learning your earning: are labor income shocks really very ......individuals make. understanding the...

Learning Your Earning: Are Labor Income Shocks ReallyVery Persistent?

By FATIH GUVENEN*

The current literature offers two views on the nature of the labor income process.According to the first view, individuals are subject to very persistent income shockswhile facing similar life-cycle income profiles (the RIP process, Thomas MaCurdy1982). According to the alternative, individuals are subject to shocks with modestpersistence while facing individual-specific profiles (the HIP process, Lee A. Lillardand Yoram A. Weiss 1979). In this paper we study the restrictions imposed by thesetwo processes on consumption data—in the context of a life-cycle model—todistinguish between the two views. We find that the life-cycle model with a HIPprocess, which has not been studied in the previous literature, is consistent withseveral features of consumption data, whereas the model with a RIP process isconsistent with some, but not with others. We conclude that the HIP model could bea credible contender to—and along some dimensions, a more coherent alternativethan—the RIP model. (JEL D83, D91, E21, J31)

When markets are incomplete, labor incomerisk plays a central role in many decisions thatindividuals make. Understanding the nature ofincome risk is thus an essential prerequisite forunderstanding a wide range of economic ques-tions, such as the determination of wealth inequal-ity (Rao S. Aiyagari 1994), the effectiveness of

self-insurance (Angus Deaton 1991), the welfarecosts of business cycles (Robert E. Lucas 2003),and the determination of asset prices (George M.Constantinides and Darrell M. Duffie 1996),among others.

The current literature offers two views on thenature of the income process. To provide aframework for this discussion, consider the fol-lowing process for log labor income of individ-ual i with t years of labor market experience:1

(1) yti � �it � zt

i,

zti � �zt � 1

i � �ti,

where �i is the individual-specific incomegrowth rate with cross-sectional variance ��

2,and �t

i is the innovation to the AR(1) process.According to the first view in the literature,

which we call the “restricted income profiles”(RIP) process, individuals are subject to largeand very persistent shocks while facing similarlife-cycle income profiles (that is, � is close to 1

* Department of Economics, University of Texas at Aus-tin, BRB 3.118, 1 University Station C3100, Austin, TX,78712 (e-mail: [email protected]). For helpful com-ments and discussions, I would like to thank Daron Ace-moglu, Joe Altonji, Orazio Attanasio, Michael Baker,Richard Blundell, Chris Carroll, Russ Cooper, MarkGertler, Jeremy Greenwood, James Heckman, LawrenceKatz, Narayana Kocherlakota, Per Krusell, Lee Ohanian,Edward Prescott, Martin Schneider, Amir Yaron, and espe-cially Mark Bils, Jonathan Parker, Gianluca Violante, NengWang, and two anonymous referees, as well as the seminarparticipants at Harvard University, MIT, Northwestern Uni-versity, NYU, University of Rochester, UCLA, Universityof Connecticut, University of Pennsylvania (Economics De-partment and Wharton School), the University of SouthernCalifornia, the University of Texas at Austin, the Universityof Western Ontario, Yale University, Koc University, Sa-banci University, the Federal Reserve Banks of Minneapolisand New York, the NBER Economic Fluctuations andGrowth Meetings in Cambridge, and the SED Conference inFlorence. I would also like to thank Dirk Krueger forproviding the CEX consumption data, and the NationalScience Foundation for financial support under grant SES-0351001. The usual disclaimer applies.

1 This income process is a substantially simplifiedversion of the models estimated in the literature, butstill captures the components necessary for the presentdiscussion.

687

and ��2 is zero). According to the alternative

view, which we call the “heterogeneous incomeprofiles” (HIP) process, individuals are subjectto shocks with modest persistence while facinglife-cycle profiles that are individual-specificand vary significantly across the population(that is, � is significantly less than one and ��

2 islarge).

A vast empirical literature has estimated var-ious versions of the specification in (1) usingrich panel data on labor income in an attempt todistinguish between these two views. While theresults of this literature are arguably best inter-preted as more supportive of the HIP model, itis fair to say that this literature has not producedan unequivocal verdict.2 Thus, to shed furtherlight on this question, we take a different route inthis paper: we use the restrictions imposed by theRIP and HIP processes on consumption data—inthe context of a life-cycle model—to bring moreevidence to bear on this important question.

Given the weakness of empirical evidencefrom labor income data in favor of the RIPview, it may seem surprising that the RIP spec-ification is overwhelmingly used as the incomeprocess in economic models. Perhaps an impor-tant reason for this preference is that the con-sumption behavior generated in response to theRIP process is consistent with important empir-ical facts. For example, Deaton and Christina H.Paxson (1994) have documented the significantrise in within-cohort consumption inequalityover the life cycle (which we reproduce in Fig-ure 1 for completeness). As conjectured bythese authors and later confirmed by KjetilStoresletten, Chris Telmer, and Amir Yaron(2004a), a life-cycle model is consistent withthis observation if idiosyncratic shocks are ex-tremely persistent, as is the case in the RIPprocess. Summarizing the existing empirical ev-idence, Lucas (2003, 10) states: “The fanningout over time of the earnings and consumptiondistributions within a cohort that Angus Deaton

and Paxson (1994) document is striking evi-dence of a sizeable, uninsurable random walkcomponent in earnings.”

A second empirical observation, documentedby Christopher D. Carroll and Lawrence H.Summers (1991), is that consumption growthparallels income growth over the life cycle.While this finding may appear to contradict theconsumption smoothing motive that underliesthe life-cycle framework, several authors haveshown that this pattern is consistent with a life-cycle model if income shocks are extremelypersistent, again, as in the RIP model (cf. Carroll1992; Pierre-Olivier Gourinchas and Jonathan A.Parker 2002).

This congruence between the RIP model’stheoretical predictions and consumption data—together with the lack of definitive evidencefrom labor income data—has made the RIPprocess the preferred choice for calibrating eco-nomic models. Perhaps surprisingly, though,there exists no corresponding study of the con-sumption behavior when individuals face theHIP process, so the implications of such a modelare largely unknown.3 The goal of this paper is to

2 A short list of these studies, which includes Thomas E.MaCurdy (1982), John M. Abowd and David Card (1989),and Robert H. Topel (1990), finds support for the RIPmodel; Lee A. Lillard and Yoram A. Weiss (1979), John C.Hause (1980), and especially the more recent studies such asMichael Baker (1997), Steve J. Haider (2001), and Guvenen(2005), find support for the HIP model.

3 Nevertheless, there has been some suggestion in theliterature that the implications of the HIP model are notlikely to be consistent with certain consumption facts. Forexample, Storesletten et al. (2001) state: “Should increasingincome inequality be attributable to heterogeneity which isdeterministic across households, many models of consump-tion choice predict that consumption inequality will notincrease with age” (416).

FIGURE 1. AGE-INEQUALITY PROFILE OF CONSUMPTION IN

THE US DATA

688 THE AMERICAN ECONOMIC REVIEW JUNE 2007

fill this gap. In particular, we study the implica-tions of the HIP and RIP processes for life-cycleconsumption behavior, and provide a systematiccomparison of these implications to the US data toassess which income process makes more accu-rate empirical predictions.

We begin with a detailed investigation of thelife-cycle HIP model, given that little isknown about this framework. This analysis isone of the main contributions of the presentpaper. Because in the HIP model individualsare ex ante heterogeneous (in their incomegrowth rates), a key question is how mucheach individual knows about his own profile.Rather than imposing a certain amount ofknowledge, we assume that individuals enterthe labor market with some uncertainty abouttheir income profile, and we infer the amountof this “prior uncertainty” from consumptiondata as described below. More specifically,we assume that individuals form a prior beliefover �i and �i (the intercept of the incomeprofile), which is updated in a Bayesian fash-ion with subsequent income realizations. Wecast the optimal learning process as a Kalmanfiltering problem, which allows us to conve-niently obtain recursive formulas for updatingbeliefs.

It is often the case with Bayesian learningthat most of the uncertainty is resolved quickly,sometimes with a handful of observations.Instead, in the HIP model learning turns out tobe very gradual, and its effects on consump-tion behavior extend throughout the life cy-cle. The key feature of the model responsiblefor this result is the presence of learningabout the growth rate of income. More gen-erally, we show in the next section thatBayesian learning about a trend parameter(such as �i in equation (1)) displays somefeatures that are inherently different from,and for our purposes more appealing than, themore standard learning about a level param-eter (such as �i).

In Section III we compare the implications ofthe HIP and RIP models to US consumptiondata. We begin by examining the rise in within-cohort consumption inequality—the first empir-ical fact mentioned above. In the HIP model, asa benchmark, we show that when the priorvariance of �i equals the population variance

(i.e., when all individuals begin life with thesame belief about their income growth pros-pects), the model generates a rise in consump-tion inequality that significantly exceeds—byabout 50 percent—what is observed in the USdata.4 This finding suggests that one way toestimate the prior variance of �i is to choose itsuch that the model exactly matches the rise ininequality in the US data. This procedure yieldsa prior variance of �i that equals 40 percent ofthe population variance of this parameter. Theinterpretation is that the remaining 60 percent ofthe variability in income growth rates across thepopulation is forecastable by individuals by thetime they enter the labor market. Thus, the HIPmodel does have the potential to generate asubstantial rise in consumption inequality, de-spite the earlier interpretation of this evidenceas providing unambiguous support to the RIPmodel.

The second fact we examine relates to theshape of the empirical age-inequality profile. Ascan be seen in Figure 1, this profile has a non-concave shape in the US data, implying that therise in consumption inequality does not slowdown as a cohort gets older (up to about age55). Several authors have emphasized this factbecause the RIP model gives rise to a concaveshape (Deaton and Paxson 1994; Storesletten etal. 2004a). Instead, we find that the HIP modelgenerates a nonconcave profile, which also pro-vides a fairly good fit to its empirical counter-part. Again, learning about the growth rate ofincome is essential for this result, as we shownin Section III.

