2.1 An Age-Dependent Income Process

Let \(y_{h,t}^{i}\) denote the log of annual earnings of individual \(i\) at age \(h\) in time \(t\). To obtain the residual income \(\tilde{y}_{h,t}^{i}\), we run cross-sectional first-stage regressions of earnings on observables. More specifically, \[ y_{h,t}^{i}=f\left (X_{h,t}^{i};\theta _{t}\right)+\tilde{y}_{h,t}^{i}. \]

The first component in this specification, \(f\), is a function of age and schooling and captures the life-cycle component of earnings that is common to all workers. \(X_{h,t}^{i}\) is a vector of observables that includes a cubic polynomial in age and education dummies for less than a high school diploma, high school diploma, and a college degree. The parameter \(\theta\) is indexed by \(t\) to allow the coefficients on age and schooling to change over time and captures changes in returns to age and schooling that took place over time.

Residual income is decomposed into a fixed effect, an \(AR(1)\) component, and a transitory component. This representation is parsimonious, yet it successfully captures the salient features of the data. Therefore, it is widely used in the literature. This paper extends the standard specification to allow for a lifetime profile in the persistence parameter, and the variance of persistent and transitory shocks:

\[ \begin{aligned} \tilde{y}_{h,t}^{i} & = & \alpha ^{i}+z_{h,t}^{i}+\phi _{t}\varepsilon _{h}^{i},\\ z_{h,t}^{i} & = & \rho _{h-1}z_{h-1,t-1}^{i}+\pi _{t}\eta _{h}^{i},\,\, z_{1,t}^{i}\sim F\left (0,\pi _{t}^{2}\sigma _{z_{1}}^{2}\right) \end{aligned} \]

Here, \(\alpha ^{i}\) is an individual-specific fixed effect that captures the variation in initial conditions such as innate ability. \(\varepsilon _{h}^{i}\) is a fully transitory component with variance \(\sigma _{\varepsilon,h}^{2}\) that encompasses both measurement error and temporary changes in earnings such as bonuses and overtime pay.²³ \(z_{h,t}^{i}\) is the persistent component of idiosyncratic income at age \(h\) that captures lasting changes in earnings such as promotions and health status. Each period, the individual is hit by a persistent shock of size \(\eta _{h}^{i}\). The magnitude of this shock is governed by the variance \(\sigma _{\eta,h}^{2}\), and the extent to which it lasts is determined by the persistence parameter \(\rho\). \(z_{1,t}^{i}\) captures the initial variation in the persistent component.⁴ The key innovation of our paper is to allow for an age profile in the variance of shocks, \(\sigma _{\eta,h}^{2}\) and \(\sigma _{\varepsilon,h}^{2}\), as well as in the durability of the persistent shocks, \(\rho _{h}\). The age profiles capture the idea that changes in earnings occur for different reasons throughout the life span.

A number of studies document the evolution of residual inequality for the United States in the past three decades (for example, Moffitt and Gottschalk (2011); Heathcote, Perri, and Violante (2010); and DeBacker, Heim, Panousi, and Vidangos (2011)). We follow Moffitt and Gottschalk (1995) and control for the change in residual inequality over time with \(\phi _{t}\) and \(\pi _{t}\), representing the time loading factors for transitory and permanent shocks, respectively.⁵

Having introduced the age-dependent income process, an immediate concern is identification. Can the variance–covariance structure of earnings data tell us how changes in earnings differ in variance and persistence over age and time together? The identification discussion allows us to connect the statistical model to the moments in the data and makes the estimation procedure meaningful.⁶ The next proposition establishes that the income process (1) is identified and provides a proof:

Proposition 1: Specification (1) is identified in levels up to the normalizations that \(\rho _{1}=\rho _{2}\), \(\pi _{1}=\phi _{1}=1,\phi _{T}=\phi _{T-1},\) and \(\sigma _{\eta,H}^{2}=\sigma _{\eta,H-1}^{2}\).

Proof: See Appendix A.

The rich panel structure of the PSID helps us to distinguish life-cycle effects from time effects: We observe individuals of a given age at different points in time and thus at a given year we observe individuals of different ages. This feature allows us to separate what effects are due to calendar time from a life-cycle phenomenon. For this particular reason, it is important to have a large number of cohorts in order to accurately separate these effects. This observation guides our sample selection process.

2.2 Sample Selection and Estimation Method

This section briefly describes the data and the variable definitions used in our empirical analysis. We use 30 waves of the PSID between 1968 and 1997. We estimate our model using both annual earnings and the average hourly wage of male heads of households.⁷ Here, we present the results for earnings data. Estimation results for wage data are reported in Appendix B.2; the results are qualitatively the same. In order to have a large number of cohorts, we include an individual in our benchmark sample if he satisfies the following criteria for three, not necessarily consecutive, years:⁸

