Nominal Rigidities and the Term Structures of Equity and Bond Returns

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Nominal Rigidities and the Term Structures of Equity and Bond Returns Pier Lopez, David Lopez-Salido, and Francisco Vazquez-Grande 2015-064 Please cite this paper as: Lopez, Pier, David Lopez-Salido, and Francisco Vazquez-Grande (2015). “Nominal Rigidities and the Term Structures of Equity and Bond Returns,” Finance and Economics Discussion Series 2015-064. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.064. 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.

Nominal Rigidities and the Term Structures of Equity and Bond Returns PierlauroLopeza,1,∗,DavidLopez-Salidob,FranciscoVazquez-Grandeb aBanquedeFrance bFederalReserveBoard Abstract A downward-sloping term structure of equity and upward-sloping termstructures of interest rates arise endogenously in a general-equilibrium model with nominal rigidities and nonlinear habits in consumption. Countercyclical marginal costs exacerbate the procyclicality of dividends after a technology shock, and hence their riskiness, and generate countercyclical inflation. Marginal costsgraduallyfallafteranegativetechnologyshockasthepricelevelincreasessluggishly,sothe payoffsofshort-durationdividendclaims(bonds)aremore(less)procyclicalthanthepayoffsof long-durationclaims(bonds). Thesimultaneouspresenceofmarketandhomeconsumptionhabits allowsforunitingnonlinearhabitsandaproductioneconomywithoutcompromisingtheabilityof themodeltofitmacroeconomicvariables. JELclassification: E43;E44;G12. Keywords: Structuraltermstructuremodeling,Equityandbondyields,Habitformation,Nominal rigidities,Macro-financemodeling. Introduction Recentevidenceshowsthattheaveragetermstructureofequityriskpremiaisdownward-sloping andstarts froma highlevel. Also, itis wellknownthat theterm structureof nominalbondsslopes upwards onaverage. These factsare of interestto financialeconomists as thematurity structureof equityandbondriskpremiarevealshowinvestorsformexpectationsaboutfuturemacroeconomic variables and their marginal utilities at different horizons. A joint general-equilibrium explanation forthisevidencehasremainedasyetelusive(e.g.,BinsbergenandKoijen,2015). ∗Correspondingauthor;Macro-FinanceDivision,BanquedeFrance,31rueCroixdesPetitsChamps,75001Paris. Emailaddresses: pierlauro.lopez@banque-france.fr(PierlauroLopez), david.j.lopez-salido@frb.gov(DavidLopez-Salido),francisco.vazquez-grande@frb.gov(Francisco Vazquez-Grande) 1Thispaperwaspreviouslycirculatedunderthetitle“Macro-financeseparationbyforceofhabit”. Wewouldlike tothankRalphKoijen,AnnaOrlikandEricSwansonforveryusefulcomments,aswellasseminarparticipantsatthe FederalReserveBoardandthe2015meetingsoftheSocietyofEconomicDynamicsforcommentsanddiscussions. TheviewspresentedherearesolelythoseoftheauthorsanddonotnecessarilyrepresentthoseoftheFederalReserve SystemortheEurosystem. Firstversion: December2014. Thisversion: June2015. Commentsaremostwelcome

0.15 0.35 dividendstrip realbond nominalbond 0.3 0.25 0.1 0.2 Equitypremium=6.4% 0.15 0.05 0.1 0.05 0 0 0 5 10 15 0 5 10 15 n (in years) n (in years) (a)Riskpremia,lnE(Re,(n)). (b)Volatilities,var(lnRe,(n)). t+1 t+1 Figure1: Averagetermstructuresofannualizedexcessreturnsandvolatilitiesforholdingforonemonththe nthzero-couponcashflowclaiminourbenchmarkmodelspecification. Differentlinesassociatewiththe termstructuresofdifferentcashflowclaims: marketequity(solid),realbonds(dotted)andnominalbonds (dash-dottedline). Thedottedlineistheaverageannualizedholding-periodequitypremium. Weproposeasimpleframeworkwithlargeandtime-varyingriskpremiathatendogenizesthe payoffofnominalbondsand thatrationalizesdividendsasaleveredversion ofconsumption. Our model links macroeconomic fluctuations with asset pricing facts in full general equilibrium and offersajointexplanationofthedocumentedtermstructureproperties. Inparticularourmodelis able to capture simultaneously the negative slope of the average term structure of dividend strip returns and volatilities as well as upward-sloping average term structures of interest rates and a positiveinflationriskpremiumatallhorizons(LettauandWachter,2011;Binsbergen,Brandtand Koijen, 2012a; Binsbergen, Hueskes, Koijen and Vrugt, 2013; Lopez, 2013). Figure 1 plots our model-impliedtermstructuresofreturnsandvolatilities. Weunite atextbook New Keynesian modeleconomy (Gal´ı,2008) andCampbell andCochrane (1999) habits in consumption. Nominal rigidities induce a time-varying labor share of output, driven by countercyclical marginal costs. The countercyclical labor share implies a procyclical dividend share as well as countercyclical inflation. It follows that dividend claims and nominal bondspayoffbadlyinatechnologicalrecessionandarethereforeriskyinvestments. Sincemarginal costsarestationary,thepayoffsoflong-durationdividends(bonds)areless(more)procyclical;the dividendshareandthepricelevelincreaseasmoreandmorefirmsareabletoadjusttheirpricesto regaintheirmarkups.2 2Arecentandrapidlygrowingliteratureisfocusingontheassetpricingimplicationsofnominalrigidities(e.g., Rudebusch and Swanson, 2008, 2012; Bekaert, Cho and Moreno, 2010; Palomino, 2012; Li and Palomino, 2014; Andreasen,2013;Campbell,PfluegerandViceira,2013;Kung,2015;GorodnichenkoandWeber,2013;Weber,2014). 2

Reconcilinghabitformationwithbusinesscyclefacts Weavoidthewell-knowndifficultiesinreconcilingbusinesscyclefactswithhabitformation models in production economies (e.g., Jermann, 1998; Lettau and Uhlig, 2000; Uhlig, 2007; Rudebusch and Swanson, 2008, 2012; Swanson, 2012) by introducing nonlinear habits in two consumptiongoods,onepurchasedinthemarketandoneproducedathome.3 Inaproductioneconomyhabitsaffectequilibriumquantitiesbytheireffectontheintertemporal rate of substitution, which drives consumption-saving and investment decisions, and by their effect ontheintratemporal rateofsubstitution, whichcontrolsthelinkbetween consumptionandlabor supply. The consequence is that equilibrium quantities (consumption, output, labor, investment and the capital stock) depend on the additional state variables that drive habit dynamics and countercyclical risk premia. The time-variation in the new state variables has quantity implications that are associated either with counterfactually large business cycle fluctuations in some real variables(suchaslabor,thecapitalstock,therealwagerate,ortherealrisk-freerate),orwithsmall riskpremiaashouseholdsabsorbaggregateshocksbyvaryinglabororinvestment. CampbellandCochrane(1999)engineeredaconsumptionhabitsensitivityfunctionthatcontrols theintertemporalconsumption-savingdecisionsandinduceslarge,volatile,andtime-varyingrisk premia with a low and stable risk-free rate. In the same spirit, we also engineer restrictions on the home-consumption habit sensitivity function to control the intratemporal effect of habits on consumption-labor decisions and thereby avoid risk-premia spillovers on macroeconomic quantities—andhencethequantitypuzzlesfirstdocumentedbyLettauandUhlig(2000). Finally, theeffectofhabitsontheconsumption-investmenttradeoffiscontrolledbythecurvatureofcapital adjustmentcosts,whichdeterminesthedependenceofinvestmentonQ;inthelimitwhencapital adjustmentisinfinitelycostlyhabitshavenoeffectoninvestment. Toprovideintuitionaboutthemodel’simplicationinassimpleasettingaspossible,wecalibrate the model to the polar case of an exact separation between risk premia and quantity dynamics that reconciles habit formation with business cycle facts, independently of the presence of other intratemporaldistortionssuchaswagerigiditiesorotherlabormarketfrictions(asconsideredfor example by Uhlig, 2007 and Rudebusch and Swanson, 2008).4 The result that such a polar case existsextendstothehabitformationsettingamacro-financeseparationresultanalogoustotheone thatTallarini(2000)describedinasettingwithEpstein-Zinpreferences(seealsoCochrane,2008). 3Wechoosetofocusonthehabitformationframeworkratherthanona‘long-runrisk’frameworkbuildingon Epstein-Zin-Weil preferences for two reasons. First, Croce, Lettau and Ludvigson (2015) show the difficulties of simultaneouslyproducinginanEpstein-Zin-Weilcontextadownward-slopingtermstructureofequityandasizeable equitypremiumforalevelofriskaversionnotexceedingthecommonlyacceptedupperboundof10(Epstein,Farhi andStrzalecki,2014). Second,theCampbell-Cochranehabitspecificationnaturallygeneratestime-variationinrisk premiawithoutrelyingoncounterfactualheteroskedasticityinmacroeconomicfundamentals(Campbell,Pfluegerand Viceira,2013). 4Theextantliteratureoffersexamplesofhabitformationinsmall-scaleproductioneconomiesbuttheyfeature habitsthatareeitherlinear(e.g.,Jermann,1998;Boldrin,ChristianoandFisher,2001;DePaoli,ScottandWeeken, 2010;ChalleandGiannitsarou,2014)orthatdepartfromtheCampbell-Cochranespecificationinordertograntexact exponential-affine term structures (e.g., Gallmeyer, Hollifield and Zin, 2005; Bekaert, Engstrom and Xing, 2009; Bekaert,ChoandMoreno,2010;Palomino,2012;Dew-Becker,2013),sothefinancialspilloversontotheintratemporal orontheintertemporalratesofsubstitutionareleftunrestrained. 3

Termstructuresofequitystripsandbondreturns Production economies provide endogenous restrictions on cashflows based on economic theory, incontrastwithareduced-formapproachthatmaybedifficulttoreconcilewithstandardmacroeconomicmodels. Additionally,jointlymodelingmacroeconomicquantitiesandassetpriceswitha structuralapproachallowsforstudyingpolicyinterventions,structuralshiftsandpotentialfeedbacks betweentherealandthefinancialsidesoftheeconomy. Sincethemacro-financeseparationensures that discount rate variation does not compromise the ability of the model to fit macroeconomic variables,wecanfocusontheassetpricingimplicationsoftherestrictionsplacedoncashflowsby theDSGEmodel. Ourframeworkpreservesall themainachievementsof CampbellandCochrane(1999), including a solution to the average equity premium puzzle, the risk-free rate and the excess volatility puzzles,andthecountercyclicalityofstockmarketreturnsandvolatility. Additionally,ournonlinearhabit model is able to explain the entire observed maturity structure of equity and bond returns andvolatilities,disentanglingtheintertemporalrisk-returntradeoffsatdifferenthorizons.5 Thekey driversofourresultsarenominalrigiditiesandsomedegreeofmeanreversioninthegrowthrateof technology. Sticky prices endogenize the payoff of nominal bonds and rationalize dividends as a levered versionofconsumption. Thepayoffofshort-termequityispositivelycorrelatedwithconsumption news but is much more volatile, and hence more risky, as long as positive short-run shocks to technologygrowthassociatewithnegativelong-runshockstotechnologygrowth. Infact,short-run shocksincreaseconsumptionanddividendsalikebutnegativelong-runshockspushdemandbelow potential,whichlowersconsumptionwhilecreatingdownwardpressureoninflationandrealwages thatraisescorporateprofits. Mean reversion in technology reduces the riskiness of dividend claims with longer duration, asapositiveexposuretolong-runtechnologygrowthriskprovidesconsumptioninsurance. Thus, themodelisabletogenerateaninitiallynegativeslopeinthetermstructureofmarketequityfora sufficiently large degree of price stickiness, capturing the evidence by Binsbergen et al. (2012a). Moreover, therole ofbonds (realand nominal)as ahedge fortransitory shocksto thegrowth rate oftechnologydoesnotproduceabondpremiumpuzzlebecausetheexposuretothestatevariable thatdrivestherisk-freerate(theconditionalmeanofthegrowthrateoftechnology)commandsa negativeprice.6 Finally,inourmodelthepriceofriskisastatevariablethathasalowunconditionalcorrelation with technology growth and its conditional mean, so investors still fear long-duration equities becausetheydopoorlyinrecessionsunrelatedonaveragewithtechnologyrisk. 5Whilethesearchforastructuralexplanationofthepositiveslopeofthetermstructureofinterestrateshasarather longhistory(see,forexample,Gu¨rkaynakandWright,2012;Duffee,2013),thesearchforastructuralexplanation forthenegativeslopeofthetermstructureofequityhasonlyrecentlyreceivedalotofattention(Croce,Lettauand Ludvigson,2015;Belo,Collin-DufresneandGoldstein,2015;LynchandRandall,2011;Ai,Croce,DiercksandLi, 2013;Marfe`,2013;Nakamura,Steinsson,BarroandUrsu´a,2013;Wachter,2013);seealsoBinsbergenandKoijen (2015). 6ThispropertymotivatesfromfirstprinciplesthedescriptivestructureassumedbyLettauandWachter(2011)that liesbehindtheirabilitytocapturetheinitialslopesofthetermstructuresofequityandrealinterestrates. 4

