1 Federal Reserve Bank of New York Staff Reports What Do Data on Millions of U.S. Workers Cycle Earnings Risk? - bout Life a Reveal Fatih Guvenen Fatih Karahan Serdar Ozkan Jae Song Staff Report No. 710 2015 February This paper presents preliminary findings and is being distributed to economists and other interested readers solely to stimulate discussion and elicit comments. s The views expressed in thi s paper are those of the author and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal s Reserve System. Any errors or omissions are t he responsibility of the author .

2 What D a bout Life - Cycle Earnings Risk? o Data on Millions of U.S. Workers Reveal Fatih Guvenen, Fatih Karahan, Serdar Ozkan, and Jae Song 710 Federal Reserve Bank of New York Staff Reports , no. 2015 February JEL classification: E24, J24, J31 Abstract over the life cycle using a large panel data We study the evolution of individual labor earnings , Using fully nonparametric set of earnings histories drawn from U.S. administrative records. methods, our analysis reaches two broad conclusions. First, earnings shocks display substantial mality — the standard assumption in the deviations from lognor incomplete markets . literature on In particular, earnings shocks display strong negative skewness and extremely high kurtosis — as that , in a given high as 30 compared with 3 for a Gaussian distribution. The high kurtosis implies but non - negligible year, most individuals experience very small earnings shocks, and a small properties vary significantly both number experience very large shocks. Second, these statistical dividuals. We also estimate impulse response over the life cycle and with the earnings level of in find important asymmetries: P functions of earnings shocks and - income ositive shocks to high individuals are quite transitory, whereas negative shocks are very persistent; the opposite is true for low - income indi viduals. Finally, we use these rich sets of moments to estimate econometric processes with increasing generality to capture these salient features of earnings dynamics. Key words: cycle earnings risk, nonparametric estimation, kurt osis, earnings dynamics, life - Gaussian shocks, normal mixture - skewness, non _________________ - mail: Guvenen: University of Minnesota, Federal Reserve Bank of Minneapolis, and NBER (e ). Karahan: Federal Reserve Bank of New York (e - mail: [email protected] [email protected] ). Ozkan: University of Toronto (e - mail: [email protected] ). Song: Social Security Administration (e - For helpful critiques and mail: [email protected]). the authors thank Joe Altonji, Andy Atkeson, Richard Blundell, Michael comments, - Keane, Giuseppe Moscarini, Fabien Postel Vinay, Kjetil Storesletten, Anthony Smith, and seminar and conference participants at various universities and research institutions. The views expressed in thi s paper are those of the author s and do not nec essarily reflect the position s of the Social Security Administration, the Federal Reserve Bank s of Minneapolis and New York, or the Federal Reserve System.

3 1 Introduction This year about 2 million young American men will enter the labor market for the first time. Over the next 40 years, each of these men will go through his unique ad- venture in the labor market, involving a series of surprises—finding an attractive career, being o ff ered a dream job, getting promotions and salary raises, and so on—as well as disappointments—experiencing unemployment, failing in one career and moving on to another one, su ff ering health shocks, and so on. These events will vary not only in their ff ects turn out to be in initial significance (upon impact) but also in how durable their e 1 the long run. An enduring question for economists is whether these wide-ranging labor market his- tories, experienced by a diverse set of individuals, display su ffi ciently simple regularities that would allow researchers to characterize some general properties of earnings dynam- ics over the life cycle. Despite a vast body of research since the 1970s, it is fair to say that many aspects of this question remain open. For example, what does the probabil- ity distribution of earnings shocks look like? Is it more or less symmetric, or does it display important signs of skewness? More generally, how well is it approximated by alognormaldistribution,anassumptionoftenmadeoutofconvenience? And,perhaps more important, how do these properties di ff er across low- and high-income workers or change over the life cycle? A host of questions also pertain to the dynamics of earnings. For example, how sensible is it to think of a single persistence parameter to characterize ff erent the durability of earnings shocks? Do positive shocks exhibit persistence that is di from negative shocks? Clearly, we can add many more questions to this list, but we have to stop at some point. If so, which of these many properties of earnings shocks are the most critical in terms of their economic importance and therefore should be included in this short list, and which are of second-order importance? One major reason why many of these questions remain open has been the heretofore 2 ffi ciently rich panel data on individual earnings histories. Against unavailability of su this backdrop, the goal of this paper is to characterize the most salient aspects of life- 1 In this paper, we focus on the earnings dynamics of men so as to abstract away from the complexities of the female nonparticipation decision. We intend to undertake a similar study that focuses on the earnings dynamics of women. 2 With few exceptions, most of the empirical work in this area has been conducted using the Panel Study of Income Dynamics (including the previous work of the authors of this paper), which contains be- tween 500 to 2,000 households per year depending on the selection criteria and su ff ers from shortcomings that are typical of survey data, such as survey response error, attrition, and so on. 1

4 cycle earnings dynamics using a large and confidential panel data set from the U.S. Social Security Administration. The substantial sample size—of more than 200 million observations from 1978 to 2010—allows us to employ a fully nonparametric approach and take what amounts to high-resolution pictures of individual earnings histories. In deciding what aspects of the earnings data to focus on, we were motivated in this paper by a growing body of theoretical work (reviewed in the next section), which attributes a central role to skewness and kurtosis of economic variables for questions ranging from the e ff ects of monetary policy to optimal taxation, and from the determi- nants of wealth inequality to asset prices. Therefore, we focus on the first four moments of earnings changes over the life cycle. This analysis reaches two broad conclusions. First, the distribution of individual earnings shocks displays important deviations from lognormality. Second, the magnitude of these deviations (as well as a host of other sta- tistical properties of earnings shocks) varies greatly both over the life cycle and with the earnings level of individuals. Under this broad umbrella of “non-normality and life-cycle variation,” we establish four sets of empirical results. First, starting with the first moment, we find that average earnings growth over the life cycle varies strongly with the level of lifetime earnings: the median individual by lifetime earnings experiences an earnings growth of 38% from ages 25 to 55, whereas for individuals in the 95th percentile, this figure is 230%; for those in the 99th percentile, 3 this figure is almost 1500%. Second, turning to the third moment (postponing the second moment for now), we see that earnings shocks are negatively skewed, and this skewness becomes more severe as individuals get older or their earnings increase (or both). Furthermore, this increasing negativity is due entirely to upside earnings moves becoming smaller from ages 25 to 45, and to increasing “disaster” risk (the risk of a sharp fall in earnings) after age 45. Although these implications may appear quite plausible, they are not captured by a lognormal specification, which implies zero skewness. Third, studying the fourth (standardized) moment, we find that earnings changes display very high kurtosis. What kurtosis measures is most easily understood by looking at the histogram of log earnings changes, shown in Figure 1 (left panel: annual change; right panel: five-year change). Notice the sharpness in the peak of the empirical density, 3 Apositiverelationshipbetweenlifetimeearningsandlife-cycleearningsgrowthistobeexpected (since, all else equal, fast earnings growth will lead to higher lifetime earnings). What is surprising is the magnitudes involved, which turn out to be hard to match standard income processes. 2

5 One-year change Five-year change 4.5 1.6 US Data US Data 4 Normal (0,0.48) Normal(0, 0.68) 1.4 Std. Dev. = 0.48 Std. Dev. = 0.68 Skewness = –1.35 3.5 1.2 Skewness = –1.01 Kurtosis = 17.80 Kurtosis = 11.55 3 1 2.5 0.8 Density 2 Density 0.6 1.5 0.4 1 0.2 0.5 0 0 -3 -2 -1 3 0 1 2 0 1 2 3 4 -4 -3 -2 -1 y y − y − y +1 t t t +5 t Figure 1 – Histogram of Log Earnings Changes. Note: The first year t is 1995, and the data are for all workers in the base sample defined in Section 2. how little mass there is on the “shoulders” (i.e., the region around ), and how long the ± tails are compared with a normal density chosen to have the same standard deviation as in the data. Thus, there are far more people with very small earnings changes in the data compared with what would be predicted by a normal density. Furthermore, this kurtosis masks significant heterogeneity across individuals by age and earnings: average prime-age males with recent earnings of $100,000 (in 2005 dollars) face earnings shocks with a kurtosis as high as 35, whereas young workers with recent earnings of $10,000 face a kurtosis of only 5. This life-cycle variation in the nature of earnings shocks is one of the key focuses of the present paper. What do these statistics mean for economic analyses of risk? Although a complete answer is beyond the scope of this paper, in Section 7 we provide some illustrative calcu- lations. They suggest that the risk premium that will be demanded to bear the measured four to twenty times larger than the one cal- earnings fluctuations can be anywhere from culated with a Gaussian distribution with the same standard deviation. Although these figures are suggestive, and a complete answer requires a fuller investigation, these back- of-the-envelope calculations provide a glimpse into the potential of these documented higher-order moments for economic analyses. Fourth, we characterize the dynamics of earnings shocks by estimating non-parametric impulse response functions conditional on the recent earnings of individuals and on the size of the shock that hits them. We find two types of asymmetries. One, fixing the shock 3

6 size, positive shocks to high-earnings individuals are quite transitory, whereas negative shocks are very persistent; the opposite is true for low-earnings individuals. Two, fixing ff ers by the size of the earnings level of individuals, the strength of mean reversion di the shock: large shocks tend to be much more transitory than small shocks. To our knowledge, both of these findings are new in the literature. These kinds of asymmetries are hard to detect via the standard approach in the literature, which relies on the au- 4 tocovariance matrix of earnings—the second In this cross-moments of the panel data. regard, our approach is in the spirit of the recent macroeconomics literature that views impulse responses as key to understanding time-series dynamics in aggregate data (e.g., et al. Christiano 2005 ), Borovicka et al. ( 2014 )). ( While this nonparametric approach allows us to establish key features of earnings dynamics in a robust fashion—which we view as the main contribution of this paper— atractableparametricprocessisindispensableforconductingquantitativeeconomic analyses. The standard approach in the earnings dynamics literature is to estimate the parameters of linear time-series models by matching the variance-covariance matrix of log earnings residuals. This approach has two di ffi culties. First, the strong deviations from lognormality documented in this paper call into question the wisdom of focusing exclusively on covariances at the expense of the rich variation in higher-order moments, which will miss key features of earnings risk faced by workers. Second, the covariance matrix approach makes it di ffi cult to select among alternative econometric processes, because it is di cult to judge the relative importance—from an economic standpoint— ffi of the covariances that a given model matches well and those that it does not. This is an important shortcoming given that virtually every econometric process used to calibrate economic models is statistically rejected by the data. With these considerations in mind, in Section 5 , we follow a di ff erent route and target the four sets of empirical moments—broadly corresponding to the first four moments of earnings changes—described above, employing a method of simulated moments (MSM) estimator. We believe this is a more transparent approach: economists can more easily judge whether or not each of these moments is relevant for the economic questions they have in hand. Therefore, they can decide whether the inability of a particular stochastic process to match a given moment is a catastrophic failure or a tolerable shortcoming. 4 These asymmetries are di ffi cult to detect because a covariance lumps together all sorts of earning changes—large, small, positive, and negative—to produce a single statistic. This approach, although economical in its use of scarce data, masks lots of interesting heterogeneity, as revealed by our analysis. 4

7 Specifically, we estimate a set of stochastic processes with increasing generality to 5 Two main findings stand out. provide a reliable “user’s guide” for applied economists. First, allowing for a rich mixture of AR(1) processes seems essential for matching the salient features of the data, especially the large deviations from normality. Second, a heterogenous income profiles (HIP) component also plays a key role in explaining data features, but only when considered together with the mixture structure. A corollary to these findings is that the workhorse model in the literature—a persistent AR(1) (or random walk) process plus a transitory shock with normal innovations—fails to match most of the prominent features of the earnings data documented in this paper. The paper is organized as follows. In Section ,wedescribethedataandtheempirical 2 approach. Section 3 presents the findings on the cross-sectional moments of earnings 4 presents the impulse response analysis. Section 5 describes the growth and Section 6 parametric estimation and Section 7 concludes. presents its results. Section Related Literature 6 Since its inception in the late 1970s, the earnings dynamics literature has worked with the implicit or explicit assumption of a Gaussian framework, thereby making no use of higher-order moments beyond the variance-covariance matrix. One of the few exceptions is an important paper by ( 2000 ), who emphasize the Geweke and Keane non-Gaussian nature of earnings shocks and fit a normal mixture model to earnings innovations. More recently, Bonhomme and Robin ( 2009 )analyzeFrenchearningsdata over short panels and model the transitory component as a mixture of normals and the dependence patterns over time using a copula model. They find the distribution of this transitory component to be left skewed and leptokurtic. In this paper, we go beyond the overall distribution and find substantial variation in the degree of non-normality with age and earnings levels. Furthermore, the impulse response analysis shows the need for a di ff erent persistence parameter for large and small shocks, which is better captured as 7 amixingofAR(1)processes—astepbeyondthenormalmixturemodel. 5 The nonparametric analysis yields more than 10,000 empirical moments of individual earnings data. It is not feasible (or sensible) to estimate every conceivable stochastic process to match combinations of these moments. However, these moments are available for download as an Excel file (from the authors’ websites), so researchers can estimate their preferred specification(s). 6 Earliest contributions include Lillard and Willis ( 1978 ), Lillard and Weiss ( 1979 ), Hause ( 1980 ), and ( 1982 ). MaCurdy 7 Geweke and Keane ( 2007 )studyhowregressionmodelscanbesmoothlymixed,andourmodeling approach shares some similarities with their framework. 5

8 Incorporating higher-order moments of earnings dynamics into economic models is still in its infancy. In an early attempt, ( 1986 ) shows that if idiosyncratic earn- Mankiw ings shocks become more negatively skewed during recessions, this could generate a large equity premium. Using nonparametric techniques and rich panel data, Guvenen et al. ) document that the skewness of individual income shocks becomes more negative 2014 ( Constan- in recessions, whereas the variance is acyclical. Building on this observation, 2014 ) show that an incomplete markets asset pricing model with tinides and Ghosh ( countercyclical (negative) skewness shocks generates plausible asset pricing implications, McKay ( 2014 )studiesaggregateconsumptiondynamicsinabusinesscyclemodel and Golosov et al. that is calibrated to match these skewness shocks. Turning to fiscal policy, ( 2014 ) show that using an earnings process with negative skewness and excess kurtosis (targeting the empirical moments reported in this paper) implies a marginal tax rate on labor earnings for top earners that is substantially higher than under a traditional 8 calibration with Gaussian shocks with the same variance. Methodologically, our work is most closely related to two important recent contri- Altonji et al. ( 2013 )estimateajointprocessforearnings,wages,hours,and butions. Browning et al. job changes, targeting a rich set of moments via indirect inference. ( 2010 )alsoemployindirectinferencetoestimateanearningsprocessfeaturing“lotsof heterogeneity” (as they call it). However, neither paper explicitly focuses on higher-order moments or their life-cycle evolution. The latter paper does model heterogeneity across individuals in innovation variances, as do we, and finds a lot of heterogeneity along that dimension in the data. In ongoing research, Arellano et al. ( 2014 ) also explore di ff erences in the mean-reversion patterns of earnings shocks across households that di ff er in their earnings histories. Using data from the Panel Study of Income Dynamics, they find asymmetries in mean reversion that are consistent with those we document in Section . 4 Relatively little work has been done on the life-cycle evolution of earnings dynamics, which is the main focus of this paper. A few papers (including Baker and Solon ( 2003 ), ( 2004 Meghir and Pistaferri Karahan and Ozkan ( 2013 ), and Blundell et al. ( 2014 )) ), allow age-dependent innovation variances but do not explore variation in higher-order moments. Our conclusion on the variance is consistent with this earlier work, indicating a decline in variance from ages 25 to 50, with a subsequent rise. 8 Higher-order moments are gaining a more prominent place in recent work in monetary economics (e.g., Midrigan ( 2011 )and Berger and Vavra ( 2011 ); see Nakamura and Steinsson ( 2013 )forasurvey) )and as well as in the firm dynamics literature (e.g., etal. ( 2011 Bloom Bachmann and Bayer ( 2014 )). 6

