Lifecycle Dynamics of Income Uncertainty and Consumption

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Lifecycle Dynamics of Income Uncertainty and Consumption James Feigenbaum and Geng Li 2008-27 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.

Lifecycle Dynamics of Income Uncertainty and Consumption James Feigenbaum and Geng Li (cid:3) y May 20, 2008 Abstract Uninsurable income risk is often cited as an explanation for empirical deviations from the Lifecycle/Permanent-Income Hypothesis such as the observation that the life-cycle pro(cid:133)le of mean consumption is humpshaped. Mostmethodsusedforestimatingincomeuncertaintyessentially measure the cross-sectional variance of a subpopulation rather than the true uncertainty or riskiness perceived by consumers. In this paper, we employ a nonparametric approach to estimate idiosyncratic income uncertainty. We measure income uncertainties as the variance of income forecasting errors at di⁄erent ages and over di⁄erent time horizons. The estimated life-cycle income uncertainty pro(cid:133)le is U-shaped and generally impliesalowerdegreeofincomeuncertaintyrelativetothepreviousliterature. We subsequently use these nonparametric estimates to calibrate a (time-inconsistent)lifecyclemodeltoassesswhetheraconsumptionhump can be generated by precautionary saving given more robust measures of incomeuncertainty. Weshowthat,withplausibleriskaversioncoe¢ cient anddiscountingfactorsandanendogenous,rarelyactiveborrowinglimit, our re(cid:133)ned measure of income uncertainty is large enough to generate a signi(cid:133)cant consumption hump that peaks around age 55 and closely matcheswiththeobservedmagnitudeoftheconsumptionhump. Wealso notice that the variation in the volatility of income shocks with respect to both age and forecast horizon has a signi(cid:133)cant impact on the size and peak age of the consumption hump. JEL Classi(cid:133)cation: E21 Keywords: consumption hump, income risk, time-inconsistent expectations, forecasting errors DepartmentofEconomics;University ofPittsburgh;4906 W.W.PosvarHall;230 South (cid:3) BouquetSt.;Pittsburgh,PA15260. E-mail: jfeigen@pitt.edu. URL: www.pitt.edu/~jfeigen. FederalReserveBoard. Emailaddress: geng.li@frb.gov. Theviewspresentedinthispaper y are those of the author(cid:146)s and are not necessarily those of the Federal Reserve Board and its sta⁄.We thank Dave DeJong, Karen Dynan, Jonathan Heathcote, Michael Palumbo, and seminar participants at the Federal Reserve Board, SUNY Stony Brook, and the University ofPittsburgh forhelpfuldiscussions and comments. Allremaining errors are ourown. 1

1 Introduction It is well documented that household nondurable-good consumption exhibits a hump-shaped pro(cid:133)le over the lifecycle. On average, consumption increases when the consumer is young, peaks in middle age, and gradually declines until retirement (Fernandez-Villaverde and Krueger (2007), Gourinchas and Parker (2002), Thurow (1969)).1 However, this pattern of consumption dynamics is not consistent with the standard Rational-Expectations Lifecycle/Permanent-Income Hypothesis (RE-LCPIH) under the assumption of complete (cid:133)nancial markets, geometric discounting, and additively separable preferences for consumption. In this framework, the lifecycle consumption pro- (cid:133)le should be monotonic (Yaari (1964)). In the case when the consumer(cid:146)s discountrateequalsthemarketinterestrate,theconsumptionpro(cid:133)leshouldbe (cid:135)at as the consumer can perfectly smooth his consumption over the lifecycle. Onepopularexplanationforahump-shapedconsumptionpro(cid:133)leisthat householdsmayfaceanincome(cid:135)owwithacomponentofuninsurablerisk(Carroll (1997), Feigenbaum (2007), Gourinchas and Parker (2002)). As shown by Leland (1968) and Sandmo (1970), a consumer with risky future income will save for precautionary reasons. This reduces current consumption and, on average,increasesfutureconsumption. Therateofconsumptiongrowthincreases withthevarianceofincomeperceivedbytheconsumer(Skinner(1988),Feigenbaum (2008b)). After all uncertainty about income is resolved, the pro(cid:133)le will concide with the path predicted by the RE-LCPIH, which will be decreasing if theinterestrateislessthanthediscountrate. Ifuncertaintydecreasesoverthe lifecycle and is su¢ ciently large at the beginning, the consumption pro(cid:133)le will be concave. Thus the dynamics of uncertainty over the lifecycle have an e⁄ect on the shape of the consumption pro(cid:133)le. The paper has two objectives. First, we construct a re(cid:133)ned measure of uncertainty over the lifecycle. Second, we assesswhetherthisre(cid:133)nedmeasureimpliesenoughvariationinlifecycleincome uncertainty to account for the observed deviation of lifecycle consumption from a monotonic pro(cid:133)le. Weproposeanovelwayofmeasuringincomeuncertaintyandapplyour estimates to a model of lifecycle consumption. Previous researchers have typically taken the cross-sectional variance of income over a subpopulation as the measure of income uncertainty. However, this approach assumes that householdsandeconometricianshaveidenticalinformationsets. Incontrast, Cunha, Heckman,andNavarro(2005)havepointedoutindividualsmayhaveadditional, private information that is relevant to predicting their future income. In other words, one has to be mindful of the distinction between income heterogeneity and uncertainty. The possibility that econometricians may have overestimated the amount of uncertainty faced by individuals creates doubt about the ability ofprecautionarymotivestoaccountfortheconsumptionhump.2 Ourapproach 1Afterretirement,asharpdeclineinnondurablesconsumptiontypicallyoccurs(Bernheim, Skinner,and Weinberg (2001)). 2It also puts into question estimates of how much precautionary motives contribute to 2

is designed to address this critique.3 Using panel data that provide information about a large number of households over many years, we estimate an income forecasting equation that includes additional information available in the data than is usually used to estimate the predictable component of income. Speci(cid:133)cally, we assume that households take into account their lagged and current income as well as impending life events when forecasting their future income. We interpret the sample variance of forecasting errors as the income uncertainty faced by households. Our approach therefore allows us to estimate a volatility matrix that summarizes the uncertainty a consumer should expect as a function of age and forecasting horizon. We also estimate the correlation between forecasting errors at various horizons. Our methodology is nonparametric in that we do not assume any functional form for the income process. We focus on the moments of the income distribution without trying to infer any underlying parameters. Wethenincorporatetheseestimatesofthevolatilityandcorrelationmatrices into a lifecycle model of consumption and saving to assess whether it can account for the hump in lifecycle consumption.4 Since we have not estimated the unconditional probability of reaching any possible history of income draws, the consumer cannot update his expectations about future income conditional on his current state using Bayes(cid:146)rule, as is normally done in the context of rational expectations. Instead, for each age we specify an income process that is consistent with our estimates of the volatility and correlation of forecast errors at each future time horizon. The consumer solves for his optimal consumption function in the present via a backwards recursion while assuming this posited income process governs his income dynamics in all future periods. However, in each ensuing period the consumer will posit a di⁄erent income process governed by the volatilty and correlation matrix estimates for that age. Thus the consumer has time-inconsistent expectations. The estimated income volatility matrix suggests that income uncertainty perceived by households evolves substantially over the lifecycle. For a given age in the future, we consistently (cid:133)nd, as one would expect, that uncertaintyaboutincomeatthatagediminishesastheconsumerapproachesthat age. For a (cid:133)xed time horizon, when consumers are young, income uncertainty gradually declines with age, presumably as decisions on career, human capital development and fertility are resolved. Income uncertainty reaches its lowest levelduringmiddleage. Afterwards,incomeuncertaintyrisesagain,potentially duetouncertaintyabouttheworkinghoursandhealthrisks. OurU-shapedlifecycle pro(cid:133)le of income uncertainty is consistent with Jaimovich and Siu (2006). aggregate saving (Carroll and Samwick (1998), Feigenbaum (2007b), Gourinchas and Parker (2001)). 3Thereisalsoaliteraturethatseekstoaddresstheseparatequestionofhowmuchincome isdispersed,irrespectiveofwhetherhouseholdsanticipatethattheirincomemaydeviatefrom the mean, so as to address issues of inequality and the sources thereof. See, for example, Heathcote, Storesletten, and Violante (2004), Huggett, Ventura, and Yaron (2007), Krueger and Perri(2001),and Storesletten,Telmer,and Yaron (2000). 4Heckman and Navarro (2005) use a similar approach to investigate the e⁄ect of income uncertainty on the decision ofhow much human capitalto accumulate. 3

Studying the CPS data, they construct the business cycle component of hours worked for workers of various age groups and (cid:133)nd a U-shaped pattern in the volatility of hours worked by age We also estimate for each age and horizon the correlation between the forecasting error in the next period (to the current age) and the error in the forecastatthathorizon. Asonewouldexpect,foragivenagethesecorrelations decrease with the forecasting horizon. Except for very young ages, our results arealsoconsistentwiththehypothesisthatthecorrelationbetweentheone-year aheadforecasterrorandtheforecasterroratsome(cid:133)xedhorizonisindependent of age. Incomparisontotheexistingliterature,wedo(cid:133)ndlessuncertaintythan has been assumed in most previous work on precautionary saving. Nevertheless, for a plausible calibration of the preference parameters, we still (cid:133)nd that precautionary saving will lead to a hump-shaped lifecycle consumption pro(cid:133)le. Indeed, one criticism of Gourinchas and Parker (2002) is that their estimates of income uncertainty imply so much precautionary saving that they require a risk aversion of one half or a discount factor of 0.84 (Feigenbaum (2007)) to obtain a lifecycle consumption pro(cid:133)le with a hump that is not signi(cid:133)cantly larger than what is seen in the data. Our more modest estimates of income uncertainty produce a consumption hump consistent with the data under more standard preference parameters. We also (cid:133)nd that the dependence of income uncertainty on both age and forecast horizon has a signi(cid:133)cant impact on the size and shape of the consumption hump. We proceed as follows. Section 2 describes the empirical methodology that we employ. Section 3 discusses the time-inconsistent consumption-saving model and our theoretical results. We also review the pertinent literature in each section. We conclude with some discussion and remarks in Section 4. 2 Measuring Income Uncertainty Over the Lifecycle 2.1 Common Practice and Some Critique Income uncertainty plays an important role in household decisions regarding consumption, saving, and investment. Traditionally, when rich micro data were not readily available, researchers had to infer the volatility of personal income from aggregate time-series data on GDP, and this volatility was taken as a proxy for income uncertainty. More recent studies of aggregate output and income such as McConnell and Perez-Quiros (2000) have documented the so-called (cid:147)Great Moderation(cid:148), a sharp decline in the volatility of GDP growth since the mid 1980s. However, aggregate data can mask important variations and correlations in income at the household level, so a decrease in the volatility of aggregate income volatility does not necessarily imply that income at the 4

