Precautionary Volatility and Asset Prices

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Precautionary Volatility and Asset Prices Andrew Y. Chen 2014-59 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.

Precautionary Volatility and Asset Prices AndrewY.Chen FederalReserveBoard andrew.y.chen@frb.gov ∗ August12,2014 Abstract Many theories of asset prices assume time-varying uncertainty in order to generate timevarying risk premia. This paper generates time-varying uncertainty endogenously, through precautionary saving dynamics. Precautionary motives prescribe that, in bad times, next period’sconsumptionshouldbeverysensitivetonews. Thistime-varyingsensitivityresultsin time-varyingconsumptionvolatility.Productionmakesthischannelvisible,andexternalhabit preferencesamplifyit. Anestimatedmodelfeaturingthischannelquantitativelyaccountsfor excessreturnanddividendpredictabilityregressions. Italsomatchesthefirsttwomomentsof excessequityreturns,therisk-freerate,andthesecondmomentsofconsumption,output,and investment. ∗ I am indebted to Aubhik Khan, Xiaoji Lin, René Stulz, Julia Thomas, and especially Lu Zhang for their training, advice, and inspiration. I would also like to thank John Cochrane, Urban Jermann, Pok Sang Lam, and Stijn Van Nieuwerburgh for comments which have significantly improved the paper, as well as seminar participants at the University of Arizona, Federal Reserve Board, and the Ohio State University. This paper was previously circulated under the titles “External Habit in a Production Economy,” and “Prudential Uncertainty Causes Time-Varying Risk Premiums.”TheviewsexpressedhereinarethoseoftheauthoranddonotnecessarilyreflectthepositionoftheBoard ofGovernorsoftheFederalReserveortheFederalReserveSystem.

1 Introduction Time-varyinguncertaintyplaysacentralroleintheoriesofaggregateassetprices. Thisfeature is frequently required in order to capture the nature of stock market volatility: stock market fluctuationshavelittlerelationshipwithmovementsinexpectedcashflowsbutarecloselylinked with movements in expected excess returns (LeRoy and Porter (1981), Shiller (1981), Cochrane (2011)). Time-varying uncertainty addresses these facts by generating fluctuations in risk which move asset prices, while leaving average cash flows unchanged. This mechanism exists in a broad array of asset pricing models, though the details of the modeling differ. In long-run risk models, time-varying uncertainty comes from the assumption of time-varying consumption volatility (Bansal and Yaron (2004), Bansal, Kiku, and Yaron (2009)). In ambiguity aversion, it is theassumptionofatime-varyingmagnitudeofambiguity(SbuelzandTrojani(2008),Bianchi,Ilut, andSchneider(2012)).Indisasters,itistheassumedtime-varyingseverityorprobabilityofdisaster (Gabaix(2012),Wachter(2012)).Inexternalhabitmodels,itistheassumptionoftime-varyinghabit sensitivity (Campbell and Cochrane (1999)). These modeling approaches provide coherent and tractable descriptions of stock market volatility, but by assuming time-varying uncertainty, they leaveunansweredthedeeperquestion: whydoesuncertaintyvaryovertime?1 This paper provides an answer to the question. I show that time-varying uncertainty is the naturalresultofafundamentaleconomicphenomenon: thedesireforprecautionarysavings. The mechanism lies in the dynamics of this motive. Precautionary motives intensify in bad times, makinginvestorshunkerdownandcutconsumption.Butinthefollowingmonth,theymayneedto hunkerdownagain. Infact,howmuchtheywillneedtohunkerdownnextmonthdependssharply on economic news. News that the bad times will get even worse would put them in austerity, leadingtoasharpdropinconsumption. Newsthattheeconomywillrecoverrelievesthemoftheir concerns, and they can relax and sharply increase consumption. While they wait for this news, next month’s consumption is very uncertain, risk is high, and asset prices are low. This volatility is absent in good times, when investors have a lot of wealth. High wealth shields investors from precautionaryconcernsandallowsthemtocontinuewiththeirplans,regardlessofeconomicnews. Iformalizethis“precautionaryvolatility”storyintwoways. Thefirstisasimpleconsumptionsavings model which allows for proofs and propositions. The model features a precautionary savings motive, that is, the agent desires savings as protection against uncertainty. This motive leads to a concave consumption function (Carroll and Kimball (1996)), as depicted in Figure 1. Concave consumption means that in bad times, consumption is very sensitive to shocks (dashed 1Early versions of these models often do not feature time-varying volatility (e.g. Bansal and Yaron (2004) model I,Rietz(1988),Barro(2006),Constantinides(1990)),thoughlaterversionsaddthisfeatureinordertoaddresstimevaryingriskpremia. 2

lines), while in good times, there is little action (dotted lines). Countercyclical sensitivity then leads to time-varying consumption volatility, even in the presence of homoskedastic shocks. I alsoexamineanamplificationmechanism:externalhabitpreferencesàlaCampbellandCochrane (1999). Investors with external habit judge their consumption relative to an aggregate reference level. Thisperspectivemakesthemfeelpoor,amplifyingprecautionarymotivesandtheconcavity of the consumption function (dash-dot line). These results are presented in a specific context for clarity, though the mechanism applies in a wide variety of settings. The model features a shock to wealth, though the underlying cause of the shock may be a shift in technology, a shift in monetary policy, or even an uncertainty shock in the sense of Bloom (2009). And though the model’s precautionary motive comes from prudent preferences (Kimball (1990)), financial constraintswouldproducesimilardynamics(Carroll(2001)). Additionally,analogouseffectsexist forfirmpoliciesunderfinancialconstraints(i.e. Almeida,Campello,andWeisbach(2004)). Figure1:PrecautionarySavingsandTime-VaryingVolatility. Thebulkofthepaperisdevotedtothesecondformalization: anestimatedgeneralequilibrium model. The general equilibrium model is a real business cycle model with capital adjustment costs and external habit preferences. Fluctuations originate from a standard AR1 productivity shock. Despite the lack of a heteroskedastic driving process, the model produces time-varying consumptionvolatilityandcapturesthenatureofstockmarketvolatility: theprice-dividendratio has little forecasting power for cash flow growth but is a good predictor of excess returns. This result is due to two key features: real investment and external habit. Real investment means that precautionary savings motives are reflected in aggregates, and thus the economy exhibits the precautionary volatility mechanism described above. External habit plays two roles. Habit amplifiesprecautionaryvolatility,andalsogivesinvestorsstrongconcernsaboutriskwhichallow 3

themodeltoaddressaggregateassetprices. The model also provides a unified description of aggregate asset prices and business cycles. Themodelcapturesalonglistofassetpricefacts: itmatchesthemean,volatility,andpersistence of excess returns, the mean, volatility, and persistence of the risk-free rate, the volatility and persistence of the price-dividend ratio, as well as the previously mentioned excess return and dividend predictability regressions using the price-dividend ratio. And these data-like asset price dynamics originate from a data-like business cycle: the model matches the volatilities of output, consumption,andinvestment,aswellastheirautocorrelationsandcross-correlations. The preferences build off of the external habit preferences of Campbell and Cochrane (1999) but feature a critical deviation. While Campbell and Cochrane assume a time-varying “habit sensitivity,” I make this preference parameter a constant. This changes the key economics. Constantsensitivitymeansthatassetpricesinmymodelaredrivenbytime-varyingconsumption volatility, rather than a time-varying sensitivity of preferences. Indeed, the model’s time-varying consumption volatility quantitatively mimics the Campbell-Cochrane habit sensitivity function. Thus, the model can be considered a way of endogenizing the Campbell-Cochrane mechanism. This description conflicts with the common wisdom that time-varying risk aversion, not timevarying volatility of habit, drives the external habit model. While the common wisdom provides an intuitive description of asset prices, it is not an entirely accurate description of the Campbell- Cochrane model. Indeed, Campbell and Cochrane (1999) explicitly state that time-varying sensitivityisrequiredfortime-varyingriskpremia(p. 211). I provide empirical support for the precautionary volatility mechanism. GARCH and GJR- GARCH estimates of U.S. data show only mild evidence of time-varying consumption volatility, andthemodelisconsistentwiththisfact. Themechanismpredictsthatlowassetpricesarelinked to high consumption volatility, and regressions of consumption volatility on the price-dividend ratioshowthatthispredictionisborneoutbythedata,consistentwithpreviouswork(Kandeland Stambaugh(1990),Lettau,Ludvigson,andWachter(2008)). Additionally,regressionsonsimulated data show that the model does a good job of describing the magnitude of this relationship. The model is able to generate significant time-varying risk premia out of mild time-variation in consumption volatility for two reasons. The first is that habit preferences make investors very sensitivetochangesinconsumptionvolatility.Asaresult,asmallchangeinconsumptionvolatility leads to a large change in risk premia. The second is that consumption volatility is persistent in boththemodelanddata.Thispersistencemakesitdifficulttodetecttime-varyingvolatilityinonly 50yearsofquarterlydata. This paper belongs to the literature which studies the macroeconomic effects of uncertainty fluctuations. Uncertainty has been linked to a broad set of aggregate economic phenomena, 4

including business cycles (Bloom (2009), Bloom et al. (2012), Gourio (2012)), aggregate stock prices (Kandel and Stambaugh (1990), Bansal and Yaron (2004), Lettau, Ludvigson, and Wachter (2008), Gabaix (2012)), and exchange rates (Lustig et al. (2011), Farhi and Gabaix (2013)). The bulkofthesepaperstakeuncertaintyfluctuationsasgivenandtheninvestigateitsconsequences. This paper differs by proposing a theory of the origins of uncertainty fluctuations. A handful of papers do provide an origin theory. Uncertainty may rise in bad times because low production makes it hard to learn about economic fundamentals (Van Nieuwerburgh and Veldkamp (2006), Fajgelbaum, Schaal, and Taschereau-Dumouchel (2012)), because bad times are a good time for experimentation (Pastor and Veronesi (2011), Bachmann and Moscarini (2011)), or because bad shocksleadtodoubtsaboutthetruemodeloftheeconomy(OrlikandVeldkamp(2013)).Adistinct channelarisingfromdisaggregatedvolatilityisproposedinCarvalhoandGabaix(2013)andKoren andTenreyro(2013)whoshowthatchangesintheabilitytodiversifyleadtochangesinvolatility. Theprecautionaryvolatilitychannelcomplementsthesetheories. Indeed,sincethevastmajority ofmodelscontainprecautionarysavingsmotives(CarrollandKimball(1996)),thischannelisactive inmanyoftheabovepapers. Thequantitativemodelbelongstotheliteratureonassetpricingmodelswithhabitformation. This literature shows that that habit formation can account for a wide variety of asset price facts,includingaggregatestockmarkets(CampbellandCochrane(1999)),termstructure(Wachter (2006),Bekaert,Engstrom,andGrenadier(2010)),foreignexchange(Verdelhan(2010)),andoption markets(BekaertandEngstrom(2010)).Theaforementionedpapersareallendowmenteconomies andsocannotaddressoutputandinvestment. Incontrast,thispapershowsthathabitformation can be successfully extended into a production economy and account for asset prices as well as quantity fluctuations. Previous habit-production papers consider a short-lived, internal habit (Jermann(1998)andBoldrin,Christiano,andFisher(2001)). Incontrast,Iconsideraslow-moving, externalhabit,inthestyleofCampbellandCochrane(1999). Withoutthismodification,themodel wouldproduceacounterfactuallyvolatilerisk-freerate.LettauandUhlig(2000)doconsideraslowmoving,externalhabitinaproductioneconomy,buttheyassumecostlessadjustmentofthecapital stock. Incontrast,Imodelconvexcapitaladjustmentcosts. Withoutsuchcosts,RBCmodelsimply acounterfactuallysmoothTobin’sQ(Boldrin,Christiano,andFisher(1999)). Thepaperproceedsasfollows.Section2examinesasimpleconsumption-savingsmodelwhich illustratesthemechanism.Sections3-6estimatesthegeneralequilibriummodelanddemonstrates thatthemechanismcanquantitativelyaccountforalonglistofassetpriceandbusinesscyclefacts. 5

2 A Two-period Model This section illustrates the precautionary volatility mechanism with a two-period model. The modelfavorssimplicityovergenerality.Itisaconsumption-savingsproblemforaprudentinvestor with external habit who faces uncertainty in the return on savings. I prove two propositions and twocorollaries. Proposition1andCorollary1showthatconsumptionvolatilityiscountercyclical. Proposition 2 and Corollary 2 show that habit amplifies this countercyclicality. Proofs are found in the Appendix. The consumption-savings problem can be formalized as a general equilibrium model with a linear technology (à la Constantinides (1990)), but the notation is simpler as a consumption-savingsproblem. Consideratwo-period-livedinvestorwithanexternalhabitstockH >0,periodutility (C−H)1−γ−1 u(C−H)= (1−γ) and γ > 0. These preferences display precautionary savings motives because they are “prudent” in the sense of Kimball (1990). An alternative approach of modeling precautionary savings motives through financial constraints would produce similar effects. Habit is there to illustrate anamplificationmechanism. Forsimplicitytheinvestorhasnotimepreference. Atdate0,theinvestorhaswealthW . Nothingoccursatdate0,butthedatehelpstoserveasa 0 referencepoint. Throughoutthissection,Idescribeavariableas‘countercyclical’ifitisnegatively related to W . At date 1, the investor has a habit level of H >0, and he receives a wealth shock 0 1 ∆W ,makinghiswealthW =W +∆W . HeconsumesC ,andsavestherestofhiswealthW −C 1 1 0 1 1 1 1 inariskyasset. Atdate2,theinvestorhasahabitlevelofH >0. ThereturnontheriskyassetR is 2 2 realized,andheconsumeshisremainingwealthR (W −C ). 2 1 1 The model comes down to finding the investor’s optimal consumption at date 1. His optimal consumptionpolicysolves C (W )=argmaxu(C −H )+(cid:69) [u(C −H )] (1) 1 1 1 1 1 2 2 {C1} s.t. C =R (W −C ) 2 2 1 1 TakingaTaylorofexpansionofC (W )aroundW ,thevolatilityofC isapproximately 1 1 0 1 σ [C (W )]≈σ [C (W )+C (cid:48) (W )∆W ] 0 1 1 0 1 0 1 0 1 =|C (cid:48) (W )|σ (∆W ) (2) 1 0 0 1 6

That is, consumption volatility is proportional to the marginal propensity to consume (MPC). Intuitively,theMPCcapturestheresponsivenessofconsumptiontoshocks.SothehighertheMPC, themoreresponsive,andthusthemorevolatile,isconsumption. Fortheremainderofthissection, Iassumethat∆W issmallenoughsothatequation(2)isagoodapproximation. 1 Unfortunately, even in a two-period consumption savings problem, the presence of precautionarysavingsmotivestypicallyprecludesclosed-formsolutions(Carroll(2001)).However, using the methods from Carroll and Kimball (1996), I can still prove that the solution exhibits countercyclicalvolatilityandthathabitamplifiesthiscyclicality.2 Proposition1. Thedate1MPCisdecreasinginwealth,thatis,C (cid:48)(cid:48) (W )<0. 1 1 The essence of the proof is that, as long as the date 2 habit is positive, the model falls into thebroadclassofmodelsforwhichconsumptionisstrictlyconcave(CarrollandKimball(1996)). CarrollandKimballdonotshowthestrictconcavityresultforthismodel,butIshowinAppendix A.1 that their proofs can be extended to include this setting. The proof is rather technical, but, intuitively, the investor saves for precautionary reasons in the presence of uncertainty. As the agentbecomeswealthier,theuncertaintybecomeslessrelevant,andthismotivedeclines,creating convexsavingsandconcaveconsumption. Corollary1. ThevolatilityofC iscountercyclical,thatis, ∂ σ [C (W )]<0. 1 ∂W0 0 1 1 TheproofisshortandillustratestheroleoftheMPC,soIstateithere. Proof. Since the MPC is positive and decreasing in wealth, the absolute value of the MPC is decreasing in wealth. Since the volatility of C is proportional to the absolute value of the MPC 1 (equation(2)),thevolatilityofC isalsodecreasinginwealth. 1 Proposition 1 and its corollary capture “precautionary volatility,” that is, the mechanism by which precautionary motives generate countercyclical volatility. Countercyclical consumption volatilitystemsfromtheeffectofthedate1wealthshockontheinvestor’sdesireforprecautionary savings at date 1. A positive shock weakens this desire, decreasing savings and boosting consumption,whileanegativeshockencourageshimtohunkerdown,withoppositeeffects. From the perspective of date 0, this uncertainty in the need for precautionary savings at date 1 causes additionalconsumptionvolatility. Sincetheneedforprecautionarysavingsintensifiesatlowlevels ofwealth,theeffectonconsumptionvolatilityiscountercyclical. Mathematically, the precautionary volatility mechanism is manifested as a strictly concave consumptionfunction. Strictlyconcaveconsumptionisanimplicationofabroadclassofmodels 2IamgratefultoPok-SangLamforteachingmehisversionoftheCarrollandKimball(1996)proof. 7

