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The Nature of Countercyclical Income Risk

This paper studies the cyclical nature of individual income risk using a confidential dataset from the U.S. Social Security Administration, which contains (uncapped) earnings histories for millions of individuals. The base sample is a nationally representative panel containing 10 percent of all U.S. males from 1978 to 2010. We use these data to decompose individual income growth during recessions into "between-group" and "within-group" components. We begin with the behavior of within-group shocks. Contrary to past research, we do not find the variance of idiosyncratic income shocks to be countercyclical. Instead, it is the left-skewness of shocks that is strongly countercyclical. That is, during recessions, the upper end of the shock distribution collapses--large upward income movements become less likely--whereas the bottom end expands--large drops in income become more likely. Thus, while the dispersion of shocks does not increase, shocks become more left skewed and, hence, risky during recessions. Second, to study between-group differences, we group individuals based on several observable characteristics at the time a recession hits. One of these characteristics--the average income of an individual at the beginning of a business cycle episode--proves to be an especially good predictor of fortunes during a recession: prime-age workers that enter a recession with high average earnings suffer substantially less compared with those who enter with low average earnings (which is not the case during expansions). Finally, we find that the cyclical nature of income risk is dramatically different for the top 1 percent compared with all other individuals--even relative to those in the top 2 to 5 percent.


Introduction
From 2007 to 2009, US male workers experienced an average decline in their annual labor earnings of 6.5 percent. 1 While this figure represents a very steep decline compared with any other post-war recession, it is dwarfed by the dispersion of income growth rates across workers during the same recession: for example, a quarter of workers saw their labor earnings rise by more than 16 percent, one in ten saw a rise of more than 65 percent, whereas another one in ten saw their incomes fall by more than 55 percent. Moreover, despite the 6.5 percent mean decline in earnings just noted, the worker with median income change actually experienced a slight rise-of 0.1%-during these two years. The goal of this paper is to understand how this dispersion of fortunes varies over the business cycle. More specifically, we seek to decompose income changes during a business cycle "episode" (i.e., recession or expansion) into a component that can be predicted based on the observable characteristics of individuals (prior to the episode) and a separate "residual" component that represents purely idiosyncratic shocks that hit individuals that are ex ante very similar.
The first one represents the "between-group" component of business cycle risk, whereas the second can be thought of as the "within-group" component.
An important advantage of our analysis is the very rich dataset that we employ. Basically, our main panel dataset is a 10 percent random sample of all US males who had a Social Security number between the ages of 25 and 60 from 1978 to 2010. This dataset has three important advantages. First, earnings records in our dataset are uncapped (no topcoding), allowing us to study individuals with very high incomes. 2 Second, the substantial sample size allows us to employ flexible non-parametric methods and still obtain extremely precise estimates. To give some idea about the size of the sample, the bulk of our analysis 1 The source for all reported statistics is authors' calculations from the SSA data described below. 2 Kopczuk et al. (2010) also employ an SSA dataset with uncapped earnings (after 1978), whereas Haider and Solon (2006), Schulhofer-Wohl (2011), Bonhomme and Hospido (2012, Spain), and Bönke et al. (2011, Germany) used datasets with earnings capped at the Social Security contribution limit.
is conducted with a sample that has about 4.5 million individuals in each year for a total of more than 165 million individual-year observations during this period. Third, thanks to their records-based nature, the data contain very little measurement error, which is a serious issue with survey-based micro datasets. One drawback is possible underreporting (e.g., cash earnings), which can be a concern at the lower end of the earnings distribution.
The panel aspect of our dataset allows us to use individuals' earnings and employment histories to construct observable characteristics as of the beginning of a business cycle episode. For example, we can ask whether individuals that entered a recession with a high average income are affected differently during the recession relative to those that entered with a low average income. How about individuals who were rising stars (i.e., had fast income growth rate) versus those whose careers were stagnant when the recession hit? Similarly, are some individuals' incomes inherently more sensitive to business cycle fluctuations than others', perhaps because of their occupations, the industry they work in, or the nature of their skill sets, etc.? Finally, how does age factor into any of these patterns? To answer these questions systematically, we group individuals along three observable dimensions at the time a business cycle episode begins: (i) age, (ii) pre-episode average income, and (iii) pre-episode income growth rate.
Our main findings can be summarized as follows. First, we study the cyclical nature of idiosyncratic shocks, once observable factors are accounted for. Contrary to past research, we find that income shock variances are not countercyclical. However, uncertainty does have a significant countercyclical component, but it comes from the left-skewness increasing during recessions. That is, during recessions, the upper end of the income shock Our results on the business cycle behavior of top incomes complement and extend the findings in Piketty and Saez (2003), Parker and Vissing-Jørgensen (2010), and Saez (2012).
In particular, these papers used repeated cross-sections to construct synthetic groups of individuals based on their income level. to the small sample size, they did not, however, investigate the cyclicality of the skewness of shocks, nor did they allow for a factor structure as we do here. Moreover, note that the question of interest is "the cyclical changes in the dispersion of income growth rates," which involves triple-differencing. It is extremely challenging to answer such a question without a very large and clean dataset.
Our findings are more in line with the way Mankiw (1986) modeled idiosyncratic shocks.
Basically, he showed that one can resolve the equity premium puzzle if idiosyncratic shocks have countercyclical left-skewness-as found in the current paper. In a related context, Brav et al. (2002) found that accounting for the countercyclical skewness of individual consumption growth helps generate a high equity premium with a low risk aversion parameter.
The spirit of our analysis is similar to the literature that decomposed wage inequality trends into between-group and within-group components (among many others, Juhn et al. (1993), Lemieux (2006), and Autor et al. (2008)). But there are several notable differences. First, our focus is on growth rates rather than levels, which is feasible with the panel dimension of our dataset. Second, we focus on business cycle variation whereas that literature examined secular trends. Third, relying on repeated cross-sections, that literature had to confine itself to the few observable characteristics that were available in the cross-section, such as gender, age, education, and, sometimes, industry. With longitudinal data, we are able to define groups of individuals based on their history, such as individuals with high versus low past average income and/or income growth rates.
Finally, in a related strand of literature, Bloom et al. (2011) fit an AR(1) process to firm-level total factor productivity (TFP) time series and allow a fixed aggregate shock and fixed firm effect. They find that the residual of the AR(1) has a larger cross-sectional dispersion during recessions. While the skewness also appears to be more negative, the difference is not statistically significant. In contrast to that paper, we do allow for a factor structure (loading factor on their aggregate shock) and allow the loading factor to vary with observables. Of course, we study individual labor income, whereas they focus on firm-level TFP, so the two sets of results are not necessarily inconsistent with each other.

