u +D π +E v +F q ) (16) n,t n n t n t n t n t where A n =fA(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) C n =fC(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) D =0 n E n =fE(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) F n =fF (A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) And the above functions are represented by fA =lnβ γ(δ+µ d )+A n−1 +E n−1 µ v +(F n−1 +γ)µ q − fC γρ du +C n−1 ρ uu +γ(1 ρ uu ) ≡− − fE E n−1 ρ vv ≡ 1 + ( γσ dd +(C n−1 γ)σ ud σ dd +(F n−1 +γ)σ qd )2 2 − − 1 + ((C n−1 γ)σ uu +(F n−1 +γ)σ qu )2 2 − 1 + ( γσ dv +(C n−1 γ)σ ud σ dv +(F n−1 +γ)σ qv +E n−1 σ vv )2 2 − − 1 fF F n−1 ρ qq +γ(ρ qq 1)+ ((F n−1 +γ)σ qq )2 ≡ − 2 8

and A =C =E =F =0. (Proof in Appendix). 0 0 0 0 Note that inflation has zero impact on real bond prices, but will, of course, affect the nominal termstructure. Becauseinterestratesareasimplelinearfunctionofbondprices,ourmodelfeatures a three-factor real interest rate model, with the consumption-dividend ratio, risk aversion, and uncertainty as the three factors. The pricing effects of the consumption-dividend ratio,capturedby the C term, arise because the lagged consumption-dividend ratio enters the conditional mean of n bothdividendgrowthanditself. Eitherofthesechannelswillingeneralimpactfutureconsumption growth given Equation (3). The volatility factor, v , has important term structure effects captured t by the fE term because it affects the volatility of both consumption growth and q . As such, v t t affectsthevolatilityofthepricingkernel,therebycreatingprecautionarysavingseffects. Intimesof high uncertainty,investorsdesire to save more but they cannot. For equilibrium to obtain, interest rates must fall, raising bond prices. Note that the second, third and fourth lines of the E terms n are positive, as is the first line if v is persistent: increased volatility unambiguously drivesup bond t prices. Thusthemodelfeaturesaclassic“flighttoquality”effect. Finally,thefF termcapturesthe effect ofthe riskaversionvariable,q , which affects bondprices throughoffsetting utility smoothing t and precautionary savings channels. Consequently, the effect of q cannot be signed and we defer t further discussion to Section 6. From Proposition 1, the price-coupon ratio of a hypothetical real consol (with constant real coupons) simply represents the infinite sum of the zero coupon bond prices. The nominal term structure is analogous to the realterm structure, but simply uses the nominal pricing kernel, m , t+1 in the recursionsunderlying Proposition1. The resulting expressionsalsolook verysimilar to those b obtained in Proposition 1 with the exception that the A and C terms carry additional terms n n reflecting inflation effects and D is non-zero6. Becausethe conditionalcovariancebetween the real n kernel and inflation is zero, the nominal short rate rf satisfies the Fisher hypothesis, t 1 rf =rrf +µ +ρ π +ρ u σ2 (17) t t π ππ t πu t − 2 ππ whererrf isthe realrate. The lasttermis the standardJensen’sinequality effectandthe previous t three terms represent expected inflation. 6The exact formulasfor the price-coupon ratio of a real consol and for a nominal zero coupon bond are given in Propositions3and4respectively, intheAppendix. 9

3.2 Equity Prices In any present value model, under a no-bubble transversality condition, the equity price-dividend ratio (the inverse of the dividend yield) is represented by the conditional expectation, ∞ n P t =E exp (m +∆d ) (18) t t+j t+j D    t n=1 j=1 X X    where Pt is the price dividend ratio. This conditional expectation can also be solved in our Dt framework as an exponential-affine function of the state vector, as is summarized in the following proposition. Proposition 4 For the economy described by Equations (9) and (10), the price-dividend ratio of aggregate equity is given by ∞ P t = exp A +C u +E v +F q (19) n n t n t n t D t n X =1 (cid:16) (cid:17) b b b b where A n =fA A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ +µ d (cid:16) (cid:17) Cbn =fC Abn−1 ,Cbn−1 ,Ebn−1 ,Fbn−1 ,Ψ +ρ du (cid:16) (cid:17) Ebn =fE Abn−1 ,Cbn−1 ,Ebn−1 ,Fbn−1 ,Ψ 1(cid:16) (cid:17) b + 2 σ d 2b d +σ d b d ( − γb)σ dd +b C n−1 − γ σ ud σ dd + F n−1 +γ σ qd (cid:18) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)(cid:19) 1 + 2 σ d 2 v +σ dv ( − γ)σ dv + C b n−1 − γ σ ud σ dv + F b n−1 +γ σ qv +E n−1 σ vv (cid:18) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)(cid:19) F n =fF A n−1 ,C n−1 ,E n−1 ,F n−1 b,Ψ b b (cid:16) (cid:17) where thbe functionsbfX(b) are bgiven ibn Proposition 1 for X (A,C,E,F) and A = C = E = 0 0 0 · ∈ F =0. (Proof in appendix) 0 It is clear upon examination of Propositions 1 and 4 that the price-couponratio of a real consol and the price-dividend ratio of an equity claim share many reactions to the state variables. This makes perfect intuitive sense. An equity claim may be viewed as a real consol with stochastic coupons. Of particularinterestinthis study is the difference inthe effects ofstate variables onthe two financial instruments. InspectionofC andC illuminates anadditionalimpact ofthe consumption-dividendratio,u , n n t onthe price-dividendratio. Thismarginaleffectdepends positivelyonρ ,describingthe feedback b du from u to the conditional mean of ∆d . When ρ > 0, a higher u increases expected cash flows t t du t 10

and thus equity valuations. Above,weestablishedthathigheruncertaintydecreasesinterestratesandconsequentlyincreases consol prices. Hence a first order effect of higher uncertainty is a positive “term structure” effect. Two channels governthe differentialimpact of v onequity prices relativeto consolprices,reflected t inthedifferencebetweenE andE . First,theterms 1σ2 and 1σ2 arisefromJensen’sInequality n n 2 dd 2 dv and tend towards an effect of higher cash flow volatility increasing equity prices relative to consol b prices. While this may seem counterintuitive, it is simply an artifact of the log-normal structure of the model. The second channel is the conditional covariance between cash flow growth and the pricing kernel, leading to the other terms on the second and third lines in the expression for E . n As in all modern rational asset pricing models, a negative covariance between the pricing kernel b and cash flows induces a positive risk premium and depresses valuation. The “direct effect” terms (those excluding lagged functional coefficients) can be signed, they are, γ(1+σ )(1 σ ) σ2 +σ2 − ud − qc dd dv (cid:0) (cid:1) Iftheconditionalcovariancebetweenconsumptiongrowthanddividendgrowthispositive,(1+σ )> ud 0, andconsumption is negatively correlatedwith q , σ <0, then the dividend streamis negatively t qc correlatedwith the kerneland increases in v exacerbatethis covariancerisk. Consequently, uncert tainty has two primary effects on stock valuation: a positive term structure effect and a potentially negative cash flow effect. Interestingly, there is no marginal pricing difference in the effect of q on riskless versus risky t coupon streams: the expressions for F and F are functionally identical. This is true by conn n struction in this model because the preference variable, q , affects neither the conditional mean nor b t volatility of cash flow growth, nor the conditional covariance between the cash flow stream and the pricing kernel at any horizon. We purposefully excluded such relationships for two reasons. First, it does not seem economically reasonable for investor preferences to affect the productivity of the proverbial Lucas tree. Second, it would be empirically very hard to identify distinct effects of v t and q without exactly these kinds of exclusion restrictions. t Finally, note that inflation has no role in determining equity prices for the same reason that it has no role in determining the real term structure. While such effects may be present in the data, we do not believe them to be of first order importance for the questions at hand. 11

3.3 Sharpe Ratios CC point out that in a lognormal model the maximum attainable Sharpe ratio of any asset is an increasing function of the conditional variance of the log real pricing kernel. In our model, this variance is given by, V (m )=γ2σ2 q +γ2(σ 1)2σ2 v t t+1 qq t qc − cc t where σ2 =(1+σ )2 σ2 +σ2 +σ2 . cc ud dd dv uu (cid:0) (cid:1) The Sharpe ratio is increasing in preference shocks and uncertainty. Thus, counter-cyclical variation in v may imply counter-cyclical Sharpe ratios. The effect of v on the Sharpe ratio is t t larger if risk aversion is itself negatively correlated with consumption growth. In CC, the kernel variance is a positive function of q only. t 4 Empirical Implementation In this section, we describe the data and estimation strategy. 4.1 Data We measure all variables at the quarterly frequency and our base sample period extends from 1927:1 to 2004:3. Use of the quarterly rather than annual frequency is crucial to help identify heteroskedasticity in the data. 4.1.1 Bond Market and Inflation We use standardIbbotson data (fromthe SBBI Yearbook)for Treasurymarketand inflationseries. The short rate, rf , is the (continuously compounded) 90-day T-Bill rate. The log yield spread, t spd , is the averagelog yield for long term governmentbonds (maturity greater than ten years)less t theshortrate. Theseyieldsaredatedwhentheyentertheeconometrician’sdataset. Forinstance, the 90-day T-Bill return earned over January-March1990 is dated as December 1989, as it entered the datasetatthe endofthatmonth. Inflation,π ,isthe continuouslycompoundedendofquarter t change in the CPI. 12

4.1.2 Equity Market Our stock return measure is the standard CRSP value-weighted return index. To compute excess equity returns, rx, we subtract the 90-day continuously compounded T-Bill yield earned over the t same period. To construct the dividend yield, we proceed by first calculating a (highly seasonal) quarterly dividend yield series as, P −1 P +D P t+1 t+1 t+1 t+1 DP = t+1 P P − P (cid:18) t (cid:19) (cid:18) t t (cid:19) where Pt+1+Dt+1 and Pt+1 areavailabledirectly fromthe CRSP datasetasthe value weightedstock Pt Pt return series including and excluding dividends respectively. We then use the four-period moving average of ln(1+DP ) as our observable series, t 1 dpf t = 4 [ln(1+DP t )+ln(1+DP t−1 )+ln(1+DP t−2 )+ln(1+DP t−3 )]. This dividend yield measure differs from the more standard one, which sums dividends over the past four quarters and scales by the current price. We prefer our filter because it represents a linear transformation of the underlying data which we can account for explicitly when bringing the model to the data. As a practical matter, the properties of our filtered series and the more standardmeasureare verysimilar with nearlyidentical means andvolatilities andanunconditional correlation between the two of approximately 0.95. For dividend growth, we first calculate real quarterly dividend growth, DP P t+1 t+1 ∆d =ln π t+1 t+1 DP P − (cid:20) t t (cid:21) Then, to eliminate seasonality, we use the four-period moving average as the observation series, 1 ∆df t = 4 (∆d t +∆d t−1 +∆d t−2 +∆d t−3 ). (20) 4.1.3 Consumption Toavoidthe look-aheadbiasinherentinstandardseasonallyadjusteddata,weobtainnominalnonseasonally adjusted (NSA) aggregate non-durable and service consumption data from the website of the Bureau of Economic Analysis (BEA) of the United States Department of Commerce for the 13