(Carroll and Kimball (1996)). Indeed, Posch (2011) shows that a standard real business cycle model generates time-varying risk premia. However, the magnitude in a standard model is too small to account for the data. The following proposition and corollary show that habit amplifies precautionaryvolatility. Proposition2. Thedate1habitincreasesthesensitivityoftheMPCtowealth,thatis, ∂ C (cid:48)(cid:48) (W )<0 ∂H 1 1 1 To understand this result, note that an investor with high habit has become accustomed to a high standard of living. He judges consumption not by its absolute level, but by how much it exceeds this standard. As a result, he has a much stronger precautionary savings motive than a investor who is accustomed to living in poverty. The mechanism driving Proposition 1 becomes stronger, and the MPC becomes even more sensitive to wealth. Mathematically, an external and additive habit acts as a reduction in income, which reduces wealth and amplifies precautionary savingseffects. Corollary2. Asthedate1habitincreases,thevolatilityofC becomesmorecountercyclical,thatis, 1 foranysetofinitialwealth(cid:87) , 0 ∂ (cid:189) (cid:190) max σ [C (W )]− min σ [C (W )] >0 ∂H 1 W0 ∈(cid:87) 0 0 1 1 W0 ∈(cid:87) 0 0 1 1 One can think of the set of initial wealth (cid:87) as the range of the business cycle states in the 0 economy. Corollary2statesthat,acrossthisbusinesscycle,therangeofconsumptionvolatilitiesis increasinginhabit. TheproofsimplyappliesthelinkbetweentheMPCandconsumptionvolatility (equation (2)) to Proposition 2. Because habit makes the MPC more countercyclical, habit also makesconsumptionvolatilitymorecountercyclical. Figures 2 and 3 illustrate the propositions of this section. The figures show results from numerical solutions of the two-period model for different levels of habit. In all solutions, I set H = H . This specification focuses on how habit “concavifies” the consumption function by 1 2 eliminatingintertemporalsubstitutioneffectswhichcomplicatethepicture. Figure 2 plots date 1 consumption and MPC as a function of wealth. The left panel shows that, as long as habit is positive, consumption is strictly concave. The right panel shows that this concavityisreflectedinanMPCwhichdecreasesinwealth(Proposition1).Figure2alsoshowsthat habit intensifies this relationship (Proposition 2). As the line gets lighter, habit increases, and the 8

Figure2: Two-periodModel: Date1ConsumptionandMPC.C (W ) is date 1 consumption as a 1 1 (cid:48) functionofdate1wealth.C (W )isitsderivative(theMPC).Thedifferentlinesshowconsumption 1 1 forvariouslevelsofhabitH . 1 Figure3: Two-periodModel: Date0ConsumptionVolatility. σ [C (W )]isthevolatilityofdate1 0 1 1 consumption, given that date 0 wealth isW . The different lines show consumption volatility for 0 variouslevelsofhabitH . 1 slopeoftheMPCgetssteeper. Intuitively, habitmakestheinvestorfeelpoorerandintensifieshis precautionarysavingsmotives. Figure3showshowtheseconsumptionpoliciesarereflectedinconsumptionvolatility. Aslong as habit is positive, consumption volatility decreases in wealth (Corollary 1). This is the essence of precautionary volatility. Because precautionary savings motives prescribe a countercyclical MPC,andbecauseconsumptionvolatilityisproportionaltotheMPC(equation(2)),consumption 9

volatility is countercyclical. The figure also shows that habit amplifies this countercyclicality (Corollary2).Asthelinegetslighter,habitincreases,andtherangethatisspannedbyconsumption volatilityincreases. Theresultsofthissectionmaycomeasasurprisesincemanymodelsinthefinanceliterature produce consumption policies which are linear in wealth. These linear consumption policies are the result of the fact that much of this literature is set in continuous time (Sundaresan (1989), Constantinides (1990)) or relies on log-linear approximations (Campbell (1994), Lettau (2003), Kaltenbrunner and Lochstoer (2010)). Continuous time allows the investor to make an infinite number of trades within any trading period. This instantaneous trading allows for consumption policieswhich,ifappliedinadiscretetimesetting,wouldimplyapositiveprobabilitythatmarginal utility becomes infinite (Brandt (2009)). It is exactly fear of hitting this condition of infinite marginalutilitywhichseemstodriveconcaveconsumptionbehavior(Attanasio(1999)).Log-linear approximations,ontheotherhand,implicitlyassumethatconsumptionpoliciesare(log)linearin state variables. The non-linearities introduced by combining power utility functions with linear budgetconstraintsareimportantforgeneratingnon-linearconsumption(Posch(2011)). The results may also appear to conflict with the common intuition that habit encourages smoothconsumption. Thisintuitionisstraightforwardininternalhabitmodels,whereanincrease in consumption has a direct effect of lowering utility in later periods (Sundaresan (1989)). In an external habit model, however, the investor by assumption does not take into account this indirect effect. Prices may encourage smooth consumption via general equilibrium effects, but thehabititselfactsverysimilarlytoareductioninincomewhichincreasesthemarginalpropensity toconsume(CarrollandKimball(1996)). 3 The General Equilibrium Production Model The previous section illustrates the intuition behind the precautionary volatility mechanism, but a quantitative evaluation requires a richer model. This section presents a richer model. The remainderofthepaperexaminesthepredictionsofthequantitativemodel. The model sticks as close to the standard real business cycle model as possible. The only features are external habit preferences and quadratic capital adjustment costs. External habit servesasanamplificationmechanism. CapitaladjustmentcostsarerequiredforavolatileTobin’s Q. There is a representative household and representative firm. Time is discrete, the horizon is infinite,andmarketsarecomplete. Fortheremainderofthepaper,Idenotelogvariableswithlowercase,i.e. z ≡logZ . t t 10

3.1 RepresentativeHousehold Thereisacontinuumofidenticalhouseholdswithexternalhabitpreferences. Eachhousehold choosesitsassetholdingstomaximize (cid:69) (cid:189) (cid:88) ∞ βt (C t −H t )1−γ−1 (cid:190) (3) 0 1−γ t=0 Where H , the level of habit, is taken as external to the household. Habit is important for both t amplifying the precautionary volatility mechanism and for providing strong concerns about risk which have the chance at being consistent with asset price facts. For simplicity, the household doesnotvalueleisureandisendowedwithaunitoflabor. Ispecifytheevolutionofhabitusingsurplusconsumption,ratherthanthelevelofhabititself. Thatis,definesurplusconsumptionas Cˆ −H Sˆ ≡ t t (4) t Cˆ t where the hats denote aggregates. This approach leads to a simple and symmetric stochastic discount factor and allows for some clean asset pricing analysis. It also eases comparison with the existing literature on external habit (Campbell and Cochrane (1999), Wachter (2006), among others). Surplusconsumptionevolvesaccordingtoanautoregressiveprocess sˆ t+1 =(1−ρ s )s¯+ρ s sˆ t +λ(cˆ t+1 −cˆ t ) (5) where λ is a constant. Endowment economy external habit models specify λ as a decreasing function of surplus consumption. This assumption builds in a countercyclical volatility of marginal utility which is essential for addressing excess market volatility. The model does not require this assumed countercyclicality because, as we will see, production and precautionary motivesendogenouslygeneratecountercyclicalconsumptionvolatility. Forcomparabilitywiththe literature,IfixλattheCampbellandCochrane(1999)steadystatevalue 1 λ= −1 (6) S¯ This modification causes the issue that habit may move negatively with consumption. However, thathabitisstillageometricaverageofthehistoryofconsumption(AppendixA.4). Thenon-linearhabitspecificationisnotcriticaltotheeconomicmechanism,butthepersistent 11

AR1specificationis. Persistenthabitisimportantforcapturingthepersistenceinaggregateasset prices such as the price-dividend ratio. This strong persistence contrasts with previous habit models in production economies (Jermann (1998) and Boldrin, Christiano, and Fisher (2001)) whichassumethathabitdependsonlyonlastquarter’sconsumption. The external nature of habit and the surplus consumption specification produce a simple stochasticdiscountfactor M t,t+1 =β (cid:181) Cˆ t+1 Sˆ t+1 (cid:182)−γ (7) Cˆ Sˆ t t (cid:179) (cid:180)−γ The stochastic discount factor has the traditional consumption growth term Cˆ t+1 , but habit (cid:179) (cid:180)−γ Cˆ t addsasurplusconsumptionterm Sˆ t+1 . ThisSDFisthesameasthatinCampbellandCochrane Sˆ t (1999)butisverydifferentfromtheEpstein-Zin-habitmodelofDew-Becker(2011). Dew-Becker’s SDF also has two components, but the two components are those generated by Epstein and Zin (1989)preferences: aconsumptiongrowthcomponentandatermrelatedtothereturnonwealth. In Dew-Becker’s model, habit enters the SDF by affecting the curvature (power parameter) of the return on wealth component. As a result, the mechanisms driving Dew-Becker’s model are very differentfromthoseinthispaper. 3.2 Representativefirm The production side of the economy is standard. The only feature is quadratic capital adjustment costs. There is a unit measure of identical firms which produce consumption using capitalK andlaborN . Productionisgivenby t t Π(K ,Z ,N )≡Z K α N1−α (8) t t t t t t whereαiscapital’sshareofoutputandZ isproductivity. t TheCobb-Douglasspecificationofproduction,combinedwiththefactthatthehouseholddoes notvalueleisureimpliesthatwagesareequaltothemarginalproductoflabor W =(1−α)Z Kˆα (9) t t t ProductivityZ followsthestandardAR(1)process t z t+1 =(1−ρ)z¯+ρ z z t +σ z (cid:178) z,t+1 (10) 12

Where(cid:178) z,t+1 isastandardnormali.i.d. shock. Thisistheonlysourceofuncertaintyinthemodel, andbyassumptionitishomoskedastic. z¯ ischosensothatthenon-stochasticsteadystatecapital stock is approximately one. The choice of z¯ does not have a material effect on the results, but keepingthesteadystatecapitalstocknearonehelpswiththecomputation. Capitalaccumulatesaccordingtotheusualcapitalaccumulationrule K t+1 =I t +(1−δ)K t (11) andfirmsfaceaconvexcapitaladjustmentcost φ(cid:181) I (cid:182)2 Φ(I ,K )= t −δ K (12) t t t 2 K t Thisformulationpunishesthefirmfordeviatingfromthenon-stochasticsteadystateinvestment rate of δ. I assume that the adjustment costs are a pure loss. They do not represent payments to labor. Adjustment costs are included because production economies produce a counterfactually smoothTobin’sQunlessoneincludesaninvestmentfriction. Thefirm’sobjectiveisstandard (cid:189)∞ (cid:190) (cid:88) max (cid:69) M [Π(K ,Z ,N )−W N −Φ(I ,K )−I ] (13) 0 0,t t t t t t t t t {It,Kt+1,Nt} t=0 It chooses investment, capital, and labor to maximize future dividends, discounted with the household’sstochasticdiscountfactor. 3.3 RecursiveCompetitiveEquilibrium Marketclearingisstandard: Cˆ +Iˆ =Z Kˆα−Φ(Iˆ,Kˆ ) (14) t t t t t t Due to the consumption externality, the welfare theorems do not hold and the equilibrium cannotbeeasilydescribedbyasocialplanner’sproblem. Thus,Idefineequilibriumcompetitively. TheaggregatestatevariablesareaggregatecapitalKˆ,surplusconsumptionSˆ,andproductivityZ. Definition. Equilibrium is a firm decision rule for investment I(K;Kˆ,Sˆ,Z), cum-dividend value function V(K;Kˆ,Sˆ,Z), law of motion for aggregate consumption Cˆ(Kˆ,Sˆ,Z), and a law of motion ofaggregatecapitalΓ(Kˆ,Sˆ,Z)suchthat 13

(i) Firmsoptimize: I(K;Kˆ,Sˆ,Z)andV(K;Kˆ,Sˆ,Z)solve (cid:189) V(K;Kˆ,Sˆ,Z)= max Π(K,Z,N)−W(Kˆ,Sˆ,Z)N−Φ(I,K)−I {N,I,K(cid:48)} (cid:90) ∞ (cid:190) + dF((cid:178)(cid:48) )M(Kˆ,Sˆ,Z;Z (cid:48) )V(K (cid:48) ;Kˆ(cid:48) ,S (cid:48) ,Z (cid:48) ) (15) −∞ subjecttocapitalaccumulation(11), competitivewages(9), theproductivityprocess(10), the habitprocess(5),theSDFisequaltothehousehold’sIMRS(7),Kˆ(cid:48)=Γ(Kˆ,Sˆ,Z),andwhereF((cid:178)(cid:48) ) isthestandardnormalCDF. (ii) Marketsclearandaggregatesareconsistentwithindividualbehavior: Cˆ(Kˆ(cid:48) ,Sˆ(cid:48) ,Z (cid:48) )=Π(Kˆ,Z,1)−Φ(I(Kˆ;Kˆ,Sˆ,Z),Kˆ)−I(Kˆ;Kˆ,Sˆ,Z) (16) Γ(Kˆ,Sˆ,Z)=(1−δ)Kˆ+I(Kˆ;Kˆ,Sˆ,Z) (17) Sincehouseholdsandfirmsareidentical,inequilibrium,Kˆ =K,Sˆ=S andCˆ=C. Thus,inwhat follows,Idropthehats. 3.4 SolutionMethod Capturingtheeconomicsofthismodelrequiresaglobalandnon-linearsolutionmethod. This is important for both external habit (Campbell and Cochrane (1999)) and precautionary savings (Carroll(1997)). TheresultisachallengingnumericalproblemwhichIdiscussinthissection. 3.4.1 ProjectionMethod I solve the model using a projection method (Judd (1992)). Specifically, I represent the law of motionforcapitalusingcubicsplinesandthenuseBroyden’smethod(aquasi-Newtonalgorithm) to find cubic spline coefficients which satisfy the firm’s Euler equation. The solution program makesextensiveuseoftheMirandaandFackler(2001)CompEcontoolbox. Theprojectionmethodisimportantfortworeasons. Thefirstisthat,duetotheconsumption externality, the welfare theorems do not hold and the model cannot be easily solved by value functioniteration.Projectionavoidsthisproblembyworkingwithequilibriumconditionsfromthe recursivecompetitiveequilibrium. Thesecondisthatprojectionproducesaglobalandnon-linear solution. These properties are important for asset pricing models in general (Cochrane (2008b)), but are particularly important for capturing the mechanisms in this model. The precautionary savings channel which drives countercyclical risk premia are related to the third derivative of 14

