A Likelihood-Based Comparison of Macro Asset Pricing Models

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Likelihood-Based Comparison of Macro Asset Pricing Models Andrew Y. Chen, Rebecca Wasyk, and Fabian Winkler 2017-024 Please cite this paper as: Chen, Andrew Y., Rebecca Wasyk, and Fabian Winkler (2017). “A Likelihood-Based Comparison of Macro Asset Pricing Models,” Finance and Economics Discussion Series 2017-024. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.024. 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.

A Likelihood-Based Comparison of Macro Asset Pricing Models Andrew Y. Chen Rebecca Wasyk FederalReserveBoard FederalReserveBoard Fabian Winkler FederalReserveBoard ∗ March2017 Abstract We estimate asset pricing models with multiple risks: long-run growth, long-run volatility, habit, and a residual. The Bayesian estimation accountsfortheentirelikelihoodofconsumption,dividends,andthepricedividendratio.Wefindthattheresidualrepresentsatleast80%ofthevarianceoftheprice-dividendratio.Moreover,theresidualtracksmostrecognizablefeaturesofstockmarkethistorysuchasthe1990’sboomandbust. Long run risks and habit contribute primarily in crises. The dominance oftheresidualcomesfromthelowcorrelationbetweenassetpricesand consumptiongrowthmoments. Wediscusstheorieswhichareconsistent withourresults. ∗WethankFrancescoBianchi,EdHerbst,FranciscoPalomino,andMissakaWarusawitharanaforadviceandcomments,andCarterBrysonandChrisCampanoforexcellentresearch assistance. Our emails are andrew.y.chen, rebecca.d.wasyk, and fabian.winkler, all @frb.gov. Theviewsexpressedhereinarethoseoftheauthorsanddonotnecessarilyreflecttheposition oftheBoardofGovernorsoftheFederalReserveortheFederalReserveSystem.

1. Introduction Models of asset prices have come a long way since Mehra and Prescott (1985). We now have several explanations of aggregate stock market fluctuations. Arguablythemostprominentarehabitformation,longrunrisks,andrare disasters. But there are more, including limited participation, intermediarybasedmodels,andlearning.1 Inthispaper,weevaluatetherelativeimportanceoftheseexplanations. The evaluation uses a model which divides the price-dividend ratio into identifiablemacrorisks(habit,longrungrowth,andlongrunvolatility),andhard-toobserve fluctuations in risk (a persistent residual). Habit and long run risks are related to consumption and dividends in the usual way (Campbell and Cochrane (1999), Bansal, Kiku, and Yaron (2012a)). The residual accounts for all other sources of stock price movements: disaster probability movements, shiftsinbeliefsaboutreturns,etc. WeestimatethemodelusingBayesianmethodsanddataonconsumption growth, dividend growth, and the price-dividend ratio. The estimation leads to a decomposition of the price-dividend ratio into contributions from each sourceofmarketvolatility. We find that the residual is the most important source of market volatility, accountingforthevastmajorityofthevarianceoftheprice-dividendratio. The residual accounts for more than 80% of the variance across a variety of priors andmodelspecifications. Moreover,thesmoothedresidualtracksmostofthe recognizablefeaturesoftheU.S.stockmarket’shistory,suchastheboomsofthe 1950sand1990s,andthebustsofthe1970sandearly2000s. Longrunvolatility, longrungrowth,andhabithavelargeeffectsintheGreatDepressionand2008 FinancialCrisis,butoveralltheydisplayalowcorrelationwithassetpricesbetween 1929 and 2014. These results show that, while long run risks and habit haveanon-negligibleeffect,somethingelseisthekeydriverofmarketvolatility. Importantly, the dominance of the residual is independent of our choice 1Herewelistjustacouplereferencesforeachliterature. ForhabitformationseeConstantinides (1990) and Campbell and Cochrane (1999). For long run growth and volatility risks see Bansal and Yaron (2004) and Bansal, Kiku, and Yaron (2012a). For rare disasters see Rietz(1988),Barro(2006),Gabaix(2012),Wachter(2013).Forlimitedparticipation,Mankiwand Zeldes(1991)andGuvenen(2009).Forintermediary-basedmodelsseeHeandKrishnamurthy (2013)and Brunnermeierand Sannikov(2014). Forlearning modelssee Adam, Marcet, and Beutel(2015)andAdam,Marcet,andNicolini(2016). 2

of target moments, as our Bayesian estimation accounts for the entire likelihood of consumption, dividends, and the price-dividend ratio. This methodologycutsthroughtheproblemofweighingdisparatepiecesofevidencefrom themomentmatchingliterature. Howimportantistheexcessivelystrongdividend predictability in long run risks models? How important is the fact that habitimpliesacounterfactuallinkbetweenassetpricesandlaggedconsumptiongrowth? Whatdowemakeofmodelsthatarenotevaluatedagainstthese particular moments? By accounting for all moments, our Bayesian approach providesasuccinctanswertothesequestions. Fluctuationsintheresidualrepresent“excess”marketvolatility: Theresidualmovescloselywithassetprices,butisunconnectedtorealeconomicgrowth and real economic volatility. This description matches several theories in the literature which fit into two broad categories: tractable models with hard-toobserveshockstoriskorbeliefs(suchasvariabledisasterrisk)andmorecomplexmodelswhichdirectlylinkexpectedreturnstoobservablesotherthanaggregate consumption (such as intermediary-based models). We discuss these theories and avenues for future research, but we cannot distinguish among thesetheoriesinthispaper. Modelswithhard-to-observeriskleadtoseveralobservationallyequivalent structuralmodels. Thisequivalencemotivatesustofocusonasemi-structural model— that is, we simply assume that the log price-dividend ratio is linear inthefourstatevariablesratherthanderivethecoefficientsfromassumptions about preferences and market structure. But there are additional considerationsthatcompelustodeviatefromthestandardapproachoflookingforequilibriumamongoptimizingagents. Thesemi-structuralmodelletsthedataspeakfreely. Itensuresthattheestimation results are due to properties of the data rather than functional form restrictions imposed by our choice of model economy. Similarly, the reduced formis muchless costlyfor thereader towork through. This isespecially importantbecauseourmodelincludesseveralsourcesofrisk. Lastly,anagnostic model seems appropriate considering the vast disagreement in the literature abouttheeconomicstructureunderlyingstockprices(see,forexample,Gabaix (2012)andCochrane(2016)forsomecontrastingperspectives). Our estimator uses the entire likelihood, but the low correlation between price/dividendsandrealgrowthorrealvolatilitydrivestheresults. Todemon- 3

strate this, we replicate our main finding with a stripped-down version of our estimator. Specifically, we extract state paths using only data on consumption and dividends. We then use OLS to regress the price-dividend ratio on thestates. Thevariancedecompositionfromtheseprocedureusesonlycovariancesbetweenprice/dividendsandthestatepaths,andthisinformationleads OLStoconcludethattheresidualexplainsthevastmajorityofmarketvolatility. Though our approach does not require choosing moment targets, we do needtotakeastandonafewmodelingandeconometricissues. Ineverycase we make choices that favor simplicity for its various scientific virtues: Simple formulationsareeasiertodissect,communicate,replicate,andextend. Indeed, thelackofreplicabilityofeconomicresearchhasbeenrecentlyhighlightedby ChangandLi(2015). Simplicityhascosts,however. Specifically,ourdesireforsimplicityrequires that we offer two caveats regarding our conclusion that the residual is dominant. Thefirstcaveatisthatourapproachmayfavortheresidualbecauseofour useofanannualmodel. Wearguethatanannualmodelisideal, notonlybecause it is the simplest approach, but also because the striking seasonality in sub-annual data suggests that risk is best understood at an annual frequency. The fact that we recover similar parameters to the cash flow only estimates ofSchorfheide,Song,andYaron(2016)’smixedfrequencyestimationsuggests thatthedatafrequencyisnotcritical. Nevertheless,somestudiessuggestthat amonthlymodelisimportant(Bansal,Kiku,andYaron(2012a),Campbelland Cochrane(2000)),andaddingthislayerofcomplexitymaydecreasetheroleof theresidual. Thesecondcaveatisthatourapproachmayfavortheresidualbecausewe use relatively simple formalizations of habit and long run risks. More subtle formalizations, such as the use of several volatility processes for long run risk (Schorfheide,Song,andYaron(2016))ortheincorporationofadditionalshocks to habit (Bekaert, Engstrom, and Xing (2009)) may also decrease the residual contribution. RelationtotheLiterature Ourpaperfitsintothegrowingliteraturethatcompares the empirical performance of macro asset pricing models. Bansal, Gallant,andTauchen(2007),BeelerandCampbell(2012),Bansal,Kiku,andYaron 4