period 1946-2004. We deflate the raw consumption growth data with the inflation series described above. We denote the continuously compounded real growth rate of the sum of non-durable and service consumption series as ∆c . From 1929-1946, consumption data from the BEA is available t only at the annual frequency. For these years, we use repeated values equal to one-fourth of the compounded annual growthrate. Because this methodology has obvious drawbacks,we repeat our analysis using an alternate consumption interpolation procedure which presumes the consumptiondividend ratio, rather than consumption growth is constant over the year. Results using this alternate method are very similar to those reported. Finally, for 1927-1929, no consumption data areavailablefromtheBEA. Fortheseyears,weobtainthegrowthrateforrealper-capitaaggregate consumption from the website of Robert Shiller at www.yale.edu, and compute aggregate nominal consumption growth rates using the inflation data described above in addition to historical population growth data from the United States Bureau of the Census. Then, we use repeated values of the annual growthrate as quarterly observations. Analogous to our procedure for dividend growth, we use the four-period moving average of ∆c as our observation series, which we denote by ∆cf. t t 4.1.4 Heteroskedasticity in Consumption and Dividend Growth Many believe that consumption growth is best described as an i.i.d. process. However, Ferson and Merrick (1987), Whitelaw (2000) and Bekaert and Liu (2004) all demonstrate that consumption growth volatility varies through time. For our purposes, the analysis in Bansal, Khatchatrian and Yaron (2005) and Kandel and Stambaugh (1990) is most relevant. The former show that pricedividend ratios predict consumption growth volatility with a negative sign and that consumption growthvolatility is persistent. KandelandStambaugh(1990)link consumptiongrowthvolatilityto three state variables, the price-dividend ratio, the AAA versus T-Bill spread, and the BBB versus AAspread. Theyalsofindthatprice-dividendratiosnegativelyaffectconsumptiongrowthvolatility. We extend and modify this analysis by estimating the following model by GMM, VAR (y )=b +b x (21) t t+1 0 1 t where y is, alternatively, ∆df or ∆cf. We explore asset prices as well as measures of the business t t t cycle and a time trend as elements of x . The asset prices include, rf , the risk free rate, dpf, t t t the (filtered) dividend yield (the inverse of the price-dividend ratio), and spd , the nominal term t spread. We alsoallowfortime-variationinthe conditionalmeanusing a linearprojectionontothe 14

consumption-dividend ratio,uf. Because consumption and dividend growthdisplay little variation t inthe conditionalmean,theresultsarequite similarforspecificationswhereintheconditionalmean is a constant, and we do not report these projection coefficients. The results are reported in Table 1. Panel A focuses on univariate tests while Panel B reports multivariate tests. Wald tests in the multivariate specification reject the null of no time variation for the volatility of both consumption and dividend growth at conventional significance levels. Moreover, all three instruments are mostly significant predictors of volatility in their own right: high interest rates are associated with low volatility, high term spreads are associated with high volatility as are high dividend yields. Hence, the results in Bansal et al. (2005) and Kandel and Stambaugh (1990) regarding the dividend yield predicting economic uncertainty are also valid for dividend growth volatility. Note that the coefficients on the instruments for the dividend growth volatility are a positive multiple,inthe5to25range,oftheconsumptioncoefficients. Thissuggeststhatonelatentvariable may capture the variationin both. We test this conjecture by estimating a restricted versionof the model where the slope coefficients are proportionalacrossthe dividend andconsumption equations. This restriction is not rejected, with a p-value of 0.11. We conclude that our use of a single latentfactorfor bothfundamentalconsumptionanddividendgrowthvolatilityis appropriate. The proportionalityconstant(notreported), is about0.08,implyingthat the dividendslopecoefficients are about 12 times larger than the consumption slope coefficients. Table 1 (Panel A) also presents similar predictability results for excess equity returns. We will later use these results as a metric to judge whether our estimated model is consistent with the evidenceforvariationintheconditionalvolatilityofreturns. Whilethesignsarethesameasinthe fundamentals’equations,noneofthe coefficientsaresignificantlydifferentfromzeroatconventional significance levels. Panel C examines the cyclical pattern in the fundamentals’ heteroskedasticity, demonstrating a strong counter-cyclical pattern. This is an important finding as it intimates that heteroskedasticity may be the driver of the counter-cyclical Sharpe ratios stressed by CC and interpreted as countercyclical risk aversion. Lettau, Ludvigson and Watcher (2006) consider the implications of a downward shift in consumption growth volatility for equity prices. Using Post-war data, they find evidence for a structural break in consumption growth volatility somewhere between 1983 or 1993 depending on the data used. Given our very long sample, the assumption of a simple AR(1) process for volatility is 15

definitely strong. If non-stationarities manifest themselves through a more persistent process than thetruemodelreflectingabreak,aregimechange,oratrend;therobustnessofourresultsisdubious and we may over-estimate the importance of economic uncertainty. Therefore, we examine various potential forms of non-stationary behavior for dividend and consumptiongrowthvolatility. We startbyexaminingevidenceofatrendinvolatility. Itisconceivable that a downward trend in volatility can cause spurious counter-cyclical behavior as recessions have become milder and less frequent over time. While there is some evidence for a downward trend (see Panel C in Table 1) in dividend and consumption growth volatility, there is still evidence for counter-cyclicality in volatility, although it is weakened for dividend growth volatility. Yet, a trend model is not compelling for various reasons. First, the deterministic nature of the model suggests the decline is predictable, which we deem implausible. Second, using post-war data there is no trend in dividend growthvolatilityand the downwardtrend for consumptiongrowthvolatility is no longer statistically significant. Finally, the models with and without a trend yield highly correlated conditional volatility estimates. For example, for dividend growth, this correlationis 0.87. A more compelling model is a model with parameter breaks. We therefore conduct Bai and Perron(1998)multiple breaktests separatelyforconsumptionanddividends basedonthe following regressionequation 2 ∆df t =a 0 +a 1 rf t−4 +a 2 dpf t−4 +a 3 spd t−4 +u t (22) (cid:16) (cid:17) andanalogouslyforconsumption. FollowingBaiandPerron(1998),wefirsttestthenullhypothesis of no structural breaks against an alternative with an unknown number of breaks. For both dividend and consumption growth volatility, we reject at the 5 percent level. Having established the presenceofabreak,weuseaBIC-basedcriteriontoestimatethenumberofbreaks. Inthecase of dividend growth, this procedure suggested one break. This break is estimated to be located at 1939Q2with a95percentconfidence intervalextending through1947Q1. For consumptiongrowth, theBICcriterionselectstwobreakswiththemostrecentoneestimatedat1948Q1witha95percent confidence interval extending through 1957Q1. Other criteria suggested by Bai and Perron (1998) also found two or fewer breaks for both series. These results are robust to various treatments of autocorrelation in the residuals and heteroskedasticity across breaks. Given that pre-war data are also likely subject to more measurement error than post-war data, we therefore consider an alternativeestimationusingpost-wardata. Consistentwiththeexistingevidence,including Lettau, Ludvigson and Watcher (2006)’s, we continue to find that dividend yields predict macro-economic 16

volatility, but the coefficients on the instruments are indeed smaller than for the full sample. 4.2 Estimation and Testing Procedure 4.2.1 Parameter Estimation ′ Our economy has five state variables, which we collect in the vector Y =[∆d ,v ,u ,q ,π ]. While t t t t t t u ,∆d andπ aredirectlylinkedto thedata,v andq arelatentvariables.We areinterestedinthe t t t t t implications of the model for seven variables: filtered dividend and consumption growth, ∆df and t ∆cf, inflation, π , the short rate, rf, the term spread, spd , the dividend yield, or dividend price t t t t ratio,dp ,andlogexcessequityreturns,rx . Forallthesevariablesweusethedatadescribedabove. t t Thefirstthreevariablesare(essentially)observablestatevariables;thelastfourareendogenousasset prices and returns. We collect all the observables in the vector W . t The relation between term structure variables and state variables is affine, but the relationship between the dividend yield and excess equity returns and the state variables is non-linear. In the Computational Appendix, we linearize this relationship and show that the approximation is quite accurate. Note that this approachis very different from the popular Campbell-Shiller (1988) linearizationmethod, whichlinearizesthereturnexpressionitselfbeforetakingthe linearizedreturn equation through a present value model. We first find the correct solution for the price-dividend ratio and linearize the resulting equilibrium. Conditional on the linearization, the following property of W obtains, t W =µw(Ψ)+Γw(Ψ)Yc (23) t t whereYc isthecompanionformofY containingfivelagsandthecoefficientssuperscriptedwith“w” t t arenonlinearfunctionsofthemodelparameters,Ψ. BecauseY followsalinearprocesswithsquaret root volatility dynamics, unconditional moments of Y are available analytically as functions of the t underlying parameter vector, Ψ. Let X(W ) be a vector valued function of W . For the current t t purpose, X() will be comprised of first, second, third and fourth order monomials, unconditional · expectations of which are uncentered moments of W . Using Equation (23), we can also derive the t analytic solutions for uncentered moments of W as functions of Ψ. Specifically, t E[X(W )]=f(Ψ) (24) t 17

where f() is also a vector valued function (subsequent appendices provide the exact formulae)7. · ThisimmediatelysuggestsasimpleGMMbasedestimationstrategy. TheGMMmomentconditions are, T 1 g (W ;Ψ )= X(W ) f(Ψ ). (25) T t 0 t 0 T − t=1 X Moreover, the additive separability of data and parameters in Equation (25) suggests a “fixed” optimal GMM weighting matrix free from any particular parameter vector and based on the data alone. Specifically, the optimal GMM weighting matrix is the inverse of the spectral density at frequency zero of g (W ;Ψ ), which we denote as S(W ). To reduce the number of parameters T t 0 T estimated in calculating the optimal GMM weighting matrix, we use a procedure that exploits the structure implied by the model, and then minimize the standard GMM objective function, as described in Appendix D. 4.2.2 Moment Conditions We use a total of 34 moment conditions, listed in the notes to Table 2, to estimate the model parameters. They can be ordered into five groups. The first set is simply the unconditional means of the W variables; the second group includes the second uncentered moments of the state variables. t In combination with the first moments above, these moments ensure that the estimation tries to match the unconditional volatilities of the variables of interest. The third set of moments is aimed at identifying the autocorrelationof the fundamental processes. The moving average filter applied to dividend and consumption growth makes it only reasonable to look at the fourth order autocorrelations. Because our specification implies complicated ARMA behavior for inflation dynamics, we attempt to fit both the first and fourth order autocorrelation of this series. The fourth set of moments concerns contemporaneous cross moments of fundamentals with asset prices and returns. As was pointed out by Cochrane and Hansen (1992), the correlation between fundamentals and asset prices implied by standard implementations of the consumption CAPM model is often much toohigh. Wealsoincludecrossmomentsbetweeninflation,theshortrate,andconsumptiongrowth to help identify the ρ parameter in the inflation equation. πu 7In practice, we simulate the unconditional moments of order three and four during estimation. While analytic solutionsareavailableforthesemoments,theyareextremelycomputationallyexpensivetocalculateateachiteration of the estimation process. For these moments, we simulate the system for roughly 30,000 periods (100 simulations perobservation)andtakeunconditonalmomentsofthesimulateddataastheanalyticmomentsimpliedbythemodel withouterror. Duetothehighnumberofsimulations perobservation, wedonotcorrectthestandard errorsofthe parameter estimates forthesimulationsamplingvariability. Tocheck thatthisisareasonablestrategy, weperform a one-time simulationat amuch higher rate (1000 simulations /observation) at the conclusion of estimation. We checkthat theidentifiedparametersproduceavaluefortheobjectivefunctionclosetothatobtainedwiththelower simulationrateusedinestimation. 18

