Life-Cycle Portfolio Choices and Heterogeneous Stock Market Expectations

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Life-Cycle Portfolio Choices and Heterogeneous Stock Market Expectations Mateo Vel´asquez-Giraldo 2024-097 Please cite this paper as: Vel´asquez-Giraldo, Mateo (2024). “Life-Cycle Portfolio Choices and Heterogeneous Stock Market Expectations,” Finance and Economics Discussion Series 2024-097. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.097. 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.

Life-Cycle Portfolio Choices and Heterogeneous Stock Market Expectations‗ Mateo Velásquez-Giraldo† BoardofGovernorsoftheFederalReserveSystem‡ December 12, 2024 Abstract Surveymeasurementsofhouseholds’expectationsaboutU.S.equityreturnsshow substantialheterogeneityandlargedeparturesfromthehistoricaldistributionofactual returns. Theaveragehouseholdperceivesalowerprobabilityofpositivereturnsand agreaterprobabilityofextremereturnsthanhistoryhasexhibited. Ibuildalife-cycle model of saving and portfolio choices that incorporates beliefs estimated to match these survey measurements of expectations. This modification enables the model to greatly reduce a tension in the literature in which models that have aimed to match riskyportfolioinvestmentchoicesbyagehaverequiredmuchhigherestimatesofthe coefficientofrelativeriskaversionthanmodelsthathaveaimedtomatchageprofiles of wealth. The tension is reduced because beliefs that are more pessimistic than the historicalexperiencereducepeople’swillingnesstoinvestinstocks. G11,G40,G51,G53,E21,D15 JELCodes: ‗ThispaperhasbeensupportedbytheAlfredP.SloanFoundationPre-doctoralFellowshipinBehavioral Macroeconomics, awarded through the NBER. I thank Christopher D. Carroll, Nicholas Papageorge, and Francesco Bianchi for their guidance and support in the development of this paper. For their helpful comments,IthankDanielBarth,MichaelKeane,EvaF.Janssens,BenceBardóczy,KevinThom,andStelios Fourakis. The paper has benefited from discussions in the Economics Department at Johns Hopkins and the Board of Governors of the Federal Reserve. Finally, I thank participants at the 2021 Summer School in Dynamic Structural Econometrics, the 2022 Cherry Blossom Financial Education Institute, the 4thBehavioralMacroeconomicsWorkshop,the2022LACEA-LAMESannualmeetings,andthe2023IAAE annualconference(whichalsosupportedthispaperwithatravelgrant). †mateo.velasquezgiraldo@frb.gov. ‡Views and conclusions stated in this paper are the responsibility of the author alone and do not necessarilyreflecttheviewsoftheFederalReserveBoardorothermembersofitsstaff.

1. Introduction When a canonical model of portfolio choice over the life cycle is calibrated to reproduce the fact that most households do not invest most of their wealth in equities, coefficients of relative risk aversion exceeding 10 are usually required.1 In contrast, the literature on consumption and saving over the life cycle has found that coefficients of relative risk aversion of 2 or less are able to fit the data comfortably when labor income uncertainty is calibrated to match facts from widely available sources of micro data.2 This discrepancy generates difficulties for studies that attempt to simultaneously reproduce both groups of facts using life cycle models. Virtually all of such studies calibrate household beliefs about equity returns to match the statistical properties of actual realized returns over long periods of history.3 The thesis of this paper is that the ability of these models to simultaneouslyreplicateportfolioallocationsandsavingsdramaticallyimproves,andthe tensionintheirparameterestimatesisgreatlyreduced,iftheyarecalibratedusingsurvey measurementsofconsumers’actualexpectationsinsteadofhistoricaldata. To evaluate this proposition, I use a model with two main blocks. The first is a measurementsystembasedonAmeriks,Kézdi,etal.(2020)thatIusetoinferthedistribution of beliefs about equity returns across U.S. households from survey measurements in the Health and Retirement Study (HRS). The second is a life cycle model of saving and portfoliochoicesthatbuildsontheworkhorsemodelbyCocco,Gomes,andMaenhout(2005). This model adds a monetary cost of entering the stock market, which is common in the literature,andaproportionaltaxonstocksalesthatrepresentsearly-withdrawalpenalties from retirement plans. It also incorporates a bequest motive and age-varying medical 1Thecanonicallife-cyclemodelofportfoliochoiceisduetoCocco,Gomes,andMaenhout(2005). While thepurposeofthatpaperisnottoreproduceempiricalportfolioshares,itshowsthat,evenwithacoefficient of relative risk aversion of 10 and a conservative equity premium, the model-implied portfolio shares are higherthanthoseempiricallyobserved. Thehighestimatedcoefficientsofrelativeriskaversioncanbeseen, forinstance,inFagereng,Gottlieb,andGuiso(2017),Catherine(2021)andthispaper,allofwhichcompare theirfullmodelswithbaselinespecificationsthatbuildonCocco,Gomes,andMaenhout(2005). 2See,forinstance,Carroll(1997),Attanasioetal.(1999),andGourinchasandParker(2002). 3Asmeasured,forexample,byMehraandPrescott(1985)andShiller(1990)sincethelateXIXcentury. 2

expense risks to replicate savings late in life.4 I estimate the life cycle model targeting theageprofilesofsavingsandportfoliosintheU.S.andcomparetheresultsusingbeliefs from survey measurements with those obtained using the standard beliefs based on historical data. The fit of the model calibrated with survey measurements of expectations is dramatically better and its parameter estimates are much more plausible. For college graduates, using the beliefs from survey measurements reduces the distance between model-implied and empirical moments from the Survey of Consumer Finances (SCF) by 46 percent5 and reduces the estimated coefficient of relative risk aversion from 11.4 to 5.1. The high risk aversion required in the baseline calibration produces extremely high precautionary savings which, to match observed wealth, must be offset with an implausibly low time-discount factor of 0.6—suggesting that consumers discount future utility by 40 percent per year. With the beliefs from survey measurements, the discount factor increases to 0.9. Finally, the estimated monetary cost of entering the stock market falls from1.0percentofannualincometo$0.6 Several features of the beliefs estimated from survey measurements contribute to the improved performance of the life cycle model. The estimates imply that only 60 percent of high-school graduates and 72 percent of college graduates think there is any equity premium at all. This helps to rationalize the limited rates of stock ownership and their education gaps, and lowers the estimated monetary costs of entry. Among stock owners, the expected risk-adjusted returns are on average 13% and 18% lower than those implied by historical calibrations for high-school and college graduates respectively, allowing the model to match moderate portfolio shares with lower risk aversion.7 The consequent weakeningoftheprecautionarymotiveletsthemodelreplicatesavingswithmorepatient 4See. e.g.,DeNardi,French,andJones(2010)andAmeriks,Briggs,etal.(2020). 5Measuredbytheobjectivefunctionofthemethodofsimulatedmomentsminimizedinestimation. See Section4fordetails. 6For high school graduates, the distance to empirical moments falls by 75 percent, the coefficient of relativeriskaversionfallsfrom8.6to4.2,theannualpreferencetime-discountfactorraisesfrom0.3to0.8, andthemonetarycostofenteringthestockmarketfallsfrom3.1to2.5percentofannualincome. 7ThesefiguresrefertotheaverageSharperatioimpliedbytheestimatedbeliefsofthosewhothinkthere isanequitypremium,comparedtothehistoricalSharperatiooftheS&P500index. 3

discountingoffutureutility. Finally,heterogeneityinthebeliefsofstockownersgenerates considerable dispersion in their portfolio shares, which is an empirical fact particularly difficulttoreproduceusinghistoricalcalibrationsofbeliefs. Among the possible ways to reduce the difficulties in modeling savings and portfolios, beliefs about returns have the virtue of being susceptible of estimation from the individual-level measurements of expectations that a growing number of household surveys now include. Manski (2018), Caplin (2021), and Almås, Attanasio, and Jervis (2023) recommend using this type of measurements to resolve the challenge of separately identifying preferences and beliefs from observed choices, which traditional portfolio-choice models circumvent by assuming that beliefs match historical data. The measurements alsoproduceadditionalempiricalfactsagainstwhichmodelscanbetested. Ofparticular importanceamongthesefactsisthatmeasuredexpectationspredictportfolioallocations, arealitydemonstratedbyavastliteratureandcorroboratedbythispaper. Because the estimated beliefs differ from the historical experience, individuals with those beliefswould suffer welfare shortfallsif equities continuedto perform as theyhave in the past. These individual welfare shortfalls would be large, quantified as a share of permanent income. They follow a hump shape across the life cycle, starting at less than 3.5 percent at age 24 and peaking at the age of retirement at averages of 8.1 percent for high-school graduates and 14.2 percent for college graduates. Those who do not own stocks due to their pessimistic beliefs suffer the greatest welfare shortfalls. I analyze the variation of these shortfalls across individuals and age groups and relate the model’s predictionstofindingsinthefinancialliteracyliterature(LusardiandMitchell2023). Beyond its effect on the fit of wealth and portfolios, the choice of how to specify beliefs has stark implications for the welfare and counterfactual questions that life cycle models answer. For example, the different preferences that these models need to fit the data under historical and survey-based beliefs translate to different valuations of social programs. Using changes to unemployment insurance and Social Security benefits as 4

illustrative examples, I show that estimates of the welfare effects of policy changes can easilydifferbyfactorsoftwototenacrossmodelswithdifferentspecificationsofbeliefs. Related literature and contributions This paper relates and contributes to various groups of studies in household finance and behavioralmacroeconomics. Thefirstgroupofrelatedstudiesexploresthereasonsforthediscrepanciesbetweenthe actual portfolio choices made by households throughout their lives and the predictions of theoretical models like Merton (1969), Samuelson (1969), Viceira (2001), and Cocco, Gomes, and Maenhout (2005). Since the detection of these discrepancies, numerous studies have attempted to address them by adding various features to their models including: moreflexiblespecificationsofhouseholds’preferences(GomesandMichaelides 2003, 2005; Wachter and Yogo 2010; Calvet et al. 2021), richer models of labor income and its risks (Chang, Hong, and Karabarbounis 2018; Catherine 2021), and the addition of different costs that could be associated with stock ownership (Khorunzhina 2013; Campanale, Fugazza, and Gomes 2015; Fagereng, Gottlieb, and Guiso 2017). My paper contributestothisliteratureshowingthatifbeliefsarealignedwithsurveymeasurements, the predictions of the model come closer to the actual choices of households. This force can complement the mechanisms identified in this group of papers; for example, my estimates show that entry costs and rebalancing frictions become more powerful when householdsexpectthelowerrisk-adjustedreturnsthattheirresponsesimply. The second group of related papers analyzes survey measurements of expectations about future stock market returns.8 In this literature, a large body of work has demonstratedthatexpectationsareheterogeneousacrosspeopleandthatdifferencesinexpectations are predictive of portfolio choices.9 These facts have been corroborated in multiple 8SeeHurd(2009)andManski(2018)forreviewsonthemeasurementofeconomicexpectationsinsurveys. 9See,e.g.:DominitzandManski(2007),Hurd,VanRooij,andWinter(2011),AmrominandSharpe(2014), Drerup,Enke,andvonGaudecker(2017),Ameriks,Kézdi,etal.(2020),Giglioetal.(2021),andCalvo-Pardo, 5

surveys,samples,andcountries,aswellasbyusingdifferentwaysofelicitingexpectations. In addition to their predictive power, other important features of measured expectations about stock returns, such as pessimism (Dominitz and Manski 2007; Hurd, Van Rooij, andWinter2011),socioeconomicgradients(Das,Kuhnen,andNagel2020),androunding (Manski and Molinari 2010; Giustinelli, Manski, and Molinari 2022) have been established. A critical characteristic for modeling these measured expectations is that most of their variation comes from cross sectional differences that persist over time—individual fixed effects. Only a small part of the variation of fixed effects across individuals is explained by sociodemographic characteristics (Giglio et al. 2021). My contribution to this literatureisamodelthatIusetoestimateaninterpretablerepresentationofthepersistent component of beliefs (structural analogues to individual fixed effects) that can be incorporated into life cycle portfolio-choice models. The model, which builds on Kézdi and Willis (2011), Ameriks, Kézdi, et al. (2020), and Giustinelli, Manski, and Molinari (2022), accounts for persistent and heterogeneous rounding patterns, and its estimates capture manyoftheempiricalfeaturesofbeliefsthathavebeenhighlightedintheliterature. The formation and dynamics of expectations are important areas that this paper does not address. It is a well established fact in this domain that experiences—recent and distant, personal and vicarious—have an effect on expectations (see, e.g., Malmendier andNagel2011,2016;AmrominandSharpe2014;GreenwoodandShleifer2014;Coibion andGorodnichenko2015;Baileyetal.2018;Bordalo,Gennaioli,Porta,andShleifer2019). However, despite the robustness of this fact and its macroeconomic significance, experiences and common belief revisions (time fixed effects) capture only a small share of the micro-level variation in measured expectations about stock returns (Giglio et al. 2021). Therefore, with the goal of modeling households’ individual choices, this paper focuses instead on the persistent components of individual expectations, which capture around halfoftheirvariation.10 Oliver,andArrondel(2022). 10Campanale (2011), Peijnenburg (2018), and Foltyn (2020) have analyzed portfolio-choice models in 6

This paper also relates to a growing literature that uses measurements of individual expectations in the estimation of structural economic models.11 Studies such as Guiso, Jappelli, and Terlizzese (1992), Dominitz and Manski (1997), Lusardi (1997, 1998), and Caplin et al. (2023) examine measurements of households’ expectations about their income dynamics, showing that the expectations differ from standard estimates that use administrative data, and using the measured expectations in models of saving decisions and job-transitions. Similarly, measurements of beliefs are increasingly used in models of other economic decisions, such as educational and occupational choices (Arcidiacono et al. 2020; Wiswall and Zafar 2021), parental investments (Almås, Attanasio, and Jervis 2023), and purchase decisions (Erdem et al. 2005). Measured expectations have also been shown to improve the performance of macroeconomic and asset pricing models (Nagel andXu2022;Bordalo,Gennaioli,andShleifer2022;Bordalo,Gennaioli,Porta,andShleifer 2024;Bordalo,Gennaioli,Porta,OBrien,etal.2024;Bianchi,Ilut,andSaijo2024). However, despitethewelldocumenteddifferencesbetweenhouseholds’measuredexpectationsand the standard historically-based calibrations, this paper is the first to use survey measurementsofexpectationsaboutequityreturnsinalifecyclemodeltoexplainthesavingsand portfoliochoicesofU.S.householdstothebestofmyknowledge. Finally, this paper contributes to the literature on wealth differences between sociodemographicgroups. Studiesinthisliteraturehaveidentifiedcross-groupdifferencesthat are difficult to explain using life cycle or permanent income models of consumption and saving. Precautionary savings, differences in time-preference rates and the differential effects of social programs on the incentive to save have been explored as explanations for these difficulties (Carroll 1994; Hubbard, Skinner, and Zeldes 1995; Cagetti 2003). More recently, Lusardi, Michaud, and Mitchell (2017) show that cross-group differences in which individuals learn about the distribution of risky returns form their experiences. While learning, coupled with participation costs, helps to replicate participation patterns, it does not amend the basic model’s prediction about conditional portfolio shares. Therefore, some of the studies rely on additional mechanismslikeambiguityaversion. 11SeeKoşarandO’Dea(2023)foranexcellentreviewofthisliteratureandManski(2018),Caplin(2021), andAlmås,Attanasio,andJervis(2023)forargumentsinfavorofthisapproach. 7

financialproficiencyhavethepotentialtoexplainalargepartoftheempiricalrelationship betweensavingsasafractionofincomeandeducationalattainment,andaccountforalarge shareofwealthinequalitybetweengroupswithdifferentlevelsofeducation. Iaddtothis literaturebydemonstratingthat,indeed,whenalifecyclemodelaccountsformeasurable differences in expectations about asset returns, it can replicate educational differences in wealthandportfolioswithmuchsmallercross-groupdifferencesinpreferences. The rest of this paper is organized as follows. Section 2 presents basic empirical facts about U.S. households’ wealth, stock holdings, and beliefs about future stock returns. Section3presentsthemodelofbeliefsandlifecyclesavingandportfoliochoices. Section 4 discusses estimation. Section 5 presents estimates and discusses their implications. Section 6 quantifies the welfare losses that individuals may suffer from misspecified beliefsaboutfuturestockreturns. Section7estimatesthewelfareeffectsofcounterfactual policychangesunderdifferentspecificationsofbeliefs. Section8concludes. 2. The Portfolios and Expectations of U.S. Households This section reviews various empirical facts about stockholding that challenge the predictions of portfolio-choice models that calibrate households’ beliefs to match historical returns. Then, using 16 years of measured expectations, I show that U.S. households’ subjectivedistributionsofstockreturnsappeartodeviatefromthehistoricaldistribution ofactualstockreturns. Thedifferencesbetweenmeasuredexpectationsandthehistorical distribution of returns have various features that could explain some of the discrepancies between the predictions offered by traditional models and U.S. households’ actual stockholdingbehaviors. 8

2.1 Aggregate Patterns of Stockholding This section examines the stock market participation and portfolio choices of U.S. households. Ihighlightpatternsthatdeviatefromthepredictionsofstandardlifecyclemodels of portfolio choice. Deviations include low participation rates and shares of wealth in stocks,andarelativelyflatshareofwealthinstocksacrossthelifecycle. To study aggregate patterns in U.S. households’ savings and stock holdings, I use the triennial (SCF). I restrict my analysis to the survey waves Survey of Consumer Finances between1995and2019.12 TheSCFprovidesacomprehensivepictureofAmericanhouseholds’balancesheets,including“summaryfiles”withusefulaggregatessuchasthetotal financial assets and stock holdings of each surveyed economic unit. These measures include stocks owned both directly and indirectly through, e.g., mutual funds and retirement accounts. Indirect stock holdings are based on respondents’ descriptions of the types of assets that a given fund or account invests in. Although account level data can offer more precise measurements of stock holdings (as noted by Parker et al. 2022), the advantage of the SCF lies in its comprehensive coverage of a nationally representative sampleofeconomicunitsandalltheirfinancialaccounts. Table 1 presents summary statistics for the main variables of interest, as well as demographic variables that describe the sample. The statistics are shown for the full set of observations and split by the respondent’s highest level of education. The average respondent is 51 years old. Out of respondents, 12% do not have a high school degree, 56% have a high school degree but no college degree, and 32% have obtained a college degree. Due to changes in educational access, those without a high school degree were born, on average, ten years earlier (1948) than those with high school or college degrees (1957and1958). Ifocusonhighschoolandcollegegraduatesonly. TherearethreemainreasonswhyI 12Thewavesbefore1995donothavethequestionofincome“innormaltimes”thatIuseinmyanalysis. Istopat2019becauseitisthelatestpre-COVIDwave. 9

Table1: Summarystatistics: mainvariablesofinterest All LessthanH.S. HighSchool College Variable Mean St. Dev. Mean St. Dev. Mean St. Dev. Mean St. Dev. Ageandeducation BirthYear 1,956.00 18.37 1,947.84 20.68 1,956.77 18.16 1,957.73 16.99 Age 51.00 17.16 58.23 18.55 50.06 17.16 49.92 15.94 LessThanH.S. 0.12 0.32 — — — — — — High-School 0.56 0.50 — — — — — — College 0.32 0.47 — — — — — — Income,wealth,andstockownership Income(1000s) 67.42 145.88 30.82 48.65 51.07 61.79 109.60 236.75 Fin. Assets(1000s) 243.02 1,645.72 42.33 324.56 109.67 775.27 550.55 2,686.12 Ownsstocks? 0.53 0.50 0.21 0.41 0.47 0.50 0.75 0.43 Stocks/Fin. Assets 0.25 0.31 0.09 0.23 0.21 0.30 0.37 0.32 Cond. stockshare 0.46 0.29 0.44 0.31 0.44 0.29 0.49 0.28 Thesummarystatisticsinthistablecomefrompoolingtheobservationsfromthe1995to2019SCFwaves. Thesampleisrestricted torespondentsabovetheageof21withnon-negativefinancialassetsandastock-shareoffinancialwealthbetween0and100%. All calculationsusesurveyweights,whicharepooledandre-scaledsothatthetotalweightofeachwaveisthesame.Theunitofanalysis inSCFitisthe“primaryeconomicunit”. Wealthandincomeareexpressedin2010U.S.dollarsandwereadjustedusingtheusing theCPIindex.Irefertoindividualsthatdonotpossesahigh-schooldiplomaorGEDas“LessthanH.S.”tothosewithahigh-school diplomaorGEDbutnocollegedegreeas“Highschool”andtothosewithacollegedegreeas“College.” excludethosewithoutahighschooldegree. First,theirnumberofavailableobservations is much lower than that of high school and college graduates. When grouped into agebins, as my analysis requires, the number of observations per group becomes too low to producesufficientlypreciseestimatesofthemomentsofinterest. Second,alargefraction of respondents without a high school degree answer “does not know/refuse” to the probabilistic questions in the that I use in my analysis. Third, Health and Retirement Study asTable1shows,householdsinwhichtherespondentdoesnothaveahighschooldegree havelowincomesandlevelsofwealth;asarguedbyHubbard,Skinner,andZeldes(1995) the saving decisions of these households are severely impacted by social programs that I donotmodelinthispaper. Table 1 reproduces the low rates of stockholding that constitute the “stockholding puzzle”(HaliassosandBertaut1995). Thetableshowsthat,outofallobservedeconomic units,only53%ownstocksdirectlyorindirectly. Thisfactisatoddswiththeprescription that all households should own stocks, which is a feature of frictionless models in which 10

everyone expects the stock market to perform in the future as it has historically. There is a positive correlation between level of education and stock market participation, with the lowest participation rate (21%) observed among those without a high school degree and the highest participation rate (75%) among those with a college degree. Multiple factors could contribute to this correlation. One explanation could be that individuals with higher education are more likely to believe in the existence of an equity premium. Another possibility could be the existence of barriers like monetary participation costs thatdisproportionatelyaffectthosewithlowereducationandincome. Thesharesofwealthinstocksamongthosewhoparticipateinthestockmarketshows aweakerrelationshipwitheducationthantherateofparticipation,buttheyarealsolower than what baseline models predict. Given the low rates of participation, the mean share ofwealthinstocksisalsolowat25%. Amoremeaningfulmeasureistheshareofwealth invested in stocks among those who participate, commonly referred to as the conditional share of wealth in stocks. College graduates have a slightly higher average conditional share of 49% compared to that of high school graduates, which is 44%. These levels are lowcomparedtotheprescriptionsoflifecycleportfolio-choicemodelslikethatofCocco, Gomes, and Maenhout (2005) which, even with a relative risk aversion coefficient of 10 andamoderateequitypremium,findsanoptimalshareofwealthinstocksrangingfrom 60to100percentdependingontheageofthehousehold. The age patterns of stock market participation and conditional wealth in stocks are similar for high school and college graduates. Figure 1 shows stock market participation ratesandquartilesofconditionalsharescalculatedonfive-yearagebins. Thefigureshows that,despitedifferencesinlevels,thestockmarketparticipationratesofbothhighschool andcollegegraduatesfollowasimilarinverted“U”shapeastheyage. Forbothgroups,the highest participation rates occur between the ages of 56 and 60, with 55% for high school graduates and 81% for college graduates. In contrast, the conditional share of wealth in stocks remains relatively stable across different age bins, showing little variation in the 11