A third fact, also documented by Carroll andSummers (1991), is that college graduates havenot only steeper income profiles than high-school graduates, but also steeper consumptionprofiles (see Figure 2). In the RIP model, theestimated persistence and innovation varianceof income shocks are similar for different edu-cation groups (R. Glen Hubbard, Jonathan Skin-ner, and Stephen P. Zeldes 1995; Carroll andAndrew A. Samwick 1997; Guvenen 2005). Asa result, both groups display similar precaution-ary savings behavior, which in turn results in

4 Uncertainty about the intercept of income, �i, turns outnot to play an important role in the model.

689VOL. 97 NO. 3 GUVENEN: ARE LABOR INCOME SHOCKS REALLY VERY PERSISTENT?

consumption profiles with similar slopes. Turn-ing to the HIP model, we find that the estimateddispersion of income growth rates among col-lege graduates is more than twice that amonghigh-school graduates. This larger dispersionresults in more prior uncertainty, more precau-tionary savings, and consequently a steeper con-sumption profile for college graduates. Theselast two examples underscore the differencesbetween the nature of labor income risk impliedby the RIP model and the HIP model withBayesian learning. We conclude from these re-sults that consumption data provide support tothe HIP model as a credible contender to theRIP model, and along some dimensions, a morecoherent alternative.

This paper is related to several other studies.Mark Huggett, Gustavo Ventura, and AmirYaron (2004) study a human capital modelwith heterogeneity in the ability to accumu-late human capital. They also find differencesin income growth rates (induced by abilityheterogeneity) to be a key element in explainingthe moments of the cross-sectional distributionof income and consumption. The difference isthat in their framework, individuals know theirincome growth rate exactly so there is no learn-ing over time. Moreover, they focus on a subsetof consumption facts studied in this paper. Inaddition to Deaton and Paxson (1994), dis-cussed above, the idea of using consumptiondata to draw inferences about the income pro-cess is also employed by Richard Blundell and

Ian P. Preston (1998) in an attempt to under-stand whether the changes in income inequalityafter 1970 were due to an increase in the vari-ance of persistent shocks or transitory shocks.In a different line of research, Jorn-SteffenPischke (1995) studies the consumption be-havior when individuals cannot distinguishbetween the aggregate and idiosyncratic com-ponents of their labor income and have to solvea signal extraction problem. Finally, NengWang (2004, 2006) extends Pischke’s insight toallow for precautionary savings, and obtainsclosed-form solutions in a continuous-timeframework.

The rest of the paper is organized as follows.In the next section, we introduce the RIP andHIP processes and examine the properties ofBayesian learning about profiles in the lattermodel. In Section II we present the life-cycleHIP and RIP models of consumption-savings.In Section III, we present the quantitative re-sults of the model. Section IV discusses possi-ble extensions and applications of the modeland presents conclusions.

I. Bayesian Learning about Income Profiles

We first specify the RIP and HIP processesand discuss the specific parameterizations weuse. Second, because individuals are ex anteheterogeneous in the HIP process, a key ques-tion is how much individuals know, and howthey learn, about their individual-specific in-come profiles. Thus, we investigate the proper-ties of optimal learning in this environment. Themain result of this section is that learning aboutincome growth rate (or a “trend variable” ingeneral) has some interesting features notpresent when individuals learn about the levelof income (or a “stationary variable” in gen-eral). This distinction is crucial and plays acentral role in the determination of consumptionand savings over the life cycle.

A. Two Stochastic Processes for LaborEarnings

We first introduce the two income processes.The general process for log earnings, yt

i, ofindividual i who is t years old is given by

FIGURE 2. AVERAGE LIFE-CYCLE PROFILE OF CONSUMPTION

BY EDUCATION IN THE US DATA


(2) yti � g��0, Xt

i� � f��i, Xti� � zt

i � �ti,

zti � �zt � 1

i � �ti, z0

i � 0,

where the functions g and f denote two separatelife-cycle components of earnings. The firstfunction captures the part of variation that iscommon to all individuals (hence, the coeffi-cient vector �0 is not individual-specific) and isassumed to be a quartic polynomial in experi-ence, t.5

The second function, f, is the centerpiece ofour analysis and captures the component oflife-cycle earnings that is individual- or group-specific. For example, if the growth rate ofearnings varies with the ability of a worker or isdifferent across occupations, this variation willbe reflected in an individual- or occupation-specific slope coefficient in f. In the baselinecase, we assume this function to be linear inexperience: f(�i, Xt

i) � �i � �it, where therandom vector �i � (�i, �i) is distributed acrossindividuals with zero mean, variances of ��

2 and��

2, and covariance of ��.6

The stochastic component of income is mod-eled as an AR(1) process plus a purely transi-tory shock. This specification is fairly commonin the literature, and despite its parsimoniousstructure, it appears to provide a good descrip-tion of income dynamics in the data (Topel1990; Hubbard et al. 1995; Robert A. Moffittand Peter T. Gottschalk 1995; Storesletten et al.

2004b).7 The innovations �ti and �t

i are assumedto be independent of each other and over time(and independent of �i and �i), with zero mean,and variances of ��

2 and ��2, respectively. The

RIP and HIP processes are distinguished bytheir assumptions about f. The HIP model refersto the general (unrestricted) process given inequation (2). The RIP model, on the other hand,refers to the same process estimated with therestriction �i � 0 imposed.

To calibrate the model we present in thenext section, we use the parameter valuesdisplayed in Table 1 taken from Guvenen(2005). The first two rows display the esti-mates for the whole population from the RIPand HIP models, respectively. The HIP modelimplies a significantly lower persistence forthe AR(1) process (0.82 compared to 0.988)and a statistically (and as shown below, quan-titatively) significant heterogeneity in incomegrowth rates (��

2 � 0.00038 with a t-value of4.9). For comparison, Table 4 in the Appen-dix (p. 710) presents the estimates from theHIP model obtained in the previous literature.Overall, the parameter values we use are con-sistent with this earlier work, with one excep-tion: the variance of the fixed effect, ��

2 , ismuch smaller in our estimates (0.02 comparedto 0.14 in Baker 1997, and 0.29 in Haider2001). In Section ID we show that using avalue of 0.20 has no appreciable effect on ourresults. Finally, the subsequent rows of Ta-ble 1 display the parameter estimates forcollege-educated and high-school-educatedindividuals that are used in Section IC. To ourknowledge, the parameter estimates of theHIP model for each education group are avail-able only in Guvenen (2005).

5 While it is also possible to include an educationdummy in g, we do not pursue this strategy in the baselinespecification. Later in the paper, we will allow for a separateincome process for each education group to control fully forthe effect of education on the life-cycle profiles as well as itseffect on the persistence and variance of income shocks.

6 The zero-mean assumption is a normalization, since galready includes an intercept and a linear term. Moreover,although it is straightforward to generalize f to allow forheterogeneity in higher-order terms, Baker (1997, 373) findsthat this extension does not noticeably affect parameterestimates or improve the fit of the model. In addition, eachterm introduced into this component will appear as anadditional state variable in the dynamic programming prob-lem we solve later. In the baseline case, that problemalready has five continuous state variables and certain non-standard features described later below, so we prefer toavoid any further complexity.

7 Alternatively, it is possible to use an unrestrictedARMA(1,1) or (1,2) process (MaCurdy 1982; Abowd andCard 1989; Moffitt and Gottschalk 1995). Although thisspecification provides more flexibility, it also introducesadditional parameters that appear as state variables in dy-namic decision problems (as the one we study in Section II)expanding the state space. Consequently, economic modelsthat use individual income processes as inputs typically optfor more parsimonious specifications similar to the one usedhere.


B. Quantifying the Heterogeneity in IncomeProfiles

While the point estimate of ��2 of 0.00038

may appear small, this value implies substantialheterogeneity in income growth rates. To seethis, we first define the income residual, yt

i �yt

i � g, and use the following equation (derivedfrom equation (2)) to decompose the within-cohort income inequality into its components:

vari �yti� � ��

2 � ��2� � �1 �2t � 1

1 �2 ��2�

� �2�� t � ��2 t2�.

The first set of parentheses contains terms thatdo not depend on age (i.e., the intercept of theage-inequality profile). The second set of paren-theses captures the rise in inequality due to theaccumulated effect of the autoregressive shock.Finally, the last set of parentheses contains twoterms that vary with age, which are due to profileheterogeneity: a decreasing linear term in t (since�� 0) and, more importantly, a quadratic termin t. It is easy to see that even when ��

2 is verysmall, the effect of profile heterogeneity on in-come inequality will grow rapidly with t2 as thecohort gets older. Table 2 illustrates this point bydisplaying the value of terms in each set of paren-theses over the life cycle. As can be seen in

column 4, the contribution of profile heterogeneityto income inequality is very small early in the lifecycle. In fact, up to age 47, more than half of thecross-sectional variance of income is generated bythe fixed effect, and by the transitory and persis-tent shocks. The effect of profile heterogeneitycontinues to rise, however, and accounts for al-most 80 percent of inequality at retirement age.

C. The Kalman Filtering Problem

The key feature of the HIP model is that indi-viduals are ex ante different in their income pro-files, which—as the analysis above illustrates—accounts for a large fraction of the rise in within-cohort income inequality. Hence, to embed theestimated income process into a life-cycle model,we need to be specific about what the individualknows about (�i, �i). A plausible scenario is onein which an individual enters the labor marketwith some prior belief about his income growthprospects. This prior could incorporate some rel-evant information unavailable to the econometri-cian, as we discuss below. Over time, a rationalindividual will refine these initial beliefs by incor-porating the information revealed by successiveincome realizations. We assume that this updating(“learning”) process is carried out in an optimal(Bayesian) fashion.