The Data
We employ a unique, confidential, and very large panel dataset on earnings histories from the U.S. Social Security Administration records. For our baseline analysis, we draw a 10 percent random sample of US males-covering 1978 to 2010-directly from the Master Earnings File (MEF) of Social Security records. 5 The Master Earnings File.
The MEF is the main source of earnings data for the Social Security Administration and grows every year with the addition of new earnings information received directly from employers (Form W-2 for wage and salary workers). 6 The MEF includes data for every individual in the United States who has a Social Security number. The dataset contains basic demographic characteristics, such as date of birth, sex, race, type of work (farm or non-farm, employment or self-employment), self-employment taxable income, and several other variables. Earnings data are uncapped (no top-coding) and include wages and salaries, bonuses, and exercised stock-options as reported on the W-2 form (Box 1). 7 For more information, see Panis et al. (2000) and Olsen and Hudson (2009). Finally, all nominal variables were converted into real ones using the Personal Consumption Expenditure (PCE) deflator with 2005 taken as the base year.
5 Our focus on males is motivated by the fact that this group had a relatively stable employment rate and labor supply during this period. In contrast, female labor participation increased substantially during this period. Because our dataset contains only labor earnings but no hours information, including women in the analysis would have introduced an important confounding factor, which we wished to avoid.
6 Although the MEF also contains income information for self-employed individuals, these data are topcoded at the taxable limit until 1994. Because of this, we do not use these data in this paper. In an earlier version, we conducted all the analysis using total labor income (and included self-employed individuals) and found no difference in our substantive conclusions. 7 Our earnings measure does not include deferred compensation, such as through 401(k), 403(b), and 457(b) plans, because information on these is not available consistently throughout the period.  Our base sample is determined as follows. For a statistic computed using data for (not necessarily consecutive) years (t 1 , t 2 , .., t n ), an individual observation is included if the following three conditions are satisfied for all these years: the individual (i) is between the ages of 25 and 60, (ii) has annual wage/salary earnings that exceeds a time-varying minimum threshold, and (iii) is not self-employed (i.e., has self-employment income less than the same minimum threshold). This minimum, denoted Y min,t , is equal to one-half of the legal minimum wage times 520 hours (13 weeks at 40 hours per week). This condition is standard in the literature (see, e.g., Autor et al. (2008)) and allows us to focus on a subset of workers with a reasonably strong labor market attachment. Finally, the MEF contains a small number of extremely high income observations each year. To avoid potential problems, we cap (winsorize) observations above the 99.999th percentile. Similarly, Figure A.1 plots the levels of labor income that correspond to selected percentiles of the income distribution in each year. The lowest income that qualifies a male worker in the top 10% (e.g., above the 90th percentile) has been steady at approximately $98,000 since year 2000. In 2011, a worker must be making more than $297,000 to be in the top 1%. This threshold was highest in 2007 when it reached $318,000.
Recessionary vs. Expansionary Episodes. Since the main focus of this analysis is on business cycle variation, we need to be clear about how a given year is classified as a recession or an expansion year. Because our labor earnings data are annual and recessions can start or end in any quarter during the year, this is not always straightforward.
The start date of a recession is determined as follows. If the National Bureau of Economic Research (NBER) peak of the previous expansion takes place in the first half of 8 Appendix A contains a more detailed comparison of inequality trends revealed by the base sample to those found in the Current Population Survey (CPS) data.
a given year, that year is classified as the first year of the new recession. If the peak is in the second half, the recession starts in the subsequent year. 9 The ending date of a recession is a bit more open to interpretation for our purposes, because the NBER "troughs" are often not followed by a rapid fall in unemployment rates and rise in individual wages.
This can be seen in Figure  Before delving into the full-blown panel data analysis in the next section, we begin by studying some simple statistics to provide a bird's eye view of the business cycle patterns in income risk. The difference of the analysis in this section from previous work (e.g., Juhn et al. (1993), Autor et al. (2008), and others) is the panel dimension of our data.
This allows us to document the moments of earnings growth at different horizons, which is informative about the short-and long-run dynamics of earnings.

Evolution of Inequality: Cross-Sectional Data
We begin by briefly establishing some facts on the evolution of cross-sectional inequality.
This analysis does not require the panel dimension and it is presented here for completeness and comparison to the existing work.
It is useful to distinguish between the changes in top and bottom end inequality. To this end, we take the log differential between the 90th and 50th percentiles of the earnings distribution (hereafter L90-50), as well as the log differential between the 50th and 10th percentiles (L50-10), and plot the 1-year change in both series in Figure 4. To reduce short-term mean reversion in inequality, the solid lines plot the 2-year difference in each inequality measure (divided by two), which is smoother. This differencing eliminates the secular trend and allows us to focus on the cyclical change in inequality.  1990s. The change in the top-end inequality is also cyclical, rising during the 1980-83 and 1991-92 recessions. Compared with the bottom-end inequality though, L90-50 rises virtually throughout the period. Overall, the combination of these two pieces shows that overall inequality (L90-10) itself is countercyclical.

