Life-Cycle Variation in the Association between Current and Lifetime Earnings. Steven Haider, Michigan State University

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1 Lfe-Cycle Varaon n he Assocaon beween Curren and Lfeme Earnngs Seven Hader, Mchgan Sae Unversy Gary Solon, Unversy of Mchgan Sepember 2005 The auhors graefully acknowledge gran suppor from he Naonal Insue on Agng (2-P01 AG 10179). They also are graeful for advce from he edor, he referees, John Bound, Charle Brown, Ar Goldberger, Nahan Grawe, Jacob Klerman, Lug Psaferr, Mahew Shapro, Mel Sephens, Bob Wlls, Jeff Wooldrdge, and semnar parcpans a he Unversy of Mchgan, Amercan Unversy, he Unversy of Calforna a Berkeley, he Unversy of Calforna a Davs, Harvard s Kennedy School, he Naonal Bureau of Economc Research, Oho Sae Unversy, Penn s Wharon School, he Socey of Labor Economss, he Unversy of Torono, and Wesern Mchgan Unversy.

2 Lfe-Cycle Varaon n he Assocaon beween Curren and Lfeme Earnngs Absrac Researchers n a varey of mporan economc leraures have assumed ha curren ncome varables as proxes for lfeme ncome varables follow he exbook errors-nvarables model. In an analyss of Socal Secury records conanng nearly career-long earnngs hsores for he Healh and Reremen Sudy sample, we fnd ha he relaonshp beween curren and lfeme earnngs depars subsanally from he exbook model n ways ha vary sysemacally over he lfe cycle. Our resuls can enable more approprae analyss of and correcon for errors-n-varables bas n a wde range of research ha uses curren earnngs o proxy for lfeme earnngs. (JEL D31, J30)

3 Lfe-Cycle Varaon n he Assocaon beween Curren and Lfeme Earnngs I. Inroducon In he year 2003 alone, he Amercan Economc Revew s refereed ssues conaned 14 arcles reporng regresson analyses nvolvng ndvdual or famly ncome varables, and he May Proceedngs ssue conaned almos ha many agan. In some cases, he ncome varables were dependen varables; n ohers, hey were regressors used o explan dependen varables rangng from chld healh n he Uned Saes o borrowng and lendng behavor n Ghana. Whou excepon, he measured ncome varables were shor-erm values even hough, n mos cases, appeared ha he relevan economc consruc was a longer-erm value. Many nfluenal economc sudes have recognzed ha he use of curren ncome as a proxy for long-run ncome can generae mporan errors-n-varables bases. Perhaps he mos famous examples are he semnal sudes by Modglan and Brumberg (1954) and Fredman (1957), whch analyzed he properes of consumpon funcons esmaed wh curren raher han permanen ncome varables as he regressors. Anoher nsance s he leraure (e.g., Lllard, 1977) suggesng ha nequaly as measured n cross-secons of annual earnngs oversaes he nequaly n lfeme earnngs. A recen offshoo of ha leraure exemplfed by Goschalk and Moff (1994), Hader (2001), and Baker and Solon (2003) has aemped o paron he upward rend n earnngs nequaly no perssen and ransory componens. Sll anoher recen example s he burgeonng leraure on nergeneraonal ncome mobly (surveyed n Solon, 1999), whch has found ha he assocaon beween parens and chldren s long-run ncome s

4 suscepble o dramac underesmaon when curren ncome varables are used as proxes for long-run ncome. Neverheless, appled researchers ofen gnore he dsncon beween curren and long-run ncome. Mos researchers who do aend o he ssue assume he exbook errors-n-varables model and mpue he nose-o-sgnal rao by esmang resrcve models of ncome dynamcs on he bass of shor panels of ncome daa spannng only a segmen of he lfe cycle. 1 In hs paper, we reconsder he appropraeness of he exbook errors-n-varables model, and we fnd ha does no accuraely characerze curren earnngs as a proxy for lfeme earnngs. Thanks o a remarkable new daa se, we are able o generae dealed evdence on he assocaon beween curren and lfeme earnngs, ncludng s evoluon over he lfe cycle, whou havng o resor o an arbrary specfcaon of he earnngs dynamcs process. Our emprcal analyss uses he Socal Secury earnngs hsores of he members of he Healh and Reremen Sudy sample. Despe some lmaons dscussed n secon III, hese daa provde nearly career-long earnngs hsores, whch are based on relavely accurae admnsrave daa and peran o a broadly represenave naonal sample. In secon II, we develop smple models o llusrae some mporan aspecs of he assocaon beween curren and lfeme earnngs and o demonsrae he mplcaons for errors-n-varables bases n appled economerc research. In secon III, we descrbe he daa se and our economerc mehods. In secon IV, we presen our evdence on he connecons beween annual and lfeme earnngs. Secon V summarzes our fndngs and llusraes her usefulness wh a bref applcaon o nergeneraonal earnngs mobly. 1 See Mazumder (2001) for a relavely sophscaed recen example. 2