(2012a), andBarroandJin(2016)usemomentmatchingmethodstocompare theempiricalperformanceofhabit,longrunrisks,andraredisasters. Thepicture that emerges from this approach is somewhat muddled, as the preferred model depends on which moments one considers important. For example, habitispreferredifoneplacesalargeweightonaccountingfortheShiller(1981) volatility puzzle. On the other hand, long run risks are preferred if one is particularlyconcernedwithmatchingtime-varyingconsumptionvolatility. AldrichandGallant(2011)clarifythepicturebyusingaBayesianframework tocomparehabit,longrunrisks,andprospecttheory. Ourresultsechotheirs: longrunrisksarecriticalforaddressingthevolatile1930s,butarelessimportant forothertimeperiods. WedifferfromAldrichandGallant,however,byallowing aresidualtodriveassetprices. Theimportanceofincludingaresidualisseeninmorerecentpaperswhich findthatneitherlongrunrisksnorhabitformationiscapableofmatchingsome interesting stylized facts. Van Binsbergen, Brandt, and Koijen (2012) examinedividendstripsandequityoptions,Dew-Becker,Giglio,Le,andRodriguez (2015) examine variance swaps, and Muir (2015) examines international wars and financial crises. We complement these papers by showing that one does notneedtointroducederivativemarketsnorinternationaldatatoempirically challengelongrunrisksandhabitformation. ThetimeseriesofU.S.consumptionandstockpricesissufficient,ifoneaccountsfortheentirelikelihoodofthe data. Our paper also owes a large intellectual debt to Schorfheide, Song, and Yaron(2016), whopioneerthe useaparticle filterandBayesian Markovchain Monte Carlo (MCMC) methods (Herbst and Schorfheide (2014)) to estimate a modelwithlongrunrisks. Wefollowtheirapproachclosely,adoptingtheirelegantstatespacesystemandfilteringprocedure. Ourresultscomplementtheirs inthatwealsofindstrongevidenceoflongrunrisksinconsumptionanddividends,andindeed,similarposteriorestimates,usinganannualmodelandannual data. We deviate from Schorfheide, Song, and Yaron (2016), however, by allowingforapersistentresidualintheprice-dividendratioequation. Thus,our estimation is a much more stringent test of the long run risks model. We also assumeasimplerversionoflongrunrisk, whichhighlightstheimportanceof themultiplevolatilitystatesintheirmodel. 5

2. Model, Estimation, and Main Results Thissectiondescribesthemodel,explainstheestimationmethod,andends withthemainresults: decompositionsoftheprice-dividendratio(Section2.6). Alongtheway,wediscussthemodelfrequency,data,andparameterestimates. 2.1. Semi-StructuralModelwithMultipleSourcesofRisk Ourkeyvariableofinterestisthelogprice-dividendratiopd . pd islinear t t infourstatevariables pd =µ +A x +A σ˜2+A s˜ +A e . (1) t pd x t V t s t e t wherex islongrungrowth,σ˜ islongrunvolatility,s˜ issurplusconsumption t t t (habit), and e is a residual. The tildes over σ˜ and s˜ indicate that these varit t t ables are demeaned (σ˜2 = σ2−(cid:69)(σ2) and s˜ = s −s). These transformations t t t t t implythatµ isthemeanlogprice-dividendratio. pd Residualsarenotusuallycalled“statevariables,”butourresidualispersistent,playsanimportantroleinaccountingforthedata,andcanbeinterpreted throughseveraleconomicmodels(seeSection5). OurgoalistoestimatethecoefficientsA ,A ,A ,A andfindthesmoothed x V s e pathsofthestates x ,σ˜2,s˜ and e . Thecoefficientsandsmoothedpathsprot t t t videasimpledescriptionoftheimportanceofeachsourceofmarketvolatility. Wedonotderive(1)fromanequilibriummodelinordertolettheestimator speakfreely. However,thereareseveralwaystoderive(1). Forexample,onecan extendYang(2016)’sEpstein-Zinhabitmodeltoincludetime-varyingdisaster probability. Allofthestatesarelatent, buttheycanbeidentifiedbytheirlinkageswith observables. Thelongrunriskstates x ,σ˜2 areidentifiedbytheirrelationship t t withconsumptionanddividendgrowth ∆c =µ +x +σ η (2) t c t−1 t−1 c,t ∆d =µ +φ x +φ σ η +ϕ σ η t d x t−1 ηc t−1 c,t d t−1 d,t η ,η ∼N(0,1) i.i.d., c,t d,t where long run growth x evolves according to the standard heteroskedastic t 6

AR(1) x =ρ x +ϕ (cid:198) 1−ρ2σ η (3) t x t−1 x x t−1 x,t η ∼N(0,1) i.i.d., x,t andlongrunvolatilityσ evolvesaccordingto t (cid:113) h =ρ h +σ 1−ρ2η (4) t h t−1 h h h,t σ =σexp(h ). t t η ∼N(0,1) i.i.d. h,t ThevolatilityspecificationborrowsatechnicalfixfromSchorfheide,Song,and Yaron(2016),butotherwisetheaboveconsumptionanddividendsareequivalenttoBansal,Kiku,andYaron(2012a)’sprocesses.2 Schorfheideetal’sfixensuresthatvolatilityisalwayspositive. Importantly, ourspecificationdoesnotincludethemultiplevolatilityprocesses of Schorfheide et al. Using a single volatility process is consistent with the bulk of the long run risk literature (for example, Bansal, Kiku, and Yaron (2012a))andmakestheestimationsimpler. However,thisassumptionisrestrictive in that it assumes that the impact of volatility on the price-dividend ratio canbeidentifiedwith“short-run”realizedconsumptionvolatility(seeequation (2)). Surplus consumption s˜ is also identified by consumption growth, but is t more“backwardlooking.” s˜ isanAR(1)-likeprocess t s˜ =ρ s˜ +λ(s˜ )(∆c −(cid:69) ∆c ) (5) t s t−1 t−1 t t−1 t  λ(s˜ )= exp(−s)(cid:112) 1−2s˜ t−1 −1, s˜ t ≤ 2 1[1−exp(2s)] t−1 0, otherwise which is equivalent to Campbell and Cochrane (1999)’s habit process. This process means that surplus consumption is the average of past consumption growth and that habit is the average of past consumption levels (Campbell (2003),Chen(2016)). Intherobustnesssection(Section4),weexamineapro- 2To see the mapping, note that σ2 −σ ≈ 2σ2h implies σ2 −σ2 ≈ ρ (cid:0)σ2 −σ2(cid:1)+ t t t h t−1 (cid:113) 2σ2σ 1−ρ2η . h h h,t 7

cessinwhichs˜ respondstoconsumptiongrowthitselfratherthaninnovations. t Thisalternativeassumptiondoesnotchangethemainresults. Unliketheotherstatevariables,theresidualisnotidentifiedbyeitherconsumptionordividends. ItissimplyanAR(1) e =ρ e +σ η (6) t e t−1 e e,t η ∼N(0,1) i.i.d. e,t andisthusidentifiedprimarilybytheprice-dividendratio. e captureseveryt thinginmarketvolatilitythatisnotlongrungrowth,longrunvolatility,orhabit. 2.2. ModelFrequencyandData Weassumethemodelfrequencyisannual,thesamefrequencyasthedata we use. This differs from the typical approach in the literature which tests monthlymodelsagainstannualdatamoments. We choose this approach for two reasons. The first is that monthly consumption and dividends exhibit stark seasonality which is entirely unaccounted for by models. The enormous end-of-quarter boosts to dividend growth and spikes in consumption at the end of the year suggest that risk is properly understood at an annual horizon. Indeed, if monthly risk is relevant toagentsintheeconomy,whywouldweobservesuchstarkseasonalityinequilibrium? Moreover,modelingthisseasonalityisnotasimpletask. Simpledeterministic month or quarter fixed effects do a poor job, leading to the Census Bureau’s sophisticated X-13ARIMA-SEATS seasonal adjustment. As discussed in FersonandHarvey(1992),theCensusBureauadjustmentsareforward-looking: Theyboostthecurrentmonth’sobservationifthefuturemonthsarehigh. This forward-lookingandsmoothedseriesisdifficulttointerpretinamodelofconsumptionrisk. The second reason we use an annual model is that the robustness of asset pricing frameworks to changes in model frequency is an interesting question in itself. The annual frequency is particularly relevant, as annual data is far moreaccessibleanduncontroversial. Indeed,monthlynondurableconsumptionisneverdirectlyobserved,andinsteadiscalculatedbyholdingfixedshares observed every five years (Wilcox (1992)). Time-aggregating a monthly model 8

totheannualhorizonispossiblebutdramaticallyincreasesthecomplexityof modelevaluation(Schorfheide,Song,andYaron(2016)). Thus, we estimate the model using annual consumption, dividend, and stockpricedatafromtheBureauofEconomicAnalysis(BEA)andtheCenterfor ResearchonSecurityPrices(CRSP).3Consumptionisrealnon-durableandservices consumption. Dividends andprices correspondto theCRSPindex. The samplerunsfrom1929to2014. 2.3. EstimationMethod Themodelcontainsanumberoflatentstatevariables, soit’simportantto useanestimationapproachthattakesfulladvantageofthedata. Tothisend,we estimatethemodelusingBayesianMCMCmethods. Suchmethodsutilizethe full likelihood of the data while maintaining computational tractability. This approachalsoavoidsthepotentiallycontentiouschoiceofmomentconditions. To evaluate the likelihood of our nonlinear model, we use a particle filter (Herbst and Schorfheide (2014)). We also take advantage of the conditionally Gaussian nature of the model to adapt the filter, following Schorfheide, Song, andYaron(2016). Toestimatethemodelparameters, weembedthefilterina standard random-walk Metropolis-Hastings algorithm. Details of the particle filterandMetropolis-HastingsalgorithmscanbefoundintheAppendix. Wefixsomeparametersoutsideoftheestimationthatareuninterestingor difficulttoidentify. The(uninteresting,forourpurposes)meansofallobservablesµ ,µ ,µ arefixedtobetheirsamplemeans. pd c d s and A are difficult to identify separately as they jointly determine the s volatility of the habit contribution to the price-dividend ratio. Thus we chose s =log0.06,closetotheCampbellandCochrane(1999)value. InSection4we estimatethisparameterandfindthatitispoorlyidentifiedbutdoesnotaffect themainresults. AssumingalternativevaluesofS¯alsodidnothaveasignificant impactonthemainresults. Similarly,σ andA jointlycontrolthevolatilityoftheresidualcontribution. e e Thus,wesetσ =1. e 3CRSP data is from Wharton Research Data Services (WRDS). wrdsweb.wharton.upenn.edu/wrds/about/databaselist.cfm. 9