household level has become less uncertain. To see this, consider a hypothetical economy populated by two consumers. Every period, each consumer receives oneunitofendowmentandthentheyengageinazero-sumgameofchancewith their endowments. After they (cid:133)nish gambling, the aggregate income of the economy remains (cid:133)xed at two with no uncertainty at all, but one cannot say the income of each individual consumer is risk-free, though their incomes are perfectly (negatively) correlated. Studying household income directly avoids this di¢ culty. To this end, large panel surveys of household income, such as the Panel Study of Income Dynamics (PSID), have become increasingly available. Previous work has routinely used either cross-sectional or time-series variances of income as a proxy forincomeuncertainty. Forexample,Dynan,ElmendorfandSichel(2007)simply focus on the standard deviations across households of the percent change in household income.5 More elaborate models often postulate some parsimoniously parameterized income process that includes a predictable part and a stochasticpart, whichinturnissomecombinationofpermanentandtransitory shocks. In practice, econometricians estimate the predictable part of income as afunctionofanagepolynomialandotherdemographicvariablesaswellasyear dummies to (cid:133)lter out economy-wide time variation (Carroll 1994, Carroll and Samwick 1997, Gourinchas and Parker 2002). The residual of observed income relative to estimated income is then interpreted as the stochastic component. Varioustechniquescanbeappliedtoinvestigatethevarianceandpersistenceof the stochastic part to quantify income riskiness. Asanalternativeapproach,insteadofestimatinganincometrend,some researchers take the mean of the income of a household over a given period as a proxy for permanent income and treat the gap between observed income and themeanasthetransitorycomponent. Inthismanner,GottschalkandMo¢ tt (1994) argued an increase in the variance of transitory earnings could explain a large portion of the widening of income inequality during the 1970s and 80s. Essentially, what these methods attempt to measure is either the crosssectionalvarianceorthetime-seriesvolatilityofaspeci(cid:133)ccomponentofincome that is orthogonal to speci(cid:133)ed information sets. Conceptually, it is true that greater income uncertainty will create larger cross-sectional variance and/or higher time-series volatility. However, it is not generically true that a larger variance or higher volatility always implies greater income uncertainty. Several important caveats must be kept in mind when interpreting such variations as a measure of the underlying income uncertainty. First, (cid:133)tting individual income with a trend driven by age and other demographiccharacteristicsrequirestheassumptionthatallindividualssharea common life-cycle income trend. This is the view introduced by, among others, MacCurdy(1982). Acompetingviewisthatindividualsfaceindividual-speci(cid:133)c income pro(cid:133)les, as proposed by Lillard and Weiss (1979). Guvenen (2007) labels the MacCurdy-type income process as a restricted income pro(cid:133)le, and 5Dynan,ElmendorfandSichel(2007)alsoprovideanelegantandcomprehensivesurveyof the literature studying household income volatilities using household data. 5

the Lillard-and-Weiss-type income process as a heterogeneous income pro(cid:133)le. He presents evidence that consumption data is more consistent with the view of the heterogeneous income process. If the underlying income process is better characterized in this way, (cid:133)tting it with a trend common to all households will increase the residual variance and hence exaggerate the apparent income uncertainty. Second, the presumed information set on which the stochastic componentisestimatedcouldbeonlyapotentiallysmallsubsetofthefullinformation set possessed by households. Consequently, what an econometrician treats as unpredictable variations may be predictable to households. If so, measured variations will also exaggerate apparent income uncertainty. Asamoreconcreteexample, considerthefollowingsimplehypothetical economy where household i(cid:146)s income at time t is given by y =e it; i;t where , the rate of income growth, is a random number assigned by nature i to household i. Suppose is known to the household but is not observed i by the econometrician. The common practice in the literature is to take the mean of over the population as the rate of predictable income growth for all i households. Ifaneconometricianmeasuresincomeuncertaintyasthedispersion of income relative to this assumed trend, he will (cid:133)nd that income uncertainty growsquadraticallywithrespecttotime, butthis is aspuriousconclusion since all households have a deterministic income process. Likewise, if the econometrician cannot identify the income trend for each household, the estimated time series volatility will also exaggerate the apparent income uncertainty. Finally, previous studies have not fully characterized how household income uncertainty evolves over the life cycle. The variance and persistence of income shocks are rarely postulated to depend on age. However, it is both theoreticallyandempiricallyappealingtostudywhethertheincomeuncertainty perceived by households does stay constant over the life cycle. Intuitively, a single 22-year old college graduate (cid:133)rst entering the labor market should have muchgreateruncertaintyabouthisincome(cid:133)veortenyearsdowntheroadthan a 40-year old with a spouse, both of whom have settled on career paths. 2.2 A Forecast-Based Non-Parametric Approach What method can more consistently and accurately measure income uncertainty, and also shed light on its evolution over the life cycle? Uncertainty stems from risk. Heuristically, greater uncertainty should make future income more di¢ cult to forecast. Intuitively then, for a given income process the income level at a remote future should bear more uncertainty than the income at a near future. Conversely, we may use the forecast accuracy to approximate and evaluate the underlying uncertainty. The larger the variances of forecast errors are the greater uncertainty a household has about its future income. Let income at age t be y . Then at age t, the s-period-ahead income t 6

can be decomposed as y =E y H +"t ; (1) t+s t+s jIt t+s whereE y H isthemathemati (cid:2) calexpec (cid:3) tationofy conditionedonage-t t+s t+s jI household information, H, and "t is an error term orthogonal to H. We (cid:2) (cid:3) It t+s I de(cid:133)ne the realized uncertainty associated with the s-period-ahead income for age-tconsumers,! ,asthevarianceofthiserrortermacrossalltheconsumers t;s with same age, t, or (cid:10)t =Var["s H]=Var[y E[y H] H]: (2) s t+sjIt t+s (cid:0) t+s jI jIt Thisapproachisnonparametricbecauseitdoesnotpresumethatincomeshocks followanyspeci(cid:133)edprocess. Wecharacterizelifecycleincomeuncertaintiesusing two L P matrices, (cid:10) and (cid:2), where we have data for L ages over P time (cid:2) horizons.. Element ! of the (cid:10) matrix is the variance of the s-year-ahead t;s forecast errors of all age-t households whereas element (cid:18) of the (cid:2) matrix is t;s the correlation between the s-year-ahead forecast error and the 1-year-ahead forecast error of age-t households. In examining the matrices (cid:10) and (cid:2), we can explicitlystudythelifecycledynamicsofincomeuncertaintyanditsimplications on consumption and asset holding patterns over the lifecycle. One obstacle to implementing this strategy is we do not know the joint distribution of y and H, and, therefore, we cannot compute E[y H] t+s It t+s jIt directly. Indeed, we do not even know exactly what H encompasses. To un- It derstand how severe this superior information problem could be and to provide some alleviation, we experiment with two speci(cid:133)cations. First, in what we call the restricted information speci(cid:133)cation (RIS), we project y conditional on t+s R, the information set that households certainly have at age t since econome- It tricians collect this data from the households at that time. Second, in what we call the augmented information speci(cid:133)cation (AIS), we project y condition t+s on the augmented information set A, where It A = R F: (3) It It [It The augmenting information set, F, contains elements that econometricians It observe as of age t + s, the target age of the forecast, but that households likely or possibly know at age t. Put di⁄erently, F approximates the supe- It rior information that households possess but econometricians do not observe concurrently. To (cid:133)x the idea, we estimate the following RIS equation y =(cid:11)+(cid:12) y +(cid:12) y +(cid:12) y +(cid:13)Z +(cid:24)Trend +"t ; (4) i;t+s 0 i;t 1 i;t 1 2 i;t 2 i;t t+s i;t+s (cid:0) (cid:0) and AIS equation y =(cid:11)+(cid:12) y +(cid:12) y +(cid:12) y +(cid:13)Z +(cid:14)Q +(cid:24)Trend +"t : (5) i;t+s 0 i;t 1 i;t 1 2 i;t 2 i;t i;t+s t+s i;t+s (cid:0) (cid:0) In the above equations, Z is a vector of variables that belong to R. This i;t It 7

includes race, education level, marital status, family size, a dummy of whether household members are currently laid o⁄or unemployed, a dummy of whether household members are self-employed, and a vector of occupation and industry dummies, all evaluated at age t, when expectations about income at t+s are formed. In addition Z includes a fourth-order polynomial of imputed years of i;t working experience, evaluated at the forecast horizon, t+s. Q is a vector of t+s variables that belong to the augmenting information set, F. We assume Q It t+s includes family size, marital status, a retirement dummy, a self-employment dummy, a vector of occupation and industry dummies, all evaluated at t+s. Besides R and A, our speci(cid:133)cation departs from most previous speci- It It (cid:133)cationsforestimatingthepredictableincomecomponentinthatweincludenot onlystandarddemographicandemploymentvariables,butalsothecurrentand lagged income in our projection equation. In principle, if we have a very long income history for a given household, even a univariate ARIMA model could potentiallyhavedecentforecastingpower. Includingsomerecentincomehistory canhelptoteaseoutinformationaboutrecentincomeshocks,andcapturepart of the individual-speci(cid:133)c component of income growth that is emphasized by Lillard and Weiss (1979) and Guvenen (2007). In our speci(cid:133)cation, we include two lags to preserve degrees of freedom. Finally, we add a simple calendar year trend to control for aggregate economic growth. Thereareseveralimportantcaveatsweneedtopointout. First,because most households do plan ahead regarding their family and career, it is not unreasonabletoassumehouseholdsknowseveralyearsaheadoftimewhattheir family size and marital status will be; whether they will be working, retired, or self-employed;andwhethertheywillchangeoccupationandindustry. However, itismoredi¢ culttojustifythathouseholdsknowallthisinformationasthetime horizon gets long.6 Therefore if s is large, we might not always have F H. It (cid:26)It Second, we try our best to project y on an information set as close to H t+s It as possible, but it is still possible that there exist some information elements (cid:19) such that (cid:19) H, but (cid:19) = R F. Third, equations (4) and (5) should 2 It 2 It [It be interpreted only as a forecasting equation, instead of a structural income equation. Finally, although we allow for age-varying income uncertainty, we computetheseforecasterrorsbyapplyingthesameparametersoftheforecasting modeltoallhouseholdsofdi⁄erentages. Asarobustnesscheck,weestimatethe forecasting model separately for each age group. Our results are qualitatively preserved, which reassures us that the age pro(cid:133)le of income uncertainty is not driven by the across-age inconsistency of the (cid:133)tting accuracy of the forecasting model. 2.3 Data Description and Sample Construction We use data from the PSID. The PSID is a nationwide household longitudinal survey conducted by the Institute of Social Research at the University of Michigan. Before 1997, the PSID was an annual survey, while after 1997 it 6Ourstudy forecasts future income up to twenty-(cid:133)ve years ahead. 8