the utility function and are not captured by traditional linearizations. Moreover, precautionary effects are particularly pronounced when the investor is threatened with infinite marginal utility (Attanasio(1999)),andfullycapturingthiseffectrequiresaglobalsolution. 3.4.2 HomotopyMethod Projectionmethodsrequireagoodinitialguessofthesplinecoefficients. Thereisnoguarantee thatageneralnon-linearequationsolverwillconvergeandthehighdimensionalityoftheproblem tendstomakethesolversunstable. Thestandardapproachistousetherealbusinesscyclemodel asaninitialguess. Unfortunately,withexternalhabitpreferences,therealbusinesscyclemodelis apoorinitialguess. Toovercomethisissue, Iuseahomotopymethod. Specifically, Imodifythefirm’sproblemso thatitdiscountsfutureprofitsusingtheSDF (cid:183) (cid:48)(cid:181) (cid:48)(cid:182)χ(cid:184)γ C S M (cid:48)=β C S Note that χ=0 corresponds to a model with no habit, and χ=1 corresponds to the full model. I begin by solving the model for χ=0, and then slowly increase χ, using the coefficients from the previousχastheinitialguessforthecurrentχ. Thishomotopyalgorithmisverycomputationallyintensive.Toaidinthespeedofcomputation, IdiscretizetheproductivityprocessusingtheRouwenhorstmethod. 3.4.3 SolvingfortheEvolutionofSurplusConsumption Anotherdifficultissuewhichariseswithexternalhabitpreferencesisthattheyresultinastate variable which is not predetermined and is endogenous. Surplus consumption tomorrow is not knowntodayanditdependsonconsumptiontomorrow,whichisendogenous. Thismakessolving forthesurplusconsumptionprocessratherdifficult. Toseethisclearly,ithelpstowritetheevolutionofsurplusconsumption(5)asfunctionsofstate variables logS (cid:48)=(1−ρ )s¯+ρ logS+λ[c(K (cid:48) ,S (cid:48) ,Z )−c(K,S,Z )] (18) s s j j (cid:48) (cid:48) Where Z is tomorrow’s discrete productivity state. Since K is predetermined, S is a function of j (cid:48) (cid:48) four variables s (K,S,Z ,Z ). Note that surplus consumption tomorrow S appears on both sides i j of this equation. Thus determining surplus consumption tomorrow requires solving this non- 15

linear equation. I once again use Broyden’s method to solve this equation. This calculation is verycomputationallyintensivesinceitmustbedoneateverycollocationnodeforeverypotential productivity shock within every iteration of the big Broyden’s method which is solving for the coefficientsofthelawofmotionofcapital. Forfurtherdetailsaboutthesolutionmethod,seeAppendixA.5. 4 Simulated Method of Moments Estimation Inordertocaptureeconomicmagnitudes,Ineedtogivetheparametersnumericalvalues.Most papersinthisliteratureuseaninformalcalibrationprocedure.Ichooseamorerigorousestimation method. Iestimatethemodelusingsimulatedmethodofmoments(SMM),thesimulatedversion of Hansen (1982)’s generalized method of moments (GMM). SMM differs from traditional GMM inthatitusessimulationtocomputemodelmomentsratherthanclosed-formexpressions,which may be unavailable for moments of interest. Duffie and Singleton (1993) derive adjustments to GMMformulasandadditionalregularityconditionsrequiredbytheuseofsimulation. 4.1 Data The estimation uses post-war (1948-2011) data from the CRSP, BEA, and BLS. Some authors argue for using the longest sample available when evaluating consumption-based asset pricing models (e.g. Bansal, Kiku, and Yaron (2009)). However, in a production economy I must also address the data on aggregate investment and output. The nature of this data is significantly affected by using pre-war data. For example, the correlation between investment growth and outputgrowthhaveahighcorrelationof0.73inthepost-warsample, buthaveamildcorrelation of0.23forthesample1929-2011. Thisconsiderationleadsmetotargetonlypost-wardata. All variables are real and per capita. Consumption is real per capita non-durable goods and services consumption. This measure excludes volatile consumer durables such as automobiles, which produce a smooth consumption flow over the life of the durable. Aggregate equity is represented by the CRSP value-weighted index, adjusted for inflation with the consumer price index. Dividendsarecalculatedwiththeassumptionthatalldividendsarereinvestedinthestock market. This method of aggregation preserves the Campbell and Shiller (1988) present value identity. The risk-free rate is a forecast of the inflation-adjusted 90-day T-bill return using the previousyear’sinflationrateandthenominal90-dayyield(followingBeelerandCampbell(2009)). Noadjustmentsaremadeforfinancialleverage. Furtherdetailsregardingthedataarefoundinthe AppendixA.2. 16

4.2 EstimationMethodandPredefinedParameters BecauseSMMiscomputationallyintensive,Isetthemoretraditionalparametersoutsideofthe estimation. I set the depreciation rate δ=0.02 to match the mean, growth-adjusted, investment rate. I set the capital share parameter α = 0.35 to match the capital share of output implied by constant returns to scale. The persistence of productivity shocks ρ = 0.979 matches the z persistence of the Solow residual with a fixed labor input. In an external habit model, the utility curvature γ and the steady state surplus consumption ratio S¯ jointly control risk aversion. As a result, it is difficult to identify these parameters separately. For ease of comparison with the literature,γ=2issetatCampbellandCochrane(1999)’svalue.3 The remaining five parameters are estimated by SMM. Explicitly, let θ represent the five parameters as a vector and Mˆ∗ represent a vector of target data moments. SMM estimates the parametersbyminimizingthedistancebetweendatamomentsandsimulatedmoments: θˆ=argmin (cid:163) Mˆ∗−M(θ) (cid:164)(cid:48) W (cid:163) Mˆ∗−M(θ) (cid:164) (19) θ∈Θ whereM(θ)isthemodel-simulatedcounterparttoMˆ∗ andW isaweightingmatrixchosenbythe econometrician. Iuseone-stage,exactly-identifiedGMM,thatis,Itargetfivedatamomentswhich are economically informative about the five parameter values. This approach has the advantage of transparency. Seeing that the empirical targets are equal to the model moments immediately verifies that the minimization algorithm is successful. Under exact identification, a consistent estimatorfortheasymptoticvarianceoftheestimatedparametervaluesis (cid:112) (cid:181) (cid:182) (cid:112) 1 V(cid:100)ar[ T(θˆ−θ 0 )]≡ 1+ [DM(θˆ) (cid:48) ] −1V(cid:100)ar( TMˆ∗ )[DM(θˆ)] −1 (20) S whereS isthenumberofsimulationsusedtocalculatemodelmoments,DM(θ)isthederivativeof (cid:112) thesimulatedmomentswithrespecttoθandV(cid:100)ar( TMˆ∗ )isaconsistentestimateoftheasymptotic varianceoftheestimateddatamoments. Intuitively, wehaveagoodestimateifthemomentsare informative about the parameter values (DM(θˆ) is large), or if we have a precise estimate of the (cid:112) moments(V(cid:100)ar( TMˆ∗ )issmall).4 I optimize using Levenberg-Marquardt, a variant of Newton’s method. I choose this method ratherthanthemorecommonly-usedsimulatedannealingalgorithmfortworeasons. Thefirstis 3CampbellandCochrane(1999)chooseγtofittheSharperatioandS¯toproduceaconstantrisk-freerate,butthat cannotbedoneherebecauseIsetλasaconstantinordereliminateexogenoustime-varyingvolatility. 4ThemomentsIusearenotmomentsinthestrictsense(i.e.theSharperatio),andIestimatetransformationsofthe parametersinordertoavoidcornersolutions. Thisleadstoslightlymorecomplicatedformulasthanthosepresented here.ThesedetailscanbefoundintheAppendixA.3. 17

that the moment function does not display an extreme number of local minima, which is where simulated annealing has an advantage. With a relatively smooth objective function, a method whichusesderivativeinformationismuchmoreefficient. AnotheradvantageofLMisrobustness. Simulated annealing tends to be sensitive to the choice of the optimization parameters (Press, Teukolsky, Vetterling, and Flannery (1992)). Additional details regarding the estimation method canbefoundinAppendixA.3. 4.3 ParameterEstimatesandMomentTargets Table1summarizestheestimation. Itshowsestimatedparametervalues,standarderrors,and targeted moments. For convenience, the predefined parameters are also shown, with standard errors omitted. The model is quarterly and all parameter values are quarterly. To eliminate seasonality in dividends, both the U.S. and simulated data are aggregated to the annual level. All momentsareannual. Preference parameters are identified with asset prices. Because time-preference β is reflected in the risk-free rate, I choose the mean 90-Day T-bill return as a target moment. The resulting β=0.970isratherlowbecausethemodelfeaturesanon-trivialprecautionarysavingsmotive,and alowlevelofpatiencehelpscounteractthatmotive. SteadystatesurplusconsumptionS¯controls the magnitude of habit, and, in effect, the degree of risk aversion in the model. I thus choose the mean Sharpe ratio of the CRSP index as a target moment. The resulting value S¯= 0.063 is close tothevaluesusedintheexternalhabitliterature(CampbellandCochrane(1999),Wachter(2006), SantosandVeronesi(2010)). Thepersistenceofsurplusconsumptionρ hasastrongeffectonthe s volatilityofthemarketreturn,andsoIchoosethevolatilityoftheexcessreturnontheCRSPindex asatarget. Theresultingvalueofρ =0.963indicatesaverypersistenthabitprocess,whichisalso s consistentwithvaluesusedintheexternalhabitliterature. Technological parameters are identified with moments of the real economy. The volatility of productivity σ = 0.014 targets the volatility of HP-filtered log GDP of 0.014. Since the data is z annual, I use the annual smoothing parameter of 6.25 advocated by Ravn and Uhlig (2002). The quadratic adjustment cost parameter φ = 75.00 targets the relative volatility of consumption to GDP (also HP-filtered). This estimated value results in mean adjustment costs as a percentage of outputoflessthat1%. 18

Table1:ParameterEstimatesandMomentTargets Themodelisquarterly,andallparametervaluesarequarterly.Empiricalfiguresareannual.Standarderrors are Newey-West with 10 lags and shown only for estimated parameters. The sample period is 1948-2011. Consumptionisnon-durablegoodsandservicesconsumption. GDPandconsumptionareloggedandHPfilteredwithasmoothingparameterof6.25.FurtherdetailsarefoundinAppendicesA.2andA.3. Parameter Estimate SE EmpiricalTarget Preferences β TimePreference 0.971 (0.009) Mean90-DayT-billReturn(%) 0.98 ρ PersistenceofSurplus 0.963 (0.006) VolatilityofExcessReturn(%) 16.07 s Consumption ofCRSPIndex S¯ SteadyStateSurplus 0.063 (0.010) MeanSharpeRatioof 0.48 Consumption CRSPIndex γ UtilityCurvature 2 (ChosentoMatch Campbell-Cochrane(1999) Technology φ AdjustmentCost 75.00 (10.65) RelativeVolatilityof 0.47 Parameter(Quadratic) ConsumptiontoGDP σ VolatilityofProductivity 0.014 (0.001) VolatilityofGDP 0.015 z ρ PersistenceofProductivity 0.979 PersistenceofSolowResidual 0.979 z α CapitalShare 0.35 Capital’sShareofGDP 0.35 δ DepreciationRate 0.02 MeanInvestmentRate 0.02 (Growth-Adjusted) 5 Matching Asset Price and Business Cycle Moments This section contains the main quantitative results. It shows that precautionary volatility channel,whenamplifiedbyexternalhabit,providesaquantitativedescriptionoftime-varyingrisk premia. Themodelalsomatchesnumerousothermoments,andprovidesaunifieddescriptionof asset prices and business cycles. I make no adjustments to account for un-modeled leverage or payoutpolicy. Thepriceofequityissimplythepresentvalueofdividendsfromtherepresentative firm,discountedwiththehousehold’sSDF. For quick navigation, this section does not discuss mechanisms. Readers focused on mechanismsmaywanttoskiptoSection6. 19

5.1 UnconditionalAssetPriceMoments Table 2 shows that the model produces a nice fit for all of the basic moments of asset prices. Asintendedbytheestimation, ithitsthreeofthesemomentsexactly: themeanrisk-freerate, the SharpeRatio,andthevolatilityofexcessreturns. Producingthislargeequityvolatilityisadifficult taskinproductioneconomies(Gourio(2010),KaltenbrunnerandLochstoer(2010),Croce(2010)). Table2:UnconditionalAssetPriceMoments Figuresareannual. Noadjustmentsaremadetoaccountforfinancialleverage. Themodelcolumnsshow meansandpercentilesacrosssimulationsofthesamelengthastheempiricalsample. r, p, andd arethe logsofreturns,prices,anddividendsfromtheCRSPvalue-weightedindex.Capitallettersshowlevelsrather thanlogs.r isaforecastoftheex-postrealreturnon90-dayT-billsfollowingBeelerandCampbell(2009).(cid:69), f σ,andAC1representthesamplemean,standarddeviationandfirst-orderautocorrelation. Furtherdetails arefoundinAppendixA.2. Data Model 1948-2011 mean 5% 50% 95% IdentifyingMoments (cid:69)(r ) (%) 0.98 0.98 -2.72 0.28 6.83 f (cid:69)(R−R )/σ(R) 0.48 0.48 0.29 0.48 0.68 f σ(r −r ) (%) 16.07 16.07 10.77 15.71 22.53 f UntargetedMoments (cid:69)(r −r ) (%) 6.47 6.77 5.03 6.96 7.88 f AC1(r −r ) -0.03 -0.07 -0.28 -0.07 0.16 f σ(r ) (%) 2.24 2.96 0.55 1.88 8.60 f AC1(r ) 0.56 0.88 0.69 0.91 0.99 f (cid:69)(p−d) 3.42 2.67 2.09 2.66 3.27 σ(p−d) 0.43 0.46 0.26 0.44 0.76 AC1(p−d) 0.95 0.90 0.79 0.91 0.96 (cid:69)(r −r ) (%) 2.03 0.93 2.10 2.92 10yr f σ(r −r ) (%) 1.42 0.62 1.37 2.44 10yr f AC1(r −r ) 0.85 0.70 0.86 0.95 10yr f The model also captures many asset market features beyond those used in the identification. Themodelproducesarisk-freeratevolatilityof2.96%whichisclosetothedataestimateof2.24%. This moment is difficult to match in habit models, which tend to produce an excessively volatile 20

risk-freerate(i.e. 11%inJermann(1998)and25%inBoldrin, Christiano, andFisher(1999)). This lowrisk-freeratevolatilityisreflectedinareasonabletermpremium. Themodelproducesamean excessreturnon10-yearbondsof2%,indicatingthatthemodel’sequitypremiumisdistinctfrom thetermpremium,asinthedata.Thevolatilityofthelogprice-dividendratiois0.46,whichisclose tothedatavalueof0.43.Thismomenthasbeendifficulttocaptureeveninendowmenteconomies (Bansal,Gallant,andTauchen(2007),Bansal,Kiku,andYaron(2009)). Additionally, the model generates data-like persistence. As in the data, the excess market returnismildlynegativelyautocorrelated,andtherisk-freerateandprice-dividendratioarehighly positivelyautocorrelated. Themodelsomewhatoverstatesthepersistenceoftherisk-freerate,but thisdeviationcontainssomeuncertaintybecauserisk-freerateisnotdirectlyobservable. 5.2 ExcessReturnandDividendPredictability The previous section shows that the model generates a volatile, data-like stock market. This large volatility does not mean that the model captures the nature of stock market fluctuations, however. The data show that these fluctuations are driven by time-varying risk premia: stock price movements have little relationship with movements in expected cash flows, but are closely linkedtomovementsinexpectedexcessreturns(LeRoyandPorter(1981),Shiller(1981),Campbell andShiller(1988), Cochrane(1992)). Inthissection, Iconsiderdividendsascashflows, butother notionsofcashflow(i.e. consumptionorprofits)producesimilarresults. Table 3 shows that the model captures key elements of time-varying risk premia. Panel A shows regressions of future dividend growth on the log price-dividend ratio at various horizons. The model predicts no relationship between the price-dividend ratio and future dividend growth at the one-year horizon, as is seen in the data. As in the data, this predictability increases with the horizon, but the coefficients remain small and statistically insignificant. The predictability is somewhatoverstatedatthe5-yearhorizon,butSection6.6showsthatthisissuecanbefixedwith aricherdividendprocesswhichaccountsfordeviationsbetweenneoclassicalfirmcashflowsand dividends. Iftheprice-dividendratiodoesnotforecastdividends,thenitmustforecastreturns(Campbell andShiller(1988),Cochrane(2008a)).PanelBshowstheabilityofthemodeltocapturethisflipside of dividend predictability. It shows regressions of future excess returns on the log price-dividend ratio. Regression coefficients, standard errors, and R2’s, are close to the data for all forecasting horizons.Thecoefficientsontheprice-dividendratioarenegative,andtheyarebotheconomically andstatisticallysignificant.Attheone-yearhorizon,themodelexactlymatchesthedatacoefficient 21

