Cyclical Dynamics in Idiosyncratic Labor Market Risk

Transcription

1 Cyclical Dynamics in Idiosyncratic Labor Market Risk Kjetil Storesletten University of Oslo and Centre for Economic Policy Researc Cris I. Telmer Carnegie Mellon University Amir Yaron University of Pennsylvania and National Bureau of Economic Researc Is individual labor income more risky in recessions? Tis is a difficult question to answer because existing panel data sets are so sort. To address tis problem, we develop a generalized metod of moments estimator tat conditions on te macroeconomic istory tat eac member of te panel as experienced. Variation in te cross-sectional variance between ouseolds wit differing macroeconomic istories allows us to incorporate business cycle information dating back to 1930, even toug our data do not begin until We implement tis estimator using ouseold-level labor earnings data from te Panel Study of Income Dynamics. We estimate tat idiosyncratic risk is (i) igly persistent, wit an annual autocorrelation coefficient of 0.95, and (ii) strongly countercyclical, wit a conditional standard Tis paper circulated previously as a component of te paper Asset Pricing wit Idiosyncratic Risk and Overlapping Generations. In addition to participants at seminars and conferences, we tank Dave Backus, Geert Bekaert, David Bowman, Martin Browning, Jon Cocrane, Wouter den Haan, Mick Devereux, Ron Gallant, René Garcia, Rick Green, Jon Heaton, Burton Hollifield, Mark Huggett, Narayana Kocerlakota, Per Krusell, Debora Lucas, Pierre Mella-Barral, Bob Miller, Micael Parkin, Victor Ríos, Amlan Roy, Tony Smit, Paul Willen, Steve Zeldes, Stan Zin, and tree anonymous referees for elpful comments and suggestions. We ave benefited from te support of National Science Foundation grant SES and te Rodney Wite Center at Warton. Valuable researc assistance was provided by Xiaorong Dong. [Journal of Political Economy, 2004, vol. 112, no. 3] 2004 by Te University of Cicago. All rigts reserved /2004/ $

2 696 journal of political economy deviation tat increases by 75 percent (from 0.12 to 0.21) as te macroeconomy moves from peak to troug. I. Introduction Te interaction between cross-sectional risk and aggregate risk plays an increasingly important role in macroeconomics and finance. For example, Bernanke and Gertler (1989), Carlstrom and Fuerst (1997), Cooley, Marimon, and Quadrini (2002), and oters argue tat financing constraints are an important propagation mecanism because tey generate asymmetric effects of aggregate socks on te te cross-sectional distribution of firms. Davis and Haltiwanger (1992), Caballero and Hammour (1994), Boeri (1996), Foote (1998), and oters examine cyclical dynamics in job creation and destruction. Rampini (2000) argues tat incentive constraints associated wit entrepreneurial activity give rise to countercyclical cross-sectional risk. Krusell and Smit (1999), Storesletten, Telmer, and Yaron (2001b), and Krebs (2002, 2003) ask if countercyclical idiosyncratic risk is important for te welfare costs of business cycles. Finally, and most closely related to tis paper, Lucas (1994), Heaton and Lucas (1996), Krusell and Smit (1997), Marcet and Singleton (1999), Balduzzi and Yao (2000), Alvarez and Jermann (2001), Lustig (2001), Storesletten et al. (2001a), Brav, Constantinides, and Geczy (2002), Cogley (2002), and Sarkissian (2003) follow Mankiw (1986) and Constantinides and Duffie (1996) and study models in wic assets wit iger expected returns do badly at times of iger crosssectional labor income risk. Te size of cyclical variation in te cross-sectional distribution of labor income risk is tus an important question. It is ard to answer, owever, because panel data sets are so sort. Even te Panel Study of Income Dynamics (PSID) te panel wit te longest time dimension covers only four or five business cycles. We address tis problem. We incorporate macroeconomic information dating back to 1930, in spite of te fact tat our panel data do not begin until To understand wat we do, consider two coorts of individuals, te first born in 1910 and te second in Everyone is subject to idiosyncratic labor market socks, some fraction of wic are igly persistent. Te conditional variance of tese socks is countercyclical, increasing during contractions and decreasing during expansions. Suppose tat we ave labor income data on eac coort wen its members are 60 years old. Tat is, we ave 1970 data on te 1910 coort and 1990 data on te 1930 coort. Wat we sall observe given te ig persistence and te countercyclical idiosyncratic risk is more cross-sectional dispersion among te 1910 coort tan among te 1930 coort. Te reason is tat te

3 cyclical dynamics 697 former worked troug more contractionary years tan te latter, including te Great Depression. More generally, we sall see variation in te cross-sectional variance between any two coorts of similar ages wo ave worked troug different macroeconomic istories. Tis is te essence of our procedure. Even toug te time dimension of our panel data is limited to , we ave a ric cross section of ages in eac year of te panel. We can terefore use macroeconomic data to caracterize te working istory of eac ouseold in te panel and ten use cross-sectional variation between coorts of similar ages to identify te cyclical idiosyncratic-risk effects we are after. More specifically, we focus on te properties of ouseold-level labor market earnings from te PSID. We model idiosyncratic risk as a class of ARMA(1,1) processes wit a regime-switcing component in te conditional variance. Te regime is identified by using macroeconomic data to classify eac year between 1930 and 1993 as eiter a contraction or an expansion. A critical feature is finiteness: we make strong assumptions about initial conditions tat allow us to base a generalized metod of moments (GMM) estimator on age-dependent moments. We find robust evidence tat idiosyncratic earnings risk is bot igly persistent and countercyclical. Estimates of annual autocorrelation range from 0.94 to Estimates of conditional standard deviations increase by rougly 75 percent (from 0.12 to 0.21) as te macroeconomy moves from expansion to contraction. We use grapical metods to make tese GMM estimates transparent. We sow tat ig autocorrelation is driven by a linearly increasing pattern of cross-sectional variance wit age. 1 Countercyclical volatility is driven by coort effects in te cross-sectional variance, someting tat is clearly evident in a simple scatter plot. Te remainder of te paper is organized as follows. Section II presents our time-series model, and Section III discusses our data. Section IV presents a grapical analysis tat sows tat most of our results can be understood visually. Section V presents GMM estimates of te model in Section II. Section VI reconciles our estimates of variation in conditional variance wit variation in te overall (age-pooled) cross section. Section VII presents conclusions. II. Time-Series Model for Idiosyncratic Risk We begin wit a time-series model for idiosyncratic labor earnings risk. Its most distinctive property wic will be empasized in estimation is tat it depends explicitly on age. We denote as te logaritm of Y t 1 Deaton and Paxson (1994) ave also documented tis evidence using a different data source, te Consumer Expenditure Survey.