5 II. Models Followng Fredman (1957), mos analyses of curren ncome varables as proxes for unobserved lfeme ncome varables have adoped he exbook errors-n-varables model (1) y = y + v where y s a curren ncome varable, such as log annual earnngs, observed for ndvdual n perod ; y s a long-run ncome varable, such as he log of he presen dscouned value of lfeme earnngs; and v, he measuremen error n y as a proxy for y, s assumed o be uncorrelaed wh y (and each of s deermnans). Ofen, he curren ncome varable y has been adjused for sage of lfe cycle wh a regresson on a polynomal n age or experence or by subracng ou he cohor mean. Throughou hs secon, we wll suppress nerceps by expressng all varables as devaons from her populaon means. The exbook errors-n-varables model n equaon (1) s effecvely a regresson model ha assumes he slope coeffcen n he regresson of y on y equals 1. One famlar mplcaon of ha resrcon s ha, f y proxes for y as he dependen varable n a lnear regresson equaon, ordnary leas squares (OLS) esmaon of ha regresson equaon conssenly esmaes he equaon s slope coeffcens. Anoher well-known mplcaon s ha, f y proxes for y as he sole explanaory varable n a smple regresson equaon, he probably lm of he OLS esmaor of he equaon s slope coeffcen equals he rue coeffcen mes an aenuaon facor equal o Var( y )/[ Var( y ) Var( v )]. + 3

6 These of-used resuls no longer apply f he exbook errors-n-varables model ncorrecly characerzes he relaonshp beween curren and lfeme ncome. In par A of hs secon, we explan our reasons for suspecng ha he slope coeffcens n regressons of curren ncome varables on lfeme varables vary sysemacally over he lfe cycle and do no generally equal 1. In par B, we show how such deparures from he exbook model aler he sandard resuls on errors-n-varables bas. A. Lfe-cycle varaon Several fragmens of evdence sugges ha he assocaon beween curren and lfeme ncome varables vares over he lfe cycle. Bjorklund (1993), he closes predecessor o our sudy, uses Swedsh ncome ax daa from o conduc a drec comparson of curren and lfeme ncome. He fnds a srong lfe-cycle paern n he correlaon beween curren and lfeme ncome. In hs words, he correlaons are que low and n some cases even negave up o around 25 years of age and are raher hgh afer 35 years of age. In general he correlaons are around 0.8 afer he age of 35. Unforunaely, he correlaons n ncome levels repored by Bjorklund do no map drecly no magnudes of errors-n-varables bases n he sors of regresson esmaon ha economss commonly do. In he nex subsecon, we develop measures of assocaon beween curren and lfeme earnngs ha do have drec mplcaons for errors-n-varables bases. Anoher ndcaon of lfe-cycled-relaed deparures from he exbook errors-nvarables model, noed by Jenkns (1987) and Grawe (forhcomng), nvolves he esmaon of nergeneraonal mobly models such as he regresson of son s log 4

7 lfeme earnngs on faher s log lfeme earnngs. If son s log annual earnngs as a proxy for he dependen varable obeyed he exbook errors-n-varables model, he esmaed nergeneraonal elascy would have he same probably lm regardless of he age a whch he son s earnngs were observed. On he oher hand, f he slope coeffcen n he regresson of son s log annual earnngs on son s log lfeme earnngs devaes from 1 n a way ha evolves over he lfe cycle, hen analyses observng sons earnngs a dfferen ages wll yeld sysemacally dfferen elascy esmaes. Solon s (1999) survey of he nergeneraonal mobly leraure reveals precsely such a paern he sudes ha esmae he smalles elasces end o be hose ha observe sons earnngs early n her careers. Correspondngly, several sudes (e.g., Revlle, 1995) ha have explcly nvesgaed he effecs of varyng he ages a whch sons earnngs are observed have found ha he esmaed nergeneraonal elasces ncrease subsanally as he sons earnngs are observed furher no her careers. Nowhsandng he srong radon of assumng ha curren ncome varables as proxes for lfeme ncome varables follow he exbook errors-n-varables model, ndcaons ha hs assumpon s false should no be surprsng. Any realsc model of ncome evoluon over he lfe cycle would conradc he radonal assumpon. As an exremely smple example, suppose ha y, he log real earnngs of worker n year of hs career, follows (2) y = α + γ where nal log earnngs α vares across he populaon wh varance earnngs growh rae γ vares across he populaon wh varance 2 σ α and he. Heerogeney n earnngs growh s a naural consequence of heerogeney n human capal nvesmen, 2 σ γ 5

