2.1 The SSA Dataset

We draw a representative 10% panel sample of the U.S. population from the Master Earnings File (MEF) of the SSA. The MEF combines various datasets that go back as far as 1978. For our purposes, the most important variables include labor income from W-2 forms (for each job held by the employee during the year), self-employment income (from the Internal Revenue Service (IRS) tax form Schedule SE), and various demographics (date of birth, sex, and race).⁸ We focus on total annual labor earnings, which is the sum of total annual wage income and the labor portion (2/3) of self-employment income.

Wage income is not top coded throughout our sample, whereas self-employment income was capped at the SSA taxable limit until 1994. Although this top coding affects only a small number of individuals, we restrict our sample to the 1994–2013 period to ensure that our analysis is not affected by this issue.⁹ The only exception is our use of the entire 1978 to 2013 period in Section 5, where we analyze long-term outcomes of workers, for which a longer time series is essential. For robustness, we impute self-employment income above the cap for the years before 1994 using quantile regressions. Only a small number of individuals who make substantial income from self-employment are affected by this imputation, so the effect on our results is minimal. The details are provided in Appendix A.1. Finally, we convert nominal values to real values using the personal consumption expenditure (PCE) deflator, taking 2010 as the base year (see Appendix A for further details on the construction of our sample and variables).

Despite the advantages noted, the dataset also has some important drawbacks, such as limited demographic information, the absence of capital income, and the lack of hours (and thus hourly wage) data. To overcome some of these limitations, we supplement our analysis with survey data whenever possible. Another important limitation is the lack of household-level data. Even though a large share of quantitative models focus on individual earnings fluctuations (which we study), household earnings dynamics are key for some economic questions, about which we have little to say about in this paper.¹⁰

2.2 Sample Selection

Our base sample is a revolving panel consisting of males with some labor market attachment that is designed to maximize the sample size (important for precise computation of higher-order moments in finely defined groups) and keep the age structure stable over time. First, in order for an individual-year income observation to be admissible to the base sample, the individual (i) must be between 25 and 60 years old (the working lifespan) and (ii) have earnings above the minimum income threshold \(Y_{\text{min},t}\), equivalent to earnings from one quarter of full-time work (13 weeks at 40 hours per week) at half of the legal minimum wage in year \(t\) (e.g., approximately $1,885 in 2010). The revolving panel for year \(t\) then selects individuals that are admissible in \(t-1\) and in at least two more years between \(t-5\) and \(t-2\). This ensures that the individual was participating in the labor market and we can compute a reasonable measure of average recent earnings—a variable widely used extensively in the paper—which we describe next.

Recent Earnings. The average income of a worker \(i\) between years \(t-1\) and \(t-5\) is given by \(\hat{Y}_{t-1}^{i}=\frac{1}{5}\sum _{j=1}^{5}\max \left \{\tilde{Y}_{t-j}^{i},Y_{\text{min},t}\right \}\), where \(\tilde{Y}_{t}^{i}\) denotes his earnings in year \(t\). We then control for age and year effects by regressing \(\hat{Y}_{t-1}^{i}\) on age dummies separately for each year, and define the residuals as recent earnings (hereafter RE), \(\bar{Y}_{t-1}^{i}\). In Sections 3 and 4, we will group individuals by age and by \(\bar{Y}_{t-1}^{i}\) to investigate how the properties of income dynamics vary over the life cycle and by income levels.

3.1 Empirical Methodology: A Graphical Construct

Our main focus is on how the moments of earnings growth vary with recent earnings and age. To this end, for each year \(t\), we divide individuals into six groups based on their age in \(t-1\) (25–29,…, 45–54), and then within each age group, sort individuals into 100 percentile groups by their recent earnings \(\bar{Y}_{t-1}^{i}\). If these groupings are done at a sufficiently fine level, we can think of all individuals within a given age/RE group to be ex ante identical (or at least very similar). Then, for each such group, the cross-sectional moments of earnings growth between \(t\) and \(t+k\) can be viewed as the properties of earnings uncertainty that workers within that group expect to face looking ahead (see Figure 2). In our figures, we plot the average of these moments for each age/RE group over the years between 1997 and 2013-\(k\) (for moments by age only, see Appendix C.6). This approach allows us to compute higher-order moments precisely as each bin contains a large number of observations (see Table A.1 for sample size statistics). To make the figures more readable, we aggregate the six age groups into three: 25–34, 35–44, 45–54.¹¹

Figure: Figure 2 – Timeline for Rolling Panel Construction

Growth Rate Measures. In our analysis we use two measures of income change, each with its own distinct advantages and trade-offs. The first measure islog growth rate of income between \(t\) to \(t+k\), \(\Delta _{\text{log}}^{k}y_{t}^{i}\equiv y_{t+k}^{i}-y{}_{t}^{i}\), where \(y_{t}^{i}=\tilde{y}_{t}^{i}-d_{t,h(i,t)}\) denote the log income (\(\tilde{y}_{t}^{i}\)) of individual \(i\) in year \(t\) at age \(h(i,t)\) net of age and year effects \(d_{t,h(i,t)}\). \(\left \{d_{t,h}\right \} _{h=25}^{60}\) are obtained by regressing \(\tilde{y}_{t}^{i}\) on a full set of age dummies separately in each year. While its familiarity makes the log change a good choice for the descriptive analysis, it has a well-known drawback that observations close to zero need to be dropped or winsorized at an arbitrary value. When we use \(\Delta _{\text{log}}^{k}y_{t}^{i}\), we drop individuals from the sample with earnings less than \(Y_{\text{min}}\) in \(t\) or \(t+k\), and lose information in the extensive margin.

Our second measure of income growth—arc-percent change—is not prone to this caveat and is commonly used in the firm-dynamics literature, where firm entry and exit are key margins (e.g., Davis et al. (1996)). We define \(\Delta _{\text{arc}}^{k}Y_{t}^{i}=\frac{Y_{t+k}^{i}-Y_{t}^{i}}{\left (Y_{t+k}^{i}+Y_{t}^{i}\right)/2}\), where earnings level \(Y_{t}^{i}=\frac{\tilde{Y}_{t}^{i}}{\tilde{d}_{t,h(i,t)}}\) is net of average earnings in age \(h\) and year \(t\), \(\tilde{d}_{t,h(i,t)}\). Because of its familiarity, we use the log change measure in this section and report the results for arc-percent change in Appendix C.2, which show qualitatively similar patterns.

Transitory vs. Persistent Income Changes. As is well understood, longer-term earnings changes (i.e., \(\Delta _{\text{log}}^{k}y_{t}^{i}\) with larger \(k\)) reflect more persistent innovations. To see this intuition, consider the commonly used random-walk permanent/transitory model in which permanent (\(\eta _{t}^{i}\)) and transitory (\(\varepsilon _{t}^{i}\)) innovations are drawn from distributions \(F_{\eta}\) and \(F_{\varepsilon},\) respectively. We denote the variance, skewness and excess kurtosis of distribution \(F_{x},x\in \{\eta,\varepsilon \}\) by \(\sigma _{x}^{_{2}}\), \(\mathcal{S}_{x}\), and \(\mathcal{K}_{x}\), respectively. Then the second to fourth moments of \(k-\)year log income growth \(\Delta _{\text{log}}^{k}y_{t}^{i}\) are given by (see Appendix B for the derivations):

\[ \begin{aligned} \sigma ^{2}(\Delta _{\text{log}}^{k}y_{t}^{i}) & = & k\sigma _{\eta}^{2}+2\sigma _{\varepsilon}^{2}, \\ \mathcal{S}(\Delta _{\text{log}}^{k}y_{t}^{i}) & = & \underbrace{\frac{k\times \sigma _{\eta}^{3}}{(k\sigma _{\eta}^{2}+2\sigma _{\varepsilon}^{2})^{3/2}}}_{<1}\mathcal{S}_{\eta},\\ \mathcal{K}(\Delta _{\text{log}}^{k}y_{t}^{i}) & = & \underbrace{\frac{k\times \sigma _{\eta}^{4}}{(k\sigma _{\eta}^{2}+2\sigma _{\varepsilon}^{2})^{2}}}_{<1}\mathcal{K_{\eta}}+\underbrace{\frac{2\times \sigma _{\varepsilon}^{4}}{(k\sigma _{\eta}^{2}+2\sigma _{\varepsilon}^{2})^{2}}}_{<1}\mathcal{K}_{\varepsilon}. \end{aligned} \]

Equation 1 shows that as \(k\) increases the variance and kurtosis of \(k-\)year log change \(\Delta _{\text{log}}^{k}y_{t}^{i}\) reflect more of the distribution of \(\eta _{t}^{i}\) than that of \(\varepsilon _{t}^{i}\). Also, skewness is solely driven by permanent changes.¹² Finally, the distribution of \(\Delta _{\text{log}}^{k}y_{t}^{i}\) is closer to normal than the underlying distributions of \(F_{\eta}\) and \(F_{\varepsilon}\), because as innovations \(\eta _{t}^{i}\) and \(\varepsilon _{t}^{i}\) accumulate, the distribution of \(\Delta _{\text{log}}^{k}y_{t}^{i}\) converges toward Gaussian, per the central limit theorem.

With these considerations in mind, we document the moments of one-year (\(k=1\)) and five-year (\(k=5\)) residual earnings growth to capture properties of transitory and persistent changes, respectively. As persistent changes have a greater effect on economic decisions compared with easier-to-insure transitory ones, we present the results for \(k=5\) in this section. The figures for \(k=1\) in Appendix C.1 show the same qualitative patterns.

(a) Second Standardized Moment

(b) P90-P10

3.2 Second Moment: Standard Deviation

Figure 3a plots the standard deviation of five-year residual earnings growth by age and recent earnings (for clarity we use one marker for every 4th RE percentile group). In the right panel, we also report the difference between the 90th and 10th percentiles of log earnings changes, denoted by P90-P10, which is robust to outliers. Both measures show a pronounced U-shaped pattern by RE for every age group. For example, for 35- to 44-year-olds, the standard deviation falls from 1.05 for the lowest RE group to 0.6 for the 90th percentile, and then rises rapidly to 1.05 for the top 1%.

Baker and Solon (2003) and Karahan and Ozkan (2013) have estimated a U-shaped lifecycle profile for the variance of persistent shocks. Our analysis reveals a more intricate lifecycle variation, as we also condition on RE: Dispersion declines with age for the bottom one third of the RE distribution, is U-shaped until the 95th percentile, and monotonically increases for the top earners. However, notice that the variation with age is quite a bit smaller compared with the RE variation. Importantly, the highest earners (the top 5% or so) are strikingly different from other high earners—even those just below the 95th percentile. The same theme emerges again in higher-order moments.

