# Supplemental Material

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### Data, Sample, Variables

We use data from the Panel Study of Income Dynamics (PSID) as prepared for and described in Pfeffer and Killewald (2018). A replication package containing the data and code used in that article is available through the PSID Public Data Extract Repository via OpenICPSR at http://doi.org/10.3886/E101094V1 and a full replication package for the animations reported here is available at http://github.com/fpfeffer/wealthmobility.

The net worth measure in the PSID is based on a set of survey questions on the ownership and value of separate asset components, including financial assets (e.g. savings, stocks, etc.), home ownership (home value minus mortgages), other real assets (e.g. business and farm wealth, real estate, etc.), and different forms of debt. Parental wealth is measured as average household net worth in survey years 1984 and 1989, which contain the first full asset survey modules. Offspring’s wealth is measured as average household net worth in survey years 2013 and 2015, the latest available PSID waves. We restrict the analytic sample to offspring aged 45-64 (N=1,936) to reduce life-cycle bias in estimates of intergenerational wealth mobility (as revealed in Pfeffer and Killewald 2018). The average parental age in this sample is 52.1 years (in 1984) and the average offspring age is 53.7 years (in 2013). Net worth quintiles are drawn based on the analytic sample just described and using the distribution of wealth for parents and children, weighted using longitudinal individual weights. The racial classification relies on the first mention of the offspring’s race and distinguishes Non-Hispanic whites from Non-Hispanic blacks.1 The sample size is too low to support separate analyses of other racial or ethnic groups.

### Additional Information on Animation 1

Higher rates of intergenerational downward social mobility for black children compared to white children have been established for at least fifty years (Blau and Duncan 1968: 208ff) and found across a wide range of indicators of socio-economic well-being (Featherman and Hauser 1978; Hertz 2005; Bloome 2014; Mazumder 2014). Our visualization confirms that these racial differences in intergenerational mobility also apply to family wealth (see also Conley and Glauber 2008; Pfeffer and Killewald 2018).

Unlike the visualization in Badger et al. (2018), which relied on full population administrative data (Chetty et al. 2018), our visualization draws on a much smaller sample from survey data. Thus, rather than displaying observed transition rates, we estimate transition probabilities more parsimoniously in a statistical model.2 Animation 1 displays predicted probabilities from an ordered logistic regression model. We can express offspring’s relative wealth position as a latent variable, $$\tilde{y}$$, that is a function of their parents' wealth position, $$\bf x$$, measured in quintiles, plus an error term, $$\epsilon$$. That is,

$${\tilde{y}=\mathbf{x}\beta+\epsilon}$$

The probability that a child's attained wealth quintile, $$y$$, is quintile $$q$$ can then be expressed as

$${Pr(y=q|\mathbf{x})=F(\tau_q-\mathbf{x}\beta) - F(\tau_{q-1}-\mathbf{x}\beta)}$$

where $$F$$ is the logistic cumulative distribution function and the $$\tau_q$$ parameters describe the cut-points of the latent measurement model. We estimate the ordered logistic regression interacting race and wealth origins to accommodate racial differences in transition rates, $$\beta$$. The regressions are estimated using PSID’s longitudinal individual weights, which partly adjust for attrition (Berglund et al. 2017a; more on weighting below).

One of the parametric assumptions of the ordered logistic model is the proportional odds assumption, which constrains all $$\beta$$s to be equal across outcome categories $$q$$. Using global significance tests (see Long and Freese 2014: 329), across all but one test we cannot reject the null hypothesis of this assumption.3

To assess robustness of the reported patterns to alternative modeling approaches with either more or less stringent constraints, we provide Animation S.1, which displays the results created under alternative statistical models. First, a slightly less parsimonious model estimates the ordered logistic regression model separately for whites and blacks, which not only accommodates different transition rates, $$\beta$$, but also allows the cut-points of the measurement model, $$\tau_q$$, to vary by race, effectively allowing the distribution of the underlying latent variable to differ by race. In these models, we find clear statistical support for the proportional odds assumption.4 Second, we include results from a multinomial logistic regression, which –- with interaction terms for race and parental wealth -– imposes no further constraints on the data (saturated model): In contrast to the ordered logistic regression model, it allows the coefficients for each independent variable to vary by outcome category and thereby corresponds to the observed probabilities.5 Third, we provide results from a stereotype logistic regression model. In terms of parsimony, this model occupies a middle ground between the ordered and multinomial model: Instead of fixing the coefficients to be equal across outcome categories (as in the ordered logistic model), it imposes a linear proportionality constraint on the associations of the independent variables with each outcome category (Anderson 1984).