8 and s emprcal mporance has been documened by Mncer (1974), Baker (1997), Hader (2001), and Baker and Solon (2003) among ohers. For smplcy, assume zero covarance beween α and γ, nfne lfemes, and a consan real neres rae Then he presen dscouned value of lfeme earnngs s r > γ. s (3) V = exp( α + γ s)(1 + r) exp( α )[(1 + r) /( r γ )], = s 0 and he log of he presen value of lfeme earnngs s hus (4) γ log V α + r log r +. r I follows ha he slope coeffcen n he regresson of curren log earnngs on he log of he presen value of lfeme earnngs s (5) 2 2 Cov(logV, y ) σ ( / r) α + σ γ λ Var(logV ) σ + ( σ / r ) α γ The man hng o noe abou hs resul s ha, conrary o he exbook errors-nvarables model, λ generally does no equal 1. Insead, sars a a value less han 1 a he ouse of he career and hen ncreases monooncally over he lfe cycle. I reaches 1 when = 1/ r and hen exceeds 1 aferwards. The nuon s ha he workers wh hgh lfeme earnngs end o be hose wh hgh earnngs growh raes. Consequenly, when comparng he curren earnngs of hose wh hgh and low lfeme earnngs, an earlycareer comparson ends o undersae her gap n lfeme earnngs, and a lae-career comparson may oversae. Noe ha he common pracce of adjusng curren earnngs for he cenral endency of earnngs growh over he lfe cycle does no undo hs resul. The resul s due o heerogeneous varaon around he cenral endency. 6

9 Of course, he exac resul n equaon (5) s parcular o he very smple assumpons of he model. A more realsc model would ncorporae many addonal feaures ncludng ransory earnngs flucuaons, nonzero covarance beween nal earnngs and earnngs growh, nonlnear growh, and shocks wh permanen effecs. Whle hese feaures would lead o a more complex relaonshp beween λ and, hey clearly would no overurn he man qualave resuls ha λ does no generally equal 1 and should be expeced o vary over he lfe cycle. Fgure 1 provdes a pcoral verson of he argumen. The fgure conans he lfe-cycle log earnngs rajecores of workers 1 and 2, wh worker 2 aanng hgher lfeme earnngs. Boh rajecores dsplay he famlar concave shape documened and analyzed by Mncer (1974), and worker 2 experences more rapd earnngs growh hrough mos of he lfe cycle. The horzonal lnes depc he log of he annuzed value of each worker s presen dscouned value of lfeme earnngs. The dfference beween he wo workers log lfeme earnngs herefore s smply he vercal dsance beween he wo horzonal lnes. Bu how well s ha dfference esmaed f s proxed by he dfference n log earnngs a a parcular age? If he worker wh hgher lfeme earnngs has a seeper earnngs rajecory, hen he curren earnngs gap beween he wo workers early n her careers ends o undersae her gap n lfeme earnngs (and could even have he oppose sgn). As he workers maure, hs downward bas becomes less severe unl age *, when he vercal dsance beween he curren earnngs rajecores equals he dsance beween he horzonal lnes. Tha s he age a whch he exbook errors-nvarables model s correc. For a leas some of he lfe cycle beyond ha age, he gap n curren earnngs ends o oversae he gap n lfeme earnngs. 7

10 B. Implcaons for errors-n-varables bases Suppose we wsh o esmae he regresson model (6) y = β X + ε where he error erm ε s uncorrelaed wh he regressor vecor X. Sarng wh he case of lef-sde measuremen error, suppose ha y s he log of lfeme earnngs, whch s no observed and hence s proxed by, log annual earnngs a age. In accordance wh he dscusson n he precedng subsecon, we do no assume he exbook errors-nvarables model n equaon (1). Insead, we generalze ha model o y = λ y + v (7) y where λ, he slope coeffcen n he lnear projecon of y on y, need no equal 1 and may vary over he lfe cycle. By consrucon, v s uncorrelaed wh y, and we wll connue o manan he exbook model s assumpon ha also s uncorrelaed wh each separae deermnan of y ( X and ε ). 2 Then, f OLS s appled o he regresson of y on X, (8) y = λ β X + ( λε + v ), he probably lm of he esmaed coeffcen vecor for X s λ β nsead of β. In he exbook case where λ = 1, measuremen error n he dependen varable does no resul n nconssen esmaon of β. More generally, however, he OLS esmaor s 2 When hs assumpon fals, as somemes does, neher he exbook analyss nor our exenson s applcable. When Var( v ) = 0, equaon (7) specalzes o he rescalng of varables ofen dscussed n nroducory economercs exbooks (e.g., secon 2.4 of Wooldrdge, 2006). See secon 4 of Angrs and Krueger (1999) for an excellen overvew of errors n varables, ncludng non-classcal measuremen error. 8

11 nconssen, and he nconssency vares as a funcon of he age a whch annual earnngs are observed. Movng on o he case of rgh-sde measuremen error, suppose ha he log of lfeme earnngs s one elemen x n he regressor vecor X. Because x s no observed, s proxed by, log annual earnngs a age. Analogously o equaon (7) x for y, we express he lnear projecon of x on x as (9) x = λ x + v where v agan s assumed o be uncorrelaed wh X and ε. If x s he only elemen n X and OLS s appled o he lnear regresson of y on x, he probably lm of he esmaed slope coeffcen s Cov( x, y ) (10) plm ˆ β = = θ β Var( x ) where (11) Cov( x, x ) λvar( x ) θ =. 2 Var( x ) λ Var( x ) + Var( v ) The nconssency facor θ, somemes referred o as he relably rao, s mos smply nerpreed as he slope coeffcen n he reverse regresson of x on x. In he exbook case where λ = 1, hs facor smplfes o he famlar aenuaon facor Var ( x ) /[ Var( x ) + Var( v )]. More generally, he facor θ also depends on he value of λ. Indeed, wh λ < 1 and suffcenly small Var v ) / Var( x ), ( θ can exceed 1 so ha he errors-n-varables bas s an amplfcaon bas raher han an aenuaon bas. 9