We employ an equally weighted minimum distance estimator. We minimize the distance between the moments of the \((T\times T)\) and \((H\times H)\) empirical variance–covariance structure of residual earnings and their theoretical counterparts implied by income process (1). In particular, we target all the variance and covariance terms over age, \(cov\left (\tilde{y}_{h}^{i},\tilde{y}_{h+n}^{i}\right)\), and over time, \(cov\left (\tilde{y}_{t}^{i},\tilde{y}_{t+n}^{i}\right)\), to which at least 150 individuals contribute. This targeting strategy leaves us with 1,067 moments.⁹ To obtain the theoretical counterpart of \(cov\left (\tilde{y}_{h}^{i},\tilde{y}_{h+n}^{i}\right)\), we average \(cov\left (\tilde{y}_{h,t}^{i},\tilde{y}_{h+n,t+n}^{i}\right)\) over \(t\). Similarly, we compute the theoretical counterpart of \(cov\left (\tilde{y}_{t}^{i},\tilde{y}_{t+n}^{i}\right)\) by averaging \(cov\left (\tilde{y}_{h,t}^{i},\tilde{y}_{h+n,t+n}^{i}\right)\) over \(h\). Due to small sample considerations explained in Altonji and Segal (1996), our minimum distance estimator employs the identity matrix as the weighting matrix.

2.3 Estimation Results

In this section, we present our estimation results. The emphasis is on the existence of a nontrivial lifetime profile.

We estimate the lifetime profile of shocks and persistence in two ways. First, we estimate a nonparametric specification, that is, we do not impose any functional form on the lifetime profiles.¹⁰ Then, we assume the life-cycle profiles follow a cubic function of age and estimate its parameters. Figure 1 shows the results for persistence. The point estimates for the nonparametric estimation are plotted in dots along with the 95 percent bootstrap confidence interval in dashed lines; the point estimates are shown in Tables 6 and 7 in Appendix B.1. We employ a block bootstrap with 150 repetitions.¹¹ The results of the cubic specification are shown in the solid blue line. The parameter estimates as well as bootstrap standard errors are reported in the left panel of Table 1.

Figure 1 reveals an interesting fact: Early in life, shocks are moderately persistent. Persistence starts at around \(0.70\) for young workers and increases with age up to unity by around age 40. The differences also appear to be economically large (although a quantitative evaluation needs to await the consumption model in Section 4). For example, more than \(70\) percent of a change in a 24-year-old’s earnings dies out in five years. This number is only around \(15\) percent for a 40-year-old worker.

Figure 1: Persistence Profile

Figure 2: Variance Profile of Persistent Shocks

The variance of persistent shocks, shown in Figure 2, follows a different pattern. It exhibits a U-shaped profile over the lifetime. Early in life, shocks are larger compared with those around age 40. The variance starts around \(0.06\), decreases to around \(0.01\) by age 35, and remains roughly flat for 10 years. Shocks toward the end of the life cycle are larger and exhibit a variance of around \(0.035\). These differences again appear to be economically large; a one-standard-deviation persistent shock implies a \(26\) percent change in earnings at age 24, whereas a one-standard-deviation shock implies only a \(12\) percent change at age 40.

Figure 3: Variance Profile of Transitory Shocks

Figure 3 plots the variance of transitory shocks. There is a sizable increase early on from around \(0.03\) to \(0.07\) by age \(35\). The profile is flat afterwards. As we will discuss in Section 2.4, this nonflat profile is not statistically significant.

What features of the data give rise to the increase in persistence early in the life cycle? In Appendix A, we argue that the ratio of two-period-ahead covariance to one-period-ahead covariance at age \(h\), corrected for fixed effects, (henceforth, \(\Phi _{h}^{21}\)) yields a consistent estimate for the persistence parameter at age \(h+1\). More specifically, abstracting from time effects, (5) implies \(\Phi _{h}^{21}=\left [cov\left (\tilde{y}_{h}^{i},\tilde{y}_{h+2}^{i}\right)-\sigma _{\alpha}^{2}\right]/\left [cov\left (\tilde{y}_{h}^{i},\tilde{y}_{h+1}^{i}\right)-\sigma _{\alpha}^{2}\right]=\rho _{h+1}\) for \(h=1,\ldots,H-2\). Alternatively, the ratio of three-period-ahead covariance to two-period-ahead covariance at age \(h\), \(\Phi _{h}^{32}\), is also a consistent estimator for \(\rho\). When correcting for fixed effects, we use our baseline estimate \((\sigma _{\alpha}^{2}=0.075)\), which is in line with the estimates in the literature. The left (right) panel of Figure 4 plots the empirical counterpart of \(\Phi _{h}^{21}\) (\(\Phi _{h}^{32}\)) in the solid line, along with the estimated persistence profile in dots. The age profile of \(\Phi _{h}^{21}\) (\(\Phi _{h}^{32}\)) closely resembles the estimate of the persistence profile: It increases from below \(0.7\) to above \(0.9\).

In general, the age profile of \(\Phi _{h}^{21}\) (\(\Phi _{h}^{32}\)) depends on the level of fixed effects. To check for robustness, we plot \(\Phi _{h}^{21}\) (\(\Phi _{h}^{32}\)) for the case where there are no fixed effects \((\sigma _{\alpha}^{2}=0)\), shown in dashed lines in the left (right) panel of Figure 4. We see that the increase in persistence is robust to the variance of fixed effects, though the steepness depends on it. Note that the estimation of an upward sloping persistence profile is a result of targeting a fairly complicated variance–covariance structure. Figure 4 confirms this increase over the lifetime from a much simpler look at the data.