Next, the fifth set of moments includes higher order moments of dividend and consumption growth. Thisiscrucialtohelpensurethatthedynamicsofv areidentifiedby,andconsistentwith, t the volatility predictability of the fundamental variables in the data, and to help fit their skewness and kurtosis. Note that there are 34 19=15 over-identifying restrictions and that we can use the standard − J-test to test the fit of the model. 5 Estimation Results This sectiondescribesthe estimationresults ofthe structuralmodel, andcharacterizesthe fit ofthe model with the data. 5.1 GMM Parameter Estimates Table 2 reports the parameter estimates. We start with dividend growth dynamics. First, u t significantly forecastsdividend growth. Consequently,asin Lettau and Ludvigson(2005)andMenzly, Santos and Veronesi (2004), there is a persistent variable that simultaneously affects dividend growthand potentially equity risk premiums. Second, the conditionalvolatilityof dividend growth, v ,ishighlypersistentwithanautocorrelationcoefficientof0.9795anditselfhassignificantvolatility t (σ , is estimated as 0.3288 with a standard error of 0.0785). This confirms that dividend growth vv volatility varies through time. Further, the conditional covariance of dividend growth and v is t positive and economically large: σ is estimated at 0.0413 with a standard error of 0.0130. dv The results for the consumption-dividend ratio are in line with expectations. First, it is very persistent,withanautocorrelationcoefficientof0.9826(standarderror0.0071). Second,thecontemporaneouscorrelationof u with ∆d is sharply negative as indicated by the coefficient σ which is t t ud estimatedat 0.9226. InlightofEquation(3),thishelpstomatchthelowvolatilityofconsumption − growth.However,because(1+σ )isestimatedto begreaterthanzero,dividendandconsumption ud growth are positively correlated, as is true in the data. Finally, the idiosyncratic volatility parameter for the consumption dividend ratio σ is 0.0127with a standarderror of just 0.0007,ensuring uu that the correlationof dividend and consumption growth is not unrealistically high. The dynamics of the stochastic preference process, q , are presented next. It is estimated to t be quite persistent, with an autocorrelation coefficient of 0.9787 (standard error 0.0096) and it has significant independent volatility as indicated by the estimated value of σ of 0.1753 (standard qq 19

error 0.0934). Of great importance is the contemporaneous correlation parameter between q and t consumption growth, σ . While σ is negative, it is not statistically different from zero. This qc qc indicates that risk is indeed moving countercyclically,in line with its interpretationas risk aversion under a habit persistence model such as that of CC. What is different in our model is that the correlation between consumption growth and risk aversion8 is 0.37 instead of 1.00 in CC. The − − impatience parameter ln(β) is negative as expected and the γ parameter (which is not the same as riskaversioninthismodel)ispositive,butnotsignificantlydifferentfromzero. Thewedgebetween mean dividend growth and consumption growth, δ, is both positive and significantly different from zero. Finally, we present inflation dynamics. As expected, past inflation positively affects expected inflation with a coefficient of 0.2404 (standard error 0.1407) and there is negative and significant predictability running from the consumption-dividend ratio to inflation. 5.2 Model Moments Versus Sample Data Table 2 also presents the standard test of the overidentifyingrestrictions, which fails to reject, with a p-value of 0.6234. However,there are a large number of moments being fit and in such cases, the standard GMM overidentificationtest is known to have low power in finite samples. Therefore,we examine the fit of the model with respect to specific moments in Tables 3 and 4. Table3focusesonlinearmomentsofthe variablesofinterest:mean,volatilitiesandautocorrelations. The model matches the unconditional means of all seven of the endogenous variables. This includes generating a realistic low mean for the nominal risk free rate of about 1% and a realistic equity premium of about 1.2% (all quarterly rates). Analogously, the implied volatilities of both the financial variables and fundamental series are within one standard error of the data moment. Finally, the model is broadly consistent with the autocorrelation of the endogenous series. The (fourth) autocorrelation of filtered consumption growth is somewhat too low relative to the data. However,inunreportedresults we verifiedthatthe completeautocorrelogramsofdividendandconsumption growth implied by the model are consistent with the data. The model fails to generate sufficient persistence in the term spread but this is the only moment not within a two standard bound around the data moment. However, it is within a 2.05 standard error bound! Asexploredbelow,thetimevaryingvolatilityofdividendgrowthisanimportantdriverofequity 8Morespecifically, theconditional correlationbetween ∆ct+1 andqt+1 whenvt andqt areattheir unconditional meanofunity. 20

returnsandvolatility,anditisthereforeimportanttoverifythatthemodel-impliednonlinearitiesin fundamentals are consistent with the data. In Table 4, we determine whether the estimated model is consistent with the reduced form evidence presented in Table 1, and we investigate skewness and kurtosis of fundamentals and returns. In Panel A, we find that the volatility dynamics for fundamentals are quite well matched. The model produces the correctsign in forecasting dividend andconsumptiongrowthvolatilitywithrespecttoallinstruments. Mostly,thesimulatedcoefficients are within or close to two standard errors of the data coefficients with the coefficient on the spread for dividend growth volatility being the least accurate (2.78 standard errors too large). However, for return volatility, the sign with respect to the short rate is incorrect. With respect to multivariate regressions (not reported), the model does not perform well. This is understandable,as it representsa very toughtest of the model. Implicitly, suchtest requires the model to also fit the correlationamong the three instruments. Panel B focuses on skewness and kurtosis. The model implied kurtosis of filtered dividend growth is consistent with that found in the data and the model produces a bit too much kurtosis in consumption growth rates. Equity return kurtosis is somewhat too low relative to the data, but almost within a 2 standard error bound. The model produces realistic skewness numbers for all three series. We conclude that the nonlinearities in the fundamentals implied by the model are reasonably consistent with the data. 6 Risk Aversion, Uncertainty and Asset Prices In this section, we explore the dominant sourcesof time variationin equity prices (dividend yields), equity returns, the term structure, expected equity returns and the conditional volatility of equity returns. We also investigate the mechanisms leading to our findings. Tables 5 and 6 contain the core results in the paper. Table 5 reports basic properties of some critical unobserved variables, including v and q . Table 6 reports variance decompositions with t t standarderrorsfor severalendogenousvariables ofinterest andessentially summarizes the response of the endogenous variables to each of the state variables. Rather than discussing these tables in turn, we organize our discussion around the different variables of interest using information from the two tables. 21

6.1 Risk Aversion and Uncertainty Table 5, Panel A presents properties of the unobservable variables under the estimated model. The propertiesof“uncertainty”,v ,whichisproportionaltotheconditionalvolatilityofdividendgrowth, t and q , which drives risk aversion,werediscussed before [see Section5]. Because localrisk aversion, t RA , in this model is given by γexp(q ), we can examine its properties directly. The median level t t ofriskaversioninthe modelis 2.52,a levelwhichwouldbe consideredperfectly reasonablebymost financial economists. However, risk aversion is positively skewed and has large volatility so that risk aversionis occasionally extremely high in this model. Panel B of Table 5 presents results for means of the above endogenous variables conditional on whether the economy is in a state of expansion or recession. For this exercise, recession is defined as one quarter of negative consumption growth. Both v and q (and hence local risk aversion)are t t counter-cyclical. 6.2 Risk Aversion, Uncertainty, and the Term Structure Panel A of Table 5 also displays the model-implied properties of the real interest rate and the real termspread. The averagerealrateis17basispoints(68annualized)andthe realinterestratehasa standard deviation of around 90 basis points. The real term spread has a mean of 38 basis points, a volatility of only 28 basis points and is about as persistent as the real short rate. In PanelB, we see that real rates are pro-cyclical and spreads are counter-cyclical, consistent with the findings in Ang, Bekaert and Wei (2007). PanelCofTable5showsthatuncertaintytends todepressrealinterestrates,while positiverisk aversion shocks tend to increase them. To gain further insight into these effects, let us derive the explicit expression for the real interest rate by exploiting the log-normality of the model: 1 rrf = E [m ] V [m ]. (26) t t t+1 t t+1 − − 2 Theconditionalmeanofthepricingkerneleconomicallyrepresentsconsumptionsmoothingwhereas the variance of the kernel represents precautionary savings effects. To make notation less cumbersome, let us reparameterize the consumption growth process as having conditional mean and 22

variance E [∆c ]=δ+µ +(ρ +ρ 1)u µ +ρ u t t+1 d du uu t c cu t − ≡ V [∆c ]=σ2 v . (27) t t+1 cc t Then the real rate simplifies to rrf = ln(β)+γ(µ µ )+γρ u +φ q +φ v (28) t c q cu t rq t rv t − − with φ = γ(1 ρ ) 1γ2σ2 and φ = 1γ2(σ 1)2σ2 . Changes in risk aversion have an rq − qq − 2 qq rv −2 qc − cc ambiguous effect on interest rates depending on whether the smoothing or precautionary savings effect dominates (the sign of φ ). At our parameter values, the consumption smoothing effect rq dominates. As discussed before, v represents a precautionary savings motive, so the correlation t between real rates and v is negative. Overall, real rates are pro-cyclical because v is strongly t t counter-cyclical. Moving to the real term spread, it displays a positive correlation with both v and q , but for t t differentreasons. Toobtainintuition,letusconsideratwoperiodbondandexploitthelog-normality of the model. We can decompose the spread into three components: 1 1 1 rrf rrf = E [rrf rrf ]+ Cov [m ,rrf ] Var [rrf ] 2,t t t t+1 t t t+1 t+1 t t+1 − 2 − 2 − 4 The first term is the standard expectations hypothesis (EH) term, the second term represents the term premium and the third is a Jensen’s inequality term (which we will ignore). Because of mean reversion, the effects of u , v , and q on the first component will be opposite of their effects on the t t t level of the short rate. Figure 1 decomposes the exposures of both the real interest rate and the spread to v and q into an expectations hypothesis part and a term premium part and does so for t t various maturities (to 40 quarters). It shows that q has a positive effect on the term spread. Yet, t the coefficient on q in the EH term is φ (ρ 1) and thus negative as φ is positive. However, t rq qq rq − it is straightforwardto show that the coefficient on q for the term premium is 1γφ σ2 and hence t 2 rq qq the term premium effect of q will counter-balance the EH effect when φ > 0. Yields at long t rq maturities feature a term premium that is strongly positively correlatedwith q because higher risk t aversion increases interest rates (and lowers bond prices) at a time when marginal utility is high, making bonds risky. 23

Increased uncertainty depresses short rates and, consequently, the EH effect implies that uncertainty increases term spreads. The effect of v on the term premium is very complex because the t correlation between q and the kernel is also driven by v . In fact, straightforward algebra shows t t that the coefficient on v is proportional to t (σ 1) σ2 (γρ +φ σ )+(1+σ ) γρ σ +φ σ (σ +1) σ2 +σ2 φ σ σ . qc − uu uc rq qc ud uc ud rq qc ud dd dv − rv vv dv (cid:2) (cid:0) (cid:0) (cid:1) (cid:1)(cid:3) Empirically,weestimatethiscoefficienttobe0.0020,sothattheEHeffectisthedominanteffect. Hence, when uncertainty increases, the term structure steepens and vice versa. In Table 6, we report the variance decompositions. While three factors (u , v , and q ) affect t t t the real term structure, v accounts for the bulk of its variation. An important reason for this fact t is that v is simply more variable than q . The most interesting aspect of the results here is that t t q contributes little to the variability of the spread, so that q is mostly a level factor not a spread t t factor,whereas uncertainty is both a leveland a spreadfactor. When we considera realconsol,we find that q dominates its variation. Because consol prices reflect primarily longer term yields, they t are primarily driven by q , through its effect on the term premium. t For the nominal term structure, inflation becomes an important additional state variable accounting for about 12% of the variation in nominal interest rates. However, inflation is an even more important spread factor accounting for about 31% of the spread’s variability. What may be surprising is that the importance of v relative to q decreases going from the real to nominal t t term structure. The reasonis the rather strong positive correlationbetween inflation and v , which t arises from the negative relation between inflation and the consumption dividend ratio, that ends up counterbalancing the negative effect of v on real interest rates. t 6.3 Risk Aversion, Uncertainty and Equity Prices Here we start with the variance decompositions for dividend yields and equity returns in Table 6. For the dividend yield, q dominates as a source of variation, accounting for almost 90% of its t variation. To see why, recall first that q only affects the dividend yield through its effect on the t termstructureofrealinterestrates(seeProposition4). UndertheparameterspresentedinTable2, the impact of q on real interest rates is positive at every horizonand therefore it is positive for the t dividend yield as well. Formally, under the parameters of Table 2, F in Proposition 4 is negative n at all horizons. b 24