3.3 Third Moment: Skewness (Asymmetry)

(a) Third Standardized Moment

(b) Kelley Measure

Figure 4a plots the skewness of five-year earnings growth, measured as the third standardized moment. First, notice that earnings changes are negatively (left) skewed at every stage of the life cycle and for (almost) all earnings groups. Second, 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. Note that the variation in skewness with age is more muted for individuals at the bottom or top of the (recent) earnings distribution (similar to the dispersion patterns above).

Is negative skewness as measured by the third central moment driven by extreme observations? While the information on tails is important (and becomes especially valuable in estimating income processes in Section 6), we also look at Kelley (1947) skewness, \(\mathcal{S_{K}}=\frac{\text{(P90-P50)}-\text{(P50-P10)}}{\text{P90-P10}}\), which is robust to observations above the 90th or below the 10th percentile of the distribution. Basically, \(\mathcal{S_{K}}\) measures the relative fractions of the overall dispersion (P90–P10) accounted for by the upper and lower tails. Specifically, \(\mathcal{S_{K}<}0\) implies that the lower tail (P50-P10) is longer than the upper tail (P90-P50).

Kelley’s skewness exhibits essentially the same pattern (Figure 4b). Thus, the asymmetry is prevalent across the entire distribution rather than being driven just by the tails. Furthermore, the magnitudes are substantial. For example, a Kelley measure of –0.44 (for 45- to 54-year-old workers at the 80th RE percentile) implies that P90-P50 accounts for 28% of P90-P10, far removed from the 50-50 of a normal distribution.

(a) P90-P50

(b) P50-P10

Another question is whether skewness becomes more negative over the life cycle because of a compression of the upper tail (fewer opportunities for large gains) or because of an expansion in the lower tail (higher risk of large declines). To answer this question, we investigate how the P90-P50 and P50-P10 change over the life cycle from their levels at ages 25–34 (Figure 5). Up until age 44, both the P90-P50 and P50-P10 decline with age across most of the RE distribution. However, the upper tail compresses more strongly than the lower tail, which leads to the increasing left skewness. After age 45, the P90-P50 keeps shrinking, but the bottom end opens up for workers with above median RE (large declines become more likely). Top earners are again an exception to this pattern: The upper tail does not compress with age, but the bottom end opens up monotonically.

A natural question is whether the negative skewness is simply due to nonemployment spells. First, notice that nonemployment can generate left skewness in earnings growth only if it has persistent effects: A transitory spell contributes one negative and one positive earnings change of similar size, leaving the symmetry unaffected (see equation 1). Jacobson et al. (1993) and Von Wachter et al. (2009) show that workers’ earnings indeed experience large scarring effects after mass layoffs. We revisit this point in Section 6, where we link earnings changes to the underlying shocks. Second, negative skewness is stronger for upper-middle-income and older workers, for whom unemployment risk is relatively small, implying that a decline in hours is not the main driver. Finally, as noted above, the shift toward more negative skewness is mostly coming from the compression of the right tail up to age 45, which is unlikely to be related to nonemployment.

3.4 Fourth Moment: Kurtosis (Peakedness and Tailedness)

We can think of kurtosis as a measure of the tendency of a density to stay away from \(\mu \pm \sigma\) (see Moors (1986)). Thus, a leptokurtic distribution typically has a sharp/pointy center, long tails, and little mass near \(\mu \pm \sigma\) (relative to a Gaussian distribution). A corollary to this description is that with excess kurtosis, the usual way we interpret standard deviation—as representing the size of the typical observation—is not very useful because most realizations will be either close to the center or out in the tails.

To illustrate this point, we calculate concentration measures for earnings growth. Table 1 reports the fraction of individuals experiencing an absolute log earnings change less than a threshold, \(|\Delta _{\log}^{1}y_{t}|\leq 0.05,0.10\), and so on. In the data 31% of workers experience an earnings change of less than 5%, whereas if innovations were drawn from a Gaussian density with the same standard deviation as the data, only 8% of individuals would experience such changes. Furthermore, extreme events are more likely in the data: A typical worker experiences a change larger than three standard deviations (153 log points) once in a lifetime—with a 2.4% annual chance—whereas this probability is almost one-ninth that size under a normal distribution. These values suggest that the Gaussian assumption vastly overstates the typical earnings growth and misses the extreme changes received by a non-negligible share of the population.

Notes: The empirical distribution used in this calculation is for 1997–98, the same as in Figure 1.

The high likelihood of extreme events in the data motivates us to take a closer look at the tails of the earnings growth distribution by examining its empirical log density versus the Gaussian log density (which is an exact quadratic). First, in line with our previous discussion, the data have much thicker and longer tails compared with a normal distribution (Figure 6). Second, the tails decline almost linearly, implying a Pareto distribution at both ends. Third, they are asymmetric, with the left tail declining much more slowly than the right, which contributes to the left skewness documented above. In fact, fitting linear lines to each tail (in the regions \(\pm [1,4]\)) yields a tail index of 1.18 for the right tail and 0.40 for the left tail—the latter showing especially high thickness. We highlight that while the Pareto tail in the earnings levels distribution is well known—indeed, going as far back as Pareto (1897)—the two Pareto tails emerge here in the earnings growth distribution. To our knowledge, the present paper is the first to document this fact.¹³

Figure 6 – Double-Pareto Tails of the U.S. Annual Earnings Growth Distribution

Next, to see how kurtosis varies by age and income, we report two statistics in Figure 7 that are analogous to the ones we used for skewness: the fourth standardized moment and the quantile-based Crow and Siddiqui (1967) measure, which is defined as \(\kappa _{\text{C-S}}=\frac{P97.5-P2.5}{P75-P25}\) and is equal to 2.91 for a Gaussian distribution. As with dispersion and skewness, kurtosis varies substantially with age and recent earnings. For example, for all age groups, the fourth standardized moment increases significantly with recent earnings from 3 (Gaussian) for low-income workers to above 14 around the 90th percentile, and then declines sharply in the top 5% of the RE distribution. Kurtosis also rises with age, especially between the first two age groups. The Crow-Siddiqui measure also shows very high kurtosis levels, indicating that the excess kurtosis is not driven by outliers.

(a) Fourth Standardized Moment

(b) Crow-Siddiqui Measure

An Alternative Measure of Persistent Changes. As we noted earlier, while the five-year income growth measure reveals a good deal about persistent changes in earnings, it still contains possible transitory innovations in years \(t\) and \(t+5\), which can potentially confound the inferences we draw about persistent changes. To check the robustness of our results, we consider an alternative measure that is based on the change between two consecutive five-year averages of earnings: \(\overline{\Delta}_{\log}^{5}(\bar{y_{t}}^{i})\equiv \log (\bar{Y}_{t+4}^{i})-\log (\bar{Y}_{t-1}^{i})\), where \(\bar{Y}_{t+4}^{i}\) is calculated the same as \(\bar{Y}_{t-1}^{i}\) but over the period \(t\) to \(t+4.\) Averaging earnings before differencing purges transitory changes and better isolates the persistent ones.

(a) Standard Deviation

(b) Skewness

(c) Kurtosis

Figure 8 plots the standardized moments of this alternative measure, which show essentially identical patterns to their counterparts using our baseline five-year growth measure (see Appendix C.4 for quantile-based moments). In fact, if anything, this measure shows a slightly larger negative skewness and a higher excess kurtosis. These results confirm our conclusion that the nonnormalities are stronger in persistent earnings changes. A more formal analysis in Section 6 will further confirm this conclusion.

3.5 Job-Stayers and Job-Switchers

Economists have documented that the earnings changes of workers who stay with the same employer (job-stayers) are notably different from the changes of workers who switch jobs (job-switchers) (see Topel and Ward (1992), Low et al. (2010), and Bagger et al. (2014)). This literature has focused on the average change, whereas we examine how the higher-order moments of earnings growth vary between job-stayers and job-switchers.

The SSA dataset contains employer identification numbers (EINs) that allow us to match workers to firms. However, the annual frequency of the data, together with the fact that some workers hold multiple jobs in a given year, poses a challenge for a precise identification of job-stayers and job-switchers. We have explored several plausible definitions for stayers and switchers and found qualitatively similar results. Here, we describe one reasonable definition: A worker is said to be a job-stayer between years \(t\) and \(t+1\) if he has a W-2 form from the same firm in years \(t-1\) through \(t+2\), and that firm is the main employer by providing at least 80% of his total annual earnings in years \(t\) and \(t+1\). A worker is defined as a job-switcher if he is not a job-stayer.¹⁴

(a) P90-10

(b) Kelley’s Skewness

(c) Crow-Siddiqui Kurtosis

We show in Figure 9 how the quantile-based second to fourth moments of annual earnings growth for stayers and switchers vary with recent earnings. Relative to job-switchers, job-stayers experience earnings changes that have a smaller dispersion (about one-third for median-income workers), and are more leptokurtic, especially for low-RE workers. Changes are symmetric or slightly right skewed for stayers and left skewed for switchers. The age profiles are broadly similar across switchers and stayers, and figures for five-year changes and standardized moments display similar patterns (Appendix C.5).

3.6 What Are the Sources of Nonnormalities in Earnings Growth?

So far, our analysis has focused on the distribution of annual earnings changes and remained silent on what may be behind the nonnormalities. For example, are the left skewness and excess kurtosis also present in the wage growth distribution? What are the lifecycle events associated with extreme income changes? The lack of information in the SSA data other than annual earnings does not allow us to investigate these questions, which, in turn, we study using the PSID.

4.1 Impulse Response Functions Conditional on Recent Earnings

In Figure 10, we show the mean reversion of different sizes of earnings changes \(y_{t}^{i}-y_{t-1}^{i}\) for prime-age workers over a 10-year period. Specifically, we plot \(\log \mathbb{E}\left [Y_{t+k}^{i}\right]-\log \mathbb{E}\left [Y_{t}^{i}\right]\) of each \(y_{t}^{i}-y_{t-1}^{i}\) quantile on the y-axis against its average on the x-axis.¹⁹ This graphical construct contains the same information as a standard impulse response function but allows us to see the heterogeneous mean reversion patterns more clearly.

We start with the median-RE group (\(\overline{Y}_{t-1}\in P46-P55\)) in Figure 10a. Even at the 10-year horizon, a nonnegligible fraction of the earnings change is still present for this group of workers, indicating a very persistent component in earnings growth. Also, negative changes tend to recover more gradually than positive ones for them. For example, workers whose earnings rise by 100 log points between \(t-1\) and \(t\) lose about 50% of this increase in the following 10 years. Almost all of this mean reversion happens after one year. Workers whose earnings fall by 100 log points recover 25% of that decline in the first year and around 50% of the total within 10 years. Finally, the degree of mean reversion varies with the magnitude of earnings changes, with stronger mean reversion for large changes: Small innovations (i.e., those less than 10 log points in absolute value) look very persistent, whereas larger earnings changes exhibit substantial mean reversion. A univariate autoregressive process with a single persistence parameter will fail to capture this behavior. In Section 6, we will show how to modify the simple income process to accommodate this variation in persistence by the size and sign of the earnings shock.