From Levels to Growth Rates
As noted earlier, the previous figure can be obtained using repeated cross-sections. Now, we turn to the properties of the income growth (or change) distribution, making use of the panel dimension of the MEF dataset. 10 Properties of Income Shocks Over Time.
It will be useful to distinguish between income growth over short and long horizons. To this end, in much of the following analysis, we examine 1-year and 5-year income growth rates and think of these as roughly corre- sponding to "transitory" and "persistent" income shocks. A more rigorous justification for this interpretation will be provided below.
The top panel of Figure  year) shocks. The striking co-movement of the L90-50 and L50-10 is clearly seen here (the correlation of the two series is -0.67), even more strongly than in the transitory shocks.
Several remarks are in order. First, the fact that L90-50 and L50-10 move in opposite directions implies that L90-10 (which measures overall dispersion of income shocks) changes little over the business cycle, because the fall in L90-50 partially cancels out the rise in L50-10. An alternative measure of shock dispersion-the standard deviation-is plotted in Figure 6 for both persistent and transitory shocks, which shows that dispersion does not increase much during recessions (notice the very small variation on the y-axis). Perhaps the only exception is the 2001-02 recession, during which time the transitory shock variance increases. In the coming sections, this point will be examined further and will be made more rigorously. This observation will provide one of the key conclusions of this paper, given how clearly it contradicts the commonly held belief that idiosyncratic income shock variances are strongly countercyclical (e.g., Storesletten et al. (2004)).
Second, looking at transitory shocks, L90-50 displays a clear downward trend during The interpretation is that the likelihood of large upward movements has become less likely during this period. We see a similar, but less pronounced, trend in the L50-10, which indicates that the likelihood of large falls has also become somewhat smaller. Overall though, both the L90-10 and the standard deviation of income growth ( Figure 6) display a clear downward trend. Notice that this conclusion is in contrast to the conventional wisdom since the 1990s that income shock variances have generally risen since the 1980s (Moffitt and Gottschalk (1995)). However, it is consistent with a number of recent papers that use administrative data (e.g., Sabelhaus and Song (2010) and others). 11 In this paper, we will not dwell much on this trend, except when it is relevant for our analysis of the cyclical changes in income risk.
Third, and finally, the L90-50 of the shock distribution falls in recessions, even though 11 Moreover, Solon and Shin (2011) investigate the robustness of the finding from the PSID that the annual income growth volatility trends up over time. They show that whether self-employment income is included in the measure of labor income makes a big difference: focusing on wages and salaries implies no rise in variance, whereas including farm and business income implies a 15 log point rise in the variance. shocks becomes smaller. This is a distinction that will be examined further below.
The finding described above-that the top end of the shock distribution compresses during recessions, while at the same time the bottom end expands-suggests that one important cyclical change could be found in the skewness of shocks. Indeed, as seen in Figure 7, both the 1-year and 5-year income growth distributions become more leftskewed (negative skewness increases) during recessions and the magnitude of change is large. Below, we return to this point and sharpen it by conditioning income changes on narrowly defined groups of individuals.

Panel Analysis
The analysis so far intended to provide a general look at how income shocks vary over the business cycle. However, one can imagine that the properties of income shocks vary systematically with individual characteristics and heterogeneity: for example, young and old workers can face different income shock distributions than prime-age workers with more stable jobs. Similarly, workers at different parts of the income distribution could experience different types of income risks. The substantial number of individuals in our sample allows us to account for such variation without making any strong parametric assumptions.

A Framework for Empirical Analysis
Let e y i t denote individual i's log labor earnings in year t, and let V i t 1 denote a vector of (possibly time-varying) individual characteristics that will be used to group individuals as of period t 1. Consider the following representation for individual income: To study between-and within-group variation over the business cycle, we take the kth difference of income in equation (1) and modify it to allow for a factor structure: The specification in (2) allows for two different types of business cycle effects. First, the factor structure-captured by the introduction of the function f 1 -allows the conditional mean of income growth to vary systematically with the business cycle across different groups of workers. Second, both types of shocks have variances that can potentially vary with the business cycle in a way that is also different across groups of workers (as captured by H and G). In our implementation, we will consider a vector V i t 1 that includes three time-varying observable individual characteristics: age, past average income, and past income growth rate as of period t 1. An assumption that will be maintained in the analysis is that these characteristics vary slowly with time, so that V i t ⇡ V i t+k for small k.
This formulation allows the effects of aggregate shocks to be transmitted differently to groups that differ in their labor market characteristics at the time a recession hits or an expansion gets underway. Of course, even individuals within these finely defined groups will likely experience different income growth rates during recessions and expansions, which will be captured by the permanent and transitory shocks above. These capture the withingroup variation in shocks and we will also quantify the cyclical nature of such shocks.

Between-Group Variation in Shocks.
It is useful to look at equation (2) again: Taking the means within each group eliminates both permanent and transitory shocks (since they average zero by assumption), yielding E(y i t+k Between any two periods t and t + k, each group has a different loading factor f 1 (V i t 1 ) on the aggregate shock ( t+k t ).
The key object of interest is f 1 , whose shape will tell us about the factor structure of income changes over the business cycle. 12 One drawback of this measure is that the left hand side of equation (3) can only be computed using individuals whose incomes are positive in year t and t + k (so that log income is finite). This restriction could bias the results if the fraction of individuals who are dropped from the sample varies systematically with V i t 1 and over the business cycle, which is quite possible. Later, in Section 5.2, we examine whether this bias could be important. As another way to address this concern, we also construct a modified measure for the left hand side of (3): We use all individuals that belong to V i t 1 to compute the average income in t and in t + k and then take the logs of these averages to compute This measure now includes both the intensive margin and the extensive margin of income changes between two periods. 13 It will be our preferred measure in Section 6, where we will also discuss the differences between the two measures more fully.
Within-Group Variation in Shocks. One focus of this analysis will be on simple measures of income shock volatility, conditional on individual characteristics. That is, fix a group of workers that have the same vector V i t 1 at time t. Computing the within-group 12 To be more precise, f 1 should have a time subscript, since we will allow it to vary over time. However, to keep the notation clean, we will suppress the subscript in this paper.
13 It is also more directly comparable to recent work that used cross-sectional data to quantify the cyclicality of incomes by constructing average incomes for synthetic groups, such as the top 1 percent, 10 percent, etc. (e.g., Parker and Vissing-Jørgensen (2010) and Saez (2012)).
variance, we get: Two points should be noted from this formula. First, as we consider longer time differences (larger k), the variance reflects more of the permanent shocks as seen by the addition of the k innovation variances and given that there are always two variances from the transitory component regardless of k. For example, computing this variance over a five-year period that spans a recession (say 1979-84 or 1989-94) would allow us to measure how the variance of permanent shocks changes during recessions. It will also contain transitory variances, but for two years that are not part of a recession (1979 and 1984, for example).
Second, looking at short-term variance, say k = 1, yields a formula that contains only one permanent shock variance and two transitory shock variances. So, as we increase the length of the period over which the variance is computed, the statistic shifts from being informative about transitory shock variances towards more persistent variation. In the analysis below, we will consider k = 1 and k = 5. And with some abuse of language, we will refer to them as transitory and persistent variances, respectively. Finally, each of these variances can also differ across groups of individuals. Below, we compute various group-specific statistics, including variances, to examine the nature of such variation.