Next, consider the effect of v on the dividend yield. Uncertainty has a “real consol effect” t and a “cash-flow risk premium” effect which offset each other. We already know that v creates a t strongprecautionarysavingsmotive,whichdecreasesinterestrates. Allelseequal,thiswillserveto increaseprice-dividendratiosanddecreasedividendyields. However,v alsogovernsthecovariance t of dividend growth with the real kernel. This risk premium effect may be positive or negative, but intuitively the dividend stream will represent a risky claim to the extent that dividend growth covaries negatively with marginal utility. In this case, we would expect high v to exacerbate this t riskiness and depress equity prices when it is high, increasing dividend yields. As we discussed in section 3, σ contributes to this negative covariance. On balance, these countervailing effects of v qc t on dividend yields largely cancel out, so that the net effect of v on dividend yields is small. This t shows up in the variance decomposition of the dividend yield. On balance, q is responsible for the t overwhelming majority of dividend yield variation, and is highly positively correlatedwith it. The negative effect of u arises from its strong negative covariance with dividend growth. t Looking back to panel C in Table 5, while increases in q have the expected depressing effect on t equity prices (a positive correlation with dividend yields), increases in v do not. This contradicts t with the findings in Wu (2001) and Bansal and Yaron (2004) but is consistent with early work by Barsky(1989)andNaik(1994). Becausetherelationisonlyweaklynegative,theremaybeinstances where our model will generate a classic “flight to quality” effect with uncertainty lowering interest rates, driving up bond prices and depressing equity prices. Next notice the determinants of realized equity returns in Table 6. First, over 30% of the variationinexcessreturnsisdrivenbydividendgrowthanddividendgrowthispositivelycorrelated with excess returns. This is not surprising in light of the fact that dividend growth enters the definitionofstockreturnsdirectlyanddividendgrowthhasalmosthalfasmuchvariationasreturns themselves. The other primary driver of stock returns is q . This is a compound statistic which t includes the effect of current and lagged q . In fact, the contemporaneous effect of q on returns is t t negative (see Table 5) as increases in q depress stock valuations. However, the lagged effect of q t t on returns is positive because, all else equal, lower lagged prices imply higher current returns. 6.4 Risk Aversion, Uncertainty and the Equity Premium We again go back to Table 5 to investigate the properties of the conditional equity premium, E [rx ]. The premium is quite persistent, with an autocorrelation coefficient of 0.9789. In t t+1 PanelB,wealsofinditishigherinrecessionswhichisconsistentwithcounter-cyclicalriskaversion. 25

Panel C shows that both v and q are positively correlated with the equity premium. The risk t t premium in any model will be negatively correlated with the covariance between the pricing kernel and returns. We already discussed how uncertainty contributes to the negative covariance between the pricing kernel and dividend growth and therefore increases risk premiums. The effect of q t comes mostly throughthe capitalgain partof the return: increases in q both raise marginalutility t and decrease prices making stocks risky. Table 6 shows that the point estimate for the share of the equity premium variation due to v is about 17% but with a standard error of 13%, with the t remainder due to q . t The fact that both the dividend yield and expected equity returns are primarily driven by q t suggest that the dividend yield may be a strong predictor of equity returns in this model. Table 7 shows that this is indeed the case, with a regressionof future returns on dividend yields generating a 1.58 coefficient. We also compare the model coefficients with the corresponding statistics in the data. It turns out that the predictability of equity returnsduring our sample period is rather weak. Table 7 reports univariate coefficients linking equity returns to short rates, dividend yields and spreads. The sign of the coefficients matches well-known stylized facts but none of the coefficients are significantly different from zero. The model produces coefficients within two standard errors of these data coefficients but this is of course a rather weak test. While it is theoretically possible to generate a negative link between current short rates and the equity premium which is observed empirically, our model fails to do so at the estimated parameters. We also report the results of a multivariate regression on the aforementioned instruments. The model here gets all the signs right and is always within two standard errors of the data coefficients. More generally, the ratio, VAR(E [rx ])/VAR(rx ),fromTables3and5,impliesaquarterlyR2oflessthanonepercent, t t+1 t+1 so the model does not generate much short term predictability of equity returns consistent with recent evidence. There is a large debate on whether predictability increases with the horizon. In our model, the variance ratio discussed above for 10 year returns equals about 12% (not reported). Whilewehavestudiedtheconditionalequitypremium,itremainsusefultoreflectonthesuccess of the model in matching the unconditional equity premium. The model also matches the low risk free rate while keeping the correlation between fundamentals (dividend and consumption growth) and returns low. In fact, the correlation between dividend growth and equity returns is 0.28 in the data and 0.33 in the model. For consumption growth, the numbers are 0.07 and 0.11 respectively. In addition, the model matches the correlation between dividend yields and consumption growth, which is 0.14 under the model and in the data. Consequently, this model performs in general − 26

better than the CC model, which had trouble with the fundamentals-return correlation. 6.5 Risk Aversion, Uncertainty, Equity Return Volatility and Sharpe Ratios To conclude, we investigate the properties of the conditional variance of equity returns, the equity SharperatioandthemaximumattainableSharperatioavailableintheeconomydiscussedinSection 2. We begin with the numbers in Panel A of Table 5. The conditional variance of excess equity returns has a mean of 0.0092, a standard deviation of 0.0070 and an autocorrelation of 0.9794. The final two columns of Table 5 report results for the conditional Sharpe ratio of equity and the maximumattainableSharperatioavailableintheeconomydiscussedinSection2. Themeanequity Sharpe ratio attains approximately three quarters of the maximum attainable value. Both Sharpe ratios are strongly persistent and possess significant time variation driven by v and q . These t t Sharpe ratios are quarterly, and so their magnitude is roughly half of annualized values. The conditional variance of equity returns is counter-cyclical. Interestingly, the increase in expected equity returns during recessions is not as large as the increase in the expected variance which contributes to the equity Sharpe ratio failing to be counter-cyclical. The maximum Sharpe ratio does display counter-cyclical behavior. Moving to Table 6, not surprisingly, the conditional volatility of equity returns is largely governed by v , which accounts for 75% of its variation with a t standard error of only 32%. Here, q contributes 25% to the total volatility variation. t 7 Robustness and Related Literature By introducing a time-varying preference shock not correlated with fundamentals (q ) and timet varying economic uncertainty (v ), our model matches a large number of salient asset price features t while keeping the correlation between fundamentals and asset returns low. Economic uncertainty acts as both a level and spread factor in the term structure, has a large effect on conditional stock market volatility, but little effect on dividend yields and the equity premium, which are primarily drivenby the preference shock. In this section, we first examine whether these results hold up if we estimate the model using post-War data, where macro-economicvolatility was decidedly less severe than pre-War. We then provide intuition on why our model yields different results than a number of well-known existing articles. 27

7.1 Post-War Estimation Results The model fits the Post-War data well and the test of the over-identifying restrictions does not reject. Table 8 summarizes some of the implications of the Post-War model. The first two columns show how the model continues to match the properties of equity returns and the nominal interest rate, but more generally it does as well as the previous model did. In particular, it also fits the low correlationbetween asset returns and fundamentals that we continue to observe. Interestingly, it does so with about the same risk aversion as for the full sample. Average risk aversion is only slightly largerthan it was for the full sample estimation, but its standarddeviationis much smaller reflectingadatasamplewithfewerextremeobservations. The largestchangeisthatv isnowmuch t lesspersistentandlessvolatilethanitwasforthefullsampleestimation. Thiswillhaveimplications for the role of v in asset pricing, but it does not materially affect the cyclicality of the endogenous t variables, or the correlations of v with observables. In fact, we do not repeat Panels B and C t from the old Table 5 because all the inference is identical for the new model with one exception: the equity Sharpe ratio is now also counter-cyclical confirming the usual finding in the literature. Instead,PanelBreportsthevariancedecompositionsforthetermstructureandequityprices. Here, the reduced persistence of v implies that the role of v is overalldiminished. Interestingly, the role t t of q as a level factor, and v as a spread factor is now even sharper, with q accounting for the t t t bulk of variation in dividend yields, equity premiums and even the conditional volatility of equity returns. Hence,iftheretrulywasapermanentstructuralbreakinmacro-economicuncertaintyafter the War, our estimation suggests that preference shocks play an even larger role in driving asset prices than was reported before. 7.2 Related Literature 7.2.1 Abel (1988) and Wu (2001) Abel (1988)creates an economy in which the effect of increasedcashflow volatility on equity prices depends on a single parameter, the coefficient of relative risk aversion. His setup is vastly different from ours. Most importantly, Abel (1988) maintains that dividends themselves are stationary and so are prices (at least on a per-capita basis). Also, there is no distinction between consumption and dividends in his model, so that the covariance of cash flows with the pricing kernel and the volatility of the pricing kernel are proportional. Finally, there is no preference shock. In the current framework, we can consider the effects of some of Abel’s assumptions by simply shutting 28

downthe dynamicsofthe consumptiondividendratio(u =0)andstochasticriskaversion(q =0). t t However, we do not implement Abel’s assumption that dividends and prices are stationary. Proposition 5 FortheeconomydescribedbyEquations(9)and(10),andtheadditionalassumption that the following parameters are zero, µ u ,µ q ,ρ du ,ρuu,ρ qq ,σ ud ,σuu,σ qc ,σ qq the equity price-dividend ratio is represented by ∞ P t = exp ←→A +←→E v n n t D t n X =1 (cid:16) (cid:17) where ←→A n =lnβ+←→A n−1 +(1 γ)µ d +←→E n−1 µ v − 1 2 1 2 ←E→ n =←→E n−1 ρ vv + 2 ←E→ n−1 +1 − γ σ d 2 d + 2 ←B→ n−1 +1 − γ σ dv +←E→ n−1 σ vv (cid:16) (cid:17) (cid:16)(cid:16) (cid:17) (cid:17) with ←→A =←E→ =0 (Proof available upon request.) 0 0 The effect ofvolatility changesonthe price dividend ratiois givenby the ←→E coefficient. When n volatility is positively autocorrelated, ρ > 0, ←→E > 0 and increases in volatility always increase vv n equity valuation, essentially because they depress the interest rate. In comparison to the effects of v inProposition4,only the Jensen’s Inequalityterms remain. There is no scope for v to alter the t t riskiness of the dividend stream beyond the real term structure effects because cash flows and the pricingkernelareproportional. Clearly,thissimplifiedframeworkistoorestrictiveforourpurposes. Wu (2001) develops a model wherein increases in volatility unambiguously depress the pricedividend ratio. The key difference between his model and ours is that Wu models the interest rate as exogenous and constant. To recover something like Wu’s results in our framework requires making the real interest rate process exogenous and maintaining the volatility process of Equation (9). Assume for example that we introduce a stochastic process x and modify the specification of t the dividend growth process to be: lnβ 1 γ ∆d t = γ + γ x t−1 + 2 v t−1 +√v t−1 εd t x t =µ x +ρ xx x t−1 +σ x εx t +σ xv √v t−1 εv t +σ xd √v t−1 εd t (29) Itis easilyverifiedthatunder these specifications andthe additionalassumptionsofProposition 5, x is equal to the one-period real risk free rate. The solution for the price-dividend ratio in this t economy is described in the following proposition 29