In order to formally define the learning prob-lem, we need to specify which components of

TABLE 1—PARAMETER ESTIMATES OF THE LABOR INCOME PROCESS FROM GUVENEN (2005)

Group Model � ��2 ��

2 �� 2 ��

2

(1) A RIP 0.988 0.058 — — 0.015 0.061(0.024) (0.011) (0.007) (0.010)

(2) A HIP 0.821 0.022 0.00038 �0.0020 0.029 0.047(0.030) (0.074) (0.00008) (0.0032) (0.008) (0.007)

(3) C RIP 0.979 0.031 — — 0.0099 0.047(0.055) (0.021) (0.013) (0.020)

(4) C HIP 0.805 0.023 0.00049 �0.0024 0.025 0.032(0.061) (0.112) (0.00014) (0.0039) (0.015) (0.017)

(5) H RIP 0.972 0.053 — — 0.011 0.052(0.023) (0.015) (0.007) (0.008)

(6) H HIP 0.829 0.038 0.00020 �0.0007 0.022 0.034(0.029) (0.081) (0.00009) (0.0012) (0.008) (0.007)

Notes: Standard errors are in parentheses. In the second column, A � all individuals, C � college-educated group, and H �high-school-educated group. Time effects in the variances of persistent and transitory shocks are included in the estimationin all rows, but are not reported to save space. The reported variances are averages over the sample period. These parameterestimates are taken from Guvenen (2005).


income are observable. If the stochastic compo-nent, zt

i � �ti, were observable in addition to yt

i,individual income profiles (�i, �i) would be re-vealed in just two periods, leaving no role forfurther learning. Although we could allow either zt

i

or �ti to be separately observable and still have

nontrivial learning, it seems difficult to make acompelling case for why one component would beobservable while the other is not. Therefore, as abenchmark case, we assume that individuals ob-serve only total income, yt

i, and not its componentsseparately.

It is convenient to express the learning pro-cess as a Kalman filtering problem using thestate-space representation. In this framework,the “state equation” describes the evolution ofthe vector of state variables that is unobservedby the decision maker:8

St � 1i

� �i

�i

zt � 1i � �

F

�1 0 00 1 00 0 �

�St

i

��i

�i

zti� �

�t � 1i

� 00

�t � 1i �.

Even though the parameters of the incomeprofile have no dynamics, including them intothe state vector yields recursive updating for-mulas for beliefs using the Kalman filter. Asecond (observation) equation expresses the ob-servable variable(s) in the model—in this case,log income—as a linear function of the under-lying hidden state and a transitory shock:9

yti � �1 t 1��i

�i

zti� � �t

i � HtSti � �t

i.

We assume that both shocks have i.i.d. Normaldistributions and are independent of each other,with Q and R denoting the covariance matrix of �t

i

and the variance of �ti, respectively.10 To capture

an individual’s initial uncertainty, we model hisprior belief over (�i, �i, z1

i ) by a multivariateNormal distribution with mean S1�0

i � (�1�0i , �1�0

i ,z1�0

i ) and variance-covariance matrix:11

P1�0 � � ��,02 ��,0 0

��,0 ��,02 0

0 0 �z,02�,

where we use the shorthand notation ��,t2 to de-

note ��,t�1�t2 . After observing (yt

i, yt�1i , ... , y1

i ),an individual’s belief about the unobserved vec-tor St

i has a normal posterior distribution with amean vector St�t

i and covariance matrix Pt�t. Sim-ilarly, let St�1�t

i and Pt � 1�t denote the one-period-ahead forecasts of these two variables,respectively. These two variables play centralroles in the rest of our analysis. Their evolutionsinduced by optimal learning are given by

(3) St�ti � St�t � 1

i � Pt�t � 1Ht�HtPt�t � 1Ht � R�1

�yti HtSt�t � 1

i �,

8 Vectors and matrices are denoted by bold lettersthroughout the paper.

9 To simplify the analysis, in the rest of the paper we setg � 0, and assume that �i and �i have mean values of �� and�� , respectively.

10 The normality assumption is not necessary for the esti-mation of the parameters of the stochastic process (2) and isnot made in Guvenen (2005) to obtain the parameters inTable 1.

11 The notation Xh2�h1denotes the forecast of (alterna-

tively, belief about) Xh2, given the information available at

time h1 if h2 � h1 (if h2 � h1).

TABLE 2—DECOMPOSING WITHIN-COHORT INCOME INEQUALITY

Age

(1) (2) (3) (4)

��2 � ��

21 �2t�1

1 �2 ��2 2��t � ��

2t2�3�

�1� � �2� � �3�

30 0.069 0.082 0.005 0.0335 0.069 0.088 0.030 0.1645 0.069 0.089 0.135 0.4655 0.069 0.089 0.315 0.6765 0.069 0.089 0.568 0.79


St � 1�ti � FSt�t

i ,

and

(4) Pt�t � Pt�t � 1 Pt�t � 1Ht�HtPt�t � 1Ht � R�1

HtPt�t � 1 ,

Pt � 1�t � FPt�tF � Q.

Notice that the covariance matrix evolvesindependently of the realization of yt

i, and is alsodeterministic in this environment since Ht isdeterministic. Moreover, one can show fromequation (4) that the posterior variances of �i

and �i are monotonically decreasing over time,so with every new observation, beliefs becomemore concentrated around the true values. (Thisis not necessarily true for �z,t

2 which may benonmonotonic, depending on the parameteriza-tion.) Finally, log income has a Normal distri-bution conditional on an individual’s beliefs:

(5)

yt � 1i �St�t

i � N�Ht � 1St � 1�ti , Ht � 1Pt � 1�tHt � 1 � R�.

D. The Speed of Resolution of ProfileUncertainty

The results presented in Section IB suggest thata substantial fraction of income differences overthe life cycle is due to HIP. Consequently, theinitial income risk perceived by an individualupon entering the labor market can be substantialif the individual is sufficiently uncertain about hisincome profile. Since, however, individuals learntheir profile over time, the contribution of profileuncertainty to perceived income risk later in thelife cycle depends on the speed of learning. It isoften the case with Bayesian learning that a largefraction of prior uncertainty is resolved quickly, soit is essential to investigate this issue in the presentframework.

As we quantify below, learning is very gradualin our model and its effects extend throughout thelife cycle for two reasons. The first and mainreason is that early in life the contribution of the �itterm to income is very small—most of the varia-tion in income is due to shocks, as can be seen in

Table 2—so income observations are not veryinformative about the growth rate of income, slow-ing down learning. Second, later in life when obser-vations become potentially more informative, themoderate persistence of shocks makes it difficultto disentangle them from the trend component,again slowing down learning. In the rest of thissection we make these points more rigorous.

We begin by defining a convenient measureof income uncertainty, the forecast variance—the mean squared error (MSE) of the fore-cast—of future income:

(6) MSEt � s�t � Et �yt � s yt � s�t �2

� Ht � sPt � s�tHt � s � R,

where

(7) Pt � s�t � FsPt�tFs � �i � 0

s � 1

FiQFi.

If individuals know their profile with cer-tainty (i.e., ��,t

2 � ��,t2 � 0), the forecast vari-

ance in equation (6) reduces to MSEt�s�tidio �

Et(zt�s � zt�s�t)2 � ��

2, where the superscriptidio indicates that the only source of risk in thiscase is idiosyncratic shocks. Notice that an in-come process with RIP is a special case of this,so the same expression characterizes the fore-cast variance for such processes. In the moregeneral case where individuals are uncertainabout their profile (��,t

2 , ��,t2 � 0), the forecast

variance can be written as

(8) MSEt � s�ttotal � MSEt � s�t

idio

� ��,t2 � 2��,t�t � s� � ��,t

2 �t � s�2 � �t� s�t ,

which is again obtained using equation (6). Thefirst term captures the risk resulting from idio-syncratic shocks as before. The remaining termsin parentheses (call it MSEt�s�t

net ) capture the netcontribution of profile uncertainty to incomerisk at different horizons (given by s) as per-ceived by an individual at age t. For a given t,the terms in the square bracket imply that theforecast variance (due to profile heterogeneity)is an increasing quadratic function of horizon(t � s). In addition, although zt

i is independent


of (�i, �i), the joint updating of beliefs naturallyinduces a correlation between these two com-ponents. The last term, �t�s�t, contains the cor-responding covariances; it is an increasingfunction of s for fixed t, but does not materiallyaffect the shape of this profile.

In the left panel of Figure 3, we plot MSEt�s�tnet ,

s � 1, 2, ... , for an individual at ages t � 25, 35,45, and 55, who faces the HIP process estimatedon row 2 of Table 1.12 The top curve (t � 25)shows that the future income risk perceived bythis individual upon entering the labor market issubstantial, as can be expected from the fact thatHIP accounts for a large fraction of incomeinequality and the individual does not initiallyknow his true profile. As the individual getsolder, the successive MSE curves shift down-ward, reflecting the resolution of profile uncer-tainty. The main point to notice in this graph isthat the resolution of uncertainty is slow: by thetime the individual is 35 years old (the secondcurve from the left), only 26 percent of incomerisk at retirement will have been resolved. Atage 45, the forecast variance of income at re-

tirement is still about 0.22. For comparison, atthe same age, the forecast variance at retirementthat is due to idiosyncratic shocks (MSE65�45

idio ) isonly 0.045.