2.4. PriorParameters Priors are chosen to be as diffuse as possible, while maintaining the economic interpretation of the model. Overall, our consumption and dividend priorsaresimilartothoseinSchorfheide,Song,andYaron(2016). However,the mainresultsarenotatallsensitivetothechoiceofprior,asweshowinSection 4. Prior distributions are independent and uniform for simplicity. Uniform priors are also useful because they imply that the posterior is simply a plot of thelikelihoodfunction.4 ThelefthalfofTable1showsthewiderangespannedbythepriors. Ourprior persistenceparametersareallverydiffuse,uniformbetween0and1. Similarly, thepriorrelativevolatilityoflongrungrowthshocks,ϕ ,isuniformbetween0 x and1. Otherpriorsarediffusetoo. Forexample,theprioronthemeanvolatility ofconsumptionshocksisbetween0.1%and4.0%annually. Priors on price-dividend coefficients are empirically motivated. The bounds allow for the possibility that each state variable can account for all of market volatility (assuming standard consumption growth parameters in the literature). For example, the upper bound on the long run growth coefficient A solves x Var(∆pd )≈A ϕ σ (7) t x x (cid:112) whereϕ =0.038andσ=0.0072× 12asinBansal, Kiku, andYaron(2012a), x andVar(∆pd )=0.23inourdatasample. Weusetheanalogousexpressionsto t equation (7) for the other state variables. The exact magnitude of the bounds isnotimportantunderuniformpriors,however. Aslongastheboundexceeds themassofthelikelihood,theposteriorislargelyindependentofthebound. The signs of the price-dividend coefficients are also restricted to be consistentwiththeory. Thatis,werestrictthecoefficientsonlongrungrowthand surplusconsumptiontobepositive,andwerestrictthecoefficientsonlongrun 4ThisisjusttheresultofBayesRuleandtheconstancyofuniformpriors p(parameters|data)=[Constants]p(data|parameters)p(parameters) =[Constants]p(data|parameters). 10

volatilityandtobenegative. 2.5. PosteriorParameterEstimates TherighthalfofTable1showstheposteriorestimates. Beginningatthetop of the table, the posteriors on simple consumption and dividend parameters are standard. The mean volatility of consumption innovations σ is about 1% per year, and dividend innovations are roughly 6 times as volatile as the consumptioninnovations. Inthelongrunriskssection,weseethattheestimatorfindsevidenceofsignificantlongrunrisksinrealeconomicgrowth—thatis,expectedconsumption growthandconsumptionvolatilitybothcontainhighlypersistentcomponents, with autocorrelations of about 0.90 annually. Since the identification of long runrisksisanimportantissueintheliterature,Figure1takesacloserlookand plotstheposteriorparameterdistributions. The figure shows that the high persistence of long run growth and volatilityareestimatedratherprecisely. Theentiredistributionoftheseparametersis above0.80. Longrunrisksvaryovertime, thatis, therelativevolatilityoflong rungrowthshocksϕ andthevolatilityofloglongrunvolatilityσ arestatisx h ticallyandeconomicallysignificant. Intermsofmagnitudes,theseparameters aresimilartoSchorfheide,Song,andYaron(2016)’sestimateswhichomitasset pricedata. MovingdownTable1,habitisestimatedtobehighlypersistentwithanautocorrelationofabout0.95,similartoCampbellandCochrane(1999)’scalibration. Thepersistenceoftheresidualisalsoabout0.95,andmoreoveritisprecisely estimated, with a lower bound of about 0.90. This high persistence illustratesacriticalproblemwithlongrunrisksandhabit: Theportionofasset pricesthattheydonotexplainisverylonglived. Now we come to the main parameters of interest: the price-dividend ratiocoefficients. Theseparametersdeterminethecontributionofeachstateto marketvolatility. Critically, the residual coefficient is estimated to be quite high at about 15% per quarter. These parameters imply that the residual component of price/dividends has an unconditional volatility of above 50%, more than 11

Table1: ParameterEstimates Thetableshowspriorandposteriorparameterestimatesforthemodel pd =µ +A x +A σ˜2+A s˜ +A e t pd x t V t s t e t ∆c t =µ c +x t−1 +σ t−1 η c,t ∆d t =µ d +φ x x t−1 +φ ηc σ t−1 η c,t +ϕ d σ t−1 η d,t (cid:113) x t =ρ x x t−1 +ϕ x 1−ρ x 2σ t−1 η x,t (cid:113) h t =ρ h h t−1 +σ h 1−ρ h 2η h,t , σ t =σexp(h t ) s˜ t =ρ s s˜ t−1 +λ(s˜ t−1 )(∆c t −(cid:69) t−1 ∆c t ) e t =ρ e e t−1 +σ e η e,t wherepd isthelogprice-dividendratio,∆c isconsumptiongrowth,∆d isdividend t t t growth, and η ’s are standard normal independent noise. Prior distributions are int dependent and uniform. Posteriors are computed using annual consumption, dividend,andstockpricesfrom1929-2014,particlefilter,andMetropolisHastings.σ =1, e exps = 0.06 are chosen outside of the estimation, as are µ , µ , and µ , which are pd c d chosentobetheirsamplemeans. Parameter Prior Posterior 0% 100% Mean 5% 50% 95% SimpleConsumptionandDividends MeanVolofConsShocks σ 0.001 0.040 0.0098 0.0047 0.0094 0.0162 DivLoadingonConsShock φ ηc 0 10 1.15 0.396 1.2 1.75 RelVolofDividendShocks ϕ 0 10 6.13 5.37 6.19 6.71 d LongRunRisks PersistenceofLRGrowth ρ 0 1 0.892 0.842 0.886 0.96 x RelVolofLRGrowthShocks ϕ 0 1 0.183 0.14 0.443 0.562 x DivLoadingonLRGrowth φ 0 10 2.37 1.85 2.38 2.84 x PersistenceofLRVol ρ 0 1 0.895 0.85 0.895 0.943 h VolatilityofLogLRVol σ 0 1.5 0.954 0.864 0.964 1.01 h HabitandResidual PersistenceofSurplusCons. ρ 0 1 0.933 0.804 0.954 0.999 s PersistenceofResidual ρ 0 1 0.939 0.891 0.94 0.983 e Price-DividendCoefficients LRGrowthCoefficient A 0 243 39.1 22.4 37.9 59.2 x LRVolCoefficient A -2.4e4 0 -87.7 -149 -84.2 -39.4 V SurplusCons. Coefficient A 0 0.93 0.282 0.0818 0.294 0.45 s ResidualCoefficient A 0 0.23 0.147 0.11 0.147 0.186 e 12

Figure1:LongRunRiskParameterEstimateDetails. Plotsshowposteriordistributions of long run risks parameters from Table 1. The estimator finds significantevidenceofpersistentchangesinexpectedgrowthandvolatility. Other posteriordistributionscanbefoundintheAppendix. 3 2.5 2 1.5 1 0.5 0 0 0.5 1 PersistenceofLongRunGrowthρ x ytisned 3 Prior Posterior 2.5 2 1.5 1 0.5 0 0 0.5 1 PersistenceofLongRunVolρ h ytisned 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 RelativeVolofLRGrowthShockϕ x ytisned 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 VolofLogLRVolatilityσ h ytisned 13

enough to account for the entire unconditional volatility of the log pricedividend ratio (roughly 40%). The large role of the residual is seen in the shrunkenposteriorsoftheotherprice-dividendcoefficients. Sincepriorswere chosen so that each state could account for all market volatility, the posterior coefficients on long run growth, long run volatility, and surplus consumption shrinkdramaticallytowardzero. 2.6. MainResult: Price-DividendRatioDecompositions Withparameterestimatesinhand,wecannowaddressthemainquestion ofthepaper: Whichsourceofmarketvolatilityisthemostimportant? Figure 2 provides an answer to this question. It plots the historical path of the price-dividend ratio and decomposes the path into contributions from each state variable (see equation (1)). The states are extracted with a particle smoother(Godsill,Doucet,andWest(2004))usingmeanposteriorparameters from Table 1. The contribution of a state is the mean smoothed state multipliedbyitsrespectivemeanposteriorcoefficient(thecontributionoflongrun growthisA x ). x t The figure shows that the residual (yellow bars) played a dominant role in market volatility between 1929 and 2014. The residual is responsible for the relativelylowassetpricesinthe1940s,theboomofthe1950s,thebearmarket ofthe1970s,theepicriseofvaluationsbetween1980and2000,andthesharp crash in the early 2000s. Indeed the residual closely tracks the price-dividend ratio(blueline)forthevastmajorityofthesample. Long run risks and habit play a non-trivial role. In particular, they weigh heavilyonassetpricesduringtheGreatDepressionandGreatRecession. Long run growth and habit also boost prices somewhat throughout the 1950s and 1960s. Compared to the residual, however, long run risks and habit are relatively unimportant. Indeed, outside of the two major crises, long run volatility has almost no effect. And while the effects of long run growth and habit are more visible, their contributions are economically small and often have the oppositepatternofthemovementsseeninassetprices. Thislastresultis,perhaps, intuitive: economicgrowthdeclinedslowlybetween1960and2014, whilethe price-dividendratiohastrendedupwardsoverall. 14

