A Parsimonious Model of Idiosyncratic Income

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) A Parsimonious Model of Idiosyncratic Income Edmund Crawley, Martin Blomhoff Holm, H˚akon Tretvoll 2022-026 Please cite this paper as: Crawley, Edmund, Martin Blomhoff Holm, and H˚akon Tretvoll (2022). “A Parsimonious Model of Idiosyncratic Income,” Finance and Economics Discussion Series 2022-026. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2022.026. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A Parsimonious Model of Idiosyncratic Income* ff Edmund Crawley Martin Blomho Holm Håkon Tretvoll April 2022 Abstract Thestandardmodelofpermanentandtransitoryincomeisknowntobemisspecified. ff Estimates of income volatility in the model di er depending on the type of data ff moments used—levels or di erences—and how these moments are weighted in the estimation. We propose two changes to the standard model. First, we account for the time-aggregated nature of observed income data. Second, we allow transitory shockstopersistforvaryinglengthsoftime. Withonlyoneadditionalparameter,our proposedmodelconsistentlyrecovertheparametersoftheincomeprocessirrespective oftheestimationmethod. Totheextentthatresearchersemploythestandardmodel, ff weadvisespecialcautionwiththeuseoffirst-di erencemoments. JEL:E21,E24,J30 Keywords: IncomeUncertainty,Inequality,HouseholdFinance *WethankDmytroHryshkoforsharingthecodefromDaly,Hryshko,andManovskii(2022)withusand SisiZhangforsharingthedataandcodefromMoffittandZhang(2018). WealsothankJohannesH.Alsvik, MichaelGraber,andRobertMoffittforvaluablecommentsanddiscussions,aswellasseminarparticipants attheFederalReserveBoard, JohnsHopkinsUniversity, andStatisticsNorway. Thisprojecthasreceived funding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No. 851891). We use the Panel Study of Income Dynamics publicusedataset,producedanddistributedbytheSurveyResearchCenter,InstituteforSocialResearch, University of Michigan, Ann Arbor, MI (2021). Viewpoints and conclusions stated in this paper are the responsibilityoftheauthorsaloneanddonotnecessarilyreflecttheviewpointsofStatisticsNorwayorthe FederalReserveBoard. Crawley: FederalReserveBoard,edmund.s.crawley@frb.gov. Holm: UniversityofOslo,m.b.holm@econ.uio.no. Tretvoll: StatisticsNorway,hakon.tretvoll@ssb.no. 1

1 Introduction [T]he key challenge for future work is to develop a specification for the wage processthatisbothparsimoniousenoughtobeusedasaninputtoincompletemarkets models, and rich enough to account empirically for the covariance ff structureofwagesinbothlevelsanddi erences. Heathcote,Perri,andViolante(2010,p. 40) What is the nature of idiosyncratic income risk faced by households, and how has it changedovertime? Therelatedeconomicliteraturecentersaroundamodelofpermanent andtransitoryshocks. Inthisstandardmodel,householdsfaceaseriesofincomeshocks. ff Some of these income shocks permanently (or persistently) a ect income, such as a job ff change or promotion. Others only have transitory e ects on income, such as a bonus or a period of sick leave. However, although widely used, this model is known to be misspecified. Intheory,ifthemodelweretrue,thepermanentandtransitorycomponents’ estimated variances will be consistently estimated regardless of how the estimation is ff performed. In practice, the choice of income moments used (levels of di erences) and the weighting applied to those moments in a minimum distance estimation (optimally, ff diagonally,orequallyweighted)oftenleadstodi erentconclusionsaboutthepersistence ofincomerisk,andhencetheassociatedwelfareconsequences. In this paper, we make two changes to the standard model of idiosyncratic income. First, we account for the time-aggregated nature of observed income data at an annual level. Implicitinthestandardmodelisthathouseholdsreceiveallshockstotheirannual incomeonJanuary1steachyear. AsfirstnotedinWorking(1960),whenincomeshocksare persistentanddistributedthroughoutthecalendaryear,observedannualincomeexhibits strong serial correlation even when the underlying income shocks are uncorrelated. We provide evidence from Norway to support our assumption that income shocks tend to occur throughout the calendar year. The implied covariance matrix for permanent ff and transitory income shocks di ers significantly under this assumption relative to the standardassumptionthatshocksoccuronlyonJanuary1st. The second change we make to the standard model is to divide transitory shocks into two flavors. “Bonus” shocks (which can include measurement error) display no persistence, while “passing” shocks persist for a stochastic period of time. We find that Norwegian administrative data is best summarized by a mixture of purely transitory and passing shocks. Furthermore, a transitory shock that persists for a stochastic period of time seems to be a more natural assumption than one in which all transitory shocks ff persistforafixedperiod. Weshowthatthefirst-di erencecovariancematrixinducedby 2

the passing shock in our time-aggregated model is similar to that induced by permanent shocks in the standard model. As a result, these passing shocks have previously been ff partlyinterpretedaspermanentshockswhenusingdi erencemomentsforestimation. Onekeyresultofthetwochangesweproposetothestandardmodelistoreconcilethe ff di erent permanent and transitory income variance estimates that arise from estimation ff using the covariance of the first di erence or the level of income. The standard model ff ∆ ,∆ , inducesacovariancestructureonbothdi erencesandlevels—cov( y y )andcov(y y ), t s t s respectively. If the model is well specified, the estimates recovered in the data for permanent and transitory income variance would not depend on the applied covariance ff structureortheweightingmatrixusedintheestimation. Inpractice,usingfirst-di erence moments usually results in significantly higher estimated permanent income variance and significantly lower estimated transitory income variance relative to estimation using ff level moments. In turn, these di erent estimates are applied to models of consumption andresultinconflictingimplicationsforconsumerbehaviorandwelfare. We simulate our proposed model and show that when we estimate the standard model on the simulated time-aggregated data, we find evidence of the same type of misspecification as in actual data. Indeed, with the most commonly used weighting— diagonally weighted minimum distance (DWMD)—the permanent income variance is higher,andthetransitoryincomevarianceislower,whenestimatingthestandardmodel ff with di erence moments than with level moments. This structure of misspecification is the same as we observe when estimating the standard model using actual data. Hence, the two adjustments we make, allowing shocks to occur during the year and enriching ffi ff the description of transitory shocks, are su cient to explain the di erences in parameter estimates obtained when estimating the standard model across moments and weighting matrices. WenextestimateourproposedmodelusingNorwegianadministrativedataanddata fromthePanelStudyofIncomeDynamics(PSID).Regardlessofthemomentorweighting matrixapplied,wefindsimilarparameterestimates. Thisstabilityofparameterestimates suggests that our proposed model does not have the misspecification problems of the standard model. Moreover, with only one extra parameter, the relatively small number of households in the PSID sample yields reasonably precise estimates. Hence, we argue thatthebenefitofreducingmisspecificationoutweighsthelossfromtheincreasedmodel complexity from adding this one extra parameter. We provide codes to make estimation ofourproposedmodelavailabletootherresearchers. Althoughthestandardmodelofincomeismisspecifiedandwerecommendusingour proposed model in most cases, we provide results on how to interpret the existing litera- 3

ture. Somecombinationsofmomentsandweightingmatricesyieldmorereliableestimates than others. Both in simulated and actual data, the combination of level moments and either the equally weighted minimum distance (EWMD) method or the DWMD method provide parameter estimates that are close to the data-generating process or the parameter estimates of the proposed model. The intuitive reason is that with level moments, ffi the permanent variance will be identified from covariances generated from su ciently ff , >> large time di erences (cov(y y ) with s t). Hence, as long as these long covariances t s have non-zero weights (e.g., with EWMD), the standard model will accurately identify thepermanentincomevariance. Wecan,therefore,summarizeourmainresultsforpractitionersasfollows. Ifthedata set contains many panel observations, use our parsimonious model of income dynamics to estimate the income process. The model is robust to using specific moments (levels ff or di erences) and to the weighting matrix used in estimation (optimally, diagonally, or equally weighted). If the data set is ‘small,’ use the standard model but estimate it using levelmomentsandtheequallyordiagonallyweightedminimumdistancemethod. Using ff di erence moments potentially biases the results significantly. Therefore, we strongly ff adviseagainstestimatingthestandardmodelusingdi erencemoments. We also use our proposed model to investigate how income risk varies by age and how the nature of income risk has changed over time. We estimate our model with both Norwegian administrative data and the Panel Study of Income Dynamics (PSID). First, regarding lifecycle income risk, we find that neither permanent nor transitory income variance vary much from age 35 to 50. This result is consistent with prior findings in the literaturewhereincomeriskdoesnotvarymuchbyage(see,e.g.,Storesletten,Telmer,and Yaron, 2004, Heathcote, Storesletten, and Violante, 2005, and Guvenen, Karahan, Ozkan, and Song, 2021). Second, we find some evidence that ‘start-of-working-life’ inequality and permanent income risk have increased over time in both the Norwegian data and in the PSID.1 We further show that the estimated time trends of income risk are similar irrespectiveofwhetheroneusedourproposedmodelorthestandardmodel. Hence,while ff the standard model tends to yield very di erent estimates of the level of risk depending onthemomentorweightingmatrixapplied,thetimetrendsinincomeriskaresimilar. 1This is a question that has been discussed extensively at least since Gottschalk, Moffitt, Katz, and Dickens(1994). RecentcontributionsincludeMoffittandGottschalk(2002),GottschalkandMoffitt(2009), Heathcote, Storesletten, and Violante (2010), Heathcote, Perri, and Violante (2010), Sabelhaus and Song (2010), Moffitt and Gottschalk (2012), Bloom, Guvenen, Pistaferri, Sabelhaus, Salgado, and Song (2017), Moffitt, Bollinger, Hokayem, Wiemers, Abowd, Carr, McKinney, Zhang, and Ziliak (2021), Carr, Moffitt, andWiemers(2020),MoffittandZhang(2020),andMcKinneyandAbowd(2020). 4

Related literature. Our paper most closely relates to Daly, Hryshko, and Manovskii (2022), who also ask why estimates of permanent and transitory shocks in the standard ff model depend on the estimation method used. However, their solution is di erent. ff They argue that di erences in the sample selection that naturally arise between level ff ff and di erence estimation in unbalanced panels can explain the di erences in estimates. They show that using a balanced panel overcomes these sample selection issues and, as a result, conclude that the standard model can fit the data well. In contrast, we show, both in theory and practice, that the results of Daly, Hryshko, and Manovskii (2022) are sensitive to the weighting matrix used in the estimation. This sensitivity is distinct but similarly concerning evidence that the standard model is misspecified. We show that ourproposedmodelisrobusttothechoiceofmomentsusedandthechoiceofweighting matrix. Nevertheless,weremainconvincedbytheargumentsaboutsampleselectionand, asaresult,restrictourselvestotheuseofbalancedpanelsintheanalysis. Severalpapersarebuildingrichmodelstomatchmoremomentsoftheincomedistribution. AcloselyrelatedpaperisGuvenen,McKay,andRyan(2022),whobuildanincome process that is rich enough to match several moments of the income distribution but is stilltractableenoughtobeincludedinmodels. OtherrecentexamplesincludeDruedahl, Graber,andJørgensen(2021),whosemodelmatchesdataonmonthlyincomeinnovations in Denmark; Guvenen, Karahan, Ozkan, and Song (2021), who build an income process to match moments from administrative data on annual income innovations in the U.S.; and Arellano, Blundell, and Bonhomme (2017), who estimate a process that allows for variationinparametersbyindividualincomelevelstomatchincomeinnovations.2 Commontothisliteratureisthatthemodelsarecomplexandrequiretheestimationofseveral additional parameters. Our contribution is that, instead of building complicated models to match the dynamics of the income innovations, we construct a parsimonious income process that is robust to known misspecification issues. Hence, it matches the data well, yetitissimpleenoughtobeincludedinmodelsandallowsestimationinrelativelysmall datasets. The time aggregation problem in the context of the standard permanent-transitory model has recently been studied in Eika (2018), Crawley (2020), Crawley and Kuchler (2022),andKleinandTelyukova(2013). However,noneofthesepapersintroducethetwo flavorsoftransitoryincomeshocksweproposeand,assuch,donotaddressthemisspec- 2Several other papers also estimate variants of non-linear income dynamics, e.g., Browning, Ejrnaes, andAlvarez(2010),Altonji,SmithJr.,andVidangos(2013),DeNardi,Fella,andPaz-Pardo(2020),Braxton, Herkenhoff,Rothbaum,andSchmidt(2021),andDeNardi,Fella,Knoef,Paz-Pardo,andVanOoijen(2021). ForNorway,Halvorsen,Holter,Ozkan,andStoresletten(2020)presentevidenceofnon-gaussianfeatures ofincomedynamics. 5