4 698 journal of political economy per capita labor earnings at time t and y it as te logaritm of labor earnings for ouseold i of age at time t. Log earnings y it is decom- posed into two components: yit p g(x it, Y t) u it. (1) Te component g(x it, Y t) incorporates aggregate earnings as well as xit, deterministic components of ouseold-specific earnings attributable to age, education, and so on. 2 Te component u it is te random compo- nent of a ouseold s earnings tat is idiosyncratic to tem. It is identified by E(u, for all dates t, were itfy t, x it) { E(u t it) p 0 Et denotes te cross-sectional mean. Te specification of g(x it, Y t) is critical for identifying uit, but beyond tat, it is not central. We defer its discussion until Section V. For u it, we use an ARMA(1,1) wit a regime-switcing conditional variance: uit p ai z it e it, 1 z it p rz i,t 1 it, (2) were ai Niid(0, j a ), eit Niid(0, j e ), it Niid(0, j t ), 2 2 je if aggregate expansion at date t j t p { j2 if aggregate contraction at date t, C 0 and z it p 0. Te variable ai is a fixed effect : a sock received once at birt and ten retained trougout life (ence te absence of a time subscript). Te variables z it and eit are persistent and transitory socks, respectively (we make age explicit only wen te conditional distribution of a variable depends on it). Wat we mean by countercyclical volatility is tat jc 1 je. We do not impose tis a priori. We include fixed effects and transitory socks so tat we can better 2 measure te primary objects of interest, j t and r. For example, te way we sall identify tese parameters involves ow te cross-sectional variance of u it increases wit age. However, muc of te overall variation in u it is common to ouseolds of all ages. Were we to exclude fixed 2 effects ai, we would overstate te magnitude of jt and, terefore, overstate te amount of idiosyncratic risk tat is relevant for economic decision making. Along similar lines, te transitory socks e it are included to mitigate te extent to wic measurement error (and true transitory 2 socks) overstates our estimate of j t. 0 Te essence of our approac lies in te initial conditions: z it p 0 and 2 ai Niid(0, j a ). Tese are strong assumptions. Tey rule out, for example, time variation in te distribution of and dependence between 2 In treating education attainment deterministically, we are probably underestimating te overall risks agents face. However, adding scooling decisions is beyond te scope of tis paper. a i

5 cyclical dynamics 699 ai and, for a given i, subsequent realizations of z it and eit. Te benefits, owever, are substantial. Te initial conditions allow us to interpret equations (2) as a collection of finite processes and terefore allow us to condition on age. Te implications are most apparent in te crosssectional variances, for eac age, tat will underly our GMM estimator: j 2 2 it a e t j E t j C jp0 Var(u ) p j j r [I j (1 I )j ], (3) were Var denotes te cross-sectional variance, It p 1 if te economy is in an expansion at date t, and It p 0 oterwise. If FrF! 1, te summation term converges to te familiar j 2 /(1 r 2 ), were j 2 is a prob- 2 2 ability weigted average of j and. For finite, owever, E jc Var(u it) is increasing in, at a rate determined by te magnitude of r. Tis, in conjunction wit a (rougly) linearly increasing age profile of sample moments for Var(u it ), is wat drives te relatively large estimate of r we sall arrive at in Section V. Age dependence in equation (3) is also critical for ow we assess countercyclical volatility. Consider, for instance, te coort of 60-yearold workers in te first year of our panel, According to equation (3), te cross-sectional variance for tis coort involves indicator variables It j dating as far back as 1930 (we assume tat p 1 corresponds to a 23-year-old worker). By classifying eac year between 1930 and 1968 as eiter an expansion or a contraction, we can form istory-dependent cross-sectional moments based on (3) tat incorporate aggregate socks dating back to 1930, far beyond te confines of our panel. Doing so across all 26 years of our panel is wat identifies je and jc. To see tis, compare te 60-year-old workers in 1968 to te 60-year-old workers in Eac as worked for 38 years. However, tose in te 1968 coort ave worked during more contractions, including te Great Depression. Te process (2) predicts tat, if r is sufficiently large and jc 1 je, ten te cross-sectional variance of a particular coort will be increasing in te number of contractionary years tat its members worked troug. 3 Terefore, inequality among te 1968 coort sould exceed tat of te 1993 coort. Tis is exactly wat we document below, and (anecdotally speaking) it is wat drives our results on countercyclical volatility. In summary, we are primarily interested in te autocorrelation, r, and countercyclical volatility, jc 1 je. Wat will be pivotal for te former is ow sample analogues of Var(u ) increase in. Tis will also determine it 3 It is not enoug for r to simply be positive. Consider, e.g., two coorts of te same age in wic one worked troug many contractionary years early in life and te oter worked troug only one contraction, but late in life. If r is sufficiently small, ten te cross-sectional variance among te latter can be greater tan among te former.