Table3:PredictingDividendsandExcessReturnswiththePrice-DividendRatio Figuresareannual.Themodelcolumnsshowmeansandpercentilesacrosssimulationsofthesamelengthas theempiricalsample.r ,p ,andd arethelog-returns,prices,anddividendsfromtheCRSPvalue-weighted t t t index. r is a forecast of the ex-post real return on 90-day T-bills following Beeler and Campbell (2009). f,t StandarderrorsareNewey-Westwith2(L−1)lags.FurtherdetailsarefoundinAppendixA.2. PanelA:Predictingdividendgrowth (cid:80)L j=1 ∆d t+j =α+β(p t −d t )+(cid:178) t+L Data Model L 1948-2011 Mean 5% 50% 95% 1 -0.03 -0.00 -0.04 -0.00 0.03 βˆ 3 0.01 0.07 0.00 0.06 0.18 5 0.03 0.13 0.02 0.12 0.30 1 0.03 0.03 0.01 0.02 0.04 SE(βˆ) 3 0.07 0.05 0.03 0.05 0.09 5 0.09 0.07 0.03 0.06 0.11 1 0.01 0.00 0.00 0.00 0.02 R2 3 0.00 0.04 0.00 0.03 0.13 5 0.00 0.10 0.01 0.09 0.26 PanelB:Predictingexcessreturns (cid:80)L j=1 r t+j −r f,t+j =α+β(p t −d t )+(cid:178) t+L Data Model L 1948-2010 Mean 5% 50% 95% 1 -0.12 -0.12 -0.23 -0.11 -0.04 βˆ 3 -0.27 -0.30 -0.55 -0.29 -0.12 5 -0.40 -0.44 -0.78 -0.44 -0.17 1 0.05 0.05 0.02 0.05 0.08 SE(βˆ) 3 0.08 0.09 0.04 0.08 0.13 5 0.12 0.10 0.05 0.10 0.17 1 0.09 0.10 0.02 0.09 0.19 R2 3 0.19 0.26 0.07 0.26 0.45 5 0.26 0.37 0.11 0.38 0.59 22

of-0.12. Tounderstandthiseconomically,recallthatthevolatilityofthelogprice-dividendratiois roughly0.40inboththemodelandthedata. Thismeansthataonestandarddeviationriseinthe price-dividendratiopredictsahuge4% reductionintheequity premiumover thenextyear. This forecastingpowerincreaseswiththeforecastinghorizon,reachingR2’sofroughly30%atthe5-year horizoninbothmodelanddata. Overall, the model captures both sides of predictability. Asset price fluctuations have little relationship with fluctuations in future dividends, but are tightly linked to fluctuations in future excessreturns. 5.3 UnconditionalBusinessCycleMoments The previous sections show that the model produces a good description of numerous asset price moments. Table 4 shows that these asset price moments come with data-like fluctuations intherealeconomy. Themodelhitsthevolatilityofoutputandrelativevolatilityofconsumption to output, as intended by the estimation. Investment is more volatile than output, as in the data. Like the data, the model moments display strong co-movement between output, consumption, and investment. Output, consumption, and investment are highly persistent, and are nearly as persistent as the data. First-differenced log consumption has a low volatility. In both the model and data this volatility is about 1% per year. Lastly, the table also shows that average adjustment costsaresmallatlessthan1%ofoutput. 6 Inspecting the Mechanism I have shown that the model provides a unified description of numerous asset price and business cycle moments, including time-varying risk premia. The remainder of the paper is devotedtoexplaininghow. Most of the new economics can be seen by examining time-varying risk premia. The model containsthekeyfeaturesofthesimplemodelofSection2,suggestingthatprecautionaryvolatility is the underlying mechanism. However, the model is closely related to Campbell and Cochrane (1999), and so a closer look is informative. Section 6.1 shows that time-varying risk premia come from time-varying consumption volatility. Section 6.2 compares this mechanism with Campbell and Cochrane (1999). Section 6.3 shows that the model’s time-varying consumption volatility is consistentwiththedata. Theotherassetpricingresultsareexaminedlaterinthissection. Thelargeandvolatileequity 23

Table4:BasicBusinessCycleMoments Figuresareannual. Themodelcolumnsshowmeansandpercentilesacrosssimulationsofthesamelength astheempiricalsample. y islogGDP,c islognondurableandservicesconsumption,andi isthelogoffixed investmentplusdurablegoodsplusgovernmentinvestment.Thesubscripthpindicatesthatthemomentis calculatedfromHP-filtereddatawithasmoothingparameterof6.25. ∆indicatesfirst-differences. Further detailsarefoundintheAppendixA.2. Data Model 1948-2011 mean 5% 50% 95% IdentifyingMoments σ(y ) (%) 1.50 1.50 1.11 1.49 1.93 hp σ(c )/σ(y ) 0.47 0.47 0.40 0.47 0.55 hp hp UntargetedMoments σ(i )/σ(y ) 2.46 3.43 3.09 3.44 3.76 hp hp ρ(c ,y ) 0.83 0.99 0.98 0.99 1.00 hp hp ρ(i ,y ) 0.85 1.00 0.99 1.00 1.00 hp hp AC1(y ) 0.12 0.21 0.01 0.21 0.38 hp AC1(c ) 0.32 0.21 0.01 0.22 0.39 hp AC1(i ) 0.27 0.20 0.01 0.21 0.38 hp σ(∆c) (%) 1.11 1.41 0.97 1.38 1.90 (cid:69)(AdjCost/Y) (%) 0.54 0.21 0.49 1.04 (cid:69)(AdjCost/I) (%) 3.21 1.13 2.70 6.88 premiumisanalyzedinSection6.4. Thelowandsmoothrisk-freerateisexaminedinSection6.5. Section 6.6 investigates cash flow dynamics and alternative dividend processes. Section 6.7 ends theinspectionbyexaminingcomparativestatics. 6.1 Time-varyingRiskPremiaandPrecautionaryVolatility Themodeladdressestime-varyingriskpremiathroughtheprecautionaryvolatilitymechanism described in Section 2. To demonstrate this, I’ll work backward, starting with time-varying risk premiaandendingwithprecautionarymotives. Themodelproducesfluctuationsinriskbyproducingtime-varyingconsumptionvolatility.This can be seen in Figure 4, which shows scatterplots of asset prices against consumption volatility 24

produced by model simulations. Consumption volatility varies over time, and both the pricedividendratioandequitypremiumarenicelylinkedtoconsumptionvolatility. Theprice-dividend ratiodeclinesinconsumptionvolatility,whiletheequitypremiumincreases. Figure 4: Time-Varying Consumption Volatility and Asset Prices. The figures show scatterplots frommodelsimulations. Consumptionvolatilityandtheequitypremiumarecalculatedusingthe model’slawsofmotion. Allvaluesareannualized. Thisresulthasaverysimpleintuition.Timesofhighconsumptionvolatilityareriskytimeswith largeriskpremiaandlowassetprices. Butalittleformalismprovidesadditionalinsightandbuilds confidenceinthemechanism. Alog-normalapproximationoftheSDFshowsthattheconditional maximumSharperatiois5 {al m las a s x ets} (cid:189)(cid:69) t (R σ t+ t ( 1 R − t+ R 1 f ) ,t+1 )(cid:190) ≈γ(λ+1)σ t (∆c t+1 ) (21) The conditional maximum Sharpe ratio is the conditional volatility of consumption growth, multipliedbypreferenceparameters:γistheutilitycurvature(3),andλistheconditionalvolatility of the habit process (5). This expression shows that consumption volatility is critical. Since consumptionvolatilityisthesoletermontheRHSwhichcanvaryovertime,itmustbethedriver oftime-varyingSharperatios. 5The log-SDF is m t+1 =logβ−γ∆s t+1 −γ∆c t+1 and the habit process is ∆s t+1 =−(1−ρ s )(s t −s¯(cid:112))+λ∆c t+1 . Plug σ th t e [m ha t+ b 1 i ] t = pr γ o ( c λ e + ss 1 i ) n σ t t o (∆ th c e t+ l 1 o ) g . -SDF,thenassumethattheSDFislog-normal,andwehave σ (cid:69) t t ( ( M M t t + + 1 1 ) ) ≈ eVart[mt+1]−1≈ 25

Time-varyingconsumptionvolatilityinturnistheresultoftheprecautionaryvolatilitychannel described in the simple model (Section 2). To see this, note that the model is driven by a single, homoskedastic productivity shock (equation (10)). Moreover, the conditional volatility of the habit process is assumed to be constant (equation (5)). Thus, there are no exogenous drivers of time-varying volatility. On the other hand, the model features the same preferences as the simple model. These preferences produce precautionary savings motives which lead to strictly concave consumption. This shape means that consumption has time-varying volatility, even thoughtheunderlyingshockishomoskedastic.Externalhabitamplifiesthischannelandproduces thequantitativeresultseeninFigure4. Unlikepreviousexternalhabitmodels,thismodelfeatures real investment, which allows the precautionary volatility channel to be visible in equilibrium quantities. 6.2 EndogenizingtheCampbell-CochraneMechanism Though the model features external habit preferences, the mechanism is distinct from the existing literature. Existing external habit models drive time-varying risk premia through a timevarying preference parameter, i.e. the Campbell and Cochrane (1999) λ(s ) function. My model t drives time-varying risk premia through time-varying consumption volatility. In this section, I show that these two channels are economically distinct, but have similar quantitative properties. Thus, my model can be considered a way of endogenizing the Campbell and Cochrane (1999) mechanism. To compare the channels, it helps to examine the maximum Sharpe ratio from Campbell and Cochrane(1999) {al m las a s x ets} (cid:40) (cid:69) t (R σ t t + ( 1 R − t+ R 1 ) t f +1 ) (cid:41) ≈γ[λ(s t )+1]σ(∆c t+1 ) (22) where (cid:112) (cid:40) 1 1−2(s −s¯)−1 if s ≤s¯+1(1−S¯2) λ(s )= S¯ t t 2 (23) t 0 if s ≥s¯+1(1−S¯2) t 2 The volatility of consumption growth σ(∆c t+1 ) is assumed to be constant. So the only way the modelcangeneratetime-varyingSharperatiosisthroughthepreferenceparameterλ(s ). Thisis t a direct contrast to my model, where time-varying Sharpe ratios must come from consumption volatility (equation (21)). λ(s ) is the “habit sensitivity function,” that is, it controls how sensitive t habit is to shocks. Existing models assume that this sensitivity declines in surplus consumption 26

(equation (23)). This assumption builds in a time-varying volatility of marginal utility and thus a time-varyingmaximumSharperatio. The above analysis also shows that the common intuition regarding the external habit mechanism is misplaced. The common intuition is that time-varying risk premia are driven by time-varying risk aversion. But time-varying risk aversion is not sufficient for generating timevarying Sharpe ratios. To see this, note that a constant λ(s ) still results in time-varying surplus t consumptionS (seeequation(5)). SinceriskaversionisapproximatelyγS −1,aconstantλ(s )still t t t produces time-varying risk aversion. But equation (22) shows that, even though risk aversion is time-varying, a constant λ(s ) implies a constant maximum Sharpe ratio. It is time-varying habit t sensitivityλ(s ),nottime-varyingriskaversion,whichdrivesthetraditionalexternalhabitmodel. t ThecriticalroleofthesensitivityfunctionisacknowledgedinCampbellandCochrane(1999). This is notto say thatthe commonintuitioncannotbe modeled. Dew-Becker (2011)usesEpstein-Zin preferencestocreateahabitmodelwhichcapturesthecommonintuition. The fact that, in traditional external habit models, time-varying habit sensitivity drives timevarying risk leaves open many questions. How do we interpret a time-varying habit sensitivity? Wheredoesthistime-varyingsensitivitycomefrom? Thismodelcanhelpaddressthesequestions since it replaces the time-varying habit sensitivity function with time-varying consumption volatility, which has a very simple interpretation. And time-varying consumption volatility originatesfromtheprecautionarysavingsmotives,thatis,badtimesmakeinvestorsunsureofhow muchtheycanconsumenextperiod. Indeed, the model’s time-varying consumption volatility does a good job of mimicking the habit sensitivity function of Campbell and Cochrane. To demonstrate this I construct an ‘implied consumption volatility’ from the Campbell-Cochrane sensitivity function by equating the two expressions for the maximum Sharpe ratio (equations (21) and (22)). I call this implied consumptionvolatility because one canreplace the Campbell-Cochrane sensitivity functionwith thisimpliedconsumptionvolatilityandgetsimilarquantitativeresults. Explicitly, (cid:181)λ(s )+1 (cid:182) ImpliedConsumptionVolatility= λ t +1 σ(∆c t+1 ) (24) Impliedconsumptionvolatilityissimplythehabitsensitivityλ(s ),rescaledbyitssteadystatevalue t λ,andmultipliedbytheunconditionalvolatilityσ(∆c t+1 ). Figure5comparestheCampbell-Cochraneimpliedconsumptionvolatilitytotheconsumption volatility of my model. It shows scatterplots of the volatilities against surplus consumption. Both are countercyclical, that is, they decline in surplus consumption. Moreover, for much of the plot, implied consumption volatility runs right through the middle of the cloud of dots representing 27

consumption volatility. The two channels are both qualitatively and quantitatively similar. Thus, this production economy could be considered a method for endogenizing the external habit mechanism. Figure5:ConsumptionVolatilityandtheCampbell-Cochraneλ(s ).Dotsrepresentconsumption t volatility from model simulations. The dashed line represents Campbell-Cochrane λ(s ) implied t consumptionvolatility(equation(24)). Figuresareannualized. 6.3 EmpiricalSupportforthePrecautionaryVolatilityMechanism I have shown that the precautionary volatility channel generates time-varying consumption volatility, and that this time-varying consumption volatility produces time-varying risk premia. This section provides direct empirical support for this mechanism. I show that the mechanism generatesmildtime-variationinconsumptionvolatility,asisseeninthedata. Ialsoshowthat,in thedata,consumptionvolatilityhasanegativerelationshipwithassetprices. Themodelcaptures boththesignandthemagnitudeofthisrelationship. Iuseanumberofmeasuresofconditionalconsumptionvolatility.Allmeasuresareconstructed by first fitting an AR(1) model to log consumption growth to remove an expected growth 28

