1. Introduction

Explanations of whether or how children's economic outcomes follow those of their parents, such as attaining similar levels of education, moving into similar occupations, or ultimately achieving similar levels of income, have received extensive attention. Few studies, however, have analyzed differences in mobility between different populations, or the structure of distributional differences across races. The nonparametric methods used in this paper serve a useful role for analyzing small samples, as in the distributions pulled from the PSID, and they also provide elements of objectivity to ongoing discussions regarding the presence and structure of nonlinearities in intergenerational outcomes. Rather than arguing for or against a specific nonlinear functional relationship in the development ofintergenerational earnings, inference via nonparametric methods allows the data to speak for itself, without researcher imposed assumptions on intergenerational structure.

To address this, research in this paper measuresdifferences in intergenerational earnings mobility ofAfrican-American and white sons across the entire distribution of fathers' economic outcomes and statistically tests between-race and within-race differences in the distribution of outcomes. In general, results indicate that African-American sons not only have lower conditional means than their white peers from similar economic backgrounds, but also that the distribution of outcomes for African-Americans repeatedly skews towards lower earnings than the outcomes of whites. The evidence suggests that African-Americans do worse than whites except among those born in the highest deciles. Furthermore, within-race comparisons of white earnings demonstrates the stochastic dominance of white sons whose fathers' earned in higher deciles over those whose fathers earned in successively lower deciles. The implication is that whites exhibit very little intergenerational mobility. This should not surprise us, since it appears to support what we already believe about slowing rates of convergence. In contrast to this, this same result does not exist for the distributions of African-Americans except for some deciles in the middle class. African-Americans in general show extensive within-race intergenerational mobility (i.e. convergence), a result that previous research in this field has not confirmed. Results throughout the paper indicate that earnings outcomes for the two races follow two distinctly different stochastic processes. While primarily descriptive in nature, the evidence is compelling enough to argue that simple point estimates of the degree of intergenerational convergence are incomplete and indeed often misleading.

The remaining sections address related literature in section 2 and transition probabilities in section 3. Issues with regard to data, including measurement error corrections, follow in section 4. Section 5 includes the analysis of intergenerational distribution dynamics. These include discussions ofstationary distributions, the stochastic dominance ofdecile-conditioned white distributions over African-American distributions, and within-race comparisons of decile conditioned distributions. Section 6 provides a brief conclusion.

2. Related Literature

Much of the empirical work on intergenerational mobility extends back to the seminal works of Sir Francis Galton[1] and[2], who hypothesized on the parental transmission of attributes such as schooling, height and earnings to children. The Galton framework of intergenerational mobility has historically estimated some form of the conditional mean. The methodology of regression analysis traces its roots to Galton's studies and attests to the age and evolution of this empirical question.

The Galton framework pursues an answer of the existence or non-existence of convergence in economic outcomes, typically modeling the intergenerational relationship as

(1)

where y_0,i, the economic outcome of parents, has mean μ₀ and variance σ². The regression error, ε, is independent of y₀ and has zero mean and variance σ² . β₁ equal to one implies perfect immobility in outcomes across generations. If β₁ equals zero, adult-child outcomes are independent of parent outcomes. The empirical work in[3] and[4] improved estimates of correlation across generations, attempting to reduce data measurement errors due to transitory income. Their research generally predicts an elasticity of earnings across generations between 0.3 and 0.5. While this level of analysis tells us something about mobility, the interpretation of the results is limited, since the conditional mean only identifies average outcomes for the entire distribution. More recent work in[5] and[6] attempt to understand whether convergence to the mean occurs at similar rates across the earnings distribution via nonparametric or semi-parametric methods. The results of these papers argue for researchers to model intergenerational relationships with specifications more complex than the linear model allows. Whites and African-Americans may actually follow different stochastic processes.

Through more comprehensive models, economists have expanded on the Galton idea, or at least built more structure around it. The framework established in[7] and[8] formalized Galton's ideas and included theoretical arguments. In the Becker and Tomes model, particular factors such as government expenditures and family background may influence intergenerational mobility. Parents maximize utility subject to a family budget constraint, ultimately leading to a result implying a parental tradeoff between own consumption and investment in children. One of the key implications is that exogenous increases in the potential income of children causes parents to substitute towards their own consumption and away from investment in children. This increase can come from a random shock to the child's ability, an increase in government subsidies (e.g. welfare or educational subsidies) or income due to marriage. The partial effects of these exogenous increases negatively effect investment in children, ultimately leading to the theoretical conclusion of regression to the mean.