6 700 journal of political economy te average of jcand je. Wat will be pivotal for te latter is variation in te cross-sectional variation between ouseolds of similar ages tat ave worked troug different macroeconomic istories. III. PSID Data We use annual PSID data, , defined at te ouseold level. We define earnings as wage receipts of all adult ouseold members, plus any transfers received suc as unemployment insurance, worker s compensation, transfers from nonouseold family members, and so on. Transfers are included because we wis to measure idiosyncratic risk net of te implicit insurance mecanisms tat tese payments often represent. Along similar lines, we study te ouseold as a single unit so as to abstract from socks tat are insured via intraouseold variation in labor force participation. We depart from te common practice of using a longitudinal panel: an equal number of time-series observations on a fixed cross section of ouseolds. A longitudinal panel necessarily features an average age tat increases wit time. Tis is problematic for our approac, wic empasizes restrictions between age and aggregate variation. For instance, were we to use a longitudinal panel for te years , a large fraction of te ouseold eads will ave been retired, or at least been in teir late earning years, during te last two of only five business cycles witnessed during A longitudinal panel will also feature a relatively small sample size and be subject to survivorsip bias in tat only te relatively stable ouseolds are likely to remain in te panel for all 26 years. We overcome tese issues by constructing a sequence of overlapping tree-year subpanels. For eac of te years , we construct a tree-year subpanel consisting of ouseolds tat reported strictly positive total ouseold earnings (inclusive of transfers) for te given year and te next two consecutive years in te survey. For example, our 1971 subpanel is essentially a longitudinal panel of 2,036 ouseolds over te years 1971, 1972, and Doing tis for all years results in a sequence of 24 overlapping subpanels. Te overall data structure contains more tan enoug time-series information to identify te parameters in equation (2), wile at te same time survivorsip bias is mitigated and te cross-sectional distribution of age is quite stable. Te mean and standard deviation of te average age in eac subpanel are 40.8 and 1.0, respectively. Te mean and te standard deviation (across panels) of te number of ouseolds are 2,449 and 322, respectively. 4 Specifically, average age would increase from 39 to 64 over te years

7 cyclical dynamics 701 Additional details, including a number of filters related to measurement error and demograpic stability, are outlined in te Appendix. IV. Grapical Analysis Most of our econometric results can be anticipated by simply looking at te data. Te important dimensions are variations in te cross-sectional variance according to age and according to time. Loosely speaking, variation across age is wat drives our estimate of autocorrelation, r, wereas variation across time, and ow it relates to age, is wat drives our estimates of jeand jc. To sow tis, we use a more reduced-form representation of te data a dummy variable regression similar to tat in Deaton and Paxson (1994) and ask wat it suggests about our model s parameters. Denoting te cross-sectional variances in equation (3) as j 2,t { Var(u it), we decompose teir variance into coort and age effects: 2 j,t p ac b e,t, (4) were c p t denotes a coort (i.e., birt year). 5 Te parameters acand b are coort and age effects, and e,t are residuals. Our model represents restrictions on te age/time/coort effects captured in te ac and b parameters. We impose and test tem in Section V. Here, we sow wat te unrestricted estimates imply about te parameters in (3). We begin wit age. Figure 1a plots te age effects, b, from te re- gression (4). 6 Te grap sows tat te variance among te young is substantial and tat it increases by a factor of 2.3 between ages 23 and 60. Moreover, te increase is approximately linear. Deaton and Paxson (1994) report similar results using data from te Consumer Expenditure Survey. Inspection of equation (3) suggests tat te initial dispersion 2 2 can identify ja je, te rate of increase can (loosely speaking) identify te average of je and jc, and te linear sape can identify r. Tis suggests a value of r very close to unity. Section V formalizes tis and sows tat te near unit root implications are robust to te incorporation of information not represented in te grap, most notably autocovariances. Figure 1b plots te coort coefficients, but in a different fasion. An implication of our model is tat, wen we control for age (as eq. [4] does), a coort tat as worked troug more contractions tan anoter 5 As is well known, an alternative decomposition is age and time effects. We coose coort effects because, as fig. 1b will sow, tey are closely associated wit te essence of our procedure. Moreover, Sec. VI deals wit time effects, albeit in a somewat different fasion. 6 We plot te variance instead of te standard deviation because it will clearly delineate te case of r p 1, wic implies linearity for te former but not for te latter. Te coefficients b are scaled so tat te grap passes troug te unconditional (across coorts) variance of agents of age 40.

8 702 journal of political economy Fig. 1. Cross-sectional variance of earnings, based on te idiosyncratic component ( u it from eq. [2]) of log labor earnings plus transfers from te PSID, a, Cross- sectional variance by age. Te variances control for coort effects by regressing coort age-specific cross-sectional variances on coort age dummy variables as in Deaton and Paxson (1994) and Storesletten et al. (in press). Te points in te grap are te age coefficients, rescaled to matc te level of variance at age 40. b, Macroeconomic istory vs. cross-sectional variance (coorts). Te panel plots te coort coefficients from te dummy variable regression underlying panel a against te fraction of contractionary years during wic te oldest members of tat coort were of working age. c, Macroeconomic istory vs. cross-sectional variance. Te panel plots te age-normalized cross-sectional variance of te persistent sock underlying eq. (5) against te fraction of contractionary years during wic members of tat age panel year worked troug. d, Cross-sectional moments by time. Te panel reports te linearly detrended cross-sectional mean of log income ( yit from eq. [1]) and te linearly detrended standard deviation of uit, pooled across all ages, for eac year Te standard deviation is additively scaled for grapical reasons. Te correlation coefficient between te two series is.74. Panel d is robust to (i) alternative metods of detrending te mean and (ii) using te coefficient of variation instead of te standard deviation. Furter details are given in Sec. III. sould, on average, exibit greater cross-sectional dispersion (as long as r is large and jc 1 je). Figure 1b sows tat tis is a caracteristic of our panel. Te coort coefficients a c are plotted against te fraction of years during te working life of eac coort tat were contractions (defined as te National Income and Product Accounts measure of gross domestic product growt being below average for te elders of eac coort). Altoug tere are a number of coorts tat worked troug