ificationissuesdiscussedabove. Ourcontributionistoconstructahigh-frequencymodel ffi that can be estimated on low-frequency data and is adjusted su ciently to overcome egregiousconflictsbetweenestimationmethods. Roadmap. The rest of the paper is structured as follows. Section 2 describes the Norwegian administrative data. Section 3 presents the standard model and illustrates the misspecification issues using Norwegian data. Section 4 presents the proposed model and Section 5 shows how estimation of the standard model on simulated data from our proposed model yields the same structure of misspecification issues as with actual data. InSection6,weestimateourproposedmodelusingNorwegianandU.S.data,illustrating that the parameter estimates are now independent of the moments or weighting matrix applied and that our proposed model is still parsimonious enough to be estimated on a smalldataset(PSID).Section7concludes. 2 Data The analysis uses Norwegian administrative data on annual income. We combine this income data with demographic information such as sex, age, country of birth, and years of education. To make our results comparable to the rest of the literature, we restrict our attention to males born in Norway with income between 1971 and 2014. In a partial analysispresentedinAppendixA,wealsoutilizeamonthlyincomedatasetcoveringthe period from 2015 to 2019. The Norwegian income data have several advantages relative to other available data sets. First, the data are administrative and, therefore, cover the entire population of Norwegian residents. Second, only a small share of the earnings are right-censored.3 Therefore, our data include precisely measured earnings for almost everyone. Moreover,thelongpanelfrom1971to2014allowsustofollowindividualsfor long periods. Indeed, the data include the complete earnings history from the first job to retirementforsomecohorts. Variabledefinitions. Ourmeasureofpre-taxearningsincludesbothlaborincome(from wages and self-employment) and work-related cash transfers such as unemployment and short-term sickness benefits. In the analysis, pre-tax earnings are deflated with the consumer price index, indexed to the 2011 Norwegian kroner. The observations we use 3Bhuller,Mogstad,andSalvanes(2017)documentthatlessthan3percentofthesampleisright-censored inanygivenyear. Theincomeseriesisavailablefrom1967,butHalvorsen,Ozkan,andSalgado(2022)show thattop-codingwasmostprevalentfrom1967to1970,andthatfrom1971lessthan1%ofobservationsare right-censoredeachyear. 6

foridiosyncraticincomeareresidualsobtainedfromaregressionoflogearningsonafull setofdummiesforyear,age,andyearsofeducation. Sample selection. We select our sample from this data according to the suggestions in Daly,Hryshko,andManovskii(2022). Theyarguefortheimportanceofusingabalanced samplesincegapsin thedatacanbiastheestimation ofincomeprocesses,andthebiasis ff ff di erentforestimationusingmomentsinlevelsanddi erences. Wefocusonindividuals in the middle of their careers, restricting attention to ages 35 to 50.4 When doing so, we include individuals if earnings observations are available for all the years from 34 to 51 to ensure a balanced sample.5 In addition, we restrict our sample by removing observationsofextremeincomechangesandobservationswheretheincomelevelisvery low. We define extreme income changes as observations where income either increased by more than 500 percent or decreased by more than 80 percent as in Daly, Hryshko, and Manovskii(2022). AverylowlevelofincomeisdefinedtobebelowtheNorwegiansocial security system’s definition of a base level (around USD 10,000 in 2011). Observations where an individual has a lower income level than this base level are considered years wheretheindividualisonlylooselyattachedtothelaborforceandistreatedasamissing observation. If an individual ever experiences an extreme income change or a very low level of income, then all observations for that individual are dropped from the sample. Thus,therequirementofabalancedpanelismaintained. With these sample selection criteria, our data include 536,399 Norwegian males distributedover27cohortsbornbetween1937and1963. 3 Problems with the Standard Model Thissectionillustratessomewell-knownmisspecificationissueswiththestandardmodel using the Norwegian income data. We first describe the model and how to identify the model parameters. Next, we present estimation results for all combinations of moments andweightingmatricestypicallyapplied. Weshowthattheresultingparameterestimates varywiththemomentsandweightingmatrix,suggestingthatthemodelismisspecified. 4We start at age 35 because the variance of income by age is decreasing with age before age 35 in the Norwegiandata,drivenbythevarianceofhigh-skilledindividuals(Blundell,Graber,andMogstad,2015). Neitherthestandardnorourproposedmodelcanaccountforsuchapattern. AppendixBprovidesmore detailonoursampleselection. 5FollowingDaly,Hryshko,andManovskii(2022),thefirst(34)andlast(51)observationsensurethatwe includeindividualswhoworkedtheentirefirst(35)andlast(50)observationinoursample. 7

3.1 The Standard Model Wewillusethe‘standardmodel’asabenchmarktohighlightsomeoftheexistingproblems in the literature and how our proposed model resolves these problems. While there are many variants of the standard model that we show here—most commonly including ff some decay in the permanent component—they all su er from the same problems that ourproposedmodelcanresolve.6 The standard model is written in the same frequency as the data, most commonly annual. Logincome, y ,iscomposedofapermanentcomponent,p ,alongwithtransitory t t ε θ shocks, ,thathavesomepersistence, . Themodelcanbewrittenas t = +ε +θε y t p t t t−1 (1) = +ν p t p t−1 t (2) ε ν where and are uncorrelated with each other and also over time, in this way, the t t permanent component of income moves as a random walk at an annual frequency and thetransitorycomponentofincomefollowsanMA(1)process. Theprocessbeginsattime zero with some existing distribution of permanent income. The parameters of interest in this model are the variance of permanent and transitory income shocks, σ2 and σ2 ν ε t t respectively, as well as the ‘start-of-working-life’ variance of permanent income, σ2 and p0 θ theMA(1)parameter . ff Researchers can use either level or di erence moments to identify the parameters in ff themodel. Iftheyusedi erencemoments,thecovariancestructureofpermanentincome is: var( ∆ p ) = σ2 (3) t ν t ∆ ,∆ = (cid:44) . cov( p p ) 0 ifs t (4) t s and the covariance structure of the transitory component is (defining the transitory com- = ε +θε ponentasq t t t−1 ) var( ∆ q ) = σ2 + (1 −θ )2σ2 +θ2σ2 (5) t ε t ε t−1 ε t−2 cov( ∆ q t ,∆ q t−1 ) = − (1 −θ ) σ2 ε t−1 +θ (1 −θ ) σ2 ε t−2 (6) cov( ∆ q t ,∆ q t−2 ) = −θσ2 ε t−2 (7) ∆ ,∆ = < − . cov( q q ) 0 ifs t 2 (8) t s 6AppendixFestimatesboththestandardandproposedmodelallowingforgeneralpersistenceandfind thesamestructureofmisspecificationissuesalsowithageneralpersistence. 8

Using the independence of the permanent and transitory components of income, the ff covariancestructureofthedi erenceinlogincomeisthesumofthecovariancestructure ofeachcomponent ∆ ,∆ = ∆ ,∆ + ∆ ,∆ . cov( y y ) cov( p p ) cov( q q ) (9) t s t s t s It is then common to apply the general method of moments on (3)-(9) and to estimate parameters by minimizing the distance between the empirically observed moments and thoseimpliedbythemodel. Theapproachissimilarwhenusinglevelmoments,andthis isdeferredtoAppendixC.1. 3.2 Results Using the Standard Model We now estimate the standard model using the Norwegian data. We first describe our estimationprocedurebeforewepresenttheestimatedparameters. ff Estimationprocedure. WestartwithMbalancedpanels,eachonestartinginadi erent year. For each panel we calculate the empirical covariance matrix for either the levels or ff di erencesofincome: 1 (cid:88)N = EmpiricalLevels t,s N y i,t y i,s (10) i=1 1 (cid:88)N = ∆ ∆ EmpiricalFD t,s N y i,t y i,s (11) i=1 where y i.t isresidualizedlogincomeofindividualiattimetasdescribedinSection2. Our ff minimumdistanceprocedurefordi erencesusesthelossfunction: (cid:88)M L = vech(EmpiricalFD − ModelFD )TΩ−1vech(EmpiricalFD − ModelFD ) j j j j j j=1 Ω andequivalentlyforlevels. Here iseitherthefulloptimalminimumdistanceweighting j matrix for panel j (OWMD), the optimal minimum distance weighting matrix along ff the diagonal with all o -diagonal elements set to zero (DWMD), or the identity matrix (EWMD). Results. Table1presentsestimatedparametersusingallcombinationsofmomentsand weighting matrix typically applied in the literature. There are several notable observa- 9

EWMD DWMD OWMD ff ff ff Level Di erence Level Di erence Level Di erence σ2 0.004 0.011 0.004 0.011 0.005 0.007 perm σ2 0.032 0.020 0.033 0.020 0.021 0.021 tran θ 0.570 0.070 0.574 0.071 0.163 0.145 σ2 0.062 (cid:55) 0.062 (cid:55) 0.059 (cid:55) init Notes: ThetablepresentsestimatedparametersusingthestandardmodelonNorwegiandata. ‘Level’and‘Difference’denotethe typeofmomentsused,correspondingto(10)and(11),respectively.‘EWMD’,‘DWMD’,and‘OWMD’isthetypeofweightingmatrix applied,correspondingtoidentity,diagonallyoptimal,andfulloptimalweightingmatrix,respectively. Thetableshowsthemeanof parameterestimatesovertimeandage. Table1: Estimatedparametersusingthestandardmodel,Norwegiandata ff tions. First, the estimated parameters di er depending on whether one estimates the ff model using level or di erence moments. For example, looking at the ‘DWMD’ column, theestimatedvarianceofpermanentincomeis0.004whenestimatedusinglevelmoments ff but almost three times as large when using di erence moments. Similarly, the transitory ff variance is 0.033 when using the levels compared with 0.020 when using the di erence ff moments. These di erences in estimated parameters are large. Given that permanent inff comeriskisofprimaryimportanceforhouseholdwelfare,thesedi erencessubstantially ff a ectshouseholdbehaviorinmodels. Hence,itisimportanttounderstandtheoriginsof thisresultandhowtoestimatepermanentandtransitoryincomevariancesconsistently. ff Second,whiletheestimatedpermanentvarianceishigherwhenestimatedusingdi erence rather than level moments, the estimated transitory variance is lower. For example, the estimation using level moments with DWMD yields a lower estimate of permanent ff riskbutahigherestimateofthevarianceoftransitoryriskthanestimationusingdi erence momentswithDWMD.Contrarymovementsinparameterestimatessuggestthatpartof thevariationinthedataisnotuniquelyallocatedtothetransitoryorpermanentvariance bythemodel. Instead,thedecompositiondependsonthemomentsused. Thethirdobservationisthatwhenusingthefulloptimalminimumdistanceweighting matrix (OWMD), one gets similar parameter estimates regardless of which moments are used. Daly, Hryshko, and Manovskii (2022) notes this and argues that this estimation method, combined with a balancing of the panels, yields consistent estimates of the permanent and transitory variance. However, although parameter estimates are similar across moments used, it does not imply that these estimates yield the correct parameter values. We have now established the main misspecification issues of the standard model. The rest of the paper aims to provide an alternative yet simple model robust to these 10

misspecificationissues. 4 The Proposed Model ff Wenowpresenttheproposedmodelfocusingonhowitdi ersfromthestandardmodel. Therearetwoinnovationstothemodel: 1. Wewriteitincontinuoustime. 2. Weallowforthreetypesofincomeshocks. This section shows that combining these two innovations can solve the identification issues in the standard model discussed in Section 3. Moreover, since we only add one additional parameter, our proposed model is parsimonious enough to be estimated in relativelysmalldatasets. 4.1 Time Aggregation Thefirstchangewemakeistowritethestandardmodelincontinuoustime. Thischange ff in timing allows us to model income shocks that occur at di erent times throughout the calendar year. This adjustment is motivated by well-known time aggregation problems with the standard model, such as the implicit assumption that shocks occur on January 1st. InAppendixA,weshow,usingmonthlyincomedatafromNorway,theunsurprising result that income innovations such as job transitions, job losses, bonus payments, and pay rises tend to occur throughout the calendar year. Although we work in continuous time, a monthly or even quarterly model would serve the same purpose and result in similar estimates to those here. The key feature is that the model frequency should ffi be su ciently higher than the frequency of the observed data. Note that the model’s frequency (continuous-time) does not dictate the frequency of income shocks—for any individual,theymayonlyarriveonaverageeverydecade. Itisthedistributionofshocks withintheyearthatmatters,nottheirfrequency. Motivatedbytheempiricalobservations inAppendixA,wemodelthearrivalofshocksasbeinguniformlydistributedthroughout thecalendaryear. 4.2 Additional Shocks The second innovation is allowing for three types of income shocks in our model rather than two in the standard model. Permanent shocks are the exact analog of those in 11