12 Two furher resuls abou rgh-sde measuremen error are worh nong. Frs, f x s jus one elemen n he regressor vecor X, he aenuaon facor for s esmaed coeffcen s he same as he las expresson n equaon (11) excep wh Var( x ) replaced by he resdual varance from he auxlary regresson of x on he oher regressors n X. Second, f he measuremen error n x as a proxy for x s reaed wh an nsrumenal varable (IV) correlaed wh x bu uncorrelaed wh ε or v, he probably lm of he convenonal IV esmaor of he coeffcen of coeffcen dvded by λ. 3 x s he The resuls presened n hs subsecon delver wo key messages. Frs, wh plausble deparures from he exbook errors-n-varables assumpons, he famlar exbook resuls abou OLS and IV esmaon are overurned. Measuremen error n he dependen varable s no nnocuous for conssency, and measuremen error n he explanaory varable can nduce eher amplfcaon or aenuaon nconssency n OLS esmaon as well as n IV esmaon. Second, some of he esmaon nconssences from usng log annual earnngs as a proxy for log lfeme earnngs can be summarzed wh jus wo smple parameers: he slope coeffcen n he forward regresson of log annual earnngs on log lfeme earnngs and he slope coeffcen n he reverse regresson of log lfeme earnngs on log annual earnngs. In secon IV, we wll esmae hose wo parameers and examne how hey vary over he lfe cycle. 3 The nconssency of convenonal IV esmaon n he presence of non-classcal measuremen error has been dscussed prevously by Kane, Rouse, and Sager (1999), Bound and Solon (1999), and Km and Solon (2005). 10

13 III. Daa and Mehods A. Daa Mos U.S. sudes of he relaonshp beween curren and lfeme ncome varables have been based on longudnal survey daa from only a lmed poron of he respondens careers. In conras, lke Bjorklund s (1993) sudy of Swedsh ncome ax daa, our sudy s based on nearly career-long earnngs hsores. Ths nformaon s now avalable for a U.S. sample because, n accordance wh an agreemen wh he Socal Secury Admnsraon, he Unversy of Mchgan s Survey Research Cener asked he parcpans n s Healh and Reremen Sudy (HRS) o perm access o her Socal Secury earnngs hsores for The HRS sample s a naonal probably sample of Amercans born beween 1931 and 1941, and abou ¾ of he respondens agreed o perm access o her Socal Secury earnngs hsores. As shown n Hader and Solon (2000), n erms of observable characerscs, he respondens ha graned access appear o be surprsngly represenave of he complee sample. The earnngs daa suppled by he Socal Secury Admnsraon round he earnngs observaons o he neares hundred dollars, wh a dsncon made beween zero and posve amouns less han $50. Our analyss s for male HRS respondens born beween 1931 and 1933, who were abou 19 years old a he begnnng of he earnngs perod and abou 59 a he end. Thus, for he 821 men n our analyss, we have annual earnngs nformaon 4 Because of he hghly confdenal naure of he daa, he earnngs hsores are no par of he HRS publc release daa ses, bu are provded only hrough specal permsson from he Survey Research Cener. For nformaon on accessng HRS resrced daa, see he HRS webse hp://hrsonlne.sr.umch.edu. For more general nformaon on he HRS, see he webse or Juser and Suzman (1995). 11

14 for every year over he major poron of her careers. 5 The oher man srengh of our daa se s ha he Socal Secury earnngs hsores end o be more accurae han he survey repors of earnngs used n mos prevous research. Indeed, Bound and Krueger s (1991) nfluenal sudy of errors n earnngs repors n he Curren Populaon Survey used Socal Secury earnngs daa as he rue values agans whch he Curren Populaon Survey measures were compared. The srenghs of he Socal Secury earnngs daa are accompaned by wo serous lmaons. Frs, he earnngs daa peran only o jobs covered by Socal Secury. Accordng o Socal Secury Admnsraon (1999, able 3.B2), he percenage of earnngs covered by Socal Secury has exceeded 80% ever snce he coverage exensons effeced by 1957 and exceeded 85% over mos of our sample perod. Beween 1951 and 1956, however, hs percenage ranged beween 66 and 79%. Accordngly, n addon o our analyses for , we also wll repor resuls for Second, he Socal Secury earnngs n our daa are measured only up o he maxmum amoun subjec o Socal Secury ax. In some years, he proporon of observaons ha are rgh-censored s que large. For he 821 men n our sample, able 1 dsplays he medan observed earnngs, he percenage n he sample wh zero earnngs, he axable lm, and he percenage wh earnngs a he axable lm for each year from 1951 o The able shows ha, n he early years, very few sample members are earnng enough o approach he axable lm. As her earnngs grow over her careers, however, he axable lm becomes more consranng, especally n he 5 Ths sample s resrced o workers wh posve earnngs n a leas 10 years durng Ths creron, whch excludes only 33 ndvduals, s less resrcve han he usual pracce n survey-based earnngs dynamcs sudes of requrng posve earnngs n every year (e.g., Abowd and Card, 1989; Baker, 1997). Whn hs sample, our man analyss ncludes years of zero earnngs, bu we also wll repor resuls from an analyss based only on he posve earnngs observaons. 12