9 2 Empirical Analysis The SSA Data 2.1 The data for this paper come from the Master Earnings File (MEF) of the U.S. Social Security Administration records. The MEF is the main source of earnings data for the SSA and contains information for every individual in the United States who was ever issued a Social Security number. Basic demographic variables, such as date of birth, place of birth, sex, and race, are available in the MEF along with several other variables. The earnings data in the MEF are derived from the employee’s W-2 forms, which U.S. employers have been legally required to send to the SSA since 1978. The measure of labor earnings is annual and includes all wages and salaries, bonuses, and exercised stock options as reported on the W-2 form (Box 1). Furthermore, the data are uncapped (no top coding) since 1978. We convert nominal earnings records into real values using the personal consumption expenditure (PCE) deflator, taking 2005 as the base year. For background information and detailed documentation of the MEF, see Panis et al. ( 2000 ) and Olsen and Hudson ( 2009 ). Constructing a nationally representative panel of males from the MEF is relatively straightforward. The last four digits of the SSN are randomly assigned, which allows us to pick a number for the last digit and select all individuals in 1978 whose SSN ends 9 with that number. This process yields a 10% random sample of all SSNs issued in the United States in or before 1978. Using SSA death records, we drop individuals who are deceased in or before 1978 and further restrict the sample to those between ages 25 and 60. In 1979, we continue with this process of selecting the same last digit of the SSN. Individuals who survived from 1978 and who did not turn 61 continue to be present in the sample, whereas 10% of new individuals who just turn 25 are automatically added (because they will have the last digit we preselected), and those who died in or before 1979 are again dropped. Continuing with this process yields a 10% representative sample of U.S. males in every year from 1978 to 2010. Finally, the MEF has a small number of extremely high earnings observations. In each year, we cap (winsorize) observations above the 99.999th percentile in order to avoid potential problems with these outliers. 9 In reality, each individual is assigned a transformation of their SSN number for privacy reasons, but the same method applies. 7

10 Figure 2 – Timeline For Rolling Panel Construction Sample selection works in two steps. First, for each year we define a Base Sample. , which includes all observations that satisfy three criteria, to be described in base sample amoment.Second,toselectthe final sample for a given statistic that we analyze below, we select all observations that belong in the base sample in a collection of years, the details of which vary by the statistic and the year for which the statistic is constructed. For a given year, the base sample is constructed as follows. First, we restrict attention to individuals between the ages of 25 and 60 to focus on working-age population. Second, we select workers whose annual wage/salary earnings exceeds a time-varying minimum threshold, denoted by Y ,definedasone-fourthofafull-yearfull-time(13weeksat40 min ,t hours per week) salary at half of the minimum wage, which amounts to annual earnings of approximately $1,885 in 2010. This condition helps us avoid issues with taking the logarithm of small numbers and makes our analysis more comparable to the empirical earnings dynamics literature, where a condition of this sort is fairly standard (see, among others, Abowd and Card ( 1989 ), Meghir and Pistaferri ( 2004 ), and Storesletten et al. ( 2004 )). Third, the base sample excludes individuals whose self-employment earnings Y exceed a threshold level, defined as the maximum of and 10% of the individual’s min ,t wage/salary earnings in that year. These steps complete the selection of the base sample. The selection of the final sample for a given statistic is described further below. 2.2 Empirical Approach In the nonparametric analysis conducted in Sections 3 and 4 , our main focus will be on individual-level log earnings changes (or growth) at one-year and five-year horizons. These earnings changes provide a simple and useful measure for discussing the dynamics of earnings without making strong parametric assumptions. In Sections 5 and 6 , we will link these “changes” to underlying “shocks” or “innovations” to an earnings process by means of a parametric estimation. To examine how the properties of earnings growth vary over the life cycle and in the 8

11 i cross section, we proceed as follows. Let ̃ y denote the log earnings of individual i who t,h is t .Foreachone-andfive-yearhorizonstartinginperiod t, we h years old in year (hereafter, RE—to be defined recent earnings group individuals based on their age and 1 .Ifthesegroupingsaredoneatasu precisely in a moment) as of time ciently t ffi fine level, we can think of all individuals within a given age/recent-earnings group to be ex ante identical (or at least very similar). Then, for each such group, we can compute t and t + k ( k =1 , 2 ... ) , which the cross-sectional moments of earnings changes between can then be viewed as corresponding to the properties of shocks that individuals within each bin can expect to face (see Figure for this rolling sample construction). This 2 approach has the advantage that we can compute higher-order moments precisely, as 10 each bin contains several hundred thousands of individuals. I reports sample (Table size statistics.) Final Sample for Cross-Sectional Moments. We implement this approach by first 1 :25–29,30–34,..., grouping workers into five-year age bins based on their age in year t 50–54, and 55–60. Then, within each age group, we select all individuals that were in 11 the base sample in and in at least two more years between t 5 and t 2 . t For 1 each one of these workers, we compute his average past earnings between years t 1 P 5 i i ̃ t , denoted with We set earnings observations below . ) ⌘ and 5 Y exp ( ̃ y 1 t s,h s t s =1 to the threshold for this computation. We also further control for age e Y ff ects, min ,t ff because even within these narrowly defined age groups, age di erences of a few years can systematically skew rankings in favor of older workers. To avoid this, we first estimate 12 age dummies, denoted d ,correspondingtotheaverageoflogearningsateachage, h P 5 exp to h 1 : and construct five-year average earnings from ages h We . 5 ( d ) s h =1 s i ̃ Y then normalize recent ects. Thus, our measure of ff with this measure to clean age e 1 t 10 Asecondpossibilityisthatthepropertiesofshocksdependmoreintimatelyonthecharacteristics ff erent types of workers—for individuals (such as the health, relationship skills, stamina, etc.), and di ff erent properties. This suggests example, identified by their lifetime earnings—might face shocks with di that perhaps we should group workers based on their lifetime earnings and study the properties of shocks for each group. We have an analogous set of results obtained by adopting this alternative perspective. It turns out that both approaches yield similar substantive conclusions, so we omit these results from the main text. These results are available upon request. 11 That is, in each of these years, the individual was in the qualifying age range with wage earnings exceeding Y ,andsatisfiedtheno-self-employmentcondition. ,t min 12 These are estimated from a pooled regression of log earnings on age and cohort dummies. Further details are given later in Section 3.1 . 9

12 Table I – Sample Size Statistics for Cross-Sectional Moments # Observations in Each RE Percentile Group Max Total (’000s) Median Age group Min 674,986 337,603 381,606 25-29 40,871 30-34 640,596 566,235 712,307 63,835 441,425 35-39 61,891 642,226 721,966 606,344 707,502 57,151 40-44 356,700 320,935 686,753 524,177 45-49 50,699 50-54 383,887 291,117 619,987 42,554 240,273 215,842 407,997 27,634 55-59 Note: Each entry reports the statistics of the number of observations in each of the 100 RE percentile groups for each age. Cross-sectional moments are computed for each year and then averaged over all years, so sample sizes refer to the sum across all years of a given age by percentile group. The last column (“Total”) reports the sum of observations across all 100 RE percentile groups for the age group indicated. (hereafter, RE) is earnings i ̃ Y t 1 i ̄ . Y ⌘ P 1 t 5 exp d ) ( h s =1 s final sample for the cross-sectional moments is then obtained as follows. We rank Our i ̄ Y individuals based on ,anddividetheminto(typically100)age-specificREpercentile 1 t groups. Within each group, we drop those individuals who fail to qualify for the base 13 + k . t Table I reports the summary statistics of the number of sample in year t or observations in each age/earnings cell (summed over all years). As seen here, the sample size is very large—the smallest cell size exceeds 200,000 observations and the average is close to 500,000—which allows us to compute all statistics very precisely. 3 Cross-sectional Moments of Earnings Growth We begin our analysis by documenting empirical facts about the first four moments of earnings growth at short (one-year) and long (five-year) horizons. For computing i y moments of earnings growth, we work with the time di ff , which is log earnings erence of t of the age e ff ect. Thus: net i i i i i y ) y y . )=( ̃ y ⌘ d d ( ) ( ̃ y k + h h k t,h k,h + k t + k,h t,h + k t t + 13 t, whereas the number Therefore, the percentile bins are constructed using information only prior to of observations within each bin also depends on being in the base sample in t and t + k. 10

13 i We compute the cross-sectional moments of y 2009 for each year, t =1980 , 1981 ,..., k t 14 and then average these across all years. First Moment: Mean of Log Earnings Growth 3.1 We begin our analysis with the first moment—average earnings growth—and examine how it varies with age (i.e., over the life cycle) and, for reasons that will become clear in a moment, across groups of individuals that di er in their lifetime earnings (and not ff recent earnings). But first, to provide a benchmark, we follow the standard procedure in Deaton and Paxson ( 1994 )) to estimate the average life-cycle profile the literature, (e.g., of log earnings. Although the procedure is well understood, its details matter for some of the discussions below, so we go over it in some detail. The average life-cycle profile is obtained from panel data or repeated cross sections by regressing log individual earnings on a full set of age and (year-of-birth) cohort dummies. 3 and represent the average The estimated age dummies are plotted as circles in Figure life-cycle profile of log earnings. It has the usual hump-shaped pattern that peaks around age 50. (On a side note, these age dummies turn out to be indistinguishable from a fourth 15 order polynomial of age, Murphy and Welch ( 1990 ) in Current apointalsoobservedby Population Survey data.) One of the most important aspects of a life-cycle profile is the implied growth in average earnings over the life cycle (e.g., from ages 25 to 55). It is well understood that the magnitude of this rise matters greatly for many economic questions, because it is a 16 In our data, this rise is about strong determinant of borrowing and saving motives. 17 80 log points, which is about 127%. Notice that feeding this life-cycle profile into a calibrated life-cycle model will imply that the median individual in the simulated sample experiences (on average) a rise of this magnitude from ages 25 to 55. One question we now address is whether this implication is consistent with what we see in the data. In other words, if we rank male workers in the U.S. data by their lifetime earnings, does the median worker experience an earnings growth of approximately 127%? 14 t We use as the first year of our analysis and therefore group individuals in 1979 based on = 1980 their recent earnings computed over 1978 and 1979. Similarly, 2009 is the last feasible year for t , which allows us to construct the moments of one-year earnings changes between 2009 and 2010. 15 Regressing the age dummies on a fourth order polynomial of age yields an average absolute deviation of only 0.3 log percent! 16 ( See ( 1991 ), Attanasio etal. ( 1999 ), and Gourinchas and Parker Deaton 2002 ), among others. 17 This figure lies on the high end of previous estimates from data sets such as the PSID, but not unseen before (cf. Attanasio etal. ( 1999 )). 11

14 Figure 3 – Life-Cycle Profile of Average Log Earnings 10.6 10.5 10.4 10.3 127% 10.2 rise 10.1 10 9.9 Average Log Earnings 9.8 9.7 9.6 40 25 35 45 55 50 60 30 Age This question can be answered directly with our data. First, we need to compute life- time earnings for each individual. For this purpose, we select a subsample of individuals that have at least 33 years of data between the ages of 25 and 60. We further restrict our sample to individuals who (i) have earnings above Y for at least 15 years and (ii) ,t min are not self-employed for more than 8 years. We rank individuals based on their lifetime earnings, computed by summing their earnings from ages 25 through 60. Earnings ob- Y are set to this threshold. For individuals in a given lifetime servations lower than ,t min j , j =1 , 2 ,..., 99 , 100 , we compute earnings (hereafter, LE) percentile group, denoted LE , h h growth in average earnings between any two ages as log ( Y and ) log ( Y ) 2 ,j 1 h h ,j 2 2 i i 2 E ( Y Y for a given individual may be zero. | where ⌘ LE j ) and Y i h,j h h Figure 4 plots the results for h .Here,thereareseveraltakeaways. =25 and h =55 2 1 First, individuals in the median lifetime earnings group experience a growth rate of 38%, about one-third of what was predicted by the profile in Figure 3 . Moreover, we have to look all the way above LE90 to find an average growth rate of 127%. However, earnings growth is very high for high-income individuals, with those in the 95th percentile experi- encing a growth rate of 230% and those in the 99th percentile experiencing a growth rate of 1450%. Although some of this variation could be expected because individuals with high earnings growth are more likely to have high lifetime earnings, these magnitudes are too large to be accounted for by that channel, as we show below. 12

15 3 Top 1%: 2.5 1500% increase 2 ) 25 Y 1.5 Income Growth from Pooled Regression 1 ) – log( 55 Y 0.5 log( 0 Median worker: -0.5 38% increase -1 80 100 0 60 40 20 Percentiles of Lifetime Earnings Distribution – Life-Cycle Earnings Growth Rates, by Lifetime Earnings Group Figure 4 – Log Earnings Growth Over Sub-Periods of Life Cycle Figure 5 By Di By Decades of the Life Cycle (a) (b) ff erent Starting Ages 3 3 Overall, 25-55 Overall, 25-55 25-35 30-55 2.5 2.5 35-45 35-55 45-55 Zero line 2 2 Zero line ) ) t t Y 1.5 1.5 Y 1 1 ) – log( ) – log( k k + t + t Y 0.5 0.5 Y log( log( 0 0 -0.5 -0.5 -1 -1 100 40 20 0 100 60 80 60 40 20 0 80 Percentiles of Lifetime Earnings Distribution Percentiles of Lifetime Earnings Distribution Earnings Growth by Decades. How is earnings growth over the life cycle distributed over di ff erent decades of the life cycle? Figure 5a answers this question by plotting, separately, earnings growth from ages 25 to 35, 35 to 45, and 45 to 55. Across the board, the bulk of earnings growth happens during the first decade. In fact, for the median LE group, average earnings growth from ages 35 to 55 is zero (notice that the solid blue line and grey line with circles overlap at LE50). Second, with the exception of those in the top 10% of the LE distribution, all groups experience negative growth from ages 45 to 55. So, the peak year of earnings is strongly related to the lifetime earnings percentile. 13

16 – Standard Deviation of Earnings Growth Figure 6 Five-Year Growth One-year Growth (b) (a) 1 1.2 25-29 25-29 30-34 30-34 35-39 35-39 1.1 ) 0.9 ) t t 40-44 40-44 y y 45-49 45-49 − − 50-54 50-54 1 +1 +5 0.8 t t y y 0.9 0.7 0.8 0.6 0.7 0.5 Standard Deviation of ( Standard Deviation of ( 0.6 0.4 0.5 40 60 80 0 20 100 100 80 40 20 60 0 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution After age 45, the only groups that are experiencing growth on average are those who are in the top 2% of the LE distribution. 5b plots How do the results change if we consider a slightly later starting age? Figure earnings growth starting at age 30 (solid blue line) and 35 (dashed red line). As can be anticipated from the previous discussion, from ages 35 to 55, average growth is zero for the median LE group and is very low for all workers below L70. Top earners still do very well though, experiencing a rise of 200 log points (or 640%) from ages 30 to 55 and a rise of 90 log points (or 146%) from ages 35 to 55. Those at the bottom of the LE distribution display the opposite pattern: average earnings drops by 70 log points (or 50%) from ages 35 to 55. 3.2 Second Moment: Variance How does the dispersion of earnings shocks vary over the life cycle and by earnings groups? To answer this question, Figure 6 plots the standard deviation of one-year and five-year earnings growth by age and recent earnings (hereafter, RE) groups (as defined above, Section 2.2 ). The following patterns hold true for both short- and long-run growth rates. First, for every age group, there is a pronounced U-shaped pattern by RE levels, implying that earnings changes are less dispersed for individuals with higher RE up to about the 90th percentile (along the x -axis). This pattern reverts itself inside the top 10% as dispersion increases rapidly with recent earnings. Second, over the life cycle, the dispersion of shocks declines monotonically up to about age 50 (with the exception 14