1. IncorporatingCampbell-Cochranehabitformationinaproductioneconomy Thissectiondescribes atextbookDSGE modelwithnominalrigiditiesthat weaugmentwith nonlinearhabitsinmarketandhomeconsumption. 1.1. Households As inGreenwood and Hercowitz (1991) ourhouseholds obtain utility over consumptionof two types of goods, nondurable goods and services purchased in the market, and goods and services producedathome. Householdsgetusedtoanaccustomedstandardoflivingasrepresentedbysome particularlevelsofconsumptionofthemarket-purchasedgoodandofthehome-producedgood. Identicalconsumersindexedby j ∈ [0,1]havepreferencescapturedbythefunction (cid:88)∞ (cid:18)[C (j)−Xc]1−γ −1 [H(j)−Xh]1−γ −1(cid:19) U (j) = E βt t t +χ t t (1) 0 0 1−γ 1−γ t=0 whereC isrealconsumption purchasedinthemarket and H denotestheconsumption produced t t at home, with production function H = A(1 − N), with N the labor choice and A aggregate t t t t t productivity. Xc and Xh representhabitlevelsthatareanonlinearfunctionofcontemporaneousand t t past consumption. Parameter β is thesubjective discount rateand parameterχ controls thesteadystateeffectofhabits,whilethecurvatureoftheutilityfunctioninmarketandhomeconsumptionis thesametoensurebalancedgrowth(seeCampbellandLudvigson,2001). Weassumeacalibration forχtoachievethesamesteadystateasunderapower-utilityspecification(Xc = Xh = 0). t t Habitsareendogenousstatevariablesthatinduceadepartureoftheequilibriumdynamicsfrom the power-utility specification. As is customary in the extant literature, we assume that the law ofmotionofhabitsisspecifiedindirectlythroughtheprocessesforsurplusmarketconsumption s ≡ ln[(C − Xc)/C ] and surplus home consumption z ≡ ln[(H − Xh)/H] in order to ensure t t t t t t t t consumption levels that never fall below their respective habit levels, and hence well-behaved marginalutilities. Thelawofmotionofthesurpluslevelsisdrivenbyaggregatemarketandhome (cid:82) (cid:82) 1 1 consumption,c ≡ ln C (j)djandh ≡ ln H(j)dj;sinceeachindividualagenthaszeromass, t t t t 0 0 she takes the habit levels thus specified as external to her consumption decisions. This structure impliesthefollowingmarginalutilitiesofmarketandhomeconsumption ∂U t = C −γS −γ ∂C t t t ∂U t = χH −γZ −γ ∂H t t t The inclusionof thetwo typesof consumptionallows usto maintainthe separabilitybetween consumption and hours, while also remaining consistent with balanced growth and keeping the elasticityofintertemporalsubstitutionasafreeparameter. Inadditiontobeingabletoreconcile our modelwith the available evidence ofan intertemporal elasticitylowerthan one, this property preserves the original implicationsof Campbell and Cochrane (1999)for the stochastic discount factor. 5

1.1.1. Habitstructure Wespecifythefollowingdynamicsforthelogarithmsofaggregatesurpluslevels:7 sˆ = ρ sˆ +Λ [sˆ](E −E )f [C ] t+1 s t c t t+1 t c t+1 (2) zˆ = ρ zˆ +Λ [zˆ](E −E )f [H ] t+1 s t h t t+1 t h t+1 where f [C ] = ln[C ], f [H] = ln[Aα/(1−α)(A −H)],andthesensitivityfunctions(Λ andΛ )and c t t h t t t t c h steadystatelevelsofthesurplusvariablesare:  √ (cid:115) Λ [sˆ] =  S−1 1−2sˆ t −1, sˆ t ≤ 1 2 (1−S2) S = γ var(εc t ) c t 0 sˆ > 1(1−S2) 1−ρ −ξ /γ t 2 s 1 (cid:32) var(εc) (cid:33)−1 Λ [zˆ] = (1−α)(1+ξ )Λ [zˆ/(1+ξ )] Z = S S +(1−S) t h t 2 c t 2 cov(εc,εh) t t The sensitivityfunctions and the steadystate levels dependon the parameterξ = [ξ ;ξ ] ∈ R2 1 2 thatcontrols thespilloverof habitsdynamicsonto theequilibrium quantities. Propositions 1and2 specifiesrestrictionsonthespilloverparameterthatgrantamacro-financeseparation. AsinCampbellandCochrane(1999)andWachter(2006),ξ controlstheeffectoftime-varying 1 risk aversion on the intertemporal rate of substitution.8 Additionally ξ controls the effect of 2 time-varying risk aversion on the intratemporal rate of substitution. Note that ξ = −1 describes 2 thecasewithconstantsurplushomeconsumption,whichisequivalenttoamodelwithouthome consumptionhabits. Themarket(home)consumptionhabitindirectlyspecifiedbythesurplusprocessisacomplex nonlinearfunctionofcurrentandpastmarket(home)consumption;however,itisapproximatelya linearhabitthatadjustsslowlytounanticipatedmovementsinmarket(home)consumption. LikeCampbellandCochrane(1999),wechoosethemarketconsumptionsensitivityfunctionto satisfythefollowingconditions: (i)themarketconsumptionhabitdoesnotproducearisk-freerate puzzle;(ii)thehabitcoincideswiththeconsumptionslevelinthelongrun;(iii)thehabitislocally predetermined;and(iv)thehabitmovesnonnegativelywithconsumptionnearthesteadystate. The firstconditionshowshowhabitscanbeengineeredinsuchawaythatthespilloveronconsumptionsaving decisions can be kept under control (via parameter ξ ); the remaining conditions can be 1 interpretedas(local)microfoundationsthataddtothewell-behavedmarginalutilitiesandthelocal slow-movingrepresentationofhabits. AppendixAprovestheseproperties.9 Inthiscontext,habits pull the real risk-free rate in offsetting directions via an intertemporal substitution motive and a 7Thenotationεx ≡ (E −E )x standsfortheone-periodaheadforecasterrorinvariable x,and‘hats’denote t t t−1 t deviationsfromsteadystate. 8Unlike Wachter (2006), we do not impose exogenously a countercyclicality in real rates; rather, we let the productioneconomyintroduceasmalldeparturefromrandom-walkconsumption(whichinheritsanear-zero,negative autocorrelationbythemeanreversionintechnology),andhencegeneraterealratemovements. 9Moreover, we achieve two additional improvements in the microeconomic properties of the habits relative to CampbellandCochrane(1999)becauseweoperateinthecontextofaproductioneconomy. First,bothhabitsmove nonnegativelywithmarketandhomeconsumption,respectively,inandaroundthesteadystate. Thispropertyowes entirelytotheendogeneizationofequilibriumconsumptionchoices;infact,intheendowmenteconomyofCampbell 6

precautionarysavingsmotive;thespilloverparameterξ controlswhetheranincreaseinsurplus 1 consumptiondrivesratesupordown. Byanalogouslogic,wechoosethehomeconsumptionsensitivityfunctionasfollows: (i)the home consumptionhabit doesnot produce aquantity puzzle; (ii) thehabit coincides withthe home consumptionlevelinthelongrun;(iii)thehabitislocallypredetermined;and(iv)thehabitmoves nonnegativelywithhomeconsumptionnearthesteadystate. Asinthecaseofmarketconsumption habits,thelastthreeconditionscanbeinterpretedaslocalmicrofoundations,andthefirstcondition shows how the habits can be engineered in such a way that the spillover on consumption-labor decisionsiscontrolled(viaparameterξ ). Ourhomeconsumptionhabitislocallypredetermined 2 in a weaker sense than the market consumption habit, as its first-order predeterminedness must deteriorateifwemovesufficientlyfarfromthebenchmarkperfectcorrelationinmarketandhome consumptioninnovations. AppendixAdiscussestheseproperties. 1.1.2. Consumers’problem Consumers maximize the intertemporal objective (1) subject to the sequence of budget constraints 1 PC (j)+ B(j) ≤ W N(j)+ B (j)+P D −T (j) t t 1+i t t t t−1 t t t t andtheappropriatetransversalitycondition,where B denotestheirtime-t holdingsofone-period t bonds discounted atthe nominal ratei, D is the dividendthey receive fromowning the aggregate t t firm,andT arelump-sumtaxesthatthegovernmentleviesonconsumerstofinancethecorrective t subsidy. 1.2. Firms Ourbenchmarkmodelconsidersthepolarcaseofamacro-financiallyseparateeconomy,which implies a deterministic capital stock. In appendix B we allow for nontrivial capital accumulation and describe one last spillover, controlled by parameter ξ ∈ R , which affects the consumption- 3 + investmenttradeoff (proposition3). Topreserve macro-financeseparationwe needadeterministic capitalaccumulationbecauseinvestmentisdeterminedbyTobin’s Q,whichisanassetpriceand thereforeachannelthatmustbreakseparation. Theproductionsideoftheeconomyischaracterizedbyaunitmassofidenticalfirmsindexed byi ∈ [0,1]thatmaximizeintertemporalprofitsandoperatewithproductiontechnology Y(i) = [eµtA(cid:101)N(i)]1−αK(i)α t t t t whereY isreal output; N isthe laborinput,which theyacquireat aunitcost equaltothe nominal t t wagerateW; K = eµt isthedeterministiccapitalstock,whichgrowsatrateµonabalanced-growth t t andCochranethereisonlyazerorelationshipbetweenhabitsandconsumptioninthesteadystateandapotentially strictlynegativerelationshipnearthesteadystate(LjungqvistandUhlig,2015;seeappendixAformoredetails). Second,CampbellandCochrane(1999)needtoassumeahighaveragerelativeriskaversioncoefficientintheir endowmenteconomyanddefendthischoiceagainsttheobjectionthattheassumedcoefficientisimplausiblylarge. Ina productioneconomy,householdscanabsorbmacroeconomicshocksalongboththeconsumptionandthelabormargin, whichdramaticallyreducestheirriskaversion(Swanson,2012). Theonlineappendixshowshowthesteady-staterisk aversioncoefficientinourcalibratedmodelisabouttheupperboundof10. 7

path;andeµtA(cid:101) denotestheexogenouslabor-augmentingtechnologylevel. Theithgoodsellsfor t (cid:82) thenominalprice P(i)and P ≡ [ 1 P(i)1−εdi]1/(1−ε) isthepriceindex. Therelationshipbetween t t t 0 productivityinmarket-andhome-producedgoodsis A = eµtA(cid:101)1−α (CampbellandLudvigson,2001). t t We also allow for Calvo-type nominal price rigidities and monopolistic competition in the market for goods. Each firm i can reset prices at any given time only with probability 1 − η and faces the demand curve for the good it produces C (i) = [P(i)/P]−εC , which arises as the t t t t cost-minimizing plan of individual consumers, j ∈ [0,1], who bundle the continuum of goods, i ∈ [0,1],via aDixit-Stiglitzaggregatorwithconstant elasticityofsubstitutionbetweengoods, ε. Thegovernmentlevieslump-sumtaxesoneachfirmtofinanceanemploymentsubsidy,τ = 1/ε, whichreducestheunitnominalcostoflaborandisinplacetooffsetanysteady-statedistortions causedbythemonopolisticcompetition.10 Finally,marketequityisthevalueoftheaggregatefirm,whichpaysoutper-periodequilibrium profitsasdividends. 1.3. Equilibrium Joint intertemporal and static optimality of market and home consumption decisions are describedbytheequations ln(1+i) = −lnE βe−γ∆ct+1 −γ∆st+1 −πt+1 (3) t t w − p = −ln(χ)+γc −γh +a +γ(sˆ −zˆ) (4) t t t t t t t withtheloglinearizedhome-productionrelation, N h = a − n (5) t t 1−N t Market clearing for each good i implies market clearing at the aggregate level, y = c, and t t thereforemarketconsumptionrelateswiththeloglinearizedproductionfunctionas c = ln(1−α)+a +(1−α)n (6) t t t A standard New Keynesian Phillips curve describes the loglinearized optimal price-setting behavior of firms as the forward-looking optimality condition linking inflation, π ≡ ln(P/P ), t t t−1 andmarginalcosts,mc ≡ (w − p)−ln(∂Y/∂N), t t t t t π t =(cid:101)βE t π t+1 +λm(cid:99)c t (7) whereλ ≡ (1−η)(1−(cid:101)βη)(1−α)/η(1−α+αε)controlsthe slopeofthecurve, with(cid:101)β ≡ βe(1−γ)µ. Inflationishighwhenfirmsexpectlong-runmarginalcostsabovetheflexible-pricelevel,inwhich case resetting firms choose a price above the index to realign their marginal costs to the desired level. 10Thisassumptioncanbeeasilyrelaxedbutsimplifiesnotation. 8

Loglinearizedequilibriumdividendscanbewrittenas 1−α d t = c t − m(cid:99)c t (8) α Corporate profits, and hence dividends, are low when marginal costs are high, holding the level of output fixed. Since marginal costs fluctuate when prices are sticky, the presence of nominal rigidities is entirely responsible for breaking down the equality between dividend growth and consumptiongrowth. Finally,firms’optimallabordemandschedulerestrictsaggregaterealwagesandmarginalcosts, w t − p t = m(cid:99)c t +c t −n t andhence,matchinglabordemandandsupplyasthelabormarketclears, γ(1−α)+α+ϕ m(cid:99)c t = 1−α (c t −cn t )−γ(sˆ t −zˆ t ) (9) whereϕ ≡ γN/(1−N)istheinversesteady-statequasi-Frisch’slaborsupplyelasticity,11 andwhere cn = a denotesoptimalconsumptionunderflexibleprices. Deviationsofaggregatemarginalcosts t t fromthedesiredlevelareassociatedwithagapinaggregateactivityrelativetotheflexible-price equilibrium. 1.3.1. Technology Thelogarithmofthegrowthrateoftechnologyevolvesaccordingtotheprocess ∆a t+1 = µ+u t +σea t+1 (cid:34) ea t (cid:35) ∼ Niid (cid:32) 0, (cid:34) 1 ρ (cid:35)(cid:33) (10) u = ρ u +φσeu eu ρ 1 t+1 u t t+1 t with[φ;σ] ∈ R2,ρ ∈ [−1,1]andρ ∈ [0,1). Thestochasticcomponentoftheconditionalmeanof + u technology growth is a mean reverting process driven by shocks eu, while shocks ea have only a t t contemporaneouseffectontechnologygrowth. Accordingly,werefertoea asashort-runshockand t toeu asalong-runshocktotechnologygrowth. Thisstructureallowsfortheunit-rootdynamicsin t cashflowsroutinelyusedintheconsumption-basedassetpricingliterature. 1.3.2. Monetarypolicy Sinceinasticky-priceenvironmentthelevelofinflationinfluencestheequilibriumallocation, wemustspecifymonetarypolicy,whichwedescribebyaTaylorrulethatreactstoinflationandthe outputgap, i = φ π +φ (c −cn) (11) t π t y t t whichgrantsdeterminacyifκ(φ −1)+(1−(cid:101)β)φ > 0,for[φ ;φ ] ∈ R2,withκ ≡ λ[γ(1−α)+α+ π y π y + ϕ]/(1−α)(Gal´ı,2008). 11TheonlineappendixderivesanddiscussesFrisch’selasticityinoursetting. 9