Figure2: DecompositionoftheHistoricalLogPrice-DividendRatio. Thefigureplotsthedecompositionimpliedbyequation(1) pd =µ +A x +A σ˜2+A s˜ +A e t pd x t V t s t e t wherepd islogprice/dividends,x islongrungrowth,σ˜ islongrunvolatility, t t t s˜ issurplusconsumption,ande istheresidual. Thestatesarethemeanstate t t foundbyparticlesmootherusingmeanposteriorparameters(Table1). CoefficientsarethemeanposteriorcoefficientsinTable1. Theresidualcontribution (yellow)isdominantandcloselytrackstheprice-dividendratio. 1 0.5 0 -0.5 -1 1930 1940 1950 1960 1970 1980 1990 2000 2010 oitaR dnediviD-ecirP goL ot snoitubirtnoC Demeaned Log Price-Dividend Long Run Growth Long Run Volatility Surplus Consumption Residual 15

Figure2iscreatedusingmeanposteriorparameters,whichdonotaccount for estimation uncertainty. Is the dominant role of the residual robust to the dispersionseenintheposteriorparameters(Table1)? Figure 3 shows that the dominant role of the residual is far too large to be due to estimation uncertainty. The figure plots a variance decomposition of the price-dividend ratio using the entire distribution of posterior parameters. Thevariancedecompositioniscalculatedbytakingthecovarianceofequation (1)withpd t Var(pd )=Cov(A x ,pd )+Cov (cid:0) A σ˜2,pd (cid:1) t x t t V t t +Cov(A s˜ ,pd )+Cov(A e ,pd ) (8) s t t e t t whichcanleadtonegativesharesifastatevariablehasanegativesamplecorrelationwithpd . t The figure shows that the residual’s share of price-dividend variance is almostentirelyabove75%. Indeed,theresidual’smeanshareisabout95%,showing that something other than long run risks and habit accounts for nearly all ofthepast85yearsofmarketvolatility. Long run growth, long run volatility, and habit have very small shares, accounting for the remaining 5% of market variance altogether. These small shares are consistent with the scarcity of crises in our 85 year sample. There issomeuncertaintyintheseestimates. Thedistributionsofthesharesforlong run growth and volatility cover up to 25% and 15% respectively. Nevertheless thesesharesseemtobenegativelycorrelated,astheresidualshareveryrarely fallsbelow75%. This accounting for market volatility is somewhat complicated by the fact that shares are sometimes negative or above 100%. These negative shares are duetothefactthatthevariancedecompositioniscomputedusingsamplecovariances(equation(8)),whichcanbenegative. Indeed,economicgrowthdeclinedoverthesamplewhileassetpricesgrew,whichintuitivelyleadstoanegative share. Habit faces a similar in-sample correlation problem. We discuss bothoftheseissuesinmoredetailinSection3.2. Thissectionillustratesthemainmessageofthepaper: whilelongrunrisks and habit play a non-trivial role in asset prices, something else is behind the vastmajorityofmarketvolatility. Thisresultisrobusttoestimationuncertainty, 16

Figure 3: Price/Dividend Variance Decomposition and Estimation Uncertainty. Sharesareinpercentandimpliedbythedecomposition Var(pd )=Cov(A x ,pd )+Cov (cid:0) A σ˜2,pd (cid:1) t x t t V t t +Cov(A s˜ ,pd )+Cov(A e ,pd ) s t t e t t wherepd islogprice/dividends,x islongrungrowth,σ˜ islongrunvolatility, t t t s˜ is surplus consumption, e is the residual, and A’s are the coefficients from t t equation(1). Thedensitiesarecomputedbydrawingparametersfromtheposterior(Table1),usingthedrawtofindsmoothedmeanstates,andcalculating variancecontributionsaccordingtotheaboveequation. Coefficientsaremean posteriors. Theplotsshowthattheresidual’sdominantroleisrobusttoestimationuncertainty. 2 1.5 1 0.5 0 -25 0 25 50 75 100 125 Long run growth share ytisneD ×10-3 0.01 Mean=7.0 0.005 0 -25 0 25 50 75 100 125 Long run volatility share ytisneD Mean=3.9 6 4 2 0 -25 0 25 50 75 100 125 Surplus consumption share ytisneD ×10-3 2 Mean=-5.7 1.5 1 0.5 0 -25 0 25 50 75 100 125 Residual share ytisneD ×10-3 Mean=94.8 17

and indeed, Section 4 shows that it is also robust to several prior and model specifications. 3. Supporting Results Wenowpresentevidenceinsupportofourmainresults. Weshowthatthe estimatedstatesareintuitive—thatis,theymatchtherelatedobservablesand narrativedescriptionsofeconomichistory. WealsopresentasimpleOLSversionofourprice-dividenddecompositionthatgeneratessimilarresults. TheOLSdecompositionalsohelpsexplainwhytheresidualissodominant. The large role for the residual comes from the low correlation between asset pricesandrealeconomicgrowthorvolatility(pastandfuture). 3.1. EstimatedStatesMatchObservables Likelihood-basedestimationsgeneratehistoricalestimatesoflatentstates. These estimates provide an intuitive check on the price-dividend decompositions. Withthemwecanask: doestheestimatordoagoodjobdescribingeconomichistory? Figures4and5showtheanswerisyes. Theseplotsshowestimatedhistoricalpathsforlongrungrowth,longrunvolatility,habit,andtheresidual. These pathsarecomputedbyusingmeanposteriorparametervalues(Table1)anda particlesmoother. ThetoppanelofFigure4showsthehistoricalpathoflongrungrowthalong with demeaned consumption growth. The long run growth path does a good job of capturing historical shifts in growth. The path identifies the Great Depression, the relatively booming 60s, as well as the productivity slowdown of the1970s. Movementsinexpectedgrowtharesmallbutpersistent,consistent withthelongrunrisksstory. ThebottompanelofFigure4showsthehistoricalpathoflongrunvolatility, alongwiththeabsolutevalueofdemeanedconsumptiongrowth. Theestimator does a good job of picking up key historical patterns: the decline in volatility afterWorldWar2,thereturnofvolatilityinthe1970s,theGreatModeration,as wellastherecentreturnofvolatilityin2008. 18

Figure 4: Smoothed States and Observables Part 1 of 2. We apply a particle smoother to data on consumption, dividends, and stock prices, using mean posteriorparameters(Table1). Scatteredx’splotobservablesforcomparison. The smoothed paths are intuitive and capture historical shifts in growth and volatility. 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 1930 1940 1950 1960 1970 1980 1990 2000 2010 htworG nuR gnoL 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 htworG noitpmusnoC mean state 90% CI demeaned consumption growth (right axis) 0.12 0.1 0.08 0.06 0.04 0.02 0 1930 1940 1950 1960 1970 1980 1990 2000 2010 ytilitaloV nuR gnoL mean state 90% CI abs(demeaned consumption growth) 19

Figure 5: Smoothed States and Observables Part 2 of 2. We apply a particle smoothertodataonconsumption,dividendsandstockpricesusingmeanposterior parameters (Table 1). x’s plot observables for comparison. Scaled consumption growth is demeaned consumption growth multiplied by the steady state λ(s ) (see equation (5)). Surplus consumption responds slowly to cont sumptiongrowth,andtheresidualcloselytracksprice-dividends. 2 1 0 -1 -2 -3 -4 1930 1940 1950 1960 1970 1980 1990 2000 2010 noitpmusnoc sulpruS mean state 90% CI scaled consumption growth 1.5 1 0.5 0 -0.5 -1 -1.5 1930 1940 1950 1960 1970 1980 1990 2000 2010 laudiseR mean state 90% CI demeaned log price-dividend Figure 5 shows the remaining two states: surplus consumption and the residual. Thetoppanelshowsthehistoricalpathofsurplusconsumptionalong with scaled consumption growth. Consumption growth is scaled by subtractingoutitsmean, andthenmultiplyingbythesteadystateλ(s ), toimitatethe t “shock”terminthehabitprocess(5). Thehistoricalpathofsurplusconsumptionisintuitive. Surplusconsumption is very persistent, and responds to slowly to changes in consumption growth. Inparticular, surplusconsumptionplummetsintheGreatRecession, asnotedinCochrane(2011). ThebottompanelofFigure5showsthehistoricalpathoftheresidual,along 20

withthelogprice-dividendratio. AsnotedinSection2.6,thesetwoseriestrack eachothercloselythroughoutmostofthesample. 3.2. OLSPrice-DividendDecomposition Using the likelihood makes the estimator comprehensive: It accounts for allmomentsoftheobservables. Butalikelihood-basedestimationisalsonontrivialtodissect. Whichmomentsaredrivingtheresults? This section helps address this question by performing a simplified, OLS version of our estimation. We extract state histories using only data on consumption and dividends (excluding asset prices). We then use OLS to regress theprice-dividendratioonthestates. TheOLSestimatesfocusonthecovariancebetweentheprice-dividendratioandexpectedconsumptiongrowth,consumptionvolatility,andpastconsumptiongrowth. ThesecovariancesleadOLS toconcludethataresidualexplainsmostofmarketvolatility. Whenextractingstatehistories,weskipestimationofparametervaluesand instead use values from Bansal, Kiku, and Yaron (2012a) and Campbell and Cochrane (1999). This approach makes clear that the estimation of cash flow parametersisnotbehindourdecompositionresults. Table2showstheparametervaluesweuse. Parametersareconvertedfromthemonthlyvaluesusedin theoriginalpapersbysimpletransformations([annualpersistence]=[monthly persistence]12). Theparametersshownalsoaccountforourmodifiedvolatility process and the functional form of the conditional volatilities (equations (3) and(4)). HistoricalpathsforlongrungrowthandlongrunvolatilityarefoundbyapplyingaparticlesmoothertothedataonconsumptionanddividendsusingparametersfromTable2. ThepathsareplottedinthetoptwopanelsofFigure6. Long run growth and long run volatility follow paths that are similar to those from the baseline, and generally pick up the same historical features (see Figure 4). These two states deviate from their historical means primarily in the Great Depression and Great Recession. Long run growth is less volatile in the 1930s compared to the baseline, as the smoother no longer ties growth to the violentassetpricemovementsinthatera. Surplus consumption is constructed by initializing at the Campbell and 21