High School College 100% 75% 50% 25% 0% 5]0]5]0]5]0]5]0]5]0]5]0] 5]0]5]0]5]0]5]0]5]0]5]0] 2 3 3 4 4 5 5 6 6 7 7 8 2 3 3 4 4 5 5 6 6 7 7 8 0,5,0,5,0,5,0,5,0,5,0,5, 0,5,0,5,0,5,0,5,0,5,0,5, 2 2 3 3 4 4 5 5 6 6 7 7 2 2 3 3 4 4 5 5 6 6 7 7 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Age tekraM kcotS etaR noitapicitraP High School College 100% 75% 50% 25% 0% 5]0]5]0]5]0]5]0]5]0]5]0] 5]0]5]0]5]0]5]0]5]0]5]0] 2 3 3 4 4 5 5 6 6 7 7 8 2 3 3 4 4 5 5 6 6 7 7 8 0,5,0,5,0,5,0,5,0,5,0,5, 0,5,0,5,0,5,0,5,0,5,0,5, 2 2 3 3 4 4 5 5 6 6 7 7 2 2 3 3 4 4 5 5 6 6 7 7 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Age fo erahS lanoitidnoC skcotS ni htlaeW Thesummarystatisticsinthistablecomefrompoolingtheobservationsfromthe1989to2019SCFwavesandgroupingtheminto5-year agebins. Fortheparticipationrate,thesolidlineandpointsdisplaysthefractionofstockmarketparticipants. Fortheconditional shareofwealthinstocks,thesolidlineandpointsdisplaythemedian,andtheshadedareasspanfromthe25thtothe75thpercentile. Thesampleisrestrictedtorespondentswithnon-negativefinancialwealthandastock-shareoffinancialwealthbetween0and100%. Allcalculationsusepooledsurveyweights.Wealthisexpressedin2010U.S.dollarsandwasadjustedusingtheusingtheCPIindex. Figure1: Stockholdingoverthelifecycle 12

25th,50th,and75thpercentilesofitsdistributions. Ineveryagegroup,thedistributionsof theconditionalshareofwealthinstocksforhighschoolandcollegegraduatesaresimilar toeachother. ThestabilityofconditionalsharesofwealthinstocksoverdifferentagebinsintheSCF isinconsistentwithtraditionalportfolio-choicemodels,whichprescribethattheseshares mustdeclinewithage. Thisprescriptioncomesfromtheassumptionthataperson’sfuture lifetimeearnings—their“ ”—actasahedgeagainststockmarketfluctuations. humanwealth Therefore,itisoptimalforayoungpersonwithhighhumanwealthtoallocatemostofhis investable wealth to stocks and to reduce his exposure as he ages and his human wealth decreases. Indeed,inCocco,Gomes,andMaenhout’s(2005)benchmarkcalibration,young agentsinvest100%oftheirwealthinstocksandgraduallylowerthisshareastheyageuntil around 60 percent. Parker et al. (2022) show that the increasing popularity of target-date fundshasbroughttheconditionalsharesofrecentcohortsmoreinlinewiththedeclining patternsprescribedbylifecyclemodels. 2.2 Households’ Expectations About Stock Returns Surveymeasurementsofpeople’sexpectationsaboutthefutureperformanceofthestock market vary substantially across individuals and deviate from historical benchmarks. Comparedtothehistoricalexperience,theaveragepersonunderestimatestheprobability of positive returns and overestimates the probability of extreme returns (positive and negative). The magnitude of these deviations decreases with education. A large fraction ofthevariationinthesesurveymeasurementscorrespondstopersistentheterogeneityin people’s expectations and this heterogeneity robustly associates with differences in their portfoliochoices. TocharacterizeU.S.households’perceptionsaboutthefutureperformanceofthestock market, I use the (HRS). The HRS is a biennial longitudinal Health and Retirement Study surveyofU.S.adultsovertheageof50thatgathersdetailedinformationonrespondents’ 13

health,financialstatus,employment,andexpectations. Irestrictmysampletothe“financial respondent” of each household: the person that answers most questions about the incomeandassetsofthehousehold.13 Since 2002, the expectations module of the HRS has included questions about the futureperformanceofthestockmarket. Iusethefollowingquestions: • [𝑃≥0] “Bynextyearatthistime,whatisthepercentchancethatmutualfundsharesinvested in blue chip stocks like those in the Dow Jones Industrial Average will be worth more than theyaretoday?” • [𝑃≥20] “Bynextyearatthistime,whatisthepercentchancethatmutualfundsharesinvested inblue-chipstockslikethoseintheDowJonesIndustrialAveragewillhavegainedinvalue bymorethan20percentcomparedtowhattheyareworthtoday?” • [𝑃≤−20] “By next year at this time, what is the percent chance that mutual fund shares invested in blue-chip stocks like those in the Dow Jones Industrial Average will have fallen invaluebymorethan20percentcomparedtowhattheyareworthtoday?” Iuse 𝑃≥0, 𝑃≥20, and 𝑃≤−20 todenote thesemeasurements. The HRSfirstmeasured 𝑃≥0 in 2002,with 𝑃≥20 and 𝑃≤−20 followingin2008.14 Responses to the questions about future stock returns are disperse and their averages deviateconsiderablyfromthehistoricalperformanceofthestockmarket. Figure2depicts the distribution of 𝑃≥0, 𝑃≥20, and 𝑃≤−20 across all survey waves and shows that, far from concentrating around estimated answers from a common subjective distribution of returns, the responses span wide ranges without signs of agreement. I compare the responsestoannualreturnsoftheS&P500indexfrom1881to2018,asreportedbyShiller (1990). During this period, the S&P 500 saw positive returns on 72% of the years, returns 13Ialsorestrictmysampletothoseabovetheageof50. Thereisasmallnumberofyoungerrespondents thatareinterviewedbecausetheyaremarriedtosomeoneabovetheageof50. 14The 2008 wave asked various different combinations of “gain/fall in value by X%” to different individuals. The “gain/fall in value by 20%” versions of the question were incorporated in 2010. I use the individualswhodrew𝑋 =20in2008toconstruct𝑃≥20and𝑃≤−20forthatwave. 14

Growth in value >0% 0.3 0.2 0.1 0.0 Growth in value >20% 0.15 0.10 0.05 0.00 Fall in value >20% 0.15 0.10 0.05 0.00 0.00 0.25 0.50 0.75 1.00 Subjective Probability .sbO fo noitcarF Mean Response Historical Benchmark Responsesareroundedtothenearestmultipleof5%andeachbarreportsthefractionofobservationscorrespondingtoeachmultiple. Thesampleconsistsofindividualsabovetheageof50whoreportbeingthefinancialrespondentofthehousehold.Thequestionabout positivegrowthinvalue(firstrow)wasaddedin2002andtheothertwowereaddedin2008;therefore,thesamplesofbothcolumns donotexactlymatch. The“historicalbenchmark”linescorrespondtothefractionofyearsbetween1881and2018thattheS&P500 indexhadreturnshigherthat0%,higherthan20%,andlowerthan-20%;thesecalculationsarebasedontheaccompanyingdatafile toChapter26ofShiller(1990). Figure2: Probabilisticassessmentsaboutstockreturns 15

Table2: Probabilisticassessmentsaboutstockreturnsandeducation Question Mean St. Dev. N.Obs Fract. DK/RF LessthanHighSchool 𝑃≥0 0.40 0.30 19,175 0.29 𝑃≥20 0.38 0.30 5,136 0.05 𝑃≤−20 0.30 0.27 5,401 0.04 HighSchool 𝑃≥0 0.45 0.26 59,273 0.14 𝑃≥20 0.34 0.25 21,341 0.03 𝑃≤−20 0.33 0.25 21,380 0.02 College 𝑃≥0 0.53 0.25 23,257 0.06 𝑃≥20 0.30 0.23 10,594 0.01 𝑃≤−20 0.31 0.21 10,391 0.01 Thesampleconsistsofindividualsabovetheageof50whoreportbeingthefinancialrespondentofthehousehold.Thequestionabout positivegrowthinvalue(firstrow)wasaddedin2002andtheothertwowereaddedin2008;therefore,thesamplesforthequestions donotexactlymatch. Furthermore,eachwavethequestions𝑃≥20and𝑃≤−20areaskedonlytoparticipantswhodonotanswer“does notknow/refused”to𝑃≥0. greater than 20% on 29% of the years, and returns below −20% on 5% of the years. The average response for the probability of positive nominal returns, which is 46%, is 26 percentage points below its historical benchmark. Conversely, the average responses for the probabilities of extreme returns, which were 33% for 𝑃≥20 and 32% for 𝑃≤−20, exceed their respective historical benchmarks by 4 and 27 percentage points. The deviations of average responses from historical benchmarks and households’ pessimism about the chances of positive returns in particular are well known facts that have found support across multiple surveys, conducted in the U.S. and abroad (see Hurd 2009; Manski 2018, forreviews). Expectations about future stock returns show a systematic relationship with educational attainment. Yet, significant variability remains among individuals with the same level of education. Table 2 presents summary statistics of 𝑃≥0, 𝑃≥20, and 𝑃≤−20 for respondents with different levels of education. While all groups are pessimistic about the probability of positive returns (𝑃≥0), the average response increases steeply with edu- 16

cation, from 40% for those without a high-school degree to 53% for college graduates. This pattern is consistent with the findings of past studies, which have shown that more educated individuals tend to have a more optimistic outlook on stock returns (Dominitz and Manski 2011; Hurd, Van Rooij, and Winter 2011; Das, Kuhnen, and Nagel 2020). The degreetowhichtheaveragerespondentoverestimatestheprobabilityofextremereturns alsovarieswitheducationalattainment,withmoreeducatedhouseholdsgenerallygiving lower responses.15 However, within-group variability is greater than cross-group differences, as within-group standard deviations are higher than 20 percentage points for all questions. Table 2 also shows the fraction of participants who refused to answer each of the questions or answered with “do not know.” The refusal/unsure rate is much higher for 𝑃≥0 thanfor𝑃≥20 and𝑃≤−20 becauseparticipantswhorefusetoanswer𝑃≥0 oranswerthis questionwith“donotknow”arenotasked𝑃≥20 or𝑃≤−20. Whiletherefusal/unsurerates for high school and college graduates are moderate, it is 29% for 𝑃≥0 for those without a high-school degree. Such a high fraction of refusal/unsure answers casts doubt on the representativeness of respondents without a high school degree who do answer the probabilistic questions. For this reason, in addition to those presented in Section 2.1, I limitthemodelingexercisesinthispapertohighschoolandcollegegraduates. Rounding is pervasive in probabilistic assessments about future stock returns. Figure 2 shows that there are large masses of answers in focal points like 0%, 50%, and 100%, and that all multiples of 10% occur more frequently than their neighboring multiples of 5%. Table 3 presents the fraction of responses that belong to different groups of frequent answers. The groups are a similar partition to the one proposed by Giustinelli, Manski, and Molinari (2022). For each question, more than 8% of the answers are multiples of 100% (0 or 100%), more that 80% are multiples of 10%, and more than 97% are multiples of 5%. Using the full expectations module of the HRS, Giustinelli, Manski, and Molinari 15Theonlyexceptiontothispatternistheaverage𝑃≤−20 ofthosewithoutahigh-schooldegree,whichis thelowestofthethreegroups. 17

Table3: Fractionsof“rounded”probabilisticresponses FractionofAnswersinGroup Question {0%,100%} 50% {25%,75%} Other×10% Other×5% Other 𝑃≥0 0.12 0.30 0.09 0.44 0.05 0.01 𝑃≥20 0.09 0.15 0.07 0.57 0.09 0.02 𝑃≤−20 0.10 0.18 0.06 0.55 0.08 0.02 Thesampleconsistsofindividualsabovetheageof50whoreportbeingthefinancialrespondentofthehousehold. Thequestion aboutpositivegrowthinvalue(firstcolumn)wasaddedin2002andtheothertwowereaddedin2008;therefore,thesamplesofthe threecolumnsdonotexactlymatch.Other×10%={10,20,30,40,60,70,80,90}%.Other×5%={5,15,35,45,55,65,85,95}%.“Other” representsanswersthatdonotfallintoanyoftheothergroups. Table4: Sourcesofvariationinprobabilisticassessments Models’ 𝑅2 Question TimeF.E. Indiv. F.E. Two-WayF.E 𝑃≥0 0.01 0.46 0.47 𝑃≥20 0.01 0.63 0.64 𝑃≤−20 0.01 0.60 0.60 Thetablereports𝑅2 statisticsfromregressingeachmeasurementonindividualfixed-effects,month-of-interviewfixed-effects,and both.Thesampleconsistsofindividualsabovetheageof50whoreportbeingthefinancialrespondentofthehousehold.Thequestion aboutpositivegrowthinvalue(firstrow)wasaddedin2002andtheothertwowereaddedin2008,thereforethesamplesofthethree rowsdonotexactlymatch. (2022) show that individuals tend to round questions in the same domain (e.g., health or finances) to consistent levels of coarseness, even though the level of rounding coarseness varies between individuals and across domains. Based on these findings, the model of beliefs that I use in this paper accounts for rounding practices that are stable over time butheterogeneousacrossindividuals.16 Alargefractionofthevariationinprobabilisticassessmentsaboutstockreturnscomes from cross-sectional differences between individuals that persist over time. In their analysis of the macroeconomic beliefs of Vanguard account holders, Giglio et al. (2021) show that, for all the measurements of expectations in their analysis, persistent differences in 16Surveyresponsesthatreflectroundingandtheuseofheuristicsarepervasiveinotherrelatedcontexts suchasthehypotheticalchoiceofretirementwealthallocations(Batemanetal.2017). 18

expectationsacrossindividuals(individualfixedeffects)explainavastlylargerfractionof variation than common movements in expectations over time (time fixed effects). I replicatetheiranalysiswiththeHRSsample,whichincludesbothinvestorsandnon-investors and spans a longer period of time. Table 4 presents the 𝑅2 statistics of regressions of the form: Indiv. F.E., 𝑃𝑥 = 𝑎 + 𝜖 𝑖,𝑡 𝑖 1,𝑖,𝑡 TimeF.E., 𝑃𝑥 = 𝑏 + 𝜖 𝑖,𝑡 𝑡 2,𝑖,𝑡 Two-wayF.E., 𝑃𝑥 = 𝑎 +𝑏 + 𝜖 , 𝑖,𝑡 𝑖 𝑡 3,𝑖,𝑡 where 𝑃𝑥 is one of the probabilistic assessments (𝑃≥0, 𝑃≥20, or 𝑃≤−20), 𝑎 are individual 𝑖 fixedeffects,𝑏 aremonth-of-interviewfixedeffects,and𝜖 aretime-specificidiosyncratic 𝑡 ·,𝑖,𝑡 errors. TheresultsareconsistentwiththefindingsofGiglioetal.(2021): individualfixed effects explain a much larger fraction of the variance in responses than time fixed effects. For the probability of positive returns 𝑃≥0, individual fixed effects capture 46% of the variance. For the probabilities of extreme returns 𝑃≥20 and 𝑃≤−20, they capture 63% and 60%ofthevariance,respectively. Timefixedeffects,incontrast,capturenomorethan1% of the variance in any of the questions. As Giglio et al. (2021) point out, the persistent cross-sectional heterogeneity in beliefs manifest in these measurements stands at odds withmanyofthemodelsusedinmacroeconomicsandfinance. Tostudytherelationshipbetweenexpectationsandportfolios,IrestricttheHRSsample furtherto keeponlyrespondents belowtheage of65. Thefinancial literacyliteraturehas found that the age around retirement is when people are best able to respond financial questions(see,e.g.,Agarwaletal.2009;LusardiandMitchell2023);Iimposethisrestriction to limit concerns about cognitive decline. My measure of equity holdings includes both direct and indirect investments.17 For financial wealth, I use the value of stocks, mutual 17Fordirectholdings,IusetheRANDHRSaggregateastck. Forindirectholdings,Iusethetotalamounts andsharesinvestedinstocksofthehouseholds’retirementaccounts. 19

Avg. Share of Participation Rate Wealth in Stocks 60% 40% 20% 0% 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 P ‡ 0 naeM lanoitidnoC )rettacS−niB( High−School College Thesampleconsistsofindividualsbetweentheagesof50and65whoreportbeingthefinancialrespondentofthehousehold.Binary participation,stock-sharesofwealth,andtheexpectationresponseareaveragedovertimeattheindividuallevel. Thefiguredisplays theconditionalmean“bin-scatter”estimatesontheresultingdatasetofoneaveragedobservationperhousehold. Figure3: Stockholdingandthesubjectiveprobabilityofpositivereturns funds, investment trusts; checking, savings, and money market accounts; fixed-income assets;andIRAandKeoghaccounts.18 An extensive literature has consistently shown that cross sectional differences in individuals’ beliefs about stock returns predict their stock-holding behavior.19 Giglio et al. (2021)showthatthepersistentcomponentofexpectations—theindividualfixedeffect—is themaindriverofthispredictiverelationship;beliefrevisionshaveamuchweakerlinkto portfolios. ToillustratethisrelationshipinmyHRSsample,Figure4displays“bin-scatter” estimatorsfortheconditionalmeanofstockmarketparticipationandtheshareofwealth instocksasafunctionofthe𝑃≥0 expectation,aggregatedattheindividuallevelbytaking their average over time. The left panel of the figure shows a steep relationship between stock market participation and the reported probability of positive returns: those with the most optimistic answers participate at roughly twice the rate of the least optimistic. 18ThisisthesumofRANDHRSaggregatesastck,achck,acd,abond,andaira. 19DominitzandManski(2007),Hurd,VanRooij,andWinter(2011),AmrominandSharpe(2014),Drerup, Enke,andvonGaudecker(2017),Ameriks,Kézdi,etal.(2020),Giglioetal.(2021),andCalvo-Pardo,Oliver, andArrondel(2022)aresomeexamples. SeeHurd(2009)andManski(2018)forreviews. 20

The right panel shows that the relationship also holds for the share of financial wealth invested in stocks. These relationships hold for both levels of education and they have similar magnitudes. A college degree shifts participation rates and portfolio shares up at everylevelofoptimism. This section reviewed several challenges faced by traditional life cycle models in explaining U.S. households’ stockholding patterns. Stockholding rates are low and correlated with education, those who own stocks do not allocate as much wealth to them asthesemodelsprescribe,andtheoptimalrelationshipbetweenportfoliosharesandage predictedbythesemodelsdoesnotmatchitsempiricalcounterpart. Theseconflictingpredictionsariseinmodelsthatcalibrateagents’expectationsaboutstocksusingthehistorical dataonactualreturns,whichisatoddswithsurveymeasurementsoftheseexpectations. Several properties of measured expectations could help bridge the gap between portfolio choice models and U.S. households’ behavior. For instance, heterogeneous beliefs correlated with education could help explain limited participation rates and pessimism could rationalize low shares of wealth in stocks. In the next sections, I evaluate this possibility, testing whether a life cycle model that matches expectations from survey measurements canfitsavingsandportfoliochoicesbetterthanitshistorically-calibratedcounterpart. 3. A Model of Beliefs and Stock-Holding The model that I propose consists of two parts. The first is a measurement system that I use to interpret the probabilistic responses, 𝑃≥20, 𝑃≥20, and 𝑃≤−20, and to estimate the distribution of beliefs about stock returns across the population. The second part is a life cycle model of saving and portfolio choices in which agents’ beliefs about asset returns arefixedandexogenous. Idiscusseachpartinturn. 21