the standard model but embedded in continuous time. Bonus shocks are the analog of transitory shocks in the standard model with no MA(1) component and can also be thought of as encompassing classical measurement error.7 Finally, in place of the MA(1) process for transitory shocks, we propose ‘passing’ income shocks persisting a stochastic periodoftime. Therestofthissubsectiondescribeseachincomeshocktypeindetail,how these shocks are identified in the data, and how the standard model may misinterpret theseshocksinthepresenceoftimeaggregation. Shock type 1: Permanent. A permanent shock to income can be thought of as a promotion, a wage rise, or a job change. In our proposed model, such shocks are equally likelytooccuratanytimewithinthecalendaryear. Figure1aillustratesanexampleofthe income flow coming from this type of process. The solid-orange line shows the income flow,whiletheblackcrossesshowobservedincome. Inthisexample,apermanentincome shock occurs about one-quarter of the way into 2017, and income remains high for the remaining years. Notice that observed income for each year smooths the shock over two years. In 2017, the worker received the higher income flow for three-fourths of the year, and his income in 2017 is, therefore, three-fourths of the shock size higher than in 2016. From 2018, she received the higher income flow for the entire year, and the observed increaserelativeto2016isequaltotheshocksize. Theobservedincomeseriesexhibitapositiveserialcorrelationduetothe‘smoothing’ of shocks over two years. This smoothing is the crucial result in Working (1960), and ff it shows up in the covariance structure of the di erence of log income, and hence feeds throughtotheestimatedvarianceofpermanentandtransitoryshocks. Mathematically, we defined the permanent income process as follows. Let P be a t martingale with unit volatility and σ2 be the volatility of permanent income at time s. perm,s An example of such a P is a Brownian motion. However, the model allows for more t general martingale processes including those with large shocks. As such, although the model is set in continuous time, shocks such as promotions or job changes may occur infrequently. Wecanthendefinethepermanentincomeflowattimetas8 (cid:90) t p = p + σ2 dP . t 0 perm,s s 0 7In contrast to the standard model, our proposed model is able to distinguish between classical measurementerror(whichwouldbeinterpretedas‘bonus’shocks)fromtransitoryincomeshocks(see,e.g.,the discussioninMeghirandPistaferri,2011). 8Inthebodyofthepaper,wefocusonversionsofthemodelwherethepermanentcomponentofincome is a unit root. In Appendix F, we estimate the models in which the permanent component is allowed to mean-revert. 12

log time of income shock x x x x income flow observed income x time 2016 2017 2018 2019 2020 (a)Incomeflow var or cov x if all shocks occur on jan1 (standard in the literature) x x x x var(Δyₜ) cov(Δyₜ,Δyₜ₋₁) cov(Δyₜ,Δyₜ₋₂) cov(Δyₜ,Δyₜ₋₃) cov(Δyₜ,Δyₜ₋₄) (b)Covariancestructurefordifferencemoments ff Figure1: Permanentshockincomeflowanddi erencecovariancestructure However, the flow of income at any instance t is not observed. We instead observe the total income received over a one-year period. For example, the observed (permanent) incomeinyearT is9 (cid:90) T pobs = p dt . (12) T t T−1 9The permanent component of income is not observable by itself, but this is the relevant permanent incomecomponentofobservableincome. 13

ff Thecovariancestructureofthesepermanentincomeshocksindi erencesis 1 1 var( ∆ pobs) = σ2 + σ2 T 3 perm,T 3 perm,T−1 1 cov( ∆ pobs,∆ pobs ) = σ2 T T−1 6 perm,T−1 cov( ∆ pobs,∆ pobs) = 0 ifS < T − 1 . T S Figure1bshowsthecovariancestructureofsuchatime-aggregatedrandomwalkasblue bars, along with the standard-model version—equivalent to a continuous-time model in which all income shocks occur on January 1st —shown as orange crosses. Failing to accountforthepossibilitythatincomeshockscanhappenatanypointinthecalendaryear ff willleadtosignificantestimationproblemsifthemodelisestimatedusingthedi erence covariancestructure. Shock type 2: Bonus. The bonus shock consists of a one-time shock to income, like a bonus that arrives on a specific date. It could be either positive or negative. Figure 2a shows this type of shock modeled as an income ‘flow’ in continuous time in the solidorange line, along with the observed income in black crosses. In this example, the shock occursone-quarterofthewaythrough2017and,becauseitisinstantaneous,itismodeled asaDiracdeltafunctiontotheincomeflow. WeletB beamartingalewithunitvolatility, t and σ2 bethevolatilityofbonusincomeattimes. Observedbonusincomeisthesum bonus,s ofallbonusincomereceivedovertheyear,definedas (cid:90) T bobs = σ2 dB . (13) T bonus,s s T−1 ff The induced covariance structure for the di erence moments, shown in Figure 2b as blue bars, is identical to that induced by a transitory shock with no MA(1) component in thestandardmodel,shownasorangecrosses. Inparticular,thetimingoftheshockwithin ff the calendar year—one of our key innovations—makes no di erence to the covariance structureforthistypeofshock. The covariance of the change in income this period with the change in income in the ∆ ,∆ previousperiod(thenegativebluebarcov( y t y t−1 ))canbeusedinthestandardmodel withnoMA(1)componenttoidentifythesizeoftransitoryvariance. Thisisbecauseinthe ∆ ,∆ standard model, the component of this covariance (cov( y t y t−1 )) coming from the permanentshocksisequaltozero. Therefore,thechangeinincomenextperiodisnegatively correlated with transitory shocks this period, but uncorrelated with permanent shocks 14

log time of income shock observed x income diracdelta function income flow x x x x time 2016 2017 2018 2019 2020 (a)Incomeflow var or cov x if all shocks occur on jan1 (makes no difference) x x x x var(Δyₜ) cov(Δyₜ,Δyₜ₋₁) cov(Δyₜ,Δyₜ₋₂) cov(Δyₜ,Δyₜ₋₃) cov(Δyₜ,Δyₜ₋₄) (b)Covariancestructurefordifferencemoments ff Figure2: Bonusshockincomeflowanddi erencecovariancestructure thisperiod,conditionsthatmakethechangeinincomenextperiodavalidinstrumentfor transitory shocks this period in the standard model. Figure 1b shows that this is not the ∆ ,∆ case in our model—the existence of permanent shocks will increase cov( y t y t−1 ) and hencedepresstheestimateoftransitoryincomevarianceunderthestandardmodel. Shock type 3: Passing. The final income shock we introduce is a ‘passing’ shock. For this shock, income flow jumps by some amount at the time of the shock arrival. Income flow then remains at that new level, returning to the old level with a fixed hazard rate. Onecanthinkofthepassingshockasrepresentinganunemploymentspelloratemporary switchtopart-timeemployment. An alternative to the passing shock is to assume that the shock decays at a constant 15

rate over time, like an AR(1) process in discrete time. This type of shock process results in an identical covariance matrix to that of the passing shock process we describe here, andhenceestimationsoftheparametersinthesetwospecificationsareisomorphic. However, we choose to describe the model as having passing shocks for two reasons. First, Druedahl, Graber, and Jørgensen (2021) estimate a monthly model where they specify a shock type that is general enough to potentially contain both the passing variant and the AR(1) variant. They reject the AR(1) version and end up with an income process that is essentially similar to the passing shock in the current paper. Second, Arellano, Blundell, and Bonhomme (2017) show that individuals with very low income realizations tend to experiencesuddenlargeincreasesinincome. Thesesuddenlargeincreasesaremoreconsistent with the end of passing-type shocks when income jumps back toward the mean, ratherthanslowlydriftingtowardthemean. Figure 3a shows two examples of how the passing shock works. Approximately one quarter through 2017, a positive income flow shock hits. The income flow remains high foraroundoneyear(thesolid-orangeline)oraroundtwoyears(thedashed-orangeline). In both these examples, observed income for 2017 is about three-fourths the size of the flow shock (similar to the permanent shock example), but the observed income in 2018 either drops back toward its 2016 level (black cross) or climbs higher to be equal to the size of the shock (gray cross). By 2020 there is no residual income flow from either shock andtheobservedincomeisbacktoits2016level. Figure3bshowsthecovariancestructureforthispassingshockinbluebars. Depending ∆ ,∆ on the half-life of the passing shock, the second blue bar (cov( y t y t−1 )) may be either positive or negative. In the example illustrated, chosen to match the estimates from the ∆ ,∆ Norwegian administrative data, cov( y t y t−1 ) is slightly positive. Compared with the orangecrosses,whichmarkthecovariancestructureoftheseshocksundertheassumption theyonlyoccuronJanuary1steachyear(equivalenttoanAR(1)discretetimeprocess),the ∆ covariance structure (except for var( y )) is relatively flat and close to zero. We propose t thatthisisthereasonthepersistenceofthesepassingshockshasnotbeennotedbefore— because of the covariance structure shown in the blue bars in Figure 3b, the shock is mistaken for a permanent shock in the estimation of the standard model. Indeed, any ff estimation method will struggle to di erentiate between this covariance structure and thatshownbytheorangecrossesinFigure1b. We let Q be a martingale with unit volatility, and σ2 be the variance of passing t passing,s income at time s. Formally, the passing shock component of income flow can be written 16

log time of income shock x observed income change from2017to2018can x bepositiveornegative x x x x x time 2016 2017 2018 2019 2020 (a)Incomeflow var or cov x if all shocks occur on jan1 x x x x var(Δyₜ) cov(Δyₜ,Δyₜ₋₁) cov(Δyₜ,Δyₜ₋₂) cov(Δyₜ,Δyₜ₋₃) cov(Δyₜ,Δyₜ₋₄) (b)Covariancestructurefordifferencemoments ff Figure3: Passingshockincomeflowanddi erencecovariancestructure as (cid:90) t σ2 = passing,s1 q t Θ ( τ ) ξ s >t−s dQ s 0 Θ τ where ( ) is a normalization factor so that the total observed passing income variance is equal to σ2 .10 For each s > 0, ξ is an exponentially distributed random variable passing,s s P ξ > τ = . τ with ( ) 05—that is, the passing shocks have a half life equal to . The observed s 10TheexpressionforthenormalizingfactorΘ(τ)canbefoundinAppendixC.2 17

passingincomeisthen (cid:90) T qobs = q dt . (14) T t T−1 4.3 Mapping Between the Standard and the Proposed Model Our proposed model has five parameters. Three have almost exact counterparts in the standard model: initial permanent income variance, permanent income variance and transitory income variance.11 The two remaining parameters, the half-life of the passing τ = shock ( ) and the fraction of the transitory variance that is of the ’bonus’ variety (b σ2 /σ2 where σ2 = σ2 + σ2 ), roughly serve the same purpose as the MA(1) bonus tran tran bonus passing parameter in the standard model in determining how persistent transitory shocks are, although they are not numerically comparable. Table 2 shows how the parameters in the twomodelscompare. ParameterDescription Proposed Standard Permanentincomevariance σ2 Var( ν ) perm Transitoryincomevariance σ2 (1 +θ2)Var( ε ) tran τ (cid:55) Halflifeof‘passing’shock ‘Bonus’fractionof σ2 b (cid:55) tran (cid:55) θ MA(1)transitorypersistence Initialpermanentincomevariance σ2 Var(p ) init 0 Table2: Parametersintheproposedandstandardmodels 5 Simulation Results Inthissection,wepresentasimulationexercisethataimstoillustratehowthetwochanges we propose to the standard model can reconcile the misspecification problems discussed in Section 3. In particular, we show that in data simulated from our proposed income process, the standard model has the same type of misspecification issues as in actual 11Werefertotheincreaseinvarianceofpermanentincomeoverayearaspermanentincomevariance,but itcouldalsobecalledthepermanentvolatilityintheproposedmodel. Thetransitoryincomevarianceis definedasthevarianceoftransitoryincomeoveroneyearthatwouldbeinducedbythecurrenttransitory incomeprocess. Thatis,inthestandardmodel,transitoryincomevarianceattimeTis(1+θ2)Var(ε ),not T Var(ε )whichdoesnothaveaclearcounterpartintheproposedmodel. T 18

data. Hence, the simulation exercise indicates that the standard model’s problems can ffi be explained by time-aggregation issues combined with an insu cient description of transitoryincomeshocks. Data-generating process. Observed income in our proposed model consists of the sum ofpermanent,bonus,andpassingincome. Thesecomefromequations(12),(13),and(14) respectively, yobs = pobs + bobs + qobs. T T T T In order to simulate data from our proposed model, we approximate the continuoustime model by discretizing each year into N subperiods which we will call months. The underlyingpermanentincomeprocessinthecontinuous-timemodelcanbeapproximated as (cid:115) N((cid:88)T−1)+i σ2 = + permε . p T,i p 0 perm,j N j=0 Where p T,i is the annualized income in month i of year T, p 0 ∼ N (0 ,σ2 init ), and ε perm,j ∼ N , ∈ N (0 1)forall j . Overayear,thesimulatedobservedpermanentincomeis 1 (cid:88)N−1 po T bs = N p T,i i=0 (cid:115) = + 1 (cid:88)N−1N((cid:88)T−1)+i σ2 permε . p 0 perm,j N N i=0 j=0 Thesimulatedobservedbonuscomponentis (cid:115) (cid:88)N−1 σ2 bo T bs = b N onusε bonus,N(T−1)+i i=0 ε ∼ N , where bonus,j (0 1)forall j. Finally,apassingshockthatoccursinmonth jcontributes ξ − + topassingincomefor years. Thatis,atmonthN(T 1) i,thepassingshockfrommonth j ξ > − + − / j will be contributing to passing income only if (N(T 1) i j) N. The annualized j 19