9 cyclical dynamics 703 relatively few contractions and neverteless exibit substantial crosssectional dispersion (i.e., te cluster of points in te soutwest corner of te grap), te positive relationsip is apparent. Te ordinary least squares slope coefficient is 0.94 wit a standard error of Tis relationsip between cross-sectional variance and macroeconomic istory is te main reason tat our conclusions regarding variation in j 2 t differ from tose of previous studies, in particular Heaton and Lucas (1996), wo found relatively small effects. In Figure 1c we move beyond te variance decomposition (4) and impose some structure from our model. Suppose tat r p 1. Ten te cross-sectional variances (3) imply j p j j ( n )j n j ,t a e,t E,t C j,t ja je p je f,t(j C j E), (5) were n,t is te number of contractions tat ouseolds of age in panel year t ave worked troug, and f,t expresses n,t as a fraction of 2 2 age. We set ja je p 0.3, using te intercept from figure 1a. Figure 1 2 1c is a scatter plot of sample moments of (j,t 0.3) versus f,t. Altoug te data are obviously muc noisier tan tose in figure 1b we are now essentially plotting te raw data from te regression (4) a positive relationsip is apparent. Te ordinary least squares slope coefficient is 0.023, wic, according to equation (5), is an estimate of 2 2 countercyclical volatility (j C j E). Tis is close to te value tat we sall estimate in Section V. Figure 1d provides one last piece of evidence on countercyclical volatility. It pools te age effects igligted in figure 1a and plots te year-by-year pooled cross-sectional standard deviation in te PSID. It also plots te cross-sectional mean. Bot te mean and te standard deviation are linearly detrended. Even at tis informal level we see striking evidence of countercyclical volatility. Te correlation between te mean and te standard deviation is.74. Te magnitude of te canges, owever, must be interpreted wit caution. In Section VI we examine te pooled cross-sectional variance in more detail. We sow tat, because it is a close cousin of te unconditional variance, it will always understate (time) variation in wat we are ultimately interested 2 in: te conditional variance. j t V. Estimation Recall tat equation (1) decomposed log earnings into two components, y p g(x, Y ) u, were g(x, Y ) captures deterministic cross-sectional it it t it it t

10 704 journal of political economy TABLE 1 Summary Statistics of Panels Constant Age Age 2 Age 3 Education Family Size Estimate Standard error Note. Entries are estimates of te parameters of te first-stage regression (eq. [6]). Te regression also includes year dummy variables, wic are not reported ere. Te regression R 2 is.23. variation xit and aggregate variation Yt, and uit captures idiosyncratic cross-sectional variation. We specify te first component as l l g(x it, Y t) p v0 v1 D(Y t) v2 x it, (6) were D(Y t) is a vector of year dummy variables, t p 1968,, 1993, and x it is a vector composed of age, age squared, age cubed, and ed- ucational attainment for ouseold i of age at date t. Educational attainment is measured as te number of scool years completed by te ouseold ead. Table 1 reports estimates of v. Te estimates and fit are quite similar to tose in existing studies, implying a concave earnings function in age and education (e.g., Hubbard, Skinner, and Zeldes 1994). Equation (6) allows us to identify te idiosyncratic component of income, u it, wic follows te process (2). We estimate its parameters using GMM. For te sake of transparency, we begin wit an exactly identified system tat corresponds closely wit te grapical analysis of te previous section. Afterward, we add overidentifying restrictions. Te cross-sectional variances, equations (3), represent one variance for eac age/time pair, (, t): [ ] j 2 2 t it a e t j E t j C jp0 Ẽ (u ) (j j ) r [I j (1 I )j ] p 0 (7) for all, t, were runs from 23 to 60 and t runs from 1968 to We do not ave sufficient data to formulate sample moments for eac, t pair. We terefore form age cells of widt tree years and interpret te age of a ouseold in a particular cell as te midpoint. Furtermore, in order to mimic te grap in figure 1a (i.e., one variance per age), we aggregate te moments (7) over time and form sample analogues: T Nt 1 { } j 2 2 (u it) (j a j e ) r [I t jje (1 I t j)j C] p 0, (8) T tp1 N ip1 jp0 t were N t is te sample size of ouseolds aged at time t, and is now to be interpreted as an age cell. Row A of table 2 reports exactly identified estimates of r, j, j, and C E