3.1 Representing the Beliefs of U.S. Households TorepresentthebeliefsaboutstockreturnsofU.S.households,Iconstructandestimatea modelthatmapstheirprobabilisticassessmentstoheterogeneoussubjectivedistributions of stock returns. The model represents beliefs in a way that can be estimated directly fromsurveymeasurementsandthenpluggedintolifecyclemodels. Iestimatethemodel usingalmosttwentyyearsoflongitudinalmeasurementsofU.S.households’expectations aboutstockreturns. Theestimatessuggestthattherearepermanentdifferencesinbeliefs aboutstockreturnsacrosspeople,thattheaveragepersonismorepessimisticaboutstocks than the historical experience would suggest, and that more educated people are more optimistic about stocks. All these features are consistent with previous findings in the literature. 3.1.1 Amodelofbeliefsandprobabilisticassessments The model is an adaptation of the ones used by Kézdi and Willis (2011) and Ameriks, Kézdi, et al. (2020). Every person believes that stock returns follow a distribution that can change from one person to the next but does not change over time. People use their subjective distributions to produce probabilistic assessments but their answers are also perturbedbytime-varyingshocksthatrepresentsurveyerrorsandshorttermfluctuations intheirbeliefs. Peoplealsoroundtheiranswerstodifferentbutpersonally-stabledegrees: someroundalltheiranswerstothenearestmultipleof5%,otherstothenearestmultiple of 10%, 25%, 50%, or 100%. I identify and estimate the distribution of the persistent part ofbeliefsacrossthepopulationusingthelongitudinalnatureofthedataandthefactthat multiplequestionsaboutstockreturnsareaskedinvarioussurveywaves. In the model, people believe that nominal stock returns follow a log-normal distribu- 22

tion20 withindividual-specificparameters𝜇 and 𝜎 : 𝑖 𝑖 ln𝑅˜ ∼ 𝑖 𝒩(𝜇 ,𝜎 ). 𝑡+1 𝑖 𝑖 The individual-specific parameters of people’s beliefs are fixed over time and follow a distribution Ω across the population, (𝜇 ,𝜎 ) ∼ Ω. This assumption is a parsimonious 𝑖 𝑖 way of modeling the fact that most of the panel variation in probabilistic assessments about stock returns comes from persistent differences across individuals, as shown in Section 2.2 and in Giglio et al. (2021). The literature has suggested various mechanisms thatcouldgenerate thispersistentheterogeneity,offering differencesinlivedexperiences (Malmendier and Nagel 2011) or in costs of and returns to learning about stocks (Kézdi and Willis 2011) as two examples. I take belief heterogeneity as given and model it using thedistributionΩ,whichIestimate. People make probabilistic assessments about stock returns using their log-normal beliefs, but their responses are subject to time- and question-specific disturbances and rounded to different degrees. Manski and Molinari (2010) and Giustinelli, Manski, and Molinari (2022) demonstrate that rounding is prevalent in the answers to probabilistic questions in the HRS, and that the degree or “coarseness” of rounding varies across respondents but is stable over time. These studies show that ignoring the rounding patterns present in the data and taking probabilistic assessments at face value can alter theestimatesofeconometricmodelsandtheirprecision. Iaccountforroundingassuming that each person 𝑖 has a “rounding type” (or rounding behavior) ℛ ∈ {5,10,25,50,100}. 𝑖 Anindividualofroundingtypeℛ = 𝑥 roundsallofhisanswerstoprobabilisticquestions 𝑖 about stock returns to the nearest multiple of 𝑥% at every point in time. Rounding types ℛ are independent of (𝜇 ,𝜎 ) and I use ℘fi = {℘ ,℘ ,℘ ,℘ ,℘ } to denote their 𝑖 𝑖 𝑖 5 10 25 50 100 20ThemainqualitativeresultsofSection5holdunderdifferentdistributionalassumptions. Forexample, ifthereturnfactorofstocksisnormal(withcensoringatzero)insteadoflog-normal,itisstillthecasethat thelifecyclemodelwithbeliefsestimatedfromsurveysfitsthedatabetterthanitshistoricalcounterpartfor bothhighschoolandcollegegraduates,withsmallerestimatesofriskaversionandentrycosts,andlarger estimatesoftime-discountfactors. 23

frequenciesacrossthepopulation. In every survey wave, a person might be asked to estimate the chances of positive returns, returns greater than 20%, or returns lower than −20%. In the model, person 𝑖’s responsestothesequestionsattime 𝑡 are: (cid:20) (cid:18)𝜇 (cid:19)(cid:21) 𝑃≥0 = Φ 𝑖 + 𝜀≥0 , 𝑖,𝑡 𝜎 𝑖,𝑡 𝑖 ℛ 𝑖 (1) (cid:20) (cid:18)𝜇 −ln1.20 (cid:19)(cid:21) (cid:20) (cid:18)ln0.8−𝜇 (cid:19)(cid:21) 𝑃≥20 = Φ 𝑖 + 𝜀≥20 , 𝑃≤−20 = Φ 𝑖 + 𝜀≤−20 , 𝑖,𝑡 𝜎 𝑖,𝑡 𝑖,𝑡 𝜎 𝑖,𝑡 𝑖 ℛ 𝑖 ℛ 𝑖 𝑖 where the operator [·] rounds its argument to the nearest multiple of 𝑥%, Φ(·) is the 𝑥 (cid:110) (cid:111)′ univariate standard normal CDF, and the random disturbances 𝜀≥0,𝜀≥20,𝜀≤−20 follow 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 the joint normal distribution 𝒩(0fi,Σ). I assume that Σ is diagonal, so that the shocks are independent. Without rounding or disturbances, Equation 1 would imply that people perfectly calculate and report the queried moments of their subjective log-normal distributioneverywave—theiranswerswouldnotchangeovertime. Thetime-specificrandom disturbancesrepresentsurveyerrorsandtheeffectsofshortterminformationthatmight shiftpeople’sresponsesbutnottheirlongtermbeliefsaboutstockreturns. 3.1.2 Theestimateddistributionofbeliefs I represent the distribution of beliefs across the population Ω using grids of (𝜇,𝜎) pairs. Thegridsarediscretizationsofbivariatenormaldistributionsthatconditionontheevent thatsubjectivestandarddeviationsmustbepositive: 𝜇  𝜈  Ψ Ψ   𝑖 discr∼etized 𝒩 (cid:169)  𝜇 ,  1,1 1,2 (cid:170) | 𝜎 > 0. (2)   (cid:173)     (cid:174) 𝑖 𝜎  𝜈   Ψ Ψ   𝑖  𝜎  2,1 2,2   (cid:171)   (cid:172) For any set of parameters {𝜈 ,𝜈 ,Ψ}, I produce a set of 25 equiprobable (𝜇,𝜎) pairs that 𝜇 𝜎 approximatetheconditionednormaldistributioninEquation2;AppendixAdiscussesthe 24

discretization procedure in detail. There are two main advantages of this representation. First, it is flexible enough to accommodate distributions where beliefs have different averages, levels of dispersion, and correlations between subjective means and standard deviations. Second, I can incorporate the resulting discrete set of estimated (𝜇,𝜎) into a lifecyclemodelasasetofpossible“belieftypes.” Iestimatethemodelbymaximumlikelihood,usingresponsestothethreeprobabilistic assessmentsintheninebiennialwavesoftheHRSbetween2002and2018. Thefullsetof parameterstoestimateinmyrepresentationofbeliefscomprisesthoseinthedistribution of 𝜇 and 𝜎 in Equation 2, the covariance matrix of random disturbances Σ, and the prevalence of different rounding types ℘fi: 𝜗B ≡ {𝜈 ,𝜈 ,Ψ,Σ,℘fi}. Appendix A describes 𝜇 𝜎 thelikelihoodfunction. Myestimation sampleconsistsofall person-yearobservationsin which: a) the person being interviewed is the financial respondent of the household, b) the person being interviewed is between 50 and 65 years old, and c) the respondent did not refuse to answer nor answer “do not know” to any of the three questions regarding future stock returns.21 These restrictions, in addition to considering only respondents withatleastahighschooldegree,yieldasampleof35,211individual-waveobservations from12,025uniqueindividuals. Because of the relationship between education and probabilistic assessments documented in Section 2.2, I estimate the model separately for those with and without a collegedegree. Thisletspeoplewithdifferentlevelsofeducationalattainmenthavedifferent average beliefs, levels of disagreement, rounding patterns, and covariance structures in the random disturbances of their responses. Previous studies have found evidence of thesedifferences(see,e.g.,KézdiandWillis2011;Das,Kuhnen,andNagel2020;Ameriks, Kézdi,etal.2020). Arelationshipbetweenbeliefsaboutstockreturnsandeducationcould explainpartoftheeducationalgradientsinstockholdingdocumentedinSection2. Theparameterestimates,whicharepresentedinTable11ofAppendixA,suggestthat 21Out of the person-year observations that satisfy a) and b), the fraction of observations that I drop for notsatisfyingc)are11.8%forthosewithahigh-schooldegreeand5.2%forcollegegraduates. 25

High School College 0.2 0.1 0.0 −0.1 −0.2 0 2 4 6 8 0 2 4 6 8 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. St.Dev. of Log−Returns, s m ,snruteR−goL fo naeM Estimated Belief Grids S&P 500 − Historical ThebeliefgridsareequiprobablediscretizationsofthejointdistributioninEquation2,seeAppendixAfordetails.TheS&P500point depictstheaverageandstandarddeviationofannuallog-returnstothatindexbetween1881and2018. Thedataforthiscalculation comesfromtheaccompanyingdatafiletoChapter26ofShiller(1990). Figure4: Estimatedbeliefgrids different degrees of rounding are prevalent in the data and that, as found by Giustinelli, Manski, and Molinari (2022), more educated individuals tend to round their answers to finer levels. The fraction of individuals belonging to the finest rounding type (multiples of 5%) is 40% for high school graduates and 51% for college graduates. Coarse rounding is non-negligible: 17% of high school graduates and 9% of college graduates round their answerstolevelscoarserthan10%(25%,50%or100%). The estimated models assign a large role to persistent differences in people’s beliefs for explaining their probabilistic assessments.22 Figure 4 depicts the estimated grids of possible (𝜇,𝜎) pairs for every level of education, showing that they span large ranges of the mean-variance space. The breadth of the grids is a consequence of the fact that, as discussed in Section 2.2, most of the variation in this data comes from persistent crosssectionaldifferences,whichinthismodelcorrespondto(𝜇 ,𝜎 ). 𝑖 𝑖 22While the functional forms that I impose assume that (𝜇 ,𝜎 ) are fixed individually, they do not limit 𝑖 𝑖 thescaleoftheirdistributionacrossthepopulation. Inprinciple,themodelcouldadjudicatethedifferences (cid:110) (cid:111)′ inresponsestothetime-specificshocks 𝜀≥0,𝜀≥20,𝜀≤−20 andfindnarrowdistributionsfor(𝜇 ,𝜎 ). 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖 𝑖 26

Table5: Summarystatisticsofestimatedbeliefsaboutstockreturns 𝜇 = 𝐸 [ln𝑅˜] 𝜎 = (cid:112) 𝑉(ln𝑅˜) 𝐸 [𝑅˜]−𝑅 Sharpe Ratio* 𝑖 𝑖 𝑖 𝑖 𝑖 Mean S.D. Mean S.D. Fract > 0 Mean S.D. Estimated Beliefs High School -0.017 0.131 0.512 0.221 0.60 0.283 0.111 College 0.020 0.118 0.424 0.172 0.72 0.266 0.169 Historical Realizations S&P 500 (1881-2018) 0.087 - 0.169 - - 0.342 - TheSharperatioanditssummarystatisticsarecomputedonlyforthosebeliefsforwhichitispositive. Allsummarystatisticsare takenoverthepointsintheestimatedbeliefsgridsforeverylevelofeducation,depictedinFigure4. TheSharperatiosarecomputed withthenominal“risk-free”returnfactorasabenchmark.Itaketheaverageyearlyrisk-freereturnfactorbetween1881and2018from theaccompanyingdatafiletoChapter26ofShiller(1990),whichis1.044. For every level of educational attainment, the estimated distributions of persistent beliefs imply that most people are more pessimistic about future stock log-returns than standardcalibrationsbasedonthehistoricalexperience. Figure4comparestheestimated distributionofbeliefswiththehistoricalmeanandstandarddeviationoflog-returnstothe S&P500 index. For both levels of educational attainment, the majority of points lie below and to the right of the S&P500. This means that most people believe log-returns to stock investments are lower on average and more volatile than those historically experienced by the S&P500. Moreover, many of the points fall below the 𝜇 = 0 line, suggesting that a considerable fraction of the population believes that average log-returns are in fact negative. As suggested by Dominitz and Manski (2007), this “pessimism” about stock returns could help explain why a fraction of U.S. households do not own stocks at all, despitehavingsubstantialwealth.23 The estimated distributions of beliefs imply a steep relationship between educational attainment and anticipated rewards from investing in stocks. Table 5 presents summary 23In other countries, studies have also shown that stock-market participation is insensitive to wealth windfalls (Andersen and Nielsen 2011; Briggs et al. 2021). These facts are difficult to accommodate for modelsthattrytoexplainnon-participationusingmonetarycosts. Forinstance,Catherine(2021)assumes a fraction of the population—which he estimates to be 46%-47%—exogenously avoids the stock market. Pessimisticbeliefscouldbebehindthispersistentnon-participation. 27

statisticsof thedistributionof persistentbeliefsfor everylevel ofeducationalattainment. Expectedlog-returns(𝜇)varywidelywithineducationalattainmentgroups,withwithingroup standard deviations of 1,300 and 1,200 basis points for high-school and college graduates,respectively. Averageexpectedlog-returnsarehigherforcollegegraduates(200 basis points) than for high school graduates (−170 basis points). Subjective assessments of volatility (𝜎) vary considerably within educational attainment groups but also show a relationship with education: both their mean and standard deviation across individuals are lower for college graduates. The average subjective standard deviation of log-returns is4,200basispointsforcollegegraduatesand5,100basispointsforhighschoolgraduates. Both values are much greater than the historical standard deviation of the S&P500’s logreturns,whichhasbeenaround1,700basispoints. The estimated distributions of beliefs also imply differences in people’s expected rewards from the risks associated with stock market participation. For high school and collegegraduates,thesedifferencesalignqualitativelywiththeirdifferinginvestmentpatterns. ThefourthcolumnofTable5showsthatnotallindividualsbelievethattheexpected returnstostocksaregreaterthanthoseofasafebond: only60%ofhighschoolgraduates and 72% of college graduates do. For those who expect a premium from stocks, the fifth and sixth column calculate the Sharpe ratio, which measures the expected excess returns per unit of risk that they believe stocks offer. The average Sharpe ratios of high school and college graduates who believe there is an equity premium are similar (0.28 and 0.27 respectively) butlower thanthe historicalSharpe ratioof theS&P500 index which,based on1881to2018datafromShiller(1990),hasbeenaround0.34. The model can produce individual-level estimates of the persistent component of beliefsthatareinterpretable,accountformeasurementerrorandrounding,andaggregate the information of the multiple probabilistic questions. Similarly to Ameriks, Kézdi, et al. (2020), the estimates of beliefs that I construct are the individual level expectation of 28

Table6: Estimatedbeliefsandportfoliochoices StockMarketParticipation StockShareofFin. Wealth ProbitMarg. ProbitMarg. Tobit Tobit 𝜇 1.66 0.85 2.08 1.14 (cid:98)𝑖 [1.31;2.01] [0.63;1.07] [1.68;2.49] [0.88;1.38] 𝜎 −1.08 −0.52 −1.24 −0.59 (cid:98)𝑖 [−1.32;−0.88] [−0.66;−0.38] [−1.47;−1.02] [−0.76;−0.46] OutcomeMean 0.45 0.45 0.27 0.27 Std. Dev. (𝜇 ) 0.08 0.08 0.08 0.08 (cid:98)𝑖 Std. Dev. (𝜎 ) 0.11 0.11 0.11 0.11 (cid:98)𝑖 Pseudo 𝑅2 0.13 0.27 0.11 0.24 Controls ✓ ✓ Num. obs. 5447 5447 5447 5447 ThesampleareHRSfinancialrespondentsbetweentheagesof50and65. Iaggregatethedataattheindividuallevelbytaking averagesoverthetimedimension. Formultinomialvariableslikethewillingnesstotakerisk,Itakethemode. Thecontrolvariables are:collegedegree,age,sex,log-income,generalwillingnesstotakerisk(10possiblelevels),andfinancialplanninghorizon(5possible levels).ThefirsttwocolumnsreportaveragemarginaleffectsfromProbitregressions.Thesecondtwocolumnsreportcoefficientsfrom Tobitregressionsthatlimittheoutcometothe[0,1]interval. Reportedconfidenceintervalsare95%coveragesetsfrom150bootstrap replications. Toaccountforthefactthat𝜇 and𝜎 aregeneratedregressors,eachbootstrapreplicationre-sampleshouseholds,re- (cid:98)𝑖 (cid:98)𝑖 estimatesthefullbeliefsmodel,re-calculates𝜇 and𝜎 foreachsampledhousehold,andre-estimatestheregressionsreportedinthis (cid:98)𝑖 (cid:98)𝑖 table. (𝜇 ,𝜎 )giventhefullsetofresponsesoftheperson: 𝑖 𝑖 (cid:20) (cid:12) (cid:21) (cid:20) (cid:12) (cid:21) (cid:110) (cid:111) (cid:110) (cid:111) (cid:98) 𝜇 𝑖 = E 𝜇 𝑖 (cid:12) (cid:12) 𝑃 𝑖 ≥ ,𝑡 0,𝑃 𝑖 ≥ ,𝑡 20,𝑃 𝑖 ≤ ,𝑡 −20 and (cid:98) 𝜎 𝑖 = E 𝜎 𝑖 (cid:12) (cid:12) 𝑃 𝑖 ≥ ,𝑡 0,𝑃 𝑖 ≥ ,𝑡 20,𝑃 𝑖 ≤ ,𝑡 −20 , (3) (cid:12) 𝑡∈𝒯 (𝑖) (cid:12) 𝑡∈𝒯 (𝑖) where𝒯 (𝑖)arethewavesinwhich 𝑖 appearssatisfyingthesamplecriteria. Theindividual-levelestimatesofbeliefsaboutstockreturnsarepredictiveofportfolio choices. Table 6 reports regressions that relate these estimates to the stock ownership andportfoliosharesoffinancialrespondentsintheHRSsample. Individualswithhigher subjectivemeansoflog-returnsparticipateathigherratesandhavehighersharesoftheir wealth invested in stocks; the converse is true for those with higher expected volatilities. Theseestimatedrelationshipsarepreciseandrobusttotheinclusionofsocio-demographic controls and measures of patience and risk preferences. The estimates imply that an individual with a subjective mean 𝜇 that is higher by 100 basis points is 0.85 percentage (cid:98)𝑖 29

points more likely to own stocks, and has an (unconditional) share of wealth in stocks that is 1.14 percentage points higher. An individual with a subjective volatility 𝜎 that (cid:98)𝑖 is higher by 100 basis points is 0.52 percentage points less likely to own stocks, and has a share of wealth in stocks that is 0.59 percentage points lower. These associations are empirically weaker than the ones implied by frictionless models—Ameriks, Kézdi, et al. (2020) call this fact “the attenuation puzzle”— however, their magnitudes are material given the wide variation in 𝜇 and 𝜎 , which have standard deviations of 800 and 1,100 (cid:98)𝑖 (cid:98)𝑖 basispointsrespectively.24 Qualitatively, the fact that not all households believe that there is an equity premium couldexplainwhysomeofthemdonotinvestinstocks. ThelowperceivedSharperatios could explain why those who do own stocks invest a limited share of their wealth in them. In addition, the educational differences in beliefs about returns are also consistent withthedifferentstockholdingpatternsofhighschoolandcollegegraduates. Differences in the share of respondents who believe there is an equity premium could explain the relationship between education and participation rates. The similarity in the subjective Sharpe ratios of those who believe there is a premium could explain why the conditional shares of wealth in stocks are similar for high school and college graduates. To evaluate these possibilities quantitatively, I now present a life cycle model of saving and portfolio choices. 3.2 Life Cycle Model of Saving and Portfolio Choices The life cycle model has several features in common with portfolio choice models in the literature (see, e.g., Cocco, Gomes, and Maenhout 2005; Gomes and Michaelides 2005; Campanale, Fugazza, and Gomes 2015; Fagereng, Gottlieb, and Guiso 2017; Catherine 2021). Householdssavetosmooththeirconsumptionagainstfluctuationsintheirincome, 24Enkeetal.(2024)showthatportfoliochoiceisonlyoneofmanyinstancesinwhichpeople’sdecisionsare empiricallylessresponsivetoeconomicfundamentalsthanwewouldexpectbasedontheoreticalmodels. Theyarguethatthis“behavioralattenuation”isduetoinformationprocessingconstraints. 30

whichcomefromdeterministicchangesastheyageandrandomshocks,bothpermanent and transitory. My model features a bequest motive and age-varying health expenditure shocks as additional reasons for saving; both are important motives in explaining postretirementwealth(DeNardi,French,andJones2010;Ameriks,Briggs,etal.2020). Agents facetwodifferentfinancialfrictionswhendecidinghowtoallocatetheirsavingsbetween assets. First,theymustpayamonetarycostbeforeowningstocksforthefirsttime. Second, theyfacea10%earlywithdrawalpenaltywhenliquidatingstocksbeforeretirement. I now discuss the main components of the model and leave its full mathematical descriptionandtreatmentforAppendixC. 3.2.1 Lifespan,utility,andmortality Time periods in the model represent a year. Agents enter the model at age 24 and can liveuptoamaximumageof100. Attheendofeveryyear,theyfaceanexogenousriskof death that becomes certain at the maximum age. The probability of surviving from age 𝑡 to 𝑡 +1isrepresentedby 𝛿 𝑡+1 andtheprobabilityofnotsurvivingis (cid:19)(cid:19)𝛿 𝑡+1 ≡ 1− 𝛿 𝑡+1 . Agents derive utility from consumption. Their utility function follows a constant relativeriskaversionspecification, 𝐶1−𝜌 𝑢(𝐶) = , (4) 1−𝜌 where 𝜌 isthecoefficientofrelativerisk-aversion. Ifanagentdiesattheendofayear,aftermakingallhischoices,hederiveswarm-glow utility from bequeathing his total wealth. The utility derived from bequeathing wealth 𝑥 is: (𝑥/b)1−𝜌 𝑥1−𝜌 B(𝑥) = b× = b𝜌 × = b𝜌 ×𝑢(𝑥), 1−𝜌 1−𝜌 where b ≥ 0 is a parameter that controls the intensity of the bequest motive. This is the samespecificationusedby,e.g.,GomesandMichaelides(2005). 31