became a biannual survey. As of year 2008, there are 34 waves of data that cover 37 years. The (cid:133)rst wave of data was collected in 1968, and the latest wave was collected in 2005. The PSID not only surveys the households in the original sample constructed in 1968, but also the households headed by the grown-up children of the original sample of households. The sample of households in the PSID has a very high retention rate. The vast majority of the households that the PSID surveys in one year will continue to participate in the next wave. There are more than 1,200 households that stayed in the survey for more than 30 years. Consequently, the sample size of the survey has grown considerably since 1968. The (cid:133)rst wave of the PSID had only 4802 households, whereas the 1994 wave had more than 10,000 households. The PSID subsequently stopped surveying households in its non-core sample. As of the most recent wave of 2005, the survey has 8002 households. Besides extensive information about work status, employment history and demographic characteristics, the PSID has detailed information on household income. In the current paper we want to study the relationship between thehousehold(cid:146)sincomeuncertaintyandconsumptiondynamicsoverthelifecycle. For our de(cid:133)nition of income, we will focus on the household total income, which is the most relevant measure of income vis-a-vis household consumption. The longitudinal structure of PSID data allows us to link a household intime ttothesamehouseholdintimet+s. Wewilltreatthesamehousehold in di⁄erent waves as independent households and not exploit the econometric properties of the longitudinal structure of the data. Assume a household was surveyed over ten years from 1971 through 1980. When we project the (cid:133)veyear-ahead income, this household renders (cid:133)ve current and future income pairs (t;t+s) = (1971;1976), ... , (1975, 1980). We will simply pool these pairs together without controlling for the household (cid:133)xed e⁄ect. Several selection rules apply when we construct the sample. First, we restrict the heads of our sample households to be those younger than 66 years old at the year to be forecasted. Thus, if we project income of year t+s as of year t, we keep only the households whose heads are younger than 66 s as of (cid:0) year t. For example, in the sample we use to forecast (cid:133)ve-year-ahead income, we restrict the heads of sample households to be younger than 61 years old. Consequently, the sample we use to study forecast errors at farther horizons is smaller than the sample used for a closer horizon. On the other side of the age restriction,weremoveallhouseholdswhoseheadsareyoungerthan23yearsold in the base year. In addition, we remove households whose heads are disabled or retired, are primarily keeping house, or are students in the base year. We further remove households whose heads report zero working hours in the base year and the previous two years or did not report valid industry or occupation information. Finally, inordertominimizethee⁄ectsofoutliers, wetrimo⁄the households with very high or very low income levels and growth rates 7. 7Wetrimo⁄thetopandbottom1%oflagged,current,andfutureincomeleveldistributions and the distribution ofincome growth between yeartand t+s. 9

Forecast Horizon 1 Year 2 Years 3 Years 4 Years 5Years Number of Observations 72905 71722 62027 59547 53479 Forecast Horizon 6 Years 7 Years 8 Years 9 Years 10 Years Number of Observations 49365 44424 40132 36049 32137 Forecast Horizon 11 Years 12 Years 13 Years 14 Years 15 Years Number of Observations 28614 25320 22348 19505 16934 Forecast Horizon 16 Years 17 Years 18 Years 19 Years 20 Years Number of Observations 14504 12491 10514 8921 7331 Forecast Horizon 21 Years 22 Years 23 Years 24 Years 25 Years Number of Observations 6034 4801 3758 2831 2003 Table 1: Number of Observations for Each Forecast Horizon Variable Mean StdDev Variable Mean StdDev Log(Family Income ) 10.21 0.62 Headage 39.81 10.68 Family Size 3.31 1.75 < High School 29.1 High School 23.3 Some College 29.4 College 18.2 White 66.6 Married 71.1 Table 2: Summary Statistics of the One-Year-Ahead Forecast Sample Inourstudy,weestimatethevarianceofforecasterrorsfortimehorizons upto25years. Becauseweincludetwolagsofincomeinourforecastingequation and the PSID started in 1968, the (cid:133)rst base year is 1970. We do not use the PSID 2003 and 2005 data because these waves used di⁄erent occupation and industry codes that cannot be mapped to those used in the previous waves.8 Table 1 lists the number of observations used in estimation for each forecast horizon. Table 2 provides summary statistics of key variables at the base year for the largest sample, the one-year-ahead forecast sample. The family income variable is de(cid:135)ated using 1982-1984 dollars. 8Allwavesbut2003and2005ofthePSIDdatahave1970censusindustryandoccupation code. The2003and2005PSID datausedtheindustryandoccupationcodederivedfrom the 2000 census. 10

2.4 Estimation Results We estimate forecast-error variances for households with heads between 23 and 64 years old. To further remove possible noise in the series, we report (cid:133)ve-year-centered moving averages for households between 25 and 62 years old. Because we have from one- to twenty-(cid:133)ve-year-ahead income projections, our income uncertainty matrix (cid:10) has dimensions 38 25 9. Figures 1 and 2 present (cid:2) several lifecycle income uncertainty pro(cid:133)les at various forecast horizons. Figure 1 plots the AIS results and Figure 2 plots the RIS results. We(cid:133)rstfocusontheAISresults. ThetoppanelofFigure1showsnearterm income uncertainties. Not surprisingly, the level of uncertainty regarding the two-year-ahead income is higher than the uncertainty associated with the one-year-ahead income. Apart from the magnitude, both uncertainty pro(cid:133)les share a similar U-shaped pattern over the lifecycle. Income uncertainty is high whenconsumersareintheirmidtolatetwenties. Incomeuncertaintycontinues to decline through the mid thirties. After that uncertainties stay at a relatively stable level before rising again in the mid forties. This rise continues into ages close to retirement. The ratio between the maximum and minimum uncertainties over the lifecycle (the max-min ratio) is 1.46 forthe one-year-ahead income and 1.37 for the two-year ahead income. Similar, but less pronounced, U-shaped pro(cid:133)les repeat in the middle panel, which shows medium-term income uncertainties. The uncertainty associated with the (cid:133)ve-year-ahead income hits bottom in the mid thirties and stays quite (cid:135)at until the mid forties. The uncertainty pro(cid:133)le of the ten-yearahead income also exhibits a U-shape, but it bottoms at a somewhat earlier age. In addition, the max-min ratio is 1.30 for the (cid:133)ve-year horizon and 1.22 for the ten-year horizon. Both are appreciably lower than the max-min ratios of uncertainties at nearer horizons. Finally, income-uncertainty pro(cid:133)les at remote horizons, e.g. (cid:133)fteen and twenty-(cid:133)ve years ahead, are plotted in the bottom panel. These curves also showappreciableU-shapedpatterns. Becauseoftheagerestrictionsweimpose, these uncertainty pro(cid:133)les cover a much shorter age-span. The max-min ratios are 1.25 for both series, lower than the ratio in the near term. Now turn to Figure 2, the RIS results. Three features of this (cid:133)gure are noteworthy. First, the contours of these life cycle income uncertainty pro(cid:133)les areverysimilartothoseintheAISresults, showninFigure1. Second, because these future income projections are conditioned on a smaller and more restrictive information set, the RIS forecast error variances are greater than the AIS variances across all forecast horizons and for consumers of all ages. Third, the discrepancy between the RIS and the AIS results widen with the forecasting horizon, s. On average, the RIS one-year-ahead uncertainty is only 4% higher than the AIS uncertainty, whereas the margin is above 30% at the twenty-(cid:133)veyear horizon. This is not surprising because the di⁄erence between R and A I I iswhathouseholdsmightknowinthebaseyearbutwhicheconometriciansonly observe after a s-year lag. If s is small, the correlation between the elements 9The lowertriangle ofthe matrix is not populated. 11

Near Term Income Uncertainty 0.150 0.140 0.130 Two Year Ahead 0.120 0.110 0.100 0.090 0.080 One Year Ahead 0.070 0.060 AGE25 AGE30 AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 Medium Term Income Uncertainty 0.260 0.240 0.220 Ten Year Ahead 0.200 0.180 0.160 0.140 Five Year Ahead 0.120 0.100 AGE25 AGE30 AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 Long Term Income Uncertainty 0.400 0.350 Twenty five Year Ahead 0.300 0.250 0.200 Fifteen Year Ahead 0.150 AGE25 AGE30 AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 Figure1: Varianceofforecasterrorsfordi⁄erentforecasthorizonsasafunction of age for the augmented information set (AIS) estimates. 12

Near Term Income Uncertainty 0.150 0.140 0.130 Two Year Ahead 0.120 0.110 0.100 0.090 0.080 One Year Ahead 0.070 0.060 AGE25 AGE30 AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 Medium Term Income Uncertainty 0.260 0.240 Ten Year Ahead 0.220 0.200 0.180 0.160 0.140 Five Year Ahead 0.120 0.100 AGE25 AGE30 AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 Long Term Income Uncertainty 0.400 0.350 Twenty five Year Ahead 0.300 0.250 Fifteen Year 0.200 0.150 AGE25 AGE30 AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 Figure2: Varianceofforecasterrorsfordi⁄erentforecasthorizonsasafunction of age for the restricted information set (RIS) estimates. 13