Table2: OLSDecomposition: CashFlowParameterChoices For the OLS decomposition, we apply a particle smoother to consumption and dividend data using parameter values from Bansal, Kiku, and Yaron (2012a) (BKY) and CampbellandCochrane(1999). Themodelisthesameasthebaseline(equations(1)- (6)).Thedata,model,andparametervaluesareannual.Parametersareconvertedfrom themonthlyvaluesusedtheoriginalpapersbyapplyingsimpletransformations: [annual persistence] = [monthly persistence]12, [annual volatility] = [monthly volatility] (cid:112) × 12. Parameterarealsotransformedtoaccountforthevolatilityspecification(see footnoteonpage7). Value Source SimpleConsumptionandDividends ConsumptionVol σ 0.0249 BKY DivLoadingonConsShock φ ηc 2.60 BKY RelativeVolofDividends ϕ 5.96 BKY d LongRunRisks PersistenceofLRGrowth ρ 0.74 BKY x RelativeVolofLRGrowth ϕ 0.17 BKY x DivLoadingonLRGrowth φ 2.50 BKY x PersistenceofLRVol ρ 0.99 BKY h VolatilityofLRVol σ 2.09 BKY h Habit PersistenceofSurplusCons. ρ 0.87 Campbell-Cochrane s SteadyStateSurplusCons. s log0.06 Campbell-Cochrane 22

Figure6: SimpleStateHistories. Long run growth and long run volatility are found by applying a particle smoother to consumption and dividend growth using parameter values from Bansal, Kiku, and Yaron (2012a) (Table 2). Surplusconsumptionisconstructedusingconsumptiongrowthdata,thesurplus consumptionprocess(5),andparametervaluesfromCampbellandCochrane (1999). Thebottompanelshowsconsumptiongrowthscaledbythesteadystate λ(s ). Thestatepathsaresimilartothosefromthebaselineestimation(Figures t 4and5). 0.02 0.01 0 -0.01 -0.02 -0.03 1930 1940 1950 1960 1970 1980 1990 2000 2010 htworG nuR gnoL 0.1 0.05 0 -0.05 -0.1 -0.15 htworG noitpmusnoC mean smoothed demeaned consumption growth (right axis) 0.12 0.1 0.08 0.06 0.04 0.02 0 1930 1940 1950 1960 1970 1980 1990 2000 2010 ytilitaloV nuR gnoL mean smoothed abs(demeaned consumption growth) 2 0 -2 -4 -6 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 noitpmusnoC sulpruS mean smoothed scaled consumption growth 23

Cochrane(1999)steadystateandthenapplyingthesurplusconsumptionprocess(5),usingourlongrungrowthestimateas(cid:69) ∆c . Weassumeρ =0.87,as t−1 t s inCampbellandCochrane(1999). Theresultingpath(bottompanelofFigure 6)lookssimilartothebaseline(seeFigure5). Likethelongrunriskstates,surplus consumption experiences large movements in the Great Depression and GreatRecession,butotherwiseisrelativelyconstant. The final step of our simplified estimator uses OLS to regress the pricedividendratioontheextractedstatehistories. Table3showstheresultingcoefficients,R2’s,andprice-dividendvariancedecomposition. Thetableshowstwo specifications: one which includes all state histories and one which includes onlyvariablesthathavethe“rightsign.” Table3: OLSEstimatesandVarianceDecomposition Wefindsmoothedpathsforlongrungrowth,volatility,andsurplusconsumptionwithoutusingassetprices(Figure6).Wethenregresslogprice-dividendsonthestatepaths. Var. shareistheshareofprice-dividendvarianceimpliedby Var(pd )=Cov(A x ,pd )+Cov (cid:0) A σ˜2,pd (cid:1) t x t t V t t +Cov(A s˜ ,pd )+Cov(A e ,pd ) s t t e t t wherepd islogprice/dividends, x islongrungrowth,σ˜ islongrunvolatility, s˜ is t t t t surplusconsumption,e istheresidual,andA’sarethecoefficientsfromequation(1). t The “Long Run Volatility Only” specification omits variables whose coefficients have theoppositesignofthatimpliedbytheory.Inbothregressions,theR2showsthatmost ofthevarianceoflogprice-dividendsisexplainedbyaresidual. LongRun LongRun Surplus Specification Growth Volatility Consumption Coefficient -128.9 -460.8 -0.15 Including s.e. (53.0) (87.3) (0.06) AllVariables R2(%) 30.0 Var. Share(%) -3.2 29.1 4.0 Coefficient -190.2 LongRun s.e. (54.5) VolatilityOnly R2(%) 12.0 Var. Share(%) 88.0 The“IncludingAllVariables”specificationshowsthatlongrungrowthand surplusconsumptionhavethe“wrongsign,”inthattheirsignsaretheopposite 24

of that implied by theory. Unlike the Bayesian estimator which can constrain thesecoefficientstobepositive,OLSignoresanypriorsandfindsthatanegative signisthebestfitforthedata. The“LongRunVolatilityOnly”specificationkeepsonlyvariablesthathave the right sign. These results should thus be fairly close to the baseline, which imposes priors consistent with theory. Indeed, the coefficient on long run volatilityof-190iswithintwostandarderrorsofthe-88valuefromthefullestimation(Table1). The “Long Run Volatility Only” specification produces an R2 of 12%, indicatingthattheresidualaccountsforthevastmajorityofthevarianceofpricedividends. Byconstruction,OLSminimizestheroleoftheresidual,leadingto aslightlysmallerresidualshareofprice-dividendvariance(88%)comparedto ourbaselineestimateof95%. ThehigherR2 of30%foundinthe“IncludingAll Variables”specificationstillmeansthattheresidualaccountsfor70%ofmarket volatility. Figure7illustrateswhytheresidualplayssuchalargeroleintheOLSestimates. The figure plots state paths along with the price-dividend ratio, with all variables standardized. Long run growth, long run volatility, and surplus consumptioncomovewiththeprice-dividendratioprimarilyincrisisperiods (Great Depression and 2008 Financial Crisis). But they fail to capture most of theotherpatternsinassetpricehistory. Indeed,acrucialfeatureofthepricedividendhistoryisitslongupwardtrendoverthepast85years,somethingthat isabsentfromallthreestatevariables. 4. Robustness As with any model-based econometrics, our method could potentially be sensitive to the model specification. The Bayesian approach raises the additionalconcernthattheresultscouldbesensitivetothechoiceofpriors. This section shows that our main result is quite robust. As long as the specification allows for the possibility of a large residual, the estimator concludes that the residual is dominant and closely follows the historical path of price/dividends. Thisresultholdsin(1)ourbaselinespecification,(2)ifweremovelongrunrisksfromthemodel,(3)ifweremovehabitfromthemodel,(4) 25

Figure 7: Simple States Histories vs the Price-Dividend Ratio. All variables are scaled to have zero mean and a standard deviation of 1. Long run growth andlongrunvolatilityarefoundbyapplyingaparticlesmoothertoconsumptionanddividendgrowthusingparametervaluesfromBansal,Kiku,andYaron (2012a) (Table 2). Surplus consumption is constructed using consumption growthdata,thesurplusconsumptionprocess(5),andparametervaluesfrom Campbell and Cochrane (1999). All states move in crises but otherwise have littlecorrelationwithprice-dividends. 3 2 1 0 -1 -2 -3 -4 -5 -6 1920 1940 1960 1980 2000 2020 stinU dezidradnatS Log Price-Dividend Long Run Growth -1*Long Run Volatility Surplus Consumption 26

ifwealterthepriorcorrelationstructure,(5)ifwespecifythathabitrespondsto consumption growth rather than innovations, (6) if we rescale price-dividend coefficients for the variance of the states. Indeed, this result holds for every specification that we have examined in the course of writing this paper (that allowsforaresidual). Table 4 summarizes the robustness results. The table shows the shares of variance (equation 8) accounted for by long run growth, long run volatility, habit, and the residual across the six model and prior specifications listed above. Underallsixspecifications, theresidualaccountsforthevastmajority ofmarketvolatility,withaminimumshareof80%. Table4: P/DVarianceSharesinAlternativeModelSpecifications Figuresshowpercentcontributionstothevarianceofthelogprice-dividendratioimpliedby Var(pd )=Cov(A x ,pd )+Cov (cid:0) A σ˜2,pd (cid:1) t x t t V t t +Cov(A s˜ ,pd )+Cov(A e ,pd ) s t t e t t where pd is log price/dividends, x is long run growth, σ˜ is long run volatility, s˜ t t t t issurplusconsumption, e istheresidual, and A’sarethecoefficientsfromequation t (1).Thetableshowsthemeanandstandarddeviation(inparentheses)oftheposterior distributionoftheshares.Sharesarecomputedusingthesmoothedstatesevaluatedat 5,000drawsfromtheposteriorparameterdistribution. Regardlessofthespecification, theresidualisthedominantsourceofmarketvolatility. (1) (2) (3) (4) (5) (6) Varianceshare Baseline noLRR nohabit alt.ϕ alt.habit Arescaled x Long-runGrowth 6.99 15.27 -1.59 -2.61 14.66 (9.83) (3.62) (8.21) (7.61) (3.26) Long-runVolatility 3.90 4.49 14.09 4.13 1.95 (3.28) (4.04) (6.35) (2.84) (2.62) SurplusConsumption -5.72 6.81 0.24 -2.30 -0.86 (3.49) (4.26) (2.28) (3.52) (1.62) Residual 94.83 93.19 80.24 87.25 100.77 84.25 (7.71) (4.26) (4.38) (9.21) (7.09) (2.85) Figure 8 plots the residual under these six model specifications. Regardless of the specification, the residual is very highly correlated with the pricedividend ratio and marks most of the key events in stock market history. Indeed,theroleoftheresidualisveryconsistent: regardlessofthespecification, 27