17 Figure 7 – Skewness (Third Standardized Moment) of Earnings Growth (b) Five-Year Change One-Year Change (a) 0.5 0.5 25-29 25-29 30-34 30-34 35-39 35-39 0 0 40-44 40-44 45-49 45-49 ) t ) 50-54 50-54 -0.5 t y y -0.5 − − +1 +5 t t -1 y y -1 -1.5 -1.5 Skewness of ( -2 Skewness of ( -2 -2.5 -2.5 -3 20 0 60 100 40 80 100 80 60 40 20 0 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution of very top earners) and then rises slightly for middle- to high-earning individuals from ages 50 to 55. ff erent for top earners who experience a monotonic The life-cycle pattern is quite di increase in dispersion of shocks over the life cycle. In particular, for one-year changes, individuals at the 95th percentile of the RE distribution experience a slight increase from 0.45 in the youngest age group up to 0.51 in the oldest group (50–54). Those in the top 1% experience a larger increase from 0.62 in the first age group up to 0.75 in the oldest. Therefore, we conclude that the lower 95 percentiles and the top 5 percentiles display patterns with age and recent earnings that are the opposite of each other. The same theme will emerge again in our analysis of higher-order moments. Standard Deviation of (Log) Earnings Levels. Although the main focus of this growth levels section is on earnings ,thelife-cycleevolutionofthedispersionofearnings has been at the center of the incomplete markets literature since the seminal paper of Deaton and Paxson ( 1994 ). For completeness, and comparability with earlier work, we have estimated the within-cohort variance of log earnings over the life cycle and report A.2 in Appendix A.1 . it in Figure 3.3 Third Moment: Skewness (or Asymmetry) The lognormality assumption implies that the skewness of earnings shocks is zero. 18 7 plots the skewness, measured here as the third standardized moment, of one- Figure 18 , , with mean μ and standard deviation X the third standard- More precisely, for random variable ⇥ ⇤ 3 3 X μ ) E ( / ized moment is . 15

18 year (left) and five-year (right) earnings growth. The first point to observe is that every graph in both panels of Figure lies below the zero line, indicating that earnings changes 7 are negatively skewed at every stage of the life cycle and for all earnings groups. The second point, however, is that skewness is increasingly more negative for individuals with higher earnings and as individuals get older. Thus, it seems that the higher an individual’s current earnings, the more room he has to fall and the less room he has left to move up. And this is true for both short-run and long-run earnings changes. Curiously, and as was the case with the standard deviation, the life-cycle pattern in skewness becomes much weaker at the very top of the earnings distribution. Another measure of asymmetry is provided by Kelly’s measure of skewness, which is defined as P P ( P 90 P 50) ( 10) 50 = S (1) , K 10 90 P P Pxy refers to percentile where of the distribution under study. Basically, S measures xy K the relative fractions of the overall dispersion (P90–P10) accounted for by the upper and lower tails. An appealing feature of Kelly’s skewness relative to the third standardized moment is that a particular value is easy to interpret. To see this, rearrange ( 1 )toget P 90 P 50 S K . . =0 5+ 90 P P 2 10 Thus, a negative value of S implies that the lower tail (P50-P10) is longer than the K upper tail (P90-P50), indicating negative skewness. Another property of Kelly’s measure is that it is less sensitive to extremes (above the 90th or below the 10th percentile of the shock distribution). Instead, it captures the shift in the weight distribution in the middling section of the shock distribution, whereas the third moment also puts a large weight on the relative lengths of each tail. (We examine the tails in more detail in the next subsection.) In the left panel of Figure 8 , we plot Kelly’s skewness, which is also negative through- out and becomes more negative with age, especially below RE60. However, it does not always get more negative with higher RE. This di ff erence from the third standardized moment (Fig. 7a )indicatesthatasREincreasesitismostlytheextremenegativeshocks (captured by the third moment) that drive the negative skewness, rather than the more middling shocks—those between P10 and P90. Figure 8b plots Kelly’s skewness for five-year changes, which reveals essentially the 16

19 Figure 8 – Kelly’s Skewness of Earnings Growth One-year Change (a) (b) Five-Year Change 25-29 0 0 30-34 35-39 40-44 ) ) t t 45-49 y y 50-54 − − +5 +1 t t -0.1 y y -0.2 -0.2 -0.4 25-29 30-34 Kelly Skewness of ( Kelly Skewness of ( 35-39 40-44 45-49 50-54 -0.3 100 0 80 40 20 60 100 80 40 20 0 60 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution 7b : each measure shows a strong in- same pattern as with the third moment in Figure crease in left-skewness with both age and earnings (except for the very-high earners). Furthermore, the magnitude of skewness is substantial. For example, the Kelly’s skew- ness for five-year earnings change of –0.35 for individuals aged 45–49 and in the 80th percentile of the RE distribution implies that the P90-P50 accounts for 32% of P90- P10, whereas P50-P10 accounts for the remaining 68%. This is clearly di ff erent from a lognormal distribution, which is symmetric—both tails contribute 50% of the total. While the preceding decomposition is useful, it does not answer a key question: is the increasingly more negative skewness over the life cycle primarily due to a compression of the upper tail (fewer opportunities to move up) or due to an expansion in the lower levels of the tail (increasing risk of falling a lot)? For the answer, we need to look at the P90-P50 and P50-P10 separately over the life cycle. The left panel of Figure plots P90- 9 P50 for di ff erent age groups minus the P90-P50 for 25- to 29-year-olds, which serves as a normalization. The right panel plots the same for P50-P10. One way to understand the link between these two graphs and skewness is that keeping P50-P10 fixed over the life cycle, if P90-P50 (left panel) declines with age, this causes Kelly’s skewness to become more negative. Similarly, keeping P90-P50 fixed, a rise in P50-P10 (right panel) has the same e ff ect. Turning to the data, up until age 45, both P90-P50 and P50-P10 decline with age (across most of the RE distribution). This leads to the declining dispersion that we have seen above. The shrinking P50-P10 would also lead to a rising skewness if it were not 17

20 Figure 9 – Kelly’s Skewness Decomposed: Change in P90-P50 and P50-P10 Relative to Age 25–30 (a) P90-P50 of Five-Year Change P50-P10 of Five-Year Change (b) 0.6 30-34 0.1 35-39 0.5 40-44 0 0.4 45-49 50-54 0.3 -0.1 0.2 -0.2 0.1 -0.3 0 -0.1 30-34 -0.4 35-39 -0.2 40-44 -0.5 45-49 -0.3 P50-P10 (Relative to P50-10 at age 25-30) P90-P50 (Relative to P90-50 at age 25-30) 50-54 -0.4 -0.6 20 0 100 80 60 40 100 80 60 40 20 0 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution for the faster compression of P90-P50 during the same time. Therefore, from ages 25 entirely to 45, the increasing negativity of skewness is due to the fact that the upper end of the shock distribution compresses more rapidly than the compression of the lower end. After age 45, P50-P10 starts expanding rapidly (larger earnings drops becoming more likely), whereas P90-P50 stops compressing any further (stabilized upside). Thus, during this phase of the life cycle, the increasing negativity in Kelly’s skewness is due to increasing downward risks and not the disappearance of upward moves. The only exception to this pattern is, again, the top earners (RE95 and above) for whom P90-P50 actually never compresses over the life cycle, whereas the P50-P10 gradually rises as they get older. Therefore, as they climb the wage ladder, these individuals do not face a tightening ceiling, but do su ff er from an increasing risk of falling a lot. 3.4 Fourth Moment: Kurtosis (Peakedness and Tailedness) It is useful to begin by discussing what kurtosis measures. A useful interpretation has been suggested by Moors ( 1986 ), who described kurtosis as measuring how dispersed a 19 probability distribution is away from . μ This is consistent with how a distribution ± with excess kurtosis often looks like: a sharp/pointy center, long tails, and little mass near ± . A corollary to this description is that for a distribution with high kurtosis, μ the usual way we think about standard deviation—as representing the size of the typical 19 Z =( x μ ) / and noting that This can easily be seen by introducing a standardized variable 4 2 2 2 2 Z )= kurtosis is var ( Z  )+ E ( Z = ) E = var ( Z ( )+1 . So  can be thought of as the dispersion of 2 +1 around its expectation, which is 1, or the dispersion of Z around Z and 1 . 18

21 Figure 10 – Kurtosis of Earnings Changes (b) Annual Change (a) Five-Year Change 35 20 25-29 25-29 30-34 30-34 35-39 30 35-39 40-44 40-44 16 45-49 45-49 ) ) 50-54 t t 25 50-54 y y − − 12 +1 +5 t t 20 y y 15 8 10 Kurtosis of ( Kurtosis of ( 4 5 0 0 20 40 60 0 80 100 80 100 40 20 0 60 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution shock—is not very useful. This is because very few realizations will be of a magnitude close to the standard deviation; instead, most will be either close to the median or in the tails. With this definition in hand, let us now examine the earnings growth data. Figure 10a plots the kurtosis of annual earnings changes. First, notice that kurtosis increases monotonically with recent earnings up to the 80th to 90th percentiles for all age groups. That is, high-earnings individuals experience even smaller earnings changes of either sign, with few experiencing very large changes. Second, kurtosis increases over the life cycle, for all RE levels, except perhaps the top 5%. Furthermore, the peak levels of kurtosis range from a low of 20 for the youngest group, all the way up to 30 for the middle-age group (40–54). To provide a more familiar interpretation of these kurtosis values, it is useful to calculate measures of concentration. The first three columns of Table II report the fraction of individuals experiencing a log earnings change (of either sign) of less than athreshold x =0 . 05 , 0 . 10 , 0 . 20 , 0 . 50 , and 1 . 00 ,underalternativeassumptionsabout i i the data-generating process. For the entire sample, the standard deviation of y y t +1 t is 0.48. Assuming that the data-generating process is a Gaussian density with this standard deviation, only 8% of individuals would experience an annual earnings change of less than 5%. The true fraction in the data is 35%. Similarly, the Gaussian density predicts a fraction of 16% when the threshold is 0.10, whereas the true fraction is 54%. As an alternative calculation, we calculate the areas under the densities in three di ff erent 19

22 Table II – Fraction of Individuals with Selected Ranges of Log Earnings Change i i i i | 1 . 50 0.023 0.002 11.5 1.00 0.94 0.98 0.96 ⇤ 1 . Notes: The empirical distribution used in this calculation is for 1995-96, the same as in Figure † The intervals are defined as follows: “Center” refers to the area inside the first intersection between the 1 : [ two densities in Figure 0 . 122 , 0 . 187] . “Tails” refer to the areas outside the intersection point at the tails: ( 1 , 1 . 226] [ [1 . 237 , 1 ) . “Shoulders” refer to the remaining areas of the densities. ranges determined by the intersections of the two densities in the left panel of Figure 1 . The center is the area inside the first set of intersections, and the Gaussian density has 25% of its mass in this area compared with 65% in the data. The shoulders are the second set of areas, marked again by the intersections, and the Gaussian density has almost three-quarters of its mass in this area, compared with only 31% in the data. Turning to the tails, the Gaussian density has only 1.3% of its mass in the tails compared with almost three times that amount in the data. Further, the last row of the right panel reports that a typical worker draws a shock larger than 150 log points (an almost five-fold increase or an 80% drop in earnings) once in a lifetime (or 2.3% annual chance), whereas this probability is 11.5 times less likely under a normal. We now take a closer look at the tails of the earnings growth distribution compared with a normal density. Figure 11 plots the log density of the one-year change in the data versus the Gaussian density. This is essentially the same as the left panel of Figure 1 but with the y -axis now in logs. The lognormal density is an exact quadratic, whereas the data display a more complex pattern. Two points are worth noting. One, the data and longer tails compared with a normal distribution, distribution has much thicker and the tails decline almost linearly, implying a Pareto distribution at both ends, with 20 Two, the tails are asymmetric, with the left tail declining significant weight at extremes. much more slowly than the right, contributing the negative third standardized moment documented above. In fact, fitting linear regression lines to each tail yields a tail index of 20 A double-Pareto distribution is one where both tails are Pareto with possibly di ff erent tail indices. 20

23 Figure 11 –TailsoftheDistributions:U.S.Datavs.NormalDensity 2 US Data 2 . 0 . ) Normal (0 48 0 -2 -4 Log Density -6 -8 -3 2 1 0 -1 -2 3 y − y t +1 t 21 2 for the right tail and 1.2 for the left tail—the latter showing especially high thickness. Overall, these findings show that earnings changes in the U.S. data exhibit important deviations from lognormality and raise serious concerns about the focus in the current literature on the covariances (second moments) alone. In particular, targeting the covari- ances alone can vastly overestimate the typical earnings shock received by the average worker and miss the substantial but infrequent jumps experienced by few. Economic Models behind Skewness and Kurtosis. While the lognormal frame- work is often adopted for technical and empirical tractability, negative skewness and excess kurtosis are naturally generated by standard structural models of job search over the life cycle. For example, job ladder models in which workers do on-the-job search and move from job to job as they receive better o ff ers and fall o ff the job ladder after un- employment not only will generate negative skewness but also will imply that skewness becomes more negative with age. This is because as the worker climbs the job ladder, the probability of receiving a wage o ff er much higher than the current wage will be de- clining. At the same time, as the worker moves higher up in the wage ladder, falling down to a flat unemployment surface (or disability) implies that there is more room to 21 Notice that although the Pareto tail in the earnings distribution is well known, here the two Pareto tails emerge in earnings changes, for which much less empirical evidence exists. 21

24 fall. Furthermore, as the attractiveness of job o ff ers declines with the current wage, more ff ers will be rejected, and therefore the frequency of job-to-job transitions will also job o decline with age, implying that most wage changes will be small (within-job) changes. This increases the concentration of earnings changes near zero, which in turn raises the kurtosis of changes. That said, the magnitudes of variation over the life cycle and by earnings levels that we documented in these moments are so large that it is an open question whether existing models of job search can be consistent with these magnitudes, and, if not, what kinds of 22 modifications should be undertaken to make them consistent. Robustness and Extensions 3.5 The results documented in the previous sections show important deviations from lognormality as well as clear patterns with age and past earnings. This raises the question of whether some of these findings are due to simple statistical artifacts—say, due perhaps to extreme shocks experienced by very few individuals—and whether the age and earnings patterns might be due to sample selection or other assumptions made in the construction of these statistics. In this section, we discuss five cases of interest and report all the A.2.1 A.2.2 . relevant figures and analysis in Appendixes to I. Decomposing Moments: Job-Stayers vs. Job-Switchers. Going back to the work of ( 1992 ), economists have found important di ff erences between Topel and Ward the earnings changes that occur during an employment relationship and those that occur across jobs (see, more recently, Low et al. ( 2010 ), Altonji et al. ( 2013 ), and Bagger et al. ( 2014 )). Therefore, it is of interest to ask how the empirical patterns we have documented so far relate to within- and between-job earnings changes. Our data set contains a unique employer identification number (EIN) for each job that a worker holds in a given year, A.2.1 ). which allows us to conduct such an analysis (see Appendix ff II. Disentangling the E In the analysis so ects of Age and Recent Earnings. far, we have grouped workers first by age, and then within each age group, we have ranked and divided them into recent earnings percentiles. The implication is that RE 22 Ahighkurtosisinearningschangescouldpartlybeduetoheterogeneityacrossindividualsin shock variances, as documented by Chamberlain and Hirano ( 1999 )and Browning etal. ( 2010 ). We have explored this possibility by estimating earnings shock variances for each individual. While we do find significant heterogeneity in variances, consistent with these papers, the remaining kurtosis is still substantial. We do model and estimate individual-specific variances in Section 5 . 22