Figure 2 Correlations of Income Uncertainties AGE30 AGE40 AGE50 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 1 5 9 13 17 21 Figure 3: AIS estimates of the correlations between one-year-ahead forecast errors and forecast errors h years ahead for ages 25, 35, and 45. in R and F is very high. That is to say the net value of adding F is quite I I I limited. If s is large, R has little predictive power on F; so introducing F I I I adds much more new information and consequently beefs up the forecasting performance.10 Besidethemagnitudesofincomeuncertaintiesoverthelifecycle,weare alsointerestedinthecorrelationsamongthestochasticcomponentsofincomeat various horizons. We compute for each age group the correlations between the one-year ahead forecast errors and the forecast errors at other horizons. Figure 3 presents the correlations of consumers that are thirty, forty, and (cid:133)fty years old in the AIS model. The chart shows that, apart from the longest forecast horizons,thecorrelationsofconsumersofvariousagesarequitesimilar. Wenote the following patterns in this graph. First, the correlations decline with the forecast horizon. Second, for all age groups, the correlations between forecast errors at one- and two-year horizons are about the same and slightly below 0.5. 10We willfurtherillustrate this e⁄ect in Figure 4 14

2.5 Robustness Test To verify whether the U-shaped uncertainty pro(cid:133)le over the lifecycle is a spurious consequence of our model speci(cid:133)cation, sample size, or sample selection, we conduct a series of robustness tests. First, we examine whether the changes in forecast-error variances are due to model misspeci(cid:133)cation. Remember we project the future income of households of di⁄erent head ages using the same set of coe¢ cients. If the projection equations should be age-speci(cid:133)c and the projection equation we use is closer to the true equations for middle-aged households than to the true equations for younger and older households, the one-size-(cid:133)t-all approach will reduce (cid:133)tness for younger and older households and arti(cid:133)cially increase income uncertainties for these age groups. We divide oursampleinto(cid:133)vesubgroupsbyheadofhouseholdageandreestimateEqs. (4) and(5)separatelyforeachsubgroup. Thenwecalculateforecast-errorvariances as we did before and we(cid:133)nd the U-shaped uncertaintypro(cid:133)les are qualitatively preserved. Second, we examine whether changes in the sample size as we vary the forecast horizon (as given inTable 1) might drive the shape of the uncertainty pro(cid:133)le. We reestimate the forecast equations using a smaller common sample and reassess income uncertainties over the lifecycle. Apart from the fact that more matrix elements cannot be estimated accurately because of the smaller samplesize,themagnitudeanddynamicsofincomeuncertaintyareverysimilar to what we presented above. Finally,wetestifourresultsaredrivenbylow-incomehouseholds,which are oversampled by the PSID. The core PSID sample consisted of two independent samples (cid:150)a nationwide representative sample and a sample of low-income families. In the (cid:133)rst wave of the survey, the nationwide representative sample has about 3,000 households and the low-income sample has about 2,000 households. We redo our analysis using only the households in the nationwide representative sample and their o⁄spring. The results are very similar to those obtained using the entire PSID core sample.11 2.6 Discussion and Comparison with Earlier Results Howsubstantivearetheinnovationswehaveintroducedintothesenonparametricmeasuresofincomeuncertainty? Howdi⁄erentareourresultscompared to previous results in the literature? We answer this question by contrasting ourresultstotheincomeuncertaintyestimatesinthein(cid:135)uentialworkofCarroll and Samwick (1997) and Gourinchas and Parker (2002). We(cid:133)rstbrie(cid:135)yreviewthemethodologyadoptedinCarrollandSamwick (1997) 12. The logarithm of income, y , is decomposed into a permanent comt ponent, p , and a transitory shock, (cid:15) , where p is further assumed to follow a t t t 11For more information about the PSID sample design, see the online documentation at http://psidonline.isr.umich.edu/Guide/Overview.html. 12We use the same notations as in theirpaper. 15

random walk with predictable income growth g such that t p =g +p +(cid:17) : (6) t t t 1 t (cid:0) and (cid:17) is the permanent income shock. Let (cid:27)2 and (cid:27)2 be the variance of the t (cid:17) (cid:15) permanent and transitory shocks. It is easy to show that Var[y y ]=d(cid:27)2+2(cid:27)2; (7) t+d (cid:0) t (cid:17) (cid:15) noting that the econometrician does not know how either y or y decomt t+d poses into their permanent and transitory parts. To estimate g , Carroll and t Samwick (1997) (cid:133)t actual income data with age, occupation, education, industry, household demographic variables, and age-interaction terms, adjusting for economy-wide growth in income. Subsequently, (cid:27)2 and (cid:27)2 can be estimated by (cid:17) (cid:15) evaluatingVar[y y ]atvarioustimehorizonsd. Theyreportthatforthefull t+d t (cid:0) sample (cid:27)2 =0:022, and (cid:27)2 =0:044. Gourinchas and Parker (2002) employ the (cid:17) (cid:15) samemethodologyandredotheestimationusinganupdatedsampleandreport almost identical results.13 We denote their results CS-GP. Figure 4 contrasts the income uncertainty implied by Eq. (7) at various future horizons using CS- GP parameters and our nonparametric estimates from both the AIS and RIS speci(cid:133)cations. To be consistent with CS-GP measures, in Fig. 4 our variances are calculated at each forecast horizon using the whole sample of households of all relevant ages. For example, the ten-year-ahead variance is computed using the forecast errors of all households whose heads are between 23 and 55 years old. The graph shows that at near horizons, the CS-GP uncertainty estimates are about 30% to 40% greater than the estimated income uncertainty under the AIS assumptions, and 15% to 20% greater than estimates under the RIS assumptions. The gap widens substantially at farther horizons. Beyond a twenty-year horizon, the CS-GP variance estimates more than double the AIS estimates and are over 50% higher than the RIS estimates. 2.7 Summary of Empirical Findings We construct a nonparametric measure of income uncertainty and study its dynamics over the lifecycle. Our estimates of income uncertainty are typically smaller than previous studies have documented. Our estimates also imply less persistence in income shocks. Over the lifecycle, we (cid:133)nd robust U-shaped patterns in the evolution of income uncertainty. Young and old consumers on average have more risky future income relative to middle-age consumers. This U-shapedpatternisrobusttoanumberofsampleandmodelspeci(cid:133)cationsand prevails at almost all horizons. 13Whenwere-estimatetheCS-GPmodelusingoursample,whichismuchlongerandallows ustoincludethe(cid:133)rst-orderdi⁄erenceof(cid:133)ttingresidualsmanyyearsapart,we(cid:133)ndsomewhat larger estimate ofvariance oftransitory shock but smaller estimate ofvariance ofpermanent shock. We interpret this as an evidence indicating that the persistent component of the income shocks are not exactly a random walk process. 16

Figure 3 Contrasting Current Results with Carroll and Samwick(1997) and Gourinchas and Parker (2002) 0.7 0.6 0.5 CCWS GGPP RIS 0.4 0.3 AIS 0.2 0.1 Years Ahead 0.0 0 5 10 15 20 25 30 Forecast Horizon (h) Figure 4: Comparison of estimates of the variance of forecast errors at di⁄erent forecast errors for the entire relevant sample using the approach of Carroll and Samwick (1997) and Gourinchas and Parker (2002) versus our approach with both the RIS and AIS speci(cid:133)cations. 17

3 A Lifecycle Model with Time-Varying Income Uncertainty 3.1 Existing Theory of the Consumption Hump ModernconsumptiontheorybeginswiththestandardRational-Expectations Lifecycle/Permanent-Income Hypothesis: if consumers are rational then they should allocate consumption over the lifecycle in such a way as to maximize lifetime utility rather than according to simple rules of thumb, such as to consume a constant fraction of disposable income. Thurow (1969) (cid:133)rst noted that empirical patterns of lifecycle consumption are hump-shaped and quite similar to the pro(cid:133)le of income over the lifecycle. On the face of it, this would appear to refute the RE-LCPIH. However, there are several modi(cid:133)cations to thestandardmodelthatcanaccountforahump-shapedconsumptionpro(cid:133)le.14 Each of these modi(cid:133)cations introduces testable predictions outside the realm of nondurable consumption. It is an open question how well competing models can simultaneously account for the mean dynamics of lifecycle consumption and the relevant data from other areas of economics, especially since di⁄erent modi(cid:133)cations do not necessarily complement each other. Often studied in conjunction with borrowing constraints, precautionary saving is the most popular explanation for the consumption hump and the one we focus on in this paper. Precautionary saving will arise if we dispense withtheassumptionofcompletemarketssoagentsareunabletoperfectlyinsure themselvesagainstidiosyncraticrisk(Leland(1968),Sandmo(1970)). Nagatani (1972)(cid:133)rstsuggestedthatprecautionarysavingwouldreduceconsumptionearly in the lifecycle, and Skinner (1988) and Feigenbaum (2008b) have (cid:135)eshed out howthegrowthrateofmeanconsumptionfromoneperiodtothenextincreases with income uncertainty. Consequently, if income uncertainty decreases over the lifecycle, this will lead to a concave consumption pro(cid:133)le. Using previous measures of uncertainty as described in Section 2.1, several researchers, includingCarroll(1997),CarrollandSummers(1991),GourinchasandParker(2002), and Hubbard, Skinner, and Zeldes (1994), have documented with calibrated partial-equilibriummodelsthatacombinationofborrowingconstraintsandprecautionary saving can produce a hump-shaped consumption pro(cid:133)le similar to the data. Feigenbaum (2007) showed that, in general equilibrium, precautionary saving could better account for the hump absent no-borrowing constraints with Gourinchas and Parker(cid:146)s (2002) speci(cid:133)cation of the income process, which in particular assumes that permanent income shocks are highly persistent and follow a unit-root process. Nevertheless, precautionary saving is far from the only explanation for a hump-shaped consumption pro(cid:133)le, but it is the only one that crucially depends on income uncertainty, which is why it is essential to document precisely how much income uncertainty consumers actually face. Continuing the explo- 14Fora more detailed review ofthis literature,see Browning and Crossley (2001). 18