The analogous impulse response functions for low-income (\(\overline{Y}_{t-1}\in P6-P10\)) and high-income (\(\overline{Y}_{t-1}\in P91-P95\)) workers (Figures 10b and 10c) show that for low-income individuals, negative changes are more short-lived, whereas positive ones are more persistent, and that for high-income individuals the opposite is true.

Extending the results to the entire distribution of recent earnings, we focus on a fixed horizon and plot the cumulative mean reversion from \(t\) to \(t+10\) for the 6 RE groups in Figure 10d. Starting from the lowest RE group (the bottom 5%), notice that negative changes are transitory, with an almost 75% mean reversion rate at the 10-year horizon. But positive changes are quite persistent, with only about a 25% 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 (top 1%), we have the opposite pattern: only 20 to 25% of earnings declines revert to the mean at the 10-year horizon, whereas around 80% of the increases do so at the same horizon. We refer to this shape as the “butterfly pattern.”

This butterfly pattern broadly resonates with the earnings dynamics in job ladder models. For high-RE workers—who are at the higher rungs of the ladder—a job loss leads to a more persistent earnings decline relative to low-RE workers because of search frictions. Similarly, for low-RE workers, large increases are likely due to unemployment-to-employment or job-to-job transitions, which have long-lasting effects on earnings.²⁰

5.1 Lifecycle Earnings Growth and Its Distribution

For the analysis in this section, we use the full length of the MEF panel, covering 1978 to 2013. We select individuals who (i) were born between 1951 and 1957 (hence for whom we have 33 years of data between ages 25 and 60), and (ii) had annual earnings above \(Y_{\text{min},t}\) in at least 15 years, thereby excluding workers with very weak labor market attachment. We take a closer look at this latter group in the next subsection. We sort individuals into 100 percentiles by their lifetime earnings (LE), computed by averaging their earnings from age 25 through 60. For each LE percentile bin, denoted \(\text{LE}j\), \(j=1,2,...,99,100,\) we compute the growth rate between ages \(h_{1}\) and \(h_{2}\) by differencing the average earnings across all workers (including those with zero earnings) in those LE and age cells; i.e., \(\text{log}(\overline{Y}_{h_{2},j})-\text{log}(\overline{Y}_{h_{1},j}),\) where \(\overline{Y}_{h,j}\equiv \mathbb{E}(\tilde{Y}_{t}^{i}|i\in \text{LE}j,h(i,t)=h)\).

(a) Lifetime Earnings Growth, \(\log (\bar{Y}_{55})-\log (\bar{Y}_{25})\)

(b) CDF of Number of Years Employed

The results in Figure 11a show that between ages 25 and 55 the median worker (by LE) experiences a smaller earnings growth—about 60%—than a 150% mean growth estimated from a Deaton-Paxson pooled regression (see Appendix C.9). More importantly, higher-LE workers experience a much higher earnings growth over the life cycle compared with the rest of the distribution. While an upward slope per se is not surprising (as it is partly mechanical—faster growth will deliver higher LE, everything else held constant), the variation at the top end is so large and the curvature is so steep, that it turns out to be difficult to capture using simple earnings processes, as we discuss in the next section. For example, average earnings grow by 1.5-fold (91 log pts) over 31 years at \(LE80\), by 4.8-fold (157 log pts) at \(LE95\), and by 27.9-fold (333 log pts) in the top 1%.²¹

One question is whether this extremely high growth rate at the top is driven by higher rates of school enrollment in these groups at age 25 (and thereby low earnings). While the lack of education data does not allow us to answer this question directly, several pieces of evidence are informative. First, about 21.7% of individuals in the \(LE100\) have earnings below the \(Y_{\text{min}}\) threshold at age 25, which is higher than the rate for half of the sample, suggesting schooling could be playing some role (see Figure C.38a). However, this rate drops quickly to 5.95% by age 30, which is one of the lowest in our sample. At the same time, earnings growth for this group between ages 25 and 30 is only slightly higher compared to that between ages 30 and 35 (2.9-fold vs. 2.6-fold), when schooling is unlikely to matter much. Similarly, looking at growth from 35 to 55, we still find a steep profile of earnings growth with respect to LE (see Figure C.37). These observations suggest that low labor supply at age 25 is not the major driver of these patterns.

Turning to the lower end, individuals below \(LE20\) see their earnings decline from age 25 to 55. How important is disability for this decline? Adding SSDI to labor earnings has virtually no effect above the 40th LE group or so (Figure C.35). But it matters at the lower end, mitigating the decline by more than 50% for LE10 and LE5.

5.2 Lifetime Employment Rate and Its Distribution

Next, we investigate the lifetime nonemployment rates across individuals. Using the same criteria as before—working life defined as the period between ages 25 and 60, and full-year nonemployed defined as annual earnings below \(Y_{\text{min}}\)—we examine the cumulative distribution of total lifetime years employed in Figure 11b (for further analysis, see Appendix C.10). The results show that, first, a large fraction of individuals are very strongly attached to the labor market: 28% of individuals were never nonemployed during their working life, and almost half (48%) were nonemployed for less than three years. But second, the distribution has a long left tail, showing a surprisingly large fraction of men who spend half of their working life or more without employment: 18.3% of men spend 18 years—or half of their working life—as full-year nonemployed, and 12.3% spend at least 24 years as nonemployed.

To understand these magnitudes, note that the employment-to-population ratio for prime-age men in cross-sectional data (such as the one the Bureau of Labor Statistics publishes monthly) has averaged around 86% during this time period, implying a monthly nonemployment rate around 14%. Getting nonemployment at annual frequency for 24 years for 12% of the male population requires an extremely high persistence of the long-term nonemployment state. As we shall see in the next section, this statistic turns out to be very hard to match with a simple earnings process with standard parameter values.

6.1 A Flexible Stochastic Process

The models we estimate are special cases of the following general framework, which includes (i) an AR(1) process (\(z_{t}^{i}\)) with innovations drawn from a mixture of normals; (ii) a nonemployment shock whose incidence probability (\(p_{\nu}^{i}(t,z_{t})\)) can vary with age or \(z_{t}\) or both, and whose duration (\(\nu _{t}^{i}\)) is exponentially distributed; (iii) a heterogeneous income profiles component (HIP); and (iv) an i.i.d. normal mixture transitory shock \((\varepsilon _{t}^{i})\):

\[ \begin{aligned} \text{Level of earnings:} & \quad \tilde{Y}_{t}^{i}=(1-\nu _{t}^{i})e^{\left (g\left (t\right)+\alpha ^{i}+\beta ^{i}t+z_{t}^{i}+\varepsilon _{t}^{i}\right)}\\ \text{Persistent component:} & \quad z_{t}^{i}=\rho z_{t-1}^{i}+\eta _{t}^{i},\\ \text{Innovations to AR(1):} & \quad \eta _{t}^{i}\sim \begin{cases} \mathcal{N}(\mu _{\eta,1},\sigma _{\eta,1}) & \text{with prob.}p_{z}\\ \mathcal{N}(\mu _{\eta,2},\sigma _{\eta,2}) & \text{with prob.}1-p_{z} \end{cases}\\ \text{Initial condition of }z_{t}^{i}\text{:} & \quad z_{0}^{i}\sim \mathcal{N}(0,\sigma _{z_{0}})\\ \text{Transitory shock:} & \quad \varepsilon _{t}^{i}\sim \begin{cases} \mathcal{N}(\mu _{\varepsilon,1},\sigma _{\varepsilon,1}) & \text{with prob.}p_{\varepsilon}\\ \mathcal{N}(\mu _{\varepsilon,2},\sigma _{\varepsilon,2}) & \text{with prob.}1-p_{\varepsilon} \end{cases}\\ \text{Nonemployment duration:} & \quad \nu _{t}^{i}\sim \begin{cases} 0 & \text{with prob.}1-p_{\nu}(t,z_{t}^{i})\\ \min \left \{1,exp\left (\lambda \right)\right \} & \text{with prob.}p_{\nu}(t,z_{t}^{i}) \end{cases}\\ \text{Prob of Nonemp. shock:} & \quad p_{\nu}^{i}(t,z_{t})=\frac{e^{\xi _{t}^{i}}}{1+e^{\xi _{t}^{i}}}\text{, where}\xi _{t}^{i}\equiv a+bt+cz_{t}^{i}+dz_{t}^{i}t. \end{aligned} \]

In equation (2), \(g(t)\) is a quadratic polynomial, where \(t=\left (age-24\right)/10\) is normalized age, that captures the lifecycle profile of earnings common to all individuals. The random vector \(\left (\alpha ^{i},\beta ^{i}\right)\) determines ex ante heterogeneity in the level and in the growth rate of earnings and is drawn from a multivariate normal distribution with zero mean and a covariance matrix to be estimated. The innovations, \(\eta _{t}^{i}\), to the AR(1) component are drawn from a mixture of two normals. An individual draws a shock from \(\mathcal{N}(\mu _{\eta,1},\sigma _{\eta,1})\) with probability \(p_{z}\) and otherwise from \(\mathcal{N}(\mu _{\eta,2},\sigma _{\eta,2})\). Without loss of generality, we normalize \(\eta\) to have zero mean (i.e., \(\mu _{\eta,1}p_{z}+\mu _{\eta,2}(1-p_{z})=0\)) and assume \(\mu _{\eta,1}<0\) for identification. Heterogeneity in the initial conditions of \(z_{t}\) is captured by \(z_{0}^{i}\sim \mathcal{N}(0,\sigma _{z_{0}})\). Transitory shocks, \(\varepsilon _{t}^{i}\), are also drawn from a mixture of two normals (eq. 6), with analogous identifying assumptions (zero mean and \(\mu _{\varepsilon,1}<0\)).

Our decision to use normal mixtures is motivated by two considerations. First, they provide a flexible way to model non-Gaussian shock distributions. In fact, by increasing the number of normals that are mixed one can approximate almost any distribution (see McLachlan and Peel (2000)). Second, solving a dynamic programming problem with normal mixture shocks requires minimal adjustments to the computational methods commonly used with Gaussian shocks. This is appealing given our stated objectives.

The last component of the earnings process—and as it turns out, a critical one—is a nonemployment shock (eq. 7) that is intended to primarily capture movements in the extensive margin. Specifically, a worker is hit with a nonemployment shock with probability \(p_{\nu}\) whose duration \(\nu _{t}>0\) follows an exponential distribution with mean \(1/\lambda\) and is truncated at 1 (corresponding to full-year nonemployment with zero annual income). This shock differs from \(z_{t}\) and \(\varepsilon _{t}\) by scaling the level of annual income—not its logarithm—which allows the process to capture the sizable fraction of workers who transition into and out of full-year nonemployment every year.²²

None of the components introduced so far depend explicitly on age or recent earnings, whereas variation along these dimensions is a key characteristic of the empirical patterns we saw. One promising way we found for introducing such variation was by making the nonemployment incidence \(p_{\nu}\) depend on age \(t\) and \(z_{t}\) through the logistic function shown in equation 8.²³ The dependence of \(p_{\nu}\) on \(z_{t}\)—which we refer to as “state dependence”—turns out to be especially important as it induces persistence in nonemployment from one year to the next (despite \(\nu _{t}\) itself being independent over time).