25 percentiles in these figures are age group dependent. Therefore, when we fix an RE percentile and examine how statistics vary with age, we are simultaneously looking at changes with earnings, since earnings vary with age. The advantage of this approach was that it ensured each RE group contained a similar number of observations, whereas grouping workers based on the RE distribution in the overall sample will result in too many younger workers appearing in lower RE percentiles and vice versa for middle-age A.2.2 A.5 and A.6 ), we plot 3-D graphs of skewness and workers. In Appendix (Figures kurtosis, where we first group workers based on the RE distribution in the overall sample, and then within each RE group, we classify workers by age. Inspecting these graphs and comparing with their counterparts above shows that the main substantive conclusions described above are robust. III. Averaging Earnings Over Neighboring Years. Recall that the statistics above ff erences between two years, t and t + k .Oneconcernis were constructed by taking di the robustness of this measure to small changes in the timing of earnings. For example, suppose that an individual’s income has been shifted from the last few months of year + into the beginning of t + k +1 . If true, this would represent an earnings fluctuation t k t that is easy to smooth, but could appear as a big negative shock between t + k .A and similar comment applies to period t .Toaddressthisissue,wehaveconstructedthesame set of statistics for the second to fourth moments by using two-year average earnings. For the short-run and long-run variations, we use, respectively i i i i i i i i i ̃ ̃ Y log( Y Y + Y =log( ) and . ) y Y =log( Y + Y ) ) log( Y + + y 5 t t +6 t +1 +3 t t t t +5 t t +1 t +2 The first measure becomes more like a two-year di ff erence, whereas the second one is ff closer to a five-year di erence as before. However, we are mostly interested in whether statistics are broadly robust and the qualitative patterns remain unchanged, so these are reasonable choices. ff IV. Di Usual Earnings. Even though we condition on recent earnings erence from over the past five years and require all individuals in the sample to be employed in year t 1 ,itisconceivablethatsomeindividualsreceivelargepositiveshocksinperiod t ,and the subsequent drop in earnings from to t + k is simply mean reversion—and not a new t shock. The same argument applies for a large negative shock in t. To see if this might be important, we have constructed the same statistics using an alternative di ff erence 23

26 measure, again for the short-run and long-run variations: i i i i i i . ) ) log( Y =log( Y log( ) and y Y ) Y y =log( short long t 1 t t 1 +5 t +1 These are longer di ff erences than before, since the base year is now centered around t 3 . V. Trimming the Tails. As noted before, earnings growth displays very long tails, and even though measurement error is unlikely to explain it, it is still of interest to know, for example, how much of the very large kurtosis and negative skewness is due to the extreme observations and how much is due to the bulk of the distribution. Since the third and fourth moments are sensitive to tails, this is worth exploring (although Kelly’s skewness is already reassuring for the skewness). To this end, we have constructed the higher-order moments under alternative assumptions about the tails: (i) by dropping the top and bottom 1% of earnings growth observations, and (ii) by changing the lower threshold for sample exclusion from Y to be individual-specific and equal to 5% of min ,t 23 each worker’s own recent earnings. In Appendix A ,wereportthefiguresanalogoustothoseaboveunderthesethree robustness checks (III to V). Although the figures are quantitatively di ff erent, the dif- ference is almost always small, and therefore the substantive conclusions of this analysis remain intact. 4 Dynamics of Earnings Akeydimensionoflife-cycleearningsriskisthepersistenceofearningschanges. Typically, this persistence is modeled as an AR(1) process or a low-order ARMA process (typically, ARMA(1,1)), and the persistence parameter is pinned down by the rate of decline of autocovariances with the lag order. The AR(1) structure, for example, pre- dicts a geometric decline and the rate of decline is directly given by the mean reversion parameter. While this approach might be appropriate in survey data with small sample sizes, it imposes restrictions on the data that might be too strong, such as the uniformity of mean reversion for positive and negative shocks, for large and small shocks, and so on. 23 Because Y does not vary with an individual’s own earnings, high-income individuals can expe- ,t min rience a larger fall in earnings and still remain in the sample, whereas low-income individuals would exit the sample with the same fall. This asymmetry might give the appearance of a more negative skewness for higher-income individuals. 24

27 Figure 12 – Impulse Responses, Prime-Age Workers with Median RE 1.5 1 0.5 t y − 0 k + t y -0.5 -1 -1.5 0 6 8 4 10 2 k Here, the substantial sample size allows us to characterize persistence without making parametric assumptions. Final Sample for Impulse Response Analysis ff The final sample for this analysis is slightly di erent from the one used in the previous section. In particular, our includes all observations that are in the base final sample 2 t 1 and in at least two more years between t 5 and sample in ,andfurthermore t satisfies the age (25–60) and no-self-employment condition in years t through t +3 , and in t +5 and t +10 . To reduce the number of graphs to a manageable level, we aggregate individuals across demographic groups. First, we combine the first two age groups (ages 25 to 34) into “young workers,” and the last four groups (ages 35 to 55) into “prime-age workers.” To this end, we rank and group individuals based on their average earnings from 5 to t 1 , then within each such group, we rank and group again by the size of t t 1 and t. Hence, all individuals within a given group the earnings change between obtained by crossing the two conditions have the same average earnings up to time t 1 and experience the same earnings “shock” from t 1 to t. For each such group of individuals, we then compute their average earnings change from to t + k, for all values of t k =1 , 2 , 3 , 5 , 10 . Specifically, we construct 21 groups based on their RE percentiles: 1–5, 6–10, 11–15, ... , 86–90, 91–95, 96–99, 100. Then, we construct 20 equally-divided groups 25

28 Figure 13 – Impulse Responses (Rotated View), Prime-Age Workers with Median RE k=1 1 k=2 Transitory k=3 k=5 k=10 0.5 t y − 0 k + t y Permanent -0.5 -1 -1 -0.5 1 0.5 0 y − y t − t 1 i i ... based on the percentiles of the shock, y ,91–95, y :percentiles1–5,6–10,11–15, 1 t t 96–100. Therefore, for every year t, we have 2 age groups, 21 RE groups and 20 groups 840 groups). As before, we construct these groups separately for shock size (for a total of t and assign workers based on these averages. Then, for workers in each group, for each h i i i i i i ,for k -year earnings growth, y E y | Y y we compute the average of log ,y 1 t t t t 1 + t k 5 , , 3 k 2 , 10 . =1 , Impulse Response Functions 4.1 t 1 ), Figure 12 plots 20 impulse For prime-age males with median RE level (as of i i response functions (one for each “shock” size, y y ). As seen here, the most positive 1 t t 5% of shocks at time t are about 100 log points and the most negative are about –125 log points. Notice that the mean reversion pattern varies with the size of the shock, with much stronger mean reversion in for large shocks and smaller reversion for smaller t +1 shocks. Furthermore, even at the 10-year horizon, a nonnegligible fraction of the shocks’ e ect is still present, indicating a permanent component to these shocks. ff i i y To illustrate these patterns more clearly, Figure y ,onthe plots the shock, 13 t t 1 i i y x -axis for y ,onthe -axis and the fraction of each shock that has mean-reverted, y t k + t the median RE group. Thus, this figure contains the same information as in Figure 12 26

29 Figure 14 – Impulse Responses, Prime-Age Workers with Low or High RE Individuals with Individuals with Y 2 [ P 6 P 10] 2 [ P P 95] Y 91 t 1 t 1 2 1.5 =1 k k =1 Transitory =2 k k =2 =3 k 1.5 =3 k 1 k =5 k =5 Transitory k =10 1 k =10 0.5 0.5 t t y y − − 0 k k 0 + + t t y y -0.5 Permanent Permanent -0.5 -1 -1 -1.5 -1.5 -1.5 -2 2 1.5 1 0.5 -1.5 -1 -0.5 0 0.5 0 1.5 -0.5 -1 1 − y y − y y 1 − t 1 − t t t ff but is reported di erently. Several remarks are in order. First, negative earnings changes tend to be less persistent than positive earnings changes. For example, a worker whose t 1 earnings rise by 100 log points between t loses about 50% of this gain in the and following 10 years. It is also interesting to note that almost all of this mean reversion happens after one year, implying that whatever mean reversion there is happens very quickly. Turning to earnings losses: a worker whose earnings fall 100 log points recovers one-third of that loss by t +1 and recovers more than two-thirds of the total within 10 years. Moreover, unlike with positive shocks, the recovery (hence mean reversion) is more gradual in response to negative shocks. Second, the degree of mean reversion varies with the magnitude of earnings shocks. This is evident in Figure 13 , where small shocks (i.e., those less than 10 log points in absolute value) look very persistent, whereas there is substantial mean reversion following larger earnings changes. A univariate autoregressive process with a single persistence parameter will fail to capture this behavior. In the next section, we will allow for multiple AR(1) processes to accommodate the variation in persistence by shock size. In the next figure ( 14 ), we plot the same kind of impulse response functions but now for workers that are in the 10th percentile (left panel) and 90th percentile (right) of RE distribution. Notice that, for low-income individuals, negative shocks mean-revert much more quickly, whereas positive shocks are more persistent than before. The opposite is true for high-income individuals. 27

30 Figure 15 – Asymmetric Mean Reversion: Butterfly Pattern 2 1 − 5% − 20 25% 1.5 45 − 50% 70 75% − 1 100% − 95 100% 0.5 t y − 0 +10 t y -0.5 -1 -1.5 -2 1 0 -1 2 -2 y − y t t − 1 We now extend the results of Figure 14 to the entire distribution of recent earnings. To make the comparison clear, we focus on a fixed horizon, 10 years, and plot the total mean reversion between and t +10 for the 6 RE groups in Figure t .Startingfromthe 15 lowest RE group (those individuals in the bottom 5% of the recent earnings distribution), notice that negative shocks are transitory, with an almost 80% mean reversion rate at the 10-year horizon. But positive shocks are quite persistent, with only about a 20% mean reversion at the same horizon. As we move up the RE distribution, the positive and negative branches of each graph start rotating in opposite directions, so that for the highest RE group, we have the opposite pattern: only 20 to 25% of negative shocks mean-revert at the 10-year horizon, whereas almost 75% of positive shocks mean-revert at the same horizon. We refer to this shape as the “butterfly pattern.” 5 Estimating Stochastic Processes for Earnings With the few exceptions noted earlier, the bulk of the earnings dynamics literature relies on the (often implicit) assumption that earnings shocks can be approximated rea- sonably well with a lognormal distribution. This assumption, combined with linear time series models (e.g., an ARMA ( p, q ) process) to capture the accumulation of such shocks, 28

31 allowed researchers to focus their estimation to match the covariance matrix of log earn- 24 erence form, ignoring higher-order moments. ings either in levels or in first di ff This approach has two problems. First, the broad range of evidence presented in the previous sections implies that this approach is likely to miss important aspects of the data and produces a picture of earnings risk that does not capture salient features of the risks faced by workers. Second, the covariance matrix estimation method makes ffi cult to select among alternative models of earnings risk, because it is di cult to it di ffi judge the relative importance—from an economic standpoint—of the covariances that a given model matches well and those that it does not. This is an especially important shortcoming given that virtually every econometric process used to calibrate economic models is statistically rejected by the data. With these considerations in mind, we propose and implement a di erent approach ff that relies on matching the kinds of moments presented above. We believe that economists can more easily judge whether or not each one of these moments is relevant for the eco- nomic questions they have in hand. Therefore, they can decide whether the inability of a particular stochastic process to match a given moment is a catastrophic failure or an acceptable shortcoming. They can similarly judge the success of a given stochastic process in matching some moments and not others. More concretely, in the first stage, we use each set of moments presented above as diagnostic tools to determine the basic components that should be included in the stochastic process that we will then fit to the data. Clearly, this stage requires exten- sive pre-testing and exploratory work. For example, to generate the life-cycle earnings growth patterns documented in Section 3.1 ,weconsideredthreebasicingredients:(i)an AR(1) process + an i.i.d. shock, (ii) growth rate heterogeneity with no shocks, and (iii) a mixture of two AR(1) processes where each component receives a nonzero innovation with a certain probability. We picked the first two ingredients because of the widespread attention they garnered in the previous literature, and the third one based on our con- jecture that it might perform well. We found that the first ingredient, on its own, could not generate the rich patterns of earnings growth revealed by the data, whereas the HIP process performed fairly well, and the AR(1) mixture process performed the best. 24 Exceptions include Browning etal. ( 2010 ), Altonji etal. ( 2013 ), and Guvenen and Smith ( 2014 ). Clearly, GMM or minimum distance estimation that is used to match such moments does not require the assumption of lognormality for consistency. But abstracting away from moments higher than covariances is a reflection of the belief that higher-order moments do not contain independent information, which relies on the lognormality assumption. 29

32 Therefore, we concluded that a stochastic process for earnings should include either (ii) or (iii) (or both) as one of its components. Although earnings growth data on their own could not determine which one of these pieces is more important, when we analyze these data along with the other moments, we will be able to obtain sharper identification of the parameters of these two components. We conducted similar diagnostic analyses on the other cross-sectional moments (stan- dard deviation, skewness, and kurtosis) as well as on impulse response moments. The variation in the second to fourth moments over the life cycle and with earnings levels seemed impossible to match without introducing explicit dependence of shocks to these two characteristics. Allowing the mixing probabilities to depend on age and earnings delivered much improved results, so we make this specification part of our benchmark. A Flexible Stochastic Process 5.1 The most general econometric process we estimate has the following features: (i) a heterogeneous income profiles (hereafter, HIP) component of quadratic form, (ii) a ,and ⌫ , where each component receives z, x mixture of three AR(1) processes, denoted by [0 2 a new innovation in a given year with probability , 1] for j = z, x, ⌫ ;and(iii)an p j i.i.d. transitory shock. Here is the full specification: 2 i i i i i i i i + = t y ↵ + z + (2) + x e + v t " + t t t t t i i i = ⇢ z ⌘ z (3) + z t zt 1 t i i i ⇢ (4) x x = ⌘ + x 1 t xt t i i i = ⌫ , ⌫ ⇢ (5) ⌘ + ⌫ 1 t t t ⌫ where for = j z, x, v : i ⇤ i i ⇤ i μ (6) , ⌘ } ) and ⌘ { I = ⌘ ⇠ N 2 ⇥ I ( s i,t p j jt j jt jt j 2 jj i 2 i log . (7) , ⇠ N ) ,j = z, x, ( ⌘ j ⌫ jj ⌫ j 2 follows a multivariate normal distribution with zero mean and a covari- ↵ , , ) ( Here, ⌘ ance matrix to be estimated. The realizations of the three innovations ( ,j = z, x, v ) jt are mutually exclusive—only one of the three shocks is received per period. This is imple- mented by first drawing a standard uniform random variable, s , for a given individual i,t 30