Proposition 6 For the economy described in Proposition 5 with the dividend process modified as in Equations (29) the equity price-dividend ratio can be expressed as ∞ P t = exp −→A +−→G x +−→E v n n t n t D t n X =1 (cid:16) (cid:17) where lnβ 1 2 −→A n =−→A n−1 + +−→G n−1 µ x +−→E n−1 µ v + −→G n−1 σ x γ 2 (cid:16) (cid:17) 1 −→G n = 1+ +−→G n−1 ρ xx − γ (cid:18) (cid:19) γ2 γ 1 1 2 −→E n = + +−→E n−1 ρ vv + ( γ+1)2+ −→G n−1 σ xv +−→E n−1 σ v − 2 2 2 − 2 (cid:16) (cid:17) with −→A =−→E =−→G =0.(Proof available upon request.) 0 0 0 By considering the expression for −→E , we see that the direct effect of an increase in v is n t 1γ(1 γ). Therefore, only when γ > 1 will an increase in volatility depress the price-dividend 2 − ratio,butthis ignoresequilibriumtermstructureeffects. Inthecontextofamodelwithanendogenous term structure, Wu’s results appear not readily generalizable. 7.2.2 Relation to Bansal and Yaron In generating realistic asset return features, Bansal and Yaron (2004) (BY henceforth) stress the importance of a small persistent expected growthcomponent in consumption and dividend growth, fluctuations in economic uncertainty and Epstein and Zin (1989) preferences, which allow for separation between the intertemporal elasticity of substitution (IES) and risk aversion. In fact, many of their salient results, including the non-trivial effects of economic uncertainty on price-dividend ratios and the equity premium rely explicitly on preferences being non-CRRA and the IES being largerthan1. Whileitisofcourseconceivablethatthepresenceofq alonedrivesenoughofawedge t betweenthe Bansal-Yaronframeworkandours,itmay stillcome as a surprisethatwe findsuchan importantrolefor economicuncertainty inwhat is essentiallya powerutility framework. Moreover, empirically, our estimation does not yield as large a role for the persistent expected cash flow and consumption growth component as in Bansal and Yaron (2004)’s calibrations. In this section, we resolve this conundrum by showing that the critical importance of the Epstein -Zin preferences in the BY framework stems from a rather implausibly strong assumption in the dynamics for the exogenous variables. To conserve space, we defer all proofs and derivations to an Appendix available upon request, and provide the key results and intuition here. BY use a log-linearized framework 30

where the pricing kernel and equity return innovations can be written as follows: m E m =λ √v εc λ √v εu λ σ εv (30) t+1 − t t+1 mc t t+1− mu t t+1− mv vv t+1 rx E (rx )=β √v εd +β √v εu +β σ εv (31) t+1 − t t+1 xd t t+1 xu t t+1 xv vv t+1 where we transformed BY’s notation as much as possible into ours. Here c, d, u and v refer respectively to the consumption growth, dividend growth and unobserved expected consumption ′ growthandvolatilityprocesses,andtheλ’s (β s)measurethe exposureofthe pricing kernel(equity return)totheseshocks. Notethattheshockstoconsumption/dividendgrowthandtothepersistent expected growth process, u , are heteroskedastic, but the shock to volatility is not. Moreover, t λ ∝[(1 1/IES) 1],λ =(1 θ)λ ,λ =(1 θ)λ whereθ =[1 γ]/[1 1/IES]andγ mc mu mu mv mv − − − − − − istheusualriskaversionparameter. Hence,foraCRRAutilityfunctionθ =1,andλ =λ =0. b b mu mv From Equations (30) and (31), the equity premium simply follows from the standard expression as the conditional covariance between the return and the kernel corrected for a Jensen’s inequality term. So one key assumption is then how the various shocks are correlated. Critically, BY assume total lack of correlation between all these shocks. Consequently, the model must generate the nontrivial correlation between consumption and dividend growth through the joint exposure to the latent u variable. Of courseu represents a persistent predictable component for which we have no t t directevidence onitsexistence. The implicationsarestark. The generalequitypremiumexpression is: 1 E [rx ]= var (rx )+β λ v +β λ σ t t+1 t t+1 xu mu t xv mv v −2 Hence, if θ = 1, as it is for CRRA utility, only the Jensen’s inequality term remains. Similarly, it is then also the case that the price dividend ratio is actually increasing in economic uncertainty. However, once you assume that dividend and consumption growth shocks are correlated, these knife-edge implications of CRRA utility disappear. As a simple example, assume that dividend growthequalsconsumptiongrowth(wearepricingaclaimtoconsumption),thenitisstraightforward to show that, under CRRA utility (θ =1): 1 E [rx ]= var (rx )+γv t t+1 t t+1 t −2 31

That is, we recover an intuitive result, also an implication of Wu (2001) ’s model, that the equity premium is proportional to the conditional variance of cash flow growth. While our model does not assume such strong correlation between consumption and dividend growth as we entertained in the above example, there is nonetheless non-trivial conditional correlation that depends on v and by itself gives rise to non-trivial pricing and premium effects. Finally, t it is also the case that, even with a BY - like structure governingthe dynamics of consumption and dividend growth, the “Moody Investor” preferences we use would still lead to non-trivial pricing effects with v and q both being priced, and the sign of v ’s effect on premiums and price-dividend t t t ratios depending intimately on the sign of σ . qc 7.2.3 Relation to Campbell and Cochrane (1999) Our model differs from CC’s along numerous dimensions, yet, we would like to discuss the implicationsoftwomajordifferencesbetweenourapproachandtheirs. Thefirstisofcoursethepresenceof the q shock,whichis notcorrelatedwith fundamentals. Asecondmajordifference lies in ourstratt egy of estimating the structural parameters while matching equity related moments, bond related moments and moments capturing features of fundamentals and their correlationwith asset returns. CC instead calibrate their economy and choose a parameterization that yields a constant interest rate. We show below that both these differences are essential in interpreting our results and they are inter-related. TomovethemodelsubstantiallyinCC’sdirection,wesetσ =0andre-estimatetheparameters. qq The model is now strongly rejected and we fail to match the high equity premium and the low interest rate9. This result may be somewhat surprising as CC appear to do very well with respect to many salient asset price features. One main reason for this result is, in fact, our estimation approach. InFigure2,wegobacktoouroriginalestimationandgraphtheGMMobjectivefunction around the estimated σ and σ (a three dimensional graph yields similar conclusions). Apart vv qq from the GMM objective function, we also show a function that aggregatesthe bond moments and one that aggregates the equity moments. You can then read off which parameters would maximize these separateobjectivefunctions. Clearly,parametersyieldingthe best“bond-fit”areprettymuch indistinguishable from the estimated parameters; however, for a good equity fit the model wants a higher σ and a lower σ . In other words, the bond moments are very informative about the qq vv 9Interestingly, an analogous resultappears in Lettau and Wachter (2007) and Santos and Veronesi (2006): when thevariablemovingthepriceofriskisperfectlynegativelycorrelatedwithconsumptiongrowth(asintheCCmodel), theirmodel’sperformancewithrespecttothecross-sectionofexpected returnsalsodeteriorates considerably. 32

structural parameters. When we let σ go to zero, it is, in fact, the fit with the term structure qq moments and the link between fundamentals and asset returns that cause the bad fit. When we restrict σ equal to 0, but only try to fit equity moments, fundamental moments and the mean qq interest rate, the model is not rejected. Our emphasis on simultaneously matching these three dimensions of the data (term structure movements, equity moments, and the correlation between fundamentals and asset returns) also distinguishesourworkfromrecentarticlesbyWachter(2006)andBuraschi-Jiltsov(2007)whoshow reasonable fits of extensions of the CC model with term structure data. Consequently, while we have formulated a consumption-based asset pricing model that successfully matches many salient asset pricing phenomena, the presence of preference shocks not correlatedwith fundamental shocks is essential to its success. 8 Conclusion This paper has attempted to sort out the relative importance of two competing hypotheses for the sourcesofthe magnitude andvariationofassetprices. First, one literature has exploredthe roleof cashflow volatility dynamics as a determinant of equity premiums both in the time series and cross section. Recent work in this area includes Bansal and Yaron (2003), Bansal, Khatchatrian and Yaron(2002),and Bansaland Lundblad (2004). A quite separateliterature has exploredshocks to investors preferences as drivers of equity prices. Prominent papers in this area include Campbell andCochrane(1999),Abel(1990,1999),andalargenumberofelaborationssuchasWachter(2006), Bekaert,EngstromandGrenadier(2004),BrandtandWang(2003),GordonandSt. Amour(2000), Menzly, Santos and Veronesi (2004), Wei (2004), and Lustig and van Nieuwerburgh (2005). With some exceptions, the focus has been on equities10. We designa theoreticalmodel andempiricalstrategywhich arecapable ofaccommodating both explanations,andthenimplementanoptimalGMMestimationtodeterminetherelativeimportance ofeachstory. Westressthatfromatheoreticalperspective,itisimportanttoconsidertermstructure effects when evaluating the effect of uncertainty on equity prices, a point prominent in the work of Abel(1988)andBarsky(1989). Weconcludethatboththeconditionalvolatilityofcashflowgrowth 10Inarecentpaper,Bansal,GallantandTauchen (2007) showthatbothaCCandaBansal-Yarontypemodelfit thedataequallywell. 33

and time varying risk aversion emerge as important factors driving variation in the term structure, dividend yields,the equity risk premium andthe conditionalvolatility ofreturns. Notsurprisingly, uncertainty is more important for volatility whereas risk aversion is more important for dividend yields and the risk premium. Ourworkisindirectlyrelatedtotwootherimportantliteratures. First,thereisalargeliterature onthe conditionalCAPMwhich predicts alinear,positive relationbetweenexpectedexcess returns on the market and the conditional variance of the market. Since the seminal work of French, Schwert and Stambaugh (1987), the literature has struggled with the identification of the price of risk, which is often negative in empirical applications (see Scruggs (1998)). Of course, in our model, there are multiple sources of time variation in risk premiums and both the price of risk and the quantity of risk vary through time. Upon estimation of our structural model, we identify a strong positive contemporaneous correlationbetween expected equity returns and their conditional volatility. However, this relationship varies through time and contains a cyclical component (see Table 5). Second, the volatility feedback literature has provided a link between the phenomenon of asymmetric volatility (or the leverage effect, the conditional return volatility and price shocks are negatively correlated)and risk premiums. It suggests that prices can fall precipitously on negative news as the conditional volatility increases and hence induces higher risk premiums (when the price of risk is positive). Hence, the literature primarily builds on the conditional CAPM literature (see Campbell and Hentschel (1992) and Bekaert and Wu (2000)). Wu (2001) sets up a present value model in which the variance of dividend growth follows a stochastic volatility process and shows under what conditions the volatilityfeedback effect occurs. There are two reasonswhy Wu’s (2001) conclusions may not be generally valid. First, he ignores equilibrium considerations—that is the discount rate is not tied to preferences. Tauchen (2005) also shows how the presence of feedback may depend on preference parameters. Second, he assumes a constant interest rate. Within our set up, we can re-examine the validity of an endogenous volatility feedback effect. We intend to explore the implications of our model for these two literatures in the near future. 34