11 cyclical dynamics 705 TABLE 2 Idiosyncratic Earnings Process: Parameter Estimates A. Exactly identified (no autocovariance).963 (.053).162 (.062) B. Exactly identified (wit autocovariances) (.058) (.034) C. Overidentified, H # T (.020) (.034) D. Overidentified, NBER indicators (.041) E. Overidentified, unemployment indicators (.011) (.026) F. Overidentified, CRSP value-weigted return (.037) (.031) indicators r j C j E j e j a p-value.088 (.048).094 (.042).125 (.044).119 (.022).138 (.034).084 (.019).562 (.091).351 (.063).255 (.021).255 (.025).257 (.020).235 (.011).448 (.094).378 (.057).386 (.064).308 (.062).423 (.019) 2 Note. Entries are GMM estimates of te process (2), based on PSID earnings data Te parameters j E and 2 j C denote te conditional standard deviation of te persistent component, conditional on an aggregate expansion, E, or contraction, C. Estimates in rows A, B, and C define expansion/contraction in terms of te growt rate of U.S. GNP being above/below its mean, tose in row D are based on te NBER definitions, tose in row E are based on U.S. unemployment data, and tose in row F are based on equity index returns. Row A uses moments (7) wit ages 25, 35, 45, and 55. Row B adds one more parameter and one more moment, te first autocovariance, eq. (10) for age 40. Rows C F use an overidentified system by using eqq. (7) (9) in addition to second-order autocovariances. Standard errors are computed using te Wite (1980) estimator and incorporate sampling uncertainty from te first-stage regression, eq. (1) (reported in table 1). Te p-value pertains to te overidentifying test of te moment conditions. Furter details, including all data sources, are available in te Appendix ja je based on equations (8), using ages 25, 35, 45, and 55. Since cross-sectional variances alone cannot distinguis jeand ja, we report teir sum. Te indicator variables I t are identified by classifying eac year, , as an expansion or contraction according to weter te growt rate in U.S. gross national product per capita was above or below its sample mean (alternative definitions are reported later). Te Appendix provides specific details of all GMM estimators, including ow we construct te weigting matrix to incorporate differing sample sizes N t, te overlapping structure of our panel, te first-stage regression, and several oter issues. Te results in table 2 accord well wit te graps in figure 1. Muc of te overall level of dispersion gets attributed to some combination of fixed effects, ja, and transitory socks/measurement error, je. We estimate tat ja je p 0.56, wic corresponds closely to te intercept 2 of 0.55 p 0.30 from figure 1a. For te persistent sock z it, we estimate an autocorrelation of r p 0.963, sligtly lower tan wat we inferred grapically. For te conditional volatilities, we estimate jc p (con- traction) and je p (expansion), wic, on average, are consistent wit te size of te increase in variance in figure 1a. 7 Viewed on teir 7 Specifically, te conditional variance in fig. 1a increases by 0.4 between ages 25 and 55. Wit a omoskedastic unit root process for te persistent sock, tis implies a conditional variance of 0.4/(55 25) p 0.013, or a standard deviation of Te average of our point estimates, j and j, is ( )/2 p p C E

12 706 journal of political economy own, te estimates indicate a substantial amount of countercyclical volatility: an increase of 84 percent from peak to troug. Next, we identify jafrom jeusing cross-sectional autocovariances anal- ogous to te variances (7): [ ] (j 1) 2 2 t it it 1 a t j E t j C jp1 Ẽuu j r r [I j (1 I )j ] p 0 (9) for all, t. Again, we aggregate tese moments over t for eac age cell and ten form sample counterparts analogous to equations (8). Row B of table 2 reports exactly identified estimates obtained by adding just one moment from equations (9) to tose underlying row A. We use p 40, but te results are not sensitive to te particular age. Te estimates of primary interest, r, jc, and je, cange only marginally relative to row A. For te oter parameters, te majority of te previously combined variance gets attributed to fixed effects: ja p 0.44 and je p 0.35 (tese add up to rougly te same variance as in row A). Te estimates in rows A and B of table 2 are transparent in te sense tat, given te grapical evidence of Section IV, it is clear wic features of te data are driving tem. Te remainder of te table examines ow robust tese findings are once we add overidentifying restrictions and climb inside te usual black box of GMM. We use sample analogues of te (, t) pair variances and first-order autocovariances in equations (7) and (9), along wit a set of second-order autocovariances analogous to (9). Tat is, we eliminate te time aggregation in equations (8) (and rows A and B) and just use te primitive moments. 8 Row C of table 2 uses te same definitions of expansion and contraction as rows A and B (based on GNP growt), wereas rows D, E, and F use definitions based on NBER business cycle dates, te unemployment rate, and te stock market. We see sligtly lower autocorrelation and, wit te exception of row F, iger conditional volatility for te persistent sock. Te iger conditional volatility comes at te expense of te transitory and fixed-effect socks, were te combined volatility drops from rougly 0.32 to te neigborood of Apparently, te additional autocovariances ave reclassified an important part of te variance from transitory/measurement error socks into persistent socks. Finally, te extent to wic te persistent sock conditional variance is countercyclical is relatively stable, ranging from a 70 percent increase (expansion to contraction) in rows C and D to a 90 percent increase in row F. 8 Specifically, rows C F use variances and first- and second-order autocovariances for ages 25, 35, 45, and 55, over te years Te ages were cosen to ensure a sample size of at least 100 ouseolds per (, t) pair. Tis makes for 312 moments and 307 degrees of freedom.

13 cyclical dynamics 707 Fig. 2. Cross-sectional variance of consumption, based on te idiosyncratic component ( u it from eq. [2]) of log consumption from te PSID, , except te years 1972, 1987, and 1988, in wic no consumption data are available. a, Cross-sectional variance by age. b, Macroeconomic istory vs. cross-sectional variance (coorts). c, Macroeconomic istory vs. cross-sectional variance. d, Cross-sectional moments by time. See also te legend to fig. 1. In summary, muc of wat we can infer grapically appears robust to te inclusion of information tat is not represented in te grap. In particular, te age pattern in te cross-sectional variances (fig. 1a) suggests near unit root beavior, and tis is robust to te inclusion of autocovariances, te moments typically used to identify persistence. A. Consumption Data To tis point we ave focused on labor earnings. For many questions, most notably asset pricing, we are equally interested in consumption and ow its cross-sectional distribution is related to aggregate variation. Figure 2 displays plots analogous to tose in figure 1 using data on food expenditure from te PSID (te only consumption data available). Using tese consumption data, table 3 replicates te estimation in table 2. Many of te patterns displayed in figure 1 for earnings seem to also appear for consumption. Figure 2a displays an almost linear rise in