Some of the changes in persistence and variance that we observe might be driven by young workers who move from part-time to full-time employment or by older workers who are heterogeneous in retirement age. To control for the effect of part-time workers, we restrict our sample to only full-time workers in Appendix B.5. The conclusion for persistence is similar: Persistence is increasing. Though, the conclusion for the variance of persistent shocks is somewhat different. It is still the case that the variance of persistent shocks for younger workers is significantly larger than the variance of shocks for middle-aged workers. However, the difference in the persistence of shocks between the middle age and old age is not significant. This lack of significance suggests that some of the increase in the variance of persistent shocks for older workers is driven by early or partial retirement.

Figure 4: Ratio of Covariances

To explore the differences in age profiles between workers with and without college degrees, we estimate the age-dependent income process on a sample of workers with a college degree and on a sample of workers without one. The results are reported in Appendix B.9, Tables 16 and 17. We find that persistence for college workers is increasing over the working life, as opposed to being hump-shaped for the non-college sample. The variance of persistent shocks is decreasing for college graduates and U-shaped for those without a college degree.

2.4 Significance Tests

We now turn to the question of statistical significance, that is, we want to see whether the nonflat pattern is statistically significant. For this purpose we consider a model in which working life is divided into three stages (age intervals): young, middle, and old ages. This model restricts the persistence and variances to be constant within an interval but allows them to differ from one to another. The age bins correspond to ages 24 to 33 (young), 34 to 52 (middle), and 53 to 60 (old).¹² These intervals allow us to identify systematic differences across age intervals.

Point estimates are shown in the first three columns of Table 2 along with bootstrap standard errors in parentheses. The results, once again, point to the same life-cycle profiles of persistence and variance of shocks. We test whether the persistence profile exhibits a hump-shaped pattern. Similarly, we investigate if the variance of persistent shocks follows a U-shape. Finally, we test whether the increase in transitory shocks is statistically significant. Formally, the null hypotheses are: \(H_{0}:\rho _{1}\geq \rho _{2}\), \(H_{0}:\rho _{2}\leq \rho _{3}\), \(H_{0}:\sigma _{\eta,1}^{2}\leq \sigma _{\eta,2}^{2}\), \(H_{0}:\sigma _{\eta,2}^{2}\geq \sigma _{\eta,3}^{2}\), \(H_{0}:\sigma _{\epsilon,2}^{2}\leq \sigma _{\epsilon,1}^{2}\), and \(H_{0}:\sigma _{\epsilon,2}^{2}\leq \sigma _{\epsilon,3}^{2}\). The results are shown in the last two columns of Table 2.

Note: [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

We find that the persistence for young workers is statistically smaller than that of middle-aged workers. However, we cannot reject the null hypothesis that the persistence in the last age bin is the same as in the second. As for the variance of persistent shocks, the parameter of the second age interval is significantly lower than that of the first and third intervals (at 5 percent significance level). However, the nonflat profile in the variance of transitory shocks is insignificant with p-values of \(0.35\) and \(0.79.\)

Note that the standard errors of linear and higher-order terms in the cubic specification are large, such that one might suspect nonflat profiles implied by this specification to be insignificant (see Table 1). However, unlike quadratic polynomials, cubic polynomials can generate hump-shaped (or U-shaped) profiles for different combinations of signs of coefficients. Indeed, the correlation structure between parameters of the cubic polynomial is such that for almost all bootstrap runs, the implied persistence profile is increasing and the variance profile for persistent shocks is U-shaped. However, for the lifetime profile of the transitory variance, a significant number of bootstrap repetitions does not imply an increase over the first 10 to 15 years. In a previous version of this paper (Karahan and Ozkan (2009)), we impose a quadratic polynomial on lifetime profiles and find that both the linear and quadratic terms are significant for persistence and for the variance of persistent shocks.

Overall, these results suggest that the persistence and the variance of persistent shocks have nonflat profiles over the life cycle but not the variance of transitory shocks. Thus, from now on in our analysis, we assume that the variance of transitory shocks is age-invariant and use estimates of the cubic specification with a constant variance of transitory shocks. The results are reported in the right panel of Table 1.

2.5 Comparison with the Literature

We now compare the age-dependent process with several contenders in the literature. We start with the age-invariant version of this specification, that is, a specification consisting of a fixed effect, an \(AR(1)\) component, and an i.i.d. transitory component, where the persistence and variance of shocks are age-invariant. The age-invariant specification is widely used in quantitative models featuring income risk. In order for these cases to be comparable, we estimate this model on the benchmark sample. The estimates are shown in dashed lines on Figures 1 to 3 as well as in Table 3. Our estimate of persistence, \(0.98\), is in line with the estimates in the literature, which range from \(0.96\) to \(1.0\). It is surprising to see that for most of the life cycle, persistence in the age-dependent process is significantly lower than the estimate of persistence for the age-invariant specification. As these examples show, such differences can be economically significant.

Note: Table reports estimates of the age-invariant process. \(\sigma ^2_{z_1}\) is the initial variance of the persistent component. Note that the estimate of \(\sigma ^2_{\alpha}\) is smaller than the estimates in the literature (see Kaplan (2010)). This difference is due to the fact that the specification also allows for a nonzero initial condition in the persistent component.