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A Propositions and Proofs A.1 Proposition 1: Real Zero Coupon Bonds FortheeconomydescribedbyEquations(9)and(10),thepricesofreal,riskfree,zerocouponbonds are given by Prz =exp(A +C u +D π +E v +F q ) (A- 1) n,t n n t n t n t n t where A n =fA(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) C n =fC(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) D =0 n E n =fE(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) F n =fF (A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) And the above functions are represented by fA =lnβ γ(δ+µ d )+A n−1 +E n−1 µ v +(F n−1 +γ)µ q − fC γρ du +C n−1 ρ uu +γ(1 ρ uu ) ≡− − fE E n−1 ρ vv ≡ 1 + ( γσ dd +(C n−1 γ)σ ud σ dd +(F n−1 +γ)σ qd )2 2 − − 1 + ((C n−1 γ)σ uu +(F n−1 +γ)σ qu )2 2 − 1 + ( γσ dv +(C n−1 γ)σ ud σ dv +(F n−1 +γ)σ qv +E n−1 σ vv )2 2 − − 1 fF F n−1 ρ qq +γ(ρ qq 1)+ ((F n−1 +γ)σ qq )2 ≡ − 2 and A =C =E =F =0. 0 0 0 0 In these equations we used the following notation saving transformations: σ =σ σ (1+σ ) qd qc dd ud σ =σ σ qu qc uu σ =σ σ (1+σ ). (A- 2) qv qc dv ud This effectively means that we are solving the model for a more general q process: q = µ + t t q ρ qq q t−1 +√v t−1 σ qd εd t +σ qu εu t +σ qv εv t +√q t−1 σ qq εq t . (cid:0) (cid:1) Proof: We start from the bond pricing relationship in equation (14) in the text: Prz =E M Prz (A- 3) n,t t t+1 n−1,t+1 where Prz is the price of a real zero coupon bo (cid:2) nd at time t w (cid:3) ith maturity at time (t+n). n,t Suppose the prices of real, risk free, zero coupon bonds are given by Prz =exp(A +C u +D π +E v +F q ) (A- 4) n,t n n t n t n t n t 1

where A n =fA(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) C n =fC(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) E n =fE(A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) F n =fF (A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ) Then we have exp(A +C u +D π +E v +F q ) n n t n t n t n t =E t exp(m t+1 +A n−1 +C n−1 u t+1 +D n−1 π t+1 +E n−1 v t+1 +F n−1 q t+1 ) { } =E exp(ln(β) γ(δ+∆u +∆d )+γ∆q t t+1 t+1 t+1 { − +A n−1 +C n−1 u t+1 +D n−1 π t+1 +E n−1 v t+1 +F n−1 q t+1 ) } After taking expectations, exploiting log-normality, we equate the coefficients on the two sides of the equation to obtain the expression given in (A-1). A.2 Proposition 2: Real Consols Under the conditions set out in Proposition 1, the price-coupon ratio of a consol paying a constant real coupon is given by ∞ Prc = exp(A +B ∆d +C u +E v +F q ) (A- 5) t n n t n t n t n t n=1 X This follows immediately from recognizing that the “normalized” consol is a package of zero coupon bonds. A.3 Proposition 3: Nominal Zero Coupon Bonds For the economy described by Equations (9) and (10), the time t price of a zero coupon bond with a risk free dollar payment at time t+n is given by Pz =exp A +B ∆d +C u +D π +E v +F q (A- 6) n,t n n t n t n t n t n t (cid:16) (cid:17) where e e e e e e 1 2 A n =fA A n−1 ,B n−1 ,C n−1 ,E n−1 ,F n−1 + D n−1 − 1 µ π + 2 D n−1 − 1 σ π 2 π (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) B =0 en e e e e e e e C en =fC A n−1 ,B n−1 ,C n−1 ,E n−1 ,F n−1 + D n−1 − 1 ρ πu (cid:16) (cid:17) (cid:16) (cid:17) Den = D n−e1 1eρ ππ e e e e − (cid:16) (cid:17) Een =fE e A n−1 ,B n−1 ,C n−1 ,E n−1 ,F n−1 (cid:16) (cid:17) Fen =fF Aen−1 ,Ben−1 ,Cen−1 ,Een−1 ,Fen−1 (cid:16) (cid:17) where theefunctionsefX()eare gieven inePropoesition 1 for X (A,B,C,E,F) and A =B =C = 0 0 0 · ∈ D =E =F =0. 0 0 0 e e e e e e 2

TheproofofProposition3followsthesamestrategyasProposition1andisommitedtoconserve space. A.4 Proposition 4: Equities For the economy described by Equations (9) and (10), the price-dividend ratio of aggregate equity is given by ∞ P t = exp A +C u +E v +F q (A- 7) n n t n t n t D t n X =1 (cid:16) (cid:17) where b b b b A n =fA A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ +µ d (cid:16) (cid:17) Cbn =fC Abn−1 ,Cbn−1 ,Ebn−1 ,Fbn−1 ,Ψ +ρ du (cid:16) (cid:17) Ebn =fE Abn−1 ,Cbn−1 ,Ebn−1 ,Fbn−1 ,Ψ 1(cid:16) (cid:17) b + 2 σ d 2b d +σ d b d ( − γb)σ dd +b C n−1 − γ σ ud σ dd + F n−1 +γ σ qd (cid:18) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)(cid:19) 1 + 2 σ d 2 v +σ dv ( − γ)σ dv + C b n−1 − γ σ ud σ dv + F b n−1 +γ σ qv +E n−1 σ vv (cid:18) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)(cid:19) F n =fF A n−1 ,C n−1 ,E n−1 ,F n−1 b,Ψ b b (cid:16) (cid:17) where thbe functionbs fX(b) arebgiven bin Proposition 1 for X (A,C,E,F) and A = C = E = 0 0 0 · ∈ F =0. 0 Proof: Let P and D be the time-t ex-dividend stock price and dividend. t t Guess n J ,E exp (m +∆d ) =exp A +C u +E v +F q n,t t t+j t+j n n t n t n t   X j=1 (cid:16) (cid:17)   b b b b Then n−1 J =E exp(m +∆d )E exp(m +∆d ) n,t t t+1 t+1 t+1 t+1+j t+1+j   j=1 X =E t [exp(m t+1 +∆d t+1 )J n−1,t+1 ]  or: exp A +C u +E v +F q n n t n t n t =E(cid:16) exp[ln(β) γ(δ+∆u (cid:17)+∆d )+γ∆q +∆d t {b b − b bt+1 t+1 t+1 t+1 +A n−1 +C n−1 u t+1 +E n−1 v t+1 +F n−1 q t+1 ] } Using the propertiebs of theblognormaldbistribution abnd equating coefficients on both sides of the equation gives us: 3

A n =fA A n−1 ,C n−1 ,E n−1 ,F n−1 ,Ψ +µ d (cid:16) (cid:17) Cbn =fC Abn−1 ,Cbn−1 ,Ebn−1 ,Fbn−1 ,Ψ +ρ du (cid:16) (cid:17) Ebn =fE Abn−1 ,Cbn−1 ,Ebn−1 ,Fbn−1 ,Ψ 1 (cid:16) (cid:17) b + 2 σ d 2b d +σ dbd ( − γb)σ dd +b C n−1 − γ σ ud σ dd + F n−1 +γ σ qd 1(cid:16) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)(cid:17) + 2 σ d 2 v +σ dv ( − γ)σ dv + Cb n−1 − γ σ ud σ dv + Fb n−1 +γ σ qv +E n−1 σ vv (cid:16) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17)(cid:17) F n =fF A n−1 ,C n−1 ,E n−1 ,F n−1b,Ψ b b (cid:16) (cid:17) where thbe functionbs fX(b) are bgiven ibn Proposition 1 for X (A,C,E,F) and A = C = E = 0 0 0 · ∈ F =0. 0 For the purposes of estimation the coefficient sequences are calculated out 200 years. If the resulting calculated value for PD has not converged,then the sequences are extended another 100 t years until either the PD value converges,or becomes greater than 1000 in magnitude. t B Log Linear Approximation of Equity Prices In the estimation, we use a linear approximation to the price-dividend ratio. From Equation (19), we see that the price dividend ratio is given by ∞ P t = q0 D n,t t n=1 X ∞ = exp b0 +b ′ Y (A- 8) n n t n=1 X (cid:0) (cid:1) and the coefficient sequences, b0 ∞ and b ′ ∞ , are given above. We seek to approximate the n n=1 { n}n=1 log price-dividend ratio using a first order Taylor approximation of Y about Y, the unconditional (cid:8) (cid:9) t mean of Y . Let t q0 =exp b0 +b ′ Y (A- 9) n n n and note that (cid:0) (cid:1) ∞ ∞ ∞ ∂ ∂ q0 = q0 = q0 b ′ (A- 10) ∂Y t n,t ! ∂Y t n,t n,t· n n=1 n=1 n=1 X X X Approximating, ∞ ∞ 1 pd t ≃ ln n=1 q0 n ! + ∞ n=1 q0 n n=1 q0 n· b ′ n ! Y t − Y X ′ X (cid:0) (cid:1) =d 0 +dY t P (A- 11) ′ where d and d are implicitly defined. Similarly, 0 ∞ ∞ P 1 gpd t ≡ ln (cid:18) 1+ D t t(cid:19) ≃ ln 1+ n=1 q0 n ! + 1+ ∞ n=1 q0 n n=1 q0 n· b ′ n ! Y t − Y ′ X X (cid:0) (cid:1) =h 0 +hY t P (A- 12) 4

′ whereh andh areimplicitlydefined. Notealsothatthedividendyieldmeasureusedinthisstudy 0 can be expressed as follows D t dp ln 1+ =gpd pd (A- 13) t t t ≡ P − (cid:18) t(cid:19) so that it is also linear in the state vector under these approximations. Also, log excess equity returns can be represented follows. Using the definition of excess equity returns, rx = rf pd +gd +π +gpd t+1 t t t+1 t+1 t+1 − − ′ ′ ′ ′ ′ (h d )+(e +e +h)Y + e + d Y ∼ 0 − 0 d π t+1 − rf − t ′ ′ =r 0 +r 1 Y t+1 +r 2 Y t (cid:0) (cid:1) (A- 14) ′ ′ where r , r and r are implicitly defined. 0 1 2 B.1 Accuracy of the Equity Approximation To assess the accuracy of the log linear approximation of the price dividend ratio, the following experiment was conducted. For the model and point estimates reported in Table 2, a simulation was run for 10,000 periods. In each period, the ‘exact’ price dividend ratio and log dividend yield were calculated in addition to their approximate counterparts derived in the previous subsection. The resulting series for exact and approximate dividend yields and excess stock returns compare as follows (quarterly rates). appx dp exact dp appx rx exact rx t t t t mean 0.0099 0.0100 0.0118 0.0119 std. dev. 0.0032 0.0034 0.0945 0.0891 correlation 0.9948 0.9853 C Analytic Moments of Y and W t t Recall that the data generating process for Y is given by, t Y t =µ+AY t−1 +(Σ F F t−1 +Σ H )ε t F =sqrt(diag(φ+ΦY )) (A- 15) t t We canshowthatthe uncenteredfirst,second,andfirstautocovariancemoments ofY aregivenby, t Y =(I A) −1µ t k − vec Y Y′ =(I A A) −1 vec µµ ′ +µY ′ A ′ +AY µ ′ +Σ F2Σ ′ +Σ Σ ′ t t k2 − ⊗ · t t F t F H H vec Y (cid:16) Y′ (cid:17) =(I A A) −1 vec (cid:16) µµ ′ +µY ′ A ′ +AY µ ′ +A Σ F2Σ ′ +Σ (cid:17) Σ ′ (A- 16) t t−1 k2 − ⊗ · t t F t F H H (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) where overbars denote unconditional means and F2 =diag φ+ΦY . t t Now consider the unconditional moments of a n-vector of observable variables W which obey t (cid:0) (cid:1) the condition W t =µw+ΓwY t−1 +(Σw F F t−1 +Σw H )ε t (A- 17) where µw is an n-vector and Σw, Σw and Γware (n k) matrices. It is straightforward to show F H × 5