Figure8: ResidualContributionstothePrice-DividendRatioUnderAlternativeSpecifications. LinesshowA e from e t pd =µ +A x +A σ˜2+A s˜ +A e t pd x t V t s t e t wherepd islogprice/dividends,x islongrungrowth,σ˜ islongrunvolatility, t t t s˜ is surplus consumption, and e is the residual. Each line shows a different t t model specification (Table 4). e is the smoothed mean residual computed at t themeanposteriorparameters,andA isthemeanposterior. x’sshowthedee meaned log price-dividend ratio for comparison. Regardless of the specification, the residual tracks recognizable features of the price-dividend ratio outsideoftheGreatDepressionand2008FinancialCrisis. 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 1930 1940 1950 1960 1970 1980 1990 2000 2010 dnediviD-ecirP gol ot snoitubirtnoC laudiseR Baseline noLRR nohabit alt. ϕ x alt. habit Arescaled demeanedlogprice-dividend theresidualdrivesthestockmarketoutsideoftheGreatDepressionandGreat Recession. Thealternativespecificationshaveintuitivemotivations. Theremainderof thissectiondiscussesthesemotivationsandsomedetailsofeachresult. Acommonthemeinourresultsisthatlongrunrisksandhabitbothcapture crises. Anaturalconcernisthatthiscorrelationpollutesourresultsregarding theresidual’sshareofmarketvolatility. ThenoLRR(Table4,Column(2))andno 28

habit (Column (3)) specifications examine this concern. Column (2) removes habitandColumn(3)removesthelong-runriskfactorsfromtheprice-dividend ratioequation(1). Table4showsthat,whenlong-runrisksareremoved,theestimationassigns avarianceshareofabout9%tohabit. Similarly, whenhabitisremoved, longrunrisksaccountsforavarianceshareofabout15%. Underbothspecification, the residual plays a dominant role, making it clear that competition between thedifferentmacro-assetpricingfactorsisnotdrivingourresults. Our baseline assumes that the prior parameters are independent, but this implies that other properties of the model are correlated. The alt. ϕ specifix cation(Column(3)ofTable4)teststheimportanceofourindependencestructure. Specifically,wereplaceEquation(3)with x =ρ x +ϕ σ η , t x t−1 x t−1 x,t thusremovingtheadjustmentterm (cid:112) 1−ρ2 fortheautocorrelationinthelongx rungrowthprocess. Wethenplaceauniformpriorontherelativevolatilityof long-rungrowth: ϕ ∼(cid:85) ([0,0.2]). Table4showsthatunderthisspecification, x thevarianceshareoflong-rungrowthissomewhatlowerthaninthebaseline, andtheshareoflong-runvolatilityishigher. Thedominantshareoftheresidual remains. Another concern readers may have is that we assume a specific relationship between long run risks and surplus consumption. We assume that habit responds to consumption innovations, which leads to the two state variables interactinginaspecificway. Thealt. Habitspecification(Table4Column(5)) shows that an alternative and intuitive specification leads to similar results. Specifically,wereplaceEquation(5)with s˜ =ρ s˜ +λ(s˜ )(∆c −µ ). t s t−1 t−1 t c Underthisalternativehabitprocess,surplusconsumptionchangesinresponse to all changes in consumption growth, not only unexpected changes. This means that long-run growth x also enters the current habit state. This alt−1 ternativespecificationintroducesastrongtheoreticalcorrelationbetweenthe long-rungrowthstateandthehabitstate,butisclosertotheoriginalformulationofCampbellandCochrane(1999). Table4showsthat,underthisalterna- 29

tive,theresidualtakesupevenmoreofthevariationintheprice-dividendratio thaninthebaseline. Finally, some readers may be concerned about our price-dividend coefficient priors, as these are not a standard type of variable to place priors over. Indeed,baselinepriorforthesevariableswaschosenforsimplicityratherthan acarefulstatisticaloreconomicargument(Section2.4). The A rescaled specification (Table 4, Column (6)) places a different prior structure on the four coefficients of the price-dividend equation (1). Specifically,wechoosethepriorssuchthatthetheoreticalvarianceofthefactorsinthe price-dividendequationareidenticallyandlog-normallydistributed. Indoing so, we avoid as much as possible that the prior favors of any state variable in the variance decomposition, which is our prime object of interest in this paper. WeconstructfourindependentrandomvariablesT ,T ,T ,T thatarelogx V s e normallydistributedwithT ∼log(cid:78) (cid:0)µ ,σ2 (cid:1) ,i = x,V,s,e.Wethenconstruct i T T the A coefficientsconditionalonthevaluesoftheremainingmodelparamex tersθ asfollows: (cid:118) (cid:118) (cid:116) T (cid:116) T A = x , A =− V , x (cid:86)[x |θ] V (cid:86)[σ˜ |θ] t t (cid:118) (cid:118) (cid:116) T (cid:116) T A = s , A = e . (9) s (cid:86)[s˜ |θ] e (cid:86)[e |θ] t t Here, (cid:86)[x |θ] etc. are the theoretical variances of the state variables condit tionalontheothermodelparameters. Notethatwerestrictthesignsofthecoefficientstoconformtoeconomicintuition. Thatis,werestrictthecoefficients onlong rungrowthand surplusconsumption tobepositive, andthat onlong runvolatilitytobenegative. Theresultofthispriorchoiceisthatthepriordistributionofthevariancesofthefactorsconditionalonanyθ aregivensimply by the T ’s, and in particularly i.i.d. among each other. We set σ2 = 2 and µ i T T such that the unconditional prior variance of the price-dividend ratio equals the observed variance in the data. Other, similarly diffuse distributions of the T ’sproduceverysimilarresults. i Table4’scolumn(6)makesitclearthatthepriorstructureonthefactorloadingsinEquation(1)donotmattermuchforthehistoricalvariancedecomposition. Theresultsareverysimilartothebaseline. 30

5. Interpretation of the Residual We’veshownthattheresidualisresponsibleforthebulkofmarketvolatility. Butwhatdoesthisresidualrepresent? Broadlyspeaking,thefluctuationsintheresidualareakindofexcessstock market volatility. The residual moves closely with the price-dividend ratio, is unrelatedtoaverageeconomicgrowth(pastorfuture),andisalsounrelatedto realvolatility. Thisdescriptionmatchesseveraltheoriesintheliterature. Thetheoriesfit intotwobroadcategories: tractablerepresentativeagentmodelswithhard-toobserveshockstorisk(suchasvariabledisasterrisk)andmorecomplexmodelsthatlinkexpectedreturnstoobservablesotherthanconsumptionanddividends (such as incomplete market models). We cannot distinguish among these theories in this paper, but this section explains how these theories are consistentwithourevidence,andsuggestsavenuesforfutureresarch. 5.1. TheResidualasaHard-to-Observe,Time-VaryingRisk As the residual represents excess volatility, it naturally maps to hard-toobservevariationsinrisk. Thiskindofmodelinghasthevirtueofbeinghighly tractable,andthusleadstoexplicitpredictionsaboutavarietyofassetmarket phenomena(TsaiandWachter(2015)). To see how the residual can be modeled as hard-to-observe variations in risk,supposeconsumptiongrowthexperiencesraredisastershocks J t ∆c =µ +ση +J (10) t c c,t t ∆d =µ +φ ση +ϕ σ η +φ J t d ηc c,t d t d,t J t  J¯, withprob e J = t (11) t 0, otherwise andthattheprobabilityofadisastere isanAR(1)process t e =e¯+ρ e +σ η . (12) t e t−1 e e,t Close the model with a representative Epstein-Zin household, and standard 31

log-linearapproachesshowthattheprice-dividendratioisapproximately pd ≈µ +A (e −e¯) (13) t pd e t exp (cid:2)(φ −γ)J¯(cid:3)−1+ γ− ψ 1 (cid:0) exp (cid:2)(1−γ)J¯(cid:3)−1 (cid:1) A =− J 1−γ . (14) e 1−κ ρ 1 e whereψandγaretheintertemporalsubstitutionandtheriskaversionparametersoftherepresentativehousehold,respectively. Equations(10)-(13)showthattheprice-dividendratiomovesaroundinresponsetoavariablee thatisalmostentirelyunconnectedtoconsumptionand t dividendgrowth. e showsupinequation(10)astheprobabilitythat J >0,but t t the rare nature of these disasters means that (10) is empirically equivalent to a process in which J = 0 all the time. More formally, simulating this model t andapplyingourBayesianestimationtothesimulateddatawouldresultinA e coefficientsthataresimilartowhatwefoundinU.S.data. Thus, the probability of disaster functions just like a residual in the pricedividendequation. Butotherkindsofhard-to-observerisksactsimilarly,forexample,thechangesinthemagnitudeofambiguity(SbuelzandTrojani(2008)) or white noise shocks to habit (Bekaert, Engstrom, and Xing (2009)). Indeed, onecouldaddhard-to-observeshockstoothermodelsofassetpricesandlikely achieveasimilarresults. The simplicity of this modeling approach means that it has the potential to be extended to generate additional quantitative predictions. In production economies,increasesinhard-to-observerisksleadtoclearlyvisibledeclinesin outputandinvestment(Gourio(2012),IlutandSchneider(2014)). Similarreal effectsareseeninresponsetochangesinhabit(Chen(2016))orbeliefs(Winkler (2016)). Whether production economies can help distinguish between these theoriesisaninterestingquestionforfutureresearch. 5.2. MoreComplexModelsoftheResidual Directlylinkingtheresidualtoobservablesotherthanaggregateconsumption is possible, but requires more complicated models. There are two kinds of complications which are consistent with our results and lead to additional observables: (1)incompletemarkets, and(2)imperfectlyrationalagents. The complexityofthesemodels,however,makesdirectmodelevaluationdifficult. 32