component: ∆c t+1 =b 0 +b 1 ∆c t +(cid:178) c,t (25) ItheneitherestimateGARCH-typemodelsontheresidual(cid:178) c,t+1 orusethemeanabsoluteresidual as a non-parametric measure of conditional volatility. I use quarterly data because it is difficult to detect time-varying volatility with the post-war annual sample of only 50 observations. This procedure follows Bansal, Khatchatrian, and Yaron (2005), Bansal, Kiku, and Yaron (2009) and BeelerandCampbell(2009). Table5comparesGARCHestimatesofconsumptionvolatilityfromthedataandmodel. Panel AshowsresultsfromaGARCH(1,1)process σ2 =ω +ω (cid:178)2 +ω σ2 (26) c,t+1 0 1 c,t 2 c,t Thedatacolumnsshowmodesttime-variationinconsumptionvolatility. ConsistentwithBansal, Khatchatrian, and Yaron (2005), the ARCH parameter ω just makes it to the 95% significance 1 level if one uses non-robust standard errors. However, using standard errors which are robust to the assumption of non-normal shocks (Bollerslev and Wooldridge (1992)), one cannot reject the hypothesisofnotime-varyingvolatility. Thismodesttime-variationinvolatilityisconsistentwith themodelcolumns. Themodel’smeanestimateoftheARCHparameterisclosetozero,and5%of simulationsproduceavalueof0. SupposingthatARCHeffectsdoexist,thedataproduceGARCH parameter estimates of ωˆ =0.79 indicating that consumption volatility is persistent. The model 2 capturesthispersistence, producingaGARCHparameterof0.76. Indeed, thishighpersistenceof consumptionvolatilitymaybethereasonwhytime-varyingvolatilityishardtodetectinthedata. PanelBcomparesestimatesofaGJR-GARCH(1,1,1)model(Glosten, Jagannathan, andRunkle (1993)) σ2 =ω +ω (cid:178)2 +ω σ2 +ω I((cid:178) >0)(cid:178)2 c,t+1 0 1 c,t 2 c,t 3 c,t c,t GJR-GARCHintroducesanadditionaltermwhichallowsnegativeshockstohavealargereffecton volatility, as predicted by the analysis of Section 2. Though the data show significant sampling uncertainty, the point estimates are consistent with the intuition. The asymmetric GJR-GARCH parameter ωˆ is much larger than the symmetric ARCH ωˆ in both the model and the data. In 3 1 terms of magnitudes, the model’s parameter estimates are all smaller than the data’s. Consistent withPanelA,themodelreproducesthemodesttime-variationinconsumptionvolatility,asseenin 29

Table5:GARCHEstimatesofTime-VaryingConsumptionVolatility Dataandfiguresarequarterly. ThistableshowsmeasuresGARCHestimatesoftheresidualfromanAR(1) modelofconsumptiongrowth ∆c t+1 =b 0 +b 1 ∆c t +(cid:178) c,t+1 GARCH estimation is done by quasi maximum likelihood. Robust standard errors use the Bollerslev- Wooldridgemethod.Themodelcolumnsshowmeansandpercentilesacrosssimulationsofthesamelength astheempiricalsample.FurtherdetailsarefoundinAppendexA.2. PanelA:GARCH σ2 =ω +ω (cid:178)2 +ω σ2 c,t+1 0 1 c,t 2 c,t Data: 1948Q1-2011Q4 Model Estimate SE RobustSE mean 5% 50% 95% ω 1.39E-06 8.51E-07 1.27E-06 9.69E-06 2.94E-07 2.65E-06 4.42E-05 0 ω 0.14 0.08 0.13 0.03 0.00 0.03 0.09 1 ω 0.79 0.09 0.15 0.76 0.00 0.91 0.99 2 PanelB:GJR-GARCH σ2 =ω +ω (cid:178)2 +ω σ2 +ω I((cid:178) >0)(cid:178)2 c,t+1 0 1 c,t 2 c,t 3 c,t c,t Data: 1948Q1-2011Q4 Model Estimate SE RobustSE mean 5% 50% 95% ω 1.97E-06 1.25E-06 2.29E-06 1.09E-05 3.19E-07 2.00E-06 4.70E-05 0 ω 0.07 0.07 0.08 0.01 0.00 0.00 0.06 1 ω 0.76 0.10 0.19 0.73 0.00 0.93 0.98 2 ω 0.14 0.11 0.20 0.05 0.00 0.06 0.16 3 thedata. There are two reasons why the model can generate substantial fluctuations in risk premia out ofmodestGARCHeffects. Thefirstisthathabitpreferencesamplifyfluctuationsinconsumption volatility. This can be seen in the surplus consumption process (5). Surplus consumption is basicallyanAR1process,wheretheshocksareconsumptiongrowthandtheconditionalvolatility λisafreeparameter. Thecalibrationchoosesalargeλ=1/0.06−1≈15whichprovidessubstantial amplification. Thus small movements in the volatility of consumption growth produce large fluctuationsinthevolatilityofmarginalutility.Thesecondisthatconsumptionvolatilitygenerated by the model is very persistent. This persistent consumption volatility makes it hard to measure time-varyingvolatilityusingonly50yearsofquarterlydata. A key prediction of the precautionary volatility channel is that consumption volatility is high 30

when asset prices are low. Table 6 shows that this prediction is supported by the data. The table shows regressions of various proxies for consumption volatility on the price-dividend ratio. The datacolumnshowsthat,usingallproxies,consumptionvolatilityandtheprice-dividendratioare negatively related. A high price-dividend ratio indicates a safe time of low volatility. The model replicates this pattern. The large standard errors in Table 5 suggest that comparing magnitudes shouldbedonewithcaution,butwiththatinmind,themodelcoefficientsareofsimilarmagnitude tothosefromthedata. 6.4 TheLargeandVolatileEquityPremium Thehighvolatilityoftheequitypremiumcomesfromhighcapitaladjustmentcostsandthelow elasticity of intertemporal substitution (EIS) of habit preferences. High capital adjustment costs meanthatproductivityshocksareabsorbedbyassetpricesratherthaninvestment. Thischannel is not new (Jermann (1998), Kogan (2004), Jermann (2010), Kogan and Papanikolaou (2012)) and so I provide only a brief discussion. The role of the low EIS is that a low EIS pins down high adjustmentcostsviaestimationandgeneralequilibriumeffects. TheimportanceoftheEISisless wellunderstoodandsoIfocusonit. The link between equity volatility and adjustment costs can be seen in the investment return - stock return identity (Cochrane (1991), Restoy and Rockinger (1994)). Since the model has a homogenousproductiontechnology,thisidentitymeanswecanexpressthestockreturnas φ R t,t+1 = α(Y t+1 /K t+1 )+(1+φ( 1 I t + +1 φ / ( K I t+ / 1 K )) ) (1−δ)+ 2 (I t+1 /K t+1 )2 (27) t t The stock return is related to the marginal product of capital α(Y t+1 /K t+1 ), the investment rate I t+1 /K t+1 , and the adjustment cost parameter φ. This equality holds state-by-state, whichmeans thatthevolatilitiesofthetwosidesareequal. This identity shows the difficulty of matching equity volatility in a production economy. The marginalproductofcapitalandtheinvestmentratehaveaverylowvolatility,lessthan2%peryear. Onthe other hand, the equity returnhas ahuge volatility of about20% per year. The only way to reconcile these two sides, then, is with capital adjustment costs. Adjustments costs increase the curvatureoftheRHS.Jensen’sinequalitythenimpliesthatadjustmentcostsincreasethevolatility of the investment return. The large discrepancy between the equity volatility and the volatility of theinvestmentrateimplyaverylargeadjustmentcostparameter. The role of the EIS comes through general equilibrium effects and standard empirical 31

Table6:RegressionsofConsumptionVolatilityonthePrice-DividendRatio Thistableshowsregressionsoftheform cvol =α+β(p −d )+(cid:178) t t t t wherecvol isameasureofconditionalconsumptionvolatility, p isthelogequityprice, andd isthelog t t t dividend. Togeneratecvol ,firstanAR(1)model(25)isrunonlogconsumptiongrowth. PanelAestimates t eitheraGARCH(1,1)orGJR-GARCH(1,1,1)modelontheresiduals. PanelBusesanon-parametricmeasure: (cid:179) (cid:180) cvol t (L)≡log (cid:80)L j=1 |(cid:178) c,t+j | ,where(cid:178) c,t+j istheresidualfromtheAR(1)model.Consumptiondataisquarterly andprice-dividendratiodataisannual,whichresultsinsomeabuseofnotation. Themodelcolumnsshow means and percentiles across simulations of the same length as the empirical sample. Further details are foundinAppendixA.2. PanelA:GARCHestimatesofconsumptionvolatility Data Model 1948Q1-2011Q4 mean 5% 50% 95% βˆ -0.43 -0.19 -0.50 -0.15 0.00 GARCH SE(βˆ) 0.14 0.05 0.00 0.05 0.13 R2 0.37 0.23 -0.01 0.18 0.61 βˆ -0.43 -0.32 -0.75 -0.32 0.00 GJR-GARCH SE(βˆ) 0.16 0.07 0.00 0.06 0.16 R2 0.31 0.33 -0.01 0.34 0.77 PanelB:Non-parametricestimatesofconsumptionvolatility (cid:179) (cid:180) cvol t (L)≡log (cid:80)L j=1 |(cid:178) c,t+j | Data Model L(qtr) 1948Q1-2011Q4 mean 5% 50% 95% 4 -0.66 -0.21 -0.01 -0.22 -0.66 βˆ 12 -0.59 -0.25 0.00 -0.24 -0.68 20 -0.53 -0.25 0.00 -0.24 -0.61 4 0.16 0.25 0.12 0.23 0.45 SE(βˆ) 12 0.12 0.19 0.08 0.17 0.36 20 0.09 0.14 0.06 0.13 0.29 4 0.22 0.02 0.00 0.01 0.06 R2 12 0.36 0.04 0.00 0.02 0.13 20 0.44 0.07 0.00 0.04 0.23 32

restrictions. General equilibrium means that both the EIS and capital adjustment costs affect consumptionvolatility. TheEISchannelworksthroughtheSDF.Sincethefirmusestheinvestor’s IMRS as an SDF, the firm is rewarded for providing consumption which matches the investor’s preferences. A low EIS results in a strong incentive to produce smooth dividends, and through marketclearing,smoothconsumption. Capitaladjustmentcostsalsoaffectconsumptionvolatility through market clearing. Large adjustment costs encourage the firm to keep investment smooth in the face of shocks. Since shocks must be absorbed by either investment or dividends, this encouragesvolatiledividends,and,throughmarketclearing,volatileconsumption. Since both the EIS and capital adjustment cost affect consumption volatility, the ubiquitous requirementthatassetpricingmodelsshouldfitconsumptiondatameansthatthesetwoelements must match each other. A low EIS requires high capital adjustment costs, and vice versa. In this model,thelargeSharperatiopinsdownalowEIS.GiventhelowEIS,thevolatilityofconsumption pins down a high adjustment cost. Note that the adjustment cost is not pinned down by equity volatility,andsoequityvolatilityformsanoveridentifiedrestrictionwhichissatisfiedbythemodel. This EIS-adjustment cost link exists in long-run risk and disaster models, but it tends to hurt ratherthanhelptheirassetpricingresults. Thevastmajorityoflong-runriskanddisasterpapers do not calibrate the EIS. However, the EIS must be larger than one in both models in order to match certain qualitative facts, and the common wisdom is that the larger the EIS, the better the asset pricing results. For long-run risk models, this is required to qualitatively match the fact that consumption volatility and the price-dividend ratio are negatively related (Bansal and Yaron (2004)). Indisastermodels,thisisrequiredtomatchtheintuitivenotionthatariseindisasterrisk lowers investment and increases excess returns (Gourio (2010)). This large EIS then implies low adjustment costs and counterfactually low equity volatility (Kaltenbrunner and Lochstoer (2010), Gourio(2010)). Given that equity volatility is high, the large equity premium is the consequence of a large Sharpe ratio. The large Sharpe ratio is simply due to the fact that habit preferences offer an additional degree of freedom for the modeler. Habit results in an additional term in the SDF (7). Themodelercanchoosethevolatilityofthistermtobehighbyadjustingthepreferenceparameter λinequation(5). Economically, λ can be interpreted as the ‘moodiness’ of the economy. The surplus consumption process says that if consumption growth goes up by 1%, surplus consumption gets boosted by λ%. The calibration chooses λ = 1/0.065−1 ∼ 15, meaning that changes in ‘mood’ are responsible for the vast majority of changes in marginal utility. Checking this magnitude by introspection is, of course, a dangerous activity. But Section 5 shows that it is consistent with numerousoveridentifyingrestrictionsregardingassetprices. 33

6.5 TheLowandSmoothRisk-FreeRate The low risk-free rate is simply the result of time preference. Time preference is effectively a free parameter which allows me to hit the low risk-free rate in the data. Economically speaking, this results in an intuitive time preference of β4 ≈ 0.90 annually, that is, consumption one year fromtodayisworth90%ofconsumptiontoday. The smooth risk-free rate comes from an interplay of intertemporal substitution and precautionary savings effects. This channel has a simple intuition. In bad times, people want to borrowinordertoconsumetoday. Butinbadtimes, theeconomyisparticularlyvolatile, andthe desireforprecautionarysavingspreventsthemfromborrowing. Thischannelissimilartothatof CampbellandCochrane(1999),butwithendogenousconsumptionvolatilitydrivingprecautionary savingseffectsratherthantheexogenoushabitsensitivityfunction. Indeed,Section6.1showsthat the model does a good job of mimicking the Campbell-Cochrane sensitivity function and so the smoothnessoftherisk-freerateshouldnotbesurprising. These intuitions are fleshed out by examining the log-normal approximation of the risk-free rate6 r t f +1 ≈−logβ+γ(λ+1)(cid:69) t (∆c t+1 )−γ(1−ρ s )(s t −s¯) (28) (cid:124) (cid:123)(cid:122) (cid:125) IntertemporalSubstitution −(1/2)γ2(λ+1)2Var t (∆c t+1 ) (cid:124) (cid:123)(cid:122) (cid:125) PrecautionarySaving The1sttermreflectstimepreference. Timepreferencehasasmallimpactonotherunconditional moments, andsoitisessentiallyafreeparameterwhichonecanusetofitthelowmeanrisk-free rateinthedata. The 2nd and 3rd terms are due to intertemporal substitution and tend to create excessive volatility in habit models (Jermann (1998), Boldrin, Christiano, and Fisher (2001)). They reflect theFriedman(1957)permanentincomehypothesis. Inbadtimes, investorswanttoborrowfrom thefutureinordertoconsumetoday. Thismotivepushesdownthepriceontherisk-freebondand pushes up the risk-free rate, leading to a countercyclical effect on a risk-free rate. Habit models implyaverystrongconsumptionsmoothingmotivewhichmakesthischannelveryvolatile. This model has the volatile intertemporal substitution effect typical of habit models, but it 6Ifm t+1 isnormal, r t f +1 =−log (cid:163)(cid:69) t (emt+1) (cid:164)=−(cid:69) t (m t+1 )− 2 1 Var t (m t+1 ) ThenjustpluginthelogSDFm t+1 =logβ−γ∆s t+1 −γ∆c t+1 andhabitprocess∆s t+1 =−(1−ρ s )(s t −s¯)+λ∆c t+1 . 34