1.3.3. Competitiveequilibrium Foranyspecifiedpolicyprocess{i}∞ andexogenousstatevector{a,u,s,z},theloglinearized t t=0 t t t t competitiveequilibriumisanallocation{c,h,d,n}∞ andapricesystem{π,w,mc}∞ satisfying t t t t t=0 t t t t=0 equations(2)to(11),andtheinitialconditionfor[a ;u ;s ;z ]. 0 0 0 0 2. Macro-financeseparationbyforceofhabit Definition(Macro-financeseparation). Anequilibrium(afeasibleallocationandapricesystemthat solve each household’s and each firm’s problem and clear markets) is macro-financially separate if the equilibrium allocation and equilibrium inflation are the same as in the model without habits (i.e.,suchthat Xc = Xh = 0atalldates). t t The ability to preserve the quantity implications of the underlying real business cycle model is a crucial diagnostic to evaluate a macro-finance model. In making this claim we are taking to the logical extreme the critique made by Lettau and Uhlig (2000), and revived by Uhlig (2007), RudebuschandSwanson(2008)andSwanson(2012),andapplyingittoDSGEmodelswithhabit formationinthespiritofCampbellandCochrane(1999).12 Inourcontext,riskpremiaaredriventofirstorderbythepriceofrisk,whosedynamicsarefully determined bysurplus consumption. Therefore, thenotion of macro-financeseparation boils down totheseparation betweenthedynamicsofrisk premiaandthedynamicsof quantities(including inflation).13 Time-varying risk aversion in turn spills over onto the flexible-price equilibrium allocation if and only if it does so onto the intratemporal rate of substitution that determines the optimal consumption-labor decisions. In a sticky-price equilibrium, however, a spillover ontoconsumption-savingdecisionswould produceanadditional departurefroma macro-finance separation. 2.1. Spilloverontotheintratemporalrateofsubstitution Thekeypropertyofour homeproductionhabitsisthat inequilibriumtheaggregateproduction functionandmarketclearingimplyatalldatesthefollowingrelationshipbetweensurpluslevels14 Z t = (cid:18) S t (cid:19)1+ξ2 Z S so their respective effects on the intratemporal marginal rate of substitution can offset for an appropriatechoiceofparameterξ . 2 12Wearenotdenyingthepossibilitythatamorevolatilediscountfactorbetterfitsquantitydynamics(inparticular hump-shapeddynamics, asarguedbyBoldrinetal.,2001). However, wearguethatthefirststepofthemodeling exerciseofincorporatingvolatilediscountfactorsinamacromodelshouldbetokeepthespilloversonquantities contained. Wecanthenallowforanarbitraryspilloverandaroleofhabitsinthedeterminationofquantitydynamics. 13More precisely, under a macro-finance separation the New Keynesian part of the model entirely determines shock-exposureelasticities,whiletheCampbell-Cochranehabitscontrolshock-priceelasticities. TheRBCparthas onlyanindirectimpactonshock-priceelasticitiesviatheprecautionaryeffectofhabits;seealsoLopezetal.(2015). 14Theequilibriumlawofmotionreducestozˆ t+1 /(1+ξ 2 )=ρ s zˆ t /(1+ξ 2 )+Λ[zˆ t /(1+ξ 2 )]εc t+1 =(cid:80)∞ j=0 ρ s jΛ[zˆ t−j /(1+ ξ )]εc ,whichimplieszˆ/(1+ξ )= sˆ intheappropriateequivalenceclassforstochasticprocesses. 2 t−j+1 t 2 t 10

Parameterξ controls thesize of the spilloverof the surplus levelsonto equilibrium quantities. 2 Infact,lettingξ = 0,theoptimalintratemporalrateofsubstitutionbetweenconsumptionandlabor, 2 ∂U /∂N A1−γCγ (cid:32) S (cid:33)γ − t t = χ t t t ∂U /∂C (1−N)γ Z t t t t reducestotheoneunderpowerutility. Homeconsumptionhabitsareneededtohavethesubstitutioneffecttowardshomeconsumption dominatetheincomeeffectofanegativemarketconsumptionshock,sohouseholdschoosenotto absorbtheconsumptionmovementbyincreasingsignificantlytheirlaboreffort. 2.2. Spilloverontotheintertemporalrateofsubstitution Time-varyingrisk aversion mayspill over onto theequilibrium allocationif itaffects therate ofsubstitutionbetweenconsumptionandsaving,r = −lnE M . InourGaussianexternal-habit t t t+1 setting,thedynamicISequationbalancesanintertemporalsubstitutionmotiveandaprecautionary savingsmotiveas15 γ(1−ρ −ξ /γ) r = −ln(β)− s 1 +γE ∆c −ξ sˆ t t t+1 1 t 2 where parameterξ controls thespillover ontoconsumption-saving decisions, as inCampbell and 1 Cochrane(1999)andWachter(2006). 2.3. Macro-financeseparation: flexibleprices Weformalizethelastresultsinthefollowingproposition:16 Proposition1. Giventhespilloverparameterξ ≡ [ξ ;ξ ] ∈ R2,foranyξ ∈ Randforanyvalue 1 2 1 of the preference parameter γ ∈ R , there is a unique value of parameter ξ = 0 such that the + 2 flexible-pricecompetitiveequilibriumismacro-financiallyseparate,forallξ ∈ R. 1 The onlineappendix provides additionaldetails on theproof of proposition1 andshowshow undermacro-financeseparationourhabitstructurecanmotivatetheoriginalconsumption-based asset pricing model of Campbell and Cochrane (1999) as the outcome of a generic production economy. 2.4. Macro-financeseparation: stickyprices Once we activate further rigidities, such as sticky prices, and we discuss the equilibrium separationrequirementsacrucialquestioniswhatistheempiricallyrelevantmonetarypolicyin place. A natural choice is a Taylor rule that responds to inflation and some detrended version of output,inwhichcasethecompetitiveequilibriumisseparatewhenevertheflexible-priceequilibrium 15Inthederivationweassumetheconditionalhomoskedasticityofconsumptiongrowth,sovar t (c t+1 ) = var(εc t ), which is consistent, for example, with a macro-finance separation or with a solution for consumption based on a first-orderapproximationofthestructuralequations. 16AppendixCandproposition4analyzetheinternalhabitspecificationandtheassociatedconditionsforseparation. 11

is, with the additional requirement of no intertemporal spillovers. The following proposition formalizes this results and extends the macro-finance separation results of proposition 1 to the sticky-pricesetting: Proposition 2. For any value of the preference parameter γ ∈ R , there is a unique value of + parameterξ = [0;0]suchthatthesticky-pricecompetitiveequilibriumismacro-financiallyseparate. Theonlineappendixdescribesthecompetitiveequilibriumandprovesproposition2. Intuitively, iftheflexible-priceequilibriumismacro-financiallyseparate,theoutputgap,c −cn,isasufficient t t statistic for aggregate marginal costs and inflation; since the only remaining source of financial spilloversisthedynamicISequation,wethenrequireazerospilloverparameterξ . 1 3. Termstructuresofequityandinterestrates We work with the cashflow processes implied by the macro-financially separate competitive equilibrium under a Taylor rule to study the asset pricing implications of our New Keynesian productioneconomywithCampbell-Cochraneexternalhabits. Theseprocessesarecharacterized bythestructural relations(7),(8),(9),andthe equilibriumprocessforthelogarithm ofaggregate consumption c = cn +ψ u, t t c t withψ = γ(1−(cid:101)βρ )/{(1−(cid:101)βρ )[γ(1−ρ )+φ ]+κ(φ −ρ )}andtheflexible-priceconsumption c u u u y π u processcn = a. t t 3.1. Equilibriumcashflows The New Keynesian framework models endogenously a difference between the real and the nominal term structures and a difference between aggregate consumption and market dividends. Table1andfigure2summarizetheirmaindifferencesbyrepresentingtheanticipatedreactionof themaincashflowprocessestomacroeconomicshocks. Four properties of the equilibrium cashflow processes are worth emphasizing. First, bond payoffsdonotdisplaymeangrowthandequitypayoffsdo. Second,short-runshockstothegrowth rateoftechnologydonothaveacontemporaneouseffectonbondpayoffsbutincreaseequitypayoffs. Third, apositivelong-runshock increasesboth consumptionandmarginalcosts, withanegative overallcontemporaneouseffectonmarketdividendsandapositiveoneoninflation. Finally,ex-ante consumptionanddividendgrowtharepositivewhentheconditionalmeanoftechnologygrowthis aboveaverageandsoistheex-antevalueofinflation. Theintuitionbehindthesepropertiesofcashflowsstemsfromtheequilibriumequation r = rn −γ(1−ρ )ψ u, t t u c t where rn represents the natural rate of real interest rates. A positive movement in expected t technologygrowthprompts households,who expectfuture growth,to commanda higherinterest on savings but the real rate increases less than the natural rate as a consequence of the monetary frictions; incentives to save remain too low, so demand and output go above potential and exert upward pressure on marginal costs. This cost effect depresses corporate profits while causing 12

Cashflow Deterministic Loadingon Loadingon Loadingon Asset process growth u σea σeu t t+1 t+1 Consumptionequity ∆c t+1 µ C c ∈ (0,1) 1 > 0 Marketequity ∆d t+1 µ > γC c 1 < 0 Nominalbond −π t+1 0 < 0 0 < 0 Realbond 0 0 0 0 0 Table1: Dynamicsofthecashflowprocessesthatdeterminethepricesoffourassets: anequityclaimtotheaggregate marketconsumptiongood(consumptionequity),anequityclaimtoaggregatedividends(marketequity),aclaimtoa unitofthenumeraire(nominalbond),andaclaimtoaunitofconsumption(realbond). Thecashflowprocessthat determinesconsumptionequityisconsumptiongrowth,marketequityisdeterminedbydividendgrowth,nominal bondsbynegativeinflationandrealbondspayaconstantrealcashflowwithtrivialdynamics. Thecashflowloadings arecalculatedforanontrivialdegreeofpricerigidities. 0.04 0.04 0.01 0.01 0 0 −0.01 −0.02 d c p −0 −0.04 −0.04 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 n(inyears) n(inyears) (a)Short-runshock-exposureelasticities. (b)Long-runshock-exposureelasticities. Figure2: Percentagechangeincashflowsoveragivenhorizonafter1standarddeviationshort-runandlong-runshocks totechnologygrowtharrivingnextmonth. inflationarypressureasfirmstrytoresetpricestorealignmarkupstothedesiredlevel. Markupsare then expected to jumpback up asthe excessive production getscorrected, and hencedividends are expectedtogrowmorethanconsumption,whilepositiveinflationpersistsforawhile. Nominalrigiditiesinturnexacerbatetheseeffects,asinflationbecomesmorestablewithstronger rigidities,andhencemarkupsmustabsorbalargershareoftheshocksthathittheeconomy,witha strongercontractionary effecton corporateprofits. Note howthe NewKeynesianmodel explains endogenouslythestylizedfactthatdividendgrowthismuchmorevolatilethanconsumptiongrowth exante. Thispropertyistruealsoex-postinthepresenceofmeanreversioninthestatesthatdrive quantities(technologyandtheconditionalmeanoftechnologygrowth). Noteinfacthowthelasttwo columnsof table1 implythata negativecorrelationbetween short-runand long-runshockswould dampenthevolatilityofconsumptiongrowthbutamplifydividendgrowthfluctuations. Therefore, wehaveanendogenousmechanismbywhichdividendsarealeveredversionofconsumption,as routinelyassumedinendowment-economyequilibriumassetpricingmodels. 13

3.2. Equilibriumtermstructures Short-run shockscommand a positive price, anda negativeenough correlation betweenshortrun and long-run shocks implies that long-run shocks commanda negative price. In this context, a cashflowexposuretolong-runshocksgreaterthanthecorrespondingdiscountrateexposureimplies an insurance effect. Therefore, table 1 shows how consumption claims and bonds (nominal and real)haveanupward-slopingtermstructureofriskpremiaanddividendclaimsadownward-sloping termstructure,allstartingfromastrictlypositivelevel. Moreover,thenegativepriceoflong-run shocksimpliesthatthereisastrictlypositiveinflationriskpremiumatallhorizons.17 Toformalizetheseresults,westartbydescribingthestochasticdiscountfactor, m = −ln(β)−γ∆c −γ∆s t+1 t+1 t+1 = −ln(β)−γE ∆c +γ(1−ρ )sˆ − x(E −E )c t t+1 s t t t+1 t t+1 where x ≡ γ[1+Λ(s)]isthepriceofrisk. Wethensolveforthetermstructuresofthedifferent t t cashflow claims by relying on the essentially-affine approximation proposed by Lopez, Lopez- SalidoandVazquez-Grande(2015),whichisolatesthefirst-ordercomponentsofequilibriumasset prices,performscomparablytonumericalsolutionmethods,andisparticularlyappropriateinour contextbecausetheunderlyingmodelofcashflowsissolvedtothefirstorderandwouldtherefore notallowforanaccuratecomputationofhigher-orderterms. WesetupthesystemintheformofLopezetal.,          ∆ ∆ d c t t + + 1 1  =  µ µ  +  C C d c  ζ t +  D D d c  ε t+1 −π 0 C D t+1 −p −p ζ = Aζ + Bε t+1 t t+1 where ζ = u and ε ∼ Niid(0,I ). We subsequently apply their essentially-affine approxt t t 2 imation to solve for the no-arbitrage price of a claim to some cashflow d that will realize in n periods, P(n) = E (M D ) and the associated holding-period expected excess return d,t t t,t+n t+n E (Re,(n) ) = E (M )E (P(n−1)/P(n)),whichtaketheapproximateequilibriumlogform t d,t+1 t t+1 t d,t+1 d,t re,(n) = E re,(n) +V ε d,t+1 t d,t+1 d,n−1,t t+1 whereddenotesthefourdifferentcashflowprocesses(consumption,corporateprofits,thenumeraire, andtheinverseofthepricelevel),andwherethestochasticvector V ≡ D + B(n−1)B +B(n−1)Λ D (12) d,n−1,t (cid:124)(cid:123)(cid:122) d (cid:125) (cid:32)d(cid:32),ζ (cid:32)(cid:32) (cid:32)d(cid:32)(cid:32),(cid:32)(cid:32)s(cid:32) (cid:32)(cid:32)t(cid:32)(cid:32)(cid:32)(cid:32) c (cid:124)(cid:123)(cid:122)(cid:125) (cid:124) (cid:123)(cid:122) (cid:125) short-run long-runcashflow habit-related cashflow risk anddiscountraterisk discountrate risk 17The online appendix shows how these results are robust to a departure from macro-financial separation that activatesnontrivialinvestmentchoices(ξ >0). 3 14