ration of incomplete markets, Fernandez-Villaverde and Krueger (2005) have established there is also a hump in durable-goods consumption, and they have found that if durable goods have to be used as collateral for loans then the need to purchase durable goods before borrowing can also produce a humpshaped pro(cid:133)le for consumption of both types of goods. Heckman (1974) and Becker and Ghez (1975) suggested that if leisure and consumption are substitutes then a hump-shaped pro(cid:133)le of wages over the lifecycle would induce a hump-shaped pro(cid:133)le of consumption. This mechanism also has the side e⁄ect of a hump-shaped pro(cid:133)le of labor hours, but Bullard and Feigenbaum (2007) recently found there are reasonable calibrations that can match both the consumption hump and the labor-hours pro(cid:133)le. Time-varying mortality risk can also explain the hump (Feigenbaum (2008a), Hansen and Imrohoroglu (2006)), though only for a speci(cid:133)c set of parameters. The di¢ culty of separating out the individual e⁄ects of age and other confounding variables means that some portion of the consumption hump may bearti(cid:133)cialandduetomeasurementerror. Attanasioetal(1999)andBrowning andEjrn(cid:230)s(2002)havearguedthatvariationsinhouseholdsizeoverthelifecycle could play such a confounding role while Aguiar and Hurst (2003, 2007) have shown that the substitution of home production for market consumption may also vary with age. Ofcourse,aconsumptionhumpcanalsoariseifweabandontheassumptionofperfectlyrationaldecision-makerswithtime-consistentmodels. Laibson (1997)proposedthathypergeometricdiscountingcouldexplainthehumpwhile Caliendo and Aadland (2004) have shown that if households can only make plans over a ten- to twenty-year horizon then this can also account for the hump. While the main purpose of this paper is to study the lifecycle dynamics of income uncertainty and its impact on precautionary saving, the theoretical modelweintroduceinthenextsectionalso,forpracticalreasons,hasanelement of time-inconsistency. 3.2 The Model Ourcharacterizationofuncertainty, detailedinSection2, isfundamentally nonparametric since we only measure moments of the income process and do not estimate a parametric speci(cid:133)cation. De(cid:133)ning uncertainty in nonparametric terms has the advantage that our results are model-independent, but it has the disadvantage that we have no ready-made model of the income process that can be incorporated into lifecycle behavioral models. If we ignore the correlation data we have collected, we could suppose that the income process is simply a sequence of independent shocks with variances given by our volatility matrix. However,itiswellestablishedthat,forplausiblecalibrationsofutility, income uncertainty will only have a signi(cid:133)cant impact on consumption and savingifincomeshocksarepersistent(Skinner(1988)). Itisnotenoughforthe realization of income to be an uncertain event. Shock in earlier periods have to reveal prior information about this income, so each piece of information the 19

household anticipates it will receive about this income carries some uncertainty also (Feigenbaum (2008b)). It is the combined e⁄ect of all these information shocksthataccountsforthelargee⁄ectsofprecautionarysavinginmodelssuch as Feigenbaum (2007) or Gourinchas and Parker (2002). In order for income in one year to convey information about income in later years, these income shocks must be dependent. In a rational expectations model, a household will, by de(cid:133)nition, use Bayes(cid:146)Ruletoupdateitsbeliefsaboutthefutureasitsincomepathisrevealed. This requires a complete speci(cid:133)cation of the probability for every possible income history. Calibrating a time-consistent model that matches our volatility and correlation matrices is a complicated, multidimensional problem that we eschew in this paper. Instead, we calibrate a separate Markov process for each age group that describes how the household believes its income will evolve from that period on. The income process for each age group is calibrated to match the moments of forecast errors for that age group, but we do not impose any restrictions on the collection of these processes that would be necessary to insure time-consistent expectations. Thus we allow households to have timeinconsistent expectations. We consider a partial-equilibrium model with a consumer who lives for T working periods and T retirement periods. For each age t = 0;:::;T w r w (cid:0) 2, the consumer believes future income is determined by a stochastic process that matches the moments obtained from age-t income forecasts, and we do not require these beliefs to be consistent with Bayes(cid:146)Law. Let E(t) be the expectation operator with respect to the consumer(cid:146)s beliefs at age t. An age-t consumer then maximizes Tw(cid:0) 1 E(t) (cid:12)s (cid:0) tu(c s (t);(cid:13))+(cid:12)TW(cid:0) tV Tw (x Tw (t)) ; " # s=t X e e where c (t) is consumption planned for period s as of age t, s c1 (cid:0) (cid:13) (cid:13) =1 u(c;(cid:13))= 1 (cid:13) 6 (8) ( ln(cid:0)c (cid:13) =1 for (cid:13) >0, x (t) is (cid:133)nancial wealth at retirement, and V (x ) is the retire- Tw TW TW ment value function.15 Tildes denote random variables. Let us suppose an age-t consumer believes income at age s t; y (t), s (cid:21) can take on one of n values Y1(t) < Y2(t) < Y3(t) < < Yn(t). We ass s s (cid:1)(cid:1)(cid:1) s sumetheprobabilitydistributionisa(cid:133)rst-orderMarkovprocesswithtransition probabilities Pr(t)[y (t)=Yj (t)y (t)=Yi(t)]=(cid:5)ij(t)>0 (9) s+1 s+1 j s s s for i;j = 1;:::;n. The expectation operator E(t) computes expectations with 15See the appendix fordetails. 20

respect to this probability distribution. To compute aggregates for the population, we must also specify the actual probability distribution for income. We assume, for 0 t < T , that w (cid:20) actual income at t; y =y (t) Y1;:::;Yn , where t t 2f t t g Yi =Yi(t) t t for i = 1;:::;n and t = 0;:::;T 1. Since the Y1 (t);:::;Yn (t) may di⁄er w (cid:0) f t+1 t+1 g from Y1 (t+1);:::;Yn (t+1) ,itisnotobviouswhatprobabilitydistribution f t+1 t+1 g for y conditional on y we should impose on the model. However, the t+1 t speci(cid:133)cation of the actual probability distribution should not have any great e⁄ect on the shape of the consumption hump.16 Since we are not concerned with replicating the wealth distribution and since we also have not estimated the actual income process, for the purpose of studying the consumption hump it is su¢ cient to simply assume the actual income distribution is governed by some(cid:133)rst-orderMarkovprocesswithanage-independenttransitionmatrixthat governs mobility between the n rankings of the income states. For example, if there are two states in each period, the probability of going from the low to the high state is age-independent, although the income values corresponding to the low and high states do depend on age. For 0 t < T , we denote the actual w (cid:20) transition matrix by Pr[y =Yj y =Yi]=(cid:5) >0 (10) t+1 t+1j t t ij for i;j =1;:::;n. Denoting the invariant probability distribution of (cid:5) by (cid:25), we assume the actual initial probability distribution is Pr[y =Yi]=(cid:25) . 0 0 i Households can reallocate income across the lifecycle using the one intertemporal asset in the economy, a risk-free bond that pays the (cid:133)xed gross interest rate R. Let b (t) denote the quantity of bonds an agent at age t s+1 plans to purchase at age s that would then pay Rb (t) at age s+1. Thus s+1 the budget constraint an agent at age t expects to face at age s is c (t)+b (t)=y (t)+Rb (t) x (t), s s+1 s s s (cid:17) where x (t) is cash on hand as de(cid:133)ned by Deaton (1991). s 16Theshapeofthemeanconsumptionpro(cid:133)leisultimatelydeterminedbythemeanrateof consumptiongrowth. Feigenbaum (2007a)andSkinner(1988)haveshownthattheexpected rateofconsumptiongrowth dependsnonlinearlyonwealth,sothemean rateofconsumption growthwilldependonthedistributionofwealth. However,thiswealthe⁄ectisproportional to the variance of income and so is second-order in terms of deviations of income from the mean. Since the (cid:133)rst-order e⁄ect of deviations from the mean will, by de(cid:133)nition, vanish after taking expectations, the lowest order e⁄ect of any dispersion in the wealth distribution is also second order. Thus the aggregate e⁄ect of deviations of wealth from mean wealth is asecond-ordercorrectiontoasecond-ordere⁄ect,makingitafourth-ordere⁄ect. Thee⁄ect of the wealth distribution on the shape of the lifecycle consumption pro(cid:133)le should therefore be negligible. 21

Since our intent is to focus on the ability of uncertainty and precautionary saving to explain the consumption hump, we allow borrowing but with full commitment to debt contracts. Since consumption is required to be nonnegative, the consumer faces the endogenous borrowing limit (Aiyagari (1994)) that he would never borrow more than the minimum possible present value of incomehemightearninthefuture,i.e. theminimumthathecouldpossiblypay backinthefuture. However,thereispotentiallyadisconnectbetweenwhatthe consumerbelievesthisborrowinglimitisandwhattheborrowinglimitactually is. At age t an agent will believe at age s that the borrowing limit he faces is Tw(cid:0) 1 Y1(t) B (t)= i : s+1 Ri s i=s+1 (cid:0) X At age t, the actual borrowing limit should be Tw(cid:0) 1 Y1 B = i . t+1 Ri t i=t+1 (cid:0) X To prevent the consumer from borrowing more than he actually can pay back, we impose the exogenous borrowing constraint17 b (t) B . t+1 t+1 (cid:21)(cid:0) Thus, given y and b , the consumer(cid:146)s problem at age t is t t Tw(cid:0) 1 f cs(t) gs T = W t(cid:0) 1 m ; f a b x s+1(t) g T s= W 1(cid:0) 1 E t " X s=t (cid:12)s (cid:0) tu(c s (t);(cid:13))+(cid:12)TW(cid:0) tv Tw (Rb Tw (t)) # (11) subject to e e c (t)+b (t)=y (t)+Rb (t) s=t;:::;T 1 s s+1 s s w (cid:0) b (t) B . t+1 t+1 (cid:21)(cid:0) To simulate the model, at age t we assume that c = c (t) and b = b (t). t t t+1 t+1 See the Appendix for details on the computational procedure. Virtually every theoretical explanation for the hump described in Section3.1canquantitativelyaccountforthehumpifboththeinterestrateR and discountfactor(cid:12) arefreeparameterstobecalibrated. BullardandFeigenbaum (2007)andFeigenbaum(2008a)haveemphasizedtheimportanceofstudyingthe consumptionhumpwithgeneral-equilibriummodelsthatputsomedisciplineon thechoiceof(cid:12) andR. Sinceouractualincomeprocessischosenforconvenience ratherthanempiricalveracityandthemarket-clearinginterestratewillbevery 17Whileimposingthisconstraintisnecessarytoensurethemodelhasawell-de(cid:133)nedsolution, the distinction between Bt+1(t) and Bt+1 is usually small, so the overwhelming majority of households willnot be a⁄ected by the constraint. 22