These earlier models focus primarily on intra-family feedbacks and individual incentives to generate the results, while neighborhood spillovers are essentially ignored. Theoretical work developed in[9],[10],[11],[12] and[13] include concepts of neighborhood spillovers in the intergenerational model, arguing that the development of poverty traps and persistent inequality across generations follows from the set of intra-neighborhood spillovers, family outcomes, peer effects and the choices of the children themselves.

This more recent strand of research implies through the mere presence of a poverty trap that nonlinear relationships exist across the distribution of outcomes. Families within the distribution follow different paths, dependent on family and neighborhood characteristics. Mobility may exist, but only within subsets of the distribution, e.g. the poor with the less poor, the rich with the less rich, the upper middle class with the lower middle class, and so on. This implies that it makes little sense for empirical work to assume that all systems obey the same process. If a nonlinear process dominates the distributions of conditional outcomes, then the restrictive assumption of a parametric linear model will lead to somewhat incomplete if not flawed conclusions regarding economic mobility.

Recent empirical work indicates the need to account for non-linearities in economic mobility. Studies such as[4],[14] and[15] each provide evidence of nonlinearities in economic outcomes. Reference[15] in particular uses nonparametric regression analysis finds evidence that indicates that parental earnings in the middle of the distribution have the greatest intergenerational effect Previously mentioned work in[5] and[6] extended this research into the comparison of racial differences in nonlinear contexts. In particular, reference[5] uses regression tree analysis and highlights how group membership plays an important role in the determination of income convergence. Essentially[5] finds little mobility for families drawn from both high and low incomes, while the middle exhibits some mobility. Race plays an additional role in the analysis at the bottom of the distribution, where African-Americans from poor backgrounds may exhibit a negative elasticity in earnings. Extending this into general nonparametric regressions,[6] finds a "...relativeinsensitivity of own income to parental income for blacks at the very bottom of the income distribution..."

3. Transition Probabilities

While the presentation of transition probabilities is not new to studies of intergenerational mobility, analysis rarely has extended beyond observation of the raw transition data matrix and has not provided supporting statistical analysis or inference. References[4],[14] and[16], for example, all show discrete transition matrices that indicate little mobility exists at the top of the earnings distribution, but we are unable to determine whether or where the distributions exhibit statistical differences. Statistical evidence remains confined to parameterized results. For example,[14] analyzes a cubic model and rejects of the hypothesis for a linear specification, also indicating that the lowest and highest quartiles have the least mobility for sons. Reference[4] also finds evidence that appear to refute arguments of substantial mobility, finding some upward mobility from the lower parts of the distribution, but top-down mobility appears less substantial.

While not analyzing intergenerational mobility,[17] analyzes a transition probability matrix generated through a stochastic kernel and indicates that countries diverge over time into polarized clubs of economic growth. This method of analyzing transition matrices removes arbitrary effects from discretizing the transition matrix and allows for the discovery of emerging twin peaks or convergence clubs in the distribution of cross-country economic growth.

Reference[17] use of empirical methods that analyzes distributional anomalies in a continuous framework serves as a starting-point to help understand the evolution of family outcomes across generations. Stochastic kernels visually help indicate the presence of poverty traps and stratification, although the usefulness of this visual inspection may be limited. If stratification exists, members of different social classes will converge to different distributions. We can, however, use the nonparametrically estimated transition matrices to construct stationary distributions and the intergenerational stochastic processes for both whites and African-Americans.

The methodology of analyzing stochastic kernels has roots in earlier models of discrete transition matrices. Although not always explicitly stated in studies of intergenerational mobility that analyze transition matrices, each build on the theory of Markov chains in discrete frameworks Examples discussing proofs and basic assumptions comprehensively appear in[18] and[19]. In this discrete framework, p_jk is a measure of the probability that the adult-child of a parent from class j is in class k. Since the system is closed, and this is a probability measure, it must be that Σ p_jk =1 across all k-classes. If we consider only families in which one son is born to one father, the class history of the family will be a Markov chain. In a real population, this is an implausible assumption where some fathers have more than one son and some fathers have none. If population size remains constant over time, the assumption that on average, each father has one son, or each married couple has two children does not seem implausible. Results from this represent an average set of outcomes for a society with zero population growth.