3.2.2 Incomeprocess Agentssupplylaborinelasticallyandretireexogenouslyattheendoftheyearinwhichthey turn 65. Their labor earnings, denoted by 𝑌 , are a product of two factors: a permanent 𝑖,𝑡 componentrepresentedby𝑃 andatransitorystochasticcomponentrepresentedby 𝜃 . 𝑖,𝑡 𝑖,𝑡 Laborearningsandtheirpermanentcomponentfollow: ln𝑌 = ln𝑃 +ln𝜃 and ln𝑃 = ln𝑃 +lnΓ +ln𝜓 , 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡−1 𝑖,𝑡 𝑖,𝑡 whereΓ isadeterministicgrowthfactorthatcaptureslifecyclepatternsinearnings,and 𝑡 ln𝜓 ∼ 𝒩(−𝜎2/2,𝜎2)isamultiplicativeshocktopermanentincome.25 𝑖,𝑡 𝜓 𝜓 The transitory component of earnings 𝜃 is a mixture that represents unemployment 𝑖,𝑡 andtemporalfluctuationsinincomethatoccurwhileemployed:  ln𝒰, Withprobability℧   ln𝜃 = 𝑖,𝑡  ln𝜃˜ ∼ 𝒩(−𝜎2/2,𝜎2), Withprobability1−℧.  𝑖,𝑡 𝜃 𝜃  ℧ denotes the probability of unemployment, and 𝒰 denotes the replacement factor of unemploymentbenefits. Thesequencesofgrowthfactors{Γ }100 differforhighschoolandcollegegraduates. I 𝑡 𝑡=25 taketheirvaluesfromCagetti(2003). Figure5displaystheincomepathsthatindividuals with different education levels would experience in the absence of shocks.26 The decline after age 65 corresponds to retirement. I also take the volatilities of transitory and permanent income shocks (𝜎 and 𝜎 ) used by Cagetti (2003), which come from Carroll and 𝜓 𝜃 Samwick’s(1997)estimates. Afterretirement,individualsarenolongersubjecttotransitoryandpermanentshocks totheirearnings. Instead,theyfaceout-of-pocketmedicalexpenditureshocks. Aspeople 25Themeanoftheshockissetsothat𝐸[𝜓 𝑖,𝑡 ]=1. 26Thisis𝑃 𝑖,24 ×(cid:206)𝑡 𝑗=25 Γ𝑖,𝑡 . 32

100 50 0 40 60 80 100 Age tnenopmoC citsinimreteD emocnI fo 0102 fo sdnasuohT( )srallod SU Education High−School College TheseestimatedtrajectoriesforthedeterministiccomponentofpermanentincomecomefromCagetti(2003). Ithanktheauthorfor sharinghisexactestimates. Figure5: Deterministiccomponentofincomebyeducationlevel High−School College 0.6 0.4 0.2 0.0 50 60 70 80 90 50 60 70 80 90 Age emocnI / .pxE lacideM POO Thefiguredepictsestimatesofthedistributionofpeople’sout-of-pocketmedicalexpendituresexpressedasaratiooftheirannual income. Ateveryage,thesolidlinerepresentsthemedian,thedarkshadedarearepresentsthe25thand75thpercentiles,andthe light-shadedarearepresentsthe10thand90thpercentiles. Figure6: Out-of-pocketmedicalexpendituresoverthelifecycle 33

age,medicalexpendituresincreaserapidly(seeFigure6). Theanticipationoftheserising expenditures has been shown to be one of the reasons why the elderly do not spend their wealth as quickly as a basic life cycle model would predict (De Nardi, French, and Jones 2010; Ameriks, Briggs, et al. 2020). The literature that specializes in the study of these expenditures has identified several important features, such as their dependence on persistent health states (Kopecky and Koreshkova 2014; Ameriks, Briggs, et al. 2020), their relationship with permanent income (De Nardi, French, and Jones 2010), and the prevalenceof“catastrophic”shocks(FrenchandJones2004). Toincorporatetheseshocks into my model, I adopt a parsimonious representation that matches the distribution of expenditures across the population and matches the fact that they increase with age and permanent income. Every year, agents draw a shock oop that represents the fraction 𝑖,𝑡 of their earnings used up by out-of-pocket medical expenses. I assume that government programscoveranyhealthexpenseaboveanagent’sincome,sothatincomenetofmedical expensescannotbenegative. Theprocessforearningsnetofcostsandpermanentincome becomes: 𝑌 = 𝑃 ×max{0.0,1−oop } 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑃 = 𝑃 Γ . 𝑖,𝑡 𝑖,𝑡−1 𝑖 The shocks oop are independent across time and follow age- and education-specific 𝑖,𝑡 distributions that approximate the patterns in Figure 6. I calibrate these distributions usingtheRANDHRSlongitudinalfile;IdescribetheprocessinAppendixB. 3.2.3 Financialassetsandfrictions Agents try to smooth their consumption by saving and they have two assets available for thispurpose. Thefirstassetisarisk-freeliquidaccountwithaconstantper-periodreturn factor𝑅. Thesecondassetisastockfundwithastochasticreturnfactor𝑅˜ thatagentsview as log-normally distributed and independent across time. I denote the dollar amounts 34

availabletoagent 𝑖 atthestartofperiod𝑡 intherisk-freeaccountandthestockfundwith 𝑀 and 𝑁 ,respectively. Theflowsbetweenthetwoassetsareoneoftheagents’control 𝑖,𝑡 𝑖,𝑡 variables and denoted with 𝐷 , with 𝐷 > 0, representing a movement of funds from 𝑖,𝑡 𝑖,𝑡 therisk-freetotheriskyaccount. Themodelhasfourdifferentfinancialfrictions. First,agentscannotshort-sellanyofthe assets or borrow against their future income. Second, agents enter the model not having access to the stock fund and must pay a one-time financial cost to access it. As in Gomes andMichaelides(2005),thecostrepresentsthemoneyandtimespentopeningabrokerage accountandgettingfamiliarizedwiththestockmarket. Thecostisproportionaltoagents’ permanent income, 𝑃 × 𝐹, where 𝐹 is a parameter to be estimated. Third, withdrawals 𝑖,𝑡 from the stock fund are taxed at a constant rate 𝜏 = 0.1 before agents retire.27 This friction represents early retirement fund withdrawal penalties and the costs associated with liquidating stock positions. Finally, agents must pay for their consumption using fundsfromtheirrisk-freeaccountsonly. As demonstrated by Campanale, Fugazza, and Gomes (2015), the combination of rebalancingpenaltiesandthefactthatconsumptionmustbepaidforusingrisk-freefunds generates a reason for young people to not allocate all of their wealth to stocks, as is predictedbystandardlifecycleportfoliomodels(seeCocco,Gomes,andMaenhout2005). An agent who anticipates the possibility of consuming part of his savings in the next period—because of an unemployment spell, for instance—might keep a buffer of riskfree funds to avoid having to pay the stock withdrawal tax if this is the case. The size of the desired buffer will depend, among other factors, on the agent’s beliefs about the equity premium and its volatility, the volatility of his income, and the magnitude of the withdrawaltax. 35

a)Agentwhohasnotpaidthestock-fundentrycost Period𝑡−1ends Period𝑡ends Period𝑡starts 𝑡(Entry) 𝑡(Cns) Period𝑡+1starts • Shocks{𝜓,𝜃}are • Agentchooseswhetherto • Agentchooses • Mortalityisrealized. realized. enterthestock-fund. consumption𝐶outof liquidassets𝑀. • Iftheagentdies,he • Incomeandassets • Ifheenters,movesto receivesutilityfrom {𝑃,𝑌,𝑀}aredetermined. 𝑡(Reb)inthetimeline bequests. belowwithliquid balances𝑀−𝑃×𝐹. • Ifhedoesnotenter, continueinthistimeline. b)Agentwhohasalreadypaidthestock-fundentrycost Period𝑡−1ends Period𝑡ends Period𝑡starts 𝑡(Reb) 𝑡(Cns) Period𝑡+1starts • Shocks{𝑅˜,𝜓,𝜃}are • Agentchooses𝐷, • Agentchooses • Mortalityisrealized. realized. rebalancinghisassets. consumption𝐶outof post-rebalancingliquid • Iftheagentdies,he • Incomeandassets • Post-rebalancingassets𝑀˜ assets𝑀˜. receivesutilityfrom {𝑃,𝑌,𝑀,𝑁}are and𝑁˜ aredetermined. bequests. determined. Figure7: Summaryoftiminginthelifecyclemodel 36

3.2.4 Timingandrecursiverepresentation Figure 7 summarizes the timing of stochastic shocks and optimizing decisions that occur within a period of the life cycle model. Agents enter the model in timeline a), not having paid the stock fund entry cost 𝑃 × 𝐹. They are presented with the option to pay the 𝑖,𝑡 cost and enter the fund every year. Once they pay the cost, they move to the portfoliorebalancing stage 𝑡(Reb) of timeline b) and remain on timeline b) for the rest of their lives. Toillustratethechoicesandconstraintsfacedbyagentssuccinctly,Equation5presents therecursivevaluefunctionofanagentwhohaspaidthefinancialparticipationcostand therefore has access to the stock-fund.28 I present the value function of the agent who has not paid the financial participation cost in Appendix C, which also discusses various alternativerepresentationsofthemodelthatIuseinitssolution. 27Thewithdrawaltaxratebecomes𝜏=0afteragentsretire. 28Individualsubindices𝑖aredroppedforsimplicity. 37

𝑉In(𝑀 ,𝑁 ,𝑃 ) = max𝑢(𝐶 )+𝛽𝛿 E (cid:2) 𝑉In (𝑀 ,𝑁 ,𝑃 ) (cid:3) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡+1 𝑡 𝑡+1 𝑡+1 𝑡+1 𝑡+1 𝐶 ,𝐷 𝑡 𝑡 + (cid:19)(cid:19)𝛿 𝑡+1 B(𝐴 𝑡 + 𝑁˜ 𝑡 ) Subjectto: −𝑁 ≤ 𝐷 ≤ 𝑀 , 0 ≤ 𝐶 ≤ 𝑀˜ 𝑡 𝑡 𝑡 𝑡 𝑡 𝑀˜ =𝑀 −𝐷 (cid:0)1−1 𝜏(cid:1) 𝑡 𝑡 𝑡 [𝐷 ≤0] 𝑡 . (5) 𝑁˜ =𝑁 +𝐷 𝑡 𝑡 𝑡 𝐴 =𝑀˜ −𝐶 𝑡 𝑡 𝑡 𝑀 =𝑅𝐴 +𝑌 𝑡+1 𝑡 𝑡+1 𝑁 =𝑅˜ 𝑁˜ 𝑡+1 𝑡+1 𝑡 𝑃 =Γ 𝜓 𝑃 𝑡+1 𝑡+1 𝑡+1 𝑡 𝑌 =𝜃 𝑃 𝑡+1 𝑡+1 𝑡+1 4. Estimation To determine whether survey measurements of beliefs can improve the life cycle model’s capacity to fit U.S. households’ savings and portfolios, I now estimate the model under alternative specifications of beliefs about stock returns. In the first specification, all the simulated agents believe that future stock fund returns will follow a distribution that approximatestheirhistoricalbehavior—thisisthecommonlyused full-informationrational- (F.I.R.E.) specification. In the second specification, the agents’ beliefs about expectations future stock returns are heterogeneous and distributed across the population following the specifications that I estimated from survey measurements in Section 3.1.2. For each level of education and each specification of beliefs, I estimate the model’s unobservable 38

parametersthatgovernagents’preferencesandbarrierstostockmarketparticipation. The estimationstrategysearchesfortheparametersthatbestreplicatethelifecycleprofilesof U.S. households’ savings, stock market participation rates, and shares of financial wealth instocks. 4.1 Data, Sample Restrictions, and Targeted Variables The estimation exercise targets the life cycle profiles of wealth-to-income ratios, stock marketparticipationrates,andthesharesofwealthinstocksconditionalonparticipation of U.S. households. I construct these targeted variables using the nine waves of the (SCF)between1995and2019. Myvariablesofinterestrelyon SurveyofConsumerFinances definitionsandcalculationsintheSCF’s ,whichproducestandardmeasures summaryfiles ofhouseholds’wealthandincomefromtherawsurvey.29 The measures the financial savings of an individual relative to wealth-to-income ratio what would be their “usual” income—their permanent income in the life cycle model. For my measure of wealth, I take the economic unit’s total financial assets (fin). For my measure of income, I take the sum of wage and salary income (wageinc) and social securityandpensionincome(ssretinc). To measure stock holdings, I use the SCF’s estimated total value of financial assets invested in stocks (equity), which includes direct and indirect investments. I define stock as a binary variable that takes the value of one if the given economic market participation unithasstockholdingsgreaterthanzero,andzerootherwise. Finally,Icalculatethe share as a unit’s stock holdings divided by their wealth; if a unit’s wealth is of wealth in stocks zero,Isetthesharetozero. To generate a sample that more closely matches the type of household represented in the model and to ensure that the variables of interest are well defined, I apply various filters to the data before using it to compute the targeted moments. First, I keep only 29Variablesinteletype fontdenotecalculationsthatarereadilyavailableinthesummaryfiles. 39

Mean Avgerage Participation Rate Wealth/Income Cond. Share 10.0 1.00 1.00 7.5 0.75 0.75 5.0 0.50 0.50 2.5 0.25 0.25 0.0 0.00 0.00 40 60 80 40 60 80 40 60 80 Age High−School College Targeted Not−Targeted EachpointrepresentstherelevantstatisticcomputedoverthegroupofSCFrespondentswiththegivenlevelofeducationalattainment andfallingina3-yearagebin. Seethemaintextfordefinitionsofthesampleandrelevantvariables. Errorbandscorrespondto95% point-wiseconfidenceintervalscalculatedusing500bootstrappedsamplesforeachlevelofeducation. Figure8: Targetedmoments economicunitswhoserespondentwasbornbetween1920and1995. Tomakethewealthto-incomeratiocomparabletoamodelanaloguethatusespermanentincome,Ionlykeep units who report that their income was “normal” in the given year.30 To ensure that the ratioisdefined,Ionlykeepunitswithpositiveincomes. Finally,Iexcludebusinessowners fromthesample.31 I group observations by the respondent’s level of education and age. For education, I split the sample into those without a high school degree (which I do not analyze), those with a high school but not a college degree, and those with a college degree. For age, I form three-year bins starting at age 24 and up to a maximum age of 80, for a total of 19 groups: {24,25,26}, {27,28,90}, ..., {75,76,77}, {78,79,80}. The moments that my estimation routine targets are summary statistics of the variables of interest calculated overtheeducation-by-agegroups. 30Thequestionofwhetherincomewasunusualwasaddedinthe1995wave. Forthisreason, Iexclude previouswavesfromthispartoftheanalysis. 31Idefinebusinessownersasthosewithbus>0. 40

Foreachgroupofobservations,themomentsthatItargetinestimationare: • : the average of the wealth-to-income ratio. The Average wealth-to-income ratio ratio can take on extreme values for agents with a low measured wage income. To limit the influence of these extreme observations, I winsorize the wealth-to-income ratiosattheirwithin-group95thpercentilebeforetakingtheiraverage. • : theaverageofthebinarystockmarketparticipation Stockmarketparticipationrate variable. • : theaverageshareofwealthinstocks Averageconditionalshareofwealthinstocks ofthosewhoparticipateinthestockmarket.32 Iusesurveyweightsforthecalculationofthesemomentsandrescaletheweightsofdifferentwavessothateachofthemhasanequalrepresentationinthemoments’calculations. Figure 8 displays each of the moments for every education-by-age group. The figure also presents 95% confidence intervals for each targeted moment. The confidence intervals come from calculating the targeted moments on 500 bootstrapped samples for each level of education. The error bars correspond to the 2.5th and 97.5th percentiles of the bootstrapped values of each moment. The bars demonstrate that the only moments with considerablesamplingvariationarethepost-retirementwealthratiosofcollegegraduates. 4.2 Objective Function and Optimization For every level of education and specification of beliefs, I estimate the preferences and participation costs that minimize the distance between the targeted moments in the SCF andtheirmodel-impliedcounterparts. 32Themodelcounterpartoftheaverageconditionalsharecanbehavepoorlyinregionsoftheparameter spacewhereparticipationisverylow(forinstance,ifentrycostsareveryhigh). Toimprovethisbehavior and ensure that the SMM loss function is defined for all parameter values, I target a weighted average of the unconditional share of wealth in stocks that gives very little weight (instead of a 0 weight) to nonparticipants. UsingShare 𝑖,𝑡 todenotetheunconditionalshare,Itarget(cid:205) 𝑖 𝜔 𝑖,𝑡 Share 𝑖,𝑡 /(cid:205) 𝑖 𝜔 𝑖,𝑡 ,where𝜔 𝑖,𝑡 =1 forparticipants(Share 𝑖,𝑡 >0)and𝜔 𝑖,𝑡 =10−3fornon-participants(Share 𝑖,𝑡 =0). Thismomentispractically 41

Table7: Non-estimatedparametervalues Symbol Interpretation Value Source b BequestIntensity 2.5 GomesandMichaelides(2005) {𝛿 }𝑇 Survivalprobabilities - S.S.A.Actuarialtables201033 𝑡 𝑡=0 {Γ𝑡 }𝑇 𝑡=0 Permanentincomedrift(educ.) - Cagetti(2003) 𝜎 ,𝜎 Volatilityofincomeshocks(educ.) - CarrollandSamwick(1997) 𝜓 𝜃 ℧ Probabilityofunemployment 0.050 - 𝒰 Unemp. benefitsreplacementfactor 0.500 Nationalmedian,Ganong,Noel,andVavra(2020) 𝜋 Log-inflationrate 0.024 MeanCPILog-Inflation. 𝑟 Logrisk-freerate(nominal) 0.043 Mean1-yearU.S.bondlog-returns. 𝑅 Risk-freereturnfactor(real) 1.019 exp{𝑟−𝜋} 𝜇𝑆𝑃500 Meanstocklog-return 0.085 S&P500Index(nominal). 𝜎𝑆𝑃500 St. Dev. stocklog-returns 0.170 S&P500Index(nominal). Parametersthatdependoneducationalattainmentaremarkedwith“(educ.).”Averagesandstandarddeviationsoffinancialvariables arealltakenoverthe1881-2018period.ThedataonfinancialassetsandtheCPIindexcomefromthe‘Chapter26’fileinRobertShiller’s website:http://www.econ.yale.edu/shiller/data.htm. The set of parameters that I estimate structurally consists of the coefficient of relative riskaversion(𝜌),thetime-discountfactor(𝛽),andthesizeofthecostofaccessingthestock fund for the first time (𝐹). I denote this set of parameters with 𝜗 ≡ {𝜌,𝛽,𝐹}. I set other parameters related to the income process and mortality to historical estimates or values fromtheliteratureandtheyremainfixedthroughouttheestimationprocess;Isummarize their values or sources in Table 7. The parameters that govern the returns to different assetsandagents’expectationsaboutthemarediscussedindetailinthenextsection. Foranygivensetofparameters,Isolvethelifecyclemodelandsimulatepopulationsof agentsthatIusetofindmodel-impliedcounterpartstothetargetedmoments. Isolvethe model by backward induction using a combination of the techniques outlined in Carroll (2006), Iskhakov et al. (2017), and Druedahl (2021); I describe the process in detail in Appendix E. I use the resulting policy functions to simulate populations of agents on which I calculate model-counterparts to the targeted empirical moments. The model is not well suited to accommodate the transitional dynamics of households’ savings and portfolios as they move into retirement; it assumes that all agents retire exogenously at identicaltotheaverageconditionalshareattheparticipationlevelsobservedinthedataandtargetedbythe model. 42

age 65. Therefore, I exclude the bins spanning ages 60 to 71, leaving a total of 15 targeted age-bins and 45 moments for each level of education.34 For a level of education 𝑒, I use 𝑚𝑒 to denote a vector of the 45 targeted empirical moments and 𝑚ˆ𝑒 (𝜗) to denote its 0 model-impliedcounterpartunderparameters 𝜗. ThelossfunctionthatIminimizeis: 𝐿𝑒(𝜗) = (cid:0)𝑚𝑒 −𝑚ˆ𝑒 (𝜗)(cid:1)′ W𝑒 (cid:0)𝑚𝑒 −𝑚ˆ𝑒 (𝜗)(cid:1) , (6) 0 0 where W𝑒 is a diagonal weighting matrix. Since the moments have different scales, I set W𝑒 so that moment deviations are expressed as fractions of the average relevant statistic acrossagegroups.35 Iobtainestimatesas: 𝜗ˆ𝑒 = argmin𝐿𝑒(𝜗). (7) 𝜗 To solve the minimization problem, I use the TikTak algorithm (Arnoud, Guvenen, and Kleineberg 2019) as implemented in the estimagic toolbox (Gabler 2022). I use 2,500 initial“explorationpoints”andallowfor10fulllocal-optimizationrunsusingtheDFO-LS algorithm (Cartis et al. 2019), which takes advantage of the least-squares structure of the optimizationproblem. 4.3 Agents’ Expectations and Returns to Financial Assets Inthemodel,returnsofthestockfundfollowadistributionthatapproximatesthehistorical behavior of the S&P500 index. In the first specification of agents’ beliefs, everyone’s expectationsareconsistentwiththisdatageneratingprocess. Inthesecondspecification, agents’expectationsmatchsurveymeasurementsinstead. 3415 age-bins times three moments of interest (median wealth-to-income ratio, participation rate, and averageconditionalshareofwealthinstocks). 35Forexample,forindividualswithacollegedegree,thediagonalpositionsofWColl.thatmultiplyerrors in the stock market participation rate are set to 1/(Pa¯rt)2 where Pa¯rt is the average of the 15 stock market participationratesin𝑚Coll.. 0 43

The simulated log-returns of the stock fund follow a normal distribution. Their mean andvariancematchthoseoftheS&P500’slog-returnsbetween1881and2018,andImake aconstantadjustmentforinflation𝜋,whichItaketobeitsaverageoverthesameperiod: ln𝑅˜ ∼ 𝒩(𝜇𝑆𝑃500 −𝜋,𝜎𝑆𝑃500). (8) Table7presentsthevaluesof𝜇𝑆𝑃500, 𝜎𝑆𝑃500,and𝜋. ThefirstspecificationofbeliefsthatIuseis (F.I.R.E.). full-informationrational-expectations Underthisspecification,agentshavecorrectbeliefsaboutthestock-fund’sreturns(Equation 8) when solving their dynamic optimization problem. For each level of education, I simulate populations of 500 agents that run for 1,000 years and use them to compute themodel’scounterpartstotargetedmoments, 𝑚ˆ𝑒 (𝜗). Thesecondspecificationofbeliefs that I use is . Under this specification, I replace every agent of the F.I.R.E. estimated beliefs simulationwith25agentswhosebeliefsaboutthestockfund’sreturnscomefromthedistributions I estimated in Section 3.1.2.36 The 𝑗th agent believes that ln𝑅˜ ∼ 𝒩(𝜇ˆ −𝜋,𝜎ˆ ), 𝑗 𝑗 where{𝜇ˆ ,𝜎ˆ }25 isthegridofestimatedbeliefsforthegivenlevelofeducation,displayed 𝑗 𝑗 𝑗=1 in Figure 4. I use the resulting populations of 12,500 agents to compute model-implied moments. 5. Life Cycle Model Estimates The results from structurally estimating the life cycle model confirm that incorporating survey measurements of beliefs improves the model’s capacity to explain the savings and portfolios of U.S. households. For both high school and college graduates, replacing model-consistent (F.I.R.E.) beliefs with a specification that fits survey measurements reduces the distance between the model’s predictions and the targeted moments of the 36The25estimated-beliefsagentssharethesameshockrealizationsoftheF.I.R.E.agenttheyarereplacing. Theydifferonlyintheirbeliefsaboutstock-fundreturns. 44