sensitive to the actual income process, we do not endogenize R. The purpose ofthetheoreticalexercisethatfollowsistoseewhether,foranyplausiblevalues of (cid:12), (cid:13), and R, there is su¢ cient uncertainty to generate enough precautionary saving to replicate the consumption hump. 3.3 Speci(cid:133)cation for the Income Process Suppose that at age t, the consumer believes for s t that (cid:21) y (t)=a p (t)z (t); (12) s s s s where a is an age-dependent factor, p (t) is a (semi)permanent shock, and s s z (t) is a temporary shock. Speci(cid:133)cally, we assume that p (t) and z (t) are s s s independent, unconditionally unit-mean processes such that corr(ln(p (t));ln(p (t)))=(cid:26)<1; s+1 s V[lnp (t) (cid:26)lnp (t)p (t)]=(cid:27)2(t), s+1 (cid:0) s j s p and V[lnz (t)]=(cid:27)2 (t): s z;s t (cid:0) Note that the income speci(cid:133)cation of (12) di⁄ers from the standard speci- (cid:133)cation of Carroll and Samwick (1997) and Gourinchas and Parker (2002) in two important respects. First, the permanent shocks do not follow a unitrootprocess,althoughthismodi(cid:133)cationislessremarkablesincemanypapersin the literature have considered an AR(1) income process, including Feigenbaum (2008b)andHuggett(1996). Thesecondandmoreimportantdi⁄erenceisthat the variance of the permanent shocks depends on the age t when forecasting occurs while the variance of the temporary shocks depends both on t and the forecasting horizon s t. (cid:0) We assume that a is consistent across di⁄erent ages since we are fot cusing on changes in the perception of the variance of income rather than on changesintheperceptionofthemean. Wealsoassumethattheautocorrelation (cid:26) is consistent across ages. This is consistent with Fig. 3, which shows that the correlation between one-year ahead forecasts and h-year ahead forecasts is essentially independent of age.18 Thus lny (t)=lna +lnp (t)+lnz (t): s s s s 18We also experimented with income speci(cid:133)cations where there is no correlation between incomeshocksandwheretheforecastedincomeatt+sisonlycorrelatedwiththeforecasted incomeatt+1. However,therewasnotenoughpersistenceinshockswiththesespeci(cid:133)cations to generate a signi(cid:133)cant hump in consumption pro(cid:133)les (Feigenbaum (2007a). 23

For h 1, (cid:21) lny (t)=lna +lnp (t)+lnz (t) t+h t+h t+h t+h h =(cid:26)hlny +lna (cid:26)hlna + (cid:26)h i(lnp (t) (cid:26)lnp (t)) t t+h t (cid:0) t+i t+i 1 (cid:0) (cid:0) (cid:0) i=1 X +lnz (t) (cid:26)hlnz (t): t+h t (cid:0) Thus 1 (cid:26)2h V[lny (t)y ]= (cid:0) (cid:27)2(t)+(cid:27)2 (t)+(cid:26)2h(cid:27)2 (t) (13) t+h j t 1 (cid:26)2 p zh z0 (cid:0) For h>1, the correlation between lny (t) and lny (t) conditional on y is t+h t+1 t corr(lny (t);lny (t)y ) (14) t+h t+1 t j (cid:26)h 1(cid:27)2(t)+(cid:26)h+1(cid:27)2 (t) = (cid:0) p z0 : 1 (cid:26)2h(cid:27)2(t)+(cid:27)2 (t)+(cid:26)2h(cid:27)2 (t) ((cid:27)2(t)+(cid:27)2 (t)+(cid:26)2(cid:27)2 (t)) 1(cid:0)(cid:26)2 p zh z0 p z1 z0 r (cid:0) (cid:16) (cid:17) Todiscretizethisprocess,werestrictlnp (t)totakeonvaluessuchthat s lnp (t) s P1;:::;Pnp (cid:27) (t) 2f g p and lnz (t) to take on values such that s lnz (t) s Z1;:::;Znz : (cid:27) (t) 2f g z;s t (cid:0) De(cid:133)ning lnp (t) Q (t)= s s (cid:27) (t) p and lnz (t) Z (t)= s ; s (cid:27) (t) z;s t (cid:0) we specify an i.i.d. probability distribution for Z (t), s Pr[Z (t)=Zi]=(cid:25)z (15) s i for i=1;:::;n ; and a Markov distribution for Q (t), z s Pr[Q (t)=Pj Q (t)=Pi]=(cid:5)p (16) s+1 j s ij for i;j = 1;:::;n and s t. Thus the total number of income states at age t p (cid:21) and horizon s is n=n n . For p z k =n (i 1)+j; (17) z (cid:0) 24

where i=1;:::;n and j =1;:::;n , the kth income state is p z Yk(t)=a exp((cid:27) (t)Pi(t))exp((cid:27) (t)Zj): s s p z;s t (cid:0) Notethatthisspeci(cid:133)esthetime-inconsistentprobabilitydistributionas perceived by the household. We must also specify the Markov process of the actual probability distribution for realized income y . As described above, we t assumethatthehousehold(cid:146)sperceptionsofitscurrenttemporaryandpermanent shocks are correct, so y =y (t)=a exp((cid:27) (t)Q )exp((cid:27) (t)Z ); t t t p t z0 t where Q =Q (t) t t Z =Z (t). t t This implies the set of actual income states for y coincides with the perceived t states at t, so Yk =Yk(t) t t for k =1;:::;n. Let (cid:25)p denote the invariant distribution of (cid:5)p. We then assume the i actual probability distribution of Q is 0 Pr[Q =Pi]=(cid:25)p 0 i for i=1;:::;n and the actual probability distribution of Z is p 0 Pr[Z =Zi]=(cid:25)z 0 i for i = 1;:::;n . For 0 t < T 1, we assume that (15) and (16) correctly z w (cid:20) (cid:0) specify the probability distribution for Q and Z , so t+1 t+1 Pr[Z =Zi]=(cid:25)z t+1 i for i=1;:::;n and z Pr[Q =Pj Q =Pi]=(cid:5)p: t+1 j t ij Note that this implies the unconditional distribution of Q is (cid:25)p for all t. t 3.4 Calibration For the income process, we assume both the permanent and temporary shocks are governed by two-state processes.19 The temporary shocks are parameterized by Z2 = Z1 = 1 and (cid:25)z = (cid:25)z = 1=2. Likewise, the permanent (cid:0) 1 2 19We considerthe robustness ofthis assumption in Section 3.6. 25

shocks are parameterized by 1 Q2 = Q1 = (cid:0) 1 (cid:26)2 (cid:0) and (cid:25)p =(cid:25)p =1=2. The transition matrixpfor the permanent shocks is 1 2 1 1+(cid:26) 1 (cid:26) (cid:5)p = (cid:0) : 2 1 (cid:26) 1+(cid:26) (cid:20) (cid:0) (cid:21) We calibrate (cid:26), (cid:27) (t), and (cid:27) (t) so as to minimize the distance of the p zh correlation and volatility matrices relative to their predicted values from (13) and(14). Speci(cid:133)cally,weparameterizethepermanent-shockstandarddeviation as a d -degree polynomial p dp (cid:27) (t)= Dpti (18) p i i=0 X and the temporary-shock standard deviation as a tensor product of dt and dhz z degree polynomials dt dh z z (cid:27) (t)= Dztihj: (19) zh ij i=0j=0 XX Then we set (cid:26), Dp, . . . , Dp , Dz , . . . , Dz , Dz , . . . , Dz , . . . , Dz , . . . , Dz to m 0 inimize th d e p sum 00 of the squa 0d re h z s of 1 t 0 he deviati 1 o d n h z s between d t t z h 0 e dtdh z z predictedvaluesofthematrixelementsandtheirmeasuredvalues. Notethat,as isstandardinthisliterature(seeforexampleFeigenbaum(2007)orGourinchas and Parker (2002)), the Bellman Eq. (20) implies that the household precisely knowsitsincomestate. Thus,insolvingthemodel,weassumetheconsumerhas additional information that the econometrician does not have when computing the volatility and correlation matrices (13) and (14), for the consumer knows how his current income breaks down into permanent versus temporary income shocks. In Section 2, we measure the volatility and correlation matrices for horizons up to H =25. For t<T H 2 we will also need to specify max w max (cid:0) (cid:0) (cid:27) (t) for h H +1;:::;T t 1 , but these standard deviations are not zh max w 2f (cid:0) (cid:0) g identi(cid:133)ed by the available data. We will consider what happens both if we linearly extrapolate (19) for h > H and if we assume a (cid:135)at extrapolation max where (cid:27) (t) = (cid:27) (t) for h > H . We also do not have information zh zHmax max about(cid:27) (T 1)sincewehavenodataatforecastsinthelastworkingperiod. z0 w (cid:0) However, we do have the volatility matrix element V[lny (T 2)y ], so it is reasonable to assume (19) will still be valid at t= T T w(cid:0) 1 1. w L (cid:0) ikew j i T s w e,(cid:0)w 2 e w (cid:0) simply extrapolate (18) to obtain (cid:27) (T 1). p W (cid:0) We consider the nonparametric estimates obtained with both the AIS and RIS speci(cid:133)cations of Section 2 with a cubic approximation that sets d = p dt = dh = 3. To assess the importance of the age and forecast-horizon dez z 26