The continuous version of the model outlined in[17] and[20] for understanding distribution dynamics defines the cross-section distribution at time g as F_g, with a probability measure of ϕ_g. A sequence of probability measures, {ϕ_g: g ≥ 0}, represents the stochastic process, where the law of motion describes the evolution of the distribution into either a point mass or stratified groups of observations.

The dynamics of the stochastic process, {ϕ_g: g ≥ 0}, satisfy the difference equation:

(2)

where P^* maps the product of probability measures and disturbance terms in the past period to the probability measures in the current period, and P^*_ug absorbs the disturbance term into the operator. Since we only observe the probability measures in each period separately, the underlying process governing these distributions remains unknown, but recovering an estimate of P^* from the data, can inform about the behavior of this underlying stochastic process. More comprehensive measure theoretic treatments can be found in[17] and[20].

4. Data and Measurement Issues

A significant barrier to the accurate analysis of economic mobility in earlier studies has been the lack of quality measures for economic outcomes. When regressors contain measurement error in a linear model, attenuation bias distorts estimates of the coefficient of interest. In the context of linear intergenerational mobility, attenuation bias implies an overestimation of economic mobility.

In the presence of measurement error in the regressor, researchers do not observe the true value of parents economic outcomes, y_0,i. Instead, one observes y_0,i^* = y_0,i + u_i. (Measurement error in y_1,i appears in the error of (1) and will bias variance estimates, but not the coefficient on intergenerational persistence.) Regressing y₁ on y₀^* through ordinary least squares generates a consistent estimate of β₁^*=κ β₁ where κ, often called the attenuation factor, is less than one. (Most recent graduate econometrics textbooks contain a proof.) This implies that the coefficient for mobility attenuates towards zero (i.e. a linear regression overestimates mobility).

The relaxation of the linearity assumption of intergenerational transmissions has led to mixed results in previous research. Responding to criticisms that the models contain intrinsic nonlinearities, many papers including notably[4] and[14] incorporate quadratic or higher-order terms in estimation. By averaging data over several years to reduce attenuation bias, these studies find that higher-order effects often exist, but the results are not conclusive or ubiquitous. Reference[21] also suggests a lack of evidence pertaining to nonlinearities, and completely discounts nonlinear intergenerational models. The discrepancies, however, may be attributable to a simpler result.

While[3] and[4] formalize the theoretical effects of measurement error in linear models, the discussion has not received the same attention in nonlinear contexts. To clarify, if a model defines regressors as polynomials, the full effect of using OLS on the coefficients is not simply κ. To see this, consider the quadratic example

(3)

In this specification, we can show that measurement error flattens the curvature of the estimated function. To illustrate, consider the quadratic model shown in (3), where the researcher observes y_0,i^*. If y_0,i^*= y_0,i+u_i, then the observed square of this must be y_0,i^*2=y_0,i²+2 y_0,i u_i +u_i². The variance of the measurement error in the squared explanatory variable is functionally related to the true value of the explanatory variable. Given this result, OLS will generate inconsistent coefficient estimates in a polynomial specification, and underestimate the curvature of the true model. (A proof of these results appears in[22].) This could lead to false negatives about the presence of nonlinearities, and could also lead to the incorrect specification of a linear model. Even if one avoids this mistake and still includes a polynomial term, the observed turning point of the function with regressors measured with error may be higher or lower than the true turning point of the quadratic function. Quite possibly, measurement error and its effect on the consistency of estimators in polynomial-specified regressions contributed to discrepancy in results.

4.1. Average Observed Outcomes

Since the work of[4] and[5], researchers have dealt with measurement errors in economic outcome variables such as earnings and income through a simple technique of averaging the outcomes over as many years as possible given limitations of data. Average earnings are calculated from T_i years in which we observe data for an individual the ages of 24 and 65. This generates the simple average

(4)

where T_i depends on data availability for each individual.