This completes the description of the benchmark process. We also estimated a 2-state process, which, as expected, fits the data better but increases the computational burden in a dynamic programming problem due to an extra state variable (see Appendix D.2).

6.2 Results: Estimates of Stochastic Processes

We now present the estimation results for six different specifications (Table IV). We start from the canonical linear-Gaussian model and add new features step by step until we reach our preferred benchmark process. We discuss along the way which aspect(s) of the data each feature helps capture. Figure 12 plots the fit of each model to the six sets of moments targeted in the estimation. We also show the fit to selected impulse response functions separately in Figure 13.²⁷ We exclude two specifications from these figures for readability and show them in Figures D.12 and D.13 in Appendix D.5.

In Model (1), we start with a simple but widely used linear-Gaussian specification: the sum of an individual fixed effect, an AR(1) process, and an i.i.d. transitory shock, all drawn from Gaussian distributions (i.e., \(\sigma _{\beta}=0\), \(p_{z}=1\), \(p_{\nu}=0\), and \(p_{\varepsilon}=1\) in equations (2) to (8)). The estimates of key parameters are unusually large—the standard deviations of the fixed effect and the transitory shock (\(\sigma _{\alpha}=1.18\) and \(\sigma _{\varepsilon}=0.70\)) are several times larger than the typical estimates in the literature (c.f., Storesletten et al. (2004) and Heathcote et al. (2010b)). The persistence parameter is slightly above 1 (\(\rho =1.005\)), implying a nonstationary process, with the effects of shocks being amplified over time.

The fact that these parameter values are quite different from previous estimates should not come as a surprise, since most of the moments targeted here have not been used in previous analyses. That said, it turns out that one set of moments is responsible for most of these differences—the CDF of lifetime employment rates, which shows that nonemployment is an extremely persistent state for a significant fraction of men.²⁸ To match this large fraction of persistently nonemployed individuals, the estimation chooses a wide dispersion of fixed effects, placing many individuals closer to the minimum threshold. Combined with the large transitory and nonstationary persistent shocks, the model manages to match the lifetime (non)employment distribution very well (Figure 12e).

However, the model fails in most of the other dimensions. First, it generates essentially zero skewness and no excess kurtosis (Figures 12b and 12c), which is not surprising given the Gaussian structure. Second, it vastly overstates lifecycle income growth—for example, implying a 3.1-fold rise at the median compared with only a 60% rise in the data (Figure 12d). Finally, it overshoots both the level of inequality and its rise over the life cycle (Figure 12f).²⁹ Overall, this process does not offer a good fit to the data.

(a) Standard Deviation

(b) Skewness

(c) Kurtosis

(d) Lifecycle Earnings Growth

(e) Lifetime Nonemployment Distr.

(f) Variance of Log Earnings

6.3 Parameter Estimates of the Benchmark Process

We now turn to the parameter estimates from the benchmark process and its fit to some untargeted moments. Starting with the AR(1) process, the persistent shock is drawn about every 2.5 years (\(p_{z}=40.7\%\)) from an “unfavorable” distribution—with a negative mean and large standard deviation (\(\mu _{\eta,1}=-0.085\) and \(\sigma _{\eta,1}=0.364\))—and in the other years from a “favorable” one—with a positive mean and small standard deviation (\(\mu _{\eta,2}=0.058\), \(\sigma _{\eta,2}=0.069\)). This mixture of normals implies that innovations to the persistent component are both strongly left skewed (skewness of \(-0.87\)) and leptokurtic (kurtosis around 6.3). In contrast, transitory shocks (\(\varepsilon _{t}\)) are typically smaller: In most years (with 87% probability), they are drawn from a tight distribution, \(\mathcal{N}(-0.041,0.037^{2})\), and every eight years or so (\(p_{\varepsilon}=13.0\%\)) from a distribution with large positive mean and dispersion, \(\mathcal{N}(0.271,0.285^{2})\). Consequently, transitory shocks feature a skewness of 3.2 and a kurtosis of 15.4. However, \(\eta _{t}\) and \(\varepsilon _{t}\) are not the only sources of higher-order moments; workers also face nonemployment risk, which allows the model to generate a leptokurtic density for arc-percent changes with spikes at both ends (Figure 14a). The state dependence in nonemployment risk also leads to age- and income-varying skewness and kurtosis in persistent earnings changes. Thus, as expected, Models (3) and (4) (without nonemployment risk) find persistent shocks to be more strongly leptokurtic and left skewed than Models (5) and (6).⁴⁰ We conclude that persistent innovations are key drivers of non-Gaussian features in the data.

The initial heterogeneity in earnings is captured by (i) the permanent fixed effect \(\sigma _{\alpha}\) (since \(\beta ^{i}\times 0=0\)), and (ii) \(\sigma _{z_{0}}\), the initial dispersion in \(z_{t}\), whose effect declines at rate \(\rho\). Since the latter is twice as large as the former (\(\sigma _{z_{0}}=0.71\) versus \(\sigma _{\alpha}=0.30\)), a large part of earnings inequality at age 25 reflects persistent but not permanent differences. Finally, our estimate \(\sigma _{\beta}\simeq 2\%\) is close to earlier estimates from the PSID (e.g., Haider (2001) and Guvenen and Smith (2014)), despite using different datasets and moments.

State-dependent Nonemployment Shocks. Almost all workers hit by nonemployment shocks experience full-year nonemployment (\(\lambda =0.0001\)).⁴¹ How often are these shocks realized? Similar to Model (2), our benchmark process also displays striking heterogeneity in nonemployment risk. The probability function \(p_{\nu}\) (equation 8) is hard to interpret on its own (see Figure D.14 for a 3D graph), so instead, we investigate this probability for various age and RE quantiles for workers who satisfy the conditions of the RE sample: Nonemployment risk declines modestly over a working life, from 6.9% over 25–34 to 6.1% in the next 10 years, and to 5.5% over 45–54. There is less variation in the data, with the corresponding figures being 6.8%, 6.2%, and 6.6%, respectively.

(a) Density of Arc-Percent Change

(b) Fraction Nonemployed in \(t\), %

Differences in nonemployment risk between income groups are much more pronounced (Figure 14b). In the bottom RE decile, workers experience nonemployment almost once every six years (with 17.6% probability). This probability declines sharply to 5.4% for the median quintile and to 0.4% for the top decile. The fit to the data is fairly good, though the RE variation is slightly less pronounced compared with the data.⁴² Perhaps surprisingly, the linear-Gaussian income process also matches this feature of the data reasonably well. As noted earlier, however, this specification’s ability to capture the nonemployment CDF comes at the cost of an implausibly steep inequality profile (Figure 12f). In fact, under a more plausible parameterization, only less than 4% of individuals ever experience a full year of nonemployment over their working life.

How does the model capture the nonlinear earnings dynamics with a single AR(1) component? The autocorrelation of persistent shocks is not precisely captured by \(\rho\), because modeling \(p_{\nu}\) as a function of \(z_{t}\) implies that nonemployment is autocorrelated even though \(v_{t}\) is drawn in an i.i.d. manner.⁴³ Moreover, since this function is highly nonlinear, how income responds on impact to a given shock \(\eta _{t}\) and how persistent this response is depend very much on the persistent component \(z_{t}\) and the sign and magnitude of the shock. This feature, along with the normal mixture shocks, generates the asymmetric mean reversion in impulse responses in Figure 13.

To illustrate its persistence, we examine the future nonemployment risk of workers who are nonemployed in \(t\). As usual, we further condition workers on their RE in \(t-1\). Nonemployment is fairly persistent overall and more so for low-income workers: Between ages 25 and 35, 48% of the workers in the bottom RE decile experience another nonemployment spell five years after the initial nonemployment (Table V). This number declines monotonically over the RE distribution to 35% for the top decile. Furthermore, nonemployment risk becomes more persistent over the working life, particularly for high earners. For example, in the top decile, the conditional probability of nonemployment in \(t+5\) increases from 35% in the first 10 years to 50% in the last 10 (see Guvenen et al. (2017) for an empirical investigation of nonemployment persistence).

Autocovariance Structure of Earnings. Another set of moments that has been widely used when estimating income processes is the autocovariance matrix of log earnings in levels or changes (e.g., Abowd and Card (1989)). An important drawback of this approach, especially through the lens of our specification, is that it ignores small earnings observations below the minimum income threshold. Our estimation targets the dynamics of earnings in a more complete manner through impulse response functions, which capture the heterogeneity in the dynamics of income changes by their size and sign as well as for workers with different earnings levels (Figure 13).

For completeness and consistency with earlier work, we report these moments and show the benchmark specification’s fit to them in Appendix D.6. In our data, consistent with earlier work using survey data (Meghir and Pistaferri (2004)), the autocovariance of earnings growth approaches zero quickly after a couple of lags. Our model generates a similar pattern, although the autocovariances for smaller lags tend to be closer to zero than in the data. Modeling the transitory component as a moving average of order \(q\) may fix this shortcoming (see Meghir and Pistaferri (2004)). Furthermore, MaCurdy (1982a) noted that if a HIP component is present (\(\sigma _{\beta}^{2}>0\)), the autocovariance of one-year log earnings growth should turn positive at longer lags. In the data, this test would not reject \(\sigma _{\beta}^{2}=0\). Interestingly, MaCurdy’s test reaches a similar conclusion in data simulated from the benchmark process, which features a sizable HIP component (see Guvenen (2009) for a discussion on the reliability of this test).

Discretizing the Benchmark Income Process. Another advantage of building on the linear-Gaussian framework is that existing methods can be adopted to discretize our income process. After all, the benchmark specification is still within the realm of first-order Markov processes, with the two important additions being the normal mixture shocks in transitory and persistent components and the state-dependent nonemployment shocks. The standard Tauchen (1986) methodology does not take a stance on the distribution of shocks. Nevertheless, there are complications that arise due to the nonnormalities. The key issue is the placement of grid points in the discretized space. The negative skewness and high kurtosis require points to be irregularly placed (see Civale et al. (2016) for the optimal grid placement).

Moreover, another crucial task is to capture the nonlinearities in the nonemployment probability. How well these are captured by a discretization depends on how fine the grid points of the persistent component are in the region where nonemployment risk is highly nonlinear. Thus a given level of precision requires many more grid points relative to the standard process. Therefore, in ongoing work (Guvenen et al. (2018)), we choose not to discretize the shock distribution and use quadrature methods instead.