The main reason for the slow learning is thatindividuals learn about a slope parameter, �i ,whose contribution to income is small whenindividuals are young, but grows monotonicallywith age. Figure 4 illustrates the implications ofthis feature for the speed of learning. Specifi-cally, the vertical axes plot (log(1/�x,t�1�t

2 ) �log(1/�x,t�t�1

2 )), for x � �i (left panel) and �i

(right panel), which can be interpreted as thepercentage improvement in precision—or equiv-alently, the percentage reduction in the posteriorvariance—at each age. In the left panel, theresolution of uncertainty about �i follows thefamiliar pattern: most of the learning takes placeearly on, and after the first five or so years, eachsubsequent observation brings little fresh infor-mation about the intercept term. In contrast, inthe right panel, the information provided about�i by each additional observation increases overtime, up to about age 50. Using the terminologyof signal extraction problems, the signal-to-noise ratio increases—resulting in faster learn-ing—as the individual gets older. In fact, thiscan be seen in Figure 3, where the MSE curvesare shifting to the right faster as the individualgets older.

It is useful to contrast the resolution of un-certainty above to the hypothetical case where

12 To calculate the MSE, we need to specify the priorcovariance matrix, P1�0. We discuss the specification of thepriors more fully below. As a simple benchmark, here weassume that the individual does not have more informationthan the econometrician so that the prior variance of eachvariable is equal to its population value (that is, ��,0

2 � ��2,

etc.).

FIGURE 3. SPEED OF RESOLUTION OF INCOME UNCERTAINTY THROUGH BAYESIAN LEARNING ABOUT PROFILES


the main source of uncertainty (and hence learn-ing) is about the level of income, �i. This com-parison is also helpful because our baselineestimate of ��

2 is around 0.02, whereas the cor-responding point estimate is 0.14 in Baker(1997) and 0.29 in Haider (2001) (see Table4). Figure 5 plots the change in precision ofbeliefs about �i when ��

2 (and correspondinglythe prior variance) is set to 0.20. The two linesplot the precision when the dispersion in �i isfixed at its baseline value (‘- ˆ’) and, alterna-

tively, when it is set to zero (‘- -’). (The twolines are almost indistinguishable in the firstfour years.) In both cases, the log precisionimproves by 130 log points with the first obser-vation, implying that the posterior variance of�i falls (by � e1.3) from 0.20 to 0.054 after thefirst year, and to below 0.04 after the third year.The reason for this fast learning is clear: since�it is very small early in life, and the stochasticshocks have much smaller variances and lowerpersistence than �i, the latter stands out (i.e., thesignal-to-noise ratio is high) and is detectedeasily. Hence, even when there is significantlymore initial uncertainty about the level ofincome, it has little effect on the behavior ofindividuals after the first few years, unlike theeffect of learning about the growth rate ofincome.

A second reason for the slow learning is themoderate persistence of income shocks. We il-lustrate this point in the right panel of Figure 3.The bottom curve plots the net forecast varianceof income at retirement by an individual who is35 years old (MSE65�35

net ) as a function of thepersistence of zt, normalized by its value at � �0. The two curves above are constructed simi-larly for t � 45 and 55, respectively. Whenconstructing these graphs, we adjust the inno-vation variance of zt as we vary � to keep theunconditional variance of the AR(1) processunchanged.

FIGURE 4. CHANGE IN THE PRECISION OF BELIEFS ABOUT � AND �

FIGURE 5. CHANGE IN THE PRECISION OF BELIEFS ABOUT �WHEN ��,0

2 IS SET TO 10 TIMES ITS BASELINE VALUE


One conclusion, which is clear from thisgraph, is that the speed of resolution is not amonotone function of persistence: as � increasesfrom zero up to about 0.85, the resolution ofuncertainty slows down (reflected in a largerforecast variance at retirement), but then speedsup again as persistence increases further towarda unit root. In particular, learning is faster whenincome shocks follow a random walk than forany other value of �.13 Interestingly, the valuesof � where learning is slowest coincide with theempirical estimates of persistence reported inTable 1 (although the figure also makes clearthat the resolution of uncertainty is not dramat-ically different for values of � roughly between0.7 and 0.9).

The second feature apparent in the right panelof Figure 3 is that the impact of persistence onthe speed of learning increases with age. Forexample, at age 35, increasing the persistencefrom zero to 0.8 results in a 30 percent rise inMSE65�35

net . At age 55, the same experiment raisesthe forecast variance by 180 percent. Thus, therelatively high persistence of income shocks inthe data is important for the slow resolution ofuncertainty, especially later in the life cycle.

Before concluding this section, it should benoted that slow learning is also important foranother reason: in the next section we inferthe amount of prior uncertainty from theconsumption-savings behavior of individualsover the life cycle. But if learning were quick,individuals’ behavior later in life would con-tain little information about their prior uncer-tainty, making this setup unsuitable for thisexercise. In other words, life-cycle behavioris informative about prior uncertainty to theextent that it is not resolved very quickly.

Stochastic Dynamics for Learning Uncer-tainty.—So far, we have considered the casewith homoskedastic Gaussian innovations to thestate vector, which implies that Pt�t�1 evolves ina deterministic fashion (equation (4)). Never-theless, it is conceivable that some componentsof the state vector (such as zt) could be subject

to conditionally heteroskedastic shocks wherethe innovation variance itself evolves in a sto-chastic manner.14 Indeed, Costas Meghir andLuigi Pistaferri (2004) report some evidence ofan ARCH structure in the variances of laborincome shocks, although they consider only RIPprocesses. In this case, Pt�t�1 would also evolvestochastically, contrary to our previous analysis.Unfortunately, solving for the consumption-savings decision in the presence of Bayesianlearning and stochastic volatility would add sig-nificant computational challenges into the presentframework, which is already quite complicated.Furthermore, for a full comparison of the HIPand RIP models, we would also need to intro-duce stochastic volatility into the latter model,which would take us well beyond the scope ofthis paper. Thus, here we briefly examine howlearning and the resolution of profile uncer-tainty are affected by the presence of stochasticvolatility, but leave a full analysis of consump-tion behavior to future work.

Consider the following GARCH(1,1) processfor the innovation variance of zt,

��2�t� � �0 � �1��

2�t 1�

� �2 ��2�t� ��2�t 1��,

where �’s are positive constants, and the termin the last set of parentheses can be thought ofas a zero-mean innovation to volatility. Theforecast of future innovation variances has asimple formula:

Et ��2�t � i�� 1 �i��

2�t� �0

1 �1�

��0

1 �1, for i � 1, ... .

To gain a better idea about the effect ofstochastic volatility on the MSE, a quantitativeexample will be useful.15 We set the persistence

13 Loosely speaking, this is because when income shocksfollow a random walk, income growth becomes very infor-mative about �i, since �yt

i � �i � (� � 1)zt�1 � �t � ��t

reduces to �i � �t � ��t in this case.

14 For example, zt would have stochastic volatility ifindividuals occasionally switch between occupations thathave different levels of idiosyncratic risks.

15 The GARCH structure modifies the equations for theMSE in two ways. First, in the Kalman recursion (4) which


of volatility, �1 � 0.95, and the size of inno-vation, �2 � 0.70, which generate fairly largeand persistent movements in volatility (the stan-dard deviation of ��

2(t) is 1.2 times its averagevalue). Finally, we set �0 � (1 � �1)��

2 so thatthe average volatility remains equal to ��

2 as inthe benchmark case. Figure 6 plots the results.Because of the stochastic evolution of the pos-terior variance, there is an entire distribution ofMSEt�s�t at each age t, depending on the historyof innovation volatilities experienced by eachindividual. To summarize this information, weplot the mean of this distribution at each age(line with circles), as well as the 95 percentconfidence bands to quantify the dispersion. Forcomparison, the dashed line plots the MSE fromthe benchmark model (taken directly from theleft panel of Figure 3).

Several points can be seen from this graph.First, the average MSE with GARCH shocks issomewhat lower than its benchmark counter-part. Moreover, the difference between the twoappears to widen slightly with age, indicating amild acceleration in the speed of learning in thepresence of stochastic volatility. In addition,there is a fair amount of variation as indicatedby the confidence bands. However, reducing the

persistence of the GARCH process, to 0.80 forexample, substantially narrows the gap betweenthe two series (not shown). Another point tonote is that the shape of the average MSE pro-file is virtually identical to that in the bench-mark case. This feature is not specific to theaverage, but is also true for each individualMSE profile in that distribution. This result isimportant because this shape determines howindividuals perceive their future income riskand, consequently, has a major effect on theirconsumption decision, as we discuss further be-low. These results suggest that introducing sto-chastic volatility in zt may not greatly affectconsumption behavior in the HIP model, espe-cially if conditional volatility is not as persistentas we assumed in this example. Nevertheless, amore definitive answer awaits a full investiga-tion of consumption behavior, which is left forfuture work.

II. Two Life-Cycle Models of Consumption andSavings

In the rest of the paper, we embed each incomeprocess (HIP and RIP) into a life-cycle model.Then, by comparing the implications of eachmodel to the US data, we assess which type ofincome process makes more accurate predic-tions about life-cycle consumption behavior.The two life-cycle models will be kept assimilar to each other as possible to highlightthe differential impact of each income processon consumption behavior. We now describeeach model in turn.

A. The HIP Model

Consider an environment where each individ-ual lives for T* years and works for the first T(�T*) years of his life, after which he retires.Individuals do not derive utility from leisure, andhence supply labor inelastically. During the work-ing life, the income process is given by the HIPprocess specified in equation (2). During retire-ment, the individual receives a pension that isdesigned to mimic the current US Social Securitysystem as described in further detail below. Thereis a risk-free bond that sells at price Pb (with acorresponding net interest rate r f � 1/Pb � 1).Individuals can also borrow at the same interest

determines Pt�t, the matrix Q needs to be replaced with itsforecast, Qt�1�t. This latter differs from Q only in its lowerdiagonal element, which now equals Et(��

2(t � 1)). Simi-larly, we replace Q in equation (7) with its forecast, Qt�1�t.