The problems of censoring and measurement error in the sons' earnings variable remain an issue worth remembering in the interpretation of results. Reference[23] addresses the problem of censoring, or exogenous selection of labor force participation in her analysis of non-parametric bounds. The research here will not address the issue of exogenous selection with respect to observed outcomes. To mitigate problems associated with measurement error and exogenous selection, an alternative measure of age-adjusted, career and education controlled earnings is generated for each worker. The discussion of this measure follows in section 4.2.

4.2. Average Age-Adjusted Outcomes

An alternative method for reducing measurement error follows from the empirical model in[23], who estimates life-cycle earnings of fathers using a fixed effect model. Following more closely the theory in the labor economics literature, especially the work of[7] and[24],[23] estimates the age-earnings profiles for various occupations to capture life-cycle parental earnings outcomes, noting that that career choice and timing of human capital investments may affect the general shape of the life-cycle profile. An important contribution of this work is the use of a fixed effect model to capture the heterogeneity across individuals and occupations in age-earnings profiles. Building on[23], I include a simple error specification change which allows for auto-correlated errors in life-cycle outcomes and facilitates correct polynomial specifications. To address problems related to auto-correlation in the estimation of life-cycle earnings, age-outcome profiles for each individual and occupation are estimated using the Baltagi-Wu estimator as outlined in[25]. Not only does this address the different timing of investment decisions across occupations, levels of education and individual specific abilities, but also allows for the auto-correlation in transitory shocks that contribute to measurement error. Occupations include: professionals, managers, clerical/sales, craftsmen, operative, laborer and other. The most general c-order polynomial model, estimated separately for each occupation and generation follows as:

, (5)

for all i in each occupation partitioned on Ω_ω. Polynomials are estimated up to a measure given by c. The error term given by ξ is an AR(1) process, while the parameters defined υ are fixed effects for each individual. (Hausman specification tests not reported indicate a preference for fixed effects over random effects models for each occupation-specific regressions.) The estimator outlined by[25] allows for unbalanced panels and unequally spaced observations. Furthermore, separate life-cycle regressions are estimated for partitions by one-digit SIC occupation code and levels of education for each generation. Using estimated parameters from these models, an age-earnings profile for each individual is constructed over the 40 year period covering the ages of 25-64. The average of this 40 year estimate constructs the measure of age-adjusted outcomes.

To formalize this measure, let each individual in the ω^th group have a fixed effect, υ_i with C polynomial coefficients (attached to age-polynomials given from the ω^th regression). This reduces to the age adjusted measure defined of i’s earnings:

(6)

4.3. PSID Data

Data used in the subsequent analyses comes from both the SRC cross-section sample and SEO component of the Michigan Panel Survey of Income Dynamics (PSID) as reported longitudinally from 1967-2003. The PSID includes a vast amount of data collected from family units and individual members of families over the course of a lengthy period of time, beginning in 1968 and continuing until the present. The PSID is particularly useful for analyzing intergenerational mobility on a national scale, since it follows adult children over consecutive years after they leave their parents' households. Ideally it also measures outcomes across generations, which can facilitate the reduction of biases generated from using only a single-year of data. Exploiting this panel structure can generate two alternative measures of earnings. The first method uses the previously discussed simple average of observed annual data, equation (4). The second measure is constructed from age-adjusted predictions as shown above in equation (6). Employment and subsequent data for individuals younger than 25 years and older than 64 is not used. Monetary values are expressed in 1982-1984 dollars measured by the consumer price index for all items. Annual earnings include self-reported money income from wages and salaries earned on all jobs by the individual. For earnings estimates, I exclude individuals with non-positive outcomes. All sons, including those from multi-son households are included in the analysis. Approximately 43% of sons in the sample come from multi-son households Given estimation issues for female earnings associated with fertility choices and sample selection, only the outcomes of sons are analyzed in this paper.

To generate age-adjusted profiles, variables are constructed as the natural log of observed earnings for individuals at each point in time. Estimates are generated for 13 separate regressions for fathers' life-cycle earnings based on occupation and education and 7 separate regressions for sons' earnings based only on occupation partitions. Among sons' life-cycle earnings regressions, no statistical difference in parameters exist based on education levels within any of the occupations, but differences exist between occupations. Individuals who indicate more than one occupation over their careers were defined within occupation in which they spent the most years given by the data. Specification tests indicate that polynomial age regressors extend to C=4 for most occupations. Results from life-cycle regressions used to generate (7) are available from the author upon request.