counters this effect with a precautionary savings effect. The precautionary savings effect runs throughthe4thterm,whichisdecreasingintheconditionalvolatilityofconsumption. Intuitively, inbadtimes,highconsumptionvolatilitycreatesadesireforsavings.Investorsbuybonds,pushing up the price and down the risk-free rate. This channel creates a procyclical effect on the risk-free rate,whichhelpscounteractthecountercyclicaleffectoftheintertemporalsubstitutionchannel. Thepreviousdiscussionshowsthat,qualitatively,precautionarysavingseffectshelpcounteract intertemporal substitution effects. Whether the quantitative effect is enough generate a smooth risk-free rate is another question. Figure 6 examines the quantitative efffect. It plots the riskfree rate decomposition (28) against surplus consumption for various levels of productivity. The solid red lines represent intertemporal substitution effects. The dashed blue lines represent the precautionarysavingseffects. Thetwoeffectsarenearmirror-imagesofeachother,showingthat, quantitatively,thechannelsbalanceeachotherquitenicely. Figure 6: Decomposition of the Risk-Free Rate. ‘Intertemporal Substitution’ represents the 2nd and 3rd terms of equation (28). ‘Precautionary Saving’ represents the last term. Computed from model’slawsofmotion. Risk-freerateisinannualized%. Capitalinallpanelsisfixedatthemean capitalstock. A nice feature of the smoothing effects in this model is that they arise endogenously. This contrasts with the risk-free rate smoothing effects of Campbell and Cochrane (1999), which are the result of a parameter choice. To see this, it helps to return to approximation of the risk-free rate (28). This expression shows that, if one allows the preference parameter λ to vary over time, then one can control the magnitude of the precautionary savings effect. Indeed, Campbell and Cochrane(1999)dojustthat,andtheychoosethemagnitudeofthechanneltoexactlycancelout the intertemporal substitution effect. This model has no such freedom. The magnitude of this 35

channel comes through the amount of countercyclicality in consumption volatility, which is the result of general equilibrium effects of investors’ preferences for precautionary savings on firms’ productiondecisions. 6.6 CashFlowDynamicsandAlternativeDividendProcesses The model has a very simple notion of dividends. Dividends are just the profits (net of investment) of a representative neoclassical firm (see equation (13)). This formulation has the advantage of transparency, but it abstracts from a number of issues which affect dividends in the data. For example, the model abstracts from debt and payout policy, as well as labor market frictions. The literature has shown that these abstractions lead to some counterfactual behavior ofdividends(Rouwenhorst(1995),Jermann(1998),KaltenbrunnerandLochstoer(2010)),andmy model is not exempted from this issue. Extending the model to incorporate richer financial and production elements would help the model fit dividend dynamics (i.e. Jermann and Quadrini (2012),Kuehn,Petrosky-Nadeau,andZhang(2012)),attheexpenseofcomplexity. Would the main results still hold under a richer and more data-like dividend process? Table 7 suggests that the answer is yes. The table shows basic properties of dividends and excess returns for alternative dividend processes. The alternative processes specify dividends as stochastically related to consumption, which allows one to capture basic empirical dividend moments. This procedure follows the endowment economy literature (Bansal and Yaron (2004), Campbell and Cochrane(1999)).Thetableshowsthat,regardlessofthedividendprocess,themodelstillproduces alargeandvolatileequitypremium. The ‘neoclassical dividend’ column shows results from the baseline model. Consistent with previous studies, this simple technology leads to some counterfactual dividend statistics. Dividendsaretoosmoothcomparedtothedata,anddividendgrowthisnegativelycorrelatedwith consumption, while the data show a mild positive correlation. Despite the negative correlation withconsumptiongrowth,aclaimontheneoclassicaldividendproducesalargeandvolatileequity premium. Onemightthinkthat,duetothisnegativecorrelation,thisclaimwouldserveasahedge forconsumptionriskandofferlowreturns. However, thevastmajorityoftheclaim’svaluecomes from its continuation value, not the next year’s dividends. A negative shock leads the firm to cut investmentandboostdividendstemporarily,butthisnegativeshockisabadsignforalongfuture ofdividends. Thisresultofalargeriskpremiumwithnegativeconsumption-dividendcorrelation isconsistentwithresultsfromprevioushabitmodels(Jermann(1998)). The‘calibrateddividend’columnpresentsresultsfromadividendprocesswhichmatchessome 36

Table7:AlternativeDividendProcesses:BasicStatistics Figures are annual. ‘Neoclassical Dividend’ is a claim on the dividends from the firm (see equation 13). ‘CalibratedDividend’isaclaimonthedividendprocess(29)calibratedtomatchU.S.data. ‘Consumption Claim’isaclaimonconsumption. ∆d isdividendgrowth, ∆c isconsumptiongrowth, andr −r isexcess f returns of the CRSP index. Dividend growth is computed using no reinvestment. Details of the data are foundinAppendixA.2. US Neoclassical Calibrated Consumption 1948-2011 Dividend Dividend Claim σ(∆d) (%) 13.11 6.21 13.10 1.20 ρ(∆d,∆c) 0.17 -0.68 0.18 1.00 (cid:69)(r −r ) (%) 6.47 6.77 8.01 6.86 f σ(r −r ) (%) 16.07 16.07 20.69 17.71 f AC1(r −r ) -0.03 -0.07 -0.06 -0.06 f basicdividenddatamoments. Thedividendprocessisasimplefunctionofconsumption d ≡ψc +σ (cid:178) (29) x,t t d d,t whereψandσ areparametersand(cid:178) isastandardnormalshock. Thisapproachcapturessome d d,t key statistical features of dividend growth and follows Campbell and Cochrane (1999) and Bansal and Yaron (2004), among others. Specifically, I choose ψ=2 and σ =0.18 to hit the correlation d betweenconsumptionanddividendgrowth,andthevolatilityofdividendgrowth. Table7showsthecalibrateddividendprocessmatchesthevolatilityofdividendgrowthandits correlation with consumption growth, as intended. It also generates a data-like equity premium. Excess returns on this claim are large on average, volatile, and mildly negatively autocorrelated. This large risk premium exists despite the low correlation between dividend and consumption growth. Once again, the vast majority of this asset’s value comes from its continuation value, not the next year’s dividends. The ‘consumption claim’ column shows further robustness of the large andvolatileequitypremium. Aclaimonconsumptionalsoproducesdata-likestockreturns. Table8showsthatalternativedividendprocesseshelpthemodelbetterfittheevidenceoftimevarying excess returns. It shows regressions of future dividend growth and future excess returns on today’s price-dividend ratio, using alternative dividend processes. As seen in Section 5, the benchmark model (‘neoclassical dividend’ column) captures the lack of dividend predictability at shorter horizons, although the amount of predictability is somewhat overstated at the 5-year 37

Table8:AlternativeDividendProcesses:PredictiveRegressions Figures are annual. ‘Neoclassical Dividend’ is a claim on the dividends from the firm (see equation 13). ‘CalibratedDividend’isaclaimonthedividendprocess(29)calibratedtomatchU.S.data. ‘Consumption Claim’isaclaimonconsumption.∆d isdividendgrowth,p −d isthelogprice-dividendratio,andr−r is t t f excessreturns. StandarderrorsareNewey-Westwith2(L−1)lags. DetailsofthedataarefoundinAppendix A.2. PanelA:Predictingdividendgrowth (cid:80)L j=1 ∆d t+j =α+β(p t −d t )+(cid:178) t+L US Neoclassical Exogenous Consumption L 1948-2011 Dividend Dividend Claim 1 -0.03 -0.00 0.03 -0.04 βˆ 3 0.01 0.07 0.04 -0.02 5 0.03 0.13 0.05 -0.01 1 0.03 0.03 0.04 0.03 SE(βˆ) 3 0.07 0.05 0.06 0.04 5 0.09 0.07 0.06 0.04 1 0.01 0.00 0.01 0.01 R2 3 0.00 0.04 0.02 0.01 5 0.00 0.10 0.02 0.01 PanelB:Predictingexcessreturns (cid:80)L j=1 r t+j −r f,t+j =α+β(p t −d t )+(cid:178) t+L US Neoclassical Exogenous Consumption L 1948-2011 Dividend Dividend Claim 1 -0.12 -0.12 -0.15 -0.15 βˆ 3 -0.27 -0.30 -0.38 -0.38 5 -0.40 -0.44 -0.56 -0.55 1 0.05 0.05 0.07 0.07 SE(βˆ) 3 0.08 0.09 0.12 0.12 5 0.12 0.10 0.15 0.15 1 0.09 0.10 0.08 0.08 R2 3 0.19 0.26 0.20 0.21 5 0.26 0.37 0.29 0.30 38

horizon. The calibrated dividend column shows that this mild overstatement is fixed with a morerealisticdividendprocess. Thecalibrateddividendclaimproducesessentiallyzerodividend predictability,withamaximumR2of0.02atthe5-yearhorizon.Thisresultisanintuitiveextension oftheconsumptionclaimresults.Asseeninthe‘consumptionclaim’column,consumptiongrowth is unpredictable at all horizons. Since the calibrated dividend claim is essentially a levered claim on consumption, it inherits this lack of predictability. This lack of dividend predictability shows upinexcessreturnforecasts. Boththecalibrateddividendclaimandconsumptionclaimproduce stronglypredictableexcessreturns. The apparent lack of cash flow predictability is due to the uninformativeness of asset prices, ratherthanafundamentallackofpredictabilityintheeconomyitself. Thiseffectofconditioning information is seen in Figure 7, which shows plots of expected consumption growth computed usingthemodel’slawsofmotion,thatis,conditioningonallinformationavailableintheeconomy. The left panel shows consumption growth against the price-dividend ratio from the benchmark ‘neoclassical’dividendprocess.Thepanelshowsacloudofconsumptiongrowthobservations,and norelationshipwithassetprices. Butthesignificantverticaldispersionofthecloudindicatesthat consumption growth is very predictable: it’s just that asset prices are not very informative about consumptiongrowth. Notethatneithertheprice-dividendrationortheconsumption-wealthratio predictsconsumptiongrowth(seethe‘consumptionclaim’columnofTable8). Ontheotherhand, the model’s state variables are informative. The middle and right panels show that consumption growth varies systematically with capital, surplus consumption, and productivity. Consumption growth is decreasing in capital and productivity, but increasing in surplus consumption. These relationships are more complicated than these panels suggest however, since the state variables arecorrelatedinsimulations. 6.7 ComparativeStatics The quantitative results come from estimated parameter values, but as with all econometric methods, the point estimates can be sensitive to choices of the econometrician. This section investigatestheeffectofchangingtheparametervalues. Thecomparativestaticsalsoconfirmthe intuitiondevelopedearlierinthepaper. Table 9 shows key moments from these comparative statics exercises. Each column examines moments generated by models where only one of the parameter values is changed from the estimation described by Table 1. Three different parameter changes are examined: lower persistenceofhabit,weakersteadystatehabit,andlowercapitaladjustmentcosts. Themagnitude of the perturbations are chosen to be the smallest change that produces a clearly recognizable 39

Figure 7: Predictability of Consumption Growth: Effect of the Conditioning Set. Figures are annualized and computed from the model’s laws of motion. Price-dividend ratio is for a claim onthefirm’sdividends. Theleftpanelshowsscatterplotsfrommodelsimulations. Themiddleand rightpanelsshowconsumptiongrowthasafunctionofstatevariables. deviation from the estimated results. This approach helps isolate the direct effect of changing a parametervaluefromitsinteractionwithotherelementsofthemodel. 6.7.1 Lowerpersistenceofhabit ThethirdcolumnofTable9examinesamodelwherethepersistenceofhabitρ isloweredfrom s theestimatedvalueof0.963to0.800.Thisparameterisimportantbecausethestrongpersistenceof habitisonewayinwhichthismodeldeviatesfrompreviousmodelswithhabitandproduction.The highestimatedpersistencemeansthathabittodaydependsonaverylonghistoryofconsumption. Incontrast,habitinJermann(1998)andBoldrin,Christiano,andFisher(2001)dependsonlyonthe lastquarter’sconsumption. The table shows the high persistence is critical. In the lower persistence model, the volatility of the risk free rate triples from the estimated value of 2.96% to 9.20%, bringing the model in line with the high risk-free rate volatility of Jermann (1998) and Boldrin, Christiano, and Fisher (2001). Figure8explainswhy. Thefigureplotstherisk-freeratedecompositionofequation(28)for the low persistence of habit model. The figure shows that the intertemporal substitution effect is still countercyclical and the precautionary saving effect is still procyclical. However, the two channels no longer cancel each other out quantitatively. The intertemporal substitution effect is extremely countercyclical and overwhelms precautionary saving effects. This change can traced 40

Table9:ComparativeStatics Figures are annual. ‘Estimated’ represents parameter values from Table 1. All other columns use the estimatedvaluesbutwithoneparameterchanged.‘Lowerpersistenceofhabit’setsρ =0.80.‘Weakersteady s statehabit’setsS¯=0.12.‘Loweradjustmentcosts’setsφ=40.DetailsofthedataarefoundinAppendixA.2. Lower Weaker Lower Data Estimated persistence steadystate adjustment 1948-2011 ofhabit habit costs IdentifyingMoments (cid:69)(r ) (%) 0.98 0.98 3.31 8.25 4.66 f (cid:69)(R−R )/σ(R) 0.48 0.48 0.39 0.31 0.38 f σ(r −r ) (%) 16.07 16.07 24.22 14.77 8.92 f σ(c )/σ(y ) 0.47 0.47 0.43 0.64 0.39 hp hp σ(y ) (%) 1.50 1.50 1.63 1.53 1.49 hp UntargetedMoments (cid:69)(r −r ) (%) 6.47 6.77 7.46 3.84 3.11 f AC1(r −r ) -0.03 -0.07 -0.12 -0.05 -0.04 f σ(r ) (%) 2.24 2.96 9.20 5.20 2.30 f AC1(r ) 0.56 0.88 0.66 0.86 0.93 f (cid:69)(p−d) 3.42 2.67 2.23 2.15 2.63 σ(p−d) 0.43 0.46 0.39 0.33 0.39 AC1(p−d) 0.95 0.90 0.78 0.89 0.92 R2from 1-year 0.08 0.10 0.07 0.04 0.06 forecastingr −r 3-year 0.19 0.26 0.14 0.10 0.17 f withp−d 5-year 0.26 0.37 0.17 0.15 0.25 to the lower persistence of habit. A low persistence of habit means that habit will strongly mean reverttomorrow. Thismeansthat,inbadtimes,thereisapronounceddesiretoborrowinorderto consumetoday. Another consequence of faster mean reversion is that precautionary saving effects become weaker.Fastermeanreversionmeansthathabitwillrecoverquicklyfrombadtimes,andthusthere islesscountercyclicaluncertaintyaboutsavings. Thisweakeningofprecautionarysavingseffects isreflectedinareductionintheamountoftime-varyingriskpremia. ThethirdcolumnofTable9 41