33 at age t and dividing the unit interval into three pieces: I =[0 ,p , ] ,I ] p =( p + ,p z x p z p z x z p Depending on which =( p . + p p , 1] , where and + p p  1 I p =1 and v z z x x z p x v ⇤ i ,andtheothersareset s interval ⌘ falls in, that innovation is set to its random draw, i,t jt equal to zero. The specification in ( 7 )impliesthattheinnovationstandarddeviationforeachAR(1) process has an individual-specific component that is lognormal, with mean and stan- j dard deviation proportional to .Toeconomizeonparameters,weassumethatthe jj permanent innovation is identically distributed across individuals. Regarding the initial i i conditions of the persistent processes, x , we assume that they are drawn from a z and 0 0 25 = ,j normal distribution with zero mean and standard deviation z, x. Finally, to 0 j, ⇢ . < ⇢ avoid indeterminacy in the estimation, without loss of generality we impose < ⇢ ⌫ x z We allow the mixing probabilities to depend on age and on the lagged idiosyncratic component of earnings: i i i (8) ,t )= a ⇥ + b p ⇥ t. ( y + c y ⇥ t + d ⇥ y j j j j j t t t 1 1 1 i Recall that in Section y 3 ,wedefined as log earnings net of age e ff ects. It can alterna- t i i i i i tively be defined as y + x ’s are probabilities, they must p + v = .Finally,since + " z j t t t t t 26 8 remain between [0,1], so equation ( )istruncatedoutsideofthisrange. 5.2 Details of Estimation Procedure We estimate the parameters of the described stochastic process using a method of simulated moments (MSM) estimator. The empirical targets are: (i) moments from the 4 and 5 ;(ii)thestandard life-cycle profile of average earnings, summarized in Figures deviation, skewness, and kurtosis of one-year and five-year earnings growth presented 3.2 to 3.4 ;and(iii)momentsdescribingtheimpulseresponsefunctionspre- in Sections sented in Section 4 . In addition, and as noted above, the within-cohort variance of log earnings levels is a key dimension of the data that has been extensively studied in previ- ous research. For both completeness and consistency with earlier work, we include these A.2 )asafourthsetofmoments. variances (summarized in Figure 25 ff ect variance The initial variance of the permanent component cannot be identified from the fixed e and hence is normalized to zero. 26 We have also considered an alternative specification where the innovation variances are functions i i i ( y b t. ⇥ of earnings and age: ,t )= a y + ⇥ ⇥ y After extensive experimentation d + t + c ⇥ j j j j j t 1 1 1 t t with this formulation, we have found it to perform very poorly. 31

34 Accounting for Zeros. Recall that in order to construct the cross-sectional moments, t or t + k so Y —in year we have dropped individuals who had very low earnings—below min 27 Although this approach made sense as to allow taking logarithms in a sensible manner. for documenting empirical facts that are easy to interpret, for the estimation exercise, we would like to also capture the patterns of these “zeros” (or very low earnings observations), given that they clearly contain valuable information. This is feasible thanks to the flexibility of the MSM approach, and in Appendix B we describe how we modify the cross-sectional moments to allow these zeros in the estimation. If we were to match all data points in all these moments (i.e, Aggregating Moments. for every RE percentile and every age group), it would yield more than 10,000 moments. Although this step is doable, not much is likely to be gained from such a level of detail, and it would make the diagnostics—that is, judging the performance of the estimation— quite di ffi cult. To avoid this, we aggregate 100 RE percentiles into 10 to 15 groups and the 6 age groups into two (ages 25–34 and 35–55). Full details of how this aggregation B .Afterthe is performed and which moments are targeted are included in Appendix aggregation procedure, we are left with 120 moments that capture average earnings over the life cycle (targeted ages are 25, 30, 35, 40, 45, 50, 55, and 60); 156 moments for cross-sectional statistics (standard deviation, skewness and kurtosis of one-year and five- year earnings growth); and 1,120 moments coming from the impulse response functions. Adding the 36 moments on the variance of log earnings, in sum, we target a total of 28 , 120 + 36 = 1 , 432 moments. 120 + 156 + 1 432 Let for n =1 ,...,N =1 , be denote a generic empirical moment, and let d ) ( ✓ m n n the corresponding model moment that is simulated for a given vector of earnings process parameters, . We simulate the entire earnings histories of 200,000 individuals who enter ✓ labor market at age 25 and work until age 60. When computing the model moments, we apply precisely the same sample selection criteria and employ the same methodology to the simulated data as we did with the actual data. To deal with potential issues that could arise from the large variation in the scales of the moments, we minimize the scaled deviation between each data target and the corresponding simulated model moment. For 27 We were able to include those below the threshold in sets (i) and (iii) because for those moments it made sense to first take averages, including zeros, and then take the logarithms of those averages. 28 The full set of moments targeted in the estimation are reported (in Excel format) as part of an online appendix available from the authors’ websites. 32

35 each moment n ,define d ( ✓ ) m n n )= F ✓ ( , n | + m | n n is simply > 0 is an adjustment factor. When is positive, =0 and m where F n n n n the percentage deviation between data and model moments. This measure becomes problematic when the data moment is very close to zero, which is not unusual (e.g., impulse response of log earnings changes close to zero). To account for this, we choose to be equal to the 10th percentile of the distribution of the absolute value of the n moments in a given set. The MSM estimator is 0 ˆ (9) F ( ✓ ) ✓ =argmin F ( ✓ ) , W ✓ where F ( ✓ ) is a column vector in which all moment conditions are stacked, that is, T ( ✓ )=[ F . ( ✓ ) ,...,F ( F ✓ )] N 1 , is chosen such that the life-cycle average earnings growth W The weighting matrix, moments and impulse response moments are assigned a relative weight of 0.25 each, the cross-sectional moments of earnings growth receive a relative weight of 0.35, and the 29 variance of log earnings is given a relative weight of 0.15. The objective function is highly jagged in certain directions and highly nonlinear in general, owing to the fact that we target higher-order moments and percentiles of the distribution. Therefore, we employ a global optimization routine, described in further detail in 2013 ), to Guvenen ( 9 ). Further details can be found in Appendix B . perform the minimization in ( Results: Estimates of Stochastic Processes 6 III reports the parameter estimates. Before delving into the discussion of these Table estimates, we begin with an overview of what each of the eight columns aims to capture. Columns (1) to (3) take the general stochastic process (equations ( 2 )to( 8 )) and restrict er in the to a unit root ( ⇢ the persistent process ⌘ 1 ). The three columns only di ff ⌫ ⌫ t order of the HIP component: quadratic, linear, and none. Notice how reducing the order from quadratic to linear causes only a small rise in the objective value (from 16.65 to 17.17), whereas eliminating the HIP component altogether (going from column to 2 to 29 More precisely, each life-cycle growth moment is weighed by 0 . / 120 , each cross-sectional moment 25 by 0 . 35 / 156 ,eachimpulseresponsemomentby 0 . 25 / 1120 , and each variance moment by 0 . 15 / 36 . 33

36 Table III – Estimates of Stochastic Process Parameters (2) (3) (4) (5) (6) (7) (8) Specification: (1) BenchmarkBestfitStandard modelmodel HIP order 21011110 22232111 AR(1) 11100000 RW yes no yes yes no yes yes yes Heterog. variances? Values Parameter 0.464 0.698 0.679 0.673 0.681 0.475 0.523 0.552 ↵ ⇥ 10 0.149 0.130 — 0.137 0.191 0.067 0.252 — 100 0.043 — — ⇥ — — — — — –0.48 –0.49 — –0.04 0.07 –0.12 0.02 — corr ↵ — — — — — — — –0.06 corr ↵ –0.44 — corr — — — — — — ⇢ 0.249 0.259 0.182 0.083 0.226 1.004 0.962 1.00 z 0.639 — 0.368 0.425 0.315 0.512 ⇢ — — x ⇤ ⇤ 1.0 ⇢ —0.796— —— — 1.0 ⌫ 0.947 0.656 0.786 0.121 0.174 0.172 ̄ 0.853 0.847 z 0.425 0.337 0.123 — — — ̄ 0.422 0.361 x 0.174 0.082 0.096 0.087 ̄ — — — — ⌫ 0.039 0.029 0.023 0.019 0.017 0.029 0.027 0.040 " 0.261 0.133 0.229 0.121 0.206 0.370 0.031 0.304 0 z 0.186 0.089 0.295 0.113 0.271 — — — x 0 ———0.110———— 0 ⌫ 0.121 0.107 0.066 0.313 — — 0.134 0.167 zz 0.170 0.294 0.142 0.194 — — 0.550 — xx μ –0.544 –0.426 –0.780 –0.518 –0.231 — — — z 0.197 0.237 0.175 — — — 0.020 0.021 μ x — 0.068 0.060 0.064 0.303 — — μ v Mixture probabilities: a ⇥ 1 0.064 0.067 0.025 0.148 0.215 — — — z b t 0.036 0.038 0.001 0.005 0.027 — — — ⇥ z –0.207 –0.203 y ⇥ — — c 0.027 –0.244 –0.293 — 1 t z 0.076 0.071 ⇥ t ⇥ y — — 0.056 — d 0.072 0.080 1 t z a ⇥ 1 0.358 0.433 0.152 0.481 — — — — x b ⇥ t –0.112 –0.222 –0.030 –0.019 — — — — x 0.044 0.072 c y — — — — –0.066 –0.045 ⇥ x 1 t d — ⇥ t ⇥ y — — — –0.328 –0.373 0.099 –0.156 1 x t 11 15 16 . 17 24 . 65 17 . 29 18 . 30 38.10 38.46 38.75 Objective value . Decomposition: 097 5 (i) Cross-section 527 5 . 901 9 . . . 448 6 . 130 19.730 19.944 19.918 5 (ii) Impulse resp. 9 . 923 10 . 134 14 . 033 8 . 812 11 . 108 17.550 17.465 17.795 965 0 (iii) Inc. growth 719 0 . 644 0 . . . 958 0 . 623 0.629 0.918 0.863 0 (iv) Inequality 0 . 484 0 . 492 0 . 020 0 . 076 0 . 435 0.194 0.131 0.179 ⇤ Note: ⇢ =1 . 0 is imposed. ⌫ 34

37 3) causes a larger jump, from 17.17 to 24.11. Consequently, in columns (4) to (6), we keep the linear HIP specification but instead vary the number of persistent components. That is, column (4) allows for three AR(1) components, achieved by relaxing the unit root constraint on and estimating its persistence ⇢ ;column(5)reducesthenumber ⌫ t ⌫ of AR(1) components to two, and column (6) reduces it to one, thereby eliminating the mixture structure. In other words, column (6) is the first case in which the process is z Gaussian, because receives a normal shock with 100% probability every period. Notice the substantial jump in the objective value when the mixture structure is eliminated going from column (5) to (6). As we will see in a moment, this result is a reflection of the very poor fit delivered by the Gaussian process. Based on the overall goodness of fit and the relative economy of its parameters (rel- ative to columns (1) and (4)), we will refer to the specification in column (2) as our benchmark model .However,itshouldbestressedthatthisisnotaverystrictpreference, and depending on the application, either one of the specifications in columns (1) or (4) could serve as a more appropriate benchmark. For quick reference, we refer to column (4) as the best fit model, since it delivers the lowest objective value. The process estimated in column (6) is quite simple relative to those that came before it, but it is still not as simple as what is used to calibrate economic models in the extant literature. Therefore, column (7) further slims down the process by shutting down the heterogeneity in innovation variances: ⌘ . Finally, column (8) shuts down the linear 0 zz HIP component yet again to obtain a process that is essentially the canonical earnings process in the current literature: an AR(1) process plus an i.i.d. shock. We refer to this last and simplest process as the standard model . With this overview, we are now ready to discuss the parameter estimates in more detail. 6.1 Parameter Estimates Because the specifications in columns (1) and (2) deliver similar objective values and parameter estimates, we begin by discussing them together and in some detail. For the estimates in the remaining columns, we will mainly point out the changes from the benchmark model. 35

38 Table IV – Mixture Probabilities of Persistent Components RE (Percentile) groups Age groups 25–29 1 10 50 90 100 35–39 55–60 45–49 Benchmark (Col. 2) ( ⇢ =0 . 26 ) p z z ( ⇢ )0.320.120.060.04 0.600. p =0 43 . x x ⇤ ( =1 . 0 )0.590.810.830.72 0.150.520.790.870.91 p ⇢ ⌫ ⌫ Best fit (Col. 4) ( ⇢ =0 . 08 ) 0.300. p z z p ( ⇢ )0.440.220.130.14 0.660.510.180.070.00 =0 . 51 x x ( =0 . 80 )0.450.820.790.67 ⇢ p ⌫ ⌫ The Life-Cycle Profile. is about 0.50, implying large permanent The estimate of ↵ 30 di ff erences in earnings across individuals. More important, earnings growth rates level display significant heterogeneity: is 0.0149 (or approximately 1.5%) in column (1), and the dispersion in the quadratic component is also sizable. The fixed e ect is negatively ff 31 correlated with the growth rate (about –0.50). The fact that the linear HIP specification in the benchmark model matches the moments nearly as well suggests that the additional ff orded by the quadratic component is not critical for the empirical facts we flexibility a study in this paper. But the quadratic specification introduces three more parameters ( ,corr , corr ) relative to the linear one, which is why the latter is likely to be ↵ preferable in most applications. The Mixture Component. We now turn to the stochastic component driving the dynamics: the most important element is the mixture component. The first AR(1) component has a low persistence of =0 . 26 but a very high innovation standard ⇢ z deviation that ranges from 0.65 for the lowest 1% of individuals to 1.08 for the top i . The second AR(1) component is only 1% (ranked by ), with a mean of ̄ =0 . 85 x z ⇢ slightly more persistent, =0 . 43 with about half the average innovation volatility of x i 36 z between 0.23 and 0.57 (for the bottom and top 1% =0 . : ,andadispersionfor ̄ x x 32 The third and final persistent component is restricted to be a random respectively). 087 walk ( =1 . 0 )andhas assumed to be common to all individuals. =0 . , ⇢ ⌫ ⌫ 30 This figure is on the high end of earlier estimates; see, for example, Haider ( 2001 ), Storesletten etal. ( 2004 ), and Heathcote etal. ( 2010 ). 31 These figures are comparable to earlier estimates. It is perhaps surprising that even though the moments targeted here include a much broader set of statistics than these previous studies, and the process considered here has many more components, the estimates of growth rate heterogeneity are not too di ff erent. 32 This sizable heterogeneity is consistent with that in the work of Browning etal. ( 2010 ). 36

39 We now turn to the mixture probabilities, which are nontrivial functions of age and past earnings. Rather than try to interpret the coe cients a ffi to d directly, ( j = z, x ) j j it is more useful to calculate the values of for di ff erent age and past earnings groups. p j IV The top panel of Table reports these probabilities for the benchmark model. As seen are realized with a low probability, in the left panel, the very large innovations to z ranging from 7% to 26%, and are flat until the middle ages and increase into old ages. The midsize innovations into are realized with a probability of 32% for the youngest x workers, declining to 4% for the oldest workers. The third process, ⌫ (the most persistent t component), receives innovation with the highest probability, ranging from 59% to 83% 33 and displaying a hump-shaped pattern over the life cycle. The top right panel of IV displays how these probabilities vary with the earnings level. The first two Table z x ) show a monotonically declining probability components with low persistence ( and 34 with earnings, implying that the permanent component displays the opposite pattern. To summarize the mixture structure, the component that receives the largest shocks displays the smallest persistence and the smallest probability of realization, whereas the component that receives the smallest shocks (about 10 times smaller than the first compo- nent) displays the highest persistence and the highest probability realization. Therefore, individuals typically receive small shocks that are quite persistent, and once in a while ⌘ they receive much larger shocks ( ⌘ )thatarelesspersistent. Itisthisstructure or z x that is behind the success of the model in generating the very high kurtosis observed in the data (as well as its variation across groups, to be analyzed in a moment). The mean value of each process is also worth commenting on. The volatile process z μ = 0 . 544 , which—together with the positive but quite has a large negative mean, t z =0 μ =0 . 02 ,μ )—is largely responsible small means of the other two components ( . 068 ⌫ x for the longer left tail of the earnings change distribution as well as the small hump on the left shoulder, in turn driving the negative skewness seen in the data. Finally, the i.i.d. shock, " ,hasaverysmallstandarddeviationof in =0 . 039 " column (1) and even smaller in some of the other specifications in the table. It seems ⌘ ), but that relatively transitory shocks exist in the data and can be very large (such as z these are not shocks that are regularly realized every period. 33 Karahan and Ozkan 2013 ) find a similar pattern in the PSID: earnings shocks are only moderately ( persistent for young workers and persistence increases with age until during the first half of the life cycle. 34 Figure A.25 provides a more detailed look at these probabilities by plotting 3D graphs of p as a j function of t and y . t 1 37