Grouping Individuals into
Let t denote the generic time period that marks the beginning of a business cycle episode.
We now describe how we group individuals based on their characteristics at time t 1.
Each individual is identified by three characteristics that can be used to form groups. Not every characteristic will be used in the formation of groups in every experiment. 14 1. Age. Individuals are divided into seven age groups. The first six groups are five-year wide (25-29, 30-34,..., 50-54) and the last one covers six years: 55-60.
2. Pre-Episode Average Income. A second dimension individuals differ along is their average income (and especially where they rank relative to others). For a given year t, we consider all individuals who were in the base sample (i) in year t 1 and (ii) in at least two more years between t 5 to t 2. For example, an individual who is 23 years old in t 5 (and hence is not in the base sample that year) will be included in the final sample for year t if he has earnings exceeding Y min in every year between t 3 and t 1.
Furthermore, as noted above, we are interested in average earnings to see how a worker ranks in the income distribution relative to his peers. But even within the narrow age groups defined above, age variation can skew the rankings in favor of older workers. For example, between ages 25 and 29, average earnings grows by 35.4% in our sample, and between 30 and 34, they grow by 18.3%. So, unless this lifecycle component is accounted for, a 29-year-old worker in the first age group would appear in a higher income percentile than the same worker when he was 25. This would confound age and income differences, which ought to be avoided.
To adjust for this, we proceed as follows. First, using all income observations from our base sample from 1978 to 2010, we run a pooled regression of log raw earnings (e y i t,h ) on age and cohort dummies (without a constant) to characterize the age profile of log earnings.
14 One observable characteristic that has often been used in the literature on wage inequality is educational attainment. The MEF does not contain any information on education, so we cannot use it in our analysis. Having said that, papers that investigated the cyclicality of the skill premium (i.e., betweeneducation-group differences) found only a modest correlation with the business cycle. For example, both Castro and Coen-Pirani (2008 , Table 2) and Balleer and van Rens (2011 , Table 1) report a correlation of skill premium with GDP and productivity close to zero (ranging from -0.15 to 0.20). Therefore, this omission is probably not an important shortcoming of our analysis.
We then scale the age dummies (denoted with d h ) so as to match the average log earnings of 25-year-old individuals used in the regression. Using these age dummies, we compute the average earnings between years t 5 and t 1 for the average worker of age h in year t . Then for a given worker i of age h in year t, we first average his earnings from t 5 to t 1 (and set earnings below Y min,t equal to the threshold) and then normalize it by the population average computed using the age dummies. This 5-year average (normalized) 3. Pre-Episode Income Growth. A third dimension is (recent) income growth. This could be an indicator of individuals whose careers are on the rise, as opposed to being stagnant, even after controlling for average income as done above. For this purpose, we compute 5 (y t 1 ) ⌘ (y t 1 y t s ) /(s 1)), where s is the earliest year after t 6 in which the individual has income above the threshold. 16