Table8: S.M.M.estimatedparametersunderdifferentbeliefmodels College High-School F.I.R.E. Est. Beliefs F.I.R.E. Est. Beliefs CRRA(𝜌) 11.396 5.114 8.607 4.231 [11.344;11.571] [5.058;5.150] [8.494;8.612] [4.191;4.252] Disc. Fact. (𝛽) 0.634 0.886 0.331 0.761 [0.621;0.642] [0.884;0.890] [0.311;0.355] [0.751;0.770] EntryCost(𝐹 ×100) 1.041 0.000 3.116 2.576 [0.485;1.825] [0.000;0.000] [2.848;3.513] [2.292;2.673] SMMLoss, 𝐿𝑒(𝜗 ˆ ) 5.264 2.857 15.984 3.998 Thebracketsundereachpointestimateare95%confidenceintervalsthatcomefromestimatingasurrogatemodelonbootstrapped moments, seeAppendixFfordetails.“F.I.R.E”standsforfull-informationrational-expectationsand“Est.Beliefs”correspondstothe heterogeneousbeliefsspecification,bothdescribedinSection4. The“SMMLoss”rowdisplaysthevalueoftheSimulatedMethodof Momentslossfunction(Equation6)attainedbythegivenparametervaluesandbeliefspecifications. data. In both cases, the improvement comes from the capacity to fit low participation rates with low participation costs and low portfolio shares with moderate levels of risk aversion. This is possible because in the estimated distribution of beliefs not everyone thinks that there is an equity premium, and those who do underestimate the risk-return compensationthatthestockmarketoffers. For both high school and college graduates, the life cycle model fits the targeted moments more closely when it uses estimated beliefs instead of F.I.R.E. beliefs. Table 8 displays the estimated parameters for each level of educational attainment and specification of beliefs, along with the loss function (Equation 7) evaluated at the estimates. The loss function aggregates squared differences between the model-implied and empirical moments; thus, it serves as an index of how well each model fits the age profiles of wealth-to-incomeratios,stockmarketparticipationrates,andconditionalsharesofwealth in stocks. Specifications that use the beliefs estimated from survey measurements have lowerlossesthantheirF.I.R.E.counterparts. Thereductionsaresubstantial: 75%forhigh schoolgraduates(15.984to3.998)and46%forcollegegraduates(5.264to2.857). Figure9comparesthepredictedmomentsofdifferentmodelspecificationswiththeir 45

High−School Average Participation Avg. Conditional Wealth/Income Rate Stock Share 1.00 1.00 3 0.75 0.75 2 0.50 0.50 1 0.25 0.25 0.00 0.00 40 60 80 40 60 80 40 60 80 Age College Average Participation Avg. Conditional Wealth/Income Rate Stock Share 1.00 1.00 7.5 0.75 0.75 5.0 0.50 0.50 2.5 0.25 0.25 0.00 0.00 40 60 80 40 60 80 40 60 80 Age Source Data (SCF) F.I.R.E. Model Estim. Beliefs Model Eachdotinthefigurerepresentstherelevantstatisticcalculatedoverathree-yearage-group.Seethemaintextforprecisedefinitions oftheagegroupsandstatistics. Ages60to71areomittedbecauseoftransitionaldynamicsintoretirementthatthemodeldoesnot accountfor. Figure9: Modelfitoftargetedmoments 46

empiricalcounterparts,revealingthesourcesoftheimprovementintheirmeasuredfit. The F.I.R.E. model does not replicate the humped-shaped participation rates of high school graduates, which peak at less than 60%. Instead, it produces participation rates thatstartat2%andincreasewithageuntiltheyreach100%. Thereasonisthatunderthis specification of beliefs every agent who saves even modest amounts wants to participate and the only way to prevent them from doing so is to impose high entry costs. The costs canonlygenerateparticipationratesthatincreasewithagebecauseonceanagenthaspaid themhecanparticipatefortherestofhislife. Therefore,togenerateaparticipationprofile with a low peak of less than 60%, the F.I.R.E. model would need high entry costs that would make participation at younger ages counterfactually low. Instead, the best-fitting entry cost of 3.1% of annual income generates participation rates that do not match the shapeofthetrueageprofilebutwhoseaverageacrossagebins(51%)isclosetothatinthe data(48%). In contrast, the model that uses the estimated beliefs specification accurately reproduces the participation rates of high school graduates before retirement. Because not all agents believe that there is an equity premium under this specification, the model generates participation rates that are moderate. This would be the case even in the absence of entry costs. The model fits increasing participation rates between ages 24 and 40 with a costof2.6%ofpermanentincome. Afterage40,themodelusesthefractionofthepopulation who does not believe in an equity premium to match the plateauing of participation rates until age 60. Without these agents, participation would continue to grow. Neither specificationofbeliefscanreplicatethedeclineinparticipationratesafterretirementthat occurs in the data. This is due to the structure of costs: since there are no per-period costs associated with owning stocks, agents who already participate have little incentive tocompletelyexitthestockmarket. For college graduates, neither specification of beliefs replicates stock market participation rates perfectly: the F.I.R.E. model overestimates them and the estimated beliefs 47

model underestimates them. As was the case with high school graduates, the F.I.R.E. modelisconstrainedbythefactthatallagentswithsufficientsavingswanttoownstocks. Forcollegegraduates,itusesacostof1%ofannualpermanentincomethatreducesearly participation, but all agents overcome this cost by age 40, leading to a 100% participation rateformostofthelifecycle. Theoppositeproblemoccursintheestimatedbeliefsmodel: the fraction of agents that think there is an equity premium is lower than the actual participation rates of college graduates aged 40 to 60. Therefore, despite its null estimated participationcost,thismodelunderestimatesparticipationratesformostofthelifecycle. Thedifferencesbetweenthewealth-to-incomeratiosandconditionalstocksharesgenerated by the two specifications of beliefs are subtler. For high school graduates, both models overestimate the wealth-to-income ratios of young agents and underestimate those of retirees; these errors are greater for the F.I.R.E. model. Similar issues appear inthewealth-to-incomeratioofcollegegraduatesbuttoalesserdegree,withbothspecifications of beliefs tracking the empirical age profile more closely. The average conditional shares of wealth in stocks produced by the models are close to the empirical age profiles forbothlevelsofeducationalattainmentandbothspecificationsofbeliefs. Theprofilesare flatterthanthosepredictedbybaselinefrictionlesscalibrationsandthisbringsthemcloser to the empirical profiles. The main noticeable discrepancy in the model with estimated beliefs is the reduction of the average conditional share of wealth in stocks of college graduates after retirement. This reduction comes from a change in the composition of participants when the rebalancing tax is removed: some agents with pessimistic beliefs (but who still think that there is an equity premium) enter the market, and they drive downtheaverageconditionalshare. While the two specifications of beliefs produce qualitatively similar wealth-to-income ratios and conditional shares of wealth in stocks, they rely on different mechanisms to generate them. The main difference is how the two specifications reduce the conditional shareofwealthinstockstothemoderatelevelsobservedinthedata. ModelswithF.I.R.E. 48

beliefs rely on high relative risk aversion coefficients of 𝜌 = 8.6 for high school graduates and 𝜌 = 11.4 for college graduates. Since all agents believe in a large equity premium underthisspecification,theonlywaytodissuadeparticipantsfromallocatinglargeshares oftheirsavingstostocksistomakethemextremelyriskaverse. Ontheotherhand,under the estimated beliefs specification, Table 5 shows how even agents who believe in an equity premium think (on average) that the risk-return trade-off offered by stocks is not as attractive as historical benchmarks suggest. This feature enables the specification with estimated beliefs to match the empirical age-profiles of the conditional shares of wealth instockswithlowerrelativeriskaversioncoefficientsof𝜌 = 4.2forhighschoolgraduates and 𝜌 = 5.1forcollegegraduates. The different relative risk aversion coefficients required by the F.I.R.E. and estimatedbeliefs specifications produce differences in how the models fit wealth-to-income ratios. High relative risk aversion coefficients increase agents’ precautionary saving, preventing the models from producing agents with low wealth. This is evident in Figure 9, where both belief specifications struggle to match the savings of younger agents, especially high school graduates. To counteract the effect of precautionary saving on the wealth of young agents, the models use lower time-discount factors (𝛽) than those typically found inthemacroeconomicsandlabor-economicsliterature.37 ThiseffectisstrongerforF.I.R.E. models because of their higher relative risk aversion coefficients; it results in 𝛽 = 0.33 for high-school graduates and 𝛽 = 0.63 for college graduates, implying that agents discount their future utility at rates of 67% and 37% per year respectively. The models that use the estimated beliefs instead can afford higher time-discount factors of 𝛽 = 0.76 for highschool graduates and 𝛽 = 0.89 for college graduates because of the lower pressure of conditionalportfoliosharesontherelativeriskaversioncoefficients. 37Moststudiesthatuseconsumption-savingmodelstomatchwealthfindannualdiscountfactors𝛽 >0.9, see e.g., Gourinchas and Parker (2002), Cagetti (2003), De Nardi, French, and Jones (2010), and Carroll, Slacalek, et al. (2017). However, because of the higher relative risk aversion coefficients needed to match portfolioshares,lowerestimateddiscountfactorsarefrequentlyfoundinthehouseholdfinanceliterature, seee.g.,Fagereng,Gottlieb,andGuiso(2017)andCatherine(2021). 49

Stock Share Stock Share Stock Share 25th ptile Median 75th ptile High−School College 1.00 0.75 0.50 0.25 0.00 1.00 0.75 0.50 0.25 0.00 40 60 80 40 60 80 40 60 80 Age skcotS ni htlaeW fo erahS Source Data (SCF) F.I.R.E. Model Estim. Beliefs Model Each marker in the figure represents the relevant statistic calculated over a three-year age-group. See the main text for precise definitionsoftheagegroups. Noneofthemomentsinthisfigureweretargetedinestimation. Ages60to71areomittedbecauseof transitionaldynamicsintoretirementthatthemodeldoesnotaccountfor. Figure10: Modelfitofnon-targetedmoments In sum, replacing F.I.R.E. beliefs with the estimated specification of beliefs produces a better fit of targeted moments, lower estimates of relative risk aversion coefficients, higherestimatesofdiscountfactors,andlowerestimatesofentrycosts. Theseconclusions are robust to the sampling variation of the targeted moments shown in Figure 8. In Appendix F, I follow Chen, Didisheim, and Scheidegger 2021 and Catherine et al. 2022 in approximating the structural life cycle models with deep neural networks. I use the approximate models to demonstrate that these qualitative conclusions would hold if the estimation exercise was repeated for 500 different bootstrapped values of the targeted parameters. The confidence intervals in Table 8 come from the distributions of these bootstrappedestimates. Figure10presentsageprofilesforthequartilesoftheunconditionalshareofwealthin 50

Table9: Averageshareofwealthinstocksbysubjectivemeanquintiles Quintileof𝜇 (cid:98)𝑖 Dataset 1 2 3 4 5 HRS 17.2% 24.5% 25.1% 32.8% 37.4% SimulatedHRS 9.3% 26.3% 34.4% 38.6% 40.9% For each dataset (true and simulated HRS), I calculate the individual-level estimate of the mean of the subjective distribution of log-returns𝜇 (seeEquation3. Iaggregatethedatasetsattheindividuallevelbytakingtheaverageoftheshareofwealthinstocks (cid:98)𝑖 acrosswaves.Then,Isplitthedatasetsbyquintilesof𝜇 andreporttheaverageshareofwealthinstocksbetweenthosequintiles.The (cid:98)𝑖 “SimulatedHRS”matchesthesamplecompositionandresponsepatternsofthetrueHRS,butusesmyestimatedstructuralmodelto produceprobabilisticresponsesandportfoliochoices. stocks,whichwerenotdirectlytargetedinestimation. Theestimatedbeliefsspecification models fit these moments better than their F.I.R.E. counterparts. The F.I.R.E. models struggle to produce agents that participate in the stock market but invest a low share of their wealth in stocks. Because they rely on the participation cost, the age profiles that they imply for unconditional shares tend to start at 0% and then jump to higher values when agents decide to enter. Additionally, they do not generate much variation in the share of wealth in stocks of those who participate: the different percentiles of the unconditional share are close at most ages. In contrast, the models with estimated beliefs generate a distribution for the unconditional share whose different percentiles followqualitativelydifferenttrajectoriesacrossthelifecycle—thelevelsandshapesofthe 25th and 75th percentiles are different. The different percentiles implied by the models with estimated beliefs track their empirical counterparts closely for both high school and collegegraduates. A core prediction of the model that is not targeted by the estimation process is the relationship between people’s beliefs and their portfolio choices. To study this relationship, I use the estimated models of beliefs and life cycle portfolio choices to simulate a synthetic dataset that resembles the HRS. For each respondent in the HRS sample38 I create a simulated counterpart with the same level of education. Then, I draw the beliefs 38IusethesamesamplefromSection3.1.2: financialrespondentsaged50to65thathadatleastahigh schooldegreeandansweredaprobabilisticquestionaboutstocks. 51

(𝜇 ,𝜎 )andlevelofroundingℛ ofeverysimulatedagentfromtheestimatedmeasurement 𝑖 𝑖 𝑖 system (Section 3.1.2). I generate the simulated responses to {𝑃≥0,𝑃≤−20,𝑃≥20} using the 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 estimated distributions for the {𝜀≥0,𝜀≥20,𝜀≤−20} shocks. Given their beliefs, I simulate 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 the savings and portfolios of each agent using the life cycle model. Finally, I drop observations from the simulated dataset to exactly reproduce non-response patterns of the HRS sample: I observe the simulated counterpart of every respondent exactly the same numberoftimes,answeringthesamequestions,atthesameages. The model does well at reproducing the relationship between the portfolios of HRS respondentsandtheirexpectedlog-returns. Iusethenoisyandroundedprobabilisticresponsesinthesimulateddatasettocalculateindividualestimatesofthemeanlog-return𝜇 (cid:98)𝑖 asIdidwiththetrueHRS(seeEquation3). Usingtheseestimatesinsteadofthesimulated 𝜇 approximates the role of noise and rounding in attenuating the relationship between 𝑖 beliefs and portfolio choices in the true HRS. Table 9 presents the average unconditional share of wealth in stocks for different quintiles of 𝜇 , comparing the true and simulated (cid:98)𝑖 HRS samples. The table shows that the true and simulated relationship between beliefs and portfolios are similar. This is notable given that this relationship was not targeted: themodelofbeliefsandtheportfoliochoicemodelwereestimatedindependently. Overall,usingthespecificationofbeliefsfromsurveymeasurementsimprovesthelife cyclemodel’sfitofU.S.households’portfolioswithmoderatelevelsofriskaversion,lower financialcostsofentry,andhighertime-discountfactorsthanitsF.I.R.E.counterparts. The modelalsoreproducesnon-targetedfeaturesofthedistributionofportfoliosharesandits relationshipwithexpectations. 6. The Welfare Costs of Misspecified Beliefs This section considers a scenario in which stocks continue to perform as they have historically. In this scenario, the beliefs estimated from survey measurements would be 52

misspecified because they differ from the historical distribution of returns. An objective observer expects lower welfare for an agent with misspecified beliefs than for a counterpart with accurate beliefs because the decisions of the former are based on an incorrect model of the world. Using the estimated life cycle model, I quantify these welfare shortfalls for agents with the beliefs estimated from survey measurements. The metric that I useisthefractionofpermanentincomethattheobjectivebenevolentobserverwouldtake fromeachagentinexchangeforcorrectingtheirbeliefs.39 Averagewelfareshortfallsstart out at less than 3.5% of permanent income for young agents but follow a hump shape that peaks around the age of retirement at 8% to 14% of permanent income, depending ontheirlevelofeducation. Since beliefs can differ from the data generating process for risky returns, the discountedwelfarethatagentsexpectcandifferfromthediscountedwelfarethatan objective —one who knew the true data generating processes and their decision rules— observer would expect them to receive. Let 𝔙 (𝑃 ,𝑀 ,𝑁 ,Paid ;𝜇,𝜎) denote the discounted 𝑡 𝑡 𝑡 𝑡 𝑡 welfare that we objectively expect an agent to receive starting from age 𝑡 and state (𝑃 ,𝑀 ,𝑁 ,Paid )40 if his beliefs about returns are (𝜇,𝜎). The function 𝔙 (·), which I 𝑡 𝑡 𝑡 𝑡 𝑡 define formally in Appendix G, uses the agent’s own preferences to discount the future and assumes that he will follow the policy functions that solve his dynamic problem according to his beliefs. The difference between 𝔙 (·) and the agent’s subjective value 𝑡 function 𝑉 (·) is that 𝔙 (·) uses the true data-generating parameters (𝜇𝑆𝑃500,𝜎𝑆𝑃500) in its 𝑡 𝑡 expectations. Thus,if(𝜇,𝜎) ≠ (𝜇𝑆𝑃500,𝜎𝑆𝑃500),then𝔙 (·;𝜇,𝜎) ≠ 𝑉 (·)ingeneral. 𝑡 𝑡 To measure the expected welfare shortfalls that misspecified beliefs cause, I use a compensating variation. The metric is the proportional reduction in permanent income (present and future) that would need to accompany the correction of an agent’s beliefs to leave him at his current level of objectively-expected welfare. Formally, for an agent with 39The objective benevolent observer is necessary because agents are not aware that their beliefs are misspecified. Theywouldnotpaytochangetheirbeliefsbecausetheythinkthattheyarecorrect. 40Paid𝑡 indicateswhethertheagenthaspaidthefixedstock-marketentrycostornot. 53

statevariables(𝑃 ,𝑀 ,𝑁 ,Paid )andbeliefs(𝜇,𝜎),Ifindthe𝜆 thatsatisfies: 𝑡 𝑡 𝑡 𝑡 (cid:16) (cid:17) 𝔙 (cid:0)𝑃 ,𝑀 ,𝑁 ,Paid ;𝜇,𝜎(cid:1) = 𝔙 (1−𝜆)×𝑃 ,𝑀 ,𝑁 ,Paid ;𝜇𝑆𝑃500,𝜎𝑆𝑃500 . (9) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 The metric 𝜆 can be interpreted as the maximum fraction of permanent income that an altruistic and objective planner would be willing to take from the agent in exchange for amending his beliefs about returns. Equation 9 cannot be solved for 𝜆 analytically, but it shows that an agent’s expected welfare shortfall is contingent on his age, permanent income,assets,whetherornothehaspaidtheentrycost,andhisbeliefs.41 I calculate the expected welfare shortfalls from the estimated heterogeneous beliefs forindividualswiththepreferencesandassetsimpliedbytheestimatedlifecyclemodel. For each level of education, I find the expected welfare functions {𝔙 (·)}100 using the 𝑡 𝑡=24 preferenceandcostparametersfromTable8fortheestimatedbeliefsspecifications. Then, I simulate the lives of agents with beliefs about stock returns drawn from the educationspecific estimated distributions. These simulations use the same population sizes and shock realizations as in the estimation process (see Section 4). Finally, I evaluate the estimatedwelfareshortfall𝜆 ofeveryagentateveryyear. The welfare shortfalls from misspecified beliefs follow a hump shape across the life cycle, starting low in agents’ youth and peaking at the age of retirement. Figure 11 presents the expected welfare shortfalls 𝜆 of high school and college graduates at every age, evaluated at their simulated assets and incomes. For both levels of education, the averageshortfallislowestattheyoungestage,withvaluesof0.9%and3.4%ofpermanent income for high school and college graduates respectively. Accurate beliefs yield their benefits later in life, when agents have accumulated more savings and they rely on them for their consumption. The impatient discounting of these future benefits results in the low initial welfare shortfalls. The shortfalls increase as agents accumulate wealth and 41Because the model is homothetic in permanent income, 𝜆 can be expressed as a function of assets normalizedbypermanentincome,andtheotherstates. 54

10% 5% 0% 40 60 80 Age noitairaV .pmoC .gvA )emocnI .mreP fo %( Education High−School College Thefigurepresentsaveragewelfarelossesfrommisspecifiedbeliefsaboutrisky-assetreturnsateveryage. Isimulatepopulationsof agentsthatbehaveaccordingtothebeliefsandpreferencesestimatedinSections3.1.2and5.Then,foreveryagent-periodobservation, I compute the expected welfare shortfall 𝜆 defined in Equation 9. I report the average of this measure for every age-education combination. Figure11: Expectedwelfareshortfallsacrossthelifecycle approachretirement,reachingtheirpeakatage64forhighschoolgraduateswithavalue of8.1%ofpermanentincomeandatage65forcollegegraduateswithavalueof14.2%of permanentincome. Theshortfallsdeclineprogressivelythereafter,asagentsdepletetheir wealth and life expectancy. Despite college graduates having beliefs closer on average to the historical benchmark (see Table 5), their average welfare shortfalls are greater than those of high school graduates at all ages. This is due to their higher estimated discount factor,higherlevelsofsavings,andlowerreplacementratesofretirementincome. Themagnitude,agepatterns,andeducationaldifferencesinthewelfareshortfallsfrom distorted beliefs are consistent with several findings from the financial literacy literature. First, Figure 11 indicates that agents would derive the greatest (discounted) benefits from correcting their misconceptions about the risky asset at ages close to retirement. Empirically,differentmeasuresoffinancialliteracyfollowasimilarhumpshapeacrossthe lifecycleofU.S.respondentsandpeakclosetothisagerange(LusardiandMitchell2023). Additionally, models that allow for endogenous accumulation of financial knowledge prescribe that it must peak around the age of retirement and at higher levels for those with more education (Lusardi, Michaud, and Mitchell 2017, 2020). While these models 55