representsthequantityofriskinthenthcashflowstrip,withcoefficients B(n) = (C −γC )(I −A)−1(I −An) d,ζ d c B(n) = γ(1−ρ )+ρ B(n−1) − 1−ρ s (cid:16) B(n−1) −γ (cid:17)2 − D c (D d + B( d n ,ζ −1)B− B( d n ,s −1)D c )(cid:48) (cid:16) B(n−1) −γ (cid:17) d,s s s d,s γ2 d,s S d,s with B(0) = 0. AppendixDsketchestheessentially-affineapproximationweuse, whichiscovered d,s extensivelybyLopezetal.(2015). Theresultingclosed-formapproximatesolutionprovidesinsightintothedeterminantsofthe term structures of risk premia on different cashflow claims. The first term in equation (12) owes entirely to the one-month ahead volatility in cashflows; the second term captures the effect that news about the conditional mean of technology growth have on tomorrow’s prices through their effectonfuture cashflowsand discountrates; the thirdtermreflectstheeffectofmovementsinrisk aversion ontomorrow’s pricesthrough theireffect onlong-run discountrates ashabits slowly grow closertoconsumption. 3.3. Calibration Table2listsalldeepparametersinthemodelandtheircalibration. Wecalibrateallparameters of the production side of the economy using standard values in the New Keynesian literature as in Gal´ı (2008). Quasi-Frisch’s elasticity of labor supply is 1, consistent with steady-state hours N = 1/3. Thelaborshareinvalueaddedis1−α = 2/3. TheelasticityofsubstitutionintheCES aggregator is ε = 6 and the average duration of prices is (1−η)−1 = 9 months. The Taylor rule coefficientsareφ = 1.5andφ = 0.5/12(Taylor,1999). Thespilloverparameterξ issettozero, π y soweworkundermacro-financeseparation. WecalibrateallparametersrelatingwiththepricingkernelasinCampbellandCochrane(1999). Likeintheoriginalanalysis,thischoiceofparametersallowstocapturetheobservedpersistencein marketdividend-price ratios,arealisticmaximum Sharperatioofaround 0.4aswellas, crucially, a reasonable equity premium of 6.5% that is nonetheless substantially lower than the premium commandedbyshort-termequities,whichcanreachupto12%onaverage,consistentwithobserved stripreturns. Thisfeatureistheactualmeaningofthedownward-slopingtermstructureofequity documentedbyBinsbergen etal.(2012a). Wecalibratethesubjectivediscountrate, β,to matchan averagemonthlyrealrater of0.94%peryear. To calibrate the remaining parameters, which pin down the model’s dynamics, we choose a parametrization that matches a few moments at annual frequency of consumption and dividend growthusingannualdataonpersonalconsumptionexpenditureinnondurablegoodsandservices and on nonfinancial corporate profits over the period 1929-2014 from the Bureau of Economic Analysis,whichweexpressinrealtermsthroughthecorePCEpriceindex.18 Westartbyestimating viamaximumlikelihoodan ARMA(1,1)structureandrecovertheimpliedmonthly ARcoefficient. 18WhileweusethesamedatasetasBansalandYaron(2004),wedifferinourchoicetomatchcorporateprofits ratherthanCRSPdividends;corporateprofitsallowtoexploitacommondatasourcewhileremainingfullyconsistent withthemodel,inwhichcorporateprofitsanddividendscoincide. Resultsaresimilarifweuseadividendseries(e.g., usingCRSPvalue-weightedreturns)instead. 15

Parameter Value NewKeynesian γ Utilitycurvatureinmarketandhomeconsumption 2 block 1/ϕ Quasi-Frisch’slaborsupplyelasticity 1 β Subjectivediscountfactor .9991 1−α Laborshareinvalueadded 2/3 ε ElasticityofsubstitutioninDixit-Stiglitzaggregator 6 1/(1−η) Averagepriceduration(inmonths) 9 φ Policyresponsecoefficienttoinflationmovements 1.5 π φ Policyresponsecoefficienttooutputmovements .5/12 y Habitblock ξ Financialspilloverontotheintertemporalrateofsubstitution 0 1 ξ Financialspilloverontotheintratemporalrateofsubstitution 0 2 ρ Habitpersistence .9940 s Exogenous µ Meantechnologygrowth .0030 block ρ Persistenceoftheconditionalmeanoftechnologygrowth .8470 u σ Conditionalvolatilityoftechnology .0181 φ Relativevolatilityoftheconditionalmeanoftechnology .1311 ρ Correlationbetweenshort-runandlong-runshocks −.9433 βmatchesanaveragerealinterestrateof.94%peryear. ρ matchesanaveragemarketequitypremiumof6.53%peryear. s µmatchesanaverageannualconsumptiongrowthof3.60%. ρ matchesanaverageARrootinanARMA(1,1)representationofannualconsumptiongrowthof.136. u [σ;φ;ρ]matchavolatilityofannualconsumptiongrowthof2.49%,avolatilityofannualdividend growthof33.07%andacorrelationbetweenconsumptionanddividendgrowthof.580. Table 2: Deep parameters and their calibration (monthly frequency). Data for real consumption growth and real dividendgrowthuseannualBEAdataovertheperiod1929-2014forpersonalconsumptionexpenditureinnondurables andservicesandfornonfinancialcorporateprofitsbeforetaxes,andaredeflatedbythecorePCEpriceindex. Monthly simulateddataareaggregatedtoanannualfrequencyandarematchedtothecorrespondingdatamoments. 16

Wepicktheremainingparameterstomatchthevolatilityofannualconsumptiongrowth(2.49%), thevolatility ofannualdividendgrowth(33.07%)andthecorrelationbetweenconsumptionand dividend growth (0.580).19 Finally, it is worth noting that our baseline calibration implies a correlation between consumption growth and inflation of −0.31 at a quarterly frequency, in line withtheempiricalmomentextractedbyBinsbergenetal.(2012b). 3.4. Results Figure3 reportstheaverageterm structure ofequilibrium riskpremia,volatilitiesandSharpe ratios of consumption and market equities and of real and nominal interest rates. Figure 1b additionallyreportsthetermstructureofhold-to-maturityreturns,whichare(conditionally)linear combinations of holding-period returns. The average premium on the market portfolio is 6.53% (annualized,monthlybasis),considerablylessthantheshortendofthetermstructureoftheequity premium,andcompareswithaslightlylowerpremiumontheconsumptionportfolioof5.97%.20 Theinflationriskpremiumissizeableandpositiveatallmaturities,startingatzeroandincreasing steadilyup toslightlymore than1%peryear ata40-year horizon. Figure3 alsoshows howonly long-durationcashflowstripsaremean-varianceefficient. Figure4plotsthetermstructuresconditionalondifferentvaluesofthestatethatdrivesthem (surplusconsumption). Badsurplusconsumptionstates,whichassociatewithhighriskaversion, scaleuplevel,slopeandcurvatureofthetermstructures. Goodsurplusconsumptionstatesassociate withvirtuallyflattermstructures. Moreover, we reproduce the main appealing properties of Campbell and Cochrane (1999), including countercyclical financial market volatility and risk premia, as well as the long-horizon predictabilityofexcessstockreturns. 3.4.1. A3-factordecomposition: Level,short-runandlong-runslope Riskpremiaaretheproductof thesystematicexposureof eachstriponthestructuralshockand thepriceofaunitexposuretothestructuralshock, x D . Wecanthereforedecomposeriskpremia t c lnE Re,(n) = x D V(cid:48) t d,t+1 t c d,n−1,t into three determinants—a level factor x D D(cid:48), a factor that controls the short end of the curve, t c d x D (B(n−1)B)(cid:48) andafactorthatcontrolsthelongendofthecurve, x D (B(n−1)Λ D )(cid:48). t c ζ t c s t c 19Anotherreasonwedepartinourbaselinecalibrationfromatrend-stationarytechnologyprocessisthatitimpliesa trivialmartingalecomponentinthediscountfactormP =0becausetherearenopermanentshockstothemarginal t+1 utilityofwealth. AlvarezandJermann(2005)arguedforcefullyforamodelofthemarginalutilityofwealthtoinclude nontrivialpermanentandtransientcomponents. 20Justliketheequitypremium,alsothevarianceriskpremiumremainsatalevelsimilartoCampbellandCochrane (1999),witha30-dayexpectedvarianceundertherisk-neutralandthephysicalmeasuresof29.6and29.3,respectively (seeLopezetal.,2015, foraclosed-formapproximateexpressionforthevarianceriskpremium, whichshowsits positivenessanditsdeterminants). Thus,despiteanimplicitvariancewiththerightmagnitude,theimpliedvariance riskpremiumof0.3(inpercentagessquared)isanorderofmagnitudesmallerthantheempiricalestimatesinthe literature(e.g.,Bollerslev,TauchenandZhou,2009). 17

Riskpremium, lnE(Re,(n)) t+1 0.1 Equitypremium=6.4% 0.05 0 0 5 10 15 20 25 30 Volatility, var(re,(n)) t+1 q 0.3 0.2 0.1 0 0 5 10 15 20 25 30 Sharpe ratio, lnE(Re,(n))/ var(re,(n)) t+1 t+1 q Hansen-Jagannathanbound=0.39 0.4 0.3 0.2 dividendstrip real bond 0.1 consumptionstrip nominal bond 0 0 5 10 15 20 25 30 n(inyears) Figure3: Unconditionaltermstructuresofequityandinterestratesundermacro-financeseparation. Different linesassociatewithtermstructuresofdifferentcashflowclaims: realbonds(dotted),nominalbonds(dashdotted),consumptionequity(dashed)andmarketequity(solidline). 18

0.15 0.15 0.12 0.12 0.08 0.08 0.04 0.04 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 n(inyears) (a)Dividendstrips. (b)Consumptionstrips. 0.15 0.15 0.12 0.12 0.08 0.08 0.04 0.04 0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 n(inyears) (c)Nominalbonds. (d)Realbonds. Figure4: Statedependenceofthetermstructuresofholding-periodriskpremia(medianandinterquartilerange). 19

Level. Theshortendofthetermstructuresdependsprimarilyontheloadingsonshort-termcashflow riskthroughvector D ,whichcontrolsthelevelofthetermstructures,whoseinitialvalueis d cov(−m ,∆d ) = x D D(cid:48) t t+1 t+1 t c d This level factor in the term structures of risk premia is depicted in figure 5a under our baseline calibration. The level of the termstructure of dividendstrips can be veryhigh because of thehigh leverage in corporate profits, which fluctuate more than consumption, as nominal rigidities force firms to act on real wages rather than on prices to absorb the economic shocks, and because of their positive correlation with the priced shock (consumption news). The first dividend strip tends to have a dramatically low payoff precisely in those states in which households are hit by negative consumption shocks. Moreover, the 1-month interest rate is strictly positive by the positive correlationbetweeninflationandconsumptionnews. Short-runslope. Aconditionalmeanoftechnologygrowthaboveaveragetomorrowsignalsgood future cashflows (which increase prices) but also lower future marginal utility (which decreases prices as households want to anticipate consumption). This discount rate effect dominates the cashfloweffectforallclaimsconsideredexceptformarketequities,whosefuturepricestherefore increaseafterapositivelong-runshockandthemoresothelongerthestripduration. Sincepositive long-run shocks tend to arrive together with bad consumption news, it follows that this effect generatesanegativeslopeinthetermstructureofequityandupwardslopesintheremainingterm structures. In particular, we are able to generate a downward-sloping short end in the term structure of marketequityforanycalibrationsuchthat B(n)Bissufficientlynegative. Infact,fordividendclaims ζ,d theexposuretolong-runshockscommandsaprice 1−ρn cov(−m ,B(n)ζ ) = x(C −γC ) u(ρ+ψ φ)φσ2 t t+1 ζ,d t+1 t d c 1−ρ c u which is a negative number under ρ < −ψ φ for a sufficiently large degree of price rigidities. c Namely, the risk premium of long-run shocks commanded by dividend strips is a negative and convexfunctionofmaturity,andtheanalogousfactorinconsumptionstripsandzero-couponbonds (realandnominal)isapositiveandconcavefunctionofmaturity,ashowninfigure5bunderour baselinecalibration. Long-run slope. The loading of tomorrow’s yields on surplus consumption captures the properties ofthepremiumcommandedbylong-durationclaims;alltermstructuresdisplayanupwardslope at the long end, a property that is driven by the perfectly negative correlation between shocks to consumptionandtothepriceofrisk. Tomorrow’spriceoflong-durationclaimsislow,andhence holding-periodreturnsarelow,preciselyinthosestatesoftheworldinwhichsurplusconsumption is low, as households forecast lower future marginal utility as their habits adjust to the lower consumptionlevelandhencerequirecompensationtoshiftresourcesforwardintime. In particular, the loadings of yields on surplus consumption converge to the positive number B(∞) = γ for any dividend process, with the exception of a knife-edge case (see Lopez et al., s,d 20