14 708 journal of political economy TABLE 3 Idiosyncratic Consumption Process: Parameter Estimates A. Exactly identified (no autocovariance).934 (.042) B. Exactly identified (wit.911 autocovariances) (.037) C. Overidentified, H # T.862 (.015) D. Overidentified, NBER.893 indicators (.021) E. Overidentified, unemployment.909 indicators (.051) F. Overidentified, CRSP.913 value-weigted return (.050) indicators r j C j E j e j a p-value.126 (.047).160 (.058).222 (.059).172 (.073).155 (.028).127 (.016).089 (.038).115 (.042).172 (.043).133 (.071).121 (.029).095 (.023).382 (.061).278 (.058).283 (.040).288 (.043).263 (.034).295 (.005).269 (.053).267 (.013).246 (.013).289 (.050).303 (.057) Note. Entries are GMM estimates, based on food consumption data from te PSID. We define a first-stage regression as in (1) except tat te dependent variable is now log food consumption. We ten employ a metodology identical to tat used in table 2, were we treat te residual as te idiosyncratic consumption process. Te PSID does not report consumption data for te years 1972, 1987, and consumption dispersion. Tis leads to large autocorrelation estimates, altoug tey are lower, ranging from 0.86 to Consistent wit te plots in figure 2, estimates of te fixed-effect variance are also lower, someting we migt expect given tat we are limited to food data (i.e., overall inequality in earnings is likely to exceed tat of food consumption). Finally, estimates of te persistent sock conditional variance still indicate countercyclical volatility. Furter, consistent wit figures 2b and c, tese estimates are lower on average, and te difference between expansion and contraction is muted, wit te peak to troug increase in te neigborood of 30 percent instead of 75 percent. Furter, te estimates are more sensitive to te definition of expansion/contraction. Te extent to wic tese estimates are indicative of te beavior of a broader measure of consumption, obviously, remains to be seen. VI. Temporal Variation in te Cross-Sectional Variance Figure 1d indicates tat, over te period , te largest cange in te age-pooled cross-sectional standard deviation of log earnings was about 10 percent. Similarly, te largest cange from one year to te next was 4.4 percent. At first blus, tis may seem inconsistent wit our estimates, wic indicate a cange of rougly 75 percent over te course of te business cycle. Te missing link is tat te standard deviation in figure 1d is closely associated wit te unconditional cross-sectional distribution of earnings, wereas te estimates in table 2 are associated wit te conditional distribution. In tis section we ask weter te two

15 cyclical dynamics 709 can be quantitatively reconciled, tereby providing anoter corroborative ceck on our estimates. Te pooled cross-sectional variance in figure 1d is a weigted average 2 2 of te conditional variances je and jc. Tis is true in two senses. First, equation (3) indicates tat, in te usual AR(1) sense, te cross-sectional variance among agents of a given age is a moving average of te variances of past innovations. Te only wrinkle is tat te terms in te moving average cange over time, depending on te macroeconomic istory tat te coort as lived troug. Second, te pooled variance is a weigted average of te coort-specific variances, were te weigts are te population sares. Algebraically, { } H j 2 2 t it a e t j E t j C p1 jp0 v { Var(u ) p J j j r [I j (1 I )j ], (10) were J are population sares, and vt denotes te pooled cross-sectional variance. Inspection of equation (10) indicates tat v t is a moving average tat varies between an upper and a lower bound. Given tat jc 1 je, te upper bound coincides wit a long sequence of aggregate contractions (i.e., I p 0 for all 0! H) and is equal to j 2 j 2 [j 2 /(1 r 2 t a e C )] for large H and r! 1. Similarly, te lower bound coincides wit a long sequence of expansions and is equal to ja je [j E /(1 r )]. Te way in wic v t varies between tese extremes depends on persistence in bot aggregate and idiosyncratic socks and te economy s demograpic structure (te J s). Figure 3 graps tis for one particular realization of te process (2), using U.S. data for te J s. It plainly illustrates te main point of tis section, tat large canges in te variance of te conditional distribution are necessarily associated wit smaller canges in te pooled cross section. We use tis fact as anoter ceck on te plausibility of our estimates. We conduct a Monte Carlo experiment in wic we obtain a large number of replications of te process (2). Eac replication is a panel wit a time dimension matcing our PSID data (26 years) and a large number of individuals i in te cross section. We define two test statistics. Te first is based on te difference between te maximum and te minimum values of v t, wic are observed in any given replication: max ( v ) min ( t v t). (11) Te second is based on te maximal absolute cange between any two adjacent years: max (F v v F). (12) t We use te parameter estimates in row C of table 2, wit aggregate t 1

16 710 journal of political economy Fig. 3. Simulated cross-sectional standard deviation. Te solid line is one particular realization of te (population) pooled cross-sectional standard deviation of income sown in eq. (10), based on te process (2) from Sec. II. Parameter estimates are taken from row C of table 2. Te upper line is te limit point, sould all idiosyncratic socks be drawn from te ig-variance (contractionary) distribution. Te lower line is te analogous lower limit point. Furter details are provided in tis section socks following te transition matrix [ 3 3; 3 3]. We obtain te following results for te two test statistics. For te first, based on equation (11), te fraction of cases in wic te maximal cange across te 26 years exceeds te 10 percent value from figure 1d is 28 percent. For te second, based on equation (12), te fraction of cases in wic te cange from one year to te next exceeds 3 percent (3.5 percent) is 5 percent (! 1 percent). In summary, te estimated large canges in te conditional variance reported in table 2 are quantitatively consistent wit te relatively small canges we see in te pooled cross section, figure 1d. Tis serves to put te validity of our estimates on firmer ground, especially to te extent tat variation in te pooled cross section is more easily and accurately measured. VII. Last Tougts Our main finding is tat PSID data on idiosyncratic labor earnings risk exibit robust evidence of ig persistence and countercyclical volatility.