High−School College 6.3% 11.8% 0.2 4.8% 6.6% 9.0% 3.6% 4.2% 2.0% 4.6% 11.0% 0 0 . . 0 1 1 0 0 .1 .9 % % 1 7 0 . . 3 9 % % 8 6 . . 7 2 % % 7 6 . . 4 1 % % 7 7 6 . . . 6 0 6 % % % 2 1 0 9 . . 0 0 % % 1 9 8 .3 .7 % % 1 1 8 9 3 .7 . . 7 9 % % % 1 1 9 5 1 . . . 2 6 9 % % % 1 1 2 1 . . 9 5 % % −0.1 10.9% 10.9% 10.9% 9.0% 20.0% 20.0% 20.0% 16.3% 10.9% 9.0% 20.0% 10.9% 20.0% −0.2 10.9% 20.0% 10.9% 10.9% 0.25 0.50 0.75 0.25 0.50 0.75 s m Avg. Comp. Variation S&P 500 Hist. (% of Perm. Income) 0% 5% 10%15%20% Thefigurepresentsaveragewelfarelossesfrommisspecifiedbeliefsaboutrisky-assetreturnsforeverysetofbeliefsconsideredinthe life-cyclemodelestimation.IsimulatepopulationsofagentsthatbehaveaccordingtothebeliefsandpreferencesestimatedinSections 3.1.2and5. Then,foreveryagent-periodobservation,Icomputetheexpectedwelfareshortfall𝜆definedinEquation9. Ireportthe averageofthismeasureforeverybelief-educationcombinationatage65. Figure12: Expectedwelfareshortfallsfromdifferentbeliefsatage65 find start-of-life welfare shortfalls of similar magnitudes to those that I calculate,42 my findingsshowthat,asapercentageofincome,theshortfallscanbefargreateraroundthe age of retirement. These patterns highlight the years preceding retirement as a potential “teachablemoment”forfinancialknowledgeinterventionssincepeoplehaveaccumulated enough wealth to put their knowledge to use and anticipate that they will become more reliant on this knowledge for their support. Identifying these “teachable moments” has beenhighlightedasacrucialdeterminantofthesuccessoftheseinterventionsinchanging downstreambehaviors(KaiserandMenkhoff2017). Workplaceinterventions,forinstance, are a modality of financial knowledge program that has gathered increasing interest (see Clark 2023; Lusardi and Mitchell 2023) and which these results favor over earlier interventions. Theshortfallsinwelfarevarysubstantiallywithindividuals’beliefsaboutstockreturns, 42Forexample,intheirmodelofcostlyinvestmentsinfinancialknowledge,Lusardi,Michaud,andMitchell (2017) find that endowing 25-year-olds with perfect financial knowledge would increase their welfare by magnitudescomparableto2-3%increasesintheirlifetimeconsumption. 56

and the largest impacts fall on those with beliefs that discourage them from ever owning stocks. Figure12displaystheaverageexpectedwelfareshortfallforsimulatedindividuals with different beliefs and levels of education at age 65, their last working year. For both levels of education, the largest welfare shortfalls occur among individuals with low subjective means and volatilities of log-returns. These individuals do not perceive an equity premium and, therefore, they refrain from ever participating in the stock market. The average welfare shortfall for these groups of non-participants is 10.9% of permanent income for high school graduates and 20.0% for college graduates. Welfare shortfalls are lower for the group of agents who perceive an equity premium, progressively increasing as beliefs move further away from the model-consistent benchmark. The average welfare shortfall among those who believe there is an equity premium is 6.1% of permanent incomeforhigh-schoolgraduatesand11.9%forcollegegraduates. The results in this section highlight considerations for the design and application of interventionsaimedatchangingfinancialknowledgeandbehaviors. LusardiandMitchell (2023) stress the importance of pinpointing the groups for which these interventions have the greatest effects and suggest that they might not be cost-effective for groups that will not use their financial knowledge in a timely manner. The foregoing analysis suggests that interventions aimed at improving the financial knowledge of young and disadvantaged individuals with low savings can have smaller effects on welfare than those targeted at wealthier and more educated individuals. This conclusion is consistent with the findings of, e.g., Lusardi, Michaud, and Mitchell (2020). This socioeconomic gradient notwithstanding, the potential welfare effects of information treatments on the most disadvantaged are large. The question of how to design interventions effective at changing both financial knowledge and behaviors remains a crucial area of ongoing research(seeKaiserandMenkhoff2017;Kaiser,Lusardi,etal.2022;Clark2023). 57

7. Misspecified Beliefs and the Value of Public Policies Theprecedingsectionsdemonstratethatdifferentassumptionsaboutbeliefscanproduce different estimates of preferences in saving and portfolio choice models. Do these differences change the answers that these models offer to counterfactual questions about the welfareimpactsofpolicychanges? Thissectiondemonstratesthattheydo,bycalculating thewelfareimpactsoftwosimplepolicychangesunderthedifferentmodelsestimatedin Section5. The first policy change that I evaluate is the elimination of unemployment insurance. I model this change by making the unemployment benefits replacement factor zero, 𝒰C.F. = 0; this makes the worst transitory income shock that working agents can draw much worse. The second policy change that I evaluate is a reduction in Social Security benefits that makes the retirement income of all agents 20% lower. I model this change by shrinking the permanent income growth factor that is applied at retirement, Γ C.F. = 0.8 × Γ . The goal of these exercises is to clearly illustrate the effects of the 𝑖,66 𝑖,66 estimated preferences, not to represent realistic policy proposals. The change in unemploymentbenefitshighlightsdifferencesintheimpliedvalueofinsuranceandthechange toretirementbenefitshighlightsdifferencesinthevalueoffutureincome. The metric that I use to represent the welfare effects of policy changes is the onetime cash transfer that, if bundled with the policy change, would leave each agent at their original level of objectively-expected welfare. I express this quantity as a fraction of permanent income. Using the objective expected welfare function 𝔙(·) introduced in Section6anddenotingitscounterpartunderthecounterfactualchangeswith𝔙 C.F.(·),the welfaremetric 𝐶𝑉 solves 𝔙 (𝑃 ,𝑀 ,...) = 𝔙 C.F.(𝑃 ,𝑀 +𝐶𝑉 ×𝑃 ,...). (10) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 58

Table10: AverageWelfareLossesfromPolicyChangesUnderDifferentModels HighSchool College Model Age30 Age60 Age30 Age60 WelfareLossfromRemovalofU.I.(as%ofayear’swages) F.I.R.E. 46% 25% 120% 10% Est. Beliefs 41% 7% 50% 4% WelfareLossfromReductionofS.S.(as%ofayear’swages) F.I.R.E. 0% 6% 1% 98% Est. Beliefs 0% 60% 8% 123% Seethemaintextforthedefinitionofthepolicychanges. Thewelfarelossesarecalculatedatthesimulatedstatesandagesofthe populationsusedintheestimationofthelife-cyclemodels. Eachrowindicateswhichpreferencesand(andcorrespondingpolicy functions)fromTable8wereusedtocalculatewelfarelosses. Lossesareexpressedasafractionofagents’permanentincome;see Equation10. I evaluate this metric for each agent and period in the simulated populations used to estimate the life cycle model (see Section 4). This gives the metric the interpretation of the losses incurred by agents at different ages and states if the policies are introduced as asurprise. The preference estimates and policy functions generated by different specifications of beliefsproduceverydifferentestimatesofthewelfareeffectsofthepolicychanges. Table 10presentstheaveragewelfarecostsofthepolicychangesforindividualsofdifferentages andlevelsofeducation,comparingtheirimpliedvaluesundertheF.I.R.E.modelsandthe modelsthatuseestimatedbeliefs. Asthetableshows,theaveragewelfarelossesestimated under the F.I.R.E. models can be multiples of those estimated under the model with heterogeneous beliefs and vice-versa. These differences come mainly from the different preference estimates that the models imply (see Table 8). The removal of unemployment benefitsproducesmuchgreaterwelfarelossesundertheF.I.R.E.model,becauseitshigher estimates of risk aversion raise the value of insurance. With a relative-risk aversion (𝜌) of11,30-year-oldcollegegraduatesintheF.I.R.E.modelvalueunemploymentbenefitsat 120% of a year’s wages; this falls to less than half with the risk aversion of 5 in the model that uses estimated beliefs. For retirement benefits that occur in the future, valuations 59

are higher under the model that uses estimated beliefs due to its higher estimates of the time-discountfactor(𝛽). Forinstance,atage60,thedifferenceindiscountfactorsforhighschool graduates make the welfare losses from the reduction in social security benefits implied by the model with estimated beliefs 10 times larger than those implied by the F.I.R.E.model. Different specifications of beliefs could simply be considered alternative strategies to explainthesamesetofempiricalfacts. Withbeliefsgiven,otherfeaturesofourmodels— preferences in this case—can change to accommodate the targeted moments. However, as the exercises in this section demonstrate, the alternative explanations of facts derived from different specifications of beliefs can have starkly different implications for the type ofcounterfactualandwelfarequestionsthatthestructureofourmodelsaffords. 8. Concluding Remarks Dominitz and Manski (2007) note that “many households (...) are not as convinced as economistsareabouttheexistenceofanequitypremium.” Sincethetheirpioneeringwork, the HRS has expanded the range of available measurements of household expectations in both time and variety, having collected nearly two decades of measurements and now including three different questions regarding equity returns. This paper uses this expandedsetofmeasurementstoshowthatmanyhouseholdsremainunconvincedofthe existence of an equity premium and that even those who seem to believe in its existence deemitsmallerinrisk-adjustedtermsthanwhateconomistsusuallyassume. These facts about measured expectations provide a qualitative explanation for why most households do not invest most of their wealth in equities. The exercises carried out in this paper quantitatively evaluate the plausibility of this explanation. They demonstrate that the explanation has several attractive features: it substantially enhances the capacity of the considered model to reproduce both portfolio choices and savings, and 60

their relationship with age and education; it brings the estimates of unobserved preference parameters to ranges more consistent with alternative sources of evidence; and it is backedbyarobustbodyofmeasurementsandempiricalresults. Important challenges and questions remain. The model put forward in this paper faces difficulties in matching the low savings of groups like young households and those without a college degree. Allowing households to borrow and modeling the social programsthatlow-wealthhouseholdsusetosmooththeirconsumptionarepossiblewaysto reduce these difficulties. Additionally, the model proposed in this paper assumes that households do not change their beliefs about equity returns or that they do not react to short-term fluctuations in their opinions. I make this assumption to replicate features of belief measurements like the “dominance of individual fixed effects” (Giglio et al. 2021) and the fact that households’ portfolios have weak responses to changes in their elicited expectations. Questions such as why households do not learn in ways that eliminate the persistent heterogeneity in their measured beliefs or why the association between changesintheirelicitedexpectationsandchangesintheirportfoliosisweakareleftunresolved. The increasing availability of individual-level measurements of expectations and portfolioscanhelpaddressthesequestions. AuthorAffiliation MateoVelásquez-Giraldo, BoardofGovernorsoftheFederalReserveSystem, DivisionofResearchandStatistics, MacroeconomicandQuantitativeStudiesSection. 61

Appendix A. Estimating the Model of Beliefs AsdiscussedinSection2.2andinpreviousstudieslikeGiustinelli,Manski,andMolinari (2022),peopleroundtheiranswerstoprobabilisticquestions. Foreachoftheprobabilistic questionsaboutstockreturns,Table3showsthefractionofallanswersthataremultiples of5%,10%,25%,50%and100%. TheTableshowsthat,foreachquestion,lessthan2.5%of the answers are not multiples of 5%. Based on this fact, 5% is the finest level of rounding in my model and I round the few answers that are not multiples of 5% to the nearest 5% multiple. A.1 The Likelihood Function The likelihood function and its derivation are similar to those in Kézdi and Willis (2011) andAmeriks,Kézdi,etal.(2020). Denotethesetofparametersofthebeliefsmodelwith 𝜗B ≡ {𝜈 ,𝜈 ,Ψ,Σ,℘fi}, 𝜇 𝜎 andletthedataconsistoftripletsofresponsestoprobabilisticquestions (cid:26) (cid:27) (cid:110) (cid:111) 𝑃≥0,𝑃≥20,𝑃≤−20 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑡∈𝒯 (𝑖) 𝑖∈ℐ where ℐ is the set of respondents and 𝒯 (𝑖) denotes the set of time periods in which individual 𝑖 answeredtheprobabilisticquestions. The first step in evaluating the likelihood function for 𝜗B is to find the equiprobable grid for (𝜇,𝜎) that is associated with 𝜗B.Grid ≡ {𝜈 ,𝜈 ,Ψ}. I construct an equiprobable 𝜇 𝜎 gridwith 𝑛2 pointsthatapproximatesdistribution2usingthefollowingsteps. 62

1. Find an equiprobable 𝑛-point grid for 𝑥 ∼ 𝒩(𝜈 ,Ψ )|𝑥 > 0. Denote the grid with 𝜎 2,2 𝜎#. 2. Forevery 𝜎 in 𝜎#,findanequiprobable 𝑛-pointgridforthedistributionof𝜇 condi- 𝑖 tionalon 𝜎 = 𝜎,whichgivenEquation2is 𝑖 (cid:32) 2 (cid:33) Ψ Ψ 1,2 1,2 𝒩 𝜈 + (𝜎 −𝜈 ), Ψ − . 𝜇 𝜎 1,1 Ψ Ψ 2,2 2,2 Denotethatgridwith𝜇#(𝜎). 3. Thejoint 𝑛2-pointgridwillbe (𝜇,𝜎)# ≡ {(𝜇,𝜎) : 𝜇 ∈ 𝜇#(𝜎),𝜎 ∈ 𝜎#}. Thelikelihoodofanagent’sresponsedependsbothontheparametersoftheirsubjectivedistributionofreturns(𝜇 ,𝜎 )andonthedegreetowhichtheyroundtheiranswers. I 𝑖 𝑖 denoteagent𝑖’slevelofroundingwithℛ andconsider5%,10%,25%,50%,and100%asthe 𝑖 possiblelevelstowhichagentsroundtheiranswers. Therefore,∀𝑖 ℛ ∈ {5,10,25,50,100}. 𝑖 The likelihood of an answer that has been rounded is that of the interval of all the realnumbersthatroundtothatanswer. Tofacilitatetherepresentationoftheseintervals, definethefollowingtwosetsoffunctions: • 𝑢 (𝑥)givesthelowestnumberin[0,1]thatroundsto𝑥whenthelevelofroundingis ℛ ℛ. For instance, 𝑢 (0.15) = 0.125, 𝑢 (0.30) = 0.25, 𝑢 (0.5) = 0.375, 𝑢 (100) = 0.75, 5 10 25 50 and 𝑢 (0.0) = 0.0. 100 • 𝑢¯ (𝑥)givesthehighestnumberin[0,1]thatroundsto 𝑥 whenthelevelofrounding ℛ is ℛ. For instance, 𝑢¯ (0.15) = 0.175, 𝑢¯ (0.30) = 0.35, 𝑢¯ (0.5) = 0.625, 𝑢¯ (100) = 1.0, 5 10 25 50 and 𝑢¯ (0.0) = 0.5. 100 63

WiththesefunctionsandtheresponsemodelfromEquation1,wecansaythatifagent 𝑖 rounds their answers to the ℛ -level and has subjective-distribution parameters (𝜇 ,𝜎 ), 𝑖 𝑖 𝑖 then (cid:18)𝜇 (cid:19) 𝑃≥0 = 𝑥 ↔𝑢 (𝑥) ≤ Φ 𝑖 + 𝜀≥0 ≤ 𝑢¯ (𝑥) 𝑖,𝑡 ℛ 𝑖 𝜎 𝑖 𝑖,𝑡 ℛ 𝑖 (11) 𝜇 𝜇 ↔Φ −1(𝑢 (𝑥))− 𝑖 ≤ 𝜀≥0 ≤ Φ −1(𝑢¯ (𝑥))− 𝑖 , ℛ 𝑖 𝜎 𝑖 𝑖,𝑡 ℛ 𝑖 𝜎 𝑖 (cid:18)𝜇 −ln1.20 (cid:19) 𝑃≥20 = 𝑦 ↔𝑢 (𝑦) ≤ Φ 𝑖 + 𝜀≥20 ≤ 𝑢¯ (𝑦) 𝑖,𝑡 ℛ 𝑖 𝜎 𝑖 𝑖,𝑡 ℛ 𝑖 (12) 𝜇 −ln1.20 𝜇 −ln1.20 ↔Φ −1(𝑢 (𝑦))− 𝑖 ≤ 𝜀≥20 ≤ Φ −1(𝑢¯ (𝑦))− 𝑖 , ℛ 𝑖 𝜎 𝑖 𝑖,𝑡 ℛ 𝑖 𝜎 𝑖 and (cid:18)ln0.8−𝜇 (cid:19) 𝑃≤−20 = 𝑧 ↔𝑢 (𝑧) ≤ Φ 𝑖 + 𝜀≤−20 ≤ 𝑢¯ (𝑧) 𝑖,𝑡 ℛ 𝑖 𝜎 𝑖 𝑖,𝑡 ℛ 𝑖 (13) ln0.8−𝜇 ln0.8−𝜇 ↔Φ −1(𝑢 (𝑧))− 𝑖 ≤ 𝜀≤−20 ≤ Φ −1(𝑢¯ (𝑧))− 𝑖 . ℛ 𝑖 𝜎 𝑖 𝑖,𝑡 ℛ 𝑖 𝜎 𝑖 Equations 11-13 and the assumption that (𝜀≥0,𝜀≥20,𝜀≤−20) ∼ 𝒩(0,Σ) allow me to compute (cid:16) (cid:17) P 𝑃 𝑖 ≥ ,𝑡 0 = 𝑥,𝑃 𝑖 ≥ ,𝑡 20 = 𝑦,𝑃 𝑖 ≤ ,𝑡 −20 = 𝑧|(cid:0)𝜇 𝑖 ,𝜎 𝑖 (cid:1) ,ℛ 𝑖 (14) as the integral of a normal density over a cube. Since I do not use observations in which theanswertoanyofthequestionsis“donotknow/refuse,”observationswhereresponses for at least one of the question is missing correspond to instances where not all questions were asked. For instance, in all observations before 2008, only 𝑃≥0 was asked. For these observations,thelikelihoodofthegivenanswersomitsthequestionsthatwerenotasked anditbecomesanintegraloverarealinterval(ifonlyonequestionisasked)orarectangle (iftwoquestions wereasked). With thisclarification,Iusethe samenotationinEquation 14forcompleteandincompletesetsofanswers. 64

Now,Icanwritethelikelihoodofobservinganindividual 𝑖 withresponses {(𝑃≥0,𝑃≥20,𝑃≤−20)} 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑡∈𝒯 (𝑖) conditionalonhisroundingtypeas (cid:18) (cid:19) (cid:110) (cid:111) ℓ (𝑃≥0,𝑃≥20,𝑃≤−20) |ℛ = 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖 𝑡∈𝒯 (𝑖) 1 (cid:213) (cid:214) (cid:16) (cid:17) (cid:169) 𝑃≥0,𝑃≥20,𝑃≤−20|(cid:0)𝜇 ,𝜎 (cid:1) ,ℛ (cid:170), 𝑛2 (cid:173) P 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖 𝑖 𝑖 (cid:174) (𝜇 ,𝜎)∈(𝜇,𝜎)# 𝑡∈𝒯 (𝑖) 𝑖 𝑖 (cid:171) (cid:172) where I have integrated over the 𝑛2 equiprobable (𝜇,𝜎) grid-points. The unconditional likelihoodfollowsfromintegratingovertheroundingtypesusingtheprior ℘fi, (cid:18) (cid:19) (cid:110) (cid:111) ℓ (𝑃≥0,𝑃≥20,𝑃≤−20) = 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑡∈𝒯 (𝑖) (cid:18) (cid:19) (cid:213) (cid:110) (cid:111) ℘ ×ℓ (𝑃≥0,𝑃≥20,𝑃≤−20) |ℛ ℛ 𝑖 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖 𝑡∈𝒯 (𝑖) ℛ∈{5,10,25,50,100} 𝑖 Finally,thelog-likelihoodfunctioncomesfromaggregatingoverindividuals (cid:18)(cid:26) (cid:27) (cid:19) (cid:18) (cid:19) (cid:110) (cid:111) (cid:213) (cid:110) (cid:111) lnℒ 𝑃≥0,𝑃≥20,𝑃≤−20 |𝜗B = lnℓ (𝑃≥0,𝑃≥20,𝑃≤−20) . 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝑡∈𝒯 (𝑖) 𝑡∈𝒯 (𝑖) 𝑖∈ℐ 𝑖∈ℐ A.2 Parameter Estimates Iestimatethebeliefsmodelbymaximumlikelihoodforeverylevelofeducation, (cid:32) (cid:33) (cid:26) (cid:27) (cid:110) (cid:111) 𝜗B = argmaxlnℒ 𝑃≥0,𝑃≥20,𝑃≤−20 |𝜗 𝐸 𝑖,𝑡 𝑖,𝑡 𝑖,𝑡 𝜗 𝑡∈𝒯 (𝑖) 𝑖∈ℐ 𝐸 where 𝐸 indexeseducationalattainmentlevels—highschoolandcollegegraduates. Table11presentstheparameterestimatesforeverylevelofeducation. Readersshould 65

Table11: Maximum-likelihoodestimatesofthebeliefsmodel HighSchool College 𝜈 −0.108 −0.071 𝜇 (0.014) (0.009) 𝜈 0.499 0.418 𝜎 (0.009) (0.008) Ψ 0.019 0.016 1,1 (0.001) (0.001) Ψ 0.011 0.008 2,1 (0.002) (0.001) Ψ 0.060 0.035 2,2 (0.005) (0.003) Σ 0.607 0.453 1,1 (0.007) (0.008) Σ 0.574 0.460 2,2 (0.010) (0.011) Σ 0.642 0.397 3,3 (0.011) (0.009) ℘5 0.403 0.512 (0.006) (0.009) ℘10 0.425 0.400 (0.006) (0.009) ℘25 0.043 0.029 (0.003) (0.004) ℘50 0.116 0.052 (0.004) (0.005) ℘100 0.013 0.007 (0.002) (0.002) Log-Likeligood −121667.124 −59763.868 N.Obs 24027 11184 N.Individuals 8463 3562 N.ExcludedDK/RFObs. 3216 613 Standarderrorscomefromtheinverseofthenegativeofthehessianofthelog-likelihoodfunction,evaluatedexactlyattheparameter estimatesusingautomaticdifferentiationtools. 66

note,however,thattheestimatesofEquation2donothavethetraditionalmean-covariance interpretation of multivariate normal parameters. The reason is that the actual distributions from which (𝜇,𝜎) are drawn condition on the event 𝜎 > 0. These estimates are presented for completeness; for interpreting and comparing the belief distributions that go into the structural model, I refer readers to the depictions of the discretized belief distributions(Figure4inthemaintext). 67