that the uncentered first, second, and first autocovariance moments of W are given by, t W =µw+ΓwY t t W W′ =µwµw′ +µwY ′ Γw′ +ΓwY µw′ +ΓwY Y′Γw′ +ΣwF2Σw′ +ΣwΣw′ t t t t t t F t F H H W W′ =µwµw′ +µwY ′ Γw′ +ΓwY µw′ +ΓwY Y′ Γw′ +Γw Σ F2Σw′ +Σ Σw′ (A- 18) t t−1 t t t t−1 F t F H H (cid:16) (cid:17) It remains to demonstrate that the observable series used in estimation obey Equation (A- 17 ). This is trivially true for elements of W which are also elements of Y such as ∆d , ∆c , π . Using t t t t t Equations (17), (A- 14 ) and (A- 11 ), it is apparent that rf, dp and rx satisfy Equation (A- 17 ) t t t as well. D GMM Estimation and Constructing S ( W ) T Armed with an estimate for S(W ), S(W ), we minimize t t b −1 ′ J W ;Ψb =g Ψ S(W ) g Ψ (A- 19) T T t T (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) in a one-step GMM procedure. b b b b To estimate S(W ), we use the following procedure. t Under the model, we can project X(W ) onto the vector of state variables Yc, which stacks the t t contemporaneoubs five state variables and a number of lags, X(W )=BYc+ε t t t where B and ε t are calculated using a standard libnear pr b ojection of X(W t ) onto Y t c. We assume the covariance matrix of the residuals, D, is diagonal and estimate it using the residuals, ε , of the t projectiobn. Thbe projection implies b b S(W )=BS(Yc)B ′ +D T T whereS(Yc)isthe spectraldensityabtfrequencybbzeroofbYc. bToestimateS(Yc),weuseastandard T t T pre-whiteningtechniqueasinAndrewsandMonahan(2004). BecauseYc containstwounobservable t ′ variabl b es, v and q , we use instead the vector Yp = ∆df,π , ∆cf,rf ,dp bf and one lag of Yp to t t t t t t t t t span Yc.11 h i t Because the system is nonlinear in the parameters, we take precautionary measures to assure that we find the global minimum. First, over 100 starting values for the parameter vector are chosenatrandomfromwithinthe parameterspace. Fromeachofthese startingvalues,weconduct preliminaryminimizations. Wediscardtherunsforwhichestimationfailstoconverge,forinstance, because the maximum number of iterations is exceeded, but retain converged parameter values as “candidate” estimates. Next, each of these candidate parameter estimates is taken as a new starting point and minimization is repeated. This process is repeated for several rounds until a globalminimizerhasbeenidentifiedastheparametervectoryieldingthelowestvalueoftheobjective function. In this process, the use of a fixed weighting matrix is critical. Indeed, in the presence of aparameter-dependentweightingmatrix,thissearchprocesswouldnotbewelldefined. Finally,we confirmtheparameterestimatesproducingtheglobalminimumbystartingtheminimizationroutine at small perturbations around the parameter estimates, and verifying that the routine returns to the global minimum. 11Strictlyspeaking,themovingaveragefiltersof∆df ∆cf anddpf wouldrequireusing3lags,butthedimensionality t t t ofthatsystemistoolarge. 6

Table 1: Heteroskedasticity in Fundamentals Panel A: Univariate Volatility Regression ∆df ∆cf rx t t t rf t−1 0.0809 0.0044 0.4574 − − − (0.0279) (0.0020) (0.3468) dpf 0.1155 0.0148 1.0851 t−1 (0.0392) (0.0061) (1.1342) spd t−1 0.1288 0.0052 0.6812 (0.0735) (0.0060) (1.2146) Panel B: Multivariate Volatility Regression baseline restricted ∆df ∆cf ∆df ∆cf t t t t rf t−1 0.0660 0.0023 0.0365 0.0031 − − − − (0.0242) (0.0010) (0.0171) (0.0011) dpf 0.0770 0.0135 0.0916 0.0077 t−1 (0.0380) (0.0058) (0.0344) (0.0035) spd t−1 0.0651 0.0043 0.0280 0.0024 (0.0473) (0.0052) (0.0349) (0.0029) joint p-val 0.01 0.04 0.03 0.02 restriction, p-val 0.11 Panel C: Cyclicality and Trend Regressions recession recession and trend ∆df ∆cf ∆df ∆cf t t t t I recess,t−1 0.0033 0.0003 0.0022 0.0003 (0.0016) (0.0001) (0.0017) (0.0001) ln(t) 0.0456 0.0027 − − (0.0252) (0.0014) 7

Table 1: Notes The symbols ∆df t ,∆cf t , and rx t refer to log filtered dividend and consumption growth and log excess equityreturns. Thetablepresentsprojectionsoftheconditionalmeanandvolatilityofthesevariablesonto aset ofinstruments; includingthelog yield ona90 dayT-bill, rf t,thefiltered logdividendyield, dpf t ,the logyield spread,spd t,adummyvariableequaltooneduringNBER-definedU.S.recessions, andalog time trend ln(t). For all estimations, thegeneric specification is, E ∆df =a +a x t t+1 0 1 1t h i VAR ∆df =b +b x t t+1 0 1 2t h i wherex 1tandx 2trefertogenericvectorsofinstrumentsfortheconditionalmeanandvolatilityequations respectively, and analogous equations are estimated for consumption growth and returns (simultaneously). Throughout, the conditional mean instrument vector, x 1t, for dividend and consumption growth includes onlytheconsumption-dividendratio,uf . Forreturns,weadditionallyallowtheconditionalmeantodepend t on rf t, dpf t , and spd t. Results from the conditional mean equations are not reported. Because both the consumption and dividendgrowth series are effectively four-quarter moving averages, we instrument for all the explanatory variables in both the mean and volatility equations with the variable lagged four quarters and also use GMM standard errors with four New-West lags. InpanelA,thevolatilityequationsareunivariate,sothatx 2t iscomprisedofonlyonevariableatatime. In panels B and C, the volatility equations are multivariate, so that x 2t contains all the listed instruments at once. In Panel B, the row labeled ”joint p-val” presents a Wald test for the joint significance of the b 1 parameters. Underthecolumns labeled ”restricted,” a restriction is imposed underwhich thevolatility parameters, b 1, are proportional across consumption and dividend growth. The bottom row, reports the p-valuefor this overidentifying restriction. 8

Table 2: Dynamic Risk and Uncertainty Model Estimation Parameter Estimates E[∆d] ρ σ σ du dd dv 0.0039 0.0214 0.0411 0.0413 (0.0011) (0.0082) (0.0116) (0.0130) E[v ] ρ σ t vv vv 1.0000 0.9795 0.3288 (fixed) (0.0096) (0.0785) ρuu σ ud σuu 0.9826 0.9226 0.0127 − (0.0071) (0.0233) (0.0007) E[q ] ρ σ σ t qq qc qq 1.0000 0.9787 5.2211 0.1753 − (fixed) (0.0096) (4.5222) (0.0934) ln(β) γ δ 0.0168 1.1576 0.0047 − (0.0042) (0.7645) (0.0011) E[π ] ρ ρ σ t ππ πu ππ 0.0081 0.2404 0.0203 0.0086 − (0.0010) (0.1407) (0.0073) (0.0017) Overidentification Test J(15) 12.7262 (pval) (0.6234) 9

Table 2: Notes Themodel is definedby theequations ∆d t =µ d +ρ du u t−1 +√v t−1 σ dd εd t +σ dv εv t v t =µ v +ρ vv v t−1 +σ vv √v t(cid:0)−1 εv t (cid:1) u t =µ u +ρuuu t−1 +σ ud (∆d t − E t−1 [∆d t ])+σuu√v t−1 εu t q t =µ q +ρ qq q t−1 +σ qc (∆c t − E t−1 [∆c t ])+σ qq √q t−1 εq t π t =µ π +ρ ππ π t−1 +ρ πu u t−1 +σ π επ t ∆c =δ+∆d +∆u t t t =(δ+µ d )+(ρ ud +ρ uu − 1)u t +(1+σ ud )√v t−1 σ dd εd t +σ dv εv t +σ uu √v t−1 εu t m =ln(β) γ∆c +γ∆q t+1 t+1 t+1 (cid:0) (cid:1) − Themoments used toestimate themodel are E ∆df,∆cf,π ,rf ,dpf,spd ,rex (7) t t t t t t t h 2 2 i 2 E ∆df , ∆cf ,(π )2,(rf )2, dpf ,(spd )2,(rex)2 (7) t t t t t t t (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) E (π t π t−1 ),(π t π t−4 ), ∆df t ∆df t−4 , ∆cf t ∆cf t−4 (4) h (cid:16) (cid:17) (cid:16) (cid:17)i E ∆df∆cf , ∆dfrf , ∆dfdp , ∆dfspd , ∆cfrf , ∆cfdp , ∆cfspd , π ∆cf ,(π rf ) (9) t t t t t t t t t t t t t t t t t t h(cid:16) 2 (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) 3 (cid:17) (cid:16) 4 (cid:17) (cid:16)3 (cid:17)4(cid:16) (cid:17) (cid:16) (cid:17) i E ∆df t ⊗ rf t−4 ,dpf t−4 ,spd t−4 , ∆df t , ∆df t , ∆cf t , ∆cf t (7) (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) ThemodelisestimatedbyGMM. DataarequarterlyUSaggregatesfrom1927:1-2004:3. ∆df t ,∆cf,π t, rf t,dpf t ,spd t,and rx t refer tofiltered log dividendgrowth, filteredlog consumption growth, log inflation, the log yield on a 90 day T-bill, the filtered log dividend yield, the log yield spread, and log excess equity returns(with respect to the90 day T-bill). See text for data construction and estimation details. 10

Table 3: The Fit of the Model: Linear Moments Simulated observable moments ∆df π ∆cf rf dpf spd rx t t t t t t t mean [0.0038] [0.0084] [0.0085] [0.0097] [0.0096] [0.0038] [0.0121] 0.0026 0.0077 0.0080 0.0094 0.0099 0.0040 0.0141 (0.0029) (0.0013) (0.0008) (0.0010) (0.0004) (0.0004) (0.0062) std.dev. [0.0291] [0.0121] [0.0068] [0.0074] [0.0035] [0.0033] [0.0967] 0.0308 0.0130 0.0075 0.0078 0.0035 0.0032 0.1085 (0.0034) (0.0015) (0.0009) (0.0007) (0.0003) (0.0002) (0.0126) ∗ ∗ ∗ autocorr [ 0.0275] [0.5837] [0.0233] [0.9170] [0.9429] [0.6840] [ 0.0071] − − 0.0699 0.6016 0.2460 0.9582 0.9347 0.8107 0.0446 − (0.0995) (0.0802) (0.2008) (0.0356) (0.1751) (0.0618) (0.1004) Simulatedmoments,insquarebrackets,arecalculatedbysimulatingthesystemfor100,000periodsusing thepoint estimates from Table2 and calculating sample momentsof thesimulated data. Autocorrelations areallatonelagexceptforseriesdenotewithanasterisk(*): dividendgrowth,consumptiongrowthandthe dividend price ratio, which are calculated at 4 lags. The second and third numbers for each entry are the sample moments and corresponding standard errors (in parentheses) computed using GMM with 4 Newey West lags. Data are quarterly US aggregates from 1927:1-2004:3. ∆df t , π t, ∆cf t ,rf t, dpf t , spd t, and rx t, refertofilteredlogdividendgrowth,loginflation,filteredlogconsumptiongrowth,thelogyieldona90day T-bill, thefiltered log dividend yield, thelog yield spread, log excess equity returns(with respect to the90 day T-bill). Seetext for data construction details. 11