40 Figure 16 – Histogram of Log Earnings Changes: US Data vs. Benchmark Model 4.5 Benchmark Model US Data 4 3.5 3 2.5 2 Density 1.5 1 0.5 0 0 3 2 1 -1 -2 -3 y y − t +1 t Mixture Model with No HIP Component (Column 3). Shutting down the HIP i component (setting 0 )leadstoaconsiderablyworsefit,almostequallydueto ⌘ apoorerfittocross-sectionalmoments(objectiverisingfrom5.901to9.097)andto impulse response moments (rising from 10.134 to 14.033). The most notable changes in parameter estimates happen in the volatilities: ̄ goes up from 0.85 to 0.95, and z more important, the permanent innovation, ̄ ,nearlydoubles,from0.087upto0.174. ⌫ However, as we elaborate below, the fit of this specification to some key moments are poor, so we do not discuss this case any further. Best Fit (Column 4). ⌫ to a random walk, both because So far, we have restricted t this is a pervasive assumption in the earnings dynamics literature and because it elimi- ⇢ nates one parameter ( ), making it potentially more convenient for use in calibration. v In column (4) we relax the unit root assumption and estimate a persistence parameter for this component of ⇢ =0 . 80 . This added flexibility leads to a non-negligible improve- ⌫ ment in the fit of the model, attaining the lowest objective value ( . 29 ), indicating that 15 the unit root restriction is not supported by the data. In the bottom panel of Table IV , we report the innovation probabilities, p for this specification. As seen here, the rough , j magnitudes of probabilities as well as their pattern of variation remain largely unchanged relative to the benchmark model in the top panel. 38

41 Simplified Benchmark (Column 5). Persistent components are expensive to in- clude in quantitative economic models because each one introduces a new state variable. Therefore, in column 4, we take the benchmark model and eliminate the random walk component, reducing the number of parameters significantly (along with losing some flexibility). Not surprisingly, the fit worsens but not by a large amount (objective rising from 17.17 to 18.30). This is also borne out in the model-data comparisons of moments in the next section. In our view, this is the simplest specification that can reasonably be considered for calibrating economic models. Models with No Mixture (Columns 6, 7, and 8). We now reduce the number of AR(1) components to 1. This also eliminates variation in moments with age and RE, since these were delivered by variations in mixing probabilities. As seen in Table III , the objective value more than doubles for all three cases. The estimated persistence is close to a unit root. And whether we include a HIP component (column 7) or allow for the innovation variance to be heterogeneous across individuals does little to improve this poor fit. Clearly, this critique applies to the standard model that is the workhorse specification in the incomplete markets literature. In light of these results, we find such overly simplified models to be unsuitable for serious quantitative economic work. 6.2 Model Fit to Key Moments Alargenumberofmomentsaretargetedintheestimation,anddiscussinghowthe estimated models fit all of them is not feasible. Therefore, in this section, we discuss the model fit to some of the key moments and relegate the remaining moments to Appendix C .BecausemanyofthetargetedmomentsdisplaypatternsbyageandRElevelsthat are visually easy to see, in Figures 16 to 18 we plot these moments as well as their 35 counterparts simulated from each of the eight estimated models. 16 y generated by the benchmark model, su- Figure plots the histogram of y t t +1 perimposed on the data counterpart. The fit is quite good, except at the center where the model misses the size of the spike in the histogram. Next, Figure 17 plots the cross- sectional moments of the data for prime-age individuals. The top, middle, and bottom panels display, respectively, the standard deviation, skewness, and kurtosis of earnings 35 Eagle-eyed readers will notice that the statistics plotted in this section look slightly di ff erent from those reported before in Section 3 . This is because here we are plotting the exact moments that are targeted in the estimation, which have been aggregated as described in the previous section and also include observations below the threshold level Y . min 39

42 Figure 17 – Fit of Estimated Model to Key Cross-Sectional Data Moments i i i i y y Std. dev. of y Std. dev. of y (a) (b) t t +1 t 5 + t US Data 2AR, 1RW, HIP(2) 1.6 2AR, 1RW, HIP(1): Benchmarkmodel 2AR, 1RW, HIP(0) 3AR, HIP(1): Best fi tmodel 1.4 1.2 2AR, HIP(1) 1AR, HIP(1) 1.2 1AR, HIP(1), no variance heter. 1 Standardmodel 1AR, HIP(0): 1 0.8 0.8 0.6 0.6 Std. Dev. of y(t+1)-y(t) Std. Dev. of y(t+5)-y(t) 0.4 0.4 0.2 0.2 0 40 60 80 100 40 20 0 100 80 60 0 20 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution i i i i y (c) y y Skewness of y Skewness of (d) t t t +1 5 + t 0.2 0.2 -0.2 -0.2 -0.6 -0.6 -1 -1 Skewness of y(t+5)-y(t) Skewness of y(t+1)-y(t) -1.4 -1.4 -1.8 -1.8 40 60 80 100 100 0 20 20 40 60 80 0 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution i i i i Kurtosis of y y y (e) y Kurtosis of (f) t t t +1 + 5 t 30 25 10 20 15 Kurtosis of y(t+1)-y(t) Kurtosis of y(t+5)-y(t) 10 5 5 0 0 20 40 60 80 100 0 20 40 60 80 100 Percentiles of Recent Earnings (RE) Distribution Percentiles of Recent Earnings (RE) Distribution 40

43 Table V – Percentiles of Earnings Growth Distribution: Benchmark Model vs. Data Percentiles Percentile P1 P25 P75 P90 P95 P99 P5 P10 –1.84 –0.35 –0.07 0.13 0.34 0.64 1.40 Data –0.75 –1.58 Model –0.34 –0.09 0.12 0.34 0.65 1.33 –0.73 Percentile Di erentials ff P95 – P5 P90 – P10 P99 – P91 P75 – P25 Data 3.24 1.39 0.69 0.20 Model 2.91 1.38 0.68 0.22 changes. The subfigures on the left are for one-year changes, and those on the right are for five-year changes. Although the graphs display data generated from each one of the eight specifications, we will focus on three important cases: (i) the benchmark model, (ii) the standard model, and (iii) column (3), which is the same as the benchmark model but without a HIP component. The first is shown with a solid black line with circles; the second one with a dashed brown line; and the last one with a thick gray solid line. The first observation is that all specifications understate the standard deviation of shocks both at both short and long horizons. We have experimented with further (un- reported) specifications but failed to match this aspect of the data. The reason seems to largely be that the earnings changes in the data display extremely long tails, and the mixture of three processes fails to completely capture this feature without compromising fit in other dimensions. This can be seen in Table V , which plots selected percentiles from the annual earnings growth distribution (top panel). As seen here, the benchmark model does an excellent job of matching the all percentiles between the 5th and 95th, but fails to generate the very low values of the lowest and highest percentiles. Consequently, the P99-P1 di ff erence (bottom panel) is 3.24 in the data but only 2.91 in the model (a 36 di erence of 33 log percents). ff That said, the benchmark model still performs better for this moment than other specifications (including the best fit model). It captures the declining pattern with RE percentiles, although fails to capture the sharp rise in volatility at the very top of the RE distribution. The no-HIP specification yields an increasing and then flat profile, whereas the standard model substantially underestimates the standard deviation (generating only 36 We suspect that mixing a fourth AR(1) process could improve fit along this dimension, but we have not tried that (demanding) specification. The current benchmark model takes more than one week to estimate on a cluster using 100+ state-of-the-art CPUs in parallel. 41

44 Figure 18 – Fit of Estimated Model, Continued (a) Life-Cycle Earnings Growth: Ages 25 to 55 Variance of Log Earnings (right) (b) 1.6 0.8 US Data 2AR, 1RW, HIP(2) 0.6 Benchmarkmodel 2AR, 1RW, HIP(1): 2AR, 1RW, HIP(0) 1.4 Best fi tmodel 3AR, HIP(1): 0.4 2AR, HIP(1) 1AR, HIP(1) 0.2 1.2 1AR, HIP(1), no variance heter. 1AR, HIP(0): Standardmodel 0 1 -0.2 -0.4 0.8 -0.6 0.6 Within-Cohort Variance of Log Earnings -0.8 Log earnings growth 25-55: Model - US Data 0.4 -1 30 35 40 45 50 80 55 60 100 40 20 0 60 25 Lifetime Earnings (LE) Percentile Age one-quarter of the empirical value for the median worker). Turning to skewness, we now see that the benchmark model performs considerably better, matching both the qualitative patterns as well as the overall level of skewness throughout the distribution for one-year changes. Furthermore, both the benchmark model and the best-fit specification perform nearly as well for skewness. However, the no-HIP model (column 3) generates less than half of the level of negative skewness at the any skewness at the five-year horizon. As could one-year horizon and fails to generate be expected, once the mixture component is shut down (columns 6, 7, and 8), the model generates zero skewness at all horizons, which is grossly inconsistent with the data. The bottom panel shows the kurtosis patterns, and the benchmark model again does well. It manages to match both the qualitative pattern of rising kurtosis with RE levels up to the 90th percentile and the subsequent decline, as well as its levels throughout the picture. As with skewness, the other specifications that allow for both HIP and a mixture structure (columns 1, 4, and even 5) perform comparably well. However, if the HIP specification is eliminated (as in column 3), this reduces the level of kurtosis at its peak by almost half; further eliminating the mixture structure reduces the kurtosis to 4.5. Finally, the standard model is Gaussian with a kurtosis of 3.0. Figure 18a plots the fit of each model to life-cycle earnings growth moments. The quadratic HIP specification in column (1) is the clear winner here, matching the patterns for all LE groups between the 5th and 90th percentiles. The benchmark model is a 42

45 close second. The best fit model does less well, overestimating growth for most income groups. However, all models miss the very high income growth for those with high lifetime earnings. This is a deficiency we have not been able to remedy. Future work should focus on understanding the earnings dynamics of top earners in the United States. plots the cross-sectional variance of log earnings by age. The 18b Finally, Figure benchmark model and best fit model perform arguably the best, matching the total rise in inequality over the life cycle. The standard model and the models in columns (3) and (7) tie for the worst performance, overestimating the rise of inequality by nearly 100%. 7 Concluding Thoughts Quantifying the Measured Earnings Risk Consider the well-known thought experiment ( ); Pratt ( 1964 )), in which Arrow ( 1965 ,bya adecisionmakerchoosesbetween(i)astaticgamblethatchangesconsumption, c ̃ called the risk premium, to avoid ) and (ii) a fixed payment ⇡ , random proportion (1 + the gamble. An expected utility maximizer solves h i ̃ (1 U ⇡ )) = E ( U ( c ⇥ (1 + c ) ⇥ . (10) A ̃ Let us compare two scenarios. In the first one, is drawn from a Gaussian distribution B ̃ has the same first two with mean zero and standard deviation of 0.10. In the second, A ̃ moments as ffi cient of 2 and a kurtosis of 30 (roughly , but it also has a skewness coe corresponding to the one-year earnings change of a 45-year-old individual at P90 of the RE distribution). Further assume that U is a CRRA utility function with a curvature of ✓ =10 10 ) is an equation in one unknown. The solution for ⇡ under di ff erent . Equation ( B ̃ VI is assumptions is displayed in Table . As seen here, the willingness to pay to avoid A ̃ 22.15% compared to 4.88% for ,anamplificationof450%. To provide a decomposition for the separate e ff ects of skewness and kurtosis, the following equation is helpful. Taking a first-order Taylor-series approximation to the left hand side and fourth-order approximation to the right hand side, we get   ✓ +2)( ✓ ( ( ✓ ✓ +1) +1) ✓ ✓ 3 4 2 ⇤ ⇥ ,  ⇥ ⇥ s ⇥ + ⇥ ⇡ ⇡ 2 24 6 |{z} | {z } } {z | variance aversion kurtosis aversion negative skewnewss aversion 43

46 Table VI – E ect of Skewness and Kurtosis on Risk Premium ff ⇡ ) Risk Premium ( 3.0 Kurtosis 30.0 ! Skewness # 0.0 4.88% 18.79% 6.34% 22.15% –2.0 s cients, respectively. This expres- where  ffi are the skewness and kurtosis coe and sion is a generalization of the well-known formula that only includes the first of these 37 terms. Among other things, it shows that the first term—what we often refer to as “risk aversion”—is perhaps more appropriately called “variance aversion,” because with higher-order moments, the individual also displays aversion to negative skewness and to excess kurtosis. These are components of the risk perceived by the individual and can significantly amplify the aversion. In particular, the coe ffi cient in front of skewness is a quadratic in ✓ and a cubic in . Therefore, keeping the skewness fixed, a higher curvature or dispersion can greatly amplify the risk premium. For kurtosis, the situation ffi ✓ and a quartic in dispersion. As seen is even more extreme: the coe cient is a cubic in ✓ in the example, even a standard deviation of 0.10 and delivers a substantially =10 amplified risk premium. Increasing from 0.10 to 0.25 makes the negative skewness aversion twice as large, and kurtosis aversion twenty times as large, as variance aversion. Clearly, these calculations are meant to be suggestive rather than conclusive. A more thorough analysis in the future should start by measuring insurance and smoothing opportunities against these earnings fluctuations as well as distinguishing between an anticipated earnings change versus an unexpected shock. Summary and Conclusions Our analysis of the life-cycle earnings histories of millions of U.S. workers has reached two broad conclusions. First, the higher-order moments of individual earnings shocks display clear and important deviations from lognormality. In particular, earnings shocks display strong negative skewness (what can be viewed as individual disaster shocks) and extremely high kurtosis—as high as 35 compared with 3 for a Gaussian distribution. The high kurtosis implies that in a given year, most individuals experience very small earnings shocks, few experience middling shocks, and a small but non-negligible number 37 See Appendix B.3 for derivation. 44

47 experience extremely large shocks. The second conclusion of this analysis is that these statistical properties of earnings shocks change substantially both over the life cycle and with the earnings level of indi- viduals. For example, our estimates of the stochastic process show that low-income indi- viduals experience very large earnings shocks with low persistence, whereas high-income individuals experience shocks that are very persistent but with much lower volatility. These properties of earnings shocks over the life cycle are likely to generate a range of interesting implications for economic behavior. We have also estimated impulse response functions of earnings shocks and found sig- nificant asymmetries: positive shocks to high-earnings individuals are quite transitory, whereas negative shocks are very persistent; the opposite is true for low-earnings indi- viduals. While these statistical properties are typically ignored in quantitative analyses of life-cycle models, they are fully consistent with search-theoretic models of careers over the life cycle. After establishing these empirical facts nonparametrically, we estimated what we think is the simplest earnings process that is broadly consistent with these salient features of the data. Abroadermessageofthispaperisacalltoreconsiderthewayresearchersapproach the study of earnings dynamics. The covariance matrix approach that dominates current work is too opaque and a bit mysterious: it is di ffi cult to judge what it means to match or miss certain covariances in terms of their economic implications. With the current trend toward the increasing availability of very large panel data sets, we believe that researchers’ priority in choosing methods should shift from e ffi ciency concerns to trans- parency. The approach adopted here is an example of the latter, and we believe that economists can better judge what each moment implies for economic questions. The nonparametric empirical facts documented in Sections 3 and 4 (some of which are reported in an online appendix to save space) add up to more than 10,000 empirical moments of individual earnings data. The richness of this information is far beyond what we are able to fully utilize in the estimation exercise in this paper. Furthermore, for di ff erent questions, it would make sense to focus on a di ff erent subset of these moments from what we have aimed for in this paper. With these considerations in mind, we make these detailed moments available online for download as an Excel file. We hope the empirical findings about the nature of earnings dynamics documented in this paper feed back into economic research and policy analyses. 45