Within-Group Shocks
We begin with the cyclicality of idiosyncratic shocks, as measured by within-group variation in income growth rates. An important question is whether or not idiosyncratic shocks have countercyclical variances. To study this question, we first condition on age and Y i t 1 . A graphical construct that will be used to study within-group variation is obtained by plotting the quantiles of Y i t 1 for a given age group on the x-axis against the entire distribution of future income growth rates for that quantile on the y-axis: F(y t+k y t |Y t 1 ). 15 We have also tried an alternative measure of average earnings that weighs each observation inversely with their distance from year t 1, to further group together individuals whose incomes were similar at more recent dates. To this end, for a given t, define the weight w i t,s = (6 s)1{ e Y i t s Y min,t }, which is zero for ineligible observations and declines with s otherwise. Using these weights to construct the average earnings made almost no change to the results reported here.
16 Information on detailed SIC codes is available, so, in principle, we could further classify individuals based on their 3-or 4-digit industry. Our preliminary results indicated that little was gained by this step, so we did not pursue this approach. and P10 of the distribution of long-run changes, y i t+5 y i t , (on the y-axis) for each percentile of Y i t 1 (on the x-axis). To compare recessions and expansions, we averaged each one of these percentiles, separately over the four recessions (lines marked with "circles") and three expansions (solid blue lines) during our sample period. Similarly, because these figures look similar across age groups, to save space we also averaged across the age groups. 17 We begin by commenting on the variation in these percentiles as we move to the right along the x-axis. Interestingly, the following pattern holds in both recessions and expansions: At any point in time, individuals with the lowest levels of past average income face the largest dispersion of income shocks (y t+k y t ) looking forward. That is, L90-10 is widest for these individuals and falls in a very smooth fashion moving to the right. Indeed, 17 For 5-year changes, recession years can be defined in a number of ways since many 5-year periods cover a given recession. We have experimented with different choices and found them to make little difference to the substantive conclusions drawn here. The reported results are for a simple definition that includes one 5-year change for each recession that starts one year before the recession begins. Specifically, the recession graph averages over four 5-year periods starting in t = 1979, 1989, 1999, and 2005 (since this is the latest possible 5-year change covering the Great Recession). Expansions average over all 5-year changes that do not coincide with a recession year. That is, periods starting in t = 1983, 1984, 1993, 1994, and 2002. workers who are between the 70th and 90th percentiles of the Y i t 1 distribution face the smallest dispersion of shocks looking ahead. As we continue moving to the right (into the top 10%), the shock distribution widens again and does so monotonically as a function of past income. Notice that the P10 and P90 of the y i t+5 y i t distribution look like the mirror image of each other relative to the median, so the variation in L90-10 as we move to the right is driven by similar variations in P90 and P10 individually.
Turning to the bottom panel, the same graph is plotted now for y t+1 y t (transitory shocks). 18 Precisely, the same qualitative features are seen here with low-and high-income individuals facing a wider dispersion of persistent shocks than those in the "safer" zonesbetween the 70th and 90th percentiles. Of course, the scales of both graphs are different: the overall dispersion of persistent shocks is much larger than that of transitory shocks, which is to be expected.
To sum up, both graphs reveal extremely strong and systematic variation in the dispersion of persistent and transitory income shocks across individuals with different past income levels. 19 Now we turn to two key questions of interest. First, what happens to idiosyncratic shocks in recessions? Are idiosyncratic shock variances countercyclical? And second, how does any potential change in the distribution of idiosyncratic shocks vary across income levels (i.e., the cross-partial derivative)? In other words, do we see the shock distribution of individuals in different income levels being affected differently by recessions?
Are Shock Variances Countercyclical? The existing literature has largely focused on the cyclicality of persistent shocks, so this is where we also start (top panel of Figure 8).
First, both P90 and P10 shift downward by similar amounts from expansion to recession.
19 This finding clearly contradicts one of the standard assumptions in the income dynamics literaturethat the variance of income shocks does not depend on the current or past level of income. We explore these implications in a separate ongoing project. (As can be anticipated from this, the gap (L90-10) changes very little over the business cycle, as we shall see momentarily.) Furthermore, following the same steps as the one used to construct these graphs, one can also compute the standard deviation of y t+5 y t conditional on Y i t 1 during recessions and expansions, which is plotted in the top panel of Figure 9. As seen here, the two graphs (for expansions and recessions) virtually overlap, over the entire range of pre-episode income levels. For transitory shocks (bottom panel), there is more of a gap, but the two lines are still quite close to each other.
To make the measurement of countercyclicality more precise, Figure 10  The ratio is 1.75 (marked on the figure for comparison) compared with 1.02 we find in this paper. The figure also plots the same two ratios (L90-10 and standard deviations) for transitory shocks. Here we see a bit more movement relative to persistent shocks: the standard deviation is higher by about 4 percent (averaged across the x-axis) and L90-10 is higher by about 6 percent. These findings suggest that to the extent that recessions involve larger dispersion of shocks, these are to be found in short-term shocks without much long-term effects. Having said that, these numbers are still very small compared with the values typically used in the literature.
A second question that was raised above was whether recessions affect the distribution of shocks differently in different parts of the income distribution. It is probably evident by now that the answer is, perhaps surprisingly, "no." Even though shocks, on average, have systematically very different dispersions in different parts of the distribution, the cyclical change in this dispersion is remarkably flat across the quantiles of Y i t 1 (i.e., the crosspartial derivative is approximately zero). This is seen in the three figures just discussed, but is most apparent in Figure 10, where the ratios are quite flat, especially for persistent shocks. Therefore, we conclude that when it comes to the variance of persistent shocks, different income groups are affected similarly by business cycle fluctuations.

Countercyclical (Left-)Skewness
The obvious question now is: Do recessions have any effect on income shocks? The answer is yes, which could already be anticipated from Figure 8, by noting that while P90 and P10 move down together during recessions, P50 (the median of the shock distribution) remains extremely stable and moves down by only a little. This has important implications: L90-50 gets compressed during recessions, whereas L50-10 expands. In other words, for every income level Y i t 1 , when individuals look ahead during a recession, they see a much smaller chance of upward movements (relative to an expansion), but a much higher chance of large downward movements. In fact, this result is not specific to using P90 or P10, but is pervasive across the entire distribution of future income growth rates. This can be seen in we see the opposite pattern: P50 falls by 5 log points, whereas P25 falls by 9, and P10 falls by 20 log points respectively. Consequently, the bottom half of the shock distribution now expands, with "bad luck" meaning even "worse luck" during recessions.
From this analysis, a couple of conclusions can be drawn. First, idiosyncratic risk is has to be careful about extreme observations. These are not likely to be outliers as with survey data, but even if they are genuine observations, we may want to be careful that a few observations do not affect the overall skewness measure. For this purpose, we use "Kelley's measure" of skewness, which relies on the quantiles of the distribution and is robust to extreme observations ( Figure 12 ). It is also very straightforward to interpret as we shall see in a moment. It is computed as the relative difference between the upper and lower tail inequalities: (L90-50 -L50-10)/L90-10. A negative number indicates that the lower tail is larger than the upper tail, and vice versa for a positive number.
Turning to Figure 12, first, notice that individuals in higher income percentiles face a more negatively skewed shock distribution, consistent with the idea that the higher an individual's income is, the more it has room to fall. Second, and more importantly, this negative skewness increases during recessions both for transitory and persistent shocks.
For example, for individuals at the median of past income distribution, Kelley's measure for persistent shocks averages -0.14 during expansions. This number has a simple interpretation. It says that the dispersion of shocks above P50 accounts for 43% of overall L90-10 dispersion. Similarly, dispersion below P50 accounts for the remaining 57% (hence (43% 57%)/100% = 0.14) of L90-10. In recessions, however, this figure falls to -0.30, indicating that L90-50 accounts for 35% of L90-10 and the remaining 65% is due to L50-10. This is a substantial shift in the shape of the persistent shock distribution over the business cycle. The change in the skewness of transitory shocks is similar, if somewhat less pronounced. It goes from -0.14 down to -0.25 at the median. As seen in the figure, the increased left-skewness during recessions is pervasive-it takes place across the entire income distribution with similar magnitudes (with the exception of very low-income individuals).
To understand how different this conclusion is from a simple countercyclical variance formulation, recall Figure 1, which plots the densities of two Normal random variables: one with zero mean and a standard deviation of 0.13 (expansion) and a second one with a mean of -0.03 and a standard deviation of 0.21 (recession; both numbers from Storesletten et al. (2004)). As seen here, the substantial increase in variance and small fall in the mean implies that many individuals will receive larger positive shocks in recessions than in expansions under this formulation. For comparison, the left panel of Figure 13 plots the empirical densities of income growth from the US data, comparing the 1995-96 period to the worst year of the Great Recession (2008-09). To highlight how the density changes, the right panel plots the difference between the two densities. As seen here, the probability mass on the right side shifts from large positive shocks to more modest ones; on the left side, it shifts from small negative shocks to even larger negative ones. Thus, recessions are times when it becomes less likely for anybody to experience large upward income changes, whereas the risk of falling off the income ladder becomes significantly higher.
Interestingly, in one of the earliest papers on cyclical changes in income risk, Mankiw (1986) postulated that in recessions, a fraction of individuals all draw the same negative shock which adds up to µ. So, ex ante, each person views a recession as a state where, with probability , their income will drop by µ/ . Thus, negative shocks are concentrated among a subset of individuals in recessions. This structure induces a left-skewness of the same sort discovered in our analysis here, unlike the countercyclical variance structure proposed by Constantinides and Duffie (1996) and others. 20