We argue that targeting the lifetime profile of residual inequality in the data results in an upward bias in persistence if one does not allow for age-specific persistence and variance. For the age profile of residual inequality, we first compute \(\widehat{var(\tilde{y}_{h,t})}\) for every year \(t\) and age \(h\). An individual in year \(t\) contributes to \(\widehat{var(\tilde{y}_{h,t})}\) if he is between ages \(h-2\) and \(h+2\).¹³ We then regress these variances on a full set of age and year dummies and report the age dummies. The resulting profile is shown in Figure 5. The rise in residual inequality over the lifetime is almost linear, if not convex. The increase is particularly steep after age \(35\).

For the age-invariant process, the corresponding theoretical variances are given by \[ var(\tilde{y}_{h})=\sigma _{\alpha}^{2}+\sigma _{\eta}^{2}\sum _{j=1}^{h-1}\rho ^{2j}+\sigma _{z_{1}}^{2}\rho ^{2h}+\sigma _{\epsilon}^{2}\,, \] where \(\sigma _{z_{1}}^{2}\) represents the initial variance of the persistent component. So long as \(\rho <1\), residual inequality has a well-defined limit, say, \(var^{*}(\tilde{y})\). It can easily be shown that \(var(\tilde{y}_{h})\) will converge to \(var^{*}(\tilde{y})\) from below, in a concave fashion. The degree of concavity is more pronounced the smaller \(\rho\) is than \(1\). In the case of a unit root, the variance profile will be linear. The empirical variance profile in Figure 5 implies that the fit would be poor if \(\rho\) is too far away from \(1\). Targeting these moments puts an upward pressure on \(\rho\) and drives it closer to \(1\).¹⁴

Figure 5: Lifetime Profile of Residual Inequality

Figure 5 also plots the inequality profile implied by the cubic specification of the age-dependent process in the dash-dotted line. The model captures the increase in lifetime inequality even if persistence for young workers is very low. This increase is captured by means of the inverse relationship between persistence and the variance of labor income shocks: When persistence goes up with age, the additional increase it induces in inequality is compensated by a decrease in the variance and vice versa.

Meghir and Pistaferri (2004) estimate a process that consists of a permanent component and an \(MA(1)\) transitory component: \(\tilde{y}{}_{iht}=\alpha _{i}+p_{iht}+e_{iht},\) where \(p_{iht}=p_{ih-1t-1}+\zeta _{iht},e_{iht}=\epsilon _{iht}+\theta \epsilon _{ih-1t-1}\) and the shocks \(\zeta _{iht}\) and \(\epsilon _{iht}\) are uncorrelated at all leads and lags. This process can also generate an increase in the ratio of two-lag covariance to one-lag covariance over the life cycle (see Figure 4), which we argue is the key to the identification of the increase in persistence. For this income process, the ratio equals: \[ \Phi _{h}^{21}=\frac{cov(\tilde{y}_{h},\tilde{y}_{h+2})-\sigma _{\alpha}^{2}}{cov(\tilde{y}_{h},\tilde{y}_{h+1})-\sigma _{\alpha}^{2}}=\frac{var(p_{h})}{var(p_{h})+\theta var(\epsilon _{h})}=\frac{1}{1+\theta \frac{var(\epsilon _{h})}{var(p_{h})}}. \] Because \(var(p_{h})\) increases with age, \(\frac{1}{1+\theta \frac{var(\epsilon _{h})}{var(p_{h})}}\) is increasing over the life cycle.

However, for this specification, the ratio of three-lag covariance to two-lag covariance is constant and equal to one, which is contradictory to the data (see Figure 4): \[ \Phi _{h}^{32}=\frac{cov(\tilde{y}_{h},\tilde{y}_{h+3})-\sigma _{\alpha}^{2}}{cov(\tilde{y}_{h},\tilde{y}_{h+2})-\sigma _{\alpha}^{2}}=\frac{var(p_{h})}{var(p_{h})}=1. \]

A process containing a random walk component and an \(AR(1)\) component with age dependence in the variance of innovations (as in Baker and Solon (2003) and Moffitt and Gottschalk (2011)) can generate most of the age dependence in the variance–covariance structure that we use to identify the age profile of persistence and variance of shocks (see Figure 4). In Appendix B.10, we estimate such a process and find that it fits the variance–covariance structure both in levels and in differences as well as the age-dependent income process (see the next section for the fit of the age-dependent income process on the moments in differences). The advantage of the age-dependent specification is that it is more suitable for use in quantitative life-cycle macro models, as it requires one fewer state variable.

Guvenen (2009) argues for the existence of heterogeneity in income growth rates. The evidence he brings forward for growth rate heterogeneity is twofold: First, he points to the convexity in the variance profile of earnings, and second, he exploits the shape of higher-order covariances, which features an increase at higher lags. It is worthwhile to note that the age-dependent income process can naturally capture these features of the data without growth rate heterogeneity. In fact, the age profile of residual inequality implied by the age-dependent process is convex for most of the life cycle.

2.6 The Fit for Income Growth Rates

It is well known in the literature that the estimates of canonical income processes using levels are strikingly different than the estimates using income growth rates, suggesting misspecification of the model (see Krueger, Perri, Pistaferri, and Violante (2010)). This section investigates the performance of the age-dependent process in fitting the variance of income growth rates as well as one-lag covariances.