Table 1. Conditional Means of Sons’ Earnings

Using equation (6), the age-adjusted measures of life-cycle earnings is

(7)

Using (4), the measures of observed life-cycle earnings are constructed by the simple average

(8)

Sample means of (7) and (8) appear in Table 1. These are conditioned on the deciles of fathers’ earnings from the combined sample of both whites and blacks. Comparisons across both measures indicate that white sons earn more across the entire distribution of their fathers’ earnings.

5. Intergenerational Distributions

While simple comparisons of sample means or even regression analysis can reveal information about the conditional mean in the outcomes of a population, the transition probability matrix provides a more generalized description of outcomes. With even more detail, an analysis of distributions generated by a stochastic kernel can reveal whether the children of the rich and poor each face the same distribution of outcomes or fall into poverty traps. The empirical methodology employed for this analysis will estimate the conditional kernels for sons’ earnings

(9)

where

(10)

and

(11)

K and h represent the Gaussian kernel and Sheather-Jones bandwidth selector, respectively. (The results presented throughout this paper are not sensitive to kernel choice, and slight differences in bandwidth selectors have negligible effects on the interpretation of future results. See [26],[27],[28] and[29] for development and discussion of the superiority of the Sheather-Jones bandwidth selectors.) The construction of these conditional kernels is similar for both life cycle measures outlined by equations (7) and (8).

Extending this vein of research, the results in this section compare the stationary distributions from race conditioned continuous transition matrices and also test stochastic dominance for the decile-conditioned distributions of white outcomes over African-American outcomes. Both will lend support to the hypothesis that nonlinear intergenerational differences exist in economic outcomes. African-Americans have worse distributional outcomes than whites. These differences are exhibited both in the decile-based comparisons and tests that follow.

5.1. Testing for Stochastic Dominance

Analyzing stochastic dominance of the current intergenerational outcomes also can lead to further insight into the structure of nonlinearities. A substantial amount of work on inequality and poverty has tested for stochastic dominance across two independent samples of economic outcomes, usually comparing income distributions at different points across time. None have analyzed outcomes from an intergenerational perspective.

Following from[30], tests for stochastic dominance begin with a definition of an operator, Ƒ₁(y, F), that integrates F to order k-1. For first orders of operation we have

(12)

To compare two distributions F_M(y) and F_N(y), it is further assumed that F_M and F_N have a common and continuous support on y.

Hypothesis tests of stochastic dominance within the context of this paper are conducted within fathers' outcome deciles (0-10, 10-20, etc.). Repeating this for different deciles, and therefore different ranges of fathers' outcomes, reveals where and to what extent racial differences in the intergenerational distribution of outcomes exist. With these estimators for test statistics and critical values, the hypothesis for stochastic dominance follows generically as:

(13)

where y_max < ∞. The alternative hypothesis implies that at some economic outcome, F_M exceeds F_N. To ascertain the difference between F_M and F_N merely equaling each other, and when F_M stochastically dominates F_N, we need to conduct the test twice. The first test should fail to reject the null of F_M weakly dominating F_N, while the second should give an outright rejection of the null that F_N weakly dominates F_M. For example, weak first order stochastic dominance of

cdf's F_M (y) over F_N (y) implies F_N (y) ≥ F_M (y) for all y. The null hypothesis therefore includes many different F_M distributions for a given F_N. For the sake of brevity, the rest of the discussion in this paper focuses on tests of first-order stochastic dominance. Results from second order tests are available from the author upon request but offer no additional evidence worthy of discussion.

To construct the Kolmogorov-Smirnov type tests as discussed in[30] and[31], generally let Y define the shared finite support for two continuous distributions F_M and F_N on a measurable space. Tests are built on assumptions that {X_i} and {Z_j} are independent random samples from distributions with cdfs F_N and F_M,. The random samples following from this have empirical distributions

(14)

To test the hypothesis that F_M stochastically dominates F_N , a Kolmogorov-Smirnov type test statistic, S_, is constructed based on a maximum difference in the empirical distributions. That is:

. (15)

For tests of first-order stochastic dominance, S follows directly as . Further details on the construction of S for higher orders of stochastic dominance appear in[30],[31] and[32].