Under incomplete markets, consumption risk is not shared efficiently, so aggregateconsumptionisnolongerrelevantforassetprices. Thisnotionhasa longhistorygoingbacktoMankiw(1986). Thesimplestwaytomodelincomplete markets is by introducing idiosyncratic income risk (for example, Constantinides and Duffie (1996)). Constantinides and Ghosh (2017) use GMM to estimate a model with idiosyncratic risk, but they do not use the correlation between asset prices and idiosyncratic risk in their estimation. Schmidt (2015)arguesthatinitialclaimsforunemploymentisareasonableproxyforidiosyncraticrisk,andfindsthatthismeasureishighlycorrelatedwiththepricedividendratio. Incomplete markets can also be modeled by focusing on institutional features,namelythefactthatfinancialintermediariesappeartoplayacriticalrole in asset prices (Muir (2015)). In such models, only a subset of agents in the economytradestocks,andtheseagentsarecapitalconstrained(HeandKrishnamurthy(2013),BrunnermeierandSannikov(2014)). Asaresultoftheseconstraints,financialsectorleveragebecomescloselytiedtotheprice-dividendratio. Asallsectorvaluationstendtomovetogether,thisproxymostcertainlyhas ahighcorrelationwiththeaggregateprice-dividendratio. ModelswithimperfectlyrationalagentsgoesbacktoDeLongetal.(1990). Most of this literature assumes irrational expectations motivated by psychology (for example, Hirshleifer, Li, and Yu (2015)). Barberis et al. (2015) apply thisapproachinaheterogeneousagentmodelthatbringsinsurveydata. Their model replicates the positive correlation between survey expectations of returns and the price-dividend ratio. This qualitative relationship is difficult to matchincompletelyrationalmodels(AmrominandSharpe(2013),Greenwood andShleifer(2014),Koijen,Schmeling,andVrugt(2015)). Amorerecentliteratureassumesagentsarerational,butformbeliefsfrom amisspecifiedlawofmotionforstockprices(AdamandMarcet(2011),Adam, Marcet,andNicolini(2016)). Sinceagentsrationallyupdatebeliefsaboutstock pricesbasedonobservables,thisapproachnaturallyleadstorelationshipsbetween the price-dividend ratio and non-consumption data. Adam, Marcet, andBeutel(2015)findthatthisapproachleadstopredictionsaboutthepricedividend ratio and past returns which are quantitatively consistent with the data. Theirmodelisalsoabletomatchtheevidenceonvaluationsandsurveys expectationsofreturns. 33

6. Conclusion We develop a model of asset prices that involves multiple sources of risk: longrungrowth,longrunvolatility,habit,andapersistentresidual. Themodel isestimatedusingBayesianmethodswhichaccountfortheentirelikelihoodof thedata. Wefindthattheresidualisthemostimportantsourceofrisk,accountingforatleast80%ofthevarianceoftheprice-dividendratio, aswellasmost recognizablehistoricalfeaturesoftheprice-dividendseries. Longrunrisksand habitplayarole,butonlyincrisisperiods. This analysis raises the bar for asset pricing models. Many macro finance models that are quite successful at matching moments struggle when confrontedwiththeentirelikelihoodofthedata. Simplyput,theconditionalcorrelationsbetweenassetpricesandrealvariablesaretoosmallfortheestimator toputalotofstockinrealfactors. Modelswithhard-to-observechangesinrisk(suchasvariabledisasterrisk) pass these tests, but only because they hide the mechanism from empirical scrutiny. Indeed,itisdifficulttofalsifyamodelinwhichassetpricesaredriven byfluctuationsintheconditionaldensityofrareevents. Morecomplexmodels canlinkriskchangestoobservables,buttypicallycanonlybeevaluatedbased ontheirqualitativepredictions. Nevertheless, the results of this paper illustrate the importance of unobservable drivers of asset price data. Policy makers, market participants, and academic economists who desire to understand why valuations are currently elevated, or why valuations have recently plummeted should be careful when attributingthesechangestomovementsinlongrungrowth,longrunvolatility, orconsumption-basedriskaversion. 34

7. Appendix 7.1. StateSpaceFormulation To estimate the model, we write it in a state space formulation following Schorfheide,Song,andYaron(2016). Intheend,wehavetransitionequations (cid:113) h =ρ h +σ 1−ρ2w (15) t h t−1 h h t m =Φ(m )m +Σ (m )η . t t−1 t−1 s t−1 t andobservationequations y =µ +Zm +Z (exp(2h )−exp(2σ2)) (16) t y t v t h where m is a vector of mean “states,” y is a vector of observables, w ,η are t t t t vectorsofstandardnormalindependentnoise,Φ(m ),Σ (m )arematricies t−1 s t−1 thatdescribetheevolutionofthemean“states,” andµ ,Z,Z arevectorsand y v matricies that map states to observables. We put quotes around “states” becausetheelementsofm includetermswhicharenotstatevariablesinthetrat ditionaleconomicsense. Theseadditional“states”helpsimplifynotation. Equations (15) and (16) are convenient forms for the asset pricing models withtimevaryingvolatilityandnormalshocks. Asthisclassofmodelsisconditionally normal, it helps to express the model as close to a state space form aspossible. Moreover,thisformulationallowsthemodeltobeextendedtoaccountformixedfrequencydata. TheseequationsaremappedtothemodelinSection2byacarefuldefinition ofvectorsandmatricies. We’llnowdefinethesevectorsandmatricies. 35

ObservablesandStates Equations(1)and(2)canbemappedintotheobservationequation(16)asfollows:         z ∆c µ 0 1 1 0 t t c x  ∆d =µ + 0 φ φ 1 t−1  t  d   x ηc η˜  pd µ [A ,A ,A ] 0 0 0  c,t t pd x s e η˜ (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) d,t yt µ y Z (cid:124) (cid:123)(cid:122) (cid:125) mt   0 + 0 (exp(2h )−exp(2σ2)).   t h A σ2 V (cid:124) (cid:123)(cid:122) (cid:125) Zv where z ≡[x ,s˜ ,e ](cid:48) t t t t η˜ =σexp(h )η c,t t−1 c,t η˜ =ϕ σexp(h )η d,t d t−1 d,t (cid:48) and indicatesatranspose. 36

State Transition Then state transitions (3)-(6) can be expressed in terms of theaugmentedstatespacetransitions(15)asfollows:    ρ 0 0 x  z t      ψ x λ(u t−1 ) ρ u 0   0 0 0    z t−1    x t−1  =   0 0 ρ e     x t−2    (cid:148) (cid:151)  η˜   1 0 0 0 0 0  η˜   c,t    c,t−1 η˜ d,t   0 0 0 0   η˜ d,t−1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) 0 0 0 0 mt mt−1 (cid:124) (cid:123)(cid:122) (cid:125) Φ(mt−1 )        ϕ σexp(h ) 0 0 x t−1 η   0  λ(u )σexp(h )  0  0  x,t    t−1 t−1     η  +   0 0 σ    c,t,  e  η   0 σexp(h ) 0 0   e,t  t−1  η 0 0 0 ϕ σexp(h ) d,t d t−1 (cid:124) (cid:123)(cid:122) (cid:125) η (cid:124) (cid:123)(cid:122) (cid:125) t Σ s (mt−1 ) where ψ is an indicator variable which depends on the habit specification x (ψ =0⇒habitrespondstoinnovationsinconsumptiongrowth). x 7.2. ParticleFilterDetails Withthestatespaceformulationinhand,wecannowwritedowntheparticlefilteralgorithminacompactform. Wefirstdescribethebigpictureofthe algorithm. Wethengoontogivethedetailsofhoweachdistributionisdefined. Foreacht =1,...,T,dothefollowingforparticlesi =1,...,M. 1. Beginwithasetofparticles[mi ,hi ]andweightsπi . t−1 t−1 t−1 2. Draw hi ∼ q(hi|pdi,hi ,mi ) for each i, where q(hi|pdi,hi ,mi ) is t t t t−1 t−1 t t t−1 t−1 aproposaldistributionwhichwe’lldescribeshortly. 3. Draw mi ∼ p(m |y ,hi,hi ,mi ) for each i, where t t t t t−1 t−1 p(m |y ,hi,hi ,mi ) is the conditional density of m . We’ll det t t t−1 t−1 t scribe how this density is computed shortly. Throughout this Appendix p(x|y)meanstheconditionaldensityof x given y. 37