17 cyclical dynamics 711 We now conclude by comparing our estimates to tose of related work and offering some economic interpretation. We estimate an autocorrelation of rougly Many oter papers also advocate ig persistence. MaCurdy (1982), Abowd and Card (1989), Carroll (1997), and Gourincas and Parker (2002) prominent papers on labor income dynamics restrict labor income to be a unit root process. Hubbard et al. (1994) do not impose a unit root, but instead estimate an autocorrelation of An exception is te paper by Heaton and Lucas (1996), wo obtain estimates closer to Wat distinguises our work is an empasis on age and its relation to crosssectional variance. Te linearly increasing pattern suggests near unit root beavior. Tis is robust to te incorporation of wat most oter studies ave empasized: autocovariances. Our findings, terefore, represent new evidence in support of te Hubbard et al. estimates as well as te restrictions imposed by te oter papers. At face value, our estimates of transitory socks seem large. We get an estimated standard deviation of rougly 0.25, wic, in 2002 dollars, translates to an annual standard deviation of $11,500 for te average worker in our panel, wo earns just over $45,000. Wile we cannot say ow muc of tis represents measurement error and ow muc represents actual socks, we can say tat it is comparable to related work. Gourincas and Parker (2002) and Hubbard et al. (1994), for example, obtain 0.21 and 0.17, respectively. Our estimates are somewat iger, te most likely reason being additional measurement error in our relatively broad selection criterion. We estimate te standard deviation of fixed effects to be rougly Te average young worker earns just over $21,000 (in 2002 dollars), so tis translates into inequality among te young wit a standard deviation of rougly $8,800. Tat tis is plausible is apparent in figure 1a, were te standard deviation among te young is 0.55 (or $12,480). We get a lower number ere because (a) te transitory and persistent socks also contribute to te initial variance (see eq. [3] wit p 1), and (b) te overidentifying moments in rows C and D of table 2 generate a reduction in overall variability, as te GMM estimator gives weigt to moments not represented in figure 1a. Our estimates of te conditional standard deviations of te persistent sock are rougly 0.21 in a contraction and 0.12 in an expansion. Te frequency-weigted average is Because te autocorrelation is almost unity, tese are approximate estimates of te standard deviations 9 Te main differences between our approac and Heaton and Lucas s is tat tey do not condition on age as we do, and tey estimate one intercept term per ouseold, wereas we make te stronger assumption tat fixed effects are drawn from a singleparameter distribution. Tey also use te more traditional longitudinal panel, wereas we use te sequence of overlapping panels described in Sec. III.

18 712 journal of political economy of labor income growt rates. Equivalently, tey translate into contraction/expansion conditional standard deviations of $9,500 and $5,300, based on te average worker s 2002 dollar earnings of $45,000. By any measure, ten, idiosyncratic risk is large. Related work reaces similar conclusions (relative to our average). Gourincas and Parker (2002) estimate 0.17 for ig scool graduates and 0.15 for teir overall sample. Carroll and Samwick (1997) obtain almost te same numbers. Hubbard et al. (1994) obtain 0.16 for ig scool graduates and 0.14 for college graduates. Our estimates, terefore, agree wit related work, on average, and also suggest tat cyclical variation is an important part of te story. Finally, ow large are te socks relative to eac oter? We present a variance decomposition based on total lifetime uncertainty. Consider an unborn individual wo is to receive lifetime labor income according to equation (1). Using a constant discount factor (for simplicity), we compute te variance of te present value of teir future income (in levels, not logs): 62 [ it ] p23 Var 1.04 exp (y ). (13) We compute tis variance by simulation. 10 We do so tree times, first wit only te fixed-effect socks set to zero, second wit only te transitory socks set to zero, and tird wit only te persistent socks set to zero. We ten compute te ratio of eac of tese tree variances to te total variance in equation (13). We find tat te fixed effects contribute 54.8 percent to te total, te persistent socks contribute 44.5 percent, and te transitory socks contribute 0.7 percent. Te size of te fixed effects, ten, is only sligtly larger tan tat of te (relatively low conditional variance) persistent socks, once te total working life is accounted for. Appendix Additional details regarding te data selection procedure, specific caracteristics of te data, and te estimation procedure based on equations (1) and (7) are as follows. Selection Criteria A ouseold is selected into a panel if te following conditions are met: (i) te ead of te ouseold is no younger tan 22 and not older tan 60, (ii) total 10 To simulate, we use te parametric specifications in eqq. (1), (2), and (6), te parameter estimates in tables 1 and 2 (row C), and a two-state Markov cain wit transition matrix [ ; ] for te process expansion/contraction needed to compute te indicator variables in eqq. (2).