5.2. Between-Race / Within-Decile Tests

The unconditional means reported in Table 1 indicate that African-Americans on average have worse outcomes than whites regardless of their fathers' earnings. While these statistically differ, we can ascertain little additional information without a more in-depth analysis. One way to accomplish this is to test for stochastic dominance as outlined in section 5.2. Tests of first-order stochastic dominance within transition matrices generate perhaps the most compelling statistical results with respect to these differences. In particular, differences in between-race earnings not only appear more complex than Table 1 indicates, but evidence supports the conclusion that white distributions stochastically dominate the distributions of African-Americans at nearly all-points between the distributional tails.

Tables 2 and 3 report the estimated p-values based on test statistics S (not shown), which we use to conduct inference on the hypothesis given by equation (13). The decision rule rejects the null if p < α. As in parametric inference, large p-values indicate a failure to reject the null. Columns labeled F_b (y) ≤ F_w (y) show p-values for the null hypothesis that African-American distributions weakly dominate white distributions. Columns labeled F_w (y) ≤ F_b (y) give p-values for the null that white distributions weakly dominate African-American distributions. Sample sizes used for tests also appear in the tables.

In Table 2 which reports tests based on age-adjusted measures of earnings, results indicate that the distribution of white earnings stochastically dominates African-Americans outcomes for nearly every decile with the exception of the poorest decile, 0-10^th and the 40-50^th decile. This conclusion is jointly supported by failure to reject the null that white earning distributions dominate African-Americandistributions, and the outright rejection of the null thatAfrican-American distributions dominate white distributions.

Table 2. Between-Race Tests of First-Order Stochastic Dominance. Age-Adjusted Measures of Earnings

The same analysis of observed earnings supports these claims throughout the middle, but the top two deciles provide no conclusive result of one distribution dominating another. Near the bottom in the 10-20^th decile and to a lesser extent, the 20-30^th decile, the same result appears. All other deciles indicate that white distributions stochastically dominate African-American distributions.

The insight provided by tests on these two measures of earnings has two strong implications. First, African-American sons whose fathers earned at the top or bottom of their own generations appear to have begun to close an earnings gap. That is, the evidence weakly fails to reject the null of an equal distribution of outcomes. The upper and lower middle classes, however, clearly have different outcomes with white earnings stochastically dominating African-American earnings. Results appear ambiguous for sons' whose fathers' earnings fell into the 40-50^th decile. More on this anomaly appears and will be discussed in the following section.

Table 3. Between-Race Tests of First-Order Stochastic Dominance. Observed Measures of Earnings

5.3. Within-Race / Between-Decile Tests

Unless strong within-race mobility exists, or mobility is nonlinear or even exhibits a negative elasticity (as indicated in[5] for poor African-Americans), then within-race distributions should sequentially increase as fathers' earning deciles increase (F₁₀ (y) ≤ F₉ (y) ≤ ▪▪▪ ≤ F₁ (y)). That is, a lack of mobility implies that the top decile (indexed by 10) should stochastically dominate the ninth decile, which should dominate the eighth decile, and so on.

Tables 4-7 report the p-values for these tests. For example, the top half of Table 4 contains p-values for the between decile tests within the African-American data-set of age-adjusted earnings. Continuing the example, the top-left cell reports the p-value (0.567) from the null hypothesis that the 90-100^th decile stochastically dominates the 0-10^th decile. The bottom half of Table 4 gives the p-value (p=0.161) for the converse hypothesis that the bottom decile stochastically dominates the top decile. To complete the inference, we need and use both results. For statistical evidence that earnings of African-Americans born into the 90-100^th decile stochastically dominate the earnings of African-Americans born into the 0-10^th decile, the p-value reported in the top half of the table should exceed our cut-off for α. In addition to this, the p-value in the bottom half of the same table should be less than α. Similar reasoning follows for other distributional comparisons. Shaded cells represent one-tailed rejection regions with p-values < 0.05. In summary, less within-race mobility exists when the bottom halves of Tables 4-7 have more shaded cells and the top halves are clear. In contrast, more mobility exists if we observe few statistical differences and where little cell shading exists in the bottom half of a table. Shaded cells in the top halves of Tables 4-7 indicates negative elasticity of intergenerational earnings.