2015), with the speed of convergence controlled by the persistence of habits. Since shouldering surplus-consumption shocks is equivalent to shouldering consumption shocks, the habit-related loadingofinfinite-durationzero-couponcashflowclaimscommandsastrictlypositiveprice cov(−m ,B(∞)sˆ ) = γ2Λ(1+Λ)(cid:107)D (cid:107)2 t t+1 s,d t+1 t t c Figure5cplotstheseloadingsunderourbaselinecalibrationlistedintable2. 3.4.2. Dynamicvaluedecomposition: Borovicka-Hansenelasticities The3-factor decompositionoftheone-monthahead volatilityinstrip returnsisdeeplylinked with the shock-exposure and shock-price elasticities proposed by Borovicka and Hansen (2014) as measures to quantify the exposure of cashflows over alternative horizons to shocks and the corresponding compensation commanded byinvestors. Inparticular,Lopez et al.(2015) show that onecanwriteholding-periodriskpremiaas lnE Re,(n) = ε(n) − ε(n) +var(m ) t d,t+1 (cid:124)(cid:123) g (cid:122) ,t (cid:125) (cid:124)(cid:123) p (cid:122) ,t (cid:125) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)t (cid:123)(cid:122) (cid:32)t(cid:32)+(cid:32)(cid:32)(cid:32)1 (cid:125) discountrateshock- discountrateshock- precautionary exposureelasticity priceelasticity motive whereε(n) andε(n) denotetheelasticitiesofexpectedfuturecashflowsandofexpectedfuturereturns g,t p,t toamarginalincreaseinexposureatt+1alongdirectionα = x D . Therefore,holding-periodrisk t t c premiaareequivalenttoastrictlypositivelevelfactor(householdsrequiresomecompensationto savewhenfacinguncertaintyaroundfuturemarginalutility),plustheelasticityoffuturedividends onpositiveconsumptionnews(cashfloweffect)lesstheelasticityoffutureinvestors’compensation onconsumptionnews(discountrateeffect). A marginal increase in exposure with the same direction as a (scaled) consumption shock recovers what movement in expected cashflows and returns associates with that shock. Figure 2 plotstheseelasticities. Ontheonehand, positiveconsumptionnewsassociateswithpositiveand partially mean-reverting dividend and consumption news as well as with disinflationary news. On the other hand, positive consumption news associate with lower marginal utility in the near futuredrivenbyhigherfuturegrowthaswellaswithhighermarginalutilityintheverylongrun owing to a habit level slowly growing towards the higher consumption level. Tomorrow’s cashflow anddiscountrateeffectscombinetoexplainholding-periodriskpremia,whichwillincreasewith positivecashflownewsanddecrease withpositive discountratenewsthat willdepresstomorrow’s prices. 3.4.3. Diagnostics To gain further insight into the properties of our model of the stochastic discount factor we studythediagnosticdecompositionsofthediscountfactorproposedbyAlvarezandJermann(2005) andHansenandScheinkman(2009). In the context of an essentially-affine approximation, Lopez et al. (2015) show how the martin- 21

0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 −0.05 −0.05 −0.05 −0.1 consumptionequity −0.1 −0.1 marketequity realbonds nominalbonds 0 5 10 15 0 5 10 15 0 10 20 30 40 50 60 70 80 90 100 (a)Levelfactor. (b)Short-runslopefactor. (c)Long-runslopefactor. 0.15 0.15 0.15 0.12 0.12 0.12 0.08 0.08 0.08 0.04 0.04 0.04 consumptionequity marketequity realbonds 0.012 nominalbonds 0 0 0 0 5 10 15 0 5 10 15 0 10 20 30 40 50 60 70 80 90 100 n(inyears) n(inyears) n(inyears) (d)Precautionarymotive. (e)Discountrateshock-exposureelasticities. (f)Discountrateshock-priceelasticities. Figure5: 3-factordecomposition(upperpanel)andBorovicka-Hansendynamicvaluedecomposition(bottompanel)ofholding-periodriskpremiaofdifferent zero-couponcashflowclaims. Thebottompanelplotsannualizedshock-exposureandshock-priceelasticitiesafteramarginalincreaseinexposurealongthe directionxD . Thinsolidlinesintheplotsofshock-priceelasticitiesrepresentstheinterquartilerangefortheelasticitiesofrealbonds. Decompositionsare t c suchthatholding-periodriskpremia(figure1a)=[(a)+(b)+(c)]=[(d)+(e)-(f)].

galecomponentofthestochasticdiscountfactoris  m t P +1 =   − − 1 2 2 1 γ x t 2 2 (cid:107) σ D 2 c   (cid:107)2 1 φ −   x (cid:48) t   D ρ 1 c ε t ρ 1 +   1 ,   1 φ   −γσ   1 φ   (cid:48) Σε t+1 i e f ls φ ew = h ξ e 1 re = 0 1−ρu 1−ρu 1−ρu whichisdiscountinuousatφ = ξ = 0,hastrivialpropertiesonlyundertrend-stationarytechnology, 1 andimpliestheapproximateentropyratio var(mP ) ω = t t+1 t var(m ) t t+1  =   1 γσ2 (cid:20) 1+ 1 2 − ρ ρ φ u + (cid:16) 1− φ ρu (cid:17)2 (cid:21) , e if ls φ ew = h ξ e 1 re = 0 (13) (1−ρs)(1−2sˆt) The martingale component of the stochastic discount factor reveals a permanent component inthemarginalutilityofconsumptionsuchthatshockstosurplusconsumption(ifφ = ξ = 0)or 1 shockstothepredictablecomponentofconsumption(ifφ (cid:44) 0orξ (cid:44) 0)haveapermanenteffect 1 onthemarginalutilityofwealtheventhoughbothriskaversionandthepredictablecomponentof consumptionarestationary. Considertwoextremecases,φ = 0(random-walktechnology)and[φ;ρ] = [1−ρ ;−1](trendu stationary technology). The case of trend-stationary technology implies mP = 0 because there t+1 arenopermanentshockstothemarginalutilityofwealth. Thecaseofrandom-walktechnology, combinedwithazerospilloverparameterξ = 0,impliesavarianceratio(13)constantatunity,and 1 hencea trivialtransient componentof thestochastic discountfactor. Thisproperty isappealing in thatitsatisfiesadiagnosticpropertyadvocatedbyAlvarezandJermann(2005);however,itwould predictanalwaysflatrealbondtermstructureaswellasnotime-variationintherelativeimportance of the permanent and transient components, which seems at odds with the return forecastability literature (see Lettau and Ludvigson, 2010; Koijen et al., 2010; and Lopez et al., 2015 for more details).21 Therefore, intermediate parametrizations that display unit-root dynamics with some amount of meanreversionproduceamodelofthestochasticdiscountfactorthatdisplaysthreekeyrealistic features: a time-varying permanent component, a time-varying transient component, and timevariationintherelativeimportanceofthepermanentandtransientcomponents. 21Iftherealbondloadingsonsurplusconsumptionareonthestablepath,itisextremelydifficulttohaveanaverage entropyratioclosetoone,asadvocatedbyAlvarezandJermann(2005);ourbaselinecalibrationproducesanentropy ratioof2.4%. ThefindinginAlvarezandJermannofrealandnominalvarianceratiosclosetoonerestshoweveron proxiesfortheunobservableinfinite-horizonzero-couponbonds; inourmodelonecanshowthatusinga20-year bondasaproxyfortheinfinite-durationbondassociateswithentropyratiosmuchclosertounity(67.5%);thehigh persistence of surplus consumption is responsible for the low speed of convergence of the loadings, as shown in figure5c. Thesameistrueifweconsidernominalpayoffsandadecompositionofthenominalstochasticdiscount factor. 23

3.4.4. Varyingthedegreeofnominalrigidity Figure 6 shows the effect of price stickiness and highlights its role in generating an initially downward-slopingterm structureofmarket equityandin flatteningthebond yieldcurve. Equilibrium riskpremia and volatilities onzero-coupon equities shiftupwards as thedegreeof nominal pricerigidityincreases,whereastheoppositeoccursforzero-couponnominalbonds. Inthelimiting caseasnominalrigiditiesdisappear(priceduration=1month)thereisnoendogenousdifference betweentherealandnominalbondtermstructuresandbetweenthetermstructuresofconsumption andmarketequity. Theeffectonthetermstructuresismainlydrivenbycashflows,asstickierpricesmakedividends more volatile (which exacerbates the negative slope in the term structure of equity) and the conditionalmeanofconsumptiongrowthandinflationmorestable(whichflattensthetermstructure ofnominalinterestratesandreducestheinflationriskpremium). Notehowasimilarflatteningof thetermstructureoccursalsoforzero-couponrealbonds,drivenbytheweakerdiscountrateeffect. Finally,itisworthnotingthehighlynonlineareffectofincreasingthedegreeofpricerigidities (orsymmetricallyofdecreasingtheanti-inflationarystance),whichstemsfromtheconvexityofthe equilibriumcoefficients(e.g.,ψ )onthekeyparameters. c 3.4.5. Varyingthemonetarypolicystance Figure7showstheendogenouseffectofmonetarypolicyonthetermstructuresofequityand interestrates. Theeffectofaweakeranti-inflationarystance(lowerTaylorrulecoefficientsφ and π φ )issimilartotheeffectoflargernominalrigidities,exceptfortheoppositeeffectontheinflation y riskpremium. Equilibriumriskpremiaandvolatilitiesonzero-couponequitiesshiftupwardsasthe policyrespondslessaggressivelytoinflation,whereastheoppositeoccursforzero-couponnominal bonds. The effect on the term structures is mainly driven by cashflows, as a less aggressive antiinflationary stance makes dividends and inflation more volatile (which exacerbates the negative slopeinthetermstructureofequityandincreasestheinflationriskpremium)andtheconditional mean of consumption growth more stable (which flattens the term structure of nominal and real interestratesviaaweakerdiscountrateeffect). 4. Conclusion We incorporate risk premia variation arising from Campbell-Cochrane external habits in a standardmacro modelwith nominalrigidities. We proposeamethod tobreak theapparent tradeoff betweeneithermatchingthedynamicsofmacroeconomicvariablesorassetpricingdynamicsin nonlinearhabitmodels. Thenotionofmacro-financeseparation(andarbitrarilysmalldepartures fromit)isshowntobeusefulforincorporatinglargediscountratevariationinaDSGEframework whilepreservingthemodel’sabilitytofitquantities. Wederivetestableimplicationsforthetermstructuresofequityandinterestratesthatconform withrecentcapitalmarketevidence,includingadownward-slopingtermstructureofequityreturns andvolatilities, upward-slopingtermstructuresofnominalandrealinterestrates,andapositive inflationriskpremium. Themodelcanbeeasilyextendedtostudythereactionofcapitalmarkets 24

Riskpremium Riskpremium Riskpremium 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 Volatility Volatility Volatility 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 Sharperatio Sharperatio Sharperatio 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 p.dur.=9m p.dur.=9m p.dur.=9m p.dur.=4m p.dur.=4m p.dur.=4m 0.1 p.dur.=2m 0.1 p.dur.=2m 0.1 p.dur.=2m p.dur.=1m p.dur.=1m p.dur.=1m 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 n(inyears) n(inyears) n(inyears) (a)Dividendstrips. (b)Nominalbonds. (c)Realbonds. Figure6: Termstructureofdividendstrips,nominalinterestratesandrealinterestratesfordifferentdegreesofpricestickinessundermacrofinanceseparation. Differentlinesrepresentdifferentcalibrationsfortheaveragepriceduration: onemonth(dotted),sixmonths(dashed),nine months(dash-dotted)andtwelvemonths(solidline).

Riskpremium Riskpremium Riskpremium 0.1 0.1 0.1 0.05 0.05 0.05 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 Volatility Volatility Volatility 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 Sharperatio Sharperatio Sharperatio 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 φ φ π π = = 1 5 .5 0.1 φ φ π π = = 1 5 .5 0.1 φ φ π π = = 1 5 .5 φπ=10 φπ=10 φπ=10 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 n(inyears) n(inyears) n(inyears) (a)Dividendstrips. (b)Nominalbonds. (c)Realbonds. Figure7: Termstructureofdividendstrips,nominalinterestratesandrealinterestratesfordifferentpolicyruleparametrizations. Differentlines representdifferentcalibrationsfortheanti-inflationarystance.

to unexpected monetary news, which our model is naturally able to address as it displays timevaryingriskpremiawithinamodelofquantitiesthatisappropriatetostudytheeffectofmonetary disturbances. Ourframeworkremainedparsimoniousalongmanydimensionsthatcaneasilybegeneralized. Inparticular,furtherworkcouldrelaxthetwo-shockstructureandtheunivariatepriceofrisk. The introduction of demand shocks may help to mitigate the correlation puzzle (e.g., Albuquerque, Eichenbaum,PapanikolauandRebelo,2015,forarecentexposition). Finally, we work under a full macro-finance separation, which is likely an unnecessarily strong requirement;namely,smalldeparturesfrommacro-financeseparationaremostlikelyempirically valid descriptions of the data, as they would for example include the case of stochastic capital accumulation. An estimated model that builds on our framework could identify such spillovers, and hencethe parametervector ξ. In thiscontext,the essentially-affineapproximation by Lopez et al. (2015) is particularly convenient in estimation in that it permits the use of linear filtering techniques. Appendix A. Campbell-Cochranehabitspecificationinaproductioneconomy ThelawofmotionofsurplusconsumptionassumedbyCampbellandCochrane(1999)intheir endowmenteconomywithrandom-walkconsumptioncanbecastinthreeequivalent specifications: sˆ = ρ sˆ +Λ(sˆ)(E −E )c (A.1a) t+1 s t t t+1 t t+1 = ρ sˆ +Λ(sˆ)(∆c −µ) (A.1b) s t t t+1 = ρ sˆ +ΛE (∆c −µ)+Λ(sˆ)(E −E )c (A.1c) s t t t+1 t t+1 t t+1 whereµ = E(∆c). Theequalitybreaksdownhoweveronceweallowforapredictablecomponent in consumption growth, consistent with a generic production economy.22 To understand what specificationweshouldretaininaproductioneconomy,notehowthereisastrongreasontoprefer specification(A.1a)owingtoitsimplicationsfortherisk-freerateandfortherelationshipbetween consumptionandthehabitlevel. A.1. Localstructureandpredeterminedness As shown by Campbell and Cochrane (1999) and Lynch and Randall (2011), specifications(A.1b)and(A.1c)implythelocalhabitstructure (cid:88)∞ xc = xc+c − ρj∆c +O((cid:107)ε(cid:107)2) t+1 t+1 s t−j+1 j=0 (cid:88)∞ = xc+ θ ∆c +O((cid:107)ε(cid:107)2) j t−j+1 j=0 22Forexample,inarecentstudyofCampbell-Cochranehabitswithnon-random-walkcashflows,LynchandRandall (2011)adoptspecification(A.1c). 27