Figure 8: Decomposition of the Risk-Free Rate: low persistence of habit. ρ = 0.80. All other s parameter values are from the estimation (Table 1). ‘Intertemporal Substitution’ represents the 2nd and 3rd terms of equation (28). ‘Precautionary Saving’ represents the last term. Computed frommodel’slawsofmotion. Risk-freerateisinannualized%. Capitalinallpanelsisfixedatthe meancapitalstock. showsthattheamountoftime-variationinriskpremiafalls. R2’sfromregressionsoffutureexcess returnsonthelogprice-dividendratiodropsignificantly. Thepersistenceofhabithasastrongeffectonmanyotherassetpricemoments. Theparameter is identified with the volatility of excess returns, and, as expected, the low persistence model stronglyoverpredictsthisvolatility.Anothereffectisthatthepersistenceoftherisk-freerateandthe price-dividendratiodrops. Intuitively,thepersistenceofpreferencesisreflectedinthepersistence ofassetprices. 6.7.2 Weakersteadystatehabit The fourth column of Table 9 raises steady state surplus consumption S¯ from the estimated value of 0.063 to 0.120. This comparative static brings the model closer to CRRA utility. Roughly speaking, S¯= 1.00 is an economy with no habit, so this model is still very far from the standard model. Weakening steady state habit lowers the volatility of marginal utility of the household (see equation(21)),andthushasthedirecteffectofloweringtheSharperatiofromtheestimatedvalue of 0.48 to 0.31. This reduced Sharpe ratio has the obvious effect of reducing the equity premium from the estimated value of 6.77% to 3.84%. The Sharpe ratio does not drop as much as one mightexpect,however,sincethevolatilityofconsumptionincreases.Thisincreaseinconsumption 42

volatilityisduetoaweakergeneralequilibriumconsumptionsmoothingeffect(seeSection6.4). Weakening steady state habit has the additional effect of weakening precautionary savings effects. With weaker habit, there is less need for precautionary savings, and the risk-free rate jumps nearly ten-fold. This weakening of precautionary savings effects can be seen in the return forecastingregressionstoo. TheR2’sfromregressionsoffuturereturnsontheprice-dividendratio alldropbymorethanhalf. 6.7.3 Loweradjustmentcosts The final column of Table 9 examines the effect of lowering capital adjustment costs. The adjustment cost parameter φ is lowered from the estimated value of 75.00 to 40.00. This comparative static helps compare my model with Lettau and Uhlig (2000), who also examine externalhabitinaproductioneconomy.Mymodeldeviatesfromtheirsintwoimportantways.The firstisthatIincludecapitaladjustmentcosts,whileLettauandUhligassumecostlessadjustment. The second is that I use a non-linear, global solution method, while Lettau and Uhlig linearize the model around the non-stochastic steady state. This comparative static shows that capital adjustmentcostsmakealargeimpactonthemodelresultsandthatloweringthecostsbringsmy modelclosertoLettauandUhlig(2000). The table shows that lowering adjustment costs reduces the relative volatility of consumption to output falls from the estimated value of 0.47 to 0.39. This is consistent with the intuition from Section 6.4. Low adjustment costs reduce the incentive for smooth investment and encourages volatileconsumption. This decrease in consumption risk is reflected in a lower Sharpe ratio. The equity premium drops by half. With lower consumption risk, the precautionary motive is weakened and the riskfreeraterisesbyafactoroffive. ThisdecreaseintheprecautionarymotiveisalsoseeninlowerR2’s fromreturnforecastingregressions. 7 Conclusion Time-varying uncertainty plays a key role in many theories of aggregate asset prices. Previous papers take time-varying uncertainty as exogenous. This paper shows that timevarying uncertainty is the natural result of a fundamental economic motive: the desire for precautionary savings. These motives make investors very sensitive to shocks in bad times, and lead to countercyclical consumption volatility. This mechanism, when amplified by external habitpreferences, quantitativelyaccountsfortheempiricalevidenceoftime-varyingriskpremia. 43

The model also provides a parsimonious description of asset prices and the real economy. The estimated model matches a long list of facts about the aggregate stock market, safe government bonds,aswellasconsumption,output,andinvestment. 44

A Appendix A.1 ProofsfortheTwo-periodModel ProofofProposition1. A change of variables shows that this model is equivalent to a standard consumption-savings problem. Shift consumption by assigning C ∗ = C −H. Then the date 1 1 consumptionrulecanbewrittenas C (W )=C ∗ (W −H ,−H )+H (30) 1 1 1 1 2 1 ∗ whereC (W,Y) solves a simple consumption-savings problem with wealthW and certain future incomeofY: C ∗ (W,Y)≡argmaxu(C)+(cid:69){u[R(W −C)+Y]} (31) C andforeaseofnotationIsuppressthesubscript2onR. ItturnsoutthatforY (cid:54)=0,R random,and ∗ CRRAutility,C (W,Y)isstrictlyconvexinW. Thatis,withCRRAutility,rateofreturnrandomness, andtheintroductionofany(evenconstant)futureincomeisasufficientconditionforgenerating strictconvexityoftheconsumptionfunction. Thisisnotoneofthesufficientconditionsshownin CarrollandKimball(1996),soIwillshowthatitissufficientinwhatfollows. TheFOCoftheshiftedproblem(31)is u (cid:48) (C ∗ (W,Y))=φ(cid:48) (W −C ∗ (W,Y)) (32) Where,forconvenience,I’vedefinedthefunction φ(S)≡(cid:69){u[RS+Y]} ∂ Taking oftheFOCandrearranginggives ∂W ∂ φ(cid:48)(cid:48) C ∗ (W,Y)= ∂W u (cid:48)(cid:48)+φ(cid:48)(cid:48) ∂ ∗ Takeanother ,dosomeseriousalgebra,andwegetanexpressionfortheconvexityofC : ∂W ∂2 (cid:34) (u (cid:48)(cid:48) )2(cid:161)φ(cid:48)(cid:48)(cid:162)2 (cid:35)(cid:34) φ(cid:48)φ(cid:48)(cid:48)(cid:48) u (cid:48) u (cid:48)(cid:48)(cid:48)(cid:35) C ∗ (W,Y)= − (33) ∂W2 u (cid:48)×[u (cid:48)(cid:48)+φ(cid:48)(cid:48) ]3 (cid:161)φ(cid:48)(cid:48)(cid:162)2 (u (cid:48)(cid:48) )2 45

Thefirstbracketisnegativesimplybecauseu (cid:48)>0andu (cid:48)(cid:48)<0. Toshowthatthesecondbracketis (cid:48) (cid:48)(cid:48)(cid:48) (strictly)positive,firstnotethat,duetotheCRRAspecification, uu =1+ 1. Iwillnowshowthat, (u(cid:48)(cid:48))2 γ φ(cid:48)φ(cid:48)(cid:48)(cid:48) due to the non-zero future income Y, > 1+ 1. This is an extension of Carroll and Kimball (φ(cid:48)(cid:48))2 γ (1996)’sLemma4. φ(cid:48)φ(cid:48)(cid:48)(cid:48) Proving >1+1 requiresthefollowingtechnicalLemma. (φ(cid:48)(cid:48))2 γ Lemma1. LetΦ fori =1,...,N be2×2symmetricmatriceswiththefollowingproperties: i • thediagonalsofeachΦ arepositive i • theoff-diagonalsofeachΦ areallnegative i • foreveryi,|Φ |=0 i • (cid:175) (cid:175) (cid:80) i N =1 Φ i (cid:175) (cid:175) =0 Thenforeachpairi,j thereissomeconstantk suchthat Φ =kΦ i j Proof. I will first show this for the case where N = 2. I’ll then use the N = 2 results to prove the (cid:195) (cid:33) (cid:195) (cid:33) p q x y generalcase. ForeaseofnotationassignΦ ≡ andΦ ≡ . Withsomealgebra,one 1 2 q r y z canshowthat (cid:112) (cid:112) (cid:112) |Φ +Φ |=|Φ |+|Φ |+[ pz− xr]2+2[ prxz−qy] (34) 1 2 1 2 Afewfactswillletussimplifythisexpressiondramatically. First|Φ |=|Φ |=|Φ +Φ |=0,sothose 1 2 1 2 termsalldropout. Thennotethat,since|Φ |=|Φ |=0,wehavepr =q2 and xz =y2,andwecan 1 2 rewrite (cid:112) (cid:113) prxz= q2y2=qy andsothelastterminequation(34)alsodropsout. Thusequation(34)impliesthatpz=xr,or p r = =k (35) x z where k is the conjectured constant of proportionality. We just need to show that q = p =k. To y x 46

showthis,plugpr =q2andxz=y2into(35)andwehave p q2/p p q = ⇒ = x y2/x x y ThiscompletestheN =2case. To show the general case, first note that if Φ and Φ satisfy the requirements of the lemma, i j then Φ +Φ also satisfies those requirements. Thus I can apply the N =2 results to the general i j case,whereonematrixisΦ i andtheothermatrixis (cid:80) j(cid:54)=i Φ i . Moreover,notethatifthereisak such thatΦ i =k (cid:80) j(cid:54)=i Φ i , thenthereism suchthat (cid:80) j Φ j =mΦ i . Applythistoalli andgetthedesired resultΦ =kΦ . i j φ(cid:48)φ(cid:48)(cid:48)(cid:48) Now, back to proving the proposition. I want to show that > 1+ 1. Suppose, for (φ(cid:48)(cid:48))2 γ φ(cid:48)φ(cid:48)(cid:48)(cid:48) contradiction,that ≤1+1. Icanwritethisexpressionusingthedeterminantofa2×2matrix (φ(cid:48)(cid:48))2 γ bydefining  (cid:113)   (cid:113)  φ(cid:48) 1+1φ(cid:48)(cid:48) Ru (cid:48) (z) 1+1R2u (cid:48)(cid:48) (z) γ γ Φ≡(cid:69) =(cid:69)  (cid:113)   (cid:113)  1+1φ(cid:48)(cid:48) φ(cid:48)(cid:48)(cid:48) 1+1R2u (cid:48)(cid:48) (z) R3u (cid:48)(cid:48)(cid:48) (z) γ γ Where,foreaseofnotation,z≡R(W−C ∗ (W,Y))+Y.Thisexpressioncannowbewrittencompactly as |Φ|≤0 Note that Φ is the weighted sum of many component  (cid:113)  Ru (cid:48) (z) 1+1R2u (cid:48)(cid:48) (z) γ matricies  (cid:113) , and that due to the CRRA specification of u, the 1+1R2u (cid:48)(cid:48) (z) R3u (cid:48)(cid:48)(cid:48) (z) γ determinantofeachcomponentmatrixiszero. ThusΦispositivesemidefinite,so|Φ|≥0. Butour assumptionforcontradictionsaysΦisnegativesemidefinite,andsoitmustbethat|Φ|=0. Now I use Lemma 1. The lemma states that if |Φ|=0, all of the component matrices must be proportionaltooneanother. Thismeansthatforanystatesi and j theratioofthediagonalterms 47

ofthecorrespondingmatricesisequal,thatis,foranyi and j, R u (cid:48) (cid:181) R (cid:182)3u (cid:48)(cid:48)(cid:48) i i = i i (cid:48) (cid:48)(cid:48)(cid:48) R u R u j j j j (cid:181) R S+Y (cid:182)−γ (cid:181) R (cid:182)2(cid:181) R S+Y (cid:182)−γ−2 ⇒ i = i i R S+Y R R S+Y j j j R S+Y R ⇒ i = i R S+Y R j j Y Y ⇒ S+ =S+ R R i j ⇒ R =R i j whichisacontradiction,sinceRisrandom.NotethatthepresenceofanonzeroincomeY iscritical becauseotherwise,Icouldnotmovefromthefourthlinetothefifthlineintheequationsabove. Therefore, φ(cid:48)φ(cid:48)(cid:48)(cid:48) >1+1,andbyequation(33),C ∗ (W,Y)isstrictlyconcaveinW,andbyequation (φ(cid:48)(cid:48))2 γ (30),C (cid:48)(cid:48) (W)<0. 1 ProofofProposition2. I first show that the transformed consumption function satisfies ∂3 C ∗ (W,Y)>0. To show this, note that C ∗ (W,0) is linear, but for any (cid:178)>0, C ∗ (W,(cid:178)) is strictly ∂W3 concave. Thuswecansignthederivative     ∂3 ∂2 C ∗ (W,(cid:178))− ∂2 C ∗ (W,0) ∂2 C ∗ (W,(cid:178)) C ∗ (W,0)=lim ∂W2 ∂W2  =lim ∂W2  <0 ∂Y∂W2 (cid:178)→0 (cid:178) (cid:178)→0 (cid:178) Assumingthat ∂3 C ∗ (W,0)iscontinuous,thismeansthatthereissomeY >0suchthatfor ∂Y∂W2 anyδ<Y, ∂3 C ∗ (W,δ)<0. ∂Y∂W2 ∗ SinceC (W,Y)isHD1,Icantakederivativestoshowthat   ∂3 C ∗ (W/δ,1)= δ3   − 1 ∂2 C ∗ (W,δ)− ∂3 C ∗ (W,δ)   ∂W3 W  δ2∂W2 ∂Y∂W2  (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) <0 <0 Now note that I am free to choose W, so this means, for any W, ∂3 C ∗ (W,1) > 0. But by ∂W3 homogeneity, ∂3 C ∗ (W,Y) = (1/Y2) ∂3 C ∗ (W,1) > 0, and thus the third derivative with respect ∂W3 ∂W3 towealthofthetransformedconsumptionfunctionisnegative(aslongasY (cid:54)=0). Nowtofinishprovingtheproposition. Usingthetransformedconsumptionfunction(31)Ican 48

∂ relate the desired derivative to the third derivative of the consumption function. First take ∂W1 twice: ∂2 C (cid:48)(cid:48) (W )= C ∗ (W −H ,−H ) 1 1 ∂W2 1 1 2 ∂ Thentake ∂H1 ∂ ∂3 C (cid:48)(cid:48) (W )=− C ∗ (W −H ,−H )<0 ∂H 1 1 ∂W3 1 1 2 1 ProofofCorollary2. LetW andW betheminimumandmaximumof(cid:87) ,respectively. 0 0 0 max σ [C (W )]− min σ [C (W )] 0 1 1 0 1 1 W0 ∈(cid:87) 0 W0 ∈(cid:87) 0 (cid:183) (cid:184) =σ (∆W ) max C (cid:48) (W )− min C (cid:48) (W ) 0 1 W0 ∈(cid:87) 0 1 0 W0 ∈(cid:87) 0 1 0 (cid:104) (cid:105) =σ (∆W ) C (cid:48) (W )−C (cid:48) (W ) 0 1 1 0 1 0 (cid:183)(cid:90) W (cid:184) =σ (∆W ) 0 dWC (cid:48)(cid:48) (W) 0 1 1 W0 (cid:34) (cid:35) =−σ (∆W ) (cid:90) W0 dWC (cid:48)(cid:48) (W) 0 1 1 W 0 Wherethe2ndlineusesequation(2),thethirdlineusesC (cid:48)(cid:48) (W)<0. 1 ∂ Thentake ∂H1   ∂ (cid:183) max σ [C (W )]− min σ [C (W )] (cid:184) =−σ (∆W )   (cid:90) W0 dW ∂ C (cid:48)(cid:48) (W)  >0 ∂H 1 W0 ∈(cid:87) 0 0 1 1 W0 ∈(cid:87) 0 0 1 1 0 1   W ∂H 1 1   0 (cid:124) (cid:123)(cid:122) (cid:125) <0(prop2) A.2 DataDetails Thedataspan1948thru2011. MacroeconomicquantitiesarefromtheBEANIPAandBEAfixed asset tables. All data is real and per-capita. Output is simply GDP. Consumption is nondurable consumptionplusservicesconsumption.Investmentandcapitalarefixedinvestmentplusdurable 49

consumption plus government investment. All quantities are constructed by dividing nominal figures (Table 1.1.5 or Table 3.9.5) by the appropriate price index (Table 1.1.4 or Table 3.9.4), and thendividingbythepopulation(fromTable2.1). Moststudiesexcludegovernmentinvestmentfromtheirdefinitionofinvestment,citingthefact thattheirmodelsdonotincludegovernment.However,thislogicwouldalsoimplythatoneshould removegovernmentpurchasesfromGDP,whichisnottypicallydone.Ichoosetokeepgovernment purchases in GDP and include government investment in investment as this approach preserves theideathatthedatacomefromasinglegeneralequilibriumsystem. Moreover, thisapproachis closertothespiritofCooleyandPrescott(1995).7 Choiceoftheempiricaldefinitionofinvestment doesnothaveasignificanteffectonanyoftheresults. Asset price data is taken from CRSP. The equity is the CRSP value-weighted index. Pricedividend ratios are computed annually using the stock market reinvestment assumption. As discussed in Cochrane (2011), this aggregation method preserves the Campbell-Shiller identity, which is useful for identifying the source of asset price fluctuations. CRSP returns are deflated using the CPI, also obtained from CRSP. The risk-free rate is computed using a forecast of the expost real return of the 90-day T-bill following Beeler and Campbell (2009). Following Beeler and Campbell (2009), the ex-post real return is calculated by deflating the 90-day nominal T-bill yield usingseasonallyadjustedCPIfromtheBLS.Theforecastisconstructedbyregressingnextquarter’s ex-post real return on today’s nominal 90-day yield and the mean inflation rate over the previous year. A.3 SimulatedMethodofMomentsDetails This section spells out details of the SMM method. I use a different notation than the brief discussioninthetextinordertobemoreprecise. A.3.1 EconometricDetails I transform the parameters so that I do not need to be concerned about corner solutions. For example,ratherthanestimateβ∈[0,1],Iestimatelogit(β)∈(cid:82). Aftertheestimationisdone,Ifind the point estimates of the original parameters by inverting the transformation, and the standard errorsbythedeltamethod. 7CooleyandPrescott(1995)saythat“Oureconomyisveryabstract:itcontainsnogovernmentsector,nohousehold productionsector,noforeignsectorandnoexplicittreatmentofinventories.Accordingly,themodeleconomy’scapital stock, K, includescapitalusedinallofthesesectorsplusthestockofinventories. Similar, output, Y, includesthe outputproducedbyallofthiscapital.” 50