Extensive Margin of Business Cycle Risk
The analysis so far in this section has been based on changes in log income, which required us to drop individuals with zero income in a given year (t or t + k). In this sense, these results characterized the income risk of individuals who spend at least part of both years 20 An interesting question is whether there are cyclical changes in moments beyond the third (skewness), for example, in the fourth moment-the kurtosis. Although the answer is yes-the kurtosis is lower in recessions compared with expansions-the differences are quite modest. We omit those results for brevity. They are available upon request from the authors. in the labor market (i.e., excludes the long-term unemployed). On average, an excluded individual will have more than 18 months of zero income (so that he appears with zero income during an entire calendar year). Given that long-term unemployment is not very common in the United States, this probability is not very high. However, it is still a real risk whose likelihood probably changes over the business cycle.
Here, we provide a measure of this risk using the same graphical construct as above. For each percentile of past average income, Figure 14 plots the rise in the risk of full year nonemployment (i.e., having income less than Y min ) during recessions relative to expansions as a function of past average income. Notice that, except for the lowest 30 percentiles, the rise is less than 1% of workers in that quantile and in fact averages close to 0.5%. It is safe to conjecture that such a small change in the composition of individuals over the business cycle is not likely to materially affect the results presented so far, which largely relied on the 90th and 10th percentiles of the y t+k y t distributions.
Second, to the extent that this compositional change does affect the results (for the said that, our conjecture is that this effect would be modest as long as one uses a robust statistic to measure dispersion and skewness.

Between-Group (Systematic) Business Cycle Risk
We now turn to the factor, or between-group, component of income risk. The goal here is to understand the extent to which income growth during a business cycle episode can be predicted by available observable characteristics prior to the episode.