The theoretical moments for the age-dependent process (abstracting from time effects) are given by:

\[ \begin{aligned} var\left (\Delta y_{i,h}\right) &= \left (\rho _{h-1}-1\right)^{2}var\left (z_{i,h-1}\right)+\sigma _{\eta,h}^{2}+\sigma _{\epsilon,h}^{2}+\sigma _{\epsilon,h-1}^{2}\\ \mbox{and \,}cov\left (\Delta y_{i,h},\Delta y_{i,h+1}\right) &= \rho _{h-1}\left (\rho _{h-1}-1\right)\left (\rho _{h}-1\right)var\left (z_{h-1}^{i}\right)+(\rho _{h}-1)\sigma _{\eta,h}^{2}-\sigma _{\epsilon,h}^{2}. \end{aligned} \]

To compute the empirical counterparts, we compute these moments for all \(\left (h,t\right)\) cells and regress them on age and year dummies. The dots in the left panel of Figure 6 plot the resulting age dummies, which reveals a U-shaped profile. Theoretical moments implied by the cubic specification, shown in the solid line, show that the age-dependent process does a good job of capturing the U-shape.¹⁵ However, the same moments implied by the age-invariant process, shown in the dashed line, cannot match this profile.

To assess how the age-dependent process fits the covariance profile of differences at one lag, we plot the empirical and theoretical counterparts of this profile on the right panel of Figure 6. In the data, the covariance profile is almost flat over the lifetime, which is very similar to what is implied by the age-dependent specification. However, the empirical covariance is closer to zero. As for the age-invariant process, the covariance profile is also flat but much further away from zero compared with the age-dependent specification.

Overall, we conclude that the age-dependent income process achieves a better fit for the moments in levels without worsening the fit for the moment structure in differences. If anything, it fits the empirical variance and one-lag covariance profiles better than the age-invariant specification.

Figure 6: Variance Profile of Income Growth Rates

3.1 A Model of Job Mobility

Our economy consists of a continuum of workers endowed with one unit of time per period. Workers maximize the present value of their lifetime earnings and discount future earnings at a constant interest rate of \(r\). They are subject to death with constant probability \(\delta\). There is a continuum of firms that have access to a constant-returns-to-scale production technology. Labor is the only input to the production.

At the beginning of a period, unemployed workers meet with firms, form a match, and draw a productivity specific to the match, \(\xi\), from a normal distribution with mean \(\mu _{\xi}\) and variance \(\sigma _{\xi}^{2}\). The match-specific productivity is not known by the firm or the worker. Output of the match, \(y_{t}\), is given by \(y_{t}=\xi +\nu _{t}\), where \(\nu _{t}\) is an i.i.d. normal random variable with mean zero and variance \(\sigma _{\nu}^{2}\). After observing the output, workers and employers update their beliefs about the match productivity in a Bayesian fashion. Because the information set of the worker and the firm are the same, their beliefs are identical. By means of normality assumptions, this belief is normally distributed as well.

Let \(m_{t|t-1}\) denote the mean of the belief of a match about \(\xi\) with tenure \(t\) conditional on all of the information up to \(t-1\), and let \(1/p_{t}\) denote the variance, thereby \(p_{t}\) denoting the precision. Similarly, \(p_{\xi}=1/\sigma _{\xi}^{2}\) and \(p_{\nu}=1/\sigma _{\nu}^{2}\) denote the precision of the distribution of \(\xi\) and \(\nu _{t}\), respectively. Finally, \(\varsigma _{t}\sim N(0,1/p_{t})\) represents the deviation of the belief from the true productivity. The law of motion for these are governed by:

\[ \begin{aligned} m_{t+1|t} & = & m_{t|t-1}\frac{p_{t}}{p_{t}+p_{\nu}}+y_{t}\frac{p_{\nu}}{p_{t}+p_{\nu}}, \\ p_{t} & = & p_{\xi}+(t-1)p_{\nu},\\ \mbox{and \,}y_{t} & = & \underbrace{m_{t|t-1}+\varsigma _{t}}_{\xi}+\nu _{t}. \end{aligned} \]

For simplicity, we assume that firms pay workers their expected productivity before production takes place \(\left (w_{t}=m_{t|t-1}\right)\). After updating his beliefs, a worker decides whether to break the match. We assume that upon breaking the match, he immediately meets another employer with a new match productivity.¹⁶

3.2 A Quantitative Evaluation of the Model

In order to evaluate the performance of this model on earnings dynamics, we calibrate the model, simulate it, and then estimate the age-dependent income process using residual wages from the simulated data. Our exercise shows that the model has the potential to replicate our empirical findings for nonflat age profiles.

3.2.2 Simulation Results

We simulate 10,000 individuals, run the first stage regressions to obtain the residuals, and estimate the nonparametric specification of the age-dependent process. Figure 7 shows the results.