incomefromthepassingcomponentofincomeinmonthiofyearTcanthenbewrittenas (cid:115) N((cid:88)T−1)+i σ2 = passing1{ξ > − + − / }ε q T,i Θ τ j (T 1) (i j) N passing,j N ( ) j=0 andtheobservedincomefrompassingincomeshocksis 1 (cid:88)N−1 qo T bs = N q T,i i=0 (cid:115) = 1 (cid:88)N−1N((cid:88)T−1)+i σ2 passing1{ξ > − + − / }ε Θ τ j (T 1) (i j) N passing,j N N ( ) i=0 j=0 ε ∼ N , ξ λ = /τ where passing,j (0 1) and j is exponentially distributed with parameter ln(2) forall j. We simulate a panel of 50,000 individual log income histories using the discretized version of our proposed model as the data-generating process over 16 years, dividing = each year into N 12 subperiods. We then calculate the covariance moments of both ff , ∆ ,∆ income levels and di erences, that is cov(y y ) and cov( y y ) for all t and s between 0 t s t s ff and 15. For each set of moments, we estimate the standard model using three di erent weightingmatrices: equallyweightedminimumdistance(EWMD),diagonallyweighted minimumdistance(DWMD),andtheoptimallyweightedminimumdistance(OWMD). Simulation results. Table 3 presents the simulation results. The column with the “True Value” shows the parameters of the data-generating process. We have not displayed the resultsofestimatingtheproposedmodelbecause,despitethedata-generatingprocessbeingadiscretizedversionoftheproposedmodel,allmethodsrecoverestimatedparameter valuesalmostindistinguishablefromthetrueparametervalues. The columns using EWMD and DWMD in Table 3 show the main misspecification ff problem: relativetoestimationusinglevelmoments,estimationusingdi erencemoments tend to overestimate permanent income variance and underestimate transitory income variance. This pattern of misspecification is the same as we found when estimating the ffi standard model in Norwegian data in Table 1. Indeed, Tables 1 and 3 are di cult to tell apart. Hence, the simulation exercise indicates that the standard model’s problems ffi canbeexplainedbytime-aggregationissuescombinedwithaninsu cientdescriptionof transitoryincomeshocks. Moreover,reflectingtheresultsinDaly,Hryshko,andManovskii(2022),andasshown 20

EWMD DWMD OWMD ff ff ff Parameter TrueValue Level Di erence Level Di erence Level Di erence σ2 0.005 0.005 0.013 0.005 0.013 0.008 0.009 perm σ2 0.038 0.031 0.017 0.032 0.017 0.021 0.020 tran τ (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) 2.0years (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) b 0.40 θ (cid:55) 0.50 0.10 0.53 0.10 0.18 0.16 σ2 0.065 0.065 (cid:55) 0.065 (cid:55) 0.064 (cid:55) init Notes:Thetablepresentsestimatedparametersusingthestandardmodelondatageneratedfromtheproposedmodel(N=1,000,000). ‘TrueValue’referstotheparametersusedinthedata-generatingprocess. ‘Level’and‘Difference’denotethetypeofmomentsused, correspondingto(10)and(11),respectively.‘EWMD’,‘DWMD’,and‘OWMD’isthetypeofweightingmatrixapplied,corresponding toidentity,diagonallyoptimal,andfulloptimalweightingmatrix,respectively. Table 3: Estimated standard-model parameters using simulated data from the proposed model ff in Section 3 when estimating the standard model on Norwegian data, there is no di erff ence between the estimates obtained using level and di erence moments when one uses OWMD.However,estimationusingOWMD—albeitsimilaracrossmomentsused—does not obtain the parameters from the data-generating process. Hence, not being sensitive tomomentsusedisnotsu ffi cienttoclaimthatthemodelnolongerismisspecified.12 Proposition1shedssomelightonwhytheoptimalweightingmatrixmayyieldsimilar resultsindependentofmomentsused. ff Proposition 1. For any data-generating process with a stationary di erence distribution, as → ∞ → ∞ T and N the optimal weighted minimum distance estimator will yield the same ff estimateirrespectiveofthetypeofmoments—levelsordi erences—used. The intuition for Proposition 1 is as follows. The OWMD estimation procedure is invariant to any invertible linear mapping of the moments used for estimation. So, if we ff had an invertible linear map from level moments to di erence moments, the parameter ff estimates from the OWMD estimation procedure would be identical in levels and di erff ences. No such linear mapping exists because the dimension of the di erence moments is less than the level moments. However, there is an invertible linear mapping between 12In Appendix H we further investigate the results presented in Daly, Hryshko, and Manovskii (2022). We follow the sample selection criteria they used for their Danish data as closely as possible to obtain a similarselectionfromtheNorwegianregistrydata. Thenwereplicatetheestimationstheypresentusing bothlevelanddifferencemomentswithOWMDandshowthesameresult: withabalancedpanelthereis nodifferencebetweenestimatesfromthetwosetsofmoments. However,wealsopresentestimatesusing EWMDandDWMDandforthoseweightingmatricestheparameterestimatesagaindependonthechoice of moments. Our simulation exercise therefore indicates that we cannot have confidence in the estimates obtainedusingOWMDeventhoughtheydonotdependonthechoiceofmoments. 21

ff the level moments and the di erence moments augmented with some additional terms. The ‘some additional terms’ relate to the covariance of time-zero income levels. But if T is large enough, the relative importance of these time-zero elements in the covariance matrix converges to zero and the estimated parameters become similar. Hence, while Daly, Hryshko, and Manovskii (2022) convincingly show the importance of using balanced panels in estimation, the results of the paper do not indicate that the standard model is well-specified. A more complete sketch proof of this proposition can be found inAppendixD.1. Notably, there are some combinations of weighting matrix and moments that yield parameter estimates that are close to the data-generating process in Table 3, even when the standard model is known to be misspecified. Indeed, the combination of EWMD or DWMD and level moments provides estimates that are close to the true permanent and transitory variances in the data-generating process.13 Proposition 2 shows that as long as the data-generating process has a random walk component and the transitory ffi income shocks are su ciently temporary, the standard model will consistently estimate the permanent income variance, and therefore also the total transitory income variance, whenusinglevelmomentsandEWMDforestimation. Proposition2. Foranydata-generatingprocessinwhichincomeismadeupfromarandomwalk τ permanentincomeprocessandtransitoryincomeshocksthatpersist periods,theequally-weighted minimumdistance(EWMD)estimatorwithlevelmomentsusingthestandardmodelconsistently estimates the permanent income variance and the transitory income variance as long as the panel >> τ lengthofthedataisT . >> , The intuitionfor Proposition 2 isas follows. Fors t,cov(y y ) doesnot depend on t s >> τ thevarianceoftransitoryincome. Therefore,ifT therewillbeenoughofthese‘long’ covariances for an unbiased estimate of permanent income variance. Transitory income varianceisidentifiedresidually,andwillthereforealsobeestimatedconsistently. Proposition 2 also sheds light on when estimation of the standard model using level moments may not be robust. In particular, if the permanent shocks decay slowly over time—as is common when estimating the standard model—the level estimation using EWMD will no longer provide unbiased estimates. The intuition is straightforward. If thepermanentcomponentoftheincomeprocessisnotaunitroot,thepermanentincome variancecannolongerbeidentifiedfromthe‘long’covariances. 13However, we note that the persistence of transitory shocks is too low. The parameters for the persistence of the transitory shock—θ in the standard model, b and τ in our proposed model—are not directly comparable. However,theθestimatedinthetablesuggeststhetransitoryshocksarelesspersistentthanin thedata-generatingprocess. 22

In Appendix G, we show the results of estimating the standard model on a shorter panelofsimulateddata,aswellasonasimulatedpanelinwhichthe‘permanent’shocks ff ff decayataslowrate. Wefindthatthesechangeshavelittlee ectonthedi erenceestimates (whicharealreadybiased)butcanintroducebiasinthelevelestimates. Inparticular,when thepermanentshocksslowlydecay,thetransitoryincomevarianceisoverestimated,and thepermanentincomevarianceisunderestimatedwhenusinglevelmoments,especially forlongerpanels. 6 Data Results We now proceed to estimate our proposed model using the Norwegian income data. We first show that the parameter estimates are relatively insensitive to the moments and weighting matrix applied.14 Hence, the two adjustments we make, addressing the time ffi aggregationissueandenrichingthedescriptionoftransitoryincomeshocks,aresu cient to significantly reduce the extent of misspecification. To illustrate that our proposed model is still parsimonious, we first estimate how income risk varies by age and time using Norwegian data and next also estimate our proposed model using data from the PanelStudyofIncomeDynamics. 6.1 Results using Norwegian Data Estimationdetails. Weallowfortheage-andtime-varyingestimatesfor σ2 and σ2 , perm tran withtherestrictionthatthevariancesareconstantthrougheachcalendaryear. Inorderto reduce oscillatory behavior in the permanent volatility estimate, we add a regularization penaltytothelossfunctionthatpenalizeschangesin σ2 fromyeartoyear. Inaddition, perm weadjustthemodeltoallowforaninstitutionalfeatureinNorwaywhereashare(approx. 10percentbuttime-varying)oflaborincomeispaidinthefollowingyearasvacationpay. Weadjustthemodel-impliedcovariancematrixtoaccountforthislag.15 Income risk. Panel A of Table 4 shows the parameter estimates using the proposed model with each of the six combinations of weighting matrix and moment choice. To easecomparison,PanelCofTable4alsoincludestheestimatesusingthestandardmodel 14Theestimationofallmodelsinthebodyofthepaperassumethatthepermanentcomponentofincome isaunitroot. InAppendixF,weshowthatallourmainresultsprevailalsoifwerelaxtheassumptionofa unitroot. 15We adjust the model-implied moments such that they are associated with (1−η)y t +ηy t−1 where η is thevacationpayshareandcomparethesewiththedata. Inpracticethisadjustmenthasonlyasmalleffect onourresults. 23

EWMD DWMD OWMD ff ff ff Level Di erence Level Di erence Level Di erence PanelA:ProposedModel σ2 0.003 0.005 0.003 0.005 0.003 0.004 perm σ2 0.039 0.038 0.038 0.039 0.039 0.036 tran τ 1.547 2.045 1.770 2.058 2.234 1.862 b 0.360 0.478 0.341 0.473 0.443 0.480 σ2 0.062 (cid:55) 0.062 (cid:55) 0.059 (cid:55) init PanelB:ProposedModel(nobonusshock) σ2 0.003 0.011 0.003 0.011 0.005 0.007 perm σ2 0.036 0.021 0.036 0.021 0.023 0.022 tran τ 0.982 0.065 1.202 0.080 0.231 0.157 b 0.000 0.000 0.000 0.000 0.000 0.000 σ2 0.064 (cid:55) 0.063 (cid:55) 0.060 (cid:55) init PanelC:StandardModel σ2 0.004 0.011 0.004 0.011 0.005 0.007 perm σ2 0.032 0.020 0.033 0.020 0.021 0.021 tran θ 0.570 0.070 0.574 0.071 0.163 0.145 σ2 0.062 (cid:55) 0.062 (cid:55) 0.059 (cid:55) init Notes:ThetablepresentsestimatedparametersusingtheproposedandstandardmodelsonNorwegiandata.‘Level’and‘Difference’ denote the type of moments used, corresponding to (10) and (11), respectively. ‘EWMD’, ‘DWMD’, and ‘OWMD’ is the type of weightingmatrixapplied,correspondingtoidentity,diagonallyoptimal,andfulloptimalweightingmatrix,respectively. Thetable showsthemeanofparameterestimatesovertimeandage. Table4: EstimatedparametersusingtheNorwegiandata (same as Table 1). Our proposed model’s estimated parameters are similar across all six combinationsofdatamomentsandweightingmatrices. Thepermanentincomevariance is between 0.003 and 0.005, while the transitory income variance is between 0.036 and 0.039. In contrast, the parameter estimates vary widely depending on the moment and methodappliedwhenestimatedusingthestandardmodel. Ourproposedmodelisconstructedbymakingtwoadjustmentstothestandardmodel, addressingtimeaggregationandenrichingthetransitoryshockprocess. Toillustratethat both adjustments are necessary, Panel B shows parameters estimated by the proposed ff modelbutwiththebonusshockturnedo . Inthiscase,wehavemadeonlyoneadjustment 24