4. Updateparticleweightsusing πi =πi [updatefactor]i (17) t t−1 (cid:20) p(hi|hi ) (cid:21) [updatefactor]i =p(y |hi,hi ,mi ) t t−1 (18) t t t−1 t−1 q(hi|pd ,hi ,mi ) t t t−1 t−1 Inasimplebootstrapparticlefilter,theupdatefactorisjustthelikelihood of y givenmi andhi. We’llexplainhowtoderivetheaboveupdatefactor t t t shortly. 5. Estimatelog-likelihoodcontribution (cid:130) (cid:140) (cid:88) logpˆ(y )=log πi [updatefactor]i (19) t t−1 i 6. Resample: if 1 <0.5redraw{πi}usingamultinomialdistribution M2(cid:80) (πi)2 t i t withprobabilities{πi}. t Since the remainder of this section discusses operations applied to every particlei,wedropthesuperscriptforeaseofreading. 7.2.1. ProposalDistributionforh ∼q(h |pd ,h ,m ) t t t t−1 t−1 We draw h based off of pd and the previous state (h ,m ). The t t t−1 t−1 basic idea is that we want to draw h as close to the true probability t p(h |pd ,h ,m )aspossibleinordertominimizeMonteCarlonoiseinthe t t t−1 t−1 particlefilter. Unfortunately,therelationshipbetweenpd andh isnonlinear t t (equations(1)and(4)). Wecan,however,usethefollowingTaylorexpansion exp(2h )≈exp(2ρ h )(1−2ρ h )+2exp(2ρ h )h (20) t h t−1 h t−1 h t−1 t whichleadstoanapproximation(16)thatislinearinh t y =µ +Zm +Z (exp(2h )−exp(2σ2)) t y t v t h ≈µ +Zm +Z exp(2ρ h )(1−2ρ h )+Z 2exp(2ρ h )h . (21) y t v h t−1 h t−1 v h t−1 t 38

Thisapproximation, combinedwiththeequation(15)andthedefinitionof y t letsuswriteaministatespacesystem pd =pd +A˜ h +σ η (22) t 0 V,t t pd pd,t (cid:113) h =ρ h +σ 1−ρ2w t h t−1 h h t whereη ∼N(0,1) i.i.d.and pd,t pd 0 = (cid:148) 0 0 1 (cid:151)(cid:2)µ y +ZΦ(m t−1 )m t−1 +Z v exp(2ρ h h t−1 )(1−2ρ h h t−1 )(cid:3) A˜ =A σ22exp(2ρ h ) V V h t−1 (cid:148) (cid:151) σ˜ pd = 0 0 1 ZΣ s (m t−1 ). In this approximation, h |pd ,h ,m is normally distributed, and a onet t t−1 t−1 step Kalman filter gives the mean and variance. We use this distribution as q(h |pd ,h ,m ). t t t−1 t−1 Inprinciple,thisapproximationcouldbeusedtogenerateaproposaldistributionforallstates[h ,s ]. Butdrawingh separatelyletsusnestSchorfheide, t t t Song,andYaron(2016)’sspecificationandhelpserrorchecking. 7.2.2. ProposalDistributionform ∼p(m |y ,h ,h ,m ) t t t t t−1 t−1 Wedrawm inasimilarwaytoh . Theonlydifferenceiswedrawm given t t t h , and thus y is linear in the unobserved m and so no approximations are t t t needed. Explicitly,givenh ,h ,thestatespacesystem t t−1 y =µ +Zm +Z (exp(2h )−exp(2σ2)) (23) t y t v t h m =Φ(m )m +Σ (m )η t t−1 t−1 s t−1 t shows that p(m |y ,h ,h ,m ) is normally distributed, and a one-step t t t t−1 t−1 Kalmanfiltergivesthemeanandvarianceofthisdistribution. Weusethisdistributiontodrawm intheparticlefilter. t 7.2.3. SimplifyingtheUpdateFactor WesimplifytheparticlefilterupdatestepbytakingadvantageoftheconditionalGaussianpropertiesofthemodelandusingBayes’theorem. 39

Thestandardgenericparticlefilterupdatefollows π =π updateweight (24) t t−1 wheretheupdateweightis p(m ,h |m ,h ) updateweight≡p(y |m ,h ) t t t−1 t−1 t t t q(m ,h |m ,h ,y ) t t t−1 t−1 t and q is the proposal distribution in the propogation step (see Herbst and Schorfheide (2014)). In our case, p(m ,h |m ,h ) and t t t−1 t−1 q(m ,h |m ,h ,y ) can be broken up into mean and volatility compot t t−1 t−1 t nents p(m |m ,h ) p(h |h ) updateweight=p(y |m ,h ) t t−1 t−1 t t−1 (25) t t t p(m |y ,h ,m ,h )q(h |y ,m ,h ) t t t t−1 t−1 t t t−1 t−1 Theaboveexpressioncanbefurthersimplified. Firstnotethat(m ,h )aresuffit t cienttodeterminethedensityofy . Similarly,h addsnoinformationregarding t t m given(m ,h ). Thus, t t−1 t−1 (cid:149) p(m |h ,m ,h ) (cid:152) p(h |h ) updateweight= p(y |m ,h ,m ,h ) t t t−1 t−1 t t−1 t t t t−1 t−1 p(m |y ,h ,m ,h ) q(h |y ,m ,h ) t t t t−1 t−1 t t t−1 t−1 (26) TheterminthebracketscanbesimplifiedusingBayes’ruletwice p(m |h ,m ,h ) p(y |m ,h ,m ,h ) t t t−1 t−1 t t t t−1 t−1 p(m |y ,h ,m ,h ) t t t t−1 t−1 p(y ,h ,m ,h ) p(m |h ,m ,h ) =p(m |y ,h ,m ,h ) t t t−1 t−1 t t t−1 t−1 t t t t−1 t−1 p(m ,h ,m ,h )p(m |y ,h ,m ,h ) t t t−1 t−1 t t t t−1 t−1 p(y ,h ,m ,h ) = t t t−1 t−1 p(m |h ,m ,h ) p(m ,h ,m ,h ) t t t−1 t−1 t t t−1 t−1 p(y ,h ,m ,h ) p(m ,h ,m ,h ) = t t t−1 t−1 t t t−1 t−1 p(m ,h ,m ,h ) p(h ,m ,h ) t t t−1 t−1 t t−1 t−1 =p(y |h ,m ,h ). (27) t t t−1 t−1 Finally, combining equations (26) and (27) gives the update factor expression intheparticlefilteralgorithm(18). 40

7.3. Particlesmootherdetails We use a variant of the backward-simulation particle smoother of Godsill, Doucet, and West (2004). Our procedure explicitly takes care of the possibilitythatthetransitionallikelihoodisdegeneratewheneverthehabitinnovation λ(u )=0iszero. t−1 Westartwithasetoffilteredparticles (cid:0) mi,hi (cid:1) withweightsπi, t =1,...,T, t t t i =1,...,M. Wethencomputeasetofsmoothedparticles (cid:0) m˜i,h˜i (cid:1) asfollows. t t • Drawk fromthedistributionπ . Set (cid:0) m˜i ,h˜i (cid:1)= (cid:128) m kiT,h kiT (cid:138) . iT t T T T T • Fort =T −1...1: (cid:128) (cid:138) 1. Checkwhetherλ u kit+1 =0: t – Ifitis,thenthefilteredparticledrawnin t +1camefromadegeneratedistribution,andso π˜k =p (cid:0) mk,hk |m˜i ,h˜i ,yo (cid:1)=I (i =k) t t t t+1 t+1 1:T whereI()isanindicatorfunction. Setk =k . it it+1 – Ifnot,then π˜k =p (cid:0) mk,hk |m˜i ,h˜i ,yo (cid:1)∼πjp (cid:0) m˜i ,h˜i |mk,hk (cid:1) t t t t+1 t+1 1:T t t+1 t+1 t t andthesedensitiesarefinite. Drawk fromthedistributionπ˜ . it t 2. Set (cid:0) m˜i,h˜i (cid:1)= (cid:128) m kit,h kit (cid:138) . t t t t Inparticular,wheneverλ (cid:128) u kit+1 (cid:138) >0wehavethatp (cid:0) m˜i ,h˜i |mk,hk (cid:1)=0for t t+1 t+1 t t allk withλ(cid:0) uk (cid:1)=0. t 7.4. BayesianMCMCMethod We wrap the filter in a standard Random Walk Metropolis-Hastings algorithminordertoderiveparameterestimates(HerbstandSchorfheide(2014)). Werunstandardinitialtuningrunsofthealgorithminordertochooseagood proposaldistribution. Thatis,webeginbyfindingalocalmaximumofthelikelihood function using numerical optimization. We then run a chain of length 5,000withasymmetricstepdirectionandusethevarianceoftheposterioras 41

thestepdirectioninthenextsteps. Wealsochoosethestepsizesuchthatthe acceptance rate is a little larger than 0.3. The final MCMC chain has length 500,000. 7.5. BaselinePosteriorDestils Thesefiguresshowthedistributionsoftheallposteriorparametersinbaselineestimation(Table1). Figure9: Baseline: PosteriorDetails1of2. 100 0.4 50 0.2 0 0 0.01 0.02 0.03 0.04 0 5 10 sigbar phietac 0.4 4 0.2 2 0 0 0 5 10 0 0.5 1 vphid rhox 4 0.4 2 0.2 0 0 0 0.5 1 0 5 10 vphix phix 4 4 2 2 0 0 0 0.5 1 0 0.5 1 1.5 rhoh sigh 42

Figure10: Baseline: PosteriorDetails2of2. 10 4 5 2 0 0 0 0.5 1 0 0.5 1 rhou rhoe ×10 -3 ×10 -3 5 4 2 0 0 0 200 400 600 -1000 -500 0 Ax AV 0.4 2 0.2 1 0 0 0 2 4 6 0 1 2 Au Ae 43

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