The top panel shows that persistence profile is increasing with age. The mechanism behind this increase can be summarized as follows. First, let us consider a worker who stays in the same job. His wage can be expressed as the sum of his previous wage and a mean-zero innovation, implying a random walk. Namely, \(w_{t}=m_{t|t-1}\). On one hand, Equation (2) implies that \(w_{t+1}=w_{t}\frac{p_{t}}{p_{t}+p_{\nu}}+y_{t}\frac{p_{\nu}}{p_{t}+p_{\nu}}=w_{t}\left (\frac{p_{t}}{p_{t}+p_{\nu}}+\frac{p_{\nu}}{p_{t}+p_{\nu}}\right)+\frac{p_{\nu}}{p_{t}+p_{\nu}}\left (\varsigma _{t}+\nu _{t}\right)=w_{t}+\chi _{t},\) where \(\chi _{t}\sim N(0,\frac{p_{\nu}}{p_{t}\left (p_{t}+p_{\nu}\right)})\). On the other hand, job switchers always get the unconditional mean of the match-specific component \(\mu _{\xi}\), implying a low correlation between current and future wages. Therefore, persistence is lower for them. The persistence of income changes in the overall sample is a combination of the persistence of these two subsamples. Over the working life, the fraction of switchers declines with age because workers settle into more-productive jobs as they age.¹⁸ Thus, they are less likely to switch to other jobs, which implies a rising persistence profile. Furthermore, the bottom panel of Figure 7 shows a decreasing variance profile for persistent shocks because both the number of stayers increases and the variance of innovations to wages declines with tenure for stayers. Namely, the variance of \(\chi _{t}\) is decreasing because \(p_{t}\) is increasing in \(t\).

Figure 7: Simulation Results for the Learning Model

This section presented a theoretical background for our empirical findings. We have illustrated that a very stylized model of learning (similar to that of Jovanovic (1979)) implies an increasing persistence profile and a decreasing variance over the working life. The mechanism discussed here is known to have empirical relevance (see Flinn (1986)). Therefore, we also view these results as complementary to our econometric analysis in Section 2, providing independent evidence for the age profiles.

4.1 Consumption Insurance against Labor Income Shocks

We now turn to the differences in consumption insurance induced by the age-dependent and the age-invariant processes. Our methodology is similar in spirit to Kaplan and Violante (2010). For each specification, we calibrate the discounting factor, \(\beta\), to match an aggregate wealth-to-income ratio of \(3\). We compute the degree of consumption insurance at age \(h\) as: \[ \phi _{h}=1-\frac{cov(\Delta c_{h}^{i},\eta _{h}^{i})}{var(\eta _{h}^{i})}, \] where \(\eta _{h}^{i}\) is the persistent shock faced by worker \(i\) at age \(h\). This equation measures the amount of change in the persistent component that does not translate into consumption growth.

Figure 8 plots \(\phi _{h}\) over the life cycle for both processes in the NBC and ZBC economies. It is clear that persistent shocks from the age-dependent process are better insured throughout the lifetime. In the NBC economy, on average, \(56\) percent of persistent shocks are insured under the age-dependent process, whereas the corresponding number for the age-invariant process is only \(40\) percent.

Strikingly, most of this difference comes from younger workers. In the age-invariant process, the profile of insurance tracks the profile of assets. Persistence is constant and high throughout the working life, and agents abstain from borrowing in response to a highly persistent bad income shock. Therefore, insurance against such shocks is mainly through assets, which there are typically fewer of in young households. The increase in assets over the lifetime allows workers to better handle highly persistent shocks, which results in an increasing consumption insurance profile.

However, for the age-dependent income process, insurance for young households is much larger, precisely because the \(AR\left (1\right)\) component is moderately persistent for them. Consumption insurance first decreases until middle age and then increases until the end of working life. The U-shape is due to the combination of two opposing effects: i) households get richer and can better insure themselves against persistent shocks as they age, and ii) shocks become more persistent, making self insurance harder for households. In the initial phase of the life cycle (ages 24 to 40) the latter effect dominates the former, and insurance decreases with age. Later on, assets are large enough that they compensate the increase in persistence. Thus, insurance increases with age.

Insurance decreases for both specifications once we impose no borrowing (right panel of Figure 8), but it is still larger under the age-dependent income process, though by a smaller margin. The decrease in the gap is due to young households, who can insure against moderately persistent shocks via borrowing in the NBC economy.

Figure 8: Insurance Against Persistent Shocks

4.2 Welfare Costs of Earnings Risk

We now turn to welfare costs of idiosyncratic risk under the two processes. Recall that the low levels of persistence under the age-dependent process are compensated by the larger variance of shocks (Figures 1 and 2). On the one hand, lower persistence implies better insurability. On the other hand, larger variance implies more instability. In order to evaluate this tradeoff quantitatively, we compute the fraction of lifetime consumption that an individual would be willing to give up in order to live in an economy without earnings risk.²⁰

Note: Column (1) shows the welfare cost of total idiosyncratic risk including risk due to fixed effects, initial variation in persistent component \((z^i_1),\) as well as life-cycle shocks. Column (2) presents welfare costs of life-cycle shocks, that is, shocks accumulated after workers enter the labor force. Column (3) shows the insurance coefficient against persistent shocks.