FIGURE 6. RESOLUTION OF UNCERTAINTY WHEN THE

VOLATILITY OF zt FOLLOWS A GARCH PROCESS


rate up to an age-specific borrowing constraintW� t�1, specified below.

The relevant state variables for this dynamicproblem are the asset level, �t

i, current log in-come, yt

i, and the last period’s forecast of thetrue state in the current period, St�t�1. In thefollowing equations, the superscript i is in-cluded in individual-specific variables to distin-guish them from aggregate variables. Thedynamic problem of a typical individual canthen be written as

Vti��t

i, yti, St�t � 1

i � � maxct

i,�t � 1i

�U�cti�

� E�Vt � 1i ��t � 1

i , yt � 1i , St � 1�t

i ��St�t � 1i

(9) s.t. cti � Pb�t � 1

i � �ti � Yt

i,

(10) �t � 1i � W� t � 1 ,

equations �3, 4�

for t � 1, ... , T � 1, where Yti � eyti is the level

of income, and Vti is the value function of a

t-year-old individual.16 The evolutions of thevector of beliefs and its covariance matrix aregoverned by the Kalman recursions given inequations (3) and (4). Moreover, given that theonly state variable that is random at the time ofdecision is next period’s income, the expecta-tion is taken with respect to the conditionaldistribution of yt�1

i given by equation (5).During retirement, pension income is constant,

and since there is no other source of uncertainty orlearning, the problem simplifies significantly:

(11) Vti��t

i, yi� � maxct

i,�t � 1i

�U�cti� � Vt � 1

i ��t � 1i , yi�

s.t. Yi � ��YTi �,

(12) equations �9, 10�

for t � T, ... T*, with VT*�1 � 0, and �(YTi ) is

a function determining the pension income dur-ing retirement.

B. The RIP Model

The second model is essentially the same as thefirst one, with the exception that the income pro-cess is now given by a RIP process. Because witha RIP process all individuals share the same life-cycle income profile (�, �), there is no learningabout individual profiles, which reduces the num-ber of state variables significantly—from five,above, to two here. Specifically, the dynamic pro-gramming problem of a typical worker is

Jti��t

i, zti� � max

cti,�t � 1

i

�U�cti� � E�Jt � 1

i ��t � 1i , zt � 1

i ��zti

s.t. equations �9, 10�

for t � 1, ... , T � 1, where Jti is the value

function of a t-year-old individual. Notice thatwe assume the worker observes the persistentcomponent of the income process, zt

i, separatelyfrom yt

i. This is the standard assumption in theexisting consumption literature which uses theRIP process, and we follow them for compara-bility. Finally, because there is no income riskafter retirement, the problem of a retiree is thesame in this model as in the HIP model givenabove in (11).

C. Baseline Parameterization

There is no analytical solution to the dynamicoptimization problems stated in the previoussection, so we resort to numerical methods.While the computational method used to solvethe RIP model is standard and has been em-ployed by several earlier studies, the numericalsolution of the HIP model is significantly com-plicated by the fact that there are five continu-ous state variables and four of them (excluding�t

i) depend on each other as a result of Bayesianlearning. In particular, this interdependencemakes the solution of the value function on arectangular grid impractical. We develop an al-gorithm to tackle these issues, which could beuseful for solving similar problems. Further dis-cussions of computational issues, as well as thedetails of our algorithm, are provided in the

16 Although, given the last two variables, one can obtainboth St�t and St�1�t using equation (3) (which means that theindividual knows the latter two vectors at the time of deci-sion), our current choice turns out to be more convenient forcomputational reasons.


computational appendix available on the au-thor’s Web site (www.eco.utexas.edu/faculty/Guvenen).

In the remainder of this section we describethe parameterization of each life-cycle model(summarized in Table 3). To make the quanti-tative exercise meaningful, we choose the samevalues for parameters that appear in both mod-els (except for one parameter as explained below).

A model period is one year of calendar time.Individuals enter the labor market (are born) atage 25, retire at 65, and die at age 95. Theperiod utility function is assumed to take theCRRA form with a relative risk aversion coef-ficient equal to two. The bond price, Pb, is setequal to 0.96, implying an annual interest rate of4.16 percent. We set the time preference factor, , to match the average wealth-to-income ratioin the US data. Santiago Budrıa Rodrıguez et al.(2002) calculate this ratio both from the Surveyof Consumer Finances and from the NationalIncome and Product Accounts, and obtainvalues between 4.14 and 5.26. It is not imme-diately clear, however, how to treat housing inthis calculation, which is included in their cal-culation but not explicitly modeled in ourframework. With this in mind, we target awealth-to-income ratio of four, which is at thelower end of these reported values. Notice thatbecause the amount of precautionary savingsdepends on the amount of uncertainty, which isdifferent in the RIP and HIP models, this pro-

cedure implies (slightly) different values for thetime preference factor in each model: HIP �0.966 and RIP � 0.964. Furthermore, when wemake comparisons across different versions ofthe model (by varying �, for example), we ad-just to keep the wealth-to-income ratio on thistarget.

The parameters of the stochastic componentof income are taken from Table 1. Although theestimation of the covariance matrix pins downthe variances of � and � (in the HIP process), itdoes not identify their means. The interceptterm, �, is simply a scaling parameter, so it isnormalized to 1.5 for computational conve-nience. The mean of � is set to the mean growthof log income in the Panel Study of IncomeDynamics (PSID) sample of Guvenen (2005): itis equal to 0.9 percent per year for the wholesample, and 0.7 percent and 1.2 percent forindividuals with low and high education levels,respectively.

Redistributive Social Security.—The U.S. re-tirement pension system features significant re-distribution, thereby providing risk-sharing notonly across cohorts but also within each cohort.The extent of this risk-sharing in turn is criticalfor the rise in consumption inequality over thelife cycle. For example, with complete risk-sharing, consumption inequality would be con-stant over the life cycle in both the HIP and RIPmodels. Thus, a realistic modeling of the retire-ment system is essential for a sound quantitativeevaluation of the consumption behavior in thesemodels.17 We adopt the following formulationfor the pension system, which captures the sa-lient features of the US Social Security systemas described in Storesletten et al. (2004a). Tosimplify notation, define YT � YT/Y� T to be anindividual’s income at age T relative to theaverage income at that age, Y� T. The retirementreplacement rate is a concave function of YTgiven by

17 Clearly, the retirement system is not the only mecha-nism providing risk-sharing among individuals. In general,the market structure—the types of assets, in addition to arisk-free bond, available to individuals for consumptionsmoothing—would also affect consumption behavior. Manysuch extensions, however, would significantly complicatethe present model and hence are left for future work.

TABLE 3—BASELINE PARAMETERIZATION

Annual model

Parameter Value

Time discount factor 0.966Pf Price of risk-free bond 0.96� Relative risk aversion 2�� Avg. inc. growth for all households 0.009�� C Avg. inc. growth for college educ. 0.012�� H Avg. inc. growth for high-school educ. 0.007T Retirement age 65T* Age of death 95P1�0 The variance of prior beliefs See text

Notes: The parameters of the income process are taken fromcorresponding rows of Table 1. The time discount rate isadjusted in each experiment to generate a wealth-to-incomeratio of 4. The value reported in the table is for a baselineHIP model with � � 0. See the text for details.


��YT � � ��

0.9YT if YT � 0.30.27 � 0.32�YT 0.3� if YT � �0.3, 20.81 � 0.15�YT 2� if YT � �2, 4.11.1 if YT � 4.1

,

where �� is a scaling parameter.18

Determining the Priors in the HIP Model.—Empirical evidence is not particularly helpfulfor setting P1�0. The difficulty is that the econo-metrician is able to measure only the populationdistribution of (�i , �i) conditional on a fewobservable characteristics. But it is conceivablethat each individual could have some informa-tion, unavailable to the econometrician, that canprovide a better prediction of his income profile.Thus, rather than imposing a certain amount ofprior knowledge on the individual, we infer itfrom the observable actions over the life cycle.

We begin by describing how an individual’sprior belief about �i is determined. Suppose thatthe distribution of income growth rates in thepopulation is generated as �i � �k

i � �ui , where

�ki and �u

i are two random variables, indepen-dent of each other, with zero mean and vari-ances of ��k

2 and ��u

2 . Clearly then, ��2 � ��k

2 ��u

2 . The key assumption we make is that indi-vidual i observes the realization of �k

i but not of�u

i (hence the subscripts indicate known andunknown, respectively). Under this assumption,the prior mean of individual i is �1�0

i � �ki ,

and the prior variance is ��,02 � ��u

2 � (1 ��)��

2, where we define � � 1 � ��u

2 /��2 as the

fraction of variance known by individuals.Two polar cases deserve special attention. If

� � 0, individuals do not have any private prior

information about their income growth rate (i.e.,��,0

2 � ��2). This case provides a useful bench-

mark (or an upper bound) to gauge how muchmileage one can get by allowing uncertaintyabout individual income profiles. On the otherhand, if � � 1, each individual observes �i

completely and faces no prior uncertainty aboutits value. This case provides a useful compari-son to illustrate how profile uncertainty andlearning alter individuals’ consumption-savingsdecision. Finally, as noted earlier, the dispersionof �i is not large according to our parameter-ization and does not materially affect the resultsof the model even when it is larger. For sim-plicity, we assume that individuals have no pri-vate prior information about their intercept, sothat ��,0

2 � ��2. The prior covariance matrix is

P1�0 � � ��2 1 �� 0.0

1 �� 1 ��2 0.0

0.0 0.0 ��2�.

To emphasize that the implications of the HIPmodel depend on the amount of prior un-certainty facing individuals, we henceforthrefer to this general framework as the HIP(�)model.