8.1 Imputation of Self-Employment Income Above SSA Taxable Limit

We restrict our main sample for cross-sectional and impulse response moments to years between 1994 and 2013 during which neither self-employment income nor wage/salary income is capped. However, this sample period–covering only 20 years–is too short to construct reliable measures of lifetime incomes of individuals. For this purpose, lifetime income moments in section 5 are computed using the whole sample that covers 36 years between 1978 and 2013. But self-employment income is capped by the SSA maximum taxable earnings limit before 1994. In this section we introduce a methodology to impute self-employment income above the top code for years before 1994, and show that imputing self-employment income has a negligible effect on our results.

Let \(y_{t}^{max}\) be the official SSA maximum taxable earnings limit in year \(t\). Our goal is to impute the uncapped (unobservable) self employment income measure, \(\tilde{y}_{i,t}^{SE}\) for individuals who have self-employment income around the maximum taxable earnings limit reported in the MEF data specified by threshold \(\chi y_{t}^{max}\) (i.e., \(y_{it}^{SE}\geq \chi y_{t}^{max}\)), where \(\chi <1\).⁴⁷

Figure A.1 – Fraction of Top-Coded Self-Employment Income Observations

We then use these regression coefficients to impute the uncapped self-employment income before 1994 for individuals who have SE income above the limit \(\chi y_{t}^{max}\) reported in the MEF data. For this purpose, we randomly assign individuals to quantiles \(\tau =1....75\) in our lifetime income sample. Then, the imputed self-employment income for an individual in age group \(h\) with quantile \(\tau\) who has recorded self-employment income above the limit \(\chi y_{t}^{max}\) in year \(t=1981,1981,...1993\) is given by the following equation:⁴⁸

Figure A.2 – Income Growth for Imputed and Nonimputed Data

Figure A.1 plots the fraction of top-coded self-employment income observations against the percentiles of lifetime earnings distribution at ages 25, 30, 35, and 40 in our imputed lifetime income sample.⁴⁹

Furthermore, Figure A.2 plots the lifetime income growth between 25 and 55 against lifetime earnings percentiles using imputed and nonimputed data, which is already shown in Figure 11a. The two series are almost indistinguishable, indicating that top coding has very little effect on lifetime income growth. This is because only a very small number of workers are affected by the top coding; those who had very high self-employment income before 1994 or when the cohort was younger than age 41.

10.1 Cross-Sectional Moments of One-Year Log Earnings Growth

Throughout the main text, we showed the cross-sectional moments of five-year (log) earnings growth. Figures C.1–C.4 show analogous features of the data for one-year earnings growth.

(a) Second Standardized Moment

(b) P90-10

(a) Third Standardized Moment

(b) Kelley Measure

(a) P90-P50

(b) P50-P10

(a) Fourth Standardized Moment

(b) Crow-Siddiqui Measure

10.2 Arc-Percent Moments

In the main text, we documented moments of log earnings changes. In doing so, we are forced to drop observations close to zero to obtain sensible statistics. However, as we discuss in Section 2, such observations contain potentially valuable information, as they inform us about very large changes in earnings caused by events such as long-term nonemployment. To complement our analysis, this section reports the cross-sectional moments of arc-percent changes defined in Section 2, which we reproduce here for convenience: \[ \text{arc percent change:}\quad \Delta _{\text{arc}}Y_{t,k}^{i}=\frac{Y_{t+k}^{i}-Y_{t}^{i}}{\left (Y_{t+k}^{i}+Y_{t}^{i}\right)/2}. \] This measure allows computation of earnings growth even when the individual has zero income in one of the two years \(t\) and \(t+k\). Section C.2.1 shows the moments of one-year arc-percent change, and C.2.2 shows the moments of five-year change.

10.2.1 Moments of Annual Arc-Percent Changes

Figures C.5a–C.8b show the standardized moments of one-year arc-percent changes.

(a) Second Standardized Moment

(b) P90-10

Figure C.5 – Dispersion of Annual Arc-Percent Earnings Change

(a) Third Standardized Moment

(b) Kelley Measure

Figure C.6 – Skewness of Annual Arc-Percent Earnings Change

(a) P90-P50

(b) P50-P10

Figure C.7 – Kelley’s Skewness Decomposed (Annual Arc-Percent Growth): Change in P90-P50 and P50-P10 Relative to Age 25–34

(a) Fourth Standardized Moment

(b) Crow-Siddiqui Measure

Figure C.8 – Kurtosis of Annual Arc-Percent Earnings Change

10.2.2 Moments of Five-Year Arc-Percent Changes

Figures C.9a–C.12b show the standardized moments of -year arc-percent changes.

(a) Second Standardized Moment

(b) P90-10

Figure C.9 – Dispersion of Five-Year Arc-Percent Earnings Change

(a) Third Standardized Moment

(b) Kelley Measure

Figure C.10 – Skewness of Five-Year Arc-Percent Earnings Change

(a) P90-P50

(b) P50-P10

Figure C.11 – Kelley’s Skewness Decomposed (5-Year Arc-Percent Growth): Change in P90-P50 and P50-P10 Relative to Age 25–34

(a) Fourth Standardized Moment

(b) Crow-Siddiqui Measure

Figure C.12 – Kurtosis of Five-Year Arc-Percent Earnings Change

10.3 Further Moments of Log Earnings Change

In this section, we report some additional figures of interest that are omitted from the main text due to space constraints. First, Figure C.13 plots selected percentiles of the annual and five-year log earnings change distribution for every RE percentile.

Figure C.13 – Selected Percentiles of Log Earnings Changes

(a) One-Year Log Changes

(b) Five-Year Log Changes

Second, Figure C.14 shows an additional measure of kurtosis proposed by Moors (1988) for one- and five-year earnings changes. Similar to the measure proposed by Crow and Siddiqui (1967), this measure is robust to outliers in the tails. Moors’ kurtosis, \(\kappa _{M}\) is defined as \[ \kappa _{M}=\frac{\left (P87.5-P62.5\right)+\left (P37.5-P12.5\right)}{P75-P25}. \] For a Gaussian distribution, Moors’ kurtosis is 1.23 (shown on dashed lines).

(a) P90-P10

(b) Kelley Measure

(c) Crow-Siddiqui Measure

10.4 An Alternative Measure of Persistent Earnings Changes

In section 3, we studied the distribution of five-year earnings changes, and explained that the five-year changes reflect more of the distribution of the persistent innovations rather than transitory innovations. We also considered an alternative measure (\(\overline{\Delta}_{\log}^{5}(\bar{y_{t}}^{i})\equiv \bar{y}_{t+4}^{i}-\bar{y}_{t-1}^{i}\), where \(\bar{y}_{t+4}^{i}\equiv \log (\bar{Y}_{t+4}^{i})\) and \(\bar{y}_{t-1}^{i}\equiv \log (\bar{Y}_{t-1}^{i})\)) to deal with the caveat that our baseline measure is contaminated by transitory changes in years \(t\) and \(t+k\). The main text showed the standardized moments of this alternative measure; Figure C.15 shows the quantile-based moments.

10.5 Cross-Sectional Moments for Job-Stayers vs. Job-Switchers

In the main text, we analyzed the properties of earnings growth separately for job-stayers and job-switchers by showing quantile-based moments of five-year earnings changes. Here, we first complement our analysis by showing several features of the data that were omitted in the main text to save space. Second, we consider an alternative definition of job-stayers and investigate the cross-sectional moments of earnings according to that definition.

Figure C.16 shows the fraction of job-stayers according to our baseline definition as a function of recent earnings and age. The probability of staying with the same employer increases with recent earnings and age. For the youngest age group (25-34), the probability of staying in the same job is around 20% at the bottom of the recent earnings distribution. This fraction increases with recent earnings and reaches a peak around 60% at the 95th percentile of the RE distribution. This pattern reverses itself slightly at the top of the RE distribution. As workers age, the probability of staying with the same employer increases across the RE distribution.

Figure C.16 – Fraction Staying at Jobs Between \(t\) and \(t+1\) (Baseline Definition)

Second, Figures C.17 and C.18 show the age profile of higher-order moments shown in Section 3.5. The age patterns are broadly similar across switchers and stayers: P90-10 declines slightly, skewness becomes more negative, and kurtosis increases for both job-stayers and job-switchers over the life cycle.

Next, we complement our analysis of job-stayers and job-switchers by investigating the standardized moments of one- and five-year earnings changes (as opposed to quantile-based moments analyzed in the main text). We plot these moments in Figure C.19.

(a) P90-P10

(b) Kelley’s Skewness

(c) Crow-Siddiqui Kurtosis

(a) P90-P10

(b) Kelley’s Skewness

(c) Crow-Siddiqui Kurtosis

(a) Std. Dev., One-year

(b) Std. Dev., Five-Year

(c) Skewness, One-Year

(d) Skewness, Five-Year

(e) Kurtosis, One-Year

(f) Kurtosis, Five-Year

The results are consistent with what one might expect. Job-stayers face earnings changes that (i) have half the dispersion of job-switchers, (ii) are less negatively skewed as opposed to job-switchers, who face very negatively skewed changes, and (iii) have a much higher kurtosis than job-switchers. In fact, kurtosis is as high as 40 for annual changes and 25 for five-year changes for job-stayers, but is less than 10 for job-switchers at both horizons.

One caveat worth emphasizing again is that constructing a clean measure of job-stayers and job-switchers is not possible in our dataset, primarily because of the annual nature of the data: We observe when employment spells begin and end only at annual frequency, and therefore cannot infer if a worker worked multiple jobs at the same time or if he switched employers at some point during the year, and if so, whether the change was a direct job-to-job switch or involved a nonemployment spell in between. In our baseline measure, we have opted to be very conservative when defining job-stayers. This measure probably understates job-stayers and overstates true switchers.

We now consider an alternative definition, which is a much more conservative definition of job-switchers. According to this definition, we call an individual a job-switcher in year \(t\) if i) the largest paying employer is different in years \(t\) and \(t+1\), ii) the largest paying employers in years \(t\) and \(t+1\) contribute to at least 75% of the workers total salary, iii) the worker either has no income in year \(t+1\) from the main employer of year t, or if he does, that income in \(t+1\) is less than 25% of what he made in \(t\) (from the same employer). Figure C.20 compares the share of job-stayers according to this alternative definition (left panel) to our baseline (right panel) and Figure C.21 compares the cross-sectional moments of one-year earnings changes. By construction, the fraction of individuals identified as job-stayers and job-switchers is quite different across the two definitions. Nevertheless, all of the substantive conclusions go through regarding the differences in the cross-sectional moments of job-stayers and job-switchers.