whereθ ≡ 1−ρj and xc ≡ ln(1−S),sotheconsumptionhabitisaslowmovingaverageofpast j s consumption growth such that consumption growth moves transitorily consumption away from habits. Specification (A.1a) implies a habit structure that also depends on what people expect to consume, (cid:88)∞ xc = xc+c − ρjεc +O((cid:107)ε(cid:107)2) t+1 t+1 s t−j+1 j=0 (cid:88)∞ (cid:88)∞ = xc+ E ∆c + θ εc +O((cid:107)ε(cid:107)2) t−j t−j+1 j t−j+1 j=0 j=0 so the consumption habit is the sum of past anticipated consumption movements and of a slow moving average ofpast consumptionshocks, which receive theirfull weightonly asymptotically (lim θ = 1); only unanticipated movements in consumption move consumption away from j→∞ j habits. Surplusconsumptionisthusbasicallydetrendedconsumption. Sinceθ = 0,thehabitlevelislocallypredetermined, xc = E xc ,underallspecifications. 0 t+1 t t+1 A.2. Relationshipbetweenconsumptionandthehabitlevel Specifications(A.1a),(A.1b)and(A.1c)implytherespectiverelationshipbetweenconsumption andthehabitlevel(seetheonlineappendixformoredetails) ∂xc Λ(sˆ ) (E −E )Mc t = 1− t−1 t t−1 t ∂c exp(−s)−1 Mc t t t ∂xc Λ(sˆ ) t = 1− t−1 ∂c exp(−s)−1 t t ∂xc Λ(sˆ ) Λ(sˆ )−Λ E Mc t = 1− t−1 + t−1 t−1 t ∂c exp(−s)−1 exp(−s)−1 Mc t t t t withMc theshadowvalueofsurplusconsumption. t Itfollowsthat,inthesteadystate,consumptionhabitsmovestrictlypositivelywithconsumption, ∂xc/∂c = 1,underspecification(A.1a)buttheyareunrelatedwithconsumption,∂xc/∂c = 0,under specifications(A.1b)and(A.1c),apropertythatleadstothecritiquebyLjungqvistandUhlig(2015), who look at the second derivative ∂2xc/∂c2 and note that in a neighborhood of the steady state t t thehabitprocesscanmovestrictlynegativelywithconsumption. Thereasonspecification(A.1a) bypassesLjungqvistand Uhlig’scritiqueisthattheequilibrium expressionfortheex-antevalue of consumption is no longer a structural relation in a production economy but an outcome of optimization. 28

A.3. Norisk-freeratepuzzle Therespectiveequilibriumrisk-freeratesunderspecifications(A.1a),(A.1b)and(A.1c)are23 r = r+γE (∆c −µ) (A.2a) t t t+1 r = r+ x E (∆c −µ) (A.2b) t t t t+1 r = r+ xE (∆c −µ) (A.2c) t t t+1 where x ≡ γ(1 + Λ) is the price of risk. As shown by equations (A.2b) and (A.2c), spect t ifications (A.1b) and (A.1c) imply a distorted dynamic IS equation relative to a power-utility specificationthatwouldimplyarisk-freeratepuzzle. Noteinfacthowalargepriceofrisk x = γ/S is necessary to generate a large equity premium; the parametrization S < 1 is the element that amplifiesthecoefficientofriskaversion(seetheonlineappendix)whileremainingneutralonthe risk-freerate,andthattherebyallowsforbreakingthetradeoffbetweensolvingtheequitypremium and the risk-free rate puzzles in the habit framework. We therefore discard specifications (A.1b) and (A.1c) on the ground that they would kill the central idea of the Campbell-Cochrane habits. Wethusretainspecification(A.1a)andtheassociateddynamicISequation(A.2a). A.4. Homeconsumptionhabits Our home consumption habits can produce a macro-finance separation, and hence break the quantitypuzzle,becausethesamestatedrivesbothsurplusmarketandhomeconsumption,sothe respectiveeffectsonconsumption-labordecisionscanoffsetoneanother. Thelocalmicrofoundationsofourhomeconsumptionhabitparallelthoseofthemarketconsumptionhabit. Wechoosethesteady-statecoefficientZ tominimizethedistance (cid:13) (cid:13) (cid:13) Z 1−S (cid:13)2 ι = Z m ∈(0 in ,1) (cid:13) (cid:13) (cid:13)1−Z S εc t+1 −εh t+1 (cid:13) (cid:13) (cid:13) of thehome consumptionhabit fromlocal predeterminednessnear thesteady state. Therefore, the homeconsumptionhabitcanbewrittenlocallyas (cid:88)∞ xh = xh+h − ρjεh +O((cid:107)ιξ,ιφ,ε(cid:107)2) t+1 t+1 s t−j+1 j=0 (cid:88)∞ (cid:88)∞ = xh+ E ∆h + θ εh +O((cid:107)ιξ,ιφ,ε(cid:107)2) t−j t−j+1 j t−j+1 j=0 j=0 with xh ≡ ln(1 − Z), where φ indexes the policy in place and is such that φ = 0 implies the flexible-price equilibrium, where ξ indexes the distance from macro-finance separation (ξ = 0). Note that the habit is locally predetermined when ι = 0 even as we move arbitrarily far from the macro-financially separate, flexible-price equilibrium; for example, ι = 0 whenever market 23Forsimplicity,weturnoffthespilloverparameterξ asitaddsnothingtotheargument. 1 29

and home consumption shocks are perfectly correlated as under the commonly used one-shock technologystructure. The home consumption habit is the sum of past anticipated home consumption movements and of a slow moving average of past home consumption innovations, which receive their full weight only asymptotically (lim θ = 1); only unexpected movements in home consumption j→∞ j movehome consumption away from habits, which coincidewith home consumption inthe long run. Surplushomeconsumptionisthusbasicallydetrendedhomeconsumption. Finally,homeconsumptionhabitsrelatewithhomeconsumptionvia ∂xh 1−N (1−α)(1+ξ )Λ(s ) (E −E )Mh t = 1+ t 2 t−1 × t t−1 t ∂h N exp(−z)−1 Mh t t t t andhencethehabitmovesstrictlypositivelywithhomeconsumptioninthesteadystate,∂xh/∂h = 1. B. Capitalaccumulation Inthissectionweallowfornontrivialcapitalaccumulationdrivenby (cid:16) I (cid:17) K = (1−δ)K +Φ t K t+1 t t K t whereδisthedepreciationrateandcapitaliscostlyadjustedaccordingtothefunction Φ (cid:16) I t (cid:17) = eµ −1+δ + (cid:101)δξ 1 3 (cid:16) I t (cid:17)1− ξ 1 3, with (cid:101)δ ≡ eµ −1+δ K 1−ξ2 1− 1 K 1+ 1 t 3 ξ3 t ξ3 The parametric form of capitaladjustment costs is standard (e.g., Jermann, 1998; Boldrin et al., 2001; Binsbergen et al., 2012b) and is calibrated to imply the steady-state relations I/K = (cid:101)δ, Φ(I/K) = I/K, Φ(cid:48)(I/K) = 1 and −Φ(cid:48)(cid:48)(I/K)I/K = 1/ξ ; note that we allow the steady-state 3 investment-capitalratiotodependon ξ toavoidanunappealingdiscontinuityatξ = 0. Through 3 3 theadjustmentcostcurvature,1/ξ ,investmentisdeterminedbyTobin’sQ,whichinourfrictionless 3 setting equalsthe expecteddiscounted valueof future marketdividends (Hayashi,1982); since the discountingis done viathe Campbell-Cochranepricing kernel, surplus consumptioncan spillover ontoinvestment. Therefore, the spillover of the time-varying risk aversion on quantities is now controlled by parameter ξ = [ξ ;ξ ;ξ ] with ξ ≥ 0. As usual, parameter ξ controls the spillover on the 1 2 3 3 1 consumption-saving tradeoff and parameter ξ controls the spillover on the consumption-labor 2 tradeoff. Additionally,parameterξ controlsthespilloveronconsumption-investmentdecisions; 3 theabsenceofthistypeofspilloverimplieszeroinvestment. Theonlyadjustmentofthebaselinemodelinthissettingistheshapeoftheshockstructurethat drivessurplushomeconsumption,whichwenowspecifyintermsofthefunction f h [H t ] = ln[Aα t (A t −H t )1−α −(cid:101)δeαµtQξ t 3K t 1−α]1− 1 α 30

Surplus home consumption is no longer driven just by shocks to home consumption but is now also driven by shocks to the market value of the capital stock owned by consumers. This specificationisarbitrarilyclosetothebaselinespecificationforacurvatureξ closetozero,anditis 3 necessarytocontrolthespilloverontheintratemporalrateofsubstitution. Infact,marketclearing, Y = C +I,andtheoptimalityconditionforinvestment t t t I =(cid:101)δQξ3K, ξ < ∞ t t t 3 implythat,inequilibrium,(1−α)(E −E )f [H ] = (E −E )c andtherefore t+1 t h t+1 t+1 t t+1 zˆ = (1+ξ )sˆ t 2 t asrequiredtocontroltheintratemporalspillover. We can therefore state in proposition 3 the requirements for macro-finance separation in the contextofnontrivialcapitalaccumulation: Proposition3. Giventhespilloverparameterξ ≡ [ξ ;ξ ;ξ ] ∈ R3,foranyvalueofthepreference 1 2 3 parameterγ ∈ R , + (a) there is a unique value of parameter [ξ ;ξ ] = [0;0] such that the flexible-price competitive 2 3 equilibriumismacro-financiallyseparate,foranyξ ∈ R; 1 (b) there is a unique value of parameter ξ = [0;0;0] such that the sticky-price competitive equilibriumunderaTaylorruleininflationandoutputismacro-financiallyseparate. Startingfromamacro-financiallyseparateequilibriumwecanthenallowforanarbitrarilysmall spilloverby varyingξ; in particular, movementsin ξ allowfor positive investment as afunction of 3 Qandforanequilibriumarbitrarilyclosetotheequilibriumwithdeterministiccapital(ξ = 0). 3 C. Internalhabits Themarginalutilitiesofconsumptionandhomeconsumptionwhenconsumersinternalizethe endogeneityofhabitsare ∂Uint. t = C −γS1−γ +C−1Λ(sˆ )(E −E )Mc ∂C t t t t−1 t t−1 t t ∂Uint. Λ(s ) t = −χA H −γZ1−γ +(1−α)(1+ξ ) t−1 (E −E )Mh ∂N t t t 2 N t t−1 t t t whereMc andMh are theshadowvalues of surplusmarket andhome consumption, respectively. t t Theonlineappendixdetailsthederivation. Ontheonehand,apositivemarket(home)consumption shock means alower marginal value of market (home)consumption; onthe other hand, a positive market (home) consumption shock increases the habit level and thereby increases the marginal valueofmarket(home)consumption. Whenhabitsareinternal,householdstakeintoaccountalso thesecondeffectandtheyalsobecomesensitivetounexpectedmovementsintheshadowvalueof thesurpluslevels,whichdependoncurrentandfuturemarketandhomeconsumption. 31

If habits are internal, people balance their static habit motives as well as their precautionary savingsandintertemporal substitutionmotivesonlyif γ = 1; aunitaryelasticityofintertemporal substitutionhastheeffectofthetwohabitsonthemarginalutilityofconsumptionexactlyoffset,so themarginalutilityofconsumption(andhencethestochasticdiscountfactor)reducestotheone underpowerutility, ∂U t = C−1 ∂C t t Proposition4. Giventhespilloverparameterξ ≡ [ξ ;ξ ;ξ ] ∈ R3,foranyvalueofthepreference 1 2 3 parameterγ ∈ R ,the Paretooptimum ismacro-financiallyseparateif andonlyifγ = 1,for any + ξ ∈ R3.24 Themacro-financiallyseparateParetooptimumdisplaysthesamelowandstableriskpremiaas undera power-utility specification; alltime-variationinrisk premiais symptomaticof thepresence ofanexternality,asParetooptimalassetpricesassociatewithlog-utilityinvestors. Itfollowsthat theParetooptimaldynamicISequationis 1−ρ −ξ rint. = −ln(β)− s 1 S2 +E ∆c t 2 t t+1 Against this background, if we take to the logical extreme the critique by Lettau and Uhlig (2000), a necessary diagnostic requirement for a model with habits (or, more generally, for any modelthatincorporatesriskpremiavariationintoamacromodel)tobedeemedadmissibleisthat it can be calibrated to display a macro-finance separation. In this context, internal habits have dramatically different asset pricing implications than external habits; namely, the internal-habit economywouldreducetothepower-utilitymodelnotonlyintermsofquantityimplicationsbutalso intermsofassetpricingimplications. Thetrivialassetpricingimplicationsofthemacro-financially separateinternal-habitspecificationarethereasonwefavortheexternal-habitspecification. D. Essentially-affineapproximation Thepracticalapproximationproceedsinthreesteps. SeeLopezetal.(2015)foratreatmentin greatergeneralityanddetail,andforacomparisonofitsqualitywithalternativesolutionmethods. D.1. Firststep Cashflows. Loglinearizethefirst-orderconditionsdrivingquantitiesandsolvefortheapproximate quantitydynamics ∆c = µ +C ζ +D ε +O((cid:107)ζ,ε (cid:107)2) t+1 c c t c t+1 t t+1 ∆d = µ +C ζ +D ε +O((cid:107)ζ,ε (cid:107)2) t+1 d d t d t+1 t t+1 24Since under γ = 1 the stochastic discount factor under internal habits reduces to the one under power-utility, parameterξ canbeleftunrestrictedtoproduceamacro-financeseparation. 3 32