As for the calibration of the borrowing con-straint, we have a few considerations in mind.First, it is desirable to impose a loose constraintso as not to confound the effects of profileuncertainty and learning—the primary focus ofthis paper—with those of borrowing frictions.The loosest constraint is implied by the condi-tion that an individual cannot have debt at thetime of death. In this case, in any given periodan individual can borrow up to the point wherehe can still pay back all of his debt, even if hehappens to face the lowest possible income re-alization for the rest of his life. In the HIPmodel, however, this condition requires thateach individual face a different natural limit,unlike in the RIP model where individualsare ex ante identical. In this case, however,the constraints themselves contain informationabout an individual’s profile, which would thenneed to be optimally incorporated into beliefs.This would further complicate the model, prob-ably without providing much additional insight.Instead, we allow all individuals to borrow up to

18 There is one difference between this specification andthe one in Storesletten et al. (2004a): � here is a function ofYT instead of the average income over an individual’s lifecycle (which would require us to track one more statevariable). However, because income shocks are not verypersistent in our model, YT is highly correlated with anindividual’s average income (correlation: 0.89), so the dif-ference may not be crucial. Moreover, because YT is 40percent higher than average income over the life cycle, weneed to multiply our pension schedule by �� 1/1.40 �0.715 to match the average level of benefits in their speci-fication.


a common borrowing limit, determined as thenatural limit of an individual based on publicprior information (that is, ��,0

2 � ��2). In other

words, this is the natural limit that credit insti-tutions would enforce on individuals when onlytime-0 public information is available. For com-parability, we impose the same borrowing con-straint in the RIP model, which almost neverbinds in our simulations. Finally, notice thatsince yt is log-normally distributed, the lowestincome realization can be arbitrarily close tozero, so we truncate the Normal distribution ofincome at 2.5 standard deviations to provide aproper lower bound.

III. Quantitative Results

In this section, we compare the life-cycleconsumption behavior implied by the HIP(�)model and the RIP model to the US data.The main goal of this exercise is to assesswhich income process provides more accu-rate predictions about life-cycle consumptionbehavior.

To simulate the HIP model, we first draw1000 (�, �) combinations (each draw corre-sponding to one type of individual) from a bi-variate Normal distribution whose covariancematrix is taken from row 2 of Table 1. Then foreach (�i , �i) type, we simulate 100 incomepaths and calculate the corresponding consump-tion and wealth paths implied by the HIP model.As for the RIP model, we simulate income pathsfor 10,000 individuals who all have the same (�,�) combination but differ in the realization oftheir idiosyncratic shocks. In all cases, we as-sume that individuals start life with no assets.The reported statistics are calculated from thesesimulated data.

A. The Age-Inequality Profile of Consumption

We first examine the implications of eachmodel for the rise in within-cohort consumptioninequality over time. As can be seen in Fig-ure 1, in the US data the cross-sectional vari-ance of log consumption increases by about 21log points over the life cycle, implying thatconsumption inequality more than doubles dur-

ing this time.19 While the mere existence offanning-out in the consumption distribution isnot surprising—as it is implied, for example, bythe permanent income theory—the large mag-nitude of the increase is. Deaton and Paxson(1994) have discussed several potential expla-nations and found the existence of persistent(uninsurable) idiosyncratic shocks to be themost promising candidate. Recently, Storeslet-ten et al. (2004a) have shown that a life-cyclemodel can quantitatively match the rise in in-equality observed in the data if income shocksare extremely persistent. It should be noted,however, that both studies have restricted atten-tion to RIP models, and neither one has allowedfor a HIP process for income. Here, we analyzeconsumption inequality in both the RIP and HIPmodels.

In Figure 7, the middle line (with circles) plotsthe cross-sectional inequality implied by thebaseline RIP model, which shows an increase of26 log points over the life cycle, even exceedingthe rise in the data. For completeness, as well asfor further comparison later, we also plot thecorresponding graph from the benchmark RIP

19 We use data on consumption expenditures from theConsumer Expenditure Survey (CE) obtained from DirkKrueger and Fabrizio Perri (2006). We choose our con-sumption measure and sample period following Deaton andPaxson (1994). Thus, consumption refers to household non-durable expenditures during the last quarter. The sampleperiod is 1980–1990, but unlike these authors who concen-trate on urban households, we include all households in oursample, which requires us to exclude the 1982–1983 period,since data on rural households are not available during thistime. We use census equivalence scales to convert house-hold consumption into per-adult-equivalent units to makethem comparable to income data from PSID, which are usedto estimate the income processes in Table 1. See Kruegerand Perri (2006) for further details of sample selection andvariable construction. To obtain the graph in Figure 1, wefollow Deaton and Paxson and regress raw variances foreach age-year cell on a set of age and cohort dummies andreport the coefficients on the age dummies. To obtain areasonably large number of observations in each age cell,we define five-year age bands. For example, to calculate thevariance at age h, we use all observations on individualsaged (h � 2) to (h � 2). The age dummies are scaled so thatthe average inequality matches that in the sample. Partly dueto these differences, the rise in inequality is 21 log points inFigure 1 compared to roughly 25 log points in Deaton andPaxson (1994). However, the general pattern of rise ininequality (including the nonconcavity, which we commenton below) is very similar in the two papers.


model of Storesletten et al. (2004a). The (top)dashed line in Figure 7 shows an even largerincrease in inequality—about 40 log points—but has an overall shape that is very similar tothe one from our baseline RIP model.20 Overall,these results reiterate the earlier findings in theliterature that a RIP model generates significantfanning out of the consumption distribution asobserved in the data.

We next turn to the HIP(�) model. To pro-vide a benchmark, we begin with the case whereindividuals have no private prior informationabout �i (that is, � � 0). The (top) solid line inFigure 8 plots the age-inequality profile, whichalso shows a substantial rise in consumptioninequality over the life cycle—roughly 32 logpoints.21 Thus, the HIP(�) model is also capableof generating a significant rise in consumptioninequality, despite the previous interpretation ofthis empirical evidence, discussed in the intro-duction, as supporting the RIP model.

Before proceeding further, it is useful to dis-cuss the mechanism behind increasing inequal-ity in each model. For the sake of this discussion,we make the following simplifying assump-tions: (a) the utility function is quadratic, and � 1 � r f � 1; (b) there is no retirement (i.e.,all individuals die at the end of working life);and (c) in the RIP model, income shocks arefully permanent (� � 1.0). Under these assump-tions, in both models consumption will be ad-justed every period by the change in perceivedlifetime income divided by the number of yearsleft in life:

(13) �ct �1

T t � 1 ��s � 0

T � t

(Et Et � 1)Yt � s�.

The left panel of Figure 9 plots the simulatedincome and consumption paths of an individualin the RIP model (who happens to have a real-ized income growth that is positive). Using thefacts that income follows a random walk (soEt(Yt�s) � Yt for all s), we have �ct � �Yt forall t. Thus, we obtain the well-known result thatin the (certainty-equivalent version of the) RIPmodel permanent income movements are fullyaccommodated. If, furthermore, initial wealth iszero, consumption equals income every period.Since this is true for every individual in thiseconomy, consumption inequality will rise onefor one with income inequality over time.

20 Storesletten et al.’s (2004b) benchmark model doesnot feature a redistributive Social Security system, whichlargely explains why the rise in inequality is very differentin their RIP model compared to ours. They do, however,introduce it as an extension, which then results in a smallerrise in inequality similar to ours (of about 25 log points).

21 Only a small fraction of this fanning-out is directlyattributable to idiosyncratic shocks: if we eliminate profileheterogeneity and consequently learning from the HIPmodel (without changing the parameter values for persis-tence, etc.), the rise in inequality would be only 7 log points.

FIGURE 7. AGE-INEQUALITY PROFILE OF CONSUMPTION:US DATA VERSUS RIP MODEL

FIGURE 8. AGE-INEQUALITY PROFILE OF CONSUMPTION:US DATA VERSUS HIP MODEL


We next turn to the HIP model (with thesimplifying assumptions). The right panel ofFigure 9 plots the income path of an individualwhose income grows by a constant amount ev-ery period (dashed line).22 We consider twoextreme scenarios to illustrate the importance ofthe speed of learning for the rise in consumptioninequality. First, suppose that the individualknows exactly his true income path from thebeginning. Clearly, he will set his consumptiononce and for all and never adjust it over hislifetime (line with circles). As a result, in aneconomy populated by individuals with differ-ent income growth rates, consumption inequal-ity will be constant as the cohort ages, despitethe fact that income inequality is monotonicallyincreasing during the same time.

Consider now the opposite case: suppose thatthe individual starts life with the belief that hisincome growth rate is equal to the populationaverage of zero, but then never updates hisbeliefs as new income observations are realized(thus, Ej(Yt) � Es(Yt) for all s, j � t). In thiscase, it is easy to see from the expression in (13)that �ct � [Yt � Et�1(Yt)/(T � t � 1)] � (Yt �

Y0/(T � t � 1)). Because the individual does notupdate his beliefs, he interprets the deviations ofincome from his expectations as transitory dif-ferences and spreads them over his lifetime.Early in life, the deviation of actual incomefrom expected income (the numerator) is smalland the planning horizon (the denominator) islarge, so consumption is adjusted by an amountmuch smaller than the actual increase in in-come. Over time, the deviations become larger(since Yt is growing) while the horizon getsshorter, resulting in larger changes in consump-tion. The solid line in the right panel shows theconsumption path in this case. The interestingpoint to note is that consumption changes laterin life significantly exceed even the changes inincome. As a result, in an economy populatedby such individuals, consumption inequalitywould not only grow, but in fact would growmore than income inequality.

Although in our baseline HIP model withBayesian learning, individuals do update theirbeliefs, as discussed in the previous sectionoptimal learning turns out to be very slow, andthe present analysis shows that this slow learn-ing provides a powerful force that amplifiesconsumption inequality over time.