Animation S.1: Transition probabilities produced under different modeling approaches

### Additional Information on Animation 2

This animation relies on the weighted cross-tabulation of parental and offspring wealth quintiles, separately for whites and blacks. To correctly represent the racial distribution across quintiles, we rescale the number of observations by multiplying all observations in these cross-tabulations by the average longitudinal individual weights of blacks and whites in the offspring generation (in 2013). These weights adjust both for the initial PSID oversample ("SEO sample") and later sample attrition; for a detailed description of the PSID weighting scheme see Berglund et al. (2017a; 2017b).

Other research on intergenerational mobility, in particular by economists following Solon (1992), has excluded the SEO oversample and instead used unweighted analyses of only the population-representative part of the original PSID sample (the “SRC sample”). The resulting lower sample size (N=1,223) increases concerns about low cell counts, particularly for blacks (N=94). The issue of low cell counts can be ameliorated by choosing broader wealth quantile categories. Animations S.2 and S.3 display the intergenerational wealth structure using quartiles and terciles, respectively, though we caution that the broader the categories, the more they hide inequality in the wealth position of black and white children within those categories. However, based on these coarser wealth categories, Animations S.2 and S.3 can also be used to display results restricted to only the population-representative SRC sample.6

Animation S.2: Wealth quartile mobility for different samples

Animation S.3: Wealth tercile mobility for different samples

### Alternative Display and Comparative Advantage of the Animation

There are many other ways to visualize mobility tables, each with different comparative advantages in highlighting certain aspects of the mobility process. Examples include overlaying the cross-tabulation with a heat map that represents variation in mobility rates with different colors, or altering the size of the cells of a mobility table to include information on generational difference in the extent of inequality (see Lawrence 2018). The approach pursued here and in Badger et al. (2018) focuses on the comparison of transitions between different population groups (Animation 1) and the interplay between structural reproduction and individual-level dynamics (Animation 2). Of course, even within that focus, other approaches are available.

A commonly used alternative to our visualization is Sankey diagrams, which are displayed for comparative purposes in Figures S.1 and S.2 (as alternatives to Animations 1 and 2, respectively). While these static versions display the same quantitative information, we believe that the animations make the information more intuitively accessible and reduce the general reader’s cognitive work in the following ways (for meta-analytic evidence on the educational benefits of animations compared to static displays see Höffler and Leutner 2007). First, the displayed movement of individuals who fall down or rise up is a fitting visual representation of the dynamics of downward and upward mobility; the Sankey diagrams are static and do not convey this movement as directly. Second, the directionality of the flow (from origins to destinations) in the animation assures that non-specialists read the information in the intended direction (since we are displaying outflow, not inflow percentages), but the direction of flow is less obvious in the Sankey diagrams. Third, the focus of this contribution is on the between-race comparison, which is directly supported by overlaying information on both racial groups in the same display. To achieve a similar goal, the Sankey diagrams display racial groups next to each other within each quintile (rather than separate Sankey diagrams for each racial group), but the direct between-race comparison is still much more immediate in our animations. Fourth, both experts and non-experts are better at reading and interpreting natural frequencies – e.g., 10 in 100 – than percentages – e.g., 10% – (see Hoffrage et al. 2000; 2015) and our visualization evokes that interpretation. Fifth, and perhaps most importantly, Animation 2 provides a visual illustration of the fundamental sociological insight that structural reproduction can coincide with ample fluctuation at the individual level. The co-existence of stability (reproduction of the racial wealth structure) and movement (individual-level dynamics) cannot be displayed as effectively in a static way.