15 years when he axable lm s low relave o he general earnngs dsrbuon. The wors year s 1965, when 62% of he sample s rgh-censored. Aferwards, he degree of censorshp lessens as he axable lm s progressvely ncreased. By 1991, only 9% of he sample s rgh-censored. Alhough some prevous sudes of curren and lfeme earnngs have used annual earnngs daa wh less severe rgh-censorshp, her observaon of earnngs usually has been lmed o relavely shor segmens of he lfe cycle. In effec, hey have used resrcve models of earnngs dynamcs o mpue mssng earnngs daa over mos years of her sample members careers. If no for he rgh-censorshp, we would follow Bjorklund s (1993) approach of drecly summarzng he observed jon dsrbuon of annual and lfeme earnngs. Because of he rgh-censorshp, however, we are forced nsead o esmae he jon dsrbuon n a way ha mpues he censored rgh als of he annual earnngs dsrbuons. We descrbe our mehods n he nex subsecon. B. Economerc mehods As explaned above n secon II.B, our ulmae goal s o summarze he assocaon beween annual and lfeme earnngs n erms of wo ypes of parameers. One s λ, he slope coeffcen n he regresson of log earnngs n year on he log of he presen value of lfeme earnngs. The oher s θ, he slope coeffcen n he reverse regresson of log lfeme earnngs on log earnngs n year. If we had complee daa, we would esmae hese parameers smply by applyng leas squares o he forward and reverse regressons of he relevan varables. 13

16 Because of he censorshp of he Socal Secury earnngs daa a he axable lm, however, we canno observe he exac value of annual earnngs n he cases where earnngs are rgh-censored and furhermore, n hose cases, we also canno compue he presen value of lfeme earnngs. We herefore apply a hree-sep procedure for esmang he λ and θ coeffcens. Frs, we use a lmed-dependen-varable model o esmae he jon dsrbuon of uncensored annual earnngs n he 41 years from 1951 hrough Second, drawng from ha esmaed jon dsrbuon, we generae a smulaed sample of uncensored earnngs hsores, for whch we can calculae he presen dscouned value of lfeme earnngs. Thrd, usng he uncensored earnngs daa for ha sample, we apply leas squares o he forward and reverse regressons o oban our esmaes of he λ and θ parameers. The key assumpon n our frs sep s ha he uncensored values of log annual earnngs over he 41 years from 1951 o 1991 follow a mulvarae normal dsrbuon. Gven hs varan of he radonal Tob assumpon for lmed dependen varables, he jon dsrbuon of he 41 annual earnngs varables can be fully characerzed by he mean and varance of log earnngs for each year and he cross-year auocorrelaons of log earnngs for every par of years. To esmae he year-specfc means and varances for our sample cohor born n , we smply apply he convenonal cross-seconal Tob maxmum lkelhood esmaor separaely for each year from 1951 o The only regressor n each year s equaon s 1, he coeffcen of whch s he nercep. The esmaed nercep s our esmae of he cohor s mean log earnngs n ha year. In our man analyss, we use a wo-lm Tob model. The rgh-censorshp hreshold s he Socal Secury axable lm 14

17 for ha year. The lef-censorshp hreshold s $50. Observaons of zero earnngs and of posve earnngs less han $50 are boh ncluded as observaons lef-censored a $50. 6 Havng esmaed each year s mean and varance n he cross-seconal Tobs, we sll need o esmae he auocorrelaons beween years. To oban hose esmaes, we apply he convenonal bvarae Tob maxmum lkelhood esmaor separaely for each of he 41 40/ 2 = 820 dsnc pars of years n our perod. Wh hose auocorrelaons esmaed along wh he mean and varance for every year, we have an esmaed verson of he enre jon dsrbuon of uncensored annual earnngs over all 41 years. In he second sep of our procedure, we use our esmaed jon dsrbuon of uncensored earnngs for o perform he followng smulaon. Frs, we ake 4,000 random draws from he esmaed jon dsrbuon of he 41 years of annual earnngs. 7 Then, for each of he 4,000 smulaed earnngs hsores, we calculae he presen dscouned value of lfeme earnngs. In he man verson of he smulaon, we perform he dscounng by (1) usng he personal consumpon expendures deflaor o conver each year s nomnal earnngs o a real value and (2) assumng a consan real neres rae of In he end, we have a smulaed sample of 4,000 observaons for whch we observe he presen dscouned value of lfeme earnngs as well as each year s earnngs. 6 In he smulaon descrbed below, our reamen of zero-earnngs observaons as lef-censored observaons from a lognormal dsrbuon causes our smulaed observaons o nclude no zeros, bu nsead small annual earnngs values less han $50. The purpose of he smulaon s o generae observaons for he presen dscouned value of lfeme earnngs. For ha purpose, he dfference beween annual earnngs of zero or a few dollars s of praccally no consequence. 7 To mplemen he smulaon, we need he esmaed auocovarance marx o be posve sem-defne (as he rue one mus be). Our procedure for mposng he resrcon of posve sem-defneness s descrbed n an appendx on he AER s webse. 15