(a) Baseline

(b) Alternative definition

(a) Standard dev.: Baseline

(b) Standard dev.: Alternative definition

(c) Skewness: Baseline

(d) Skewness: Alternative definition

(e) Kurtosis: Baseline

(f) Kurtosis: Alternative definition

10.6 Cross-Sectional Moments by Age without Sample Selection

When choosing the sample for cross-sectional moments, we required an individual to have an earnings level above the minimum threshold in \(t-1\) and in at least two more years between \(t-5\) and \(t-2\). Figure C.22 shows that these conditions result in a substantial share of the initial sample being dropped from the analysis. This large selectivity opens up the possibility that some of our results might be specific to our final sample. Here, we relax the selection criteria and include any person for whom earnings change can be computed. Figures C.23–C.28 show the standardized and quantile-based moments of one-year and five-year earnings changes. We find that our substantive conclusions are unchanged: Dispersion of earnings changes declines with age for most of the life cycle, and earnings changes become more negatively skewed and more leptokurtic.

Figure C.22 – Fraction of Observations Dropped by Age

(a) Second Standardized Moment

(b) P90-P10

(a) Third Standardized Moment

(b) Kelley Measure

(a) Fourth Standardized

(b) Crow-Siddiqui Measure

(a) Second Standardized Moment

(b) P90-P10

(a) Third Standardized Moment

(b) Kelley Measure

(a) Fourth Standardized

(b) Crow-Siddiqui Measure

10.7 Survey Data and Higher-Order Moments

10.8 The Role of Disability Income

In this section, we investigate the robustness of our findings to the inclusion of income from Social Security disability benefits (SSDI). This is particularly relevant for thinking about the tails of the distribution of earnings changes. To this end, we link to our dataset information about disability benefits from the SSA records. We then define a measure of “total income” as the sum of labor income and annual disability income. Section C.8.1 compares the cross-sectional moments of earnings changes to our baseline and Section C.8.2 does the same for lifetime income growth.

10.8.1 Cross-sectional Moments

Figures C.29–C.34 show several moments of one-year and five-year earnings changes. In each figure and for each age group, we show these moments for labor income and total income separately, where total income is labor income plus disability benefits. For each measure of income, recent earnings are re-calculated using that measure. Otherwise, these graphs are calculated in a fashion analogous to that in Section 3. The main finding here is that the inclusion of disability income has little effect on cross-sectional moments, even at low levels of earnings.

(a) Second Standardized Moment

(b) P90-P10

Figure C.29 – Dispersion of One-Year Log Earnings Growth

(a) Second Standardized Moment

(b) P90-P10

Figure C.30 – Dispersion of Five-Year Log Earnings Growth

(a) Third Standardized Moment

(b) Kelley Measure

Figure C.31 – Skewness of One-Year Log Earnings Growth

(a) Third Standardized Moment

(b) Kelley Measure

Figure C.32 – Skewness of Five-Year Log Earnings Growth

(a) Fourth Standardized

(b) Crow-Siddiqui Measure

Figure C.33 – Kurtosis of One-Year Log Earnings Growth

(a) Fourth Standardized

(b) Crow-Siddiqui Measure

Figure C.34 – Kurtosis of Five-Year Log Earnings Growth

(a) 25–55

(b) 30–55

Figure C.35 – Lifecycle Earnings Growth Rates, by Lifetime Earnings Group

10.9 Additional Figures on Lifecycle Patterns of Earnings

To provide a benchmark for the analysis in Section 5, we estimate the average lifecycle profile of log earnings using a standard pooled regression of log individual earnings on a full set of age and (year-of-birth) cohort dummies using the admissible observations (as defined in Section 2) between 1994-2013.⁵¹

Figure C.36 – Lifecycle Profile of Average Log Earnings

(a) By Different Starting Ages

(b) By Decades of the Life Cycle

10.10 Concentration of Nonemployment by Lifetime Earnings Group

In this section, we investigate how concentrated (full-year) nonemployment is. We rank individuals by their lifetime earnings and group them into percentiles. For each lifetime earnings group, we compute what fraction of full-year nonemployment at a given age is accounted for by that group. These are shown in Figure C.38. For example, the bottom 1% of the lifetime earnings distribution accounts for around 50% of total nonemployment at ages 25–30.

(a) Fraction Nonemployed wrt Labor Income

(b) Fraction Nonemployed wrt Total Income

(a) Average Employment by Decades wrt Lifetime Employment

(b) Fraction Nonemployed over the Life Cycle wrt Number of Years Nonemployment

11.1 Moment Selection and Aggregation

Accounting for zeros. Recall that in order to construct the cross-sectional moments of log growth, we have dropped individuals who had very low earnings—below \(\overline{Y}_{\text{min}}\)—in year \(t\) or \(t+k\) so as to allow taking logarithms in a sensible manner. Although this approach made sense 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. Targeting log growth moments also creates technical issues with the optimization due to (little) jumps in the objective function as workers cycle in and out of employment. To this end, instead of targeting moments of log earnings change, we target moments of arc percent change, as defined in Section 3. According to this measure, an income change from any positive level to 0 corresponds to an arc-percent change of -2, whereas an income change from 0 to any positive level indicates an arc-percent change of 2.

Aggregating moments. If we were to match all data points for every RE percentile and every age group, it would yield more than 10,000 moments. Although such an estimation is not infeasible, 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 difficult. To avoid this, we aggregate the 100 RE percentiles and the 6 age groups into fewer homogeneous groups. We now describe the details of this aggregation and the resulting list of moments targeted in our estimation.

1. Cross-sectional moments of earnings changes. 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 \(t-1\). The RE percentiles are grouped as follows: 1, 2–10, 11–0, 21–30,…, 81–90, 91–95, 96–99, 100. Thus, we compute the three moments of the distribution of one- and five-year earnings changes for \(6\times 13=78\) different groups of workers. We then aggregate the 6 age bins into 3 age groups, \(A_{t-1}^{i}\), by taking an average of moments within each age group. The first age group is defined as young workers between ages 25 through 34, the second is between ages 35-44, and the third age group is defined as workers between the ages of 35 and 54. Consequently, we target three standardized moments (i.e., standard deviation, skewness, and kurtosis) of one- and five-year arc-percent change for three age and 13 recent earnings groups, giving us \(3\times 2\times 3\times 13=234\) cross-sectional moments. These moments are shown in Figure D.1.

(a) St. Deviation of One-Year Earnings Growth

(b) St. Deviation of Five-Year Earnings Growth

2. Lifecycle earnings profile. The second set of moments captures the heterogeneity in log earnings growth over the working life across workers who are in different percentiles of the LE distribution. We target the average dollar earnings at 8 points over the life cycle: ages 25, 30,…, and 60 for different LE groups. We combine LE percentiles into larger groups to keep 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 \(8\times 15=120\).

3. Impulse response functions. We target average arc-percent changes in earnings over the next \(k\) years for \(k=1,2,3,5,10\) conditional on groups formed by crossing age, recent earnings \(\overline{Y}_{t-1}\), and earnings change between \(t-1\) and \(t\Delta _{arc}^{1}Y_{t-1}^{i}\): \(\mathbb{E}[\Delta _{arc}^{k+1}Y_{t-1}^{i}\mid age,\overline{Y}_{t-1},\Delta _{arc}^{1}Y_{t-1}^{i}]\).⁵²

Within each age \(h\) and RE group \(j\), we then estimate our targets for the persistence of earnings growth. For each \(k\)-year expected future earnings growth we create a piecewise linear function of arc-percent growth between \(t\) and \(t-1\), i.e., \(\mathbb{E}_{h,j}[\Delta _{arc}^{k+1}Y_{t-1}^{i}\mid \Delta _{arc}^{1}Y_{t-1}^{i}]=f_{h,j}^{k}(\Delta _{arc}^{1}Y_{t-1}^{i})\). For this purpose, we condition workers in the data into 23 groups with respect to \(\Delta _{arc}^{1}Y_{t-1}^{i}\). We first group all workers who are full-year nonemployed in \(t\) in the first bin. Then we rank the rest of the workers into the following percentiles: 1–2, 3–5, 6–10, 11-15, 16-20, 21-25,…, 91–95, 96–98, 99–100. Then the piecewise linear function \(f_{h,j}^{k}(\Delta _{arc}^{1}Y_{t-1}^{i})\) for year-\(k\) is determined by the linear interpolation of 23 data points of average earnings growth between \(t-1\) and \(t\), \(\mathbb{E}\left [\Delta _{arc}^{1}Y_{t-1}^{i}\right]\) and their corresponding k-year future expected growth, \(\mathbb{E}[\Delta _{arc}^{k}Y_{t}^{i}]\).

For the model-simulated data, we group workers into \(2\times 8\) age and RE groups—similar to the data moments. But within each age \(h\) and RE group \(j\) we now rank workers into 10 \(\Delta _{arc}^{1}Y_{t-1}^{i}\) groups defined by the following percentiles: 1–2, 3–5, 6–10, 11–30, 31–50, 51–70, 71–90, 91–95, 96–98, 99–100. In the estimation of income processes we minimize the distance between 10 different values of \(\mathbb{E}_{h,j}^{model}[\Delta _{arc}^{k+1}Y_{t-1}^{i}\mid \Delta _{arc}^{1}Y_{t-1}^{i}]\) (for each age and RE group and \(k\)-year expectation) from the model and its corresponding data moment from the piecewise linear function, \(f_{h,j}^{k}(\Delta _{arc}^{1}Y_{t-1}^{i})\). As a result, we have a total of \(2\times 8\times 5\times 10=800\) moments based on impulse responses.

(a) Workers with Median RE

(b) Workers at the 90th Percentile of RE

The impulse response functions targeted in the estimation are plotted in Figures D.2a–D.2d (to keep the figures similar to their counterparts in Section 4, we plot \(\mathbb{E}[\Delta _{arc}^{k+1}Y_{t-1}^{i}\mid \Delta _{arc}^{1}Y_{t-1}^{i}]-\mathbb{E}[\Delta _{arc}^{1}Y_{t-1}^{i}]\)). More specifically, Figure D.2a plots for prime-age workers with median recent earnings the mean reversion patterns at various horizons. Figures D.2b and D.2c do the same for workers at the 90th and 10th percentiles of the recent earnings distribution, respectively. Lastly, Figure D.2d shows the variation of these impulse response functions with recent earnings.

4. Age profile of within-cohort variance of log earnings. Although the main focus of this section is on earnings growth, the lifecycle evolution of the dispersion of earnings levels 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 by controlling for cohort dummies in a sample of cross-sectional moments in the data (Figure D.3). In our estimation, we compute this set of moments for a sample with income observations above the minimum income threshold. We have a total of 36 moments based on the variance of log earnings, one for each age.

5. CDF of Employment over the Life Cycle. We target the distribution of total number of years employed (\(\tilde{Y}_{t,h}^{i}\geq \overline{Y}_{\text{min},t}\)) over the life cycle. In particular, we target the cumulative distribution of total lifetime years employed as shown in Figure 11b. Thus, in total there are 35 such moments targeted in our estimation.