to the standard model: addressing the time aggregation issue. The parameter estimates vary widely also in this case. Furthermore, the half-life of the passing shock is estimated ff ff (using di erences and EWMD or DWMD) to be close to zero—that is, the di erence estimationreallywantstheretobeabonusshockandreducesthedurationofthepassing shockuntilitapproximatesabonusshock. Hence,thestabilityofourparameterestimates do not come only from addressing the time aggregation issue. Instead, both adjustments tothestandardmodelareimportanttoreducemodelmisspecification.16 Similar to Daly, Hryshko, and Manovskii (2022), we also find that the parameter estimatesoftransitoryandpermanentvariancebecomesimilarifoneestimatesthestandard modelusingtheoptimallyweightedminimumdistancemethod. However,asshownusing simulated data, this does not suggest that the estimated variances are ‘more correct’. Indeed,theestimatedpermanentvarianceusingthestandardmodelinthesimulateddata wasupwardbiased. InTable4,wefindthesamepattern. TheestimatedpermanentvarianceishigherwhenusingthestandardmodelandOWMDthanwhenusingtheproposed modelandOWMD.Onereasonwhythishappensistheintroductionofthepassingshock. The covariance structure of the passing shock is similar to that of the permanent shock. Hence, when not including the passing shock, the model allocates part of the variation comingfromthepassingshockaspermanentincomevariance. In two of the six combinations of moments and weighting matrix in Table 4, the standard model provides similar permanent income variance estimates as the proposed model. Theproposedmodel’spermanentincomevarianceisbetween0.003and0.005. The standard model estimated using the equally or diagonally weighted minimum distance methodwithlevelmomentsprovidesparameterestimatesofpermanentincomevariance within the bounds of the proposed model. These are the same combination of moments and weighting matrix that provided estimates close to the data-generating process in the simulation exercises. Hence, while we express caution about using the standard model, our results indicate that if one has to use it (for example if the sample is too small), one should estimate the standard model using level moments and the diagonally or equally weighted minimum distance method. However, we warn that the transitory income variance is always lower when estimated using the standard model compared with the estimatesusingtheproposedmodel. 16Theotheralternativeintermediatemodelistoestimatethestandardmodelwithtwotypesoftransitory income shocks. However, as discussed for example in Meghir and Pistaferri (2011), it is not possible to estimate the standard model with a measurement error (or bonus) shock and an MA(1) income process becausetheincomeprocessisthenunderidentified. 25

0.0200 0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.0000 35 38 40 42 45 48 50 Age ecnairaV kcohS tnenamreP 0.06 Proposed, level Proposed, difference Standard, level 0.05 Standard, difference 0.04 0.03 0.02 0.01 0.00 35 38 40 42 45 48 50 Age ecnairaV kcohS yrotisnarT 2.5 2.0 1.5 1.0 0.5 0.0 35 38 40 42 45 48 50 Age noitaruD kcohS gnissaP detcepxE 1.0 0.8 0.6 0.4 0.2 0.0 35 38 40 42 45 48 50 Age ro noitcarF sunoB , level , difference Bonus, level Bonus, difference Figure4: Age-varyingestimates,Norwegiandata Income risk by age and time. To investigate whether our proposed model yields different trends in income risk from the standard model, we provide age-varying and timevarying estimates of income risk using both models. Figure 4 presents age profiles of income risk estimated using the Norwegian data. We restrict attention in this section to results using the diagonally weighted minimum distance method. Figure 4 thus disff plays the results for four di erent combinations of models (standard and proposed) and ff moments(levelsanddi erences). The main takeaway from Figure 4 is that the misspecification of the standard model mightmisleadresearcherstospuriousconclusionsaboutage-patterns. Thisobservationis illustratedinthetoprightpanelshowingtheestimatesofthetransitoryincomevariance. First,themisspecificationofthestandardmodelisvisibleasthelargediscrepancybetween the estimates using the standard model. Moreover, the age profiles estimated using the 26

0.0175 0.0150 0.0125 0.0100 0.0075 0.0050 0.0025 0.00010970 1980 1990 2000 2010 Year ecnairaV kcohS tnenamreP P P r r o o p p o o s s e e d d , , l f e d vels 0.06 Standard, levels Standard, fd 0.05 0.04 0.03 0.02 0.01 0.010970 1980 1990 2000 2010 Year (a)Permanentincomevariance ecnairaV kcohS yrotisnarT 0.080 0.075 0.070 0.065 0.060 0.055 Proposed, levels 0.050 Proposed, fd Standard, levels 0.045 Standard, fd 0.040 1975 1980 1985 1990 1995 Start-of-working-life Year (b)Transitoryincomevariance ecnairaV efil-gnikrow-fo-tratS Proposed, levels Standard, levels (c)Start-of-working-lifevariance Figure5: Time-varyingvarianceestimates,Norwegiandata ff standard model di er depending on the moments used. With level moments, transitory ff incomeriskdecreasesinage,whileitisapproximatelyflatifoneestimatesusingdi erence moments. Incontrast,ourproposedmodelislesssensitivetothetypeofmomentsusedin estimation. Indeed, irrespective of the moments used in estimation, the proposed model suggests a similar age pattern: transitory income variance declines slightly by age. The relativelyflatageprofilesofbothtransitoryandpermanentincomevarianceareconsistent withthefindingsinBlundell,Graber,andMogstad(2015). Figure5showshowestimatesofincomeriskhaschangedovertimeinNorway. Again, ff the figure highlights how estimation using the standard model may yield di erent estimated variance depending on the type of moments used. However, the time-trends are relatively similar across specifications. The permanent income variance has increased slightly across time, and the transitory income variance, while more noisy, has remained relatively stable across time. Moreover, start-of-working-life income variance has increased from around 0.05 to almost 0.08 from 1972 to 1998. Again, the two models find similartrends.17 Hence,ourresultssuggestthatalthoughthestandardmodelismisspecff ified in the sense that it provides very di erent estimates of the level of income risk, the time-trendsofincomeriskseemrelativelysimilaracrossmodels. 6.2 Results using the Panel Study of Income Dynamics Above, we illustrated that our proposed model provides stable parameter estimates of income risk in the Norwegian administrative data. In this section, to illustrate that the ffi modelissu cientlyparsimonious,weshowthatourproposedmodelperformswellalso 17Weonlyshowtheestimatesusinglevelmomentsbecausethedifferenceestimationmethodisunable toidentifythisparameter. 27

EWMD DWMD OWMD ff ff ff Level Di erence Level Di erence Level Di erence PanelA:ProposedModel σ2 0.010 0.008 0.008 0.010 0.006 0.007 perm σ2 0.065 0.080 0.066 0.074 0.053 0.051 tran τ 1.407 1.216 1.432 1.374 1.368 1.370 b 0.264 0.347 0.307 0.417 0.373 0.472 σ2 0.078 (cid:55) 0.092 (cid:55) 0.105 (cid:55) init PanelB:StandardModel σ2 0.012 0.021 0.010 0.019 0.009 0.010 perm σ2 0.047 0.044 0.054 0.042 0.034 0.036 tran σ2 0.069 (cid:55) 0.086 (cid:55) 0.105 (cid:55) init Notes:ThetablepresentsestimatedparametersusingtheproposedandstandardmodelsonthePSID.‘Level’and‘Difference’denote thetypeofmomentsused,correspondingto(10)and(11),respectively. ‘EWMD’,‘DWMD’,and‘OWMD’isthetypeofweighting matrixapplied,correspondingtoidentity,diagonallyoptimal,andfulloptimalweightingmatrix,respectively. SincethePSIDdata onlycontaineven-yearobservations,wedonotidentifyθinthestandardmodel. Thetableshowsthemeanofparameterestimates overtime. Table5: EstimatedparametersusingPSIDdata when using much smaller sample sizes in the Panel Study of Income Dynamics (PSID). We first describe the data source and sample selection before we estimate our proposed modelusingdatafromthePSID. PSID. The PSID has been the main source of data from which to estimate the idiosyncratic income process of households in the United States. In our analysis, we follow the ffi data selection criteria of Mo tt and Zhang (2018), “the dataset consists of male heads from 1970 to 2014, 30-59 years old who were not full-time students, had positive weeks worked and wage and salary earnings, and which excludes non-sample men and all in PSID over-samples.” Moreover, we only consider even year observations so that our estimationconsistentlyusesdataeverytwoyears.18 TheimportanceofestimatingusingabalancedpanelislaidoutinDaly,Hryshko,and Manovskii(2022). Takingthislessononboard,wecreate16balancedpanels,coveringdifferenttimeperiods,fromourunderlyingdata. Eachpanelspans14years(8observations, each 2 years apart). The first spans the years 1970 to 1984, the next 1972 to 1986, all the 18ThePSIDwasrunannuallyuntil1997,afterwhichishasbeenrunonlyeveryotheryear. 28

waytothelastpanelfor2000to2014. Theideaisthat14yearsislongenoughtoestimate the model, while the requirement that panels be balanced means there would be too few observationsinlongerpanels. ThesizeofourpanelsinthePSIDisfarsmallerthanthose from the Norwegian registry data. As a result, we do not allow the parameters to be age varying. Reassuringly, the estimates from Norwegian data imply little age variation. θ Furthermore, since the data is only even-year observations, we do not identify in the standardmodel. Incomerisk. Table5presentstheestimatedparametersusingtheproposedandstandard models in PSID data.19 There are four main takeaways. First, as was illustrated in the ff Norwegian data, the estimated parameters di er depending on the moments used when estimating using the standard model. For example, focusing on DWMD, permanent ff variance is almost twice as large when estimated using di erence moments compared with level moments. Parameter estimates from our proposed model are more similar. For example, the estimated permanent income variance varies between 0.006 and 0.010, as compared with 0.009 and 0.021 in the standard model. Second, despite the two-year gap between surveys, using the proposed model we find estimates for the half-life of passing shocks to be well over one year. We also find that these passing shocks make up more than half of the observed transitory income variance. Third, as shown by Daly, Hryshko, and Manovskii (2022), the standard model parameters are almost independent of moments used when using the optimal weighing matrix, although it does not imply thattheparameterestimatesusingthestandardmodelandtheoptimalweightingmatrix are more ‘correct’. Fourth, the parameter estimates from the standard model with level momentsareclosesttotheestimatesfromtheproposedmodel,suggestingthattheuseof levelmomentsisthebestavailableoptionifonehastorelyonstandardmodelestimates. ff ff Estimationofthestandardmodelusingdi erencemomentstendtoproduceverydi erent parameterestimates. Incomeriskovertime. Figure6displaystheestimatedpermanent,transitory,andstartof-working-lifevarianceovertimeestimatedusingPSIDdata. Themaintakeawayhereis thesameasfortheNorwegiandata. Whilethestandardmodeltendstoproduceestimates ff thatdi erfromtheproposedmodelinlevelsofincomerisk,thetimetrendsarerelatively similar. Estimates using both the standard and the proposed models suggest an increase inallthreecomponentsofincomeriskovertime. 19AppendixEprovidesbootstrappedconfidenceintervalsforthePSIDestimates. Thelargesamplesize of the Norwegian data makes the parameter estimate confidence bands small, and negligible next to the largemodeluncertainty. 29

0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.0010970 1980 1990 2000 2010 2020 Year ecnairaV kcohS tnenamreP Proposed, levels Proposed, fd 0.14 Standard, levels Standard, fd 0.12 0.10 0.08 0.06 0.04 0.02 0.010970 1980 1990 2000 2010 2020 Year (a)Permanentincomevariance ecnairaV kcohS yrotisnarT 0.18 Proposed, levels Proposed, fd Standard, levels 0.16 Standard, fd 0.14 0.12 0.10 0.08 0.06 0.04 1975 1980 1985 1990 1995 2000 2005 Start-of-working-life Year (b)Transitoryincomevariance ecnairaV efil-gnikrow-fo-tratS Proposed, levels Standard, levels (c)Start-of-working-lifevariance Figure6: Time-varyingvarianceestimates,PSIDdata 7 Conclusion Thestandardpermanent-transitoryincomeprocesshastwowell-knownproblems. First,it ff su ersfromtime-aggregationproblems. Second,itismisspecifiedbecausetheparameter estimates depend on the type of moments and weighting matrix applied. This paper proposesaparsimoniousmodelofincomedynamicsthat(i)tacklesthewell-knowntimeaggregation problem and (ii) is robust to the choice of moments or weighting matrix. The model only requires one additional parameter compared with the standard model. Hence,onecanestimateourproposedmodelusingrelativelysmalldatasetsandinclude itinheterogeneous-agentmodelswithoutneedingseveraladditionalstatevariables. We reiterate our conclusion for practitioners here. If the data set contains many panel observations, use our parsimonious model of income dynamics to estimate the income ff process. The model is robust to using specific moments (level or di erence) and the weighing matrix (optimally, diagonally, or equally weighted) applied. If the data set is ‘small,’ use the standard model, but estimate it using level moments and the equally or ff diagonally weighted minimum distance method. Using di erence moments potentially biases the results significantly. Therefore, we strongly advise against estimating the ff standardmodelusingdi erencemoments. 30