Measuring the Prior Uncertainty.—The pre-ceding discussion suggests one way to pin

22 The graphical intuition is more easily explained as-suming a constant “amount of growth,” rather than a con-stant “growth rate.” The basic argument, however, appliesequally well to the latter case.

FIGURE 9. INCOME AND CONSUMPTION MOVEMENTS IN THE RIP AND HIP MODELS


down the value of � from the data. To seethis, note that a higher � means less prioruncertainty, which in turn implies less profileuncertainty over the life cycle, and thus asmaller rise in consumption inequality. Giventhat the HIP model with � � 0 generates moreinequality than in the data, we could keepincreasing � to the point where the modelmatches its empirical counterpart (of 21 logpoints). This procedure yields �* � 0.62. InFigure 8, the middle line (marked with dia-monds) plots the implied age-inequality pro-file. Notice that although � was chosen tomatch the total rise in inequality, the overallshape of the resulting profile also provides anice fit to its empirical counterpart. We returnto this point in the next section.

The estimated value of � implies that 62percent of the variability in income growth ratesis forecastable by individuals at the time theyenter the labor market.23 While this estimateindicates a great deal of initial predictability inincome profiles, the remaining uncertainty isstill economically substantial: the standard de-viation of the prior belief about �i is ((1 �0.62) � 0.00038)1/2 � 0.012, compared to amean of 0.009, implying, for example, that anindividual who is one standard deviation abovethe mean would earn roughly 2.6 times moreincome at retirement age than another individ-ual who is one standard deviation below themean.

There are several issues related to the inter-pretation of the estimate of �. First, recall that inequation (2) we did not include an educationdummy in the common life-cycle profile g, soany variation in income growth rates betweeneducation groups is also captured in ��

2. As aresult, �* also contains any forecastability in �i

that is due to differences in education level,making it a useful summary measure relevant

for models that do not distinguish among in-dividuals based on education. However, it isalso of interest to know the degree of foreca-stability conditional on education, and alsoallow for the fact that forecastability mightvary by education.

To this end, we follow the same procedure asabove to estimate a separate � for each group.We first construct empirical age-inequality pro-files from the CEX for each group displayed inFigure 10. As can be seen here, the cross-sectional variance of log consumption increasesby 26 log points in the college sample (leftpanel), compared to 17 log points in the high-school sample (right panel). We then solve theHIP model for each education group, using the(HIP) income processes reported in rows 4 and6 of Table 1. Then, for each group, we choose aseparate � so that the rise in inequality in eachmodel matches its empirical counterpart. Thisprocedure yields �*C � 0.55 and �*H � 0.32,implying that the degree of forecastability ofincome profiles is increasing with education,partly offsetting the significantly higher profileheterogeneity among the college-educated group.Overall, however, the amount of prior uncer-tainty (that is, (1 � �)��

2) is increasing witheducation, which results in a larger rise in in-equality in the college sample. The resultingage-inequality profiles are plotted in Figure10 (lines marked with circles). For the collegesample, the model does a fairly good job ofcapturing the shape of the empirical profile,while for the high-school sample the empiricalprofile exhibits more convexity in the earlyparts of the life cycle compared to the model.

Finally, the simulated data for each educationgroup can be combined to provide an alternativeway to construct the age-inequality profile ofconsumption for the whole population, whichcan then be compared to its empirical counter-part previously analyzed in Figure 8. In Figure8 the line marked with circles displays the re-sult. Consumption inequality in the combineddata (overall sample) now rises by 20 logpoints—compared to 21 log points in thedata—and provides an arguably better fit to theshape of the empirical profile than the baselineHIP model (marked with diamonds). It is reas-suring that even though the estimated HIP pro-cesses differ in important ways across education

23 In a different context, Flavio Cunha, James J. Heck-man, and Salvador Navarro (2005) study students’ school-ing choice, in a complete markets setting, to infer theamount of earnings variability that is forecastable by thetime students decide to go to college. They estimate thatabout 60 percent of variability in returns to schooling isforecastable. Navarro (2004) extends this analysis by intro-ducing credit constraints and consumption choice, and findsthat schooling returns remain largely forecastable.


groups (and compared to the whole population),the implied consumption behavior is consistentwith the evidence for the overall population.

These results show that the HIP model easilygenerates a substantial rise in consumption in-equality. This is true even when a sizeable frac-tion of profile heterogeneity is forecastable. Ofcourse, the fact that the rise in inequality (Fig-ure 8) exactly matches that in the data does notprovide additional evidence in favor of the HIPmodel, since we explicitly chose � to match thistarget as part of our calibration strategy.

Can the HIP Model with � � 1 Provide aPlausible Benchmark?—Before closing thissection, it is useful to examine what happens towithin-cohort consumption inequality when� � 1, in which case there is no prior uncer-tainty and, therefore, no Bayesian learningabout income profiles. As a result, this versionof the model is substantially easier to solve, andit is of interest to know if such a simple modelcould provide a reasonable benchmark.

The dashed line in Figure 8 plots consump-tion inequality in this case, which increases byabout 14 log points over the life cycle. Althoughthis is still a sizeable increase in inequality—2⁄3of the rise in the US data—the reason is quitedifferent than before. In particular, when � � 1,the only source of uncertainty is the idiosyn-

cratic shock process, which does not provide astrong precautionary savings motive given itslow persistence, and cannot generate a large risein inequality on its own (see footnote 21). In-stead, the main reason for the rise in inequalityhere is the existence of borrowing constraints,which now bind very often, given the absenceof a strong precautionary savings motive. Be-cause these constrained individuals are not ableto smooth consumption effectively, consump-tion inequality rises together with income in-equality over the life cycle.

The described mechanism has several un-appealing implications, however. For exam-ple, without the uncertainty about incomeprofiles, individuals with a high �i save less(or borrow more) compared to those with alow �i (since the former have a larger fractionof their lifetime income in the future). Thisgives rise to several counterfactual results,such as a strong negative cross-sectional corre-lation between income and wealth, inconsistentwith empirical evidence.24 Furthermore, indi-

24 Specifically, the correlation between an individual’swealth and his �i starts from �0.88 at age 25, and while itgradually increases over time, it remains negative until age60, with an average value of �0.58. Similarly, the correla-tion of wealth with the level of the income profile, �i � �it,is also negative, averaging �0.40 over the life cycle. In

FIGURE 10. AGE-INEQUALITY PROFILE OF CONSUMPTION BY EDUCATION: US DATA AND GROUP-SPECIFIC HIP MODELS


viduals who are borrowing-constrained are pre-dominantly those who are income-rich and,therefore, have high consumption. In particu-lar, the average consumption of borrowing-constrained individuals is 107 percent higherthan the average of nonconstrained individualsover the life cycle.

We conclude from these results that uncer-tainty about income profiles is an essential com-ponent of the HIP model, which otherwisedelivers counterfactual implications for con-sumption and savings behavior. Therefore, inthe rest of the paper we focus on the baselineversion with profile uncertainty as calibratedabove.

B. The Nonconcavity of the Age-InequalityProfile of Consumption

A second feature of the age-inequality profilethat is apparent in Figure 1 is its nonconcaveshape, up to about age 55. We also plot 95percent confidence bands (obtained by boot-strap) around the graph, which helps to showthat this shape is fairly accurately determined.Nevertheless, based on this information alone, itstill seems hard to rule out a concave profile thatmight (perhaps barely) fit through the confi-dence bands. Thus, to tackle this question moredirectly, we characterized the sampling distri-bution of the curvature of the age-inequalityprofiles. We first drew 500 bootstrap sampleswith replacement and estimated the age-ine-quality profile from each sample. Then we fitteda second order polynomial to each profile fromage 25 to 55. The estimated coefficients on thequadratic term were all positive—ranging from2.6 � 10�4 to 3.2 � 10�3 with a mean value of1.44 � 10�3—confirming that the slightly con-vex shape of the empirical inequality profile isrobust to sampling uncertainty. It is now ofinterest to see whether each model is consistentwith this empirical fact.

Earlier studies have found it hard to reconcilethe RIP model with this evidence. For example,using the certainty equivalent version of the

permanent income model, Deaton and Paxson(1994) have shown that the inequality profilewill be concave if the income process has alarge persistent component. More recently,Storesletten et al. (2004a) have studied a moregeneral model with a rich set of realistic fea-tures and found concavity to be a robust impli-cation of the RIP model. The same result holdstrue in our baseline RIP model despite severaldifferences in our specifications. This concavitycan be seen in Figure 7, which plots the age-inequality profile from our baseline RIP model,as well as from Storesletten et al. (2004a) forcomparison.

In the HIP(�) model, instead, inequality risesin a slightly convex/linear fashion up to aboutage 55. The main driving force behind thisresult is Bayesian learning about income growthrates. The main intuition could partly be antic-ipated from the discussion in the previous sec-tion (and shown in Figure 9), which showed anincrease in the size of consumption adjustmentsas the cohort ages. We now show that this resultholds more generally—when individuals updatetheir beliefs optimally in a Bayesian fashion andr f � 0. The mechanism can be explained in thecertainty-equivalent version of the permanentincome model (i.e., assuming quadratic utilityand (1 � r f) � 1, and no retirement). In thiscase, optimal consumption choice implies

(14) �ct �1

�t�(1 �) �

s � 0

T � t

�s(Et Et � 1)Yt � s�,

where � � 1/(1 � r f), and �t � (1 � �T� t�1)is the annuitization factor. To simplify the prob-lem further, assume that income (instead of logincome) is a linear function of experience withi.i.d. innovations: Yt

i � �i � �it � �ti.25

First, consider the case where individualslearn about their intercept �i but know �i ex-actly. It can easily be shown that (14) reduces to�ct � �t�t � �t�1�t�1. Because the right-handside of this expression is shrinking with age dueto learning, the age-inequality profile will be

contrast, Budrıa Rodrıguez et al. (2002) calculate the cor-relation between individual income and wealth to be 0.47using data from the Survey of Consumer Finances.