Model (cid:26) AIS TIME CON 0.910 AIS TIME INC 0.958 RIS TIME CON 0.925 RIS TIME INC 0.964 Table 3: Correlation (cid:26) for both the time-inconsistent (TIME INC) and timeconsistent (TIME CON) income processes and both the AIS and RIS estimates of the volatility and correlation matrices. pendence of uncertainty we also consider a time-consistent calibration of the incomeprocesswhere(cid:27) and(cid:27) areconstantsindependentoftandh. Thusthe p z time-consistent income process falls into the class of Markov income processes that have previously been studied in the literature (for example in Feigenbaum (2008b)andHuggett(1996)). Thetime-consistentcalibrationisobtained,both for the AIS and RIS speci(cid:133)cations, as above but with d =dt =dh =0. p z z Thetime-inconsistentandtime-consistentcalibrationsof(cid:27) (t)areplotp ted as a function of age t in Fig. 5 for both the AIS and RIS speci(cid:133)cations. Likewise, the four calibrations of (cid:27) (t) are plotted as a function of age for repzh resentative horizons in Fig. 6. The correlation for each calibration is given in Table 3. The variance of permanent income shocks is uniformly larger for the time-consistent calibration than the time-inconsistent calibration, and for most horizons the time-consistent variance is twice as large as the time-inconsistent variance. The correlations for the time-consistent calibrations are modestly smaller than the corresponding time-inconsistent calibrations. At short time horizons the variance of temporary income shocks is comparable between the time-consistent and time-inconsistent calibrations, but the variance of temporary income shocks increases with the forecast horizon in the time-inconsistent modelwhilenecessarilyremainingconstantinthetime-consistentmodel. Thus permanentincomeshockswillhavegreateremphasisinthetime-consistentmodels whereas temporary income shocks will have more emphasis in the timeinconsistent models. For the AIS estimates, the root-mean-squared deviation between the time-inconsistent model(cid:146)s predictions for the volatility and correlation matrices and the corresponding empirical estimates is 0.022. For the time-consistent model, the root-mean-squared deviation is 0.051. Fig. 7 shows how the variances of forecast errors for these two models compare to the nonparametric estimates as a function of age at di⁄erent forecast horizons. Since the timeconsistentmodelassumesthatthestandarddeviationoftemporaryandpermanent income shocks is independent of age, its variance graphs are (cid:135)at whereas the time-inconsistent model is able to capture the U-shaped dependence of the varianceswithrespecttoage. Forshorthorizonsofonetotwoyears, thetimeconsistent model overpredicts the uncertainty at all ages. For long horizons, the time-consistent model signi(cid:133)cantly underpredicts the uncertainty. Consistentwiththeroot-mean-squareddeviations,thetime-inconsistentmodelalmost 27

s p 0.25 0.2 0.15 AIS TIME INC AIS TIME CON RIS TIME INC RIS TIME CON 0.1 0.05 0 25 30 35 40 45 50 55 60 65 Age Figure 5: Permanent shock standard deviation (cid:27) (t) as a function of age t for zh the time-inconsistent (TIME INC) and time-consistent (TIME CON) income processes with both the RIS and AIS estimates of the volatility and correlation matrices. 28

s s z1 z5 b) 0.25 a) 0.4 0.35 0.2 0.3 0.15 0.25 0.2 0.1 0.15 0.1 0.05 0.05 0 25 30 35 40 45 50 55 60 65 0 25 30 35 40 45 50 55 60 65 Age Age s s z15 c) z25 d) 0.5 0.6 0.45 0.4 0.5 0.35 0.4 AIS TIME INC 0.3 AIS TIME CON 0.25 0.3 R RI I S S T T I IM M E E I C N O C N 0.2 0.15 0.2 0.1 0.1 0.05 0 25 30 35 Age 40 45 50 0 25 30 Age 35 40 Figure 6: Temporary shock standard deviation (cid:27) (t) as a function of age t for zh horizons h of (a) one year, (b) (cid:133)ve years, (c) (cid:133)fteen years, and (d) twenty-(cid:133)ve years for both the time-inconsistent (TIME INC) and time-consistent (TIME CON) income processes and both the AIS and RIS estimates of the volatility and correlation matrices. 29

Short Term Income Uncertainty 0.14 0.12 0.1 0.08 Long Term Income Uncertainty 0.06 1 year ahead data 0.4 1 year ahead time inc 0.04 2 year ahead data 0.35 2 year ahead time inc 0.02 1 year ahead time con 0.3 2 year ahead time con 0 0.25 25 30 35 40 45 50 55 60 65 Age Medium Term Income Uncertainty 0.2 15 year ahead data 0.25 0.15 15 year ahead time inc 25 year ahead data 0.1 25 year ahead time inc 0.2 15 year ahead time con 0.05 25 year ahead time con 0 0.15 25 30 35 40 45 50 Age 5 year ahead data 0.1 5 year ahead time inc 10 year ahead data 0.05 10 year ahead time inc 5 year ahead time con 10 year ahead time con 0 25 30 35 40 45 50 55 60 Age Figure7: Varianceofforecasterrorsfordi⁄erentforecasthorizonsasafunction of age for the augmented information set (AIS) estimates and both the timeconsistent and time-inconsistent income processes. uniformlydoesbetteratmatchingtheforecast-errorvariances,althoughitdoes underpredict the variance at the two-year horizon. The comparison is similar for the RIS speci(cid:133)cation. Fig. 8showshowthecorrelationsbetweenone-yearaheadforecasterrors and h-year ahead forecast errors compare at ages 30, 40, and 50 between the nonparametric estimates and the time-inconsistent income process under the AIS speci(cid:133)cation. Fig. 9 shows the same comparison for the time-consistent income process. The time-inconsistent model matches the correlations slightly betterasthetime-consistentmodeloverpredictsthecorrelationatshorthorizons and underpredicts it at long horizons. In addition to the income process, we also have to calibrate the preferenceparameters(cid:12) and(cid:13),andsincethisisapartial-equilibriummodelthegross interest rate R. Following Feigenbaum (2008b), we set the discount factor and interest rate to common values from the literature: (cid:12) = 0:96 and R = 1:035. Feigenbaum (2007) found that a risk aversion of (cid:13) = 3 could best account for the lifecycle consumption pro(cid:133)le under Gourinchas and Parker(cid:146)s (2002) income 30

Time Inconsistent Correlations 0.6 0.5 0.4 age 30 data age 30 model age 40 data 0.3 age 40 model age 50 data age 50 model 0.2 0.1 0 0 5 10 15 20 25 30 Forecast Horizon (h) Figure8: Correlationofone-yearaheadforecasterrorsandh-yearaheadforecast errorsasafunctionofforecasthorizonhatages30,40,and50fortheaugmented information set (AIS) estimates and the time-inconsistent income process. 31

Time Consistent Correlations 0.6 0.5 0.4 age 30 data age 40 data 0.3 age 50 data model 0.2 0.1 0 0 5 10 15 20 25 30 Forecast Horizon (h) Figure9: Correlationofone-yearaheadforecasterrorsandh-yearaheadforecast errorsasafunctionofforecasthorizonhatages30,40,and50fortheaugmented information set (AIS) estimates and the time-consistent income process. 32

Model E[ctmax ] t (+25) E[c0] max GP CONS. DATA 1.15 45 AIS TIME INC 1.15 56 RIS TIME INC 1.25 60 AIS TIME CON 1.28 61 RIS TIME CON 1.43 60 GP TIME CON 2.04 57 Table 4: Ratio of peak to initial consumption and peak age for the lifecycle consumption pro(cid:133)le as measured by Gourinchas and Parker (2002) and as predicted by the model with time-inconsistent (TIME INC) and time-consistent (TIME CON) income processes calibrated for both the RIS and AIS speci(cid:133)cations. For comparison, we also include an income process similarto Gourinchas and Parker(cid:146)s (2002) baseline income process. process, so we maintain this value. 3.5 Theoretical Predictions for Consumption Hump Fig. 10showslifecyclepro(cid:133)lesformeanconsumption(normalizedbymean initial income) for the time-inconsistent and time-consistent calibrations under boththeAISandRISspeci(cid:133)cationsalongwithGourinchasandParker(cid:146)sempirical measurements of the mean consumption pro(cid:133)le20.21 For comparison with the previous literature, we also include the results for a time-consistent model with Gourinchas and Parker(cid:146)s estimates of the shock variances: (cid:27)2 = 0:0212 p and (cid:27)2 = 0:0440. In their model, the permanent income shocks follow a unitz root process. Since a unit-root process cannot be nested in our model, we set (cid:26) = 0:99. In Table 4, we also report the peak to initial consumption of the lifecycle pro(cid:133)le of mean consumption and the age of peak consumption. All(cid:133)vemodelsproduceahump-shapedlifecycleconsumptionpro(cid:133)lefor our chosen calibration of the preference parameters and the interest rate. The consumption-saving model with the time-inconsistent income process based on the AIS speci(cid:133)cation most closely matches Gourinchas and Parker(cid:146)s estimates of mean consumption as a function of age, although this ranking might change with other calibrations. Our primary interest here lies in the degree to which di⁄erent measures of uncertainty impact the consumption pro(cid:133)le for a given calibration. Since (cid:12)R < 1, in the absence of uncertainty the lifecycle consumption pro(cid:133)le should be monotonically decreasing with a peak at the initial age of 25, so the peak to initial consumption ratio should be 1. Thus the peak to initial consumption 20ThegraphsinFig. 10andensuing(cid:133)gureswereobtainedbysimulatingonemillionlifecycle paths perage group. 21Theresultsforthetime-inconsistentmodelswereobtained with a linearextrapolation of thetemporaryshockvariancesasafunctionoftheforecasthorizon. Resultsobtainedwitha (cid:135)at extrapolation are given in Section 3.6. 33

1.5 1.4 1.3 1.2 GP DATA 1.1 RIS TIME CON E[c] AIS TIME CON t AIS TIME INC E[y ] 1 RIS TIME INC 0 GP TIME CON 0.9 0.8 0.7 0.6 25 30 35 40 45 50 55 60 65 70 75 80 Age Figure 10: Mean consumption pro(cid:133)le (normalized by mean initial income) as measured by Gourinchas and Parker (2002) and as predicted by the model with time-inconsistent (TIME INC) and time-consistent (TIME CON) income processes calibrated for both the RIS and AIS speci(cid:133)cations. For comparison, we also include an income process similar to Gourinchas and Parker(cid:146)s (2002) baseline income process. 34