Table 4. Between-Decile Tests of Stochastic Dominance for African Americans, Age-Adjusted Measures of Earnings Cells contain p-values for between decile tests of the nullƑ(y;FM) ≤ Ƒ(y;FN) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 0.57 0.98 0.13 0.55 0.16 0.81 0.25 0.11 0.48 20th 0.90 0.97 0.30 0.78 0.25 0.95 0.34 0.24 - 30th 0.90 1.00 0.68 0.86 0.70 1.00 0.86 - - 40th 0.71 1.00 0.34 0.59 0.67 0.95 - - - 50th 0.67 0.59 0.01 0.05 0.00 - - - - 60th 0.45 0.73 0.06 0.32 - - - - - 70th 0.68 0.98 0.27 - - - - - - 80th 0.79 0.95 - - - - - - - 90th 0.54 - - - - - - - - Cells contain p-values for between decile tests of the nullƑ(y;FN) ≤ Ƒ(y;FM) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 0.16 0.24 0.87 0.63 0.10 0.02 0.64 0.83 0.74 20th 0.23 0.18 0.85 0.73 0.25 0.06 0.81 0.98 - 30th 0.07 0.07 0.57 0.54 0.13 0.01 0.41 - - 40th 0.09 0.12 0.64 0.54 0.12 0.00 - - - 50th 0.91 0.61 0.98 0.91 0.39 - - - - 60th 0.08 0.06 0.87 0.32 - - - - - 70th 0.17 0.18 0.86 - - - - - - 80th 0.05 0.09 - - - - - - - 90th 0.79 - - - - - - - -

Table 6. Between-Decile Tests of Stochastic Dominance for Whites, Age-Adjusted Measures of Earnings Cells contain p-values for between decile tests of the nullƑ(y;FM) ≤ Ƒ(y;FN) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 0.98 0.93 0.95 0.98 0.98 0.91 0.97 0.98 0.95 20th 0.84 0.91 0.79 0.92 0.55 0.50 0.63 0.91 - 30th 0.85 0.92 0.60 0.60 0.59 0.08 0.62 - - 40th 0.76 0.86 0.32 0.34 0.23 0.03 - - - 50th 0.99 0.98 0.98 0.97 0.97 - - - - 60th 0.98 0.95 0.57 0.56 - - - - - 70th 0.89 0.95 0.43 - - - - - - 80th 0.90 0.98 - - - - - - - 90th 0.54 - - - - - - - - Cells contain p-values for between decile tests of the nullƑ(y;FN) ≤ Ƒ(y;FM) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 20th 0.00 0.00 0.03 0.00 0.02 0.71 0.03 0.07 - 30th 0.02 0.00 0.08 0.20 0.45 0.69 0.28 - - 40th 0.01 0.00 0.02 0.09 0.18 0.57 - - - 50th 0.00 0.00 0.04 0.01 0.03 - - - - 60th 0.16 0.04 0.41 0.24 - - - - - 70th 0.08 0.04 0.62 - - - - - - 80th 0.11 0.06 - - - - - - - 90th 0.94 - - - - - - - -

Table 5. Between-Decile Tests of Stochastic Dominance for African Americans, Observed Measures of Earnings Cells contain p-values for between decile tests of the nullƑ(y;FM) ≤ Ƒ(y;FN) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 1.00 1.00 0.85 0.95 1.00 0.82 0.84 0.74 0.84 20th 1.00 1.00 0.51 0.86 1.00 0.92 0.48 0.53 - 30th 1.00 1.00 0.77 0.67 0.98 0.69 0.75 - - 40th 1.00 1.00 0.79 0.76 0.99 0.81 - - - 50th 1.00 1.00 0.30 0.88 0.78 - - - - 60th 0.97 0.90 0.12 0.62 - - - - - 70th 1.00 0.68 0.08 - - - - - - 80th 0.99 0.97 - - - - - - - 90th 0.98 - - - - - - - - Cells contain p-values for between decile tests of the nullƑ(y;FN) ≤ Ƒ(y;FM) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 0.03 0.02 0.19 0.00 0.00 0.00 0.06 0.07 0.02 20th 0.02 0.12 0.68 0.01 0.07 0.40 0.47 0.65 - 30th 0.04 0.16 0.71 0.02 0.05 0.30 0.50 - - 40th 0.03 0.16 0.81 0.03 0.09 0.22 - - - 50th 0.03 0.30 0.81 0.09 0.46 - - - - 60th 0.04 0.24 0.95 0.47 - - - - - 70th 0.02 0.34 0.84 - - - - - - 80th 0.04 0.30 - - - - - - - 90th 0.15 - - - - - - - -