References A ltonji , J. G., A. A. S mith J r ., and I. V idangos (2013): “Modeling Earnings Dynamics,” Econometrica,81(4),1395–1454. A rellano , M., R. B lundell , and S. B onhomme (2017): “Earnings and Consumption Dynamics: ANonlinearPanelDataFramework,”Econometrica,85(3),693–734. B huller , M., M. M ogstad , and K. G. S alvanes (2017): “Life-Cycle Earnings, Education Premiums,andInternalRatesofReturn,”JournalofLaborEconomics,35(4),993–1030. B loom , N., F. G uvenen , L. P istaferri , J. S abelhaus , S. S algado , and J. S ong (2017): “The greatmicromoderation,”Workingpaper. B lundell , R., M. G raber , and M. M ogstad (2015): “Labor income dynamics and the insurancefromtaxes,transfers,andthefamily,”JournalofPublicEconomics,127,58–73. B raxton , J. C., K. F. H erkenhoff , J. L. R othbaum , and L. S chmidt (2021): “Changing Income Risk across the US Skill Distribution: Evidence from a Generalized Kalman Filter,”WorkingPaper29567,NationalBureauofEconomicResearch. B rowning ,M.,M.E jrnaes ,andJ.A lvarez (2010): “Modellingincomeprocesseswithlots ofheterogeneity,”ReviewofEconomicStudies,77(4),1353–1381. C arr , M. D., R. A. M offitt , and E. E. W iemers (2020): “Reconciling Trends in Volatility: EvidencefromtheSIPPSurveyandAdministrativeData,”WorkingPaper. rawley C ,E.(2020): “Insearchoflosttimeaggregation,”EconomicsLetters,189,108998. C rawley , E., and A. K uchler (2022): “Consumption Heterogeneity: Micro Drivers and MacroImplications,”AmericanEconomicJournal: Macroeconomics,forthcoming. D aly ,M.,D.H ryshko ,andI.M anovskii (2022): “ImprovingtheMeasurementofEarnings Dynamics,”InternationalEconomicReview,63(1),95–124. D e N ardi ,M.,G.F ella ,M.K noef ,G.P az -P ardo ,andR.V an O oijen (2021): “Familyand government insurance: Wage, earnings, and income risks in the Netherlands and the US,”JournalofPublicEconomics,193,104327. D e N ardi , M., G. F ella , and G. P az -P ardo (2020): “Nonlinear household earnings dynamics, self-insurance, and welfare,” Journal of the European Economic Association, 18(2), 890–926. D ruedahl , J., M. G raber , and T. J ørgensen (2021): “High Frequency Income Dynamics,” WorkingPaper. ika E ,L.(2018): “Incomedynamicswhenshocksoccurduringtheyear,”EconomicsLetters, 173,27–29. 31

G ottschalk , P., and R. M offitt (2009): “The rising instability of US earnings,” Journal of EconomicPerspectives,23(4),3–24. G ottschalk ,P.,R.M offitt ,L.F.K atz ,andW.T.D ickens (1994): “Thegrowthofearnings instability in the US labor market,” Brookings Papers on Economic Activity, 1994(2), 217– 272. G uvenen ,F.,F.K arahan ,S.O zkan ,andJ.S ong (2021): “WhatDoDataonMillionsofUS WorkersRevealAboutLifecycleEarningsDynamics?,”Econometrica,89(5),2303–2339. G uvenen , F., A. M c K ay , and C. R yan (2022): “A Tractable Income Process for Business CycleAnalysis,”WorkingPaper. H alvorsen , E., H. H olter , S. O zkan , and K. S toresletten (2020): “Dissecting IdiosyncraticEarningsRisk,”WorkingPaper. H alvorsen ,E.,S.O zkan ,andS.S algado (2022): “EarningsDynamicsandItsIntergenerationalTransmission: EvidencefromNorway,”Workingpaper. H eathcote , J., F. P erri , and G. L. V iolante (2010): “Unequal we stand: An empirical analysis of economic inequality in the United States, 1967–2006,” Review of Economic dynamics,13(1),15–51. H eathcote ,J.,K.S toresletten ,andG.L.V iolante (2005): “Twoviewsofinequalityover thelifecycle,”JournaloftheEuropeanEconomicAssociation,3(2-3),765–775. (2010): “ThemacroeconomicimplicationsofrisingwageinequalityintheUnited States,”JournalofPoliticalEconomy,118(4),681–722. K lein , P., and I. A. T elyukova (2013): “Measuring high-frequency income risk from lowfrequencydata,”JournalofEconomicDynamicsandControl,37(3),535–542. M c K inney , K. L., and J. M. A bowd (2020): “Male Earnings Volatility in LEHD before, during,andaftertheGreatRecession,”WorkingPaper. M eghir , C., and L. P istaferri (2011): “Earnings, consumption and life cycle choices,” in Handbookoflaboreconomics,vol.4,pp.773–854.Elsevier. offitt ollinger okayem iemers bowd arr c inney M , R., C. B , C. H , E. W , J. A , M. C , K. M K , S. Z hang , and J. Z iliak (2021): “Reconciling Trends in US Male Earnings Volatility,” WorkingPaper. M offitt ,R.,andS.Z hang (2018): “IncomeVolatilityandthePSID:PastResearchandNew Results,”AEAPapersandProceedings,108,277–80. M offitt , R. A., and P. G ottschalk (2002): “Trends in the transitory variance of earnings intheUnitedStates,”TheEconomicJournal,112(478),C68–C73. M offitt , R. A., and P. G ottschalk (2012): “Trends in the Transitory Variance of Male EarningsMethodsandEvidence,”JournalofHumanResources,47(1),204–36. 32

M offitt , R. A., and S. Z hang (2020): “Estimating Trends in Male Earnings Volatility with thePanelStudyofIncomeDynamics,”WorkingPaper. S abelhaus ,J.,andJ.S ong (2010): “Thegreatmoderationinmicrolaborearnings,”Journal ofMonetaryEconomics,57(4),391–403. S toresletten ,K.,C.I.T elmer ,andA.Y aron (2004): “Consumptionandrisksharingover thelifecycle,”JournalofMonetaryEconomics,51(3),609–633. orking ff W ,H.(1960): “NoteontheCorrelationofFirstDi erencesofAveragesinaRandom Chain,”Econometrica,28(4),916–18. 33

Online Appendix A Within-year Distribution of Income Shocks Thissectionpresentsevidenceonhigh-frequencyincomeinnovationsusingmonthlylabor income data from Norway. We first describe the data sources before providing evidence suggestingthatincomeinnovationsoccurthroughouttheyear. Data. For this part of the analysis, we use monthly matched employer-employee data maintained by Statistics Norway from 2015-2019. The data contains information on hoursworked,hourlywage,andsalaryperemployer-employee-month. Further,thedata includeadetaileddecompositionoflaborincomeintocontractedwage,bonuspayments, overtime,andhourlywagecontracts. noitcarF 51 01 5 0 Job change (job−to−job transitions) 1 2 3 4 5 6 7 8 9 10 11 12 Month noitcarF 51 01 5 0 Bonus payment 1 2 3 4 5 6 7 8 9 10 11 12 Month noitcarF 51 01 5 0 Job loss 1 2 3 4 5 6 7 8 9 10 11 12 Month noitcarF 51 01 5 0 Pay rise (within firm increase in wage) 1 2 3 4 5 6 7 8 9 10 11 12 Month Notes:Ajobchangeisdefinedasatransitionofaworkerfromamainemployertoadifferentmainemployer.Ajoblossisdefinedas atransitionfromanemploymentrelationshiptonoemploymentrelationship. Apayriseisdefinedasanincreaseinthecontracted wagebetween0.5%and10%withinthesameworker-firmrelationship. Duetotheprevalenceofvacationpaythateitherispaidin JuneorJuly,wecannotconfidentlyidentifypayrisesinthesemonthsandsetthemtomissing. FigureA.1: EvidencefortheDistributionofIncomeShocksWithintheCalendarYear 34

Within-yearincomeinnovations. ThestandardmodeldescribedinSection3implicitly assumesthatincomeshocksoccuronJanuary1steachyear. Toinvestigatethevalidityof this implicit assumption and motivate our proposed model, we use Norwegian monthly data on job events to determine if they are clustered at particular times of the year. The results for job changes, bonus payments, job losses, and pay rises are shown in the four panelsofFigureA.1. Thefiguremakesclearthatincomechangestendtooccurthroughout theyear. However,thereissomeclusteringofeventsaroundspecificdates. Forexample, job changes and job losses are more prevalent in January. Bonus payments tend to be paid out in March and December, and pay rises are more common in the second half of theyear. However,althoughthereissometemporalclustering,incomeshockstendtobe occur in all months and assuming that shocks are approximately uniformly distributed acrosstimewithinayearseemsreasonable. B Further detail on sample selection WehavechosentofocusouranalysisonNorwegianmalesinthemoststablepartoftheir working life—ages 35 to 50. This age restriction is somewhat more restrictive than other studies in the literature. There are a number of reasons for this restriction. First, our large sample size allows us to restrict our analysis to the prime of working life without sacrificing accuracy. Second, the main focus of ourpaper is to reconcile known problems withthestandardmodelandthisisbestdonewithoutintroducingtoomanyage-varying complicationstothemodel. Third,weseestrongevidenceintheNorwegiandatathatthe type of model we analyze here, both the standard model and our proposed model, does not fit our data for the young and the old. Figure A.2 shows how the variance of income changes with age. In our models, this variance is expected to grow with age, which it does in the age range we restrict to. However, in the 30-35 age range, we see income variance declining. Furthermore, we have found this cannot be explained by a reduction intransitoryvariance. Rather,itappearstocomefrommean-reversingpermanentshocks astheyoungconvergetowardstablejobsfrominformalorpart-timework. Inaddition,the agegroupabove50showsanincreaseinpermanentshocksthatappearstobeassociated withearlyretirements. Furtherworkinthesedirectionsmaybefruitfulbutisbeyondthe scopeofthispaper. 35

0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 30 35 40 45 50 55 60 Age noitubirtsiD emocnI fo ecnairaV Variance of Income Distribution by Age Notes:Thefigureshowsthemeanvarianceofincomeacrosscohortsforateachage. FigureA.2: Thevarianceofincomebyage C Moments C.1 Estimating the Standard Model using Level Moments If they use level moments, the covariance structure of the permanent component is, for < t s, (cid:88)t cov(p , p ) = var(p ) = σ2 + σ2. t s t p0 ν i i=1 and the covariance stucture of the transitory component is (define the transitory compo- = ε +θε nentasq t t t−1 ) var(q ) = σ2 +θ2σ2 t ε t ε t−1 cov(q t , q t−1 ) = θσ2 ε t−1 , = < − . cov(q q ) 0 ifs t 1 t s Using the independence of the permanent and transitory components of income, the covariancestructureoflogincomeisthesumofthecovariancestructureofeachcomponent , = , + , . cov(y y ) cov(p p ) cov(q q ) t s t s t s Onecanthenapplythegeneralmethodofmomentsandfindtheparametersthatminimize thedistancebetweentheempiricallyobservedmomentsandthoseimpliedbythemodel. 36

C.2 Passing Shock Moments The following definitions will be useful for calculating the model implied passing shock moments: (cid:90) 1 Λ 0,θ = θ e −θxdx 0 = − −θ 1 e (cid:90) 1 Λ 1,θ = x θ e −θxdx 0 1 1 = − −θ + e (1 ) θ θ (cid:90) 1 Λ 2,θ = x2θ e −θxdx 0 2 2 1 = − −θ + + . e (1 ) θ2 θ θ2 Nowconsiderthevarianceofthepassingshockforincomebetweentime0andtime1. It can broken down into components that come from shocks between time 0 and 1, time -1 and0,time-2and-1,etc. Thecomponentthatcomesfromshocksthattakeplacebetween time 0 and 1 is again broken into two parts: shocks that pass before time 1 and shocks that persist past time 1. Shocks that pass before time 1 and last for a period x contribute x2 to the variance. Shocks that arrive at time 0 < t < 1 and persist past time 1 contribute (1 − t)2 tothevariance. Summingthesetwopartsupwedefine: (cid:90) 1 (cid:32)(cid:90) 1−t (cid:90) ∞ (cid:33) Θ 0,θ,var = x2θ e −θxdx + (1 − t)2θ e −θxdx dt 0 0 1−t (cid:90) (cid:32) (cid:33) 1 2(1 − t) 2 = − (1 − t)2e −θ(1−t) − e −θ(1−t) + (1 − e −θ(1−t)) + (1 − t)2e −θ(1−t) dt θ θ2 0 1 (cid:16) 1 (cid:17) = −Λ − Λ . θ2 1 1,θ θ 0,θ 37