B. Calibration of the Medical Expenditures Process Theshocksoop representtheratioofanagent’sincomethatisusedupbyout-ofpocket 𝑖,𝑡 medical expenditures in a year. To approximate their distribution at different ages and foragentswithdifferentlevelsofeducation,IusetheRANDHRSlongitudinalfile,which constructsvariousvariablesofinterestinamannerthatisconsistentacrossHRSwaves. I start by defining a measure of what would be a retiree’s household’s income. The measurethatIuseincludesthehousehold’searnings,incomefrompensionsandannuities, income from Social Security Disability and Supplemental Security Income, and income fromSocialSecurityretirement. ThiscorrespondstothesumoftheRANDHRSvariables IEARN, IPENA, ISSDI, and ISRET. It also corresponds to their measure of “total income” minusgovernmenttransfers,capitalincome,andincomefrom“othersources.” Starting with the third wave of the HRS, The RAND HRS longitudinal file includes a measure of out of pocket medical expenses over the previous two years at the time of the interview, OOPMD.43 I divide this measure by two to obtain an estimate of medical expenses over a year at the household level. The ratio between this measure and the previously defined income of the respondent’s household—for households with strictly positive income—is what I take as the ratio of out-of-pocket medical expenditures to income over a year; I denote it with oop. Figure 6 depicts the distribution of oop for individualsofdifferentagesandlevelsofeducation. To construct discrete distributions that approximate the variability of oop, I start by groupingobservationsaccordingtotheirlevelofeducationandage. Iuse5-yearagebins [66,70],[71,75],...,[85,90]andafinal[91,100]bin. Foreachcombinationofage-groupand education,Iconstructdiscreteequiprobabledistributionsusingquantilesoftheempirical distributionofoop. First,Isplitthe[0,1]intervalin 𝑛 intervalsofthesamelength,where 𝑛 is the number of points of the discrete approximation—in my case, 𝑛 = 7. Then, I 43Themeasureisnotconstructedforthefirstwave. Inthesecondwave,thequestionthatisusedtobuild themeasurehadadifferenttimehorizonandthereforeIexcludeit. 68

Table12: Discreteapproximationsofmedicalexpenditures/incomeratios EquiprobablePoints AgeGroup High-School [50,55] 0.000 0.004 0.010 0.019 0.033 0.064 0.205 (55,60] 0.000 0.005 0.013 0.023 0.040 0.077 0.245 (60,65] 0.000 0.008 0.018 0.032 0.055 0.104 0.290 (65,70] 0.001 0.011 0.023 0.038 0.064 0.111 0.264 (70,75] 0.002 0.014 0.028 0.046 0.074 0.126 0.293 (75,80] 0.001 0.015 0.031 0.053 0.084 0.143 0.346 (80,85] 0.001 0.016 0.033 0.059 0.096 0.168 0.433 (85,90] 0.000 0.016 0.036 0.066 0.110 0.229 0.849 (90,100] 0.000 0.011 0.034 0.069 0.131 0.301 1.479 College [50,55] 0.000 0.003 0.007 0.012 0.021 0.039 0.121 (55,60] 0.001 0.005 0.010 0.016 0.027 0.049 0.163 (60,65] 0.002 0.007 0.014 0.023 0.040 0.078 0.227 (65,70] 0.003 0.010 0.019 0.031 0.050 0.089 0.227 (70,75] 0.004 0.013 0.024 0.039 0.060 0.103 0.262 (75,80] 0.004 0.015 0.028 0.047 0.074 0.123 0.294 (80,85] 0.004 0.017 0.033 0.054 0.089 0.155 0.410 (85,90] 0.002 0.015 0.033 0.057 0.100 0.191 0.719 (90,100] 0.000 0.015 0.039 0.075 0.160 0.389 1.485 Thetablepresentsapproximationstothedistributionofhealthexpenditureshocksasafractionofincome.Iapproximatededistribution oftheseshocksforeachagegroupandlevelofeducationwithanequiprobablediscretedistribution. Eachrowdisplaystheseven pointsusedtoapproximatethedistributionforeachagegroupandlevelofeducation.Eachpointhasaprobabilityof1/7.Seethetext foradescriptionofhowIobtainthepoints. take the midpoint of each interval and denote with 𝑄 the set of midpoints—for 𝑛 = 7, 𝑄 = {0.071,0.214,...,0.786,0.929}. Finally, for every 𝑞 ∈ 𝑄, I obtain the 𝑞 quantile of the empiricaldistributionofoopforthegivengroup. Myapproximationofthedistributionof oopisadiscreterandomvariablewherethepossibledrawsarethe 𝑛 previouslyobtained quantiles and each of them occurs with probability1/𝑛. Table 12 displays the points that Iuseinthemodelforeveryagegroupandlevelofeducation. 69

C. Recursive Formulation and Normalization of the Model Individualsubscriptsaredroppedforsimplicitythroughoutthissection. An agent starts his life not having paid the stock-market entry cost. In periods when the cost has not been paid, the agent observes his risk-free resources and his permanent income,andthendecideswhethertoenterthestockmarketornot. Hisvaluefunctionis 𝑉Out(𝑀 ,𝑃 ) = max{𝑉Stay(𝑀 ,𝑃 ), 𝑉In(𝑀 −𝐹 ×𝑃 ,0,𝑃 )}, 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 where 𝑉Stay(·) is the value function of an agent who stays out of the stock market and 𝑡 𝑉In(·)isthevaluefunctionofanagentwhohasalreadypaidthestock-marketentrycost. An agent who has just decided not to pay the stock-market entry cost decides how much to consume out of his assets, knowing that in the next time period he will have the opportunitytoenterthestockmarketagain. Hisvaluefunctionis 𝑉 𝑡 Stay(𝑀 𝑡 ,𝑃 𝑡 ) = max𝑢(𝐶 𝑡 )+𝛽𝛿 𝑡+1 E 𝑡 (cid:2) 𝑉 𝑡 O + u 1 t(𝑀 𝑡+1 ,𝑃 𝑡+1 ) (cid:3) + (cid:19)(cid:19)𝛿 𝑡+1 B(𝐴 𝑡 ) 𝐶 𝑡 Subjectto: 0 ≤𝐶 ≤ 𝑀 𝑡 𝑡 . 𝐴 =𝑀 −𝐶 𝑡 𝑡 𝑡 𝑀 =𝑅𝐴 +𝑌 𝑡+1 𝑡 𝑡+1 𝑃 =Γ 𝜓 𝑃 𝑡+1 𝑡+1 𝑡+1 𝑡 𝑌 =𝜃 𝑃 𝑡+1 𝑡+1 𝑡+1 Finally,anagentwhohasalreadypaidthestock-marketentrycostobserveshisbalances in both the risky and risk-free assets and his permanent income, and then decides how to reallocate his balances and how much to consume. He forms expectations about the 70

futureknowingthathewillnotneedtopaytheentrycostagain. Hisvaluefunctionis 𝑉In(𝑀 ,𝑁 ,𝑃 ) = max𝑢(𝐶 )+𝛽𝛿 E (cid:2) 𝑉In (𝑀 ,𝑁 ,𝑃 ) (cid:3) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡+1 𝑡 𝑡+1 𝑡+1 𝑡+1 𝑡+1 𝐶 ,𝐷 𝑡 𝑡 + (cid:19)(cid:19)𝛿 𝑡+1 B(𝐴 𝑡 + 𝑁˜ 𝑡 ) Subjectto: −𝑁 ≤ 𝐷 ≤ 𝑀 , 0 ≤ 𝐶 ≤ 𝑀˜ 𝑡 𝑡 𝑡 𝑡 𝑡 𝑀˜ =𝑀 −𝐷 (cid:0)1−1 𝜏(cid:1) 𝑡 𝑡 𝑡 [𝐷 ≤0] 𝑡 . 𝑁˜ =𝑁 +𝐷 𝑡 𝑡 𝑡 𝐴 =𝑀˜ −𝐶 𝑡 𝑡 𝑡 𝑀 =𝑅𝐴 +𝑌 𝑡+1 𝑡 𝑡+1 𝑁 =𝑅˜ 𝑁˜ 𝑡+1 𝑡+1 𝑡 𝑃 =Γ 𝜓 𝑃 𝑡+1 𝑡+1 𝑡+1 𝑡 𝑌 =𝜃 𝑃 𝑡+1 𝑡+1 𝑡+1 I assume that the utility function 𝑢(·) and the bequest function B(·) are homothetic of the same degree (1 − 𝜌). With this assumption, the problem can be normalized by permanent income, following Carroll (2022). Using lower case variables to denote their upper-case counterparts normalized by permanent income (𝑥 = 𝑋 /𝑃 ) and defining 𝑡 𝑡 𝑡 Γ ˜ = Γ 𝜓 ,wecanwritenormalizedversionsofthepreviousvaluefunctionsas 𝑡 𝑡 𝑡 𝑣Out(𝑚 ) = max{𝑣Stay(𝑚 ), 𝑣In(𝑚 −𝐹,0)}, (15) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 71

(cid:104) (cid:105) 𝑣 𝑡 Stay(𝑚 𝑡 ) = max𝑢(𝑐 𝑡 )+𝛽𝛿 𝑡+1 E 𝑡 Γ ˜1 𝑡+ − 1 𝜌 𝑣 𝑡 O + u 1 t(𝑚 𝑡+1 ) + (cid:19)(cid:19)𝛿 𝑡+1 B(𝑎 𝑡 ) 𝑐 𝑡 Subjectto: 0 ≤𝑐 ≤ 𝑚 𝑡 𝑡 , (16) 𝑎 =𝑚 − 𝑐 𝑡 𝑡 𝑡 𝑅 𝑚 = 𝑎 +𝜃 𝑡+1 ˜ 𝑡 𝑡+1 Γ 𝑡+1 and (cid:104) (cid:105) 𝑣 𝑡 In(𝑚 𝑡 ,𝑛 𝑡 ) = max𝑢(𝑐 𝑡 )+𝛽𝛿 𝑡+1 E 𝑡 Γ ˜1 𝑡+ − 1 𝜌 𝑣 𝑡 I + n 1 (𝑚 𝑡+1 ,𝑛 𝑡+1 ) + (cid:19)(cid:19)𝛿 𝑡+1 B(𝑎 𝑡 +𝑛˜ 𝑡 ) 𝑐 ,𝑑 𝑡 𝑡 Subjectto: −𝑛 ≤ 𝑑 ≤ 𝑚 , 0 ≤ 𝑐 ≤ 𝑚˜ 𝑡 𝑡 𝑡 𝑡 𝑡 . 𝑚˜ =𝑚 − 𝑑 (cid:0)1−1 𝜏(cid:1) 𝑡 𝑡 𝑡 [𝑑 ≤0] 𝑡 𝑛˜ =𝑛 + 𝑑 𝑡 𝑡 𝑡 𝑎 =𝑚˜ − 𝑐 𝑡 𝑡 𝑡 𝑅 𝑚 = 𝑎 +𝜃 𝑡+1 ˜ 𝑡 𝑡+1 Γ 𝑡+1 𝑅˜ 𝑛 = 𝑡+1 𝑛˜ 𝑡+1 ˜ 𝑡 Γ 𝑡+1 Itcanbeshownthat 1−𝜌 𝑉Out(𝑀 ,𝑃 ) =𝑃 𝑣Out(𝑚 ), 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑉Stay(𝑀 ,𝑃 ) =𝑃 1−𝜌 𝑣Stay(𝑚 ), 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 1−𝜌 𝑉In(𝑀 ,𝑁 ,𝑃 ) =𝑃 𝑣In(𝑚 ,𝑛 ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 72

andthatthepolicyfunctionsthatsolveeachoftheproblemsarerelatedthrough 𝐶Stay(𝑀 ,𝑃 ) = 𝑃 𝑐Stay(𝑚 ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝐶In(𝑀 ,𝑁 ,𝑃 ) = 𝑃 𝑐In(𝑚 ,𝑛 ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝐷 (𝑀 ,𝑁 ,𝑃 ) = 𝑃 𝑑 (𝑚 ,𝑛 ). 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 Therefore,Isolvethenormalizedproblemandre-scaleitssolutionstoobtaintheoriginal problem’ssolutions. C.1 Partition Into Stages An additional insight that facilitates solving the dynamic problem of the agent who has paid the stock-market entry cost is that the two decisions that he takes in a period (rebalancing his assets and consuming) can be seen as happening sequentially. This is convenient because the sequential sub-problems are easier to solve than the multi-choice fullproblem. Tore-expresstheproblem,Itaketheorderofthedecisionstobe: firstrebalanceassets, then consume. I denote the stages at which these decisions are taken with Reb and Cns. I willuse 𝑣Reb(·)and 𝑣Cns(·)torepresenttherespective stagevaluefunctions . Inowpresenteachstageindetail,workingbackwardsintime. C.1.1 Consumptionstage,Cns The important fact to realize at this stage is that the first thing that the agent will do in period 𝑡 +1 is make his asset-rebalancing decision. Therefore, that is the value function aboutwhichtheagentformsexpectations. 73

Theconsumptionstageproblemis (cid:104) (cid:105) 𝑣 𝑡 Cns(𝑚˜ 𝑡 ,𝑛˜ 𝑡 ) = max𝑢(𝑐 𝑡 )+𝛽𝛿 𝑡+1 E 𝑡 Γ ˜1 𝑡+ − 1 𝜌 𝑣 𝑡 R + e 1 b(𝑚 𝑡+1 ,𝑛 𝑡+1 ) + (cid:19)(cid:19)𝛿 𝑡+1 B(𝑎 𝑡 +𝑛˜ 𝑡 ) 𝑐 𝑡 Subjectto: 0 ≤𝑐 ≤ 𝑚˜ 𝑡 𝑡 (17) 𝑎 =𝑚˜ − 𝑐 𝑡 𝑡 𝑡 𝑅 𝑚 = 𝑎 +𝜃 𝑡+1 ˜ 𝑡 𝑡+1 Γ 𝑡+1 𝑅˜ 𝑛 = 𝑡+1 𝑛˜ 𝑡+1 ˜ 𝑡 Γ 𝑡+1 C.1.2 Rebalancingstage,Reb Thefirstdecisionthatanagenttakesishowtoreallocatehisassets. Hispayoffisgivenby the subsequent consumption problem’s value function, evaluated at his post-rebalancing assets. 𝑣Reb(𝑚 ,𝑛 ) =max𝑣Cns(𝑚˜ ,𝑛˜ ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑑 𝑡 Subjectto: (18) −𝑛 ≤𝑑 ≤ 𝑚 𝑡 𝑡 𝑡 𝑚˜ =𝑚 − 𝑑 (cid:0)1−1 𝜏(cid:1) 𝑡 𝑡 𝑡 [𝑑 ≤0] 𝑡 𝑛˜ =𝑛 + 𝑑 𝑡 𝑡 𝑡 74

D. First Order Conditions and Value Function Derivatives Thecomputationalsolutionofthemodelusesthefirstorderconditionsoftheoptimization problems and the derivatives of the value functions defined above. This appendix writes thefirstorderconditionsandvaluefunctionderivativesexplicitly. D.0.1 Agentwhoisstayingoutofthestockmarket,Stay ThefirstorderconditionofthemaximizationprobleminEquation16is (cid:20) 𝜕𝑣Out (cid:21) 𝑢′(𝑐 𝑡 ) = 𝛽𝑅𝛿 𝑡+1 E 𝑡 Γ ˜− 𝑡+ 𝜌 1𝜕𝑚 𝑡+1 + (cid:19)(cid:19)𝛿 𝑡+1 B′(𝑎 𝑡 ). (19) 𝑡+1 The condition, while necessary, is not sufficient because 𝑣Out(·) is not concave. Therefore, IusetheDC-EGMmethod(Iskhakovetal.2017)tosolvethissub-problem. D.0.2 Consumptionstage,Cns Thefirstorderconditionforaninteriorsolution(𝑐 < 𝑚˜)oftheconsumptionstageproblem (Equation17)is (cid:34) (cid:35) 𝜕𝑣Reb 𝑢′(𝑐 𝑡 ) = 𝛽𝑅𝛿 𝑡+1 E 𝑡 Γ ˜− 𝑡+ 𝜌 1𝜕𝑚 𝑡+1 + (cid:19)(cid:19)𝛿 𝑡+1 B′(𝑎 𝑡 +𝑛˜ 𝑡 ) (20) 𝑡+1 Thederivativesofthestagevaluefunctionare 𝜕𝑣Cns(𝑚˜ ,𝑛˜ ) 𝑡 𝑡 𝑡 = 𝑢′(𝑐 ) (21) 𝑡 𝜕𝑚˜ 𝑡 (cid:34) (cid:35) 𝜕𝑣Cns(𝑚˜ ,𝑛˜ ) 𝜕𝑣Reb 𝑡 𝜕𝑛˜ 𝑡 𝑡 = 𝛽𝛿 𝑡+1 E 𝑡 𝑅˜ 𝑡+1 Γ ˜− 𝑡+ 𝜌 1𝜕𝑛 𝑡+1 + (cid:19)(cid:19)𝛿 𝑡+1 B′(𝑎 𝑡 +𝑛˜ 𝑡 ) (22) 𝑡 𝑡+1 75

D.0.3 Rebalancingstage,Reb Thefirstorderconditionforasolutionofthetype 𝑑 ∈ [(−𝑛,0)∪(0,𝑚)]intherebalancing stageproblem(Equation18)is 𝜕𝑣Cns 𝜕𝑣Cns (cid:0)1−1 𝜏(cid:1) 𝑡 = 𝑡 , (23) [𝑑 𝑡 ≤0] 𝜕𝑚˜ 𝜕𝑛˜ 𝑡 𝑡 andanecessaryconditionforasolutionofthetype 𝑑 = 0is 𝜕𝑣Cns 𝜕𝑣Cns 𝜕𝑣Cns (1−𝜏) 𝑡 ≤ 𝑡 ≤ 𝑡 (24) 𝜕𝑚˜ 𝜕𝑛˜ 𝜕𝑚˜ 𝑡 𝑡 𝑡 Thederivativesofthestagevaluefunctionare 𝜕𝑣Reb(𝑚 ,𝑛 ) (cid:26)𝜕𝑣Cns 𝜕𝑣Cns(cid:27) 𝑡 𝑡 𝑡 = max 𝑡 , 𝑡 (25) 𝜕𝑚 𝜕𝑚˜ 𝜕𝑛˜ 𝑡 𝜕𝑣Reb(𝑚 ,𝑛 ) (cid:26) 𝜕𝑣Cns 𝜕𝑣Cns(cid:27) 𝑡 𝑡 𝑡 = max (1−𝜏) 𝑡 , 𝑡 (26) 𝜕𝑛 𝜕𝑚˜ 𝜕𝑛˜ 𝑡 𝑡 𝑡 76

Table13: GridsandDiscretizations Symbol #Points Type of grid/Discretization Gridsfor𝑎,𝑚,𝑛,𝑚˜,𝑛˜ 𝑎#, 𝑚#, 𝑛#, 𝑚˜#, 101 Equispaced in logs be- 𝑛˜# tween 1𝑒 −6 and 5𝑒3, with 0added. Perm. Inc. Shock 𝜓 5 Equiprobable. Trans. Inc. Shock 𝜃˜ 5 Equiprobable. Riskyreturn 𝑅˜ 5 Equiprobable. E. Numerical Solution of the Life-Cycle Model. E.1 Grids and discretizations The solution of the model uses various discrete grids over state variables and discretizations of stochastic variables. Table 13 summarizes the grids and discretization schemes that I use for every variable and shock. The only shock discretization not addressed in Table 13 is the out-of-pocket medical expenditure shock, which I discuss in detail in AppendixB. E.2 Transformed-space interpolation In my solution, I treat the continuous choice variables 𝑐 and 𝑑 as continuous, instead of discretizing them. Because of this decision and the multiple shocks in the model, I must evaluatevaluefunctionsandtheirderivativesonvaluesofthestatevectorthatarenoton mygrids. Intheseinstances,Iinterpolate(andextrapolate)usingon-gridvalues. Toimprovemyapproximationofpropertiesofthevalueandmarginal-valuefunctions, such as their curvature, and the fact that they approach −∞ or ∞ as wealth approaches zero,Iperformmyinterpolationsandextrapolationsina“transformed”space. Thetrick, discussed in Carroll (2022), consists in finding a transformation 𝑇 : R → R such that 77

𝑇(𝑓(·)) behaves more like an affine function than 𝑓(·), the function that we are trying to approximate. Then, with our chosen transformation, we create an interpolator 𝑔ˆ(·) for 𝑇(𝑓(·)). When asked to approximate 𝑓(𝑥) for some off-grid 𝑥 we return𝑇−1(𝑔ˆ(𝑥)), where 𝑇−1(·)istheinverseof𝑇(·). I apply this trick when constructing interpolators for value functions and marginal valuefunctions. Forvaluefunctions,Iuse 𝑥1−𝜌 𝑇(𝑥) = 𝑢−1(𝑥) = (cid:0)(cid:0)1−𝜌(cid:1) × 𝑥(cid:1) 1− 1 𝜌 , 𝑇−1(𝑥) = 𝑢(𝑥) = . 1−𝜌 Formarginalvaluefunctions,Iuse 𝑇(𝑥) = 𝑢′−1(𝑥) = 𝑥 − 𝜌 1 , 𝑇−1(𝑥) = 𝑢′(𝑥) = 𝑥−𝜌. E.3 Solving the Consumption Stage, Cns For this stage, I use the method of endogenous gridpoints (Carroll 2006) over risk-free resourcesatdifferentfixedlevelsofriskyresources. First,forevery 𝑛˜ intherisky-assetbalancesgrid 𝑛˜#, • Apply the endogenous gridpoint method using Equation 20 over the grid 𝑎# for end-of-period risk-free assets. The result is a set of optimal consumption points on anendogenousgridofpost-rebalancingrisk-freeassets 𝑚˜#−endog(𝑛˜), 𝑡 𝑐∗(𝑚˜,𝑛˜)for 𝑚˜ ∈ 𝑚˜#−endog(𝑛˜). 𝑡 𝑡 • Denote the endogenous 𝑚˜ associated with 𝑎 = 0 by 𝑚˜ (𝑛˜). This is the point where 𝑡 0 the liquidity constraint stops binding. If 𝑚˜ (𝑛˜) > 0, then add 𝑚˜ = 0, 𝑐∗(0,𝑛˜) = 0 to 0 𝑡 thesetofendogenousrisk-freeassetsandoptimalconsumptionpoints. 𝜕𝑣Cns 𝜕𝑣Cns • Use the optimal consumption points to find 𝑣Cns, 𝑡 , 𝑡 at (𝑚˜,𝑛˜), for 𝑚˜ ∈ 𝑡 𝜕𝑚˜ 𝜕𝑛˜ 78