Table 7. Between-Decile Tests of Stochastic Dominance for Whites, Observed Measures of Earnings Cells contain p-values for between decile tests of the nullƑ(y;FM) ≤ Ƒ(y;FN) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 1.00 1.00 1.00 1.00 1.00 0.81 0.68 0.36 0.28 20th 1.00 0.98 0.98 0.95 0.98 0.93 0.86 0.64 - 30th 1.00 1.00 0.98 0.98 0.88 0.85 0.79 - - 40th 1.00 1.00 1.00 0.98 1.00 0.59 - - - 50th 1.00 1.00 1.00 0.99 0.98 - - - - 60th 0.98 1.00 1.00 0.68 - - - - - 70th 0.98 0.96 0.97 - - - - - - 80th 0.96 0.56 - - - - - - - 90th 0.98 - - - - - - - - Cells contain p-values for between decile tests of the nullƑ(y;FN) ≤ Ƒ(y;FM) FN FM 100th 90th 80th 70th 60th 50th 40th 30th 20th 10th 0.00 0.00 0.00 0.00 0.00 0.03 0.02 0.64 0.32 20th 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.33 - 30th 0.00 0.00 0.00 0.00 0.00 0.05 0.05 - - 40th 0.00 0.00 0.00 0.00 0.01 0.76 - - - 50th 0.00 0.00 0.00 0.00 0.01 - - - - 60th 0.00 0.07 0.02 0.55 - - - - - 70th 0.00 0.06 0.01 - - - - - - 80th 0.00 0.92 - - - - - - - 90th 0.00 - - - - - - - -

Following with this basic understanding, evidence for age-adjusted earnings indicates that African-American distributions do not exhibit the sequential ordering of stochastic dominance implied from a linear and positive elasticity of mobility. To some extent, the 40-50^th decile also appears to exhibit dominance over several other deciles, including those representing higher earnings. This result echoes between-race results presented in section 5.3, as well as regression tree results from research in[5]. Using the measure of observed earnings, the 50-60^th and 60-70^th deciles dominate the lowest deciles, but oddly the children of the highest earners do not demonstrate this effect.

Tests comparing age-adjusted and observed earnings distributions within the white samples provide strong evidence that higher deciles dominate most deciles below them. That is, mobility is less substantial for whites relative to other whites. White sons of high earning fathers not only have higher average earnings, but the entire distribution of their outcomes usually exceeds the distribution of white sons with lower earning fathers.

The general conclusion from these within-race comparisons implies that white outcomes follow a more rigid, nearly linear process with little within-race mobility. African-American outcomes contain more non-linearity and substantially more mobility, particularly for sons of the middle class and lower classes. African-American sons lucky enough to have high earning fathers appear somewhat resistant to mean regression.

6. Conclusions

Extending research on intergenerational earnings mobility, the results presented in this paper use nonparametric methods to analyze distributions of sons' earnings. Results indicate that the stationary distribution of whites stochastically dominates the stationary distribution of African-Americans. Additional statistical tests following methods outlined in[30] indicate that the decile-conditioned distributions of white sons' earnings stochastically dominate the distributions of African-Americans from similar economic backgrounds. The distributions of white sons' earnings also improve as fathers' earnings improve. This result does not always exist for African-Americans, who only appear to have distributional dominance from the middle class.

Although convergence clubs appear not to exist across the distribution of earnings, African-American and white sons clearly have different outcomes. While not providing a reason for these differences, this descriptive paper adds insight into the complex structure of a seemingly simple but often discussed subject. The two races follow two different stochastic processes in intergenerational earnings.