Variation Across Y
i t 1 Quantiles

Recessions
We begin with a simple and useful benchmark. Consider a specific episode such as the Great Recession, with an aggregate shock of ( 2010 2007 ), which is a constant scalar.
Suppose further that V i t 1 is a grouping based on age and pre-episode average income: so for each age group, we rank individuals by Y  Figure 15 plots the estimated function f 2 for the 2007-10 period, for each of the six age groups defined above. 22 Several results can be seen here. First, f 2 looks quite different for the youngest two groups compared to the rest-but is virtually identical for all four age groups between ages 35 and 54. Motivated by this finding (which is also true in other periods we study), from now on we combine individuals aged 35 to 54 into one group and refer to them as "prime-age males." For brevity, we also combine the first two age groups into one and refer to them as "young workers" (ages 25 to 34).
Prime-age Males. Second, and substantively more important, for prime-age males the f 2 function is upward sloping and rises almost linearly up to about the 90th percentile (with a mild concavity in the lowest 20 percentiles), subsequently tapers off, and then falls 21 These statements implicitly assume that shocks do not exhibiting mean reversion. If z t is a meanreverting process, f 2 will be downward sloping in Y 2006 in the absence of any factor structure. So any sign of upward slope (overcoming this potential downward bias) would be an indication of a factor structure. 22 The 55-60 year-old age group is excluded because many of these individuals are older than 60 by 2010. We conjecture that this has more to do with the fact that for the 1979-83 recession, we were limited to using only income in 1978 to form groups (rather than taking 5-year averages as we did for other periods), which led to a higher degree of mean reversion than would otherwise have been the case. 25 These two recessions last half as long as the other two longer recessions, so the slope of these graphs should be interpreted in this context. However, normalizing total income growth (the vertical axis in these graphs) by the duration of each recession is not necessarily a satisfactory solution, because even during longer recessions, the largest income falls have been concentrated within one-or two-year periods (2008-09, for example). Inspecting the behavior of f 2 above the 90th percentile reveals an interesting pattern.
For the first two recessions in our sample period, very-high-income individuals fared better than anybody else in the population, whereas for the latest two recessions, there has been a remarkable reversal of these fortunes and the highest-income workers suffered the most.
Finally, we construct the alternative measure of average income growth, f 1 , described above. Recall that f 1 differs from f 2 in two important ways. First, f 1 excludes individuals with zero income either in year t or t + k. Because the probability of full-year non-employment rises in recessions most strongly for low-income individuals (Figure 14), dropping them will tend to increase f 1 below the median relative to f 2 . Second, because f 1 is based on the average of log earnings, whereas f 2 is based on the log of average earnings, the latter will tend to be higher within quantiles that have a wider dispersion of income growth rates (due to Jensen's inequality). So, we would expect this force to raise f 2 relative to f 1 below the median level of Y t 1 where the variance of shocks is higher, as well as at the very top end for the same reason. shows that the two measures reveal the same qualitative patterns. The clear upwardsloping factor structure is there for all recessions. Quantitatively, the slope is somewhat smaller-10 log points difference between 90th and 10th percentiles during the Great Recession vs 17 log points under f 2 . Inspecting the two graphs shows that the difference mainly comes from the steeper drop in f 2 between the 20th and 1st percentiles, probably due to the increased chance of unemployment in this range mentioned above. Between the 20th and 90th percentiles the two graphs look very similar. The other recessions show slopes that are also slightly lower than before. Another difference to note is that under f 1 , the 1980-83 recession looks less favorable to individuals in the top 10 percent-their income growth pattern resembles the recent recessions more closely. This suggests that the strong performance of this group revealed by f 2 was affected by some large gains at the right tail which dominated the mean income measure for these groups in 1983.
Overall, the two measures are quite comparable. In the rest of the paper, we will continue to mainly focus on f 2 (but comment on differences from f 1 when applicable) so as to capture the total income risk, which includes the risk of long-term unemployment rising during recessions.
Young Males. Before concluding, it is interesting to compare these patterns with those for younger workers: during the 2007-10 recession, f 2 is virtually flat for this group, indicating no significant between-group differences (and even a small rise for those in the 90th percentile and above). In other words, for young workers, the aggregate shock seems to have hit all income groups more or less with the same force (no clear factor structure).
The same is true for the milder recessions of 1990-92 and 2000-02, which also show a generally flat shape between the 10th and 90th percentiles. The only deviation is for the lowest incomes during the 1979-83 recession-although the caveat noted above applies   1983−1990 1992−2000 2002−2007 catching up for all workers below the median. 26 To summarize these patterns, Figure 21 aggregates f 2 across all age groups and combines separate recessions and expansions. As seen here, there is a U-shape emerging during expansions-indicating a compression of the income distribution at the bottom and expansion at the top. In contrast, recessions reveal an upward sloping figure, indicating a widening of the entire distribution except at the very top (above the 95th percentile).
Thus, the main cyclical impact of business cycles is felt below the median, which expands during recessions and compresses during expansions. The same pattern also emerges at the top-inside the top 10 percent of the income distribution. Note: Recession graph has been scaled upward by 6.5 log points to be tangent to the Expansion graph.

Rising Stars versus Stagnant Careers
We now control for three characteristics-age, Y where d j Y is a dummy variable that equals one if the group on the left hand side belongs in the j th quantile of the Y t 1 distribution and zero otherwise. The dummy d p y i is defined analogously for the quantiles of 5 (y i t 1 ). The 90 dummies are estimated via OLS.
The main finding from this analysis is that pre-episode income growth has a significant effect on future income growth. This is shown in Figure 22, which plots average income growth during expansions (blue line with circle markers) and recessions (red line with square markers). While mean reversion is apparent in both cases, the gap between the two graphs is smallest in the middle and expands at both ends. The implication is that workers with the highest and lowest income growth rates prior to an episode do better 27 Because the two variables can be correlated, there is no presumption that every cell will contain the same number of observations (unlike the previous experiment with a single characteristic). Therefore, we drop cells that have less than 30% of the maximum number of observations. during expansions than recessions. This is related to the fact documented earlier that the top of the income shock distribution collapses during recessions. Consequently, those individuals whose income would have grown faster actually slow down during a recession. 28

The Top 1 Percent
We now take a closer look at the business cycle variation in the incomes of high earners.
As before, we define groups of individuals based on their pre-episode average earnings and examine how they fare during that episode. This allows us to control for compositional changes that are likely to occur during severe recessions or over long time periods (which we will explore below). In the rest of this section, we focus on prime-age males. To understand the differences and similarities within the top 1 percent, we divide this group into 10 quantiles and focus on the 1st, 5th, 8th, and 10th quantiles. We refer to each quantile by the middle point: P99.05, P99.45, P99.75, and P99.95.
28 Incidentally, controlling for past income growth has virtually no effect on the relationship between the quantiles of average income and future income growth documented above. Thus, further conditioning does not alter the relationship documented so far. These figures are available upon request. First, notice that the four groups move quite closely to each other until the late 1980s, after which point a clear ranking emerges: higher income quantiles become more cyclical than lower ones. In particular, individuals in higher quantiles have seen their income plummet in recessions relative to lower quantiles, but did not see their income bounce back more in the subsequent expansion, which would have allowed them to catch up. In fact, during expansions, the average income in each group grew by similar amounts. The implication is that these "differential losses" during recessions across income quantiles are also very persistent (Figure 24 Year GDP Growth and SP500 Annual Return (t to t + 1) 0.5 × stock returns (t to t + 1) GDP growth (t to t + 1) Unemployment rate (males) Average income growth, P99.95 Vissing-Jørgensen (2010) and Saez (2012) who used repeated cross-sections to study the same question. The former paper also finds that the differential cyclicality of incomes at the top has increased after the mid-1980s as we find in this paper. 29 While the recession/expansion classification used so far is simple and has proved useful, it is also true that there is a lot of variation in aggregate outcomes from year to year during a given expansion (and same for a recession). Thus, it is informative to compare the average income changes directly to some key variables that are well-known to co-move with the business cycle. In Figure 25, we plot three such variables-the annual growth in real GDP per capita, the annual real aggregate return on the US stock market, and the annual male unemployment rate-against 1-year average income growth for individuals in P99.95 (reproduced from the previous figure). 30 The co-movement of income growth for 29 Recall that f 2 averages each group's income before taking the logs, which could be affected by few very large income levels. Alternatively, we can construct f 1 which is the mean of log income changes, which is less sensitive to this problem. We include the resulting graph in the appendix, which shows the same qualitative patterns documented here, and reveals even larger losses for the top income groups-about 60 log points loss for individuals in P99.95 in each of the last three recessions. 30 The data for stock market return are taken from Robert Shiller's web site at Yale University.
top earners with GDP growth and stock returns is quite striking. In fact, the correlation with all three series exceed 0.6 in absolute value. Furthermore, regressing average income growth separately on each one of these series also reveals significant cyclical sensitivity.
For example, post-1985 a 1 percentage point rise in the male unemployment rate has been accompanied with an average income decline of 6.87 percent for individuals that were in P99.9 before the shock. Similarly, a 1 percent slowdown in per capita GDP growth implies a 4.55% decline in the income of the same individuals. 31 For comparison, the corresponding numbers for individuals at the median of the income distribution is 1.08 and -1.77.