𝑚˜#−endog(𝑛˜)usingEquations17,21,and22. 𝑡 The result is a set of (𝑚˜,𝑛˜) points on which we know the consumption, value, and marginal value functions. This set of points is not a rectangular grid because the values of 𝑚˜ are different for every 𝑛˜. The next step is to use the current points to obtain an approximationofthefunctionsoverarectangulargrid. Istartwithanexogenousgridforpost-rebalancingrisk-freeassets𝑚˜#whichIaugment by adding the points where the liquidity constraint stops binding, {𝑚˜ (𝑛˜) : 𝑛˜ ∈ 𝑛˜#}. 0 Denotetheaugmentedgridwith 𝑚˜#+. Forevery 𝑛˜ intherisky-assetbalancesgrid 𝑛˜#, • Use the values of 𝑐∗, 𝑣Cns, 𝜕𝑣 𝑡 Cns , 𝜕𝑣 𝑡 Cns calculated at {(𝑚˜,𝑛˜) : 𝑚˜ ∈ 𝑚˜#−endog(𝑛˜)} to ap- 𝑡 𝑡 𝜕𝑚˜ 𝜕𝑛˜ 𝑡 𝜕𝑣Cns 𝜕𝑣Cns proximatethevalue of 𝑐∗, 𝑣Cns, 𝑡 , 𝑡 at{(𝑚˜,𝑛˜) : 𝑚˜ ∈ 𝑚˜#+}usingtransformed- 𝑡 𝜕𝑚˜ 𝜕𝑛˜ spacelinearinterpolation(andextrapolation). This process yields approximations of the functions of interest on a rectangular grid, {(𝑚˜,𝑛˜) : 𝑚˜ ∈ 𝑚˜#+ and 𝑛˜ ∈ 𝑛˜#}. I use these approximations to construct bilinear 𝜕𝑣Cns 𝜕𝑣Cns transformed-spaceinterpolatorsfor 𝑐∗, 𝑣Cns, 𝑡 , 𝑡 . 𝑡 𝑡 𝜕𝑚˜ 𝜕𝑛˜ E.4 Solving the Rebalancing Stage, Reb Inthisstage,Ilookfortheoptimaldeposit/withdrawalfunction 𝑑∗(𝑚,𝑛). 𝑡 Istartbydefiningthefollowingconvenienttransformationoftheoptimaldeposit/withdrawal  𝑑∗(𝑚,𝑛)/𝑚 𝑑∗(𝑚,𝑛) > 0   𝑡 𝑡     𝔡 𝑡 (𝑚,𝑛) = 𝑑∗(𝑚,𝑛)/𝑛 𝑑∗(𝑚,𝑛) < 0 𝑡 𝑡      0 𝑑∗(𝑚,𝑛) = 0.  𝑡  The transformation simply re-scales the deposits or withdrawals by the balance of the fund that they are coming from, so that 𝔡 = 1 corresponds to moving all the risk-free 𝑡 balances to the risky stocks fund, and 𝔡 = −1 corresponds to withdrawing all balances 𝑡 fromthestocksfund. 79

I search for the optimal 𝔡 in a rectangular exogenous grid {(𝑚,𝑛) : 𝑚 ∈ 𝑚# and 𝑛 ∈ 𝑡 𝑛#}. The search uses the first order conditions in Equations 23 and 24 and proceeds as follows. Forevery(𝑚,𝑛)intherectangulargrid, 𝜕𝑣Reb(𝑚,𝑛) 𝜕𝑣Reb(𝑚,𝑛) • Evaluate 𝑡 and 𝑡 . 𝜕𝑚˜ 𝜕𝑛˜ 𝜕𝑣Cns(𝑚,𝑛) 𝜕𝑣Cns(𝑚,𝑛) 𝜕𝑣Cns(𝑚,𝑛) • If(1−𝜏) 𝑡 ≤ 𝑡 ≤ 𝑡 ,then𝔡 (𝑚,𝑛) = 0. 𝜕𝑚˜ 𝜕𝑛˜ 𝜕𝑚˜ 𝑡 𝜕𝑣Cns(𝑚,𝑛) 𝜕𝑣Cns(𝑚,𝑛) • If 𝑡 < (1−𝜏) 𝑡 𝜕𝑛˜ 𝜕𝑚˜ – We know that the solution involves withdrawing funds, 𝔡 𝑡 (𝑚,𝑛) < 0. We have tocheckthecornersolution𝔡 (𝑚,𝑛) = −1. 𝑡 If – 𝜕𝑣Cns(𝑚 +(1−𝜏)𝑛,0) 𝜕𝑣Cns(𝑚 +(1−𝜏)𝑛,0) 𝑡 < (1−𝜏) 𝑡 , 𝜕𝑛˜ 𝜕𝑚˜ thenset𝔡 (𝑚,𝑛) = −1. 𝑡 Otherwise,usebisectionsearchtofindthe 𝑑∗ ∈ (−1,0)thatsolves – 𝜕𝑣Cns(𝑚 −(1−𝜏)𝑑∗,𝑛 + 𝑑∗) 𝜕𝑣Cns(𝑚 −(1−𝜏)𝑑∗,𝑛 + 𝑑∗) 𝑡 = (1−𝜏) 𝑡 𝜕𝑛˜ 𝜕𝑚˜ andset𝔡 (𝑚,𝑛) = 𝑑∗/𝑛. 𝑡 𝜕𝑣Cns(𝑚,𝑛) 𝜕𝑣Cns(𝑚,𝑛) • If 𝑡 < 𝑡 𝜕𝑚˜ 𝜕𝑛˜ – We know that the solution involves depositing funds, 𝔡 𝑡 (𝑚,𝑛) > 0. We also know that the corner solution 𝔡 (𝑚,𝑛) = 1 is not optimal because it leaves the 𝑡 agentwithoutfundstoconsume. Usebisectionsearchtofindthe 𝑑∗ ∈ (0,1)thatsolves – 𝜕𝑣Cns(𝑚 − 𝑑∗,𝑛 + 𝑑∗) 𝜕𝑣Cns(𝑚 − 𝑑∗,𝑛 + 𝑑∗) 𝑡 𝑡 = 𝜕𝑚˜ 𝜕𝑛˜ 80

– Set𝔡 𝑡 (𝑚,𝑛) = 𝑑∗/𝑚. The result of this process is a rectangular grid of asset-combinations and their associated optimal rebalancing solutions. I use these points to construct a bilinear interpolator for𝔡 (·,·). Then,Iusethefactthat(fromEquation18), 𝑡 𝑣Reb(𝑚 ,𝑛 ) =𝑣Cns(𝑚˜ ,𝑛˜ ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 Where:     𝔡 𝑡 (𝑚,𝑛)×𝑚 𝑡 , If𝔡 𝑡 (𝑚,𝑛) ≥ 0 𝑑 = 𝑡  𝔡 (𝑚,𝑛)×𝑛 , If𝔡 (𝑚,𝑛) < 0  𝑡 𝑡 𝑡  𝑚˜ =𝑚 − 𝑑 (cid:0)1−1 𝜏(cid:1) 𝑡 𝑡 𝑡 [𝑑 ≤0] 𝑡 𝑛˜ =𝑛 + 𝑑 𝑡 𝑡 𝑡 tocalculate 𝑣Reb(·,·)anditsderivativeswhenevertheyareneeded. 𝑡 E.5 Solving the Problem of the Agent Staying Out, Stay Agents who have not paid the one-time entry cost must decide whether to pay it at the startofeveryperiod. Theydothisbycomparingthevalueofstayingoutandnotpaying, versusenteringandpaying,asshowninEquation15. Anagentwhoenterspassesontothe asset-rebalancing stage, Reb. An agent who does not enter must choose his consumption knowing that next period he will start outside of the stocks-fund again. I use the “DC- EGM”method(Iskhakovetal.2017)tosolvethisproblem. I start with an exogenous grid for end-of-period risk-free assets, 𝑎#. I apply the endogenous-gridpoint method inversion over 𝑎# using the first-order condition in Equation 19. The result is a set of candidate endogenous consumption and beginning-of- 81

period-assetspoints,associatedwiththeexogenousend-of-period-assetspoints, (cid:8)(cid:0)𝑐𝑒 (𝑎),𝑚𝑒 (𝑎)(cid:1) : 𝑎 ∈ 𝑎#(cid:9) . 𝑡 𝑡 AsarguedbyIskhakovetal.(2017),thesepointswillnotnecessarilybeoptimal. Thefuture discrete decision of whether to pay the cost or not makes the value function not-concave and therefore points that satisfy the first order condition are not necessarily optimal. I calculatethediscountedutilityassociatedwiththepointsfromtheendogenous-gridpoint inversion, (cid:104) (cid:105) 𝑣 𝑡 𝑒(𝑎) = 𝑢(𝑐 𝑡 𝑒(𝑎))+𝛽𝛿 𝑡+1 E 𝑡 Γ ˜1 𝑡+ − 1 𝜌 𝑣 𝑡 O + u 1 t(𝑎) + (cid:19)(cid:19)𝛿 𝑡+1 B(𝑎 𝑡 ) 𝑅 𝑚 = 𝑎 +𝜃 . 𝑡+1 ˜ 𝑡+1 Γ 𝑡+1 Then,Iapplytheupper-envelopealgorithminIskhakovetal.(2017)tothecandidatepoints (cid:8)(cid:0)𝑐𝑒 (𝑎),𝑚𝑒 (𝑎),𝑣𝑒 (𝑎)(cid:1) : 𝑎 ∈ 𝑎#(cid:9) toeliminatenon-optimalpoints. Iaddthe“kinkpoints” 𝑡 𝑡 𝑡 ofthevaluefunctiontothegrid. Theresultisasetofoptimalconsumptionandvaluepoints overarefinedendogenousgridforstart-of-periodassets𝑚∗#, (cid:8)(cid:0)𝑐∗(𝑚),𝑣∗(𝑚)(cid:1) : 𝑚 ∈ 𝑚∗#(cid:9) . 𝑡 𝑡 𝑡 Iusethesepointstocreatelineartransformed-spaceinterpolatorsfor𝑣Stay,𝑐Stay,and 𝜕𝑣 𝑡 Stay . 𝑡 𝑡 𝜕𝑚 82

F. Surrogate Bootstrap The estimated parameters of the structural life cycle models are functions of the targeted moments. Uncertainty about these moments generates uncertainty about the best-fitting valuesoftheparameters. Iquantifythisuncertaintybycalculatingthetargetedmoments on bootstrapped samples, estimating an approximate—“surrogate”—model on each set of moments and presenting various summary statistics of the resulting distribution of estimatedparameters. The bootstrapped targeted moments come from education-specific re-samplings of the SCF analytical sample defined in Section 4.1. I divide the sample into high-school and college graduates, and draw 500 bootstrapped samples for each level of education usingthere-scaledsurveyweights. Then,Icalculatethetargetedmomentsoneachofthe bootstrapped samples. This results in 500 vectors of 45 targeted moments each for both levelsofeducation,{𝑚HS}500 and{𝑚 College }500. 𝑏,𝑘 𝑘=0 𝑏,𝑘 𝑘=0 To find the estimated parameters that would be result from each vector of moments, I use accurate surrogate models that approximate the relationship between parameters and moments embedded in the true structural models. Estimating the structural models 500 times for each level of education would come at a high computational arising mainly fromtheirsolutionandsimulationateachcandidatevectorofparameters. Recentstudies like Chen, Didisheim, and Scheidegger (2021) and Catherine et al. (2022) show that these costly evaluations can be avoided using accurate approximations of the structural model that can be constructed using known parameter-moments pairs. These approximations are know as “surrogate models.” Denoting with (cid:2) the space of admissible parameter values and with M the set of possible values for targeted moments, a model is a function ˆ 𝑓 : (cid:2) → M and a surrogate model is a different function 𝑓 : (cid:2) → M that approximates the truemodel 𝑓,butwhichisideallymuchfastertoevaluate. The surrogate models that I use to approximate the true structural models are deep 83

Table14: Root-Mean-Squared-ErrorsofSurrogateModelsoverTargetedMoments High-School College Sample F.I.R.E. Est. Beliefs F.I.R.E. Est. Beliefs Training 5×10−3 2×10−3 4×10−3 2×10−3 Validation 1×10−2 9×10−2 1×10−2 3×10−3 neural networks. The networks have 3 inputs (the parameters {𝜌,𝛽,𝐹}) and 45 outputs (thetargetedmoments),and4hiddenlayerswith192neuronseach. Iusesigmoid-linearunit“SiLU”activationfunctionsforthehiddenlayers;fortheoutputlayer,Iuse“softplus” functionsforpositivemoments(likethewealthratio)andsigmoidfunctionsformoments that are shares (like conditional stock-shares and participation rates). I use a different networkforeachcombinationofeducationalattainmentandspecificationofbeliefs(high school,college,andF.I.R.E.,Est. Beliefs). As suggested by Catherine et al. (2022), I train and validate the surrogate models using the parameter-moment points that I evaluate when estimating the true models. The optimization routine outlined in Section 4.2 evaluates each structural model at at least 2,500 points of the space of admissible parameter values (cid:2). The initial 2,500 points come from a Sobol sequence that covers (cid:2) well. The local optimization runs of the TikTak algorithm generate additional evaluations, which concentrate around the best-fitting parameter values. I save the parameter-moments pair of every one of these evaluations for each specification of the model. Then, I randomly split the points into 90% training and 10% validation samples. I train the deep networks using the “Adam” algorithm (Kingma and Ba 2017) to minimize the root-mean-squared-error (RMSE) over the targeted moments. Table 14 presents the RMSEs of every surrogate model at the end of estimation, confirming that they do a good job of approximating the predictions of the true models bothin-andout-of-sample. Foreachvectorofbootstrappedmoments,Ifindtheinputparametersofitscorrespond- 84

Table15: DifferentPercentilesofBootstrappedEstimates Percentiles F.I.R.E. Est. Belifs Difference Parameter 𝑃 𝑃 𝑃 𝑃 𝑃 𝑃 Frac. > 0 5 50 95 5 50 95 HighSchool CRRA(𝜌) 8.51 8.56 8.61 4.20 4.22 4.25 1 Disc. Fac(𝛽) 0.31 0.33 0.35 0.75 0.76 0.77 0 EntryCost(𝐹 ×100) 2.87 3.11 3.42 2.33 2.49 2.64 1 MSMLoss 15.45 16.01 16.69 3.69 4.04 4.44 1 College CRRA(𝜌) 11.36 11.48 11.55 5.07 5.10 5.14 1 Disc. Fac(𝛽) 0.62 0.63 0.64 0.88 0.89 0.89 0 EntryCost(𝐹 ×100) 0.51 0.57 1.75 0.00 0.00 0.00 1 MSMLoss 4.60 5.33 6.13 2.27 2.94 3.73 1 ing surrogate model that minimizes the SMM loss function using the same optimization routinethatIuseforthemainestimates. The𝑘−𝑡ℎsetofbootstrappedparameterestimates forthepairofeducationandbeliefs-specification(𝑒, )is b (cid:16) (cid:17)′ (cid:16) (cid:17) 𝜗ˆ𝑒,b = argmin 𝑚𝑒 − 𝑓 ˆ𝑒,b(𝜗) W𝑒 𝑚𝑒 − 𝑓 ˆ𝑒,b(𝜗) , (27) 𝑘 𝑏,𝑘 𝑏,𝑘 𝜗 where 𝑓 ˆ𝑒,b is the surrogate model for education level 𝑒 and belief specification b. The resultsaresetsofbootstrappedestimates {𝜗ˆHS,F.I.R.E.}500, {𝜗ˆCollege,F.I.R.E. }500, and{𝜗ˆHS,Est. Beliefs}500, {𝜗ˆCollege,Est. Beliefs }500. 𝑏,𝑘 𝑘=0 𝑏,𝑘 𝑘=0 𝑏,𝑘 𝑘=0 𝑏,𝑘 𝑘=0 I additionally store and present the values of the loss functions associated with each parameterestimate. Table15presentsthe5th,50th,and95thpercentilesofeveryparameterforeverycombination of educational attainment and belief specification, in addition to the minimized lossfunction“MSMLoss.” Thetableshowsthatalltheparameterestimatesandattained 85

lossesaretightlydistributedaroundthemainestimatesreportedinTable8. Foreachvector of bootstrapped moments 𝑚𝑒 , I find the difference between the parameter estimates 𝑏,𝑘 and attained losses under the F.I.R.E. and “Est. Beliefs” specifications. The last column in Table 15 presents the fraction of moment vectors for which this difference is positive. This column shows the robustness of the conclusions that, for both levels of education, models that use the estimated beliefs improve upon the fit of F.I.R.E. models and do so with lower levels of relative risk aversion, higher discount factors, and lower entry costs. These conclusions are true for each of the 500 bootstrapped vectors of moments for each levelofeducation. The95%confidenceintervalspresentedinTable8correspondtothe2.5-thand97.5-th percentilesofthebootstrappedvaluesofeachparameter,foreachmodelspecification. 86

G. The expected welfare function The welfare calculations presented in Section 6 rely on the calculation of individuals’ objectively expected welfare, which can differ from their subjective expectations due to their misspecified beliefs. This section defines my measure of expected welfare and discusseshowIcalculateit. Foragivensetofbeliefsaboutriskyreturnsdenotedwithℬ = (𝜇,𝜎)andotherparameters,IsolvethelifecyclemodelanditscomponentsdescribedinAppendixC.Denotethe resulting policy functions for every age 𝑡 with 𝐶Stay(·; ℬ), 𝐷 (·; ℬ), and 𝐶In(·; ℬ). I cal- 𝑡 𝑡 𝑡 culatefunctions𝔙 (·)thatallowmetofindtheexpectedlifetimewelfarethatanobjective 𝑡 observer would expect an agent to derive from his remaining years of life if he behaved accordingtothepolicyfunctionsassociatedwithhisbeliefsℬ. Thesefunctionsare 𝔙 S 𝑡 tay(𝑀 𝑡 ,𝑃 𝑡 ; ℬ) = 𝑢(𝐶 𝑡 )+𝛽𝛿 𝑡+1 E 𝑡 (cid:2) 𝔙O 𝑡+ ut 1 (𝑀 𝑡+1 ,𝑃 𝑡+1 ; ℬ) (cid:3) + (cid:19)(cid:19)𝛿 𝑡+1 B(𝐴 𝑡 ) Where: (28) 𝐶 =𝐶Stay(𝑀 ,𝑃 ; ℬ) 𝑡 𝑡 𝑡 𝑡 𝐴 = 𝑀 −𝐶 , 𝑀 = 𝑅𝐴 +𝑌 , 𝑃 = Γ 𝜓 𝑃 , 𝑌 = 𝜃 𝑃 𝑡 𝑡 𝑡 𝑡+1 𝑡 𝑡+1 𝑡+1 𝑡+1 𝑡+1 𝑡 𝑡+1 𝑡+1 𝑡+1 87

foragentswhohavenotpaidtherisky-assetentrycostanddecidetonotpayitin 𝑡,and 𝔙In(𝑀 ,𝑁 ,𝑃 ; ℬ) = 𝑢(𝐶 )+𝛽𝛿 E (cid:2) 𝔙In (𝑀 ,𝑁 ,𝑃 ; ℬ) (cid:3) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡+1 𝑡 𝑡+1 𝑡+1 𝑡+1 𝑡+1 + (cid:19)(cid:19)𝛿 𝑡+1 B(𝐴 𝑡 + 𝑁 𝑡 +𝐷 𝑡 ) Where: (29) 𝐷 = 𝐷 (𝑀 ,𝑁 ,𝑃 ; ℬ), 𝐶 = 𝐶In(𝑀 ,𝑁 ,𝑃 ; ℬ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝐴 = 𝑀 −𝐷 (cid:0)1−1 𝜏(cid:1) −𝐶 , 𝑀 = 𝑅𝐴 +𝑌 𝑡 𝑡 𝑡 [𝐷 ≤0] 𝑡 𝑡+1 𝑡 𝑡+1 𝑡 𝑁 = 𝑅˜ ×(𝑁 +𝐷 ), 𝑃 =Γ 𝜓 𝑃 , 𝑌 = 𝜃 𝑃 𝑡+1 𝑡+1 𝑡 𝑡 𝑡+1 𝑡+1 𝑡+1 𝑡 𝑡+1 𝑡+1 𝑡+1 for agents who have already paid the entry cost. Equations 28 and 29 differ from the value functions defined in Appendix C because the expectations are taken using the true distribution of risky asset returns. Numerically, I construct interpolators for these functions iterating backwards, using the solved policy functions and the same grids and discretizationsdescribedinAppendixE. ThesimplifiednotationthatIusefor𝔙 (·)inthemaintextiscorrespondsto 𝑡 𝔙 (𝑃 ,𝑀 ,𝑁 = 0,Paid = 0; ℬ) ≡ max (cid:8) 𝔙 Stay(𝑀 ,𝑃 ; ℬ),𝔙In(𝑀 −𝐹 ×𝑃 ,0,𝑃 ; ℬ) (cid:9) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝔙 (𝑃 ,𝑀 ,𝑁 ,Paid = 1; ℬ) ≡ 𝔙In(𝑀 ,𝑁 ,𝑃 ; ℬ). 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 88

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