To be explicit, let ζ ∈ Θ be the vector of original parameters and ψ−1(ζ) = θ ∈ (cid:82) by the K transformedparameters. Then (cid:112) (cid:112) T(ζˆ −ζ )= T[ψ(θˆ )−ψ(θ )] (36) T 0 T 0 (cid:112) ≈ T[Dψ(θ ) (cid:48) (θˆ −θ )] (37) 0 T 0 sotheasymptoticvarianceoftheoriginalparameterscanbeestimatedby (cid:112) (cid:112) V(cid:100)ar[ T(ζˆ T −ζ 0 )]=Dψ(θˆ T ) (cid:48) V(cid:100)ar[ T(θˆ T −θ 0 )]Dψ(θˆ T ) (38) I also use the delta method on the moment errors, because some of the “moments” I use are not moments in the traditional sense (i.e. the Sharpe Ratio) but are more formally described as transformationsofmoments. Thatis,lettheGMMobjectivebe θˆ =argminQ (θ)=G (θ) (cid:48) W G (θ) T T T T T θ∈(cid:82) whereG (θ)aresomemomenterrors. T S G (θ)≡S −1 (cid:88) h[f¯s(θ)]−h[f¯∗ ] T T T s=1 where f¯s(θ)≡T −1 (cid:88) fs(θ) [M×1] isasetofmomentsfromsimulations T t t h:(cid:82)M −→(cid:82)N transformsthemomentsintosomeotherstatistic andS isthetotalnumberofsimulations. ThishfunctionhasnoeffectontheGMMformulaforthe asymptoticvariance (cid:112) V(cid:100)ar[ T(θˆ T −θ 0 )]≡[DG T W T DG T (cid:48) ] −1[DG T W T Xˆ T W T DG T (cid:48) ][DG T W T DG T (cid:48) ] −1 where DG ≡DG (θˆ ) T T T (cid:112) Xˆ T ≡V(cid:100)ar( TG T (θ 0 )) 51

(cid:112) butitdoeshaveaneffectontheestimatorofthevarianceofthemomenterrorsV(cid:100)ar( TG T (θ 0 )). (cid:181) (cid:182) 1 Xˆ ≡Dh(f¯∗ ) (cid:48) 1+ Σ Dh(f¯∗ ) T T S T T Σ ≡ (cid:88) k (cid:181) k−|j|(cid:182) T −1 (cid:88) T eˆ eˆ (cid:48) T k t t−j j=−k t=1 T eˆ ≡ f ∗−T −1 (cid:88) (f ∗ ) t t s s=1 Thatis,wesimplyadjusttheDuffieandSingleton(1993)estimatorbyapplyingthedeltamethodto thespectraldensity. Toseehowthisworks,notethatthemomenterrorcanbeapproximatedby G (θ )=S −1 (cid:88) h(f¯s(θ ))−f¯∗ T 0 T 0 T s ≈S −1 (cid:88)(cid:163) Dh(f¯∗ (θ ))[f¯s(θ )−f¯∗ ] (cid:164) T 0 T 0 T s (cid:183) (cid:184) =Dh(f¯∗ ) S −1 (cid:88) f¯s(θ )−f¯∗ T T 0 T s ThuswejustneedtoadjustthetraditionalNewey-WestestimatorbyDh(f¯∗ ). T A.3.2 NumericalDetails Tocalculatethemomenterrors,Isimulatethemodel1000times(S=1000). Theinitialstateis settobethemedianstateforalongsimulationclosetotheestimatedparametervalues.Derivatives arecomputedwithatwo-sidedfinitedifference. I optimize using Levenberg-Marquardt (LM). I choose this method rather than the more commonly used simulated annealing method for two reasons. The first is that the moment functiondoesnotdisplayanextremenumberoflocalminima,whichiswheresimulatedannealing has an advantage. With a relatively smooth objective function, a method which uses derivative information is much more efficient. Derivative information is particularly helpful in the GMM setting with small residuals, since the Gauss-Newton method provides a quick positive semi definiteapproximationoftheHessian. Speedisimportantastheestimationtakesaround24hours withavery goodguess. AnotheradvantageofLMisrobustness. Simulatedannealingtendstobe very sensitive to the choice of the annealing schedule (Press, Teukolsky, Vetterling, and Flannery (1992). IbeginwithanLMparameterof1000,anduseasimplealgorithmforadjustingtheparameter: ifthenewfunctionvalueisagoodenoughimprovement,IdecreasetheLMparameterbyafactor 52

of 10. Otherwise, I set it back to 1000. “Good enough” is judged by the difference between the improvementintheobjectiveandthepredictedimprovementaccordingtoaquadraticmodel. Since I use exact identification, the maximum moment error provides a clean convergence criterion. Iconsiderthealgorithmconvergedifthemaximummomenterrorislessthan1.0E-4. A.4 TheInterpretationofHabit Here I describe how deviating from Campbell and Cochrane (1999)’s specification of λ raises somequestionsregardingtheinterpretationofhabitinthemodel.Thekeyissueisthattheconstant λofmymodelcanmakehabitdecreaseinresponsetoanincreaseinconsumption. Thisviolates sometraditionalnotionsofhabit. This issue can be illustrated by taking the derivative of log habit h t+1 with respect to log consumptionc t+1 : dh t+1 =1− λ dc t+1 S − t+ 1 1 −1 ThusifS t+1 islargeenough, d d h ct t + + 1 1 willbenegative. ThepreferencesofthispaperstillpreservethestandardnotionofhabitinthatH isageometric t average of previous consumption. This can be seen by following the analysis of Campbell and Cochrane (1999) found in Campbell (2003). I can log-linearize the log surplus consumption ratio aroundthesteadystate: s =log[1−exp(h −c )]≈κ−λ−1(h −c ) (39) t t t t t Pluggingthisintothedefinitionofthehabitprocess(5),Ifindthelinkbetweenhabitandhistorical consumption h t+1 ≈(Constants)+ρ s h t +(1−ρ s )c t (40) ∞ =(Constants)+(1−ρ s ) (cid:88) ρ s j c t−j (41) j=0 This informal demonstration is verified by simulated data. In the simulated data, habit is highly correlatedwithconsumption. Thecontemporaneouscorrelationis0.978andthecorrelationwith lagged consumption is 0.983. Habit growth and consumption growth are moderately correlated. Thecorrelationbetween∆h and∆c is0.411. t t That habit should move non-negatively with consumption everywhere is not required if one 53

entertains a very slow-moving, historical average of consumption as responsible for our current referencepointforconsumption. Moreover, CampbellandCochrane(1999)’sspecificationisalso vulnerable to this this issue. Ljungqvist and Uhlig (2009) show that while habit moves positively withsmallmovementsinconsumption,itcanmovenegativelywithlargemovements. The issue illustrated in this section is related to Campbell and Cochrane (1999)’s three requirements on λ(s ). They require (i) the risk-free rate is constant, (ii) habit is predetermined t atthesteadystatesurplusconsumption,and(iii)habitispredeterminednearthesteadystate. The first assumption is not critical for making habit move non-negatively with consumption. In my model, (ii) is satisfied, but (iii) is not. (iii), in combination with Campbell and Cochrane (1999)’s specificationforλ(s )resultsin dht+1 ≥0foralls . t dct+1 t A.5 SolutionMethodDetails Euler Equation To be explicit about the firm’s Euler equation, let π (Z ,Z ) be the transition Z i j matrix for the discretized productivity process. The firm’s problem is to find capital policy K (cid:48) = G(K;Kˆ,S,Z )tosolve i V(K;Kˆ,S,Z )= max (cid:169)Π(K,Z ,N)−W(Kˆ,S,Z )N−Φ(I,K)−I i i i K(cid:48),I,N (cid:41) + (cid:88) π (Z ,Z )M(Kˆ,S,Z ;Z )V(K (cid:48) ;Kˆ(cid:48) ,S (cid:48) ,Z ) Z i j i j j Zj subjectto K (cid:48)=I+(1−δ)K TheFOCforinvestmentandtheenvelopeconditionare: 1+D Φ(I,K)= (cid:88) π (Z ,Z )M(Kˆ,S,Z ;Z )D V(K (cid:48) ;Kˆ(cid:48) ,S (cid:48) ,Z ) 1 Z i j i j 1 j Zj D V(K;Kˆ,S,Z )=D Π(K,N,Z )+(1+D Φ(I,K))(1−δ)−D Φ(I,K) 1 i 1 i 1 2 whichtogetherproducetheEulerequation 1+D Φ(I,K)= (cid:88) π (Z ,Z )M(Kˆ,S,Z ;Z )[D Π(K (cid:48) ,Z ,N (cid:48) )+(1+D Φ(I ,K (cid:48) ))(1−δ)−D Φ(I ,K (cid:48) )] 1 Z i j i j 1 j 1 j 2 j Zj 54

Imposethefactthatthehouseholddoesnotvalueleisureandconsistency,andwehave (cid:88) 1= π (Z ,Z )M(Kˆ,S,Z ;Z )Rˆ (Kˆ,S,Z ;Z ) (42) Z i j i j I i j Zj D Π(Kˆ(cid:48) ,Z ,1)+(1+D Φ(Iˆ ,Kˆ(cid:48) ))(1−δ)−D Φ(Iˆ ,Kˆ(cid:48) ) Rˆ (Kˆ,S,Z ;Z )≡ 1 j 1 j 2 j I i j 1+D Φ(Iˆ,Kˆ) 1 Where (cid:181)C S (cid:182)−γ M(Kˆ,S,Z ,Zj)=β j j (43) i C S C =Π(Kˆ(cid:48) ,Z ,1)−Φ(Iˆ ,Kˆ(cid:48) )−Iˆ j j j j C =Π(Kˆ,Z ,1)−Φ(Iˆ,Kˆ)−Iˆ i Kˆ(cid:48)=Gˆ(Kˆ,S,Z ) i Iˆ=Gˆ(Kˆ,S,Z )−(1−δ)Kˆ i Iˆ =Gˆ(Kˆ(cid:48) ,S (cid:48) ,Z )−(1−δ)Kˆ(cid:48) j j andtheevolutionofsurplusconsumptionsatisfies s =(1−ρ )s¯+ρ s+λ(c −c) (44) j s s j The projection algorithm looks for cubic spline coefficients which solve equations (42), (43), and (44). Solving for asset prices To find asset prices, the firm’s Bellman equation, with optimal values pluggedin,is: V(K;Kˆ,S,Z )=Π(K,Z ,1)−W(Kˆ,S,Z )−Φ(I,K)−I i i i + (cid:88) π (Z ,Z )M(Kˆ,S,Z ;Z )V(K (cid:48) ;Kˆ(cid:48) ,S (cid:48) ,Z ) Z i j i j j Zj Inequilibrium,W(Kˆ,S,Z )=(1−α)AZ Kˆα ,K =Kˆ,I =Iˆ,andK (cid:48)=Kˆ(cid:48) ,so i i V(Kˆ;Kˆ,S,Z )=αAZ Kˆα−Φ(Iˆ,Kˆ)−Iˆ i i + (cid:88) π (Z ,Z )M(Kˆ,S,Z ;Z )V(Kˆ(cid:48) ;Kˆ(cid:48) ,S (cid:48) ,Z ) Z i j i j j Zj 55

Let’sdefineVˆ(Kˆ,S,Z )≡V(Kˆ;Kˆ,S,Z ). TheaboveequationsuggeststhatVˆ(Kˆ,S,Z )canbefound i i i by repeatedly applying the above equation as an operator (using the law of motion for capital Gˆ(Kˆ,S,Z )). i A.5.1 Approximations Iapproximatetheautoregressiveprocessforproductivityz witha13pointMarkovChainusing t the Rouwenhorst method. I approximate the law of motion for capital in the K and S directions usingatwo-dimensionalcubicspline.Thesplineisof6thdegreeintheK directionand14thdegree intheS direction. Thesplinebreakpointsarelog-spacedinboththeK andS directions. Ifindthat increasing the degree to the 14th in the Kˆ direction has no material impact on the quantitative results. Inaprojectionmethod,onemustdefinewhatitmeanstosatisfytheEulerequation. Iusethe collocation method, which specifies that the Euler equation should hold exactly at a set of points (collocation nodes) in the K and S domain. I choose these nodes to be the standard nodes for splinesusingknotaveraging.Iconsiderthealgorithmconvergedif,acrossthesecollocationnodes, the maximum absolute Euler equation residual, expressed as 1−E[M (cid:48) R (cid:48) ], is less than 1.0E-8. I I searchforsplinecoefficientswhichsatisfythisconditionbyusingBroyden’smethod. A.6 AdjustmentCostsandAccounting Manypapersspecifyadjustmentcostsinthefollowingmanner: K (cid:48)=(1−δ)K +φ(I/K)K (e.g. Jermann(1998),Gourio(2009),KaltenbrunnerandLochstoer(2010),Guvenen(2009),among others). ThisformulationpreservesthetraditionalCobb-Douglasformulationofoutput: Y =ZK α N1−α=C+I Thisformulation,however,deviatesfromthestandardaccountingtreatmentofinvestmentand capital. Thestandardtreatmentspecifiesthatend-of-periodcapitalisbeginning-of-periodcapital plusinvestmentlessdepreciation. Withgeometricdepreciation, thistranslatesintothestandard, adjustment-cost-freeformulationofcapitalaccumulation: K (cid:48)=(1−δ)K +I 56

Ichoosetopreservethisaccountingidentity. Asaresult,capitaladjustmentcostsarepushedinto output: Y =ZK α N1−α−[AdjCost]=C+I Fortunately,bothchoicesresultinthesamecapitalandconsumptionallocations. Forexample, Gourio(2009)uses K ∗(cid:48)=(1−δ)K +φ∗ (I ∗ /K)K η φ∗ (x)=x− (x−δ)2 2 Fromtheseexpressions,wehavecapitalevolutionandconsumption η K ∗(cid:48)=(1−δ)K +I ∗− (cid:161) I ∗ /K −δ(cid:162)2 K 2 C =ZK α N1−α−I ∗ whichisidenticaltomyformulationwith η I ≡I ∗− (cid:161) I ∗ /K −δ(cid:162)2 K 2 57

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