Finally, we point to two aspects of what we consider purposeful pedagogical design: The underlying transition estimates are clearly too numerous to display at once (5 origin quintiles * 5 destination quintiles * 2 races = 50 transitions). We therefore decided to direct readers’ attention in two ways: First, we have made Animation 1 interactive, purposefully forcing the reader to look at only one origin quintile at once (Figure S.1 seeks to do the same by highlighting quintiles upon hovering over connected paths). This is important because the visual impression should focus on the cross-race comparison within quintiles. Second, to help focus readers’ attention in Animation 2, we decided to initialize the visualization. That is, rather than displaying the full flow at once (as in Animation 1), the animation reveals important information step-by-step (for other examples of the effectiveness of “chunking” strategies see e.g. Rieber 1991). The animation is also timed to correspond to readers’ cognitive processing, which we imagine to unfold as follows:

• Start – Second 8: Observation of overrepresentation of black children at the bottom of the distribution;
• Seconds 8-13: Observation of ample intergenerational fluctuation;
• Seconds 13-22: Observation that the overrepresentation of black children at the bottom of the distribution has not changed dramatically;
• Second 22-End: Closer investigation (based on the now-revealed estimates of the racial composition in the destination distribution) reveals some equalization; but the overall visual impression remains.

Figure S.1: Sankey diagram as static version of Animation 1

Figure S.2: Sankey diagram as static version of Animation 2

### References

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Badger, Emily, Claire Cain Miller, Adam Pearce, and Kevin Quealy. 2018. Extensive Data Shows Punishing Reach of Racism for Black Boys. The New York Times, March 19. https://www.nytimes.com/interactive/2018/03/19/upshot/race-class-white-and-black-men.html

Berglund, Patricia A., Wen Chang, Steven G. Heeringa, Katherine McGonagle, Charles Brown, and David Johnson. 2017a. Construction and Evaluation of the Longitudinal Sample Weights 2015. PSID Technical Report.

Berglund, Patricia A., Wen Chang, Steven G. Heeringa, and Katherine McGonagle. 2017b. PSID Cross-Sectional Individual Weights, 1997-2015. PSID Technical Report.

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1. Drawing on up to four mentions of race and counting as white only those who do not report any classification other than white across all mentions leads to the reassignment of only 29 cases from the white to non-white category.

2. This illustrates the well-known trade-off between data and assumptions. One of the advantages of full population data is that they often allow the display of observed relationships without further parametric assumptions, eliminating the need for additional explanation.

3. Only the Wald test –- but not other statistical tests (likelihood ratio test, score test, Wolfe-Gould test, and Brant test; see Buis 2013) –- rejects the proportional odds assumption $$p<0.1$$. Fit statistics that take into account model parsimony (e.g., BIC) indicate that the ordered logistic model is preferred over the more flexible generalized ordered logistic model (see Long and Freese 2014: p.371ff).

4. For both whites and blacks, different tests (see footnote 3) fail to reject the null hypothesis that coefficients are equal across outcome categories and the ordered logistic model is also indicated as preferred over the generalized ordered logistic model based on BIC. Because the very low number of black children in the top two quintiles results in situations of perfect prediction, the statistical tests of the proportional odds assumption for black children have to be restricted to the bottom three quintiles. Naturally, the near absence of black children from the top 40% of the wealth distribution is not merely a methodological or modeling challenge, but follows from a fundamental empirical fact, namely, the severely racialized structure of wealth holdings in the United States, which is the focus of Animation 2.

5. These probabilities also reveal the issue of empty cells in the underlying cross-tabulation for black children who grow up in the top two quintiles of the wealth distribution. We believe that the lack of any black children from these quintiles who also end up in the top wealth quintile themselves is a feature of the small sample. The model-based probabilities produced under the other statistical models represent our best estimates of the share of cases we would observe if we observed the full population.

6. The use of only the SRC sample portion of the PSID prohibits the use of any weights (as the weights were constructed based on both the SEO and SRC sample). For this reason, this sensitivity analysis does not use weights at any stage of the process (i.e., neither in the construction of the quantiles nor in calculating the cross-tabulations).