19 cyclical dynamics 713 earnings are positive in eac of te sample years, (iii) total earnings growt rates are no larger tan 20 and no lower tan 1/20 in any consecutive years, and (iv) te ouseold was not part of te Survey of Economic Opportunity. Individual Data Te individual earnings data are based on te Family Files of te PSID. We use te PSID s Individual Files to track individuals across te different years of te Family Files. Te definition of earnings includes wage earnings by te ead of te ouseold plus female wage earnings plus total transfers to te ouseold. Total transfers include unemployment insurance, worker s compensation, transfers by nonouseold family members, and several additional (minor) categories. Total earnings are ten deflated to common 1968 dollars using te consumer price index. Earnings are ten converted to rates per ouseold member by dividing total earnings by family size. Consumption data are based on total food consumption (te sum of expenditures of food for meals at ome, meals outside te ome, and te value of food stamps used). Tese questions were not available for panel years 1973, 1988, and Tree-Year Repeated Panels Agents are selected into a panel if tey meet te selection criteria above for te two years following te base year of te panel. We start wit te 1969 PSID file (and terefore follow agents troug PSID files from 1970 and 1971), wic we denote panel 1968, since eac PSID file provides information on te previous year s income. Our last tree-year panel, wic we denote as te 1991 panel, starts wit te 1990 PSID file and ends wit te 1994 PSID file. Hence, we end up wit 24 tree-year repeated panels. Table A1 reports summary statistics of te tree-year repeated panels we use in our estimations. Aggregate Data Te aggregate GNP levels used to define te indicator functions described in (7) were derived by merging GNP data from Gordon (1986) for te years wit Citibase data for te years Tese levels were converted to 1968 dollars using te CPI and to per capita terms by dividing by total population figures from Citibase. Te NBER-based business cycles are derived from te montly NBER definitions of contractions and expansions. We converted tese montly definitions into yearly ones. Our criterion, corresponding to te NBER definition, is to define years of recession as tose years for wic te majority of te monts are contractionary. For cases in wic contractions spanned less tan 12 monts but more tan six monts over two calendar years, we designated te first year as a recession. Our results are robust to close alternatives. Te unemployment definition of cycles is based on annual unemployment rates. Tis is constructed from two sources. For te years we used Historical Statistics of te United States (p. 135). For , we used te Economic Report of te President (February 1999, p. 376). To convert tese rates into contractionary years, we used information about te levels of unemployment as well as direction of canges. Recessions are defined as years for

20 714 journal of political economy Year TABLE A1 Summary Statistics of Panels Number of Houseolds Mean Age of Head Houseold Standard Deviation , , , , , , , , , , , , , , , , , , , , , , , , Note. Te entries correspond to tree-year panels. Te entries under te year column correspond to te base year of eac panel. Te age statistics correspond to te ead of te ouseold. wic te unemployment rate was greater tan 7.5 percent. If te rate was greater tan 7.5 percent but fell more tan 3 percent relative to te previous year, it was not counted as a recession year. In addition, years for wic te unemployment rate ad risen by more tan 3 percent were defined as recession years. Te returns definition is based on te real (CPI-deflated) value-weigted return from te Center for Researc in Security Prices (CRSP). We define contractionary years to be ones in wic te real return is negative. Estimation Procedure Our estimation procedure as two distinct steps. In te first stage we estimate equation (1). In te second stage we estimate te system given in (7). To recover uit in te first stage, we need to specify g(y t, x it). We let g(7) be represented by a simple linear regression using as regressors year dummies (corresponding to Y t ) and age, age squared, age cubed, and education (cor responding to individual-specific attributes x it ). Let te parameters of tis first- stage regression be summarized by v. Tat is, w (y, Y, x, v ) { y v [1, Y, 1 1 i,t t it 1 it 1 t

21 cyclical dynamics 715 x. All te coefficients are significant, and te 2 it] R is.23. Tese are displayed in table 1. Next, let te parameters of te system in (7) be denoted by v2 (i.e., r, je, j, j, and j ). Te joint system we estimate can be written compactly as C e a w 1(y it, Y t, x it, v 1) [ w (y, Y, v, v )] 2 it t 1 2 E p 0, (A1) were w1 and w2 are te moment conditions corresponding to (1) and (7), respectively. Te triangular structure of te moment condition allows us to get consistent estimates of v1 using only w1. Te weigting matrix is trivially te identity since tis is always an exactly identified system. We ten estimate v 2 using moment conditions w 2. Tis second step incorporates te standard errors in estimating v 1 using te standard two-step GMM procedure as in Ogaki (1993). Tere are two additional complications tat arise in our setup: te overlapping structure of our repeated panels wen we use autocovariances and te nonbalanced panel. For eac moment condition based on, say, panel year t, an MA(2) correction is added to te estimate of te covariance matrix associated wit moment conditions w 2. Specifically, define te empirical residuals of te moment conditions to be w 2 ( { }) 1 1,0,t j 2 2 2,i,t i,t a e t j C t j E jp0 w { (u ) j j r [I j (1 I )j ], ( { }) ( { 1 jp1 }),1,t 1,t 2 2(j 1) 2 2 w 2,i,t{ ui,tui,t 1 ja r r [I t jjc (1 I t j)j E], jp1,2,t 2,t 2 2 2(j 2) 2 2 w 2,i,t{ ui,tui,t 2 ja r r [I t jjc (1 I t j)j E], were te superscript t in u denotes te base year of te panel from wic tis agent is selected. Te superscript j denotes weter te moment is based on variances ( j p 0) or first and second autocovariances ( j p 1, 2). By assumption,,j,l w2,i,t is not correlated wit w2,i,t k for all k ( 0 and j, l p 0, 1, 2. It can be easily sown tat, because of te overlap of te sample, for eac t, w,0,,1 2,i,t w2,i,t, and,2 w 2,i,t are correlated. We stack te repeated tree moment conditions and use sample counterparts to estimate tese covariance terms; te covariance matrix is block-diagonal, and eac 3 # 3 block as nonempty off-diagonal elements. In practice, our results do not seem to be sensitive to tis correction. Because of te block-diagonal structure described above, te nonbalanced panel allows for te construction of standard GMM statistics. Let N be te minimum sample size across all moments, and let oter sample moments, say moment j, be of size kj times N. Te standard GMM asymptotics follow from assuming tat sample sizes grow in tese ratios. Te covariance matrix used is ten constructed using eac sample moment scaled by its k j. References Abowd, Jon M., and David Card On te Covariance Structure of Earnings and Hours Canges. Econometrica 57 (Marc):