wherecislogconsumption,d isthelogofanarbitrarycashflowprocess,andwherethestateζ that t drivesquantitiesfollowstheVAR(1)process ζ = Aζ + Bε t+1 t t+1 withε ∼ Niid(0,I)avectorofshocks. t Discountrates. Thestochasticdiscountfactoris m = ln(β)−γµ −γC ζ +γ(1−φ)sˆ − x D ε +O((cid:107)ζ,ε (cid:107)2) t+1 c c t t t c t+1 t t+1 1 = −r − x2(cid:107)D (cid:107)2 − x D ε +O((cid:107)ζ,ε (cid:107)2) t 2 t c t c t+1 t t+1 where theresidual term comes fromthe approximate equation forconsumption growth andthe last equalityisbytheno-arbitragerelationr = −lnE M . Thetime-varyingpriceofriskfollowsa t t t+1 nonlinearprocess x : sˆ (cid:55)→ x(sˆ) = γ(1−2sˆ)1/2 thatisresponsiblefortheabsenceofaclosed-form t t S t solution to the problem, which would otherwise take an exponential-affine form. We therefore approximatetheendogenousandnonlineardynamicsofthepriceofriskas x = x(0)+ x(cid:48)(0)sˆ +O((cid:107)sˆ(cid:107)2) (D.1) t t t x2 = x(0)2 +2x(0)x(cid:48)(0)sˆ +O((cid:107)sˆ(cid:107)2) (D.2) t t t where, since in the Campbell-Cochrane specification x(cid:48)(cid:48)(0)x(0) + x(cid:48)(0)2 = 0, the residual in equation(D.2)isexactlyzero. Thusapproximated,thepriceofriskhasanessentially-affineformandtherebyallowsforan exponential-affinesolution forequilibriumyields, sinceall sourcesofstochastic volatilityowe to thetime-varyingpriceofriskandsincetherisk-freerateisexactlyaffineinthestatevector. D.2. Secondstep Guesstheexponential-affinesolutionforyieldsy(n) ≡ −1 ln(P(n)/D), d,t n t t 1 1 1 y(n) = − A(n) − B(n)ζ − B(n)sˆ +O((cid:107)ζ,sˆ,ε (cid:107)2) d,t n n ζ t n s t t t t+1 andverifyitbythefundamentalno-arbitragepricingformula0 = lnE (M Ri ). t t+1 t+1 Infact,giventheGaussianityoflogreturnsr(n) ≡ ∆d −(n−1)y(n−1) +ny(n),wehave d,t+1 t+1 d,t+1 d,t 1 0 = E m +dp(n) −E dp(n−1) +E ∆d + var(m −dp(n−1) +∆d ) t t+1 t t t+1 t t+1 2 t t+1 t+1 t+1 = ln(β)+µ −γµ −A(n) +A(n−1) +[C −γC − B(n) + B(n−1)A]ζ +[γ(1−φ)− B(n) + B(n−1)φ]sˆ d c d c ζ ζ t s s t 1 1 + (cid:107)D (cid:107)2x2 + V V(cid:48) − x D V(cid:48) +O((cid:107)ζ,sˆ,ε (cid:107)2) 2 c t 2 n−1,t n−1,t t c n−1,t t t t+1 33

whereV = D + B(n−1)B− B(n−1)D + x B(n−1)D /γ. Therefore,usingequations(D.1)and(D.2), n−1,t d ζ s c t s c 1 0 = ln(β)+µ −γµ −A(n) +A(n−1) + (cid:107)V − x(0)(D −V )(cid:107)2 d c 0,n−1 c 1,n−1 2 +[γ(1−φ)− B(n) + B(n−1)φ+ x(0)x(cid:48)(0)(cid:107)D −V (cid:107)2 − x(cid:48)(0)V (D −V )(cid:48)]sˆ s s c 1,n−1 0,n−1 c 1,n−1 t +[C −γC − B(n) + B(n−1)A]ζ +O((cid:107)ζ,sˆ,ε (cid:107)2) d c ζ ζ t t t t+1 whichidentifiestheexponential-affinesolutionasthesolutiontotheRiccatiequations 1 A(n) = A(n−1) +ln(β)+µ −γµ + (cid:107)V − x(0)(D −V )(cid:107)2 d c 0,n−1 c 1,n−1 2 B(n) = B(n−1)A+C −γC ζ ζ d c B(n) = B(n−1)φ+γ(1−φ)+ x(0)x(cid:48)(0)(cid:107)D −V (cid:107)2 − x(cid:48)(0)V (D −V )(cid:48) s s c 1,n−1 0,n−1 c 1,n−1 with V = D + B(n−1)B− B(n−1)D 0,n−1 d ζ s c 1 V = B(n−1)D 1,n−1 γ s c These closed-form expressions allow for computing the entire term structure of yields, y(n), d,t fromasimulatedpathofthestatevector[ζ;sˆ]uptoaremainderoforderatleastO((cid:107)ζ,sˆ,ε (cid:107)2). t t t t t+1 D.3. Thirdstep Finally,wecanusethelognormalno-arbitragepricingformulatocompute lnE Re,(n) = x D V(cid:48) t t+1 t c n−1,t 1 E re,(n) = x D V(cid:48) − V V(cid:48) t t+1 t c n−1,t 2 n−1,t n−1,t Tosimulate x andtherebyasamplepathforriskpremiaandreturnvolatilitiesweusetheexact t dynamics x(sˆ). t 34

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ONLINEAPPENDIX I. Flexible-priceequilibria ThissectioncharacterizestheParetooptimumandtheflexible-priceequilibrium. I.1. Paretooptimum(internalhabits,flexibleprices) TheParetooptimumcanbecharacterizedasthesolutiontoasocialplannerproblem. However,weappealtothe welfaretheoremsanddecentralizetheeconomytobuildintuitionandgaininsightintotheconsumptionandlabor margins. I.1.1. Consumers Internal-habitconsumersmaximizetheintertemporalobjective (cid:88)∞ (cid:32) (C −Xc)1−γ−1 (H −Xh)1−γ−1 (cid:33) max U = E βt t t +χ t t 0 0 1−γ 1−γ t=0 subjecttothebudgetconstraintandthestructuralhabitequations,H = A(1−N),and t t t C t −X t c =C t S t , sˆ t+1 =ρ s sˆ t +Λ(sˆ t )(E t+1 −E t )ln(C t+1 ) H t −X t h = H t Z t , zˆ t+1 =ρ s zˆ t +(1−α)(1+ξ 2 )Λ[zˆ t /(1+ξ 2 )](E t+1 −E t )ln(A t+1 −H t+1 ) Optimalityrequiresthatthejointevolutionoftheprocessessatisfies ∂Uint. Λ(s ) t =C−γS1−γ+ t−1 (E −E )Mc (I.1) ∂C t t C t t−1 t t t Mc =C1−γS1−γ+βEMc [ρ +Λ(cid:48)(s)εc ] (I.2) t t t t t+1 s t t+1 ∂Uint. Λ(s ) t =−χAH−γZ1−γ+(1−α)(1+ξ ) t−1 (E −E )Mh (I.3) ∂N t t t 2 N t t−1 t t t Mh =χH1−γZ1−γ+βEMh [ρ +Λ(cid:48)(s)εc ] (I.4) t t t t t+1 s t t+1 whereMcandMhareLagrangemultipliersassociatedwiththemarketandhomeconsumptionhabitequationsthat t t affectthemarginalutilityofwealthwithatime-varyingloading. I.1.2. Firms Firmsmaximizeperiodprofits,Y −WN,subjecttotheproductiontechnology,Y = AN1−α,whichresultsinthe t t t t t t optimalitycondition Y W =(1−α) t t N t I.1.3. Equilibrium Afterimposingmarketclearing,Y = C,wecancharacterizetheParetooptimumbytheequalitybetweenthe t t intratemporalrateofsubstitutionandthemarginalproductoflabor, ∂Uint./∂N C − t t =(1−α) t ∂U t int./∂C t N t 37

It is straightforward to verify how a unitary elasticity of intertemporal substitution, γ = 1, produces constant shadowvaluesofsurplusmarketandhomeconsumption. Underthisparametrizationwehave Mint. =β (cid:16)C t+1 (cid:17)−1 t+1 C t ∂Uint./∂N χC − t t = t ∂U t int./∂C t 1−N t soallintertemporalandintratemporaleffectsofthehabitareabsent. Theconditionγ=1isthereforesufficienttogrant amacro-financeseparationwhenhabitsareinternal,foranyvalueofthespilloverparameterξ . 2 Moreover,theconditionξ = 1issufficienttograntamacro-financeseparationtoafirst-orderapproximation 2 underbalancedgrowth,foranyvalueoftheelasticityofintertemporalsubstitution. I.2. Flexible-priceequilibrium(externalhabits) Optimalityrequiresthatthejointevolutionoftheprocessessatisfies ∂U t =C−γS−γ ∂C t t t ∂U t =−χAH−γZ−γ ∂N t t t t ∂U/∂N C − t t =(1−α) t ∂U/∂C N t t t Thus,thecompetitiveequilibriumischaracterizedby χ N t Sˆ−γξ2 =(1−α) (cid:16)C t (cid:17)1−γ (1−N)γ t A t t uptoanirrelevantconstant. Thecompetitiveequilibriumismacro-financiallyseparateifandonlyifξ =0. 2 II. CompetitiveequilibriumunderaTaylorrule Thefullmodeldrivingquantitiesis,toafirst-orderapproximation, π t =(cid:101)βE t π t+1 +κ(c t −cn t )−γλξ 2 (s t −sn t ) r t −r t n =γE t (∆c t+1 −∆cn t+1 )−ξ 1 (s t −sn t ) r t −r t n =i t −r t n−E t π t+1 i =φ π +φ (c −a)+φ sˆ t π t y t t s t =rn+φ π +φ (c −cn)+φ (s −sn)+v t π t y t t s t t t wherev =−rn+φ snandwhereweallowforahypotheticalreactiontoriskpremiabymonetarypolicytogainbetter t t s t insightintotheroleoftheTaylorrule. Thestateequationsare a t+1 =a t +µ+u t +σea t+1 u t+1 =ρ u u t +φσeu t+1 s t+1 =(1−ρ s )s+ρ s s t +Λ t (E t+1 −E t )c t+1 Toverifytheguessedstationarityofm(cid:99)c t andhenceof[π t ;c t −cn t ], posethelinearparametricformsc t −cn t = 38

ψ (s −sn)+ψ u andπ =ψ (s −sn)+ψ u,andverifythemas cs t t c t t πs t t π t λγξ (φ −ρ )−(ξ +φ )(1−(cid:101)βρ ) ψ = 2 π s 1 s s cs (1−(cid:101)βρ )[γ(1−ρ )+φ ]+κ(φ −ρ ) s s y π s (II.1) λγξ [γ(1−ρ )+φ ]+(ξ +φ )κ ψ =− 2 s y 1 s πs (1−(cid:101)βρ )[γ(1−ρ )+φ ]+κ(φ −ρ ) s s y π s and γ(1−(cid:101)βρ ) ψ = u c (1−(cid:101)βρ )[γ(1−ρ )+φ ]+κ(φ −ρ ) u u y π u γκ ψ = π (1−(cid:101)βρ )[γ(1−ρ )+φ ]+κ(φ −ρ ) u u y π u whichistheuniquesolutionofthemodeleconomyaslongasthesystemisdetermined. Macro-financeseparation holdsonlyifφ =−ξ andξ =0,inwhichcasesolution(II.1)reducestoψ =ψ =0. s 1 2 cs πs Consumptionequitycashflows. Equilibriumaggregateconsumptiongrowsatrate ∆c t+1 =µ+[1−(1−ρ u )ψ c ]u t +σea t+1 +ψ c φσeu t+1 =µ+C u +σea +ψ φσεu c t t+1 c t+1 whereC ≡[φ (1−(cid:101)βρ )+κ(φ −ρ )]/{[γ(1−ρ )+φ ](1−(cid:101)βρ )+κ(φ −ρ )}∈(0,1). c y u π u u y u π u Marketequitycashflows. Equilibriumaggregateprofits,PD = PC −WN,whichfirmspayoutasdividends,are,up t t t t t t toafirst-orderapproximationaroundtheundistortedsteadystate, 1−α d t =c t − α m(cid:99)c t whereaveragemarginalcostsaretheinverseofaveragemarkups. Corporateprofitsincreasewithoutputanddecrease withmarginalcosts. Therefore, γ(1−α)+α+ϕ ∆d t+1 =∆c t+1 − α [∆c t+1 −∆a t+1 ] γ(1−α)+ϕ =µ+C u +σea − ψ φσeu d t t+1 α c t+1 whereC ≡1+γ2(2/α−1)(1−ρ )(1−(cid:101)βρ )/{[γ(1−ρ )+φ ](1−(cid:101)βρ )+κ(φ −ρ )}>1. d u u u y u π u Nominalbondcashflows. Thepayoffattimet+nforan-periodzero-couponnominalbondisaunitofmoney,whose realvalueis1/P t+n ,i.e.,thedividendgrowsatrate∆d t+1 =ln(1/P t+1 )−ln(1/P t )=−π t+1 with −π t+1 =−ψ π ρ u u t −ψ π φσeu t+1 κ =C u − ψ φσeu −p t 1−(cid:101)βρ c t+1 u withC ≡−γκρ /{[γ(1−ρ )+φ ](1−(cid:101)βρ )+κ(φ −ρ )}<0. −p u u y u π u Realbondcashflows. Thepayoffattimet+nforan-periodzero-couponrealbondisaunitofnumeraire,i.e.,thelog dividendisd =0. t 39