Table 4: The Fit of the Model: Nonlinear Moments Panel A: Univariate Volatility Regression ∆df ∆cf rx t t t rf t−1 [ 0.0298] [ 0.0020] [0.3969] − − 0.0809 0.0044 0.4574 − − − (0.0279) (0.0020) (0.3468) dpf [0.0439] [0.0019] [1.3486] t−1 0.1155 0.0148 1.0851 (0.0392) (0.0061) (1.1342) spd t−1 [0.3332] [0.0179] [2.9831] 0.1288 0.0052 0.6812 (0.0735) (0.0060) (1.2146) Panel B: Skewness and Kurtosis ∆df ∆cf rx t t t skew [ 0.2250] [ 0.4574] [0.1494] − − 0.3287 0.7537 0.1254 − − (0.6339) (0.4450) (0.7228) kurt [10.0250] [10.1726] [5.4295] 7.9671 6.4593 9.7118 (1.3668) (0.9673) (2.0755) PanelArepeatstheregressionmodelofTable1andalsoreportsanalogoussimulatedstatisticsgenerated bythemodelestimatedinTable2. PanelBreportsunconditionalskewnessandkurtosisforthevariablesin each column. In each panel, the simulated moments (50,000 observations) are reported in square brackets and the corresponding data statistics and standard errors are reported below, with the standard errors in parentheses. 12

Table 5: Dynamic Properties of Risk, Uncertainty and Asset Prices Panel A: Unconditional Simulated unobservable univariate moments v q RA rrf rspd E [rx ] V [rx ] S MaxS t t t t t t t+1 t t+1 t t mean 1.0090 1.0097 7.06 0.0017 0.0038 0.0121 0.0092 0.1396 0.2075 median 0.3611 0.7784 2.52 0.0037 0.0034 0.0103 0.0070 0.1320 0.2095 std.dev. 1.6063 0.9215 36.34 0.0093 0.0028 0.0075 0.0083 0.1265 0.0491 autocorr 0.9788 0.9784 0.9212 0.9784 0.9777 0.9789 0.9794 0.5384 0.9653 Panel B: Cyclicality of Means Simulated unobservable univariate means v q RA rrf rspd E [rx ] V [rx ] S MaxS t t t t t t t+1 t t+1 t t Expansion 0.8665 0.9893 6.73 0.0024 0.0035 0.0117 0.0085 0.1406 0.2053 Recession 2.7195 1.2544 10.96 0.0064 0.0076 0.0171 0.0183 0.1283 0.2349 − Panel C: Correlations with v and q t t Simulated correlations between v , q and observables t t rrf rspd rf dp rx E [rx ] V [rx ] t t t t t t t+1 t t+1 v 0.9232 0.9562 0.5163 0.1835 0.1470 0.3428 0.8799 t − − − q 0.4687 0.1756 0.5375 0.9215 0.1071 0.8943 0.3758 t − Simulatedmomentsarecalculatedbysimulatingthesystemfor100,000periodsusingthepointestimates from Table 2 for a number of variables including: v t, dividend growth volatility, q t, the log inverse consumptionsurplusratio,RA t,local riskaversion whichisγexp(q t ). Thevariablesrrf tandrspd trepresent the real short rate and real term spread respectively, and E t [rx t+1 ] and V t [rx t+1 ] denote the conditional mean and conditional variance of excess stock returns. S tdenotes the conditional Sharpe ratio for equity. MaxS tdenotes the maximum attainable Sharpe ratio for any asset in the economy which is given by the quantity,[exp(V t (m t+1 )) 1]1/2 . − InPanelB,meansofsimulateddataconditionalonabinaryrecession/expansion variablearepresented. Recessions are defined in the simulated data as periods of negative real consumption growth. Recessions represent approximately 8% of all observations in the simulated data. In Panel C, the simulated unconditional correlations among v t, q tand other endogenous variables are reported. 13

Table 6: Variance Decompositions Fraction of variance due to variation in each state element ∆d π u v q t t t t t rrf [0.0000] [0.0000] [0.0999] [0.7239] [0.1761] t 0.0000 0.0000 0.1154 0.1472 0.0698 h i h i h i h i h i rspd [0.0000] [0.0000] [0.0752] [0.8653] [0.0596] t 0.0000 0.0000 0.01010 0.0943 0.0510 h i h i h i h i h i cprcons [0.0000] [0.0000] [0.0502] [0.2299] [0.7199] t 0.0000 0.0000 0.0630 0.1041 0.1296 h i h i h i h i h i rf [0.0000] [0.1230] [0.0904] [0.5010] [0.2856] t 0.0000 0.0765 0.2222 0.1216 0.1796 h i h i h i h i h i spd [0.0000] [0.3148] [0.0035] [0.6019] [0.0797] t 0.0000 0.3407 0.0599 0.3413 0.0745 h i h i h i h i h i dpf [0.0000] [0.0000] [0.0655] [0.0544] [0.8801] t 0.0000 0.0000 0.0901 0.0798 0.0627 h i h i h i h i h i rx [0.3605] [0.0091] [ 0.1593] [0.1640] [0.6257] t − 0.0733 0.0036 0.0401 0.0895 0.1397 h i h i h i h i h i E [rx ] [0.0000] [0.0000] [ 0.0167] [0.1665] [0.8502] t t+1 − 0.0000 0.0000 0.0146 0.1281 0.1182 h i h i h i h i h i V [rx ] [0.0000] [0.0000] [0.0000] [0.8029] [0.1971] t t+1 0.0000 0.0000 0.0000 0.2229 0.2229 h i h i h i h i h i Thesymbols, rrf t,rspd t,andcprconsrefertothetheoreticalreal shortrate, realtermspread,andthe coupon-price ratio of a real consol. The table reports the fraction of variation of selected variables due to variation in elements of the state vector. Thevariableineachrowcanbeexpressedasalinearcombinationofthecurrentstateandlaggedvector. Generally, underthemodel in Table 2, for the row variables, x t, x =µ+Γ ′ Yc t t where Y t c is the ‘companion form’ of the N − vector,Y t; that is, Y t c is comprised of ‘stacked’ current and laggedvaluesofY t. µandΓareconstantvectorsimpliedbythemodelandparameterestimatesofTable2. Let Var(Yc)be the variance covariance matrix of Yc. Based on µ and Γ, the proportion of the variation t t of each row variable attributed to thenth element of thestate vector is calculated as Γ ′ Var(Yc)Γ(n) t Γ′Var(Yc)Γ t whereΓ(n) isacolumnvectorsuchthat Γ(n) = Γ fori=n,N+n,...andzero elsewhere. Standard i { }i errors are reported below in angle brackets and are calculated from the variance covariance matrix of the (cid:8) (cid:9) parameters in Table 2 using the ∆-method. 14

Table 7: Model Implied Reduced Form Return Predictability Excess Returns Parameter estimates multivariate univariate β [ 0.0037] 0 − 0.0358 − (0.0256) β [ 0.3097] [0.2192] 1 − 0.1669 1.1651 − − (0.7464) (0.7839) β [2.0695] [1.5770] 2 3.7980 3.7260 (2.0231) (1.9952) β [0.7668] [1.2389] 3 3.5376 3.4728 (1.8900) (1.8728) Thepredictability model for excess returnsis definedas, rx =β +β rf +β dpf +β spd +ε t+1 0 1 t 2 t 3 t t+1 and is estimated by GMM. Data are quarterly US aggregates from 1927:1-2004:3. The symbols rf t, dpf t , spd t, and rx t refer to the log yield on a 90 day T-bill, the filtered log dividend yield, the log yield spread, and log excess equity returns (with respect to the 90 day T-bill). Simulated moments, in square brackets, are calculated by simulating the model for 100,000 periods using the point estimates from Table 2 and estimating the above model on the simulated data. The second and third numbers for each entry are the sample moments and corresponding standard errors (in parentheses). 15

Table 8: Selected Post-War Estimation Results Panel A: Unconditional Moments Simulated univariate moments rf rx v q RA rrf E [rx ] V [rx ] S t t t t t t t t+1 t t+1 t mean 0.0118 0.0122 1.0038 1.0072 7.70 0.0022 0.0123 0.0073 0.1440 median 0.0114 0.0121 0.8989 0.8898 4.50 0.0020 0.0113 0.0067 0.1411 std.dev. 0.0066 0.0861 0.6128 0.7354 13.48 0.0065 0.0053 0.0036 0.0275 autocorr 0.9028 0.0018 0.9072 0.9854 0.9690 0.9643 0.9827 0.9670 0.9756 Panel B: Variance Decompositions Fraction of var. due to state element v q t t rrf [0.2825] [0.5008] t 0.1566 0.1347 h i h i rspd [0.9252] [0.0638] t 0.0700 0.0478 h i h i dpf [0.0035] [0.9960] t 0.0102 0.0125 h i h i E [rx ] [0.0480] [0.9523] t t+1 0.0540 0.0538 h i h i V [rx ] [0.2843] [0.7157] t t+1 0.1754 0.1754 h i h i This table reports results from estimation of the structural model using data from 1946Q1 through 2004Q3, the post-war era. Panel A reports results analagous to Table 5 (see Table 5 notes) and Panel B reports results similar to Table 6. 16

Figure 1: Term Structure Determinants Effect on Yields of a 1 std. dev. Increase in v t 20 15 10 5 0 −5 −10 −15 −20 0 5 10 15 20 25 30 35 40 horizon (qtr) spb Effect on Yields of a 1 std. dev. Increase in q t 20 EH Term Prem 15 10 5 0 −5 −10 −15 −20 0 5 10 15 20 25 30 35 40 horizon (qtr) spb Effect on Spreads of a 1 std. dev. Increase in v t 20 15 10 5 0 −5 −10 −15 −20 0 5 10 15 20 25 30 35 40 horizon (qtr) spb Effect on Spreads of a 1 std. dev. Increase in q t 20 15 10 5 0 −5 −10 −15 −20 0 5 10 15 20 25 30 35 40 horizon (qtr) spb Underthe model of Table 2, real risk free yieldsof horizon, h, havesolutions of theform, ′ rrf =a +A Y h,t h h t where the coefficients above are functions of the ‘deep’ model parameters. This figure shows the effect on theseyieldsandtheassociatedspreads(relativetothe1periodyield)ofastandarddeviationchanges,inthe latentfactors, v t andq t ,usingthepointestimatesinTable2. Athorizonsgreaterthan1,theseeffectscan be further decomposed into parts corresponding to the expectations hypothesis (EH), and term premiums, which are drawn in circle and star respectively. 17

Figure 2: Stock and Bond Moment Sensitivity to Volatility Parameters 0.1 0.08 0.06 0.04 0.02 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 .tcnuf .jbo σ Varying v 2 1.5 1 0.5 0 0.168 0.17 0.172 0.174 0.176 0.178 0.18 0.182 0.184 .tcnuf .jbo all moments equity+fund. bond+fund. σ Varying q ThelinesplottheshapeoftheGMMobjectivefunctionfromthestructuralmodelforsmallperturbations ofσ qq andσ vv abouttheirestimatedvaluesinTable2,holdingallotherparametersatexactlytheirestimated values. Thebluetriangle lineplotsthevaluesfor theoverall GMM objective function. Thered stars lines plot values for thequadratic, gbonds(Ψ) S(Wbonds) −1 gbonds(Ψ) ′ T t 1T (cid:16) (cid:17) asafunctionofσ qq andσ vv (holdingallotherpa b rametersattheirestimatedvalues). The‘bonds’superscript denotesthatwearerestrictingattentiontothe10momentswhichinvolveeithertheshortrateortermspread. The black circle lines plots thelocation of theanalogous quadratic for theequitymoments, gequity(Ψ) S(Wequity) −1 gequity(Ψ) ′ T t 1T (cid:16) (cid:17) where we nowrestrict attention thethe7 mombentswhich involveeitherthe dividendyield or excess equity return. 18