18 Fnally, for hs sample of 4,000 ndvduals, we apply OLS o he regresson of each year s log annual earnngs on he log of he presen value of lfeme earnngs, and hereby produce a for each year from 1951 o Smlarly, we oban a for each λˆ year by applyng OLS o he reverse regresson of he log of he presen value of lfeme earnngs on each year s log annual earnngs. Plong each of hese coeffcen esmaes over me depcs he lfe-cycle rajecory of he assocaon beween curren and lfeme earnngs n a way ha ranslaes drecly no mplcaons for errors-n-varables bases. θˆ IV. Emprcal Resuls In he frs sep of our esmaon procedure, he Tob analyss descrbed above resuls n a esmaed auocovarance marx for log annual earnngs from 1951 o The full marx s avalable n an appendx on he AER s webse. Table 2 shows he esmaed auocorrelaons for , a perod when our cohor born n s beween he ages of abou 43 and 52. As shown n he second column of able 3, he frs-order auocorrelaons over hs perod average o 0.89, he second-order auocorrelaons average o 0.82, he hrd-order auocorrelaons average o 0.78, and so forh. Table 3 also dsplays esmaed auocorrelaons from wo oher sudes of admnsrave daa. The mos comparable resuls repored n Baker and Solon s (2003) sudy of Canadan ncome ax daa are he auocorrelaons over he perod for he cohor born n Ther average auocorrelaons, shown n he hrd column, are farly smlar o ours, bu somewha lower. As shown n he fourh column, he esmaes from Bohlmark and Lndqus (2005), a replcaon of our sudy based on Swedsh ncome ax daa, are closer o ours han o Baker and Solon s. Noe ha hs 16

19 resemblance beween oher sudes esmaes and ours occurs even hough he oher sudes use uncensored daa and herefore can esmae he auocorrelaons drecly whou mposng dsrbuonal assumpons. 8 We fnd reassurng ha, despe he omsson of earnngs no covered by Socal Secury and he mpuaon of rgh-censored values, our auocorrelaon esmaes are smlar o hose from oher daa ses. Mos of hese esmaed auocorrelaons are somewha hgher han hose repored by Baker (1997) and Hader (2001) n her analyses of he Panel Sudy of Income Dynamcs, bu he survey-based esmaes may be based downward by reporng error. Anoher relevan comparson s o an alernave earnngs varable avalable for our sample for For hose years, n addon o he Socal Secury earnngs daa, he Survey Research Cener also has obaned earnngs daa from employers W-2 repors o he Inernal Revenue Servce. Unlke he Socal Secury daa, he W-2 varable ncludes earnngs no covered by Socal Secury, and s rgh-censored (for confdenaly reasons) a $125,000, whch s far less consranng han he Socal Secury axable lms lsed n able 1. On he oher hand, he W-2 varable leaves ou self-employmen earnngs and earnngs allocaed o 401(k) pensons. As shown n he ffh column of able 3, when we use he W-2 daa o reesmae our Tobs for , he frs-order auocorrelaons average o 0.89, he second-order auocorrelaons average o 0.83, and he hrd-order auocorrelaons average o As shown n he las column, he correspondng average auocorrelaons for he Socal Secury earnngs varable over he same perod are 0.91, 0.85, and The dosyncrases of he 8 Also lke hese oher sudes, we fnd ha earnngs auocorrelaons are somewha smaller early n he lfe cycle. 17

20 alernave earnngs measures do no appear o generae major dscrepances n he esmaed perssence of earnngs. 9 In he second sep of our esmaon procedure, we perform he smulaon n whch we ake 4,000 draws from he esmaed jon dsrbuon of he 41 years of annual earnngs. Then, usng he resulng sample of 4,000 uncensored earnngs hsores, our hrd sep summarzes he connecon beween annual and lfeme earnngs by esmang he forward and reverse regressons beween he logs of annual and lfeme earnngs. The op poron of fgure 2 plos our esmaes of λ, he slope coeffcen n he regresson of log annual earnngs a me on he log of he presen value of lfeme earnngs. To focus on he lfe-cycle varaon n λ, we express on he horzonal axs as year mnus 1932, whch gves he approxmae age of our cohor n each year. 10 In conras o he exbook assumpon ha λ equals 1 hroughou he lfe cycle, λˆ begns a 0.24 a age 19, ncreases seadly unl rses o abou 1 a age 32, and hen declnes some n he lae fores. The man mplcaon s ha, conrary o he exbook errors-n-varables model, usng log curren earnngs o proxy for log lfeme earnngs as he dependen varable can nduce an errors-n-varables bas. Mos mporanly, usng curren earnngs n he wenes causes a large aenuaon bas. A consrucve mplcaon s ha he bas s small f one uses curren earnngs beween he early hres and he md fores, when he exbook assumpon ha λ = 1 s reasonably accurae. 9 Perhaps he smlary of he auocorrelaon esmaes should no be a surprse. If one hnks of he log of covered earnngs as he sum of log oal earnngs and he log of he proporon covered, one would expec he auocorrelaon of log covered earnngs o be approxmaely a weghed average of he auocorrelaons for log oal earnngs and log coverage. Presumably, boh of hese auocorrelaons are hghly posve. If hey are no very dfferen from each oher, hen averagng n he coverage auocorrelaon wll no produce a large bas n esmang he earnngs auocorrelaon. 10 The pon esmaes ploed n fgures 2 and he assocaed sandard error esmaes are abulaed n our elecronc appendx, whch also descrbes our boosrap procedure for esmang he sandard errors. 18