In sum, we target a total of \(J=234+120+800+36+35=1,227\) moments in our estimation.⁵³

11.2 2-State Process

We also estimate a more flexible income process which has 2 AR(1) components, denoted by \(z_{1}\) and \(z_{2}\), each subject to innovations from a mixture of two normals with age and income dependent shock probabilities. Here is the full specification where \(t=\left (age-24\right)/10\) denotes normalized age and for \(j=1,2\):

\[ \begin{aligned} Y_{t}^{i} & =(1-\nu _{t}^{i})\exp \left (g\left (t\right)+\alpha ^{i}+\beta ^{i}t+z_{1,t}^{i}+z_{2,t}^{i}+\varepsilon _{t}^{i}\right)\\ z_{1,t}^{i} & =\rho _{1}z_{1,t-1}^{i}+\eta _{1t}^{i}\\ z_{2,t}^{i} & =\rho _{2}z_{2,t-1}^{i}+\eta _{2t}^{i},\\ \text{Innovations to AR(1):} & \quad \eta _{j,t}^{i}\sim \begin{cases} \mathcal{N}(\mu _{z_{j},1},\sigma _{z,1}) & \text{with pr.}p_{z_{j},t}\\ \mathcal{N}(\mu _{z_{j},2},\sigma _{z,2}) & \text{with pr.}1-p_{z_{j},t} \end{cases}\\ \text{Initial value of AR(1) process:} & \quad z_{j0}^{i}\sim \mathcal{N}(0,\sigma _{j0}),\text{}j=1,2.\\ \text{Nonemployment shocks:} & \text{equation 7}\\ \text{Transitory shock:} & \text{equation 6} \end{aligned} \]

Each AR(1) component, \(z_{1}\) and \(z_{2}\), receives a shock drawn from a mixture of two Gaussian distributions as in our benchmark specification. We again normalize the mean of innovations to the persistent components to zero; i.e., \(\mu _{z_{j},1}p_{z_{j}}+\mu _{z_{j},2}(1-p_{z_{j}})=0\).⁵⁴

Figure D.3 – Within-Cohort Variance of Log Earnings

The age and income dependence of moments is captured by allowing the mixture probabilities to depend on age and the sum of persistent components (\(z_{1}+z_{2}\)):⁵⁵

\[ \begin{aligned} p_{z_{j},t}^{i} & = & \frac{\exp \left (\xi _{j,t-1}^{i}\right)}{1+\exp \left (\xi _{j,t-1}^{i}\right)},\\ \xi _{jt}^{i} & = & a_{z_{j}}+b_{z_{j}}\times t+c_{z_{j}}\times \left (z_{1,t}^{i}+z_{2,t}^{i}\right)+d_{z_{j}}\times \left (z_{1,t}^{i}+z_{2,t}^{i}\right)\times t, \end{aligned} \]

for \(j=1,2.\) The equation for \(p_{\nu t}\) is the same as (16) but \(\xi _{j,t-1}^{i}\) is replaced with \(\xi _{jt}^{i}\). This completes the description of the 2-state process.

This 2-state process provides a significantly better fit to the targeted moments and matches top income inequality as well as the income variation in nonemployment risk (Figures D.4b and D.4a). We find that the two AR(1) components are quite different from each other, especially in terms of their persistence with \(\rho _{2}=0.98\) vs. \(\rho _{1}=0.79\) (Table D.1). We report the probability of drawing a nonemployment shock for various age and RE percentile groups for workers who satisfy the conditions of the base sample in Table D.2. The composition of large negative shocks changes from (hard-to-insure) more persistent innovations to the less persistent ones over the life cycle. The probability of receiving at least one large shock to one of the two AR(1) components or a nonemployment shock in a given year is declining in recent earnings, ranging from 29% at the low end to 9% for individuals above the 90th percentile. Finally, the age and income variation of nonemployment risk in the 2-state specification is qualitatively similar to that in the benchmark process.

(a) Fraction Nonemployed in \(t\), %

(b) Log Density of Earnings

11.3 Numerical Method for Estimation

Objective function. Let \(d_{j}\) for \(j=1,...,J=1227\) denote a generic empirical moment, and let \(\tilde{d}_{j}(\theta)\) be the corresponding model moment that is simulated for a given vector of earnings process parameters, \(\theta\). We simulate the entire earnings histories of 100,000 individuals who enter the 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 with 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 arc-percent deviation between each data target and the corresponding simulated model moment. For each moment \(j\), define

\[ m_{j}(\theta)=\frac{\tilde{d}_{j}(\theta)-d_{j}}{0.5\left (|\tilde{d}_{j}(\theta)|+|d_{j}|\right)+\psi _{j}}, \]

where \(\psi _{j}>0\) is an adjustment factor. When \(\psi _{j}=0\) and \(d_{j}\) is positive, \(m_{j}\) is simply the (arc) 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 arc-percent earnings changes close to zero). To account for this, we choose \(\psi _{j}\) to be equal to the 10th percentile of the distribution of the absolute value of the moments in a given set. The MSM estimator is \[ \hat{\theta}=\arg \min _{\theta}\boldsymbol{m}(\theta)'W\boldsymbol{m}(\theta), \] where \(\boldsymbol{m}(\theta)\) is a column vector in which all moment conditions are stacked, that is, \[ \boldsymbol{m}(\theta)=\left [m_{1}(\theta),...,m_{J}(\theta)\right]^{'}. \]

We choose a weighting matrix that corresponds to essentially first averaging the moments within each of the seven sets, and then assigning equal weight (1/7) to each set of moments. For example, each of the 117 cross-sectional moments (standard deviation, skewness, kurtosis) of one-year earnings growth receives a weight of \(1/(7\times 117)\), each of the 480 short-term impulse response moments receives a weight of \(1/(7\times 480)\), and so on. Recall again that the seven sets of moments are as follows: (i) the standard deviation, skewness, and kurtosis of one-year and (ii) five-year earnings growth; (iii) impulse response moments over short (at one-, two-, and three-year) horizons and iv) long (at five- and ten-year) horizons; (v) average earnings of each LE group over the life cycle; (vi) the cumulative distribution of nonemployment; and (vii) the age profile of the within-cohort variance of log earnings.

Numerical method. The global stage is a multi-start algorithm where candidate parameter vectors are uniform Sobol (quasi-random) points. We typically take about 250,000 initial Sobol points for pre-testing and select the best 1,000 (i.e., ranked by objective value) for the multiple restart procedure depending on the number of parameters to be estimated. For processes with a large number of parameters to be estimated (e.g., the benchmark process or the 2-state process), we also tried using 300,000 initial Sobol points and used the best 2,000 of them. We found this wider search for parameter values to be inconsequential for our estimates. The local minimization stage is performed with a mixture of Nelder-Mead’s downhill simplex algorithm (which is slow but performs well on difficult objectives) and the DFNLS algorithm of Zhang et al. (2010), which 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.

11.4 Model Selection

Clearly, income processes with more parameters deliver a better fit to the data. To what extent should we prefer the richer parameterized processes in Table IV? To answer this question, we now implement a procedure for model selection to the specifications in Table IV. Specifically, we carry out two tests.

Test 1. The first procedure tests the null hypothesis that a given specification is the true data-generating process. It does so, as in existing specification tests in the literature, by using the asymptotic distribution of the objective value—the J-statistic in the GMM context. Our objective value is different than the traditional J-statistic, since we do not use the efficient weighting matrix, and therefore it does not follow the chi-squared distribution. Therefore, we first derive analytically the asymptotic distribution of our objective value, which we label as the pseudo-J statistic.

Let \(W\) denote the weighting matrix and \(\boldsymbol{m}(\theta)\) the moment conditions defined by \(\tilde{d}(\theta)\) and \(d\) (equation 17) for sample size \(N.\) Let the matrix \(L\) be the Cholesky decomposition of the variance-covariance matrix of moment conditions, \(S=\mathbb{E}\boldsymbol{m}\boldsymbol{m}'\) so that \(LL'=S\).⁵⁶

\[ \begin{aligned} pseudo-J_{n} & =N\boldsymbol{m}'W\boldsymbol{m}\\ & =\left (\sqrt{N}L^{-1}\boldsymbol{m}\right)^{'}L'WL\left (\sqrt{N}L^{-1}\boldsymbol{m}\right) \\ & \sim z'L'WLz,\ \ \ \ z\sim \mathcal{N}(0,I) \end{aligned} \]

The last line holds because \(\sqrt{N}L^{-1}\boldsymbol{m}\rightarrow _{d}\mathcal{N}(0,I)\) (per the central limit theorem). (If \(W\) is the efficient weighting matrix; that is, \(W=\left [\mathbb{E}\boldsymbol{m}\boldsymbol{m}'\right]^{-1}\), equation (19) boils down to the commonly used chi-squared J-statistic in Hansen (1982).)

To calculate the distribution of this statistic, we take 5,000 draws of the moment conditions (\(\boldsymbol{m}\)) from the benchmark process by repeatedly simulating data with different seeds of random variables. We use these draws to compute \(S=\mathbb{E}\boldsymbol{m}\boldsymbol{m}'\), which is then used to construct \(L\). Then, we simulate random draws from a standard normal distribution, compute pseudo-J for each draw using (19) and use these to obtain the distribution of pseudo-J (see Figure D.5a). Let \(F_{1}\) denote the CDF of this distribution.

Our test computes the probability that the pseudo-J obtained from a given specification, denoted as \(\zeta\), comes from this distribution; that is, \(1-F_{1}(\zeta)\). We apply this test to the 8 specifications in the main part of the draft and, as explained in Section 6.2, reject all of them (see the bottom panel of Table IV).

Test 2. We develop a second procedure that tests a specification against a benchmark (in this case column (8) in Table IV). First, we obtain the distribution of the objective values that can be attained from the 1-state distribution via Monte Carlo methods. Specifically, we draw 100 seeds of random variables and estimate our benchmark process by running a local minimization around the current estimates. We then use these objective values to construct the nonparametric distribution denoted by \(F_{2}\) (see Figure D.5b). Our test compares the objective value of the specification at hand (\(\zeta\)) against this distribution and reports \(1-F_{2}(\zeta)\) (see the bottom panel of Table of IV).

To sum up, our investigation reveals that the benchmark process offers the best fit to the data in a statistical sense. The data reject the hypothesis that the simpler versions analyzed in this paper can provide a similar fit.

(a) Test 1: Distribution of Pseudo J-statistic, \(F_{1}\)

(b) Test 2: The Distribution of Objective Values, \(F_{2}\)

Bootstrap Standard Errors. The last column in Table IV reports the standard errors of our benchmark process using a parametric bootstrap. In calculating the bootstrap standard errors we first simulate data using the parameter estimates reported in Table IV and create moments from simulated data. We then run the estimation for 100 different seeds of random variables by targeting these moments obtained in the previous step. For each seed of random variables we run the estimation once by employing a simplex algorithm around the original parameter estimates. We compute the standard errors using the resulting 100 parameter vectors.