Conclusions
This paper has studied between-and within-group variation in income growth rates over the business cycle. Using a very large and confidential panel dataset with little measurement error, we document three sets of empirical facts.
Our first set of findings concerns the cyclical nature of idiosyncratic shocks. During recessions, the upper end of the shock distribution collapses-i.e., large upward wage movements become less likely-whereas the bottom end expands-i.e., large drops in income become more likely. Moreover, the center of the shock distribution (i.e., the median) is very stable and moves very little compared to either tail. 32 What does change (more significantly) is the behavior of the tails, which swing back and forth in unison over the business cycle. These swings lead to cyclical changes in skewness, but not so much in overall dispersion. We conclude that recessions are best viewed as a small negative shock to the median and a large negative shock to the skewness of the idiosyncratic income shock distribution, with little change in the variance. 31 The corresponding figures for the whole sample period are 4.76 percent and 3.07 percent, respectively. 32 This is also related to another fact not studied in detail in this paper: earnings changes are extremely leptokurtic. That is, most income changes from year to year are very small, but once in a while there is a substantial change in earnings. Consequently, in a given year, most observed income changes are small and change little from recession to expansions, which gives the distribution its very stable center.
Second, we examined the systematic component of business cycle risk. First, we found important age variation between young (25-34) and prime-age (35-54) male workers, but not much variation within each group. Second, and more importantly, the pre-episode average income level turns out to be an excellent predictor of a worker's income growth during business cycle episodes. The magnitudes are large and the documented patterns are simple (straight lines, or U-shapes, etc). Third, the one deviation we find from these simple patterns is a remarkable non-linearity for individuals who enter a recession with very high incomes-those in the top 1%. During the last two recessions, these individuals have experienced enormous and persistent income losses (about 30 log points), which dwarfs the losses of individuals even with slightly lower incomes. In fact, individuals who entered the last three recessions in the top 99.9th percentile of the income distribution had incomes five years later that was at least 50 log points lower than their pre-recession levels.
The analysis in this paper was intended to be as non-parametric as possible, which was made feasible with the very large sample size. However, an alternative approach that has been used in the extant literature relies on fitting a separate AR(1) process to each individual's time series of earnings (see, e.g., Bloom et al. (2011)). We have implemented different versions of this method as well. In the most general case, we started by first running the following regression: to account for a factor structure (first term; where y A t is average log earnings) and an individual-specific lifecycle component (terms in square brackets). We then fit an AR (1) to the residual income, ⇠ i t,h , and studied the cyclical properties of the estimated innovation series. This analysis yielded the same substantive conclusions as those reported in this paper. For example, the behavior of the skewness of these innovations mirrored those reported in Figure 7 almost exactly. One drawback of this analysis is that we can only track a handful of cohorts with very similar ages over time (since we need to run the regression in (6) with a sufficiently long time series). Thus, age effects are confounded with time effects, which makes us less comfortable about drawing strong conclusions. Partly due to this concern and for sake of brevity, we did not include these results in the paper. A.2 Comparison to CPS Data Figure A.2 plots the log differential between the 90th and 50th percentiles of the labor income distribution, as well as the log differential between the 50th and 10th percentiles (hereafter abbreviated as L90-50 and L50-10, respectively). A couple of remarks are in order. First, it is useful to compare this figure to the Current Population Survey (CPS) data, which has been used extensively in the previous literature to document wage inequality trends. An important point to keep in mind is that studies that used the CPS have typically focused on hourly wage inequality, whereas our dataset only contains information on annual (wage and labor) earnings. With this difference in mind, note that Autor et In both datasets, the rise in L90-50 is secular and is remarkably stable over three decades. 33 Thus, even though the difference between hourly wage and annual income matters for the levels, it has little effect on the secular trend during this period.
Second, turning to the bottom end, the CPS data shows slightly different patterns, depending on whether one uses CPS March weekly wages or May/ORG hourly data. But the general pattern is a rapidly widening L50-10 gap from 1978 to 1987, which then stays flat or declines, depending on the dataset. In our case, the rise in L50-10 happens between 1979 and 1983, which then stays relatively flat until 2000, after which time it starts rising again.
It seems safe to conjecture that labor supply heterogeneity could be more important at the bottom end and could account for some of the gap between the two datasets. Another source of difference could be underreporting of income in our administrative dataset, or 33 Fitting a quadratic polynomial to the L90-50 reveals a very small negative curvature, indicating an ever so slight slowdown in the rate of increase of inequality at the top. over-reporting in the CPS. Some papers on measurement error adopt this latter interpretation (e.g., Gottschalk and Huynh (2010)). Notice also that the level of L50-10 is much higher in our sample-about 125 log points in 1978 compared with 65 log points in the CPS, which again can be explained by a combination of labor supply heterogeneity and under-or over-reporting. 34 Overall, the two datasets reveal the same pattern at the top end, while having similar but slightly different behavior at the bottom. 34 In our sample, the average wage income at the 10th percentile is $8,520 per year. If an individual works 52 weeks a year at a wage of $5.85 per hour (legal minimum wage in 2007), he has to work 28 hours per week, which does not appear to be an unreasonable figure.