ratiocanbeviewedhereasameasureofhowmuchprecautionarysavingcauses these models to deviate from the LCPIH. Since we are also holding the degree ofriskaversion(cid:133)xed,thevariationinprecautionarysavingoverdi⁄erentincome processesshouldre(cid:135)ecttheamountofuncertaintyfacedoverthelifecycleunder each process. Not surprisingly, since the AIS speci(cid:133)cation assumes households have more information than the RIS speci(cid:133)cation, the two AIS models have smaller peaktoinitialconsumptionratiosthantheircorrespondingRISmodels. Moreover,theGourinchasandParker-basedmodelwithitsnearunit-rootprocesshas a substantially larger peak to inital consumption ratio than the four models we estimatesinceithasthemostuncertainty. Indeed,for(cid:12) =0:96andR=1:035, Gourinchas and Parker (2002) have to dial the risk aversion all the way down to one half to get a lifecycle consumption pro(cid:133)le that resembles the data, but this would not be necessary for our more robust uncertainty measure. For both the RIS and AIS speci(cid:133)cations, we also (cid:133)nd that the timeinconsistent model has a signi(cid:133)cantly smaller peak to initial consumption ratio than the corresponding time-consistent model. This can be explained in terms of Figs. 5 and 6. Because the time-consistent model assumes a constant variance for the permanent and temporary shocks, independent of age and forecast horizon, the time-consistent model needs a large permanent shock variance to best match the volatility and correlation matrices. In contrast, the time-inconsistent model has a smaller permanent shock variance and exploits its ability to increase the temporary shock variance at longer forecast horizons to better match the forecast-error moments. As Constantinides and Du¢ e (1996) argued, permanent shocks will have a substantially larger impact on the behavior of consumers, and Feigenbaum (2007,2008b) con(cid:133)rm that more persistent income shocks lead to greater precautionary saving. This intuition is further corroborated here. Thus, failing to account for the time and forecasthorizondependenceofuncertaintycanbiasupwardestimatesoftheimportance of precautionary saving. 3.6 Robustness Checks OnecauseforconcernaboutthecalibrationoftheincomeprocessinSection 3.4 is that we have no data on the variance and correlation of forecast errors beyond a twenty-(cid:133)ve-year horizon, but the model requires us to specify the household(cid:146)s beliefs about income at all future horizons. In Section 3.5, we reported results for the baseline case where we assume a linear extrapolation of Eq. (19) for h H . In Fig. 11, we consider what happens with the max (cid:21) alternative assumption of a (cid:135)at extrapolation such that (cid:27) (t) = (cid:27) (t) zh z;Hmax for h H . For both the AIS and RIS speci(cid:133)cations, we (cid:133)nd a negligible max (cid:21) di⁄erence between the two extrapolations. The e⁄ect of a temporary income shock twenty-(cid:133)ve years in the future is going to be heavily discounted, so a change in the temporary shock variance at such large horizons has little to no e⁄ect on consumption behavior. Anotherpotentialcauseforconcernaboutourresultsisthatweuseonly 35

1.4 1.2 1 0.8 GP DATA E[c] AIS LINEAR t RIS LINEAR E[y ] AIS FLAT 0 0.6 RIS FLAT 0.4 0.2 0 25 30 35 40 45 50 55 60 65 70 75 80 Age Figure 11: Mean consumption pro(cid:133)le (normalized by mean initial income) as measuredbyGourinchasandParker(2002)andaspredictedbythemodelwith time-inconsistent income processes calibrated for both the RIS and AIS speci(cid:133)cations with linear and (cid:135)at extrapolations of the temporary shock variances as a function of time horizon. 36

two-stateprocessestomodelboththepermanentandtemporaryincomeshocks. Would a (cid:133)ner distribution produce di⁄erent results? Feigenbaum (2008b) suggeststhatonlythesecond-ordermomentsshouldhaveasigni(cid:133)cante⁄ectonthe shape of the mean consumption pro(cid:133)le. Given the common assumption that shocks are normally distributed, the next moment of interest is the kurtosis. For a normal distribution, the kurtosis should be 3. However, an unskewed two-state process must have the minimum kurtosis of 1. For the temporary shock process, we can easily replicate both the variance and kurtosis of a normal distribution with a three-state process characterized by Z3 = Z1 =1 and (cid:0) Z2 = 0 with (cid:25)z = (cid:25)z = 1=6 and (cid:25)z = 2=3. We can also obtain an uncondi- 1 3 2 tional distribution of permanent shocks that matches the variance and kurtosis of a normal distribution in this way, but with (cid:26) 1 we cannot do the same for (cid:25) the conditional distribution of permanent shocks, which is what should matter most for precautionary saving. With three states, the best we can do is choose a transition matrix that minimizes the kurtosis of the conditional distribution. If the unconditional kurtosis is K , the minimum conditional kurtosis goes as p 1:5K =(1 (cid:26)) in the limit as (cid:26) 1. p (cid:0) ! In Fig. 12, we plot the mean consumption pro(cid:133)le for the AIS speci(cid:133)cation of the time-inconsistent model for various choices of K and the kurtosis p of the temporary shock distributions. To get a sense of scale, we also plot the empiricalpro(cid:133)leofGourinchasandParker(2002). Threeofthemodelpro(cid:133)les, for which the unconditional kurtoses of both shocks are between 1 and 3, are virtually indistinguishable. The ratio of peak consumption to initial consumption for all three curves is between 1.125 and 1.146. Only when we increase K to 10 do we get any signi(cid:133)cant departure, and even then the peak to initial p ratio only increases to 1.17. Comparing Fig. 12 to Fig. 10, we see that the impact of using a (cid:133)ner distribution is much smaller than the impact of ignoring the time-dependence of uncertainty or adding more information to the model. As a (cid:133)nal robustness check, let us consider how sensitive the model is to the risk aversion (cid:13). In Fig. 13 and Table 5 we replicate Fig. 10 and Table 4 but for (cid:13) = 2. We still get hump-shaped consumption pro(cid:133)les, but not surprisingly with less risk aversion there is less precautionary saving so the humps are more modest in size. Interestingly, while the consumption pro(cid:133)le for the AIS speci(cid:133)cation of the time-inconsistent model now has a peak to initial consumption ratio smaller than Gourinchas and Parker (2002) (cid:133)nd, this calibration does match the age of the peak. 4 Conclusion We introduce a new method of measuring income uncertainty and apply the estimates obtained via this approach to investigate the extent to which variations in income uncertainty over the lifecycle could be responsible for the hump-shapedconsumptionpro(cid:133)le. Ourmeasurementtechniquearticulatesthe distinction between income heterogeneity and uncertainty, and acknowledges 37

1.4 1.3 1.2 1.1 PERM 1 TEMP 3 E[c] PERM 3 TEMP 3 t 1 PERM 10 TERM 3 E[y ] GP DATA 0 PERM 1 TEMP 1 0.9 0.8 0.7 0.6 25 30 35 40 45 50 55 60 65 70 75 80 Age Figure 12: Mean consumption pro(cid:133)le (normalized by mean initial income) as measured by Gourinchas and Parker (2002) and as predicted by the model with time-inconsistent income processes calibrated for the AIS speci(cid:133)cations with three-state processes for the permanent (PERM) and temporary (TEMP) shocks with various kurtoses. Model E[ctmax ] t (+25) E[c0] max GP DATA 1.15 45 AIS TIME INC 1.07 46 RIS TIME INC 1.13 55 AIS TIME CON 1.16 59 RIS TIME CON 1.28 58 GP TIME CON 1.58 53 Table5: Ratioofpeaktoinitialconsumptionandpeakageforthelifecycleconsumptionpro(cid:133)leasmeasuredbyGourinchasandParker(2002)andaspredicted by the model with time-inconsistent (TIME INC) and time-consistent (TIME CON)incomeprocessescalibratedforboththeRISandAISspeci(cid:133)cationswhen (cid:13) =3. Forcomparison,wealsoincludeanincomeprocesssimilartoGourinchas and Parker(cid:146)s (2002) baseline income process. 38

1.4 1.2 1 AIS TIME INC 0.8 E[c] GP DATA t RIS TIME INC E[y ] AIS TIME CON 0 RIS TIME CON 0.6 GP TIME CON 0.4 0.2 0 25 30 35 40 45 50 55 60 65 70 75 80 Age Figure 13: Mean consumption pro(cid:133)le (normalized by mean initial income) as measured by Gourinchas and Parker (2002) and as predicted by the model with time-inconsistent (TIME INC) and time-consistent (TIME CON) income processes calibrated for both the RIS and AIS speci(cid:133)cations with (cid:13) = 2. For comparison, we also include an income process similar to Gourinchas and Parker(cid:146)s (2002) baseline income process. 39

that households may have superior information to econometricians. Our estimation reveals an income uncertainty level that is lower than what has been presented in the existing literature and shows that income uncertainty does evolve over the lifecycle in a fashion consistent with conventional wisdom. In addition, we show that for a plausible calibration of the preference parameters, our estimate of income uncertainty does imply a hump-shaped lifecycle consumption pro(cid:133)le that matches consumption data very well. Two lines of research are worth pursuing in the future. First, it is important to study whether we can (cid:133)nd a time-consistent, parametric income process consistent with our nonparametic estimates. This is necessary in order to study what happens in general equilibrium for this model. Second, we will study more carefully the source of the variation of income uncertainty over the lifecycle. 40

A Computational Procedure Wecanwritetheproblem(11)asarecursivesequenceofBellmanequations as follows. Let us denote the state variable I (t) such that y (t) = YIs(t)(t). s s s For each base age t 0;:::;T , the consumer solves for the age-t perceived 2 f g value functions V (x ;I (t);t) at current and future ages s=t;:::;T 1. For s s s w (cid:0) s=t;:::;T 1, the Bellman equation is w (cid:0) V (x ;I (t);t)= max u(c (t);(cid:13))+(cid:12)E(t)[V (YIs+1(t)(t)+Rb (t);I (t);t)I (t)] s s s cs(t);bs+1(t) s s+1 s e +1 s+1 s+1 j s (20) e subject to c (t)+b (t)=x s s+1 s For the case when s=t, we impose the additional constraint b (t) B . t+1 t+1 (cid:21)(cid:0) Theretirementvaluefunctionthatterminatesthesequenceofvaluefunctionsis simply the perfect-foresight utility of an agent who will live for T periods with r no further income. With CRRA utility (8), this is V (x )= (cid:30) (cid:0) Tr (cid:0) 1 (cid:13) u(x ;(cid:13)); (21) Tw Tw (cid:30) 1 1 Tw (cid:18) (cid:0) (cid:0) (cid:19) where (cid:30)=((cid:12)R1 (cid:13)) 1=(cid:13) (cid:0) (cid:0) istheinversemarginalpropensitytoconsumeinthelimitoflargelifetimes. The sequence of Bellman equations (20) can easily be translated into the sequence of Bellman equations solved in Feigenbaum (2008b), and the computational procedure we use here is described in the technical appendix for that paper.22 Note that only the policy functions c (x ;I ;t) for t=0;:::;T 1 are actually t t t w (cid:0) relevant to the behavior of the model. 22This is available online at http://www.pitt.edu/jfeigen/infoshockappendix.pdf. 41

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