The upper panel of Table 5 shows the results for the NBC economy. Column (1) shows the total welfare cost of idiosyncratic risk, that is, welfare costs of income risk for a person who has not yet entered the labor force. The age-dependent income process delivers higher welfare costs. These higher welfare costs are not surprising as the level of inequality both in the beginning and at the end of the life cycle is higher for the age-dependent specification (see Figure 5). What we want to emphasize is not the welfare cost of inequality but the welfare cost of the rise in inequality over the life cycle, that is, the welfare cost of labor income shocks accumulated over the life cycle. For this purpose, column (2) reports the welfare costs of labor income shocks for a person with the average fixed effect \((\alpha ^{i}=0)\) and the average initial persistent component \((z_{1}^{i}=0)\). In the NBC economy, a worker subject to the age-invariant income process is willing to give up \(4.76\) percent of his consumption every period in order to have perfect insurance against labor income shocks. The same number is only \(3.13\) percent for a worker subject to the age-dependent specification.²¹

The bottom panel of Table 5 presents the results for the ZBC economy.²² As expected, welfare costs have increased compared with the NBC economy for both specifications. Note that the increase in the welfare cost of labor income shocks accumulated over the lifetime is larger for the age-dependent process (from \(3.13\) percent to \(5.80\) percent for the age-dependent process and from \(4.76\) percent to \(6.80\) percent for the age-invariant process). Thus, the difference between the two processes decreases from \(1.63\) percent in the NBC economy to \(1\) percent in the ZBC economy (see column 2). We conclude that the welfare cost of labor income risk is substantially different for the two processes; however, the margin depends on the amount of borrowing allowed.

7.1 Parameter Estimates for the Nonparametric Specification of the Age-Dependent Income Process

Here, we present the point estimates of the nonparametric specification as well as their bootstrap standard errors. These results are plotted in Figures 1 through 3.

7.2 Results with Wage Data

Here, we report the estimation results of the age-dependent income process on wage data. Wage is defined as annual labor income divided by hours worked. Table 8 reports the results.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \epsilon ^i_{h,t},\) where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t},\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h}),\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.3 Transitory Component Modeled as \(MA(1)\)

Table 9 reports the results of the estimation of a process where the transitory component is modeled as an \(MA(1)\) process instead of a fully transitory process.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \varepsilon ^i_{h,t},\) where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t},\varepsilon ^i_{h,t}=\epsilon ^i_{h,t}+\theta \epsilon ^i_{h-1,t-1}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.4 External Estimate of Measurement Error

Table 10 reports the results of the estimation of a process where the transitory component is modeled as an \(MA(1)\) process instead of a fully transitory process. We also allow for measurement error, which is modeled as a fully transitory component. Its variance is assumed to be constant over the working life and is taken from Bound, Brown, Duncan, and Rodgers (1994).

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \varepsilon ^i_{h,t} + \nu ^i_{h,t}\), where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t},\varepsilon ^i_{h,t}=\epsilon ^i_{h,t}+\theta \epsilon ^i_{h-1,t-1}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\), \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\), and \(\nu ^i_{h,t}\) is classical measurement error with \(\sigma ^2_{\nu}=0.015\) (see Bound, Brown, Duncan, and Rodgers (1994)). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.5 Full-Time Sample

This section investigates if the finding of age-dependent persistence and variances are due to young workers frequently changing between full-time and part-time work. For this purpose, we restrict our sample to contain only full-time workers. Table 11 presents the results.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \epsilon ^i_{h,t}\), where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.6 Sample with Consecutive Observations

In this section, we require individuals in our dataset to have at least three consecutive income observations, subject to conditions explained in Section 2.2. Table 12 contains the estimation results of the age-dependent income process.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \epsilon ^i_{h,t}\), where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.7 Sample with At Least 10 Years of Observations

To further test the robustness of our results, we require individuals in the sample to have at least 10 (not necessarily consecutive) observations, subject to conditions explained in Section 2.2. Table 13 contains the estimation results of the age-dependent income process.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \epsilon ^i_{h,t}\), where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.8 Estimates of the Age-Dependent Process with Some Age-Invariant Parameters

7.9 Results for College and Non-College Samples

In this section, we investigate what the profiles of persistence and variance of shocks look like for households with a college degree and those without one. Tables 16 and 17 report the results for these two samples.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \epsilon ^i_{h,t}\), where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

Note: Estimated process: \(\tilde{y}^{i}_{h,t} = \alpha ^{i}+z^{i}_{h,t} + \epsilon ^i_{h,t}\), where \(z^i_{h,t}=\rho _{h-1} z^i_{h-1,t-1}+\eta ^i_{h,t}\), \(\eta ^i_{h,t}\sim (0,\sigma ^2_{\eta, h})\) and \(\epsilon ^i_{h,t}\sim (0,\sigma ^2_{\epsilon, h})\). [24,33], [34,52], [53,60] are age intervals in which the persistence and variances are assumed to be constant. Point estimates are shown in the first three columns along with bootstrap standard errors in parentheses. The last two columns report p-values for bootstrap significance tests.

7.10 An Income Process with a Random Walk and an \(AR(1)\) Component

A process containing a random walk component and an \(AR(1)\) component with age dependence in the variance of innovations (as in Baker and Solon (2003) and Moffitt and Gottschalk (2011)) can generate most of the age dependence in the variance–covariance structure that we use to identify the age profile of persistence and variance of shocks (see Figure 4).