25 These assumptions are rather innocuous in this con-text. See the working paper version for a justification of thisassertion.


unambiguously concave. Now consider the op-posite case where individuals learn about �i butknow �i with certainty. When an individualupdates his beliefs in period t, the revision inexpected future income is (Et � Et�1)Yt�s �(�t�t � �t�1�t�1)(t � s). Substituting this ex-pression into (14) and after performing somealgebra, one can show that

(15) �ct � ��

1 �� t (T � 1)�T � t�1

1 �T � t�1 �(16) � ��t�t �t � 1�t � 1�.

For a range of plausible values for r f and T, theterm in the square bracket is an approximatelylinear (and slightly convex) increasing functionof t. Moreover, recall—from the graph in theright panel of Figure 4—that the speed of learn-ing about �i increases over time so the absolutevalue of (�t�t � �t�1�t�1) is getting larger onaverage up to age 50. Therefore, consumptionchanges will be larger in absolute value as thecohort ages, implying a convex shape for theage-inequality profile. While in the baselineHIP model there is learning about both �i and�i , the former happens very quickly and thushas no significant effect on the shape. Instead,the shape is mainly determined by learningabout �i, which gives it the nonconcave form.

C. The Co-movement of Consumption andIncome over the Life Cycle

A third well-documented empirical finding isthat consumption growth parallels incomegrowth over the life cycle (Carroll and Sum-mers 1991). While at first glance this findingmay appear to contradict the consumption-smoothing motive inherent in permanent in-come theory, several authors have shown that itis possible to generate this pattern if individualsdisplay precautionary savings behavior (i.e., ifthey have CRRA preferences, for example) andif income shocks are extremely persistent as inthe RIP model. In this case, individuals reducetheir consumption early in life to build a bufferstock wealth for self-insurance. As they getolder, persistent shocks have fewer periods left

to affect income, effectively resulting in lessuncertainty. In response, individuals reducetheir savings rate, allowing consumption to risealong with income, generating the empiricalco-movement (Carroll 1992; Orazio P. Attana-sio et al. 1999).

Another finding documented by Carroll andSummers, however, poses a challenge to thisexplanation. In particular, these authors foundthat consumption also tracks income within ed-ucation groups: that is, college-educated indi-viduals have not only steeper income profiles,but also steeper consumption profiles than high-school-educated individuals. In Figure 2 wereplicate the same finding using data from ourCEX sample: consumption grows by 74 percentbetween ages 25 and 55 for the college group,compared to 36 percent for the high-schoolgroup.26

To explain this observation by precautionarysavings alone, as in the RIP model, would re-quire the former group to face income shocksthat are either more persistent or more volatilethan the latter group,27 neither of which weseem to find in the data. For example, the esti-mates reported in rows 3 and 5 of Table 1 reveallittle difference between the RIP processesfaced by each group, which is consistent withearlier studies in the literature (Hubbard et al.1995; Carroll and Samwick 1997). Using theseestimated processes, we solve the RIP modelseparately for each education group. The rightpanel in Figure 11 displays the consumptionprofiles of each group normalized to be one atage 25. It can be readily seen that consumptiongrowth over the life cycle is virtually thesame—43.4 percent versus 42.7 percent—forthe two groups, inconsistent with empiricalevidence.

We next turn to the HIP model. First, noticethat the estimates of the HIP process reveal an

26 Similarly, Jesus Fernandez-Villaverde and Krueger(2005) employ nonparametric methods and document thattotal consumption expenditures (as opposed to our use ofnondurable expenditures) increase by about 50 percent forthe college group and by about 10 percent for the high-school group.

27 Clearly this is because, without income shocks, bothgroups should have the same slope of the income profilesunless they differ systematically in some other respect.


important difference between the two groups(rows 4 and 6 of Table 1): while the persistenceand innovation variance are still similar, theheterogeneity in �i is dramatically different,with college graduates facing a much widerdispersion of income growth rates (��

2 �0.00049) than less educated individuals (��

2 �0.00020).28 To solve the HIP model for eachgroup, we also use the values of � that werecalibrated in the section on “Measuring thePrior Uncertainty” to match the rise in con-sumption inequality for each group: �*C � 0.55and �*H � 0.32. The left panel of Figure 11 plotsthe resulting average consumption profile, whichshows a 53 percent increase for the collegegroup compared to a 29 percent increase for thehigh-school group. Therefore, although the con-sumption growth of both groups is understatedcompared to the data, the HIP model does implya steeper consumption profile for individualswith higher education. Notice that this differ-ence would have been even larger if it were notfor the offsetting effect of higher forecastabilityfor the high-education group. Overall, this anal-ysis shows that a basic version of the HIP model

is consistent with differences in consumptionprofiles between education groups, while thebaseline RIP model is not. However, the RIPmodel can also be made consistent with thisfact by introducing two additional features—systematic differences between education groupsin their demographics characteristics and in un-observable dimensions such as their time dis-count factors—as shown by Attanasio et al.(1999).

IV. Conclusion

In this paper, we have studied the implica-tions of the HIP and RIP models for life-cycleconsumption behavior, and compared these im-plications to the US data to assess which type ofincome process implies more accurate empiricalpredictions. While the RIP model has been stud-ied extensively in the previous literature, thishas not been the case for the HIP model. Onecontribution of the present paper, therefore, is toprovide a detailed investigation of a life-cycleHIP model. An important finding from thisanalysis is that the uncertainty about incomegrowth rates is resolved very gradually, and wehighlighted two channels that were responsiblefor this slow learning. Moreover, because ofslow learning, consumption behavior over thelife cycle is informative about initial uncer-tainty, which we used to estimate the prior

28 Note that these different estimates of ��2 also imply

that within-cohort income inequality should rise more overtime among college workers compared to noncollege work-ers. As we document in Guvenen (2005, fig. 2), this isindeed the case in the US data.

FIGURE 11. AVERAGE LIFE-CYCLE PROFILE OF CONSUMPTION BY EDUCATION: HIP MODEL VERSUS RIP MODEL


information individuals have about their incomegrowth rate.

The HIP and RIP models display some sim-ilarities, as well as differences, regarding theirconsumption implications. For example, bothmodels are consistent with the substantial rise inconsumption inequality over the life cycle ob-served in the US data. However, while the shapeof this profile is also consistent with the data inthe HIP model, this is not true in the RIP model.Furthermore, the HIP model generates steeperconsumption profiles for college-educated indi-viduals compared to lower education individu-als as in the data, which again was not the casein the RIP model. As shown by Attanasio et al.(1999), one needs also assume systematic dif-ferences across education groups in preferencesand demographics to explain these facts in theRIP model.

Another conclusion that we draw is that ifindividuals face the HIP process without uncer-tainty about income growth rates, the resultingconsumption behavior is counterfactual: anindividual’s income and wealth become neg-atively correlated; borrowing-constrained in-dividuals are the income-rich and, as a result,the average consumption of constrained indi-viduals is twice that of unconstrained individu-als. Therefore, it is important not to misinterpretthe results of this paper as providing unqualifiedsupport to the HIP model. The HIP incomeprocess has plausible consumption implicationsonly when combined with uncertainty about in-come profiles that resolves slowly over time dueto learning.

An appealing feature of the present frame-work with slow learning is that it could providea setup for estimating � from a broad set ofeconomic actions of individuals over the lifecycle. For example, the discussion of equation(15) shows that the dynamic response of con-sumption to income shocks contains useful in-formation about profile uncertainty. Similarly,one could augment this model with other eco-nomic decisions, such as labor supply and/orportfolio choice, to bring a wide range of evi-dence to bear on the estimation of �. We intendto pursue these issues in future research.

APPENDIX: ESTIMATES OF THE HIP MODEL IN THE

LITERATURE

Table 4 presents the estimates of the HIPmodel from the US data in the previous liter-ature. As can be seen here, the estimates of ��

2

range from 0.00018 in Lillard and Weiss(1979) to 0.00041 in Haider (2001). Theformer paper estimates a separate income pro-cess for each finely defined occupation cate-gory (such as chemists, psychologists, etc.),which could be partly responsible for thesmaller estimate of profile heterogeneity. Allthe estimates of ��

2 are statistically signifi-cant, however, and the latter two papers’point estimates are rather close to each other.Baker also reports estimates as high as0.00082; his lowest estimate is 0.00031. Sec-ond, the persistence parameter in these studiesis around 0.6 to 0.7, indicating significantlylower persistence than a unit root.

TABLE 4—ALTERNATIVE ESTIMATES OF THE HIP MODEL

Source � ��2 ��

2 ��

Stochasticprocess

Time effects invariances?

Lillard and Weiss (1979) 0.707 0.0305 0.00018 0.00076 AR(1) � i.i.d. No(0.073) (0.0015) (0.00004) (0.0001)

Baker (1997) 0.674 0.139 0.00039 �0.004 ARMA(1, 2) Yes(0.050) (0.069) (0.00013) (0.003)

Haider (2001) 0.639 0.295 0.00041 �0.00827 ARMA(1, 1) Yes(0.077) (0.137) (0.00012) (0.0036)

Notes: Lillard and Weiss’s data are biannual from the National Science Foundation’s Register of Scientific and TechnicalPersonnel covering 1960–1970. The reported estimates are from table 7 of their paper, which has the most similarspecification to ours. Baker uses PSID data for 1967–1986, and this result is from table 4, row 6, which has the best overallfit. Haider’s data are also from PSID covering 1967–1992, and the results are from table 4.


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