48 References Abowd, J. M. Card, D. (1989). On the covariance structure of earnings and hours and Econometrica , (2), 411–45. changes. 57 Smith, A. A. Vidangos, I. (2013). Modeling earnings dynamics. , and Altonji, J. 81 (4), 1395–1454. Econometrica , Blundell, R. and Bonhomme, S. (2014). Household Earnings and , Arellano, M. . mimeo, University College London. Consumption: A Nonlinear Framework (1965). Aspects of the Theory of Risk Bearing ,YrjöJahnssonlectures,Yrjo Arrow, K. Jahnssonin Saatio, Helsinki. Attanasio, O. P. Banks, J. , Meghir, C. and Weber, G. (1999). Humps and bumps , Journal of Business & Economic Statistics , in lifetime consumption. (1), 22–35. 17 Bachmann, R. and Bayer, C. (2014). Investment dispersion and the business cycle. American Economic Review , 104 (4), 1392–1416. Bagger, J. , Fontaine, F. , Postel-Vinay, F. and Robin, J.-M. (2014). Tenure, experience, human capital and wages: A tractable equilibrium search model of wage dynamics. , 104 (6), 1551–1596. American Economic Review and (2003). Earnings dynamics and inequality among Canadian Baker, M. Solon, G. men, 1976–1992: Evidence from longitudinal income tax records. Journal of Labor , 21 Economics (3), 289–321. Berger, D. and Vavra, J. (2011). Dynamics of the U.S. Price Distribution .Working paper, Yale University. Bloom, N. , Floetotto, M. , Jaimovich, N. , Saporta-Eksten, I. and Terry, S. J. Real ly Uncertain Business Cycles .Workingpaper,StanfordUniversity. (2011). Blundell, R. , Graber, M. and Mogstad, M. (2014). Labor income dynamics and the insurance from taxes, transfers, and the family. Journal of Public Economics . Bonhomme, S. and Robin, J.-M. (2009). Assessing the equalizing force of mobility using short panels: France, 1990 a 2000. Review of Economic Studies , 76 (1), 63–92. 46

49 Borovicka, J. , and Sheinkman, J. A. (2014). Shock Elasticities and Hansen, L. P. . Working paper, University of Chicago. Impulse Responses Ejrnaes, M. and Alvarez, J. (2010). Modelling income processes Browning, M. , , ,1353–1381. Review of Economic Studies 77 with lots of heterogeneity. Hirano, K. (1999). Predictive distributions based on longitu- Chamberlain, G. and , 55-56 ,211–242. Annales d’Économie et de Statistique dinal earnings data. , Eichenbaum, M. and Evans, C. L. Christiano, L. J. (2005). Nominal rigidities and the dynamic e ects of a shock to monetary policy. Journal of Political Economy , ff (1), 1–45. 113 and Ghosh, A. (2014). Asset Pricing with Countercyclical Constantinides, G. M. . Working paper, University of Chicago. Household Consumption Risk (1991). Saving and liquidity constraints. Econometrica , 59 (5), 1221–48. Deaton, A. and Paxson, C. (1994). Intertemporal choice and inequality. Journal of Political — Economy , 102 (3), 437–67. Geweke, J. and (2000). An empirical analysis of earnings dynamics among Keane, M. Journal of Econometrics 96 ,293–356. men in the PSID: 1968-1989. , and — (2007). Smoothly mixing regressions. Journal of Econometrics , 138 — ,252–290. Golosov, M. , Troshkin, M. and Tsyvinski, A. (2014). Redistribution and Social Insurance . Working paper, Princeton University. Gourinchas, P.-O. and Parker, J. A. (2002). Consumption over the life cycle. Econo- metrica 70 (1), 47–89. , Guvenen, F. (2013). Quantitative economics with heterogeneity. Book Manuscript, University of Minnesota. — , Ozkan, S. and Song, J. (2014). The nature of countercyclical income risk. Journal of Political Economy 122 (3), 621–660. , — and Smith, A. A. (2014). Inferring labor income risk and partial insurance from economic choices. Econometrica . 47

50 Haider, S. J. (2001). Earnings instability and earnings inequality of males in the united Journal of Labor Economics , (4), 799–836. states: 1967-1991. 19 (1980). The fine structure of earnings and the on-the-job training hypoth- Hause, J. C. esis. , 48 (4), 1013–1029. Econometrica Storesletten, K. and Heathcote, J. (2010). The macroeco- , Violante, G. L. Journal of Political nomic implications of rising wage inequality in the united states. Economy , 118 (4), 681–722. Karahan, F. Ozkan, S. (2013). On the persistence of income shocks over the and life cycle: Evidence, theory, and implications. , 16 (3), Review of Economic Dynamics 452–476. Lillard, L. A. Weiss, Y. (1979). Components of variation in panel earnings data: and Econometrica , 47 American scientists 1960-70. (2), 437–454. — and Willis, R. J. (1978). Dynamic aspetcs of earnings mobility. Econometrica , 46 (5), 985–1012. Low, H. , and Pistaferri, L. (2010). Wage risk and employment risk over Meghir, C. American Economic Review 100 (4), 1432–1467. the life cycle. , MaCurdy, T. E. (1982). The use of time series processes to model the error structure Journal of Econometrics , of earnings in a longitudinal data analysis. (1), 83–114. 18 Mankiw, N. G. (1986). The equity premium and the concentration of aggregate shocks. Journal of Financial Economics , 17 (1), 211–219. McKay, A. (2014). Time-Varying Idiosyncratic Risk and Aggregate Consumption Dy- namics .Workingpaper,BostonUniversity. and Pistaferri, L. (2004). Income variance dynamics and heterogeneity. Meghir, C. Econometrica 72 (1), 1–32. , Midrigan, V. (2011). Menu costs, multiproduct firms, and aggregate fluctuations. Econometrica , 79 (4), 1139–1180. Moors, J. J. A. (1986). The meaning of kurtosis: Darlington reexamined. The Ameri- can Statistician , 40 ,283–284. 48

51 Murphy, K. M. and (1990). Empirical age-earnings profiles. Journal of Welch, F. , 8 Labor Economics (2), 202–29. Steinsson, J. (2013). Price rigidity: Microeconomic evidence and and Nakamura, E. Annual Review of Economics , 5 (133-163). macroeconomic implications. and Hudson, R. (2009). Social security administration’s master earnings Olsen, A. Social Security Bulletin file: Background information. 69 (3), 29–46. , Bradley, M. , , Panis, C. , Euller, R. , Peterson, C. E. , Hirscher, R. Grant, C. and Stinberg, P. (2000). SSA Program Data User’s Manual . Baltimore, MD: Social Security Administration. Pratt, J. W. Econometrica , (1964). Risk aversion in the small and in the large. (1/2), 122–136. 32 , Telmer, C. I. and Yaron, A. Storesletten, K. (2004). Cyclical dynamics in idiosyncratic labor market risk. Journal of Political Economy , 112 (3), 695–717. Topel, R. H. and Ward, M. P. (1992). Job mobility and the careers of young men. The Quarterly Journal of Economics 107 (2), 439–79. , Zhang, H. , Conn, A. R. and Scheinberg, K. (2010). A derivative-free algorithm for least-squares minimization. SIAM Journal on Optimization , 20 (6), 3555–3576. 49

52 Supplemental Online Appendix NOT FOR PUBLICATION 50

53 A Appendix: Additional Figures Further Figures A.1 In this section, we report some additional figures of interest that are omitted from the A.1 plots the selected percentiles of the main text due to space constraints. First, Figure future earnings change distribution for every RE percentile. Second, Figure A.1 plots the cross- sectional variance of log earnings over the life cycle, constructed along the lines described in Deaton and Paxson ). ( 1994 Robustness and Extensions A.2 A.2.1 Decomposing Moments: Job-Stayers vs. Job-Switchers In this section, we present the the cross-sectional statistics analyzed in Section 3 by first splitting the sample in each year depending on whether a worker switched employers or whether he stayed at the same job. One challenge we face is that many workers hold multiple jobs in a given year, which requires us to be careful about how to think of job changes. We have explored several plausible definitions and found broadly very similar results. Here we describe one reasonable classification. A worker is said to be a “job-stayer” between years t and +1 if a given EIN (employer identification t number) provides the largest share of his annual earnings (out of all his EINs in that year) in t 1 through t +2 , and if the same EIN provides at least 90% of his total annual earnings years 38 t and t +1 . A worker is defined as a “job-switcher” if he is not a job-stayer. in years As seen in Figure A.4 , the results are consistent with what we might expect. Job-stayers (i) face a dispersion of earnings changes that is less than half that of job-changers, (ii) face shocks that have zero or slightly positive skewness as opposed to job-switchers who face shocks that are very negatively skewed, and (iii) experience shocks with much higher kurtosis than job-switchers. In fact, kurtosis is as high as 43 for annual changes and 28.5 for five-year changes for job-stayers, but is less than 10 for job-switchers at both horizons. Before concluding this section, we examine how the tails of the earnings change distribution ff ect the computed statistics and also examine how this e ff a ect varies by stayers and switchers. Unlike with survey-based data, here we are not too concerned that these tails might be domi- nated by measurement error, as most of these changes are likely to be genuine. Instead, we are simply interested in understanding what parts of the distribution are critical for the di ff erent moments that we estimated so far. Table A.1 reports the 2nd to 4th moments for the original sample used so far (left panel) as well as for a sample where we drop extreme observations, defined as those in the top or bottom 1% of the earnings change distribution. As expected, discarding the tails reduces all statistics, but does not change the ranking between stayers and switchers. 38 Clearly, this classification is quite stringent for classifying somebody as a job stayer, meaning that some individuals will be classified as job switchers even though they did not change a job. An alternative definition we have explored defines a job switcher directly as somebody who has an EIN that provides more than 50% of his annual earnings in year t and provides less than 10% of his annual earnings in year t +1 ; and also has an EIN that provides less than 10% of his earnings in t and more than 50% in t +1 . The results were very similar to those reported here. 51

54 Figure A.1 – Quantiles of Earnings Changes Quantiles of One-Year Earnings Changes (a) 2.5 P95 P90 P75 P50 P25 P10 P5 P99 1.5 t y − 0.5 +1 t y -0.5 Percentiles of -1.5 100 80 60 40 20 0 Percentiles of 5-Year Average Income Distribution 3 P90 P75 P50 P99 P25 P10 P5 P95 2 t y − 1 +5 t y 0 Percentiles of -1 -2 80 60 40 20 0 100 Percentiles of 5-Year Average Income Distribution (b) Quantiles of Five-Year Earnings Changes 52

55 Figure A.2 – Within-Cohort Variance of Log Earnings 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 Cross Sectional Variance of Log Earnings 0.55 55 35 60 25 50 45 40 30 Age –FractionStayingJobsBetween t and t +1 Figure A.3 0.8 0.7 t y and 0.6 +1 t y 0.5 0.4 0.3 Fraction of Stayers between 0.2 25-35 36-45 46-55 0.1 60 70 80 90 100 0 10 20 30 40 50 Percentiles of Recent Earnings Distribution 53

56 Figure A.4 – Second to Fourth Moments of Annual Earnings Growth: Stayers vs Switchers Std. Dev., Five-Year (b) (a) Std. Dev., One-year 1.2 Switchers 25-35 Stayers 25-35 ) ) 1 t t Switchers 36-45 y y Stayers 36-45 1 − − Switchers 46-55 +1 +5 t Stayers 46-55 t y y 0.8 0.8 0.6 0.6 0.4 0.4 Standard Deviation of ( Standard Deviation of ( 0.2 0.2 100 80 40 20 0 60 100 80 60 40 20 0 Percentiles of RE Distribution Percentiles of RE Distribution (c) Kelly’s Skewness, One-Year Kelly’s Skewness, Five-Year (d) ) 0.1 0.1 t ) t y y − − 0 0 +5 t +1 t y y -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 Stayers 25-35 Stayers 36-45 -0.4 -0.4 Stayers 46-55 Zero line Kelley’s Skewness of ( Switchers 46-55 -0.5 -0.5 Kelley’s Skewness of ( Switchers 25-35 Switchers 36-45 -0.6 -0.6 0 0 60 100 20 100 80 60 40 20 40 80 Percentiles of RE Distribution Percentiles of RE Distribution Kurtosis, One-Year Kurtosis, Five-Year (f) (e) 40 40 Switchers 25-35 Switchers 36-45 35 35 Switchers 46-55 Stayers 25-35 Stayers 36-45 ) ) 30 30 t t Stayers 46-55 y y − − 25 25 +5 +1 t t y y 20 20 15 15 Kurtosis of ( Kurtosis of ( 10 10 5 5 0 0 60 40 20 0 0 100 80 100 80 60 40 20 Percentiles of RE Distribution Percentiles of RE Distribution 54

57 Table A.1 – Moments with Trimmed Data Trim top/bot. 1% Original Std. dev. Skew. Kurtosis Std. dev. Skew. Kurtosis One-Year 0.49 –1.49 18 . All 0.36 –0.89 8 . 91 35 0.31 –1.51 42 . 81 0.20 –0.51 13 . Stayers 50 Switchers –1.09 9 . 69 0.56 –0.85 6 . 17 0.67 Five-Year 0.66 –1.06 12 . 16 0.53 –0.60 6 . 58 All Stayers 0.41 –1.04 28 . 54 0.29 0.10 7 . 16 Switchers 0.78 86 0.65 –0.67 5 . 61 –0.92 8 . ff ects of Age and Recent Earnings (Case II) A.2.2 Disentangling The E and A.6 A.5 Figures plot the 3D graphs of skewness and kurtosis that jointly conditions on age and recent earnings as described in Section 3.5 . – Skewness of One-year Earnings Change, Jointly Conditioning on Age and Figure A.5 Recent Earnings 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 0 0 2 20 40 4 Age group 60 6 80 RE Percentiles 8 100 A.2.3 Cross-Sectional Facts This section reports the analogs of figures in Section 3 by considering three additional robustness exercises. In particular, case III averages earnings over two consecutive years before 55

58 Figure A.6 – Kurtosis of One-year Earnings Change, Joint Conditioning on Age and Recent Earnings 30 25 20 15 10 5 100 0 80 8 60 6 40 4 Age Group 20 2 RE Percentiles 0 0 computing statistics: i i i i i i i i i ̃ ̃ Y Y + ) log( Y . = log( + Y Y Y log( ) and ) y ) = log( Y y + Y + t 5 t +5 +6 +1 t t t +2 t +3 t +1 t t t Case IV takes the di ff erence between earnings in year t + k and usual earnings: i i i i i i y log( Y ) ) and = log( Y = log( Y y ) ) log( Y . long short 1 +1 t t t 1 t +5 Case V considers two methods of trimming the tails. The first method excludes the top and bottom 1% of earnings growth observations (referred to as Case V.A in figures). The second method changes the lower threshold for sample exclusion from Y to be individual-specific ,t min and equal to 5% of each worker’s own recent earnings (Case V.B). For all case numbers indicated below, see Section 3.5 in the main text contains more detailed descriptions. 56