11.5 Additional Estimation Results

This section contains estimation results not reported in the main text. We first report the estimates of additional specifications. Then, for all of the estimated income processes, we report some of the parameters that were not reported in Table IV. A comprehensive set of parameters for all income processes are available online for download as an Excel file on authors’ websites.

Model fit: Additional figures. Figure D.6 plots the fit on cross-sectional moments of one-year earnings changes against recent earnings, averaging over the life cycle. Figures D.7 and D.8 show how the estimated models fit the lifecycle variation in the cross-sectional moments of one- and five-year earnings changes (averaging over recent earnings). Figures D.9 and D.10 show the fit on the variation by both recent earnings and age.

(a) Std. dev.

(b) Skewness

(c) Kurtosis

(a) Standard deviation of \(\Delta _{\text{arc}}^{1}Y_{t}^{i}\)

(b) Skewness of \(\Delta _{\text{arc}}^{1}Y_{t}^{i}\)

(c) Kurtosis of \(\Delta _{\text{arc}}^{1}Y_{t}^{i}\)

(a) Standard deviation of \(\Delta _{\text{arc}}^{5}Y_{t}^{i}\)

(b) Skewness of \(\Delta _{\text{arc}}^{5}Y_{t}^{i}\)

(c) Kurtosis of \(\Delta _{\text{arc}}^{5}Y_{t}^{i}\)

(a) Standard deviation of \(\Delta _{\text{arc}}^{1}Y_{t}^{i}\)

(b) Skewness of \(\Delta _{\text{arc}}^{1}Y_{t}^{i}\)

(c) Kurtosis of \(\Delta _{\text{arc}}^{1}Y_{t}^{i}\)

(a) Standard deviation of \(\Delta _{\text{arc}}^{5}Y_{t}^{i}\)

(b) Skewness of \(\Delta _{\text{arc}}^{5}Y_{t}^{i}\)

(c) Kurtosis of \(\Delta _{\text{arc}}^{5}Y_{t}^{i}\)

Deviations of estimated models from targeted values. In the main text, we compared the model counterparts of targeted moments to the data. In this section, we show how the fit looks through the lens of our objective function in (17). Figure D.11 shows these for several key moments. More specifically, we plot equation 17 for each set of moments, with the exception of income growth moments. Recall that in our estimation we target the levels of income at various ages of different LE percentiles and not the lifecycle growth rates.

Models (3) and (4)’s fit to the data. Figures D.12 and D.13 show how models (3) and (4)—that were omitted in the main text—fit selected moments of the data.

(a) Standard Deviation

(b) Skewness

(c) Kurtosis

(d) Lifecycle Earnings Growth

(e) Nonemployment Distribution

(f) Variance of Log Earnings

(a) Standard Deviation

(b) Skewness

(c) Kurtosis

(d) Lifecycle Earnings Growth

(e) Lifetime Nonemployment Distr.

(f) Variance of Log Earnings

Additional parameter estimates. Table D.3 contains several parameters that were not reported in the main text in Table IV due to space constraints. These parameters include deterministic lifecycle profiles and the coefficients on age and income in the probability functions. Since it is difficult to interpret the magnitudes of the coefficients on shock probabilities, in Figure D.14 we show the 3D figure for the estimated relationship between the nonemployment shock probability and age and the persistent component for the benchmark specification.

Results for uniform nonemployment risk and non-Gaussian Persistent Shocks. Table D.4 shows estimates of two specifications that appeared in the previous version of the paper. Model (1b) is an intermediate case between Models (1) and (2): It adds uniform nonemployment risk to the Gaussian process in (1). The estimates from this model implies that 2.1% of workers are hit with a nonemployment shock each year, and about 42% of those experience full-year nonemployment. The nonemployment shocks soaks up some of the transitory variation in earnings, especially in the tails, in turn reducing the estimated standard deviation of \(\varepsilon\) relative to Model (1). The improvement in the objective value is quite limited (73.39 versus 74.87), mainly because this model manages to generate some excess kurtosis but very little negative skewness, and it largely misses the age and income variation in the moments. Furthermore, the estimated persistence is even higher (\(\rho =1.015\)) than in Model (1), moving the model further away from stationarity. it also implies an unusually large initial heterogeneity (\(\sigma _{\alpha}=1.26\)). As a result the fit deteriorates slightly for the impulse response and lifecycle income growth moments.

Model (3b) investigates the relative importance of having a mixture component in only persistent innovations. In that sense it serves as a bridge between models (1) and (3), which features normal mixtures for both transitory and persistent shocks. The objective value falls from 74.9 to 62.1, about two thirds of the improvement of Model (3) relative to (1), indicating that the data demands non-Gaussian features in persistent innovations more than in transitory ones.

Figure D.14 – Benchmark Process: 3-D Plot of Nonemployment Shock Probability \(p_{\nu}\)

Results for additional specifications. Table D.5 presents estimates from an earlier version of the paper that used a slightly different weighting matrix in the estimation. In particular, the weighting matrix used in this table first assigns 15% relative weight to the employment CDF moments. The rest of the moments share the remaining 85% weight according to the following scheme: the cross-sectional moments (standard deviation, skewness, kurtosis) collectively receive a relative weight of 35%, the lifecycle earnings growth moments and impulse response moments each receive a weight of 25%, and the variance of log earnings by age receives a weight of 15%.

The columns 7 to 15— report the estimates of 8 different specifications that are not reported in the main text. Columns (7), (8), and (9) are different versions of Column (5) of Table (IV). Namely, we model nonemployment probability as a logistic function of a number of combinations of individual fixed effect \(\alpha\) and persistent component \(z\) (similar to equation (8)). In particular, in column (7) nonemployment probability is a quadratic function of \(\alpha\). In column (8) nonemployment probability \(p_{\nu}\) is assumed to be a linear function of \(\alpha +z\) and age and their interaction. In column (9) nonemployment probability depends linearly on \(\alpha\) and \(z\) and their interaction.

Columns (10) and (11) are similar to our benchmark specification. Again they only differ in how the nonemployment shock probability is modeled. In column (10) \(p_{\nu}\) is a quadratic function of \(z\). In column (11) \(p_{\nu}\) depends on \(z\), \(z^{2}\) and age and the interaction of \(z\) and age.

In column (12) we introduce variance heterogeneity to the 1-state benchmark process. In particular, we allow the variance of each innovation from \(\mathcal{N}(\mu _{z,1},\sigma _{z,1}^{i})\) to the persistent component be individual-specific, with a lognormal distribution with mean \(\overline{\sigma}_{z,1}\) and a standard deviation proportional to \(\widetilde{\sigma}_{z,1}\), i.e., \(\text{log}\left (\sigma _{z,1}^{i}\right)\sim \mathcal{N}(\log \overline{\sigma}_{z,1}-\frac{\widetilde{\sigma}_{z,1}^{2}}{2},\widetilde{\sigma}_{z,1})\).

In the next income process (column (13)), in the 1-state benchmark process we incorporate age and income dependence into the mixture probability in innovations to the persistent component similar to equation (16).

In column (14) we introduce ex ante variance heterogeneity in the 2-state benchmark process. Thus the variance of each innovation from \(\mathcal{N}(\mu _{z_{j},1},\sigma _{z_{j},1}^{i})\) to the persistent component \(j\) is individual-specific, with a lognormal distribution with mean \(\overline{\sigma}_{z_{j},1}\) and a standard deviation proportional to \(\widetilde{\sigma}_{z_{j},1}\), i.e., \(\text{log}\left (\sigma _{z_{j},1}^{i}\right)\sim \mathcal{N}(\log \overline{\sigma}_{z_{j},1}-\frac{\widetilde{\sigma}_{z,1}^{2}}{2},\widetilde{\sigma}_{z_{j},1})\).

The last column (column (15)) shows the parameter estimates for the specification presented in column (3) but without imposing a lower bound for the mean of persistent shocks.

11.6 Autocovariance Structure of Earnings

Another set of moments that has been widely used for decades to estimate income processes is the autocovariance of earnings in level and changes (e.g., Abowd and Card (1989)). In this section, for completeness with the earlier literature we document these moments from data along with their simulation counterparts from our benchmark specification. Tables D.6 and D.7 show autocovariance matrix of log earnings over the life cycle in levels and changes, respectively.

Following Abowd and Card (1989); MaCurdy (1982a), in figure D.15, we show the autocovariance of 1-year log earnings growth at several lags:

\[ \operatorname{cov}\left [\log \left (y_{h+1}\right)-\log \left (y_{h}\right),\log \left (y_{h+k+1}\right)-\log \left (y_{h+k}\right)\right]\quad \text{for}\quad k>1, \]

where \(\log \left (y_{h}\right)\) is log earnings at age \(h\). As usual we only include observations that are above the minimum income threshold, \(Y_{min}.\) Left panel of Figure D.15 shows that in our data, consistent with earlier work using survey data (Meghir and Pistaferri (2004)), the autocovariance of earnings growth is small and approaches to zero quickly after a couple of lags. Our model generates a similar pattern, although the autocovariance for \(k=1\) tend to be smaller and the lags \(k>1\) approach to zero (right panel of Figure D.15). Similar patterns are also clearly seen in autocorrelations of earnings from the data and our benchmark process (Figure D.16). Both in the data and in our benchmark process, the autocorrelations of earnings growth starts around -0.20 and approaches quickly to zero after a couple of lags (in the benchmark process for \(k>1\)). The small discrepancy between our process and the data for shorter lags \(k\) can be addressed by modeling the transitory component as a moving average of order \(q\) (MA(q)) process (see Meghir and Pistaferri (2004)).

(a) Data

(b) Benchmark Specification

MaCurdy (1982a) noted that if a HIP component is present (\(\sigma _{\beta}^{2}>0\)), the autocovariance of one-year log earnings growth should turn positive at longer lags. Figure D.15 shows that in the data the autocovariance of log earnings growth does not increase above zero even after 35 years (left panel). Therefore, this test would not reject \(\sigma _{\beta}^{2}=0\) in the data. Interestingly, MaCurdy’s test reaches a similar conclusion when applied to data simulated from our benchmark specification, which features a sizable HIP component (right panel of Figure D.15).

(a) Data

(b) Benchmark Specification

Guvenen (2009) discusses why MaCurdy (1982a) may reject \(\sigma _{\beta}^{2}>0\) even if the true process features a HIP component. If the variance of the persistent component is large enough the autocovariance of earnings growth may not be significantly greater than zero even after 20–30 years. This is indeed the case in our model. The theoretical autocovariance of earnings growth given by equation 20 for the estimated parameter values of our benchmark process becomes positive, and barely so, only with a 35 year lag. Furthermore, as Karahan et al. (2019); Guvenen (2009) show, the HIP component may have a Pareto distribution, which would imply that \(\beta\) heterogeneity is negligible for most of the population, but the top of the distribution has a much larger \(\beta\) than the rest.