Nowconsidershocksthatarrivebetweentime-1and0. Again,thesecanbedividedinto twoparts,sowedefine: (cid:90) 1 (cid:32)(cid:90) 1 (cid:90) ∞ (cid:33) Θ −1,θ,var = e −θ(1−t) x2θ e −θxdx + θ e −θxdx dt 0 0 1 (cid:90) (cid:32)(cid:90) (cid:90) (cid:33) 1 1 1 = e −θ(1−t) x2θ e −θxdx + 1 − θ e −θxdx dt 0 0 0 1 = Λ −θ +Λ . θ 0,θ(e 2,θ) − − + ≥ Nowforshocksthatarrivebetweentime N and N 1wehaveforN 1: (cid:90) 1 (cid:32)(cid:90) 1 (cid:90) ∞ (cid:33) Θ −N,θ,var = e −θ(N−t) x2θ e −θxdx + θ e −θxdx dt 0 0 1 = e −θ(N−1)Θ −1,θ,var . Define the variance of a passing shock assuming the underlying martingale has variance 1: ∞ (cid:88) Υ θ = Θ . ( ) −N,θ,var N=0 Θ τ θ = /τ Notethisexpressionalsogivesusthenormalizingfactor ( ),substituting log(2) τ forthehalf-life : Θ τ = Υ /τ . ( ) (log(2) ) (A.1) Assumingconstantvariancebetweeneachintegertimeinterval,thevarianceofthepassingshockattimeT is: (cid:88) ∞ σ2 = T−N,passingΘ . Var T,passing Υ θ −N,θ,var (A.2) ( ) N=0 Similarly for the covariance of income between time 1 and 2 with that between 0 and 1, firstconsiderthecomponentthatcomesfromshocksbetween0and1anddefine: (cid:90) 1 (cid:32)(cid:90) 2−t (cid:90) ∞ (cid:33) Θ 0,θ,cov1 = (1 − t)(x − (1 − t)) θ e −θxdx + (1 − t) θ e −θxdx dt 0 1−t 2−t −θ 1 e = Λ2 + Λ . θ 1,θ θ 1,θ 38

Nowthecomponentthatcomesfromshocksbetween-1and0: (cid:90) 1 (cid:32)(cid:90) 3−t (cid:90) ∞ (cid:33) Θ −1,θ,cov1 = (1 − t)(x − (2 − t)) θ e −θxdx + (1 − t) θ e −θxdx dt 0 2−t 3−t e −θ e −2θ = Λ Λ + Λ θ 0,θ 1,θ θ 1,θ − − + > andhencethecomponentthatcomesfromshocksbetween N and N 1forN 1is: Θ −N,θ,cov1 = e −(N−1)θΘ −1,θ,cov1 . Thecovarianceoftime-aggregatedincomewithitslagisthen: (cid:88) ∞ σ2 = T−N−1,passingΘ . Cov T,T−1,passing Υ θ −N,θ,cov1 ( ) N=0 Finally,thecovarianceoftime-aggregatedincomewithitsMthlagisthen: (cid:88) ∞ σ2 Cov T,T−M,passing = T−N Υ −M θ ,passing e −θ(M−1)Θ −N,θ,cov1 . (A.3) ( ) N=0 Equations A.2 and A.3 give the model covariance matrix in level for the passing shocks. ff Infirstdi erences,wehave: ∆ ,∆ = , − , − , + , cov( y T y TS ) cov(y T y T−S ) cov(y T−1 y T−S ) cov(y T y T−S−1 ) cov(y T−1 y T−S−1 ) (A.4) ff fromwhichwecancalculatethemodelcovariancematrixinfirstdi erences. D Proofs D.1 Sketch Proof of Proposition 1 The aim of this proof is to show that using the optimal minimum distance estimator will ff give the same parameter estimates regardless of using level or di erence moments when ffi there are su cient panel observations of each individual. There are two steps in the proof. We first show that the optimal minimum distance estimator gives the same result when the moments are transformed by an invertible linear mapping. We next show that one can construct such an invertible linear mapping that transforms level moments into ff di erence moments, but it includes some additional terms. However, as we add more 39

panelobservations,thecontributionoftheseadditionaltermsconvergestozerosuchthat the optimal minimum distance estimator will yield the same result regardless of which momentsareused. = ,..., AssumethatY (y y )isarandomvariable,that f(Y)isafunctionthatgenerates 1 T θ data moments from Y, and that g( ) is a function generating model moments for a set of θ parameters . We can then formulate the original optimal minimum distance problem (OMD)as argmin E (f(Y) − g( θ )) (cid:48)Ω−1E (f(Y) − g( θ )) θ Ω = E −E (cid:48)E −E where (f(Y) (f(Y))) (f(Y) (f(Y))). Thisproblemisequivalentto argmin E (Af(Y) − Ag( θ )) (cid:48)Ω˜−1E (Af(Y) − Ag( θ )) θ foranyinvertiblelinearmapAofdimension T(T+1) where Ω˜ = E (Af(Y) −E (Af(Y))) (cid:48)E (Af(Y) − 2 E θ (Af(Y))). Hence, solving the optimal minimum distance problem of f(Y) and g( ) is θ equivalenttosolvingthesameproblemforAf(Y)andAg( ). ff To show that the level and di erence moments give the same estimates under the optimal minimum distance estimator, we have to show that there exists such an A that ff transformslevelmomentsintodi erencemoments. Thelevelmomentsaredefinedas = (cid:48) f(Y) vech(Y Y) ff andthedi erencemomentsaredefinedas ∆ = ∆ (cid:48) ∆ . f( Y) vech(( Y) ( Y)) ∆ − ∆ ff SinceYisoflengthTand YisoflengthT 1, f(Y)and f( Y)havedi erentdimensionso = ∆ there does not exist an invertible A such that Af(Y) f( Y). However, we can construct aninvertiblelinearmappingAwithsomeextraterms: Af(Y) = [y2, y ∆ y , y ∆ y ,..., y ∆ y , f( ∆ Y)] . 1 1 1 1 2 1 T ∆ ∆ → ∞ Given that Y is a stationary process, y y goes to zero in expectation as n . Thus, 1 n → ∞ as T , the extra terms play no role in the minimization so long as the covariance ∆ matrixof Y isafunctionoftheparameterofinterest. 40

D.2 Sketch Proof of Proposition 2 Suppose the data-generating process for income consists of initial permanent income, permanentincomethatfollowsarandomwalk,andtransitoryshocksthatpersistnomore = , ,..., than N periods. Assume that Y (y y y ) is random variable of length T generated 0 1 T fromthisincomeprocess. Thecovariancematrixforthisprocesscanbesummarizedas: var(y ) = σ2 + t σ2 +σ2 (A.5) t init perm trans , ≤ cov(y t y t+s )isunconstrained ifs N (A.6) cov(y t , y t+s ) = σ2 init + t σ2 perm ifs > N (A.7) Thestandardmodelmomentsare: var(y ) = σ2 + t σ2 +σ2 (A.8) t init perm trans cov(y t , y t+s ) = σ2 init + t σ2 perm ifs ≤ N (A.9) cov(y t , y t+s ) = σ2 init + t σ2 perm ifs > N (A.10) ff Giventhatthetransitoryvarianceonlya ectsthevalueofA.8,givenadiagonalweighting matrix(eitherEWMDorDWMD)theestimationmethodologywillchoose σ2 toexactly trans match the model and empirical var(y ). As a result, A.8 will have no role to play in the t estimation of σ2 and σ2 . Indeed, these will be entirely determined by A.10 as T → ∞ perm init → ∞ because the relative weight assigned to A.9 will go to zero (T but N is fixed, and , the weight on cov(y t y t+s ) will tend to a fixed positive number under both EWMD and DWMD). As A.10 is identical to A.7, the estimation of σ2 and σ2 will be consistent perm init resultingthereforeinanconsistentestimateof σ2 fromA.8. trans E Bootstrapped PSID confidence intervals Table A.1 shows the PSID estimation results for both the proposed and standard model, ff levelanddi erencemoments,alongwiththeirbootstrappedconfidenceintervalsinparenthesesbelow. 41

EWMD DWMD OWMD Level Difference Level Difference Level Difference PanelA:ProposedModel σ2 0.010 0.008 0.008 0.010 0.006 0.007 perm (0.008,0.012) (0.005,0.012) (0.007,0.010) (0.006,0.014) (0.005,0.007) (0.005,0.008) σ2 0.065 0.080 0.066 0.074 0.053 0.051 tran (0.057,0.075) (0.068,0.088) (0.057,0.073) (0.056,0.079) (0.035,0.050) (0.036,0.051) τ 1.407 1.216 1.432 1.374 1.368 1.370 (1.258,1.440) (1.152,1.399) (1.262,1.463) (1.356,1.452) (1.255,1.471) (1.279,1.463) b 0.264 0.347 0.307 0.417 0.373 0.472 (0.243,0.288) (0.314,0.440) (0.297,0.329) (0.359,0.502) (0.316,0.451) (0.352,0.554) σ2 0.078 (cid:55) 0.092 (cid:55) 0.105 (cid:55) init (0.067,0.098) (cid:55) (0.079,0.107) (cid:55) (0.088,0.114) (cid:55) PanelB:StandardModel σ2 0.012 0.021 0.010 0.019 0.009 0.010 perm (0.010,0.014) (0.016,0.025) (0.008,0.011) (0.015,0.022) (0.007,0.009) (0.008,0.011) σ2 0.047 0.044 0.054 0.042 0.034 0.036 tran (0.040,0.053) (0.036,0.051) (0.046,0.058) (0.032,0.047) (0.022,0.033) (0.025,0.034) σ2 0.069 (cid:55) 0.086 (cid:55) 0.105 (cid:55) init (0.058,0.088) (cid:55) (0.074,0.101) (cid:55) (0.068,0.090) (cid:55) TableA.1: PSIDestimationwithbootstrappedstandarderrors F Persistent (but not permanent) shocks Thepaperrestrictsattentiontoestimatingincomeprocesseswherethepermanentcomponent is a unit root. In this appendix, we present the paper’s main results when we allow thepersistentpartoftheincomeprocesstohaveanarbitrarypersistence. Specifically,weadjustthestandardincomeprocesstobe = +ε +θε y p (A.11) t t t t = ρ + p t p t−1 v t (A.12) ρ where the adjustment is to allow for a that is not equal to 1. Similarly, we adjust the persistentpartoftheproposedmodeltobe = ρ − +σ dp log( )(p p )dt dP t t 0 perm t ρ = wheretheadjustmentisthatlog( ) 0intheversionofthemodelinthepaper,whilewe ρ allowforageneral inthisappendix. 42

EWMD DWMD OWMD ff ff ff Level Di erence Level Di erence Level Di erence PanelA:ProposedModel σ2 0.004 0.006 0.004 0.006 0.003 0.005 perm σ2 0.038 0.034 0.041 0.035 0.039 0.033 tran τ 1.494 1.810 1.328 1.815 2.241 1.640 b 0.390 0.540 0.407 0.526 0.443 0.525 ρ 0.991 0.979 0.982 0.985 1.000 0.989 σ2 0.063 (cid:55) 0.063 (cid:55) 0.059 (cid:55) init PanelB:StandardModel σ2 0.010 0.014 0.010 0.015 0.011 0.010 perm σ2 0.022 0.017 0.022 0.016 0.019 0.019 tran θ 0.216 0.026 0.233 0.000 0.088 0.092 ρ 0.923 0.894 0.921 0.809 0.926 0.952 σ2 0.063 (cid:55) 0.062 (cid:55) 0.059 (cid:55) init Notes:ThetablepresentsestimatedparametersusingtheproposedandstandardmodelsonNorwegiandata.‘Level’and‘Difference’ denote the type of moments used, corresponding to (10) and (11), respectively. ‘EWMD’, ‘DWMD’, and ‘OWMD’ is the type of weightingmatrixapplied,correspondingtoidentity,diagonallyoptimal,andfulloptimalweightingmatrix,respectively. Thetable showsthemeanofparameterestimatesovertimeandage. TableA.2: Estimationwithpersistent,butnotpermanentshock TableA.2presentstheestimatedincomeprocessesonNorwegiandatawhenweallow ρ ≤ for a 1. The proposed model provides consistent estimates of the model parameters even in this more general case. In particular, across all combinations of moments and ρ weightingmatricesapplied,theestimatesof arecloseto1,andtheestimatedpermanent andtransitoryvariancesarestable. In contrast, estimates of the standard model vary more depending on moments, and ρ . . weighting matrices applied. For example, the estimates of vary from 081 to 095. The estimates of the permanent variance also vary more. However, when adjusted for the persistence of the permanent component (annual variance = σ2 / (1 −ρ2)), they are perm only slightly higher than the permanent variance estimates from the proposed model. Moreover,whenweestimateusingtheoptimalweightedminimumdistancemethod,we getsimilarresultsregardlessofthemomentsused,asdiscussedinProposition1. TableA.3alsopresentstheresultswhenwesimulatetheproposedmodelandestimate thestandardmodel. Again,wedonotprovidetheestimatesoftheproposedmodelsinceit isthedata-generatingprocess. Comparedwiththedata-generatingprocess,thestandard 43