Specifically,

\[ \begin{aligned} \tilde{y}_{h}^{i} & = & \alpha _{i}+p_{h}^{i}+z_{h}^{i}+\nu _{h}^{i},\\ p_{h}^{i} & = & p_{h-1}^{i}+\zeta _{h}^{i}, \\ z_{h}^{i} & = & \rho _{h-1}z_{h-1}^{i}+\eta _{h}^{i}, \\ \mbox{and \,}\eta _{h}^{i} & \sim & N\left (0,\sigma _{\eta,h}^{2}\right)\,\,\zeta _{h}^{i}\sim N\left (0,\sigma _{\zeta,h}^{2}\right),\,\,\nu _{h}^{i}\sim N\left (0,\sigma _{\nu}^{2}\right). \end{aligned} \]

The ratio of n-year ahead covariance to (n-1)-year ahead covariance is given by:

\[ \bar{\Phi}_{h}^{n,n-1}=\frac{cov(\tilde{y}_{h}^{i},\tilde{y}_{h+n}^{i})-\sigma _{\alpha}^{2}}{cov(\tilde{y}_{h}^{i},\tilde{y}_{h+n-1}^{i})-\sigma _{\alpha}^{2}}=\frac{var\left (p_{h}^{i}\right)+\rho ^{n}var\left (z_{h}^{i}\right)}{var\left (p_{h}^{i}\right)+\rho ^{n}var\left (z_{h}^{i}\right)}=\frac{1+\rho ^{n}\frac{var\left (z_{h}^{i}\right)}{var\left (p_{h}^{i}\right)}}{1+\rho ^{n-1}\frac{var\left (z_{h}^{i}\right)}{var\left (p_{h}^{i}\right)}}=\frac{1+\rho ^{n}\kappa _{h}}{1+\rho ^{n-1}\kappa _{h}}. \]

As long as \(\rho <1,var\left (z_{h}^{i}\right)\) will increase in a concave fashion, whereas \(var\left (p_{h}^{i}\right)\) increases linearly. Therefore, \(\kappa _{h}\) will be decreasing in \(h\), thereby generating an increasing profile in the ratio in (9). Note that this result holds for \(n=1\) and \(n=2\) as well, which is consistent with what we report in Figure 4.

For estimates of this process, persistence is moderate. Therefore, the concavity of \(var\left (\tilde{y}_{h}^{i}\right)\) will be quite pronounced as long as \(\sigma _{\eta,h}^{2}\) and \(\sigma _{\zeta,h}^{2}\) are constant over \(h\). Therefore, in order to give this specification a chance so it matches the almost convex increase in residual inequality in the data, variances of the permanent and transitory shocks should be allowed to vary by age. We have estimated the income process in (8) with age-varying variances on our benchmark sample. In Figure 9, we plot \(\bar{\Phi}_{h}^{21}\) for these estimates (shown in the diamond-marked line). The process in (8) fits the ratio of covariances equally well.

We now turn to the fit of the process in (8) on the covariance structure of income growth rates. Figure 10 plots the life-cycle profile of the variance of income growth rates in the left panel and the covariance at 1 lag in the right panel, as well as their data counterparts. Overall, the fit of the age-dependent income process is as good as the fit of the process containing a random walk and an \(AR(1)\) component, provided that one allows for age dependence in the variance of innovations to the random walk and \(AR(1)\) components. We conclude that the age-dependent specification is a serious contender of this process. An advantage of the age-dependent income process over the process in (8) is that it is more suitable for use in the life-cycle macro models, as it requires one fewer state variable.

Figure B.1: Ratio of Covariances: $ _ h ^ 21 = cov( y _ h ^ i , y _ h+2 ^ i )- _ ^ 2 cov( y _ h ^ i , y _ h+1 ^ i )- _ ^ 2 $, \(h=23, ,50\)

Figure B.2: Variance Profile of Income Growth Rates

8.1 Value Functions

Let \(V_{h}\left (a_{h}^{i},\alpha ^{i},z_{h}^{i},\epsilon _{h}^{i}\right)\) denote the value function of an agent at age \(h\leq R\), with asset holdings \(a_{h}^{i}\), fixed effect \(\alpha ^{i}\), persistent component of labor income \(z_{h}^{i}\), and transitory component of income \(\epsilon _{h}^{i}\). The agent’s programming problem can be written recursively as

\[ \begin{aligned} V_{h}^{i}\left (a_{h}^{i},\alpha ^{i},z_{h}^{i},\epsilon _{h}^{i}\right) &= \max _{a_{h+1}^{i},c_{h}^{i}}u\left (c_{h}^{i}\right)+\beta EV_{h+1}\left (a_{h+1}^{i},\alpha ^{i},z_{h+1}^{i},\epsilon _{h+1}^{i}\right)\\ s.t. & & (\mbox{3})\,\,\mbox{and}\\ \log \left (y_{h}^{i}\right) &= \beta _{0}+\beta _{1}h+\beta _{2}h^{2}+\beta _{3}h^{3}+\alpha ^{i}+z_{h}^{i}+\epsilon _{h}^{i}\\ z_{h+1}^{i} &= \rho _{h}z_{h}^{i}+\eta _{h}^{i}\\ a_{h+1}^{i} & \geq & -\bar{A}_{h+1}. \end{aligned} \]

Upon retirement, the agent has a constant stream of income from social security and faces no risk. His problem is given by:

\[ \begin{aligned} V_{h}^{i}\left (a_{h}^{i},\alpha ^{i},z_{R}^{i}\right) &= \max _{a_{h+1}^{i},c_{h}^{i}}u\left (c_{h}^{i}\right)+\beta V_{h+1}\left (a_{h+1}^{i},\alpha ^{i},z_{R}^{i}\right)\\ s.t. & & (\mbox{3})\\ & lny_{h}^{i}=\Phi (\alpha ^{i},z_{R}^{i})\\ & a_{h+1}^{i}\geq -\bar{A}_{h+1}. \end{aligned} \]