III. Relationshipbetweenmarketandhomeconsumption,andthehabitlevels III.1. Marketconsumptionhabits AsshownbyCampbellandCochrane(1999),wecanwritethederivativeofutilitywithrespecttoconsumptionas ∂U t =C−γFc ∂C t t t Fc =S−γ (cid:104) 1−E (cid:88)∞ βj (cid:16)C t+j S t+j (cid:17)−γ ∂X t c +j (cid:105) (III.1) t t t CS ∂C j=0 t t t =S−γ (cid:104) 1− ∂X t c(cid:105) −βE (cid:16)C t+1 (cid:17)−γ(cid:104)∂X t c +1 + ∂X t c +1 ∂X t c(cid:105) Wc (III.2) t ∂C t C ∂C ∂Xc ∂C t+1 t t t t t with Wc =S−γ+βE (cid:16)C t+1 (cid:17)−γ∂X t c +1Wc t t t C ∂Xc t+1 t t whereweused ∂X t c +j = dX t c +1 (cid:89)j ∂X t c +h ∂C dC ∂Xc t t h=2 t+h−1 dXc ∂Xc ∂Xc ∂Xc t+1 = t+1 + t+1 t dC ∂C ∂Xc ∂C t t t t becausethelawofmotionofsurplusconsumptiondefinesanimplicitfunctionX t c +1 = X t c +1 (X t c,C t+1 ,C t ),soC t affects Xc directlyandviaXc. Fromthelawofmotion t+1 t (cid:16) Xc (cid:17) (cid:16) Xc(cid:17) (cid:104) (cid:16) Xc(cid:17)(cid:105)(cid:16) (cid:17) ln 1− C t t + + 1 1 =ρ s ln 1− C t t +Λ ln 1− C t t lnC t+1 −E t lnC t+1 wecaneasilyderive ∂X t c +1 = C t+1 S t+1[ρ +Λ(cid:48)(s)εc ] (III.3) ∂Xc CS s t t+1 t t t ∂X t c +1 =−(1−S ) C t+1 S t+1[ρ +Λ(cid:48)(s)εc ] (III.4) ∂C t CS s t t+1 t t t andwecanverifythatMc = C1−γS Wc inequation(I.2). Thus,wecanplugequations(III.3)and(III.4)inexprest t t t sion(III.2)and,since∂Uint./∂C =C−γFc,weuseequation(I.1)todeduce t t t t ∂xc Λ(s ) (E −E )Mc t =1− t−1 × t t−1 t (III.5) ∂c exp(−s)−1 Mc t t t Itfollowsthatconsumptionhabitsmovestrictlypositivelywithconsumptioninthesteadystate,∂xc/∂c=1. 40

III.2. Homeconsumptionhabits Analogouslyfortheinternalhomeconsumptionhabit, ∂U t =χH−γFh ∂H t t t Fh =Z−γ (cid:104) 1−E (cid:88)∞ βj (cid:16)H t+j Z t+j (cid:17)−γ ∂X t h +j (cid:105) t t t HZ ∂H j=0 t t t =Z−γ (cid:104) 1− ∂X t h(cid:105) −βE (cid:16)H t+1 (cid:17)−γ(cid:104)∂X t h +1 + ∂X t h +1 ∂X t h(cid:105) Wh t ∂H t H ∂H ∂Xh ∂H t+1 t t t t t with Wh =Z−γ+βE (cid:16)H t+1 (cid:17)−γ∂X t h +1Wh t t t H ∂Xh t+1 t t Fromthelawofmotion (cid:16) Xh (cid:17) (cid:16) Xh(cid:17) (cid:104) (cid:16) Xh(cid:17)(cid:105)(cid:104) (cid:105) ln 1− H t t + + 1 1 =ρ s ln 1− H t t +Λ l ln 1− H t t ln[Aα t+ / 1 (1−α)(A t+1 −H t+1 )]−E t ln[Aα t+ / 1 (1−α)(A t+1 −H t+1 )] wecaneasilyderive ∂X t h +1 = H t+1 Z t+1[ρ +Λ(cid:48)(s)εc ] ∂Xh HZ s t t+1 t t t ∂X t h +1 =−(1−Z) H t+1 Z t+1[ρ +Λ(cid:48)(s)εc ] ∂H t HZ s t t+1 t t t andwecanverifythatMh =χH1−γZWhinequation(I.4). Thus,since∂Uint./∂H =χH−γFh,weuseequation(I.3)to t t t t t t t t find ∂xh 1−N (1−α)(1+ξ )Λ(s ) (E −E )Mh t =1+ t 2 t−1 × t t−1 t ∂h N exp(−z)−1 Mh t t t t It follows that home consumption habits move strictly positively with home consumption in the steady state, ∂xh/∂h=1. IV. Coefficientofriskaversion WefollowSwanson(2012)andcomputethecoefficientofriskaversionofaconsumerfacedwithamean-zero, variance-σ,state-independentgamble,whichshecanavoidbypayingaone-timefeeµ(σ). Inourcontext,theindirect t utilityfunctionofaconsumerwithgenericbudgetconstraintA t+1 = (1+r t )A t +W t N t +D t −C t ,whereA t isthe household’sbeginning-of-periodasset,is (C −Xc)1−γ−1 χ[A(1−N)−Xh]1−γ−1 V(A t ;ζ t )= max t 1− t γ + t 1 t −γ t +βE t V(A t+1 ;ζ t+1 ) [Ct;Nt]: Ct +A t+1 =(1+rt)A t +WtNt +Dt whereζ denotesallstatevariablesthatdrivestheeconomy. Swanson(2012)showshowinacontextofexpectedutility t thehousehold’scoefficientofabsoluteriskaversiontothegamble,R(A;ζ)≡lim µ(A t;ζt,σ),equals t t σ→0 σ2/2 R(A;ζ)= −E t V 11 (A t+1 ;ζ t+1 ) t t E t V 1 (A t+1 ;ζ t+1 ) 41

IV.1. Externalhabits Bytheoptimalityconditionsandtheenvelopetheorem,thesteady-statecoefficientofabsoluteriskaversionis −V γ r/C 11 = V 1 S 1+χ (cid:0)HZ(cid:1)1−γ CS Relativetothecasewithouthabits(S =Z =1),thecoefficientofriskaversionscalesupdramatically,asS <1. Relativetocaseofanendowementeconomy(χ=0),thecoefficientscalesdownaspeoplecanusethelabormarginto absorbeconomicshocks(Swanson,2012). Moreover,wecanexpressthecoefficientofrelativeriskaversionaseither −V W γ 1 11 = V 1 S 1+χ (cid:0)HZ(cid:1)1−γ CS whereW t = E t ( (cid:80)∞ j=0 M t,t+j C t+j )definesthewealthportfolio. Inourbaselinecalibration,thesteady-statecoefficientof relativeriskaversionis10,areasonableamountthatstandsincontrastwiththevalueof35intheoriginalcalibration byCampbellandCochrane(1999). Themainreasonforsuchalowerriskaversioncoefficientisthefactthatina productioneconomypeoplecanusethelabormargintoabsorbeconomicshocks,whichscalesdownthesteady-state priceofriskγ/S. Importantly,however,thispropertydoesnotcompromisethemodel’sabilitytocapturelargeaverage riskpremia. IV.2. Internalhabits Bytheoptimalityconditionsandtheenvelopetheorem,wehave −V r/C r/C 11 =γ >γ V 1 1+χ (cid:0)HZ(cid:1)1−γ 1+χ (cid:0)H(cid:1)1−γ CS C whichislargerthanthesteady-statecoefficientofabsoluteriskaversioninthemodelwithouthabitsbecauseofthe typicalratioZ/S >1. Fortypicalcalibrations,however,thedifferenceisimmaterialandinanyeventthecoefficientof riskaversionwhenhabitsareinternalissubstantiallysmallerthanthecoefficientunderexternalhabitformation. V. Frisch’selasticityoflaborsupply Theoptimallaborchoice(4)impliestheelasticityofhoursworkedtothewagerate,givenaconstantmarginal utilityofwealth, ∂ln(N t ) (cid:12) (cid:12) (cid:12) = Z t 1−N t (cid:12) ∂ln(W t /P t ) V1,t γ N t Thus,inthesteadystateFrisch’selasticityscalesdownbyafactorZ ∈(0,1)relativetotheno-habitcase,whereas overtimetheelasticitydropsinarecession(Z =S1+ξ2)asifpeoplebecameveryaversetofluctuationsinlaborduring t t adownturn;thispropertyfollowsfromthefactthatinadownturnpeoplebecomeparticularlysensitivetofluctuations inbothmarketandhomeconsumption. Inthetextwealsodefinethequasi-Frisch’selasticityastheelasticityofhoursworkedtothewagerate,givena constantmarginalutilityofwealthandaconstantsurplushome-consumptionratio, ∂ln(N t ) (cid:12) (cid:12) (cid:12) = 11−N t (cid:12) ∂ln(W t /P t ) V1,t,Zt γ N t 42

VI. ReplicatingCampbell-Cochraneinaproductioneconomy InlinewithCampbellandCochrane(1999),considerflexiblepricesandarandom-walkspecification,φ=0,in ourexternal-habitframeworkundermacro-financeseparation,ξ =0. Theabsenceofinvestmentandtherandom-walk specificationoftechnologyreducetheproductioneconomytoaparticularlysimplestructure: ∆c t+1 =µ+σea t+1 γ(1−ρ ) r =−ln(β)+γµ− s t 2 m t+1 =ln(β)−γµ+γ(1−ρ s )sˆ t −x t σea t+1 wherex =γ[1+Λ(sˆ)]. t t TheoutcomeisobservationallyequivalenttothemodelbyCampbellandCochrane. VII. Homeproductionvs. standardleisure Thepresenceofpreferencesthatincludehomeproductionratherthanstandardleisureinducesadifferencebetween thetextbookNewKeynesianmodelinGal´ı(2008)andourspecification,whichreducestoascalefactorinconsumption:   Gal´ı(2008): c t =  a t + (1−(cid:101)βρ )[γ(1 γ − (1 ρ − ) (cid:101) + βρ φ u ) ]+κ(φ −ρ ) u t γ(1− 1 α + )+ ϕ α+ϕ withγ=1 u u y π u   LLV2015: c t =  a t + (1−(cid:101)βρ )[γ(1 γ − (1 ρ − ) (cid:101) + βρ φ u ) ]+κ(φ −ρ ) u t  u u y π u Thisdifferenceisnecessarytoallowtheeconomytobecompatiblewithbalancedgrowthforanyparameterγ>0 inthecasewithhomeproduction. Allfirst-orderdifferencesdissappearunderthechoiceofaunitEIS,γ=1. VIII. Robustnessofresultstonontrivialcapitalaccumulation Nontrivial capital accumulation, flexible prices. The volatility of dividends relative to consumption is increasing intheinvestment-outputratio,whichinturnisincreasingintheadjustmentcostcurvature,ξ . Withflexibleprices, 3 weareunabletoproduceenoughleverageindividendswhilesimultaneouslymatchingthevolatilityofrealprivate nonresidentialfixedinvestment(BEA-NIPA).Thevolatilityofdividendsistoolowforalevelofadjustmentcosts thatproducesarealisticvolatilityininvestmentgrowth,whichisjustanotherwaytostatethequantitypuzzle;wecan generateenoughvolatilityincorporateprofitsonlyifcapitaladjustmentcostsaresufficientlylow(ξ islarge)and 3 thereforeifthespilloverofsurplusconsumptiononquantitiesislarge. Suchalargedeparturefrommacro-finance separationinducesalargefinancialspilloveroninvestmentviatheelasticityofinvestmenttoQ. Moreover,evenifweweretoacceptsuchacounterfactuallylargevolatilityofinvestment,themerepresenceof nontrivialcapitalcannotdeliverthedesiredtermstructureproperties. Addingcapitalwithoutstickypricesaddsastate thatcouldintheoryprovideinsurance,andsoexplainthetermstructureofequitybutonlyforlowadjustmentcosts (largeξ ). However,capitalaccumulationbreaksmacro-financeseparationandalsoaddssurplusconsumptionasastate 3 thatdrivesquantities;foralowdegreeofadjustmentcosts,dividendsbecomemuchriskierandtheinsuranceeffectof capitalisoverwhelmedbytheriskeffectofsurplusconsumption,sothetermstructureofequityslopesupwards. Nontrivialcapitalaccumulation,stickyprices. Thecontinuityofthemodel’ssolutioninξ impliesthatwhenadjustment 3 costs are high the documented properties of the term structures of equity and interest rates are robust. We match a volatility of annual investment growth 3 times as large as that of consumption growth for a coefficient ξ = .3. 3 FigureH.8plotstheequilibriumtermstructures. Wecanproducealargervolatilityininvestmentforloweradjustment costsbutthepositivespilloverofsurplusconsumptiononinflationbecomesparticularlyimportantforlargervalues ofξ andthetermstructureofnominalbondswouldslopedownwards. Moreover,notethatthenonlinearitiesthat 3 surplusconsumptioninjectsinquantitiesbecomelargerwithξ aswemoveawayfrommacro-financeseparation,so 3 theaccuracyoftheloglinearizedsolutiondeteriorates. 43

0.15 0.15 dividendstrip consumptionstrip realbond nominalbond 0.1 0.1 0.05 0.05 0 0 0 5 10 15 0 5 10 15 n(inyears) n(inyears) (a)Holding-periodriskpremia. (b)Hold-to-maturityriskpremia. FigureH.8: Average term structures of risk premia (in percent per year) with sticky prices and stochastic capitalaccumulation(ξ = .3). 3 IX. Pricingleveredconsumption Analternativedefinitionofmarketequitycanbeformulatedintermsofaclaimtoaleveredversionofconsumption, d =const.+(cid:96)c t t TableI.3showshowthistypeofequitydisplaysadownward-slopingtermstructureofreturnsfor(cid:96)≥γ. Theclaim isparticularlyriskybecauseofitsperfectcorrelationwithconsumption,whilethelong-durationclaimscontainan insurancecomponent,asthecashfloweffectoftheloadingonlong-runtechnologydominatesthediscountrateeffectof long-runtechnology. Cashflow Deterministic Loadingon Loadingon Loadingon Asset process growth u σea σeu t t+1 t+1 Unleveredconsumption ∆c t+1 µ C c ∈ (0,1) 1 > 0 Corporateprofits ∆d t+1 µ > γC c 1 < 0 Leveredconsumption (cid:96)∆c t+1 µ > γC c (cid:96) > 0 TableI.3: Dynamicsofthecashflowprocessesthatdeterminethepricesofthreetypesofequity: aclaimtoaggregate marketconsumption,aclaimtoaggregatecorporateprofits,andaclaimtoleveredmarketconsumption. Thecashflow loadingsarecalculatedforaleverageparameter(cid:96)>γandforanontrivialdegreeofpricerigidities. 44