EWMD DWMD OWMD ff ff ff Parameter TrueValue Level Di erence Level Di erence Level Di erence σ2 0.005 0.008 0.015 0.009 0.015 0.013 0.013 perm σ2 0.038 0.026 0.015 0.027 0.015 0.017 0.017 tran τ (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) 2.0years (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) b 0.40 θ (cid:55) 0.38 0.07 0.39 0.07 0.12 0.11 ρ 1.00 0.97 0.93 0.97 0.93 0.94 0.95 σ2 0.065 0.062 (cid:55) 0.063 (cid:55) 0.064 (cid:55) init Table A.3: Estimation of simulated data on the standard model with persistent, but not permanentshock model, in this case, tends to underestimate the persistence of the permanent component. Furthermore, the standard model tends to overestimate permanent variance somewhat and underestimate the transitory variance, suggesting that some of the transitory shocks inthemodelaremisrepresentedaspermanentshockswhenestimatedusingthestandard model. Thispatternofmisrepresentedshockswasalsopresentinthesimulationexercise inthetext. G Further Simulation Results Tables A.4, A.5, and A.6 show estimation results for the standard model for simulation = results varying the length of the panel (T 16 or 5) and introducing some decay in the ρ = . permanent shock (we set 097 in the proposed model data-generating process). The ρ = estimationresultsreportedareforthestandardmodelwith 1. = Two results are noteworthy. First, reducing the panel size to T 5 results in the level ff estimation mildly underestimating the transitory income variance, while the di erence estimates are unchanged and remain far from the true parameter values. Second, if the ‘permanent’ income shock decays even slowly over time, estimating the standard model ρ = with 1, especially for long panels, results in an overestimation of the transitory variance andan underestimationof the permanentvariance whenusing level moments. The ff di erenceestimatesarelittlechanged. Thisbiasinthelevelmomentsisalsofoundwhen ρ = we estimate the proposed model with 0 and is caused by the model misinterpreting thedecayingpermanentshocksastransitory. ρ Inthepaperwedonotallowforpermanentshockstodecay. Whenweestimated in theNorwegiandata,wefoundittobeequalto0. 44

EWMD DWMD OWMD ff ff ff Parameter TrueValue Level Di erence Level Di erence Level Di erence σ2 0.005 0.005 0.012 0.005 0.012 0.008 0.012 perm σ2 0.038 0.027 0.017 0.027 0.017 0.021 0.018 tran τ (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) 2.0years (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) b 0.40 θ (cid:55) 0.32 0.10 0.32 0.10 0.18 0.12 σ2 0.065 0.073 (cid:55) 0.073 (cid:55) 0.070 (cid:55) init TableA.4: Estimatedstandard-modelparametersusingdatasimulatedfromtheproposed = model(T 5) EWMD DWMD OWMD ff ff ff Parameter TrueValue Level Di erence Level Di erence Level Di erence σ2 0.005 0.003 0.012 0.003 0.012 0.005 0.008 perm σ2 0.038 0.038 0.017 0.039 0.017 0.023 0.021 tran τ (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) 2.0years (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) b 0.40 θ (cid:55) 0.71 0.10 0.76 0.10 0.21 0.18 σ2 0.065 0.048 (cid:55) 0.048 (cid:55) 0.041 (cid:55) init TableA.5: Estimatedstandard-modelparametersusingdatasimulatedfromtheproposed = ρ= model(T 16, 0.97) EWMD DWMD OWMD ff ff ff Parameter TrueValue Level Di erence Level Di erence Level Di erence σ2 0.005 0.003 0.012 0.003 0.012 0.006 0.011 perm σ2 0.038 0.029 0.017 0.029 0.017 0.023 0.018 tran τ (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) 2.0years (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) (cid:55) b 0.40 θ (cid:55) 0.36 0.11 0.36 0.11 0.21 0.12 σ2 0.065 0.060 (cid:55) 0.060 (cid:55) 0.056 (cid:55) init TableA.6: Estimatedstandard-modelparametersusingdatasimulatedfromtheproposed = ρ= model(T 5, 0.97) 45

H Replicating DHM on Norwegian data As mentioned in Section 1, the paper most closely related to ours is Daly, Hryshko, and Manovskii (2022) (DHM). Therefore, this section replicates their results on Norwegian ff data by comparing estimates using moments in levels and di erences. They use registry data from both Denmark and Germany, and here we aim to mimic as closely as possible ff thethreedi erentsampleselectioncriteriathattheyapplytotheDanishdata. We again use the Norwegian registry data described in section 2 and restrict our attention to males born in Norway. As DHM do in the Danish data, we further restrict attentiontothosebornintheyearsfrom1951to1955andonlyusewagedatafrom1981to 2006. Wealsodropindividualswhoseeducationalstatushaschangedduringtheirlongest spell(discussedfurtherbelow). Outliersaredefinedasyear-to-yearearningsincreasesof more than 500 percent or a decrease of more than 80 percent. Individuals with earnings outlierswithintheirlongestspellaredropped. There are two selection criteria that DHM apply to the Danish data that we cannot replicateexactlyintheNorwegiandata: 1)Theydroprecordswhereindividualsworked less than 10 percent of the year as a full-time employee, and 2) they remove individuals who were ever self-employed. We handle both of these by referring to the Norwegian social security system’s definition of a base level of income (“grunnbeløp” which is abbreviated to ‘G’), which is used as a basis for calculations of various social security and pensionbenefits. Thefirstcriteriamentionedaboveishandledbydroppingobservations whereincomeisbelow1G.Thisshouldcaptureindividualswhoareonlylooselyattached tothelaborforceduringtheyear. Thesecondcriteriadropsobservationswherebusiness incomeisabove1Gwhichshouldcapturethosewhoareself-employed.20 DHM’s focus is on the importance of using a balanced sample, and to highlight this ff they contrast estimates obtained from three di erent samples in their paper. The first sample (“Balanced”) only keeps individuals where observations are available for all 26 years they consider. The second sample (“9 consec.”) constructs spells of consecutive observationsforeachindividualandonlykeepsthoseindividualswherethelongestspell contains at least 9 consecutive observations. Only the observations within that longest spell are kept, so that there are no gaps in the resulting data set. The third sample (“20 not nec. consec.”) keeps individuals where at least 20 income observations are available but does not require that these are consecutive. So in this sample, the “longest spell” is not relevant and the data can contain gaps. Table A.7 shows the number of individuals ff we obtain in the three di erent samples and compares them to the numbers obtained by 20Notethatthemeasureofbusinessincomeisonlyavailablefrom1993. 46

Sample Norwegiandata Danishdata(fromDHM) Sample1-Balanced 71,825 67,008 Sample2-9consec. 98,078 102,825 Sample3-20notnec. consec. 90,305 90,668 TableA.7: NumberofindividualsinthethreesamplesinNorwegianandDanishdata DHMintheDanishdata. Tables A.8, A.9, and A.10 present results from estimating the standard model from ff section3.1usingbothmomentsinlevelsanddi erencesforeachofthethreesamples. Table A.8 shows that we get similar results to DHM when we follow their approach and use the optimal weighting matrix—that is, the inverse of the variance-covariance matrix of the data moments—in the estimation. Columns (1)—(4) show results for the samples with 9 or more consecutive observations and with 20 or more observations that are not necessarily consecutive. For both of these samples, we see the same patterns that DHMreportfortheDanishdata: Theestimatedpersistenceandvarianceofthepermanent shock as well as the estimated persistence of the transitory shock are higher when using ff di erence moments, while the estimated variance of the transitory shock is higher when ff usingmomentsinlevels. Columns(5)and(6)showthatthesedi erencesdisappearwhen usingthebalancedsamplewhereindividualsareonlyincludedifdataisavailableforall of the 26 years. Thus the estimation on Norwegian registry data give results very similar totheonesobtainedbyDHMforDanish(andGerman)data. ff Tables A.9 and A.10 show that using a di erent weighting matrix in the estimation— respectivelytheinverseofadiagonalweightingmatrixusingonlythevarianceofthedata moments and the identity matrix—does not yield the same results as in DHM. In both ff of those estimations, we get that using di erence moments leads to a higher estimated variance of the permanent shock and a lower estimated variance of the transitory shock even in the balanced sample. The estimated values of persistence also depend on the ff momentsused,buttherankingdi ersdependingontheweightingmatrix. ff That the choice of weighting matrix a ects the estimation results in this way is yet another indication that the standard model is misspecified and that simply using a balanced panel does not fix the issue. We remain convinced that using a balanced panel is important for the reasons pointed out by DHM, but the observation that the estimated values in columns (5) and (6) of table A.8 are the same does not necessarily show that these estimates are correct. As shown in the estimation on simulated data presented in ff table3,obtainingthesameestimatedvaluesusingmomentsinlevelsanddi erencesdoes 47

9consec. 20notnec. consec. Balanced ff ff ff Levs. Di s. Levs. Di s. Levs. Di s. ρ 0.952 0.990 0.967 0.981 0.970 0.975 (0.001) (0.0003) (0.0006) (0.0005) (0.0006) (0.0008) σ2 0.010 0.015 0.008 0.013 0.006 0.006 perm (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) θ 0.221 0.250 0.195 0.263 0.273 0.272 (0.002) (0.003) (0.003) (0.003) (0.003) (0.003) σ2 0.017 0.009 0.019 0.010 0.009 0.009 tran (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) (0.0001) σ2 0.025 — 0.027 — 0.025 — init (0.0004) — (0.0004) — (0.0004) — χ2 8,312 5,259 8,970 8,705 8,335 7,228 (d.f.) 346 321 346 321 346 321 Table A.8: Estimates of the earnings process in Norwegian administrative data using = = DHM’ssampleselectioncriteria. Vmatrix DHM( optimalweightingmatrix) notguaranteethattheestimationsrecoverthetrueparametervalues. 48

9consec. 20notnec. consec. Balanced ff ff ff Levs. Di s. Levs. Di s. Levs. Di s. ρ 0.968 0.990 0.977 0.889 0.979 0.828 (0.0002) (0.0007) (0.0002) (0.0037) (0.0002) (0.0064) σ2 0.008 0.017 0.006 0.017 0.005 0.011 perm (0.00003) (0.0002) (0.00003) (0.0002) (0.00002) (0.00016) θ 0.477 0.233 0.440 0.206 0.737 0.211 (0.012) (0.003) (0.010) (0.004) (0.044) (0.004) σ2 0.023 0.009 0.026 0.008 0.012 0.007 tran (0.0003) (0.0001) (0.0003) (0.0001) (0.0006) (0.0001) σ2 0.025 — 0.029 — 0.026 — init (0.0001) — (0.0001) — (0.0001) — χ2 11,883 6,147 18,423 7,643 13,985 5,993 (d.f.) 346 321 346 321 346 321 Table A.9: Estimates of the earnings process in Norwegian administrative data using = DHM’ssampleselectioncriteria. Vmatrix diagonal. 9consec. 20notnec. consec. Balanced ff ff ff Levs. Di s. Levs. Di s. Levs. Di s. ρ 0.969 0.992 0.981 0.987 0.983 0.988 (0.279) (4.65) (0.293) (5.38) (0.336) (8.71) σ2 0.007 0.018 0.006 0.018 0.005 0.011 perm (0.046) (0.668) (0.040) (0.670) (0.040) (0.669) θ 0.466 0.239 0.460 0.239 0.726 0.245 (14.6) (14.3) (12.3) (14.8) (52.5) (16.6) σ2 0.024 0.010 0.028 0.010 0.014 0.008 tran (0.348) (0.439) (0.343) (0.443) (0.745) (0.442) σ2 0.026 — 0.031 — 0.027 — init (0.124) — (0.126) — (0.127) — χ2 0.0064 0.0012 0.0119 0.0017 0.0079 0.0013 (d.f.) 346 321 346 321 346 321 Table A.10: Estimates of the earnings process in Norwegian administrative data using = DHM’ssampleselectioncriteria. Vmatrix identity. 49