59 0.8 ) t y 25-29 0.7 − 30-34 +1 t 35-39 y 40-44 0.6 45-49 50-54 0.5 0.4 Standard Deviation of ( 0.3 60 20 80 100 0 40 Percentiles of 5-Year Average Income Distribution (a) Case III: Bi-Annual 1.1 ) t 1 y 25-29 − 30-34 +1 0.9 t 35-39 y 40-44 45-49 0.8 50-54 0.7 0.6 0.5 Standard Deviation of ( 0.4 80 100 0 20 40 60 Percentiles of 5-Year Average Income Distribution (b) Case IV: Relative to Usual Figure A.7 – Standard Deviation of Annual Earnings Growth 57

60 1 ) t 0.9 y 25-29 − 30-34 +5 t 35-39 y 0.8 40-44 45-49 50-54 0.7 0.6 0.5 Standard Deviation of ( 0.4 80 20 100 0 60 40 Percentiles of 5-Year Average Income Distribution Figure A.8 – Standard Deviation of Bi-Annual Five-Year Earnings Growth, Case III 1.2 ) 1.1 t y 25-29 − 30-34 1 +5 t 35-39 y 40-44 0.9 45-49 50-54 0.8 0.7 0.6 0.5 Standard Deviation of ( 0.4 100 0 20 40 60 80 Percentiles of 5-Year Average Income Distribution Figure A.9 – Standard Deviation of Five-Year Earnings Growth, Usual, Case IV 58

61 Kelly’s measure Third standardized moment 0.1 0.5 0 ) 0 t y ) − t y +1 0.1 − t − 0.5 − y +1 t y 0.2 − 1 − 25-29 25-29 − 0.3 − 1.5 30-34 30-34 35-39 Skewness of ( 35-39 40-44 Kelly Skewness of ( 0.4 − 40-44 2 − 45-49 45-49 50-54 50-54 0.5 − − 2.5 40 60 80 100 0 20 20 100 80 60 40 0 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.10 – Skewness of (Bi-Ann) Annual Earnings Growth, Case III Third standardized moment Kelly’s measure 0.5 0 ) 25-29 25-29 t 0 y 0.5 − ) 30-34 30-34 t − y 35-39 35-39 +1 1 − − t 40-44 40-44 y +1 t 45-49 45-49 y − 1.5 50-54 50-54 − 2 − 0.2 − 2.5 Skewness of ( 3 − Kelly Skewness of ( 3.5 − 0.4 4 − − 0 20 40 60 80 100 20 80 100 0 40 60 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.11 – Skewness of Annual Earnings Growth, Usual, Case IV 59

62 Third standardized moment Kelly’s measure 0.5 0.1 0 0 ) ) t t y y − 0.5 − − − 0.1 +5 +5 t t y y − 0.2 1 − − 0.3 25-29 1.5 − 25-29 30-34 30-34 − 0.4 Skewness of ( 35-39 35-39 40-44 − 2 40-44 Kelly Skewness of ( 0.5 − 45-49 45-49 50-54 50-54 − 0.6 2.5 − 20 0 100 40 60 80 80 60 40 100 0 20 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution – Skewness of (Bi-Ann) Five-Year Earnings Growth, Case III Figure A.12 Kelly’s measure Third standardized moment 0.2 0.5 0 0 25-29 ) 30-34 t y 0.5 − 35-39 − 0.2 − 40-44 +5 t 45-49 − 1 y 50-54 1.5 − − 0.4 25-29 30-34 2 − 35-39 Skewness of ( 0.6 − 40-44 − 2.5 45-49 50-54 − 0.8 3 − 60 80 100 0 20 40 80 0 20 40 60 100 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.13 – Skewness of Five-Year Earnings Growth, Usual, Case IV 60

63 Kelly’s measure Third standardized moment 0.2 0.5 0 ) t 25-29 y 0.1 ) 30-34 t − y − 0.5 35-39 +1 − t 40-44 y +1 0 45-49 t − 1 y 50-54 − 1.5 0.1 − 25-29 2 − 30-34 Skewness of ( 35-39 0.2 − Kelly Skewness of ( 40-44 2.5 − 45-49 50-54 0.3 − 3 − 60 0 20 0 80 20 40 40 100 100 80 60 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.14 – Skewness of One-Year Earnings Growth, Trimming Tails, Case V.A Third standardized moment Kelly’s measure 0.5 0 ) t y ) t − 0.5 − y +5 − 0.2 − t y +5 t y 1.5 − − 0.4 25-29 25-29 30-34 30-34 2.5 − Skewness of ( 0.6 − 35-39 35-39 Kelly Skewness of ( 40-44 40-44 45-49 45-49 50-54 50-54 0.8 − − 3.5 80 100 20 40 60 80 100 0 0 20 40 60 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.15 – Skewness of Five-Year Earnings Growth, Trimming Tails, Case V.A 61

64 Kelly’s measure Third standardized moment 0.2 0.5 0 ) t 25-29 y 0.1 ) 30-34 t − y − 0.5 35-39 +1 − t 40-44 y +1 0 45-49 t − 1 y 50-54 − 1.5 0.1 − 25-29 2 − 30-34 Skewness of ( 35-39 0.2 − Kelly Skewness of ( 40-44 2.5 − 45-49 50-54 0.3 − 3 − 60 0 20 0 80 20 40 40 100 100 80 60 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.16 – Skewness of One-Year Earnings Growth, Trimming Tails, Case V.B Third standardized moment Kelly’s measure 0.5 0 ) t y ) t − 0.5 − y +5 − 0.2 − t y +5 t y 1.5 − − 0.4 25-29 25-29 30-34 30-34 2.5 − Skewness of ( 0.6 − 35-39 35-39 Kelly Skewness of ( 40-44 40-44 45-49 45-49 50-54 50-54 0.8 − − 3.5 80 100 20 40 60 80 100 0 0 20 40 60 Percentiles of 5-Year Average Income Distribution Percentiles of 5-Year Average Income Distribution Figure A.17 – Skewness of Five-Year Earnings Growth, Trimming Tails, Case V.B 62

65 Figure A.19 – Kurtosis of Annual Earnings Change, Usual, Case IV 32 28 25-29 30-34 ) 24 t y 35-39 − 40-44 20 +1 45-49 t y 50-54 16 12 Kurtosis of ( 8 4 0 80 60 40 20 0 100 Percentiles of 5-Year Average Income Distribution – Kurtosis of Bi-Annual Earnings Change, Case III Figure A.18 22 18 ) t y − 14 +1 t y 10 Kurtosis of ( 25-29 6 30-34 35-39 40-44 45-49 50-54 2 100 80 60 40 20 0 Percentiles of 5-Year Average Income Distribution 63

66 Figure A.20 – Kurtosis of Five-Year Earnings Change, Bi-Annual, Case III 16 14 ) t 12 y − +5 t 10 y 8 6 Kurtosis of ( 25-29 30-34 35-39 4 40-44 45-49 50-54 2 80 60 40 20 0 100 Percentiles of 5-Year Average Income Distribution Figure A.21 – Kurtosis of Five-Year Earnings Change, Usual, Case IV 20 16 ) t y − 12 +5 t y 8 25-29 30-34 Kurtosis of ( 35-39 4 40-44 45-49 50-54 0 0 60 100 80 40 20 Percentiles of 5-Year Average Income Distribution 64

67 Figure A.22 – Kurtosis of One-Year Earnings Change, Trimming Tails, Case V.A 15 ) t y 10 − +1 t y 25-29 5 30-34 Kurtosis of ( 35-39 40-44 45-49 50-54 0 60 0 100 80 40 20 Percentiles of 5-Year Average Income Distribution Figure A.23 – Kurtosis of Five-Year Earnings Change, Trimming Tails, Case V.A 10 8 ) t y − 6 +5 t y 4 25-29 30-34 Kurtosis of ( 35-39 2 40-44 45-49 50-54 0 20 80 100 0 40 60 Percentiles of 5-Year Average Income Distribution 65

68 Figure A.24 – Kurtosis, Robustness (a) Kurtosis of One-Year Earnings Change, Excluding those with Earnings lower than 5% of RE, Case V.B 25 20 ) t y − 15 +1 t y 10 25-29 30-34 Kurtosis of ( 35-39 5 40-44 45-49 50-54 0 20 0 100 80 60 40 Percentiles of 5-Year Average Income Distribution 12 10 ) t y 8 − +5 t y 6 25-29 4 30-34 Kurtosis of ( 35-39 2 40-44 45-49 50-54 0 60 80 100 0 20 40 Percentiles of 5-Year Average Income Distribution (b) Kurtosis of Five-Year Earnings Change, Excluding those with Earnings lower than 5% of RE, Case V.B 66

69 B Details of Estimation Method Moment Selection and Aggregation B.1 3 , we have dropped 1. 1. Cross-sectional moments of earnings changes. In Section y + ) < Y individuals with low earnings ( )in t or t exp( k (or both) when constructing the it min moments of the cross-sectional distribution. We now modify this procedure slightly so as to allow the incorporation of these previously dropped observations. The basic idea is to transform these observations by mapping them into earnings values that are slightly above Y . Specifically, min the transformed earnings values are obtained as ! Y min y + =ln ̃ ) y (exp Y x + 10 , i,t min i,t Y 10 + 10 min where ⇠ U (0 , 1) follows a uniform distribution. x In order to have a smooth earnings change distribution, we replace earnings observations lower than our minimum earnings threshold by mapping them to levels above. This imputation has the following three goals: First, there is a mass of individuals with an earnings observation of zero, which would otherwise be dropped when constructing log earnings changes. We would like to use these observations as they contain information about the nature of earnings risk. Second, we would like to avoid having any mass in the distribution to avoid possible jaggedness in the objective function. Third, we want to preserve the ranking among individuals with less than threshold earnings. In order to capture the variation in the cross-sectional moments of earnings changes along the age and recent earnings dimensions, we condition the distribution of earnings changes on these variables. For this purpose, we first group workers into 6 age bins (five-year age bins between 25 and 54) and within each age bin into 13 selected groups of RE percentiles in age 1 . The RE percentiles are grouped as follows: 1, 2–10, 11–0, 21–30, ..., 81–90, 91–95, 96–99, t 100. Thus, we compute the three moments of the distribution of one- and five-year earnings changes for 6 ⇥ 13 = 78 di ff erent groups of workers. We aggregate these 6 age bins into 2 age i groups, A . The first age group is defined as young workers between ages 25 through 34, 1 t whereas the second age group is defined as prime-age workers between the ages of 35 and 54. 2 ⇥ 2 13 ⇥ 3 = 156 cross-sectional moments. Consequently, we end up with ⇥ The second set of moments captures the heterogeneity 2. Mean of log earnings growth. in log earnings growth over the working life across workers that are in di ff erent percentiles of the LE distribution. We target the average dollar earnings at 8 points over the life cycle: ages 25, 30, ..., and 60 for di erent LE groups. We combine LE percentiles into larger groups to keep ff the number of moments at a manageable number, yielding 15 groups consisting of percentiles of the LE distribution: 1, 2–5, 6–10, 11–20, 21–30,..., 81–90, 91–95, 96–97, 98–99, and 100. The total number of moments we target in this set is ⇥ 15 = 120 . 8 3. Impulse response functions. We target average log earnings growth over the next k years for k =1 , 2 , 3 , 5 , 10 ,thatis, E [ y , conditional on groups formed by crossing ] y t k + t i i age, y y , and Y . We impute income observations lower than our minimum income 1 t t t 1 threshold following the same procedure as in the smoothing of the cross-sectional moments. 67

70 We aggregate age groups into two: young workers (25-34) and prime-age workers (35-55). In each year, individuals are assigned to groups based on their ranking in the age-specific RE distribution. The following list defines these groups in terms of the RE distribution: 1–5, 6–10, 11–30, 31–50, 51–70, 71–90, 91–95, 96–100. We then group workers based on the percentiles of i i y :1–2,3–5,6–10,11–20,21–30, the age- and RE-specific income change distribution, y t 1 t 31–40, 41–50, 51–60, 61–70, 71–80, 81–90, 91–95, 96–98, 99–100. As a result, we have a total ⇥ 14 ⇥ 5 = 1120 moments based on impulse response. 2 ⇥ of 8 We target the life cycle profile of the variance of log earnings 4. Variance of log earnings. for income observations larger than our minimum income threshold. As a result, we have a total of 36 moments based on the variance of log earnings. Numerical Method for Estimation B.2 The estimation objective is maximized as described now. The global stage is a multi-start algorithm where candidate parameter vectors are uniform Sobol’ (quasi-random) points. We typically take about 10,000 initial Sobol’ points for pre-testing and select the best 2000 points (i.e., ranked by objective value) for the multiple restart procedure. The local minimization stage is performed with a mixture of Nelder-Mead’s downhill simplex algorithm (which is slow but Zhang et al. performs well on di 2010 ), which ffi cult objectives) and the DFNLS algorithm of ( is much faster but has a higher tendency to be stuck at local minima. We have found that the combination balances speed with reliability and provides good results. B.3 Derivation of the Risk Premium ⇣ ⌘ ̃ )) = E U U ( c (1 + (1 ) ⇡ ( c Taking a first-order Taylor-series approximation to the left hand side and fourth order approx- imation to the right hand side we get: ◆ ✓ 1 1 1 3 2 4 2 4 000 0 0000 3 00 0 ̃ ̃ ̃ ̃ ( c ) c ) + U U ( ( c ) c ( c ) + c . U ⇡ ( c ) c = E U + U ( U c ( c ) c )+ U c 24 6 2 ̃ , )=0 Observing that the second term on the right hand side is zero when and rearranging E ( yields: 00 000 0000 ( 1 ) c c ) c 1 ( u u ( c ) c 1 u c , m ⇥ ⇡ (11) ⇥ m = + m ⇥ 2 4 3 00 000 0 ( ) c u ) ( c ) ( u 3 u 2 4 c th ̃ denotes the n where central moment of m . To convert these into statistics that are reported n 2 3 ⇥ in the paper, we write cient, and ,m ffi = s is the skewness coe m = s , where 3 2 4 is kurtosis. With this notation, and assuming a CRRA utility function = k k ⇥ m , where 4 with curvature ✓ , we get: ✓ ✓ + 2)( ( ✓ + 1) ✓ + 1) ✓ ( ✓ 3 4 2 ⇤ ⇥ s ⇥ ⇥ = k + ⇡ , ⇥ ⇥ 24 6 2 68

71 which can also be written as:  ◆ ✓ ✓ 1 1 2 2 ⇤ 1+ ⇥ ( ✓ = s ⇥ + . ⇥ ( ✓ + 2) k ⇡ ⇥ + 1) 4 3 2 C Estimation Results Figure A.25 – 3-D Plot of Mixing Probabilities p for z (a) Mixing Probability (b) Mixing Probability x for p t z x t 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 60 60 55 55 50 50 6 6 45 45 4 4 2 2 40 40 0 0 35 35 Age Age -2 -2 30 30 -4 -4 ̃ ̃ y y 25 25 (c) Mixing Probability p for ⌫ ⌫ t 1 0.8 0.6 0.4 0.2 0 60 55 50 6 45 4 2 40 0 35 Age -2 30 -4 y ̃ 25 C.1 Model Fit: Additional Figures 69

72 Figure A.26 – Fit of Estimated Model to Key Impulse Response Data Moments for the 8 RE Groups 1 1 t y − 0 0 +10 t y -1 -1 RE Pct: 1–5% RE Pct: 5–6% -1 -1 1 0 1 0 1 1 t y − 0 0 +10 t y -1 -1 RE Pct: 6–10% RE Pct: 10–11% 1 0 -1 -1 0 1 1 1 t y − 0 0 +10 t y -1 -1 RE Pct: 30–31% RE Pct: 11–30% 1 -1 0 -1 1 0 1 1 t y − 0 0 +10 t y -1 -1 RE Pct: 50–51% RE Pct: 31–50% 1 -1 1 0 0 -1 y y − y − y t 1 t t t − − 1 Data Model 70

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