21 The lower poron of fgure 2 shows he esmaed lfe-cycle rajecory of he relably rao θ, he relevan parameer for assessng errors-n-varables bas from usng log annual earnngs o proxy for log lfeme earnngs as he explanaory varable n a smple regresson. θˆ begns a only abou 0.2, ncreases o a farly fla peak averagng abou 0.65 beween he lae wenes and md fores, and hen decreases. Our dscusson n secon II.B showed ha heorecally he errors-n-varables bas could be eher an aenuaon bas or an amplfcaon bas. Our emprcal resuls, however, confrm he convenonal presumpon ha usng curren earnngs o proxy for lfeme earnngs as a regressor nduces an aenuaon bas. The bas s especally large f curren earnngs are measured early n he lfe cycle. There s a wde age range n md-career when he errorsn-varables bas says abou he same, bu remans que subsanal even n ha range. To check he robusness of our man resuls, we have carred ou a seres of sensvy analyses, he resuls of whch are dsplayed n fgure 3. The frs s movaed by he queson of how o rea years of zero earnngs. Our man resuls are based on wo-lm Tob esmaes ha rean observaons of zero earnngs n he analyss. Because mos prevous analyses of earnngs dynamcs, however, have excluded observaons of zero earnngs, we supplemen our man analyss wh anoher ha excludes he zeros, codes posve earnngs less han $50 as $25, and esmaes one-lm Tobs wh only rgh-censorshp. As shown n able 1, zero earnngs are especally prevalen n he early years of our sample, boh because many of our sample members are no ye workng for pay and because he Socal Secury sysem s coverage s less exensve before We herefore conduc hs analyss only for Excludng he zeros changes he esmaes of he varances and auocovarances n log 19

22 annual earnngs, bu because hose changes are roughly proporonal, he esmaed auocorrelaons are smlar o hose n he man analyss. Accordngly, he new esmaes of λ and θ shown n fgure 3 are smlar o he esmaes from our man analyss repeaed from fgure 2. Our second and hrd robusness checks explore he sensvy of our resuls o our choce of neres rae seres. In our man smulaon, we calculaed he presen dscouned value of lfeme earnngs by (1) usng he personal consumpon expendures deflaor o conver each year s nomnal earnngs o a real value and (2) assumng a consan real neres rae of Our hrd supplemenary analyss uses a real neres rae of 0.04, and our fourh dscouns nomnal earnngs by a nomnal neres rae seres, he annual one-year T-noe neres raes. 11 The resuls shown n fgure 3 are que smlar o hose based on our orgnal neres rae seres. Fourh, we have checked wheher our resuls are affeced by he Healh and Reremen Sudy s oversamplng of blacks, Hspancs, and resdens of Florda. To do so, we have reesmaed he jon dsrbuon of earnngs wh a Tob quas-maxmum lkelhood procedure ha weghs each observaon s conrbuon o he lkelhood funcon by s nverse probably of selecon no he sample. The resulng Tob esmaes are very smlar o hose from our orgnal unweghed analyss, and consequenly he new esmaes of λ and θ are agan very smlar o he man esmaes. Fnally, fgure 3 ncludes he resuls of Bohlmark and Lndqus s (2005) replcaon of our man analyss based on Swedsh ncome ax daa. Ths comparson s 11 Ths seres s avalable only back o For , we added o he neres raes for hreemonh T-blls. Ths adjusmen was based on he relaonshp beween he one-year and hree-monh raes observed for

23 parcularly neresng because Bohlmark and Lndqus s daa are largely free of he censorshp and coverage ssues ha afflc our U.S. Socal Secury earnngs daa. As a resul, Bohlmark and Lndqus esmae λ and θ drecly wh he forward and reverse regressons nvolvng log curren and lfeme earnngs whou havng o resor o our more complex esmaon procedure based on he mulvarae normaly assumpon. Ther esmaes of λ n he wenes are somewha hgher han ours, bu sll much less han 1. In general, he paerns of he Swedsh and U.S. resuls are srkngly smlar. V. Summary and Dscusson All of our analyses ell he same sory: conrary o he exbook errors-n-varables model usually assumed n appled research, he slope coeffcen n he regresson of log curren earnngs on log annual earnngs vares sysemacally over he lfe cycle and s no generally equal o 1. We can llusrae he usefulness of our resuls by applyng hem o he nergeneraonal mobly regresson n whch son s log of lfeme earnngs s he dependen varable and faher s log of lfeme earnngs s he explanaory varable. As summarzed n Solon (1999), mos recen research n ha leraure has devoed consderable aenon o he rgh-sde measuremen error from usng shor-run proxes for faher s lfeme earnngs. Our esmaes of θ shown n fgures 2 and 3 confrm he leraure s presumpon ha rgh-sde measuremen error causes an aenuaon nconssency n OLS esmaon of he nergeneraonal elascy Many researchers have aemped o reduce he aenuaon nconssency by averagng faher s log earnngs over mulple years. In an analyss summarzed n our elecronc appendx, we repea our esmaon of θ excep ha he new esmaes are for fve-year averages of log annual earnngs, raher han for sngle years. Our resuls srongly suppor he concluson of Mazumder (2001, 2005) ha even 21

24 The leraure, however, has gven much less aenon o he lef-sde measuremen error from usng shor-run proxes for son s lfeme earnngs. Presumably, hs neglec reflecs an assumpon by researchers ha, n accordance wh he exbook errors-n-varables model, lef-sde measuremen error s nnocuous for conssency. All our esmaes of λ sugges ha assumpon would be farly well founded f sons earnngs were measured beween he early hres and md fores. Many nergeneraonal mobly sudes, however, have measured sons earnngs a earler ages, and hs has subsanally affeced he fndngs. Revlle (1995), for example, esmaes nergeneraonal elasces of abou 0.25 when he measures he sons earnngs n her wenes, bu hs esmaes sar approachng 0.5 when he observes he sons well no her hres. Ths s jus he paern one should expec from he rajecores of n fgures 2 and 3. An mporan mplcaon s ha many esmaes of he nergeneraonal earnngs elascy have been subjec o subsanal aenuaon nconssency from lef-sde measuremen error n addon o he well-known nconssency from rgh-sde measuremen error. Of course, nerpreng evdence on nergeneraonal earnngs mobly s jus one example of how our resuls mgh be appled. We advse readers, however, o exercse due cauon n mporng our esmaes of λ and θ o oher earnngs daa. We already have menoned ssues of comparably beween admnsrave and survey daa. Furhermore, he lfe-cycle rajecores for our U.S. cohor born n may dffer from hose for oher cohors and oher counres. In addon, as emphaszed n Solon (1992), sample selecon crera ha affec he sample s dsperson n earnngs also affec λˆ esmaes based on fve-year averages of he earnngs varable for fahers are subjec o a subsanal errorsn-varables bas. 22

25 he measuremen error properes of curren earnngs as proxes for lfeme earnngs. Neverheless, akng accoun of our evdence on deparures from he exbook errors-nvarables model should enable beer-nformed analyses of esmaon bases n a wde varey of research ha uses curren earnngs varables as proxes for long-run earnngs. 23

26 Table 1. Descrpve Sascs for Nomnal Annual Earnngs Covered by Socal Secury Year Medan Percen wh Zero Earnngs Taxable Lm Percen a Taxable Lm , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

27 Table 2. Esmaed Auocorrelaons n Log Annual Earnngs, Year (.02) (.02) (.02) (.02) (.02).69 (.02).66 (.02).64 (.02) (.02).68 (.02).69 (.02) (.02).70 (.02).69 (.02) (.02).71 (.02) (.02) Noe: Numbers n parenheses are esmaed sandard errors. 25

28 Order of Auocorrelaon Table 3. Average Esmaed Auocorrelaons from Varous Sudes Our Table 2 Baker and Solon (2003), Table 3 Bohlmark and Lndqus (2005), Table 2 Our W-2 Daa for Our Socal Secury Daa for

29 Fgure 1. Illusrave Example of Log Annual Earnngs and Log Annuzed Lfeme Earnngs Log Earnngs * ** =age Worker 1 Worker 2 Noes: The doed lnes are for worker 1, and he sold lnes are for worker 2. For each worker, he upward-slopng lne depcs log annual earnngs by age, and he horzonal lne depcs log annuzed lfeme earnngs. 27

30 Fgure 2. Man Esmaes of λ and θ Esmaes of λ Esmaes 95% CI =age Esmaes of θ Esmaes 95% CI =age Noes: The sold lnes graph he parameer esmaes, and he doed lnes are 1.96 esmaed sandard errors above and below he sold lnes. 28

31 Fgure 3. Alernave Esmaes of λ and θ Esmaes of λ Man (1) (2) (3) (4) (5) =age Esmaes of θ Man (1) (2) (3) (4) (5) =age Noes: The ploed esmaes are from fve dfferen analyses: Man man esmaes coped from fgure 2 (1) same as man, bu droppng zeros and esmang one-lm Tobs (2) same as man, bu usng 0.04 real neres rae (3) same as man, bu dscounng wh one-year T-noe neres raes (4) same as man, bu weghng by nverse probables of selecon (5) Swedsh esmaes from Bohlmark and Lndqus (2005) 29

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