Social Mobility in a High-Inequality Regime

Abstract

Are opportunities to get ahead growing more unequal? Using data from the General Social Survey (GSS), it is possible to provide evidence on this question, evidence that is suggestive but must be carefully interpreted because the samples are relatively small. The GSS data reveal an increase in class reproduction among young and middle-age adults that is driven by the growing advantage of the professional-managerial class relative to all other classes. This trend is largely consistent with our new “top-income hypothesis” that posits that rising income inequality registers its effects on social mobility almost exclusively in the divide between the professional-managerial class and all other classes. We develop a two-factor model in which the foregoing effects of the inequality takeoff are set against the countervailing effects of the expansion of mass education. As the model implies, the trend in intergenerational association takes on a convex shape in the younger age groups, with the change appearing to accelerate in the most recent decade. These results suggest that the takeoff in income inequality may account in part for the decline in mobility.

Keywords

intergenerational mobility trends social mobility class mobility income inequality takeoff educational expansion top-income hypothesis professional-managerial class inequality of opportunity

At several critical junctures in U.S. history, scholars and other commentators have become concerned that opportunities to get ahead may be growing more unequal, a hypothesis that was prominent during the Depression years, the postwar period, and then again in the 1950s (Hertzler 1952; Sibley 1942). Although these concerns have never been borne out, the recent takeoff in income inequality has revived them yet again (e.g., Chetty et al. 2014; DeParle 2012; The Economist 2010; Franke-Ruta 2012; Pfeffer and Hertel 2015; Putnam 2015). There is rather less evidence than one might imagine on whether such concerns are warranted.

The main goal of this article is to eke out as much evidence on these concerns as the available data will allow. This descriptive objective might at first blush seem easily achieved. To the contrary, a host of methodological problems immediately emerge in attempting to establish recent trends in social mobility, not least of which is that the takeoff in income inequality has not been in play long enough to affect the upbringing of all that many current workers. It is equally problematic that the available survey samples are too small to reliably detect anything but the broadest trends. We overcome these obstacles in this article by searching for trend among those age groups and social classes that are most likely to evince trend.

If our article is largely descriptive in focus, this is not to suggest that we lack hypotheses about what might be driving trends. The key explanation lurking behind our descriptive interests is that the takeoff in income inequality may be producing a historic decline in social fluidity. We will not be able to directly assess this hypothesis. We will, however, provide indirect evidence by developing a framework that has changes in social fluidity resulting from the operation of just two forces: educational expansion and the inequality takeoff. This framework allows us to advance expectations about the pattern of trend that exploit variation in educational upgrading and the experience of income inequality across age-gender groups and time periods.

The article is organized as follows. We lead off our discussion by reaffirming the normative implications of social mobility against old and new objections that stress the role of factors other than opportunity in generating the data in a mobility table. In the sections that follow, we present our analytical framework, review the (scarce) evidence available on recent mobility trends, and introduce a key revision of the conventional inequality hypothesis that better reflects the likely effects of rising income inequality on social mobility. This discussion is followed by a section on data and measures and on the empirical strategy and models we employ. We then turn to our results and their implications for the simple hypothesis that income inequality is reducing social fluidity.

We find a substantial decline in social fluidity that may be attributable to the takeoff in income inequality. The estimates presented here, which are sometimes only at borderline levels of significance, must be interpreted cautiously, but it is striking that our results are inconsistent with the prevailing view that class mobility has been increasing (e.g., Pfeffer and Hertel 2015) and that economic mobility has remained stable (e.g., Chetty et al. 2014).

Why Should We Care about Social Mobility?

Because the study of intergenerational mobility had, until very recently, fallen out of fashion among U.S. sociologists, it is perhaps necessary to remind ourselves why we have historically cared about it. There are all manner of motives underlying the social scientific interest in mobility (see Grusky and Cumberworth [2010] for a review), but we mainly care about it because it speaks to the extent to which life chances depend on social class origins. The mobility table is accordingly valued for the evidence it provides on the extent to which equal opportunity is achieved. When the mobility table reveals a strong association between the social class of parents and their children, the departure from equality of opportunity is assumed to be substantial as well.

This conventional interpretation might be disputed insofar as class origins are also associated with innate talents and preferences that affect people’s life chances (e.g., Breen 2010b). If, for example, the daughters and sons of musicians tend to be bestowed at birth with a “great musical ear” and to become musicians themselves, then some scholars would not treat the resulting immobility as an expression of unequal opportunity. It is of course unclear whether an innate “musical ear,” in and of itself, is all that important in determining musical success. Indeed, because most capacities require considerable cultivation to be parlayed into a job, the intergenerational association has to be understood, in part, as reflecting a collective decision to allow unequal innate capacities to be realized rather than attenuated through compensatory training. Moreover, insofar as genetic differences are associated with class origins, this association again reflects the operation of complex social processes that played out in the prior generation (e.g., assortative mating). It follows that even the “genetic lottery” is shaped by class-based unequal opportunities in a fundamental way (cf. Swift 2005).

It is likewise problematic to treat preferences as wholly exogenous to one’s class origins. For example, those with a preference for “delayed gratification” have likely developed that preference, at least in part, because of their social class environment. In some social classes, we may find cultural rules of thumb implying that one should “live for the day,” prescriptions that may have developed in part because long-term investments in training tend not to pay off within those classes. The intergenerational association generated by these and other similarly class-dependent preferences is relevant to judgments about equality of opportunity.

We cannot discuss here the lengthy literature on the relationship between equality of opportunity, talents, and preferences (see Roemer 2012, 2004, 1998). It suffices to reiterate that, even if part of the origin-destination association is the result of differences in innate talents and preferences, this does not automatically make that association less informative with respect to concerns about equality of opportunity. Although we do not deny that there may be preferences that are correlated with the class of origin, that influence the class of destination, and that are irrelevant for equality of opportunity, our arguments do at least suggest that the share of the origin-destination association attributable to that type of preferences is likely to be very small. The implication is that any observed changes in the association between origins and destinations are largely determined, directly or indirectly, by true differences in opportunities. Without a doubt, there is much more that needs to be known about equality of opportunity than is revealed by mobility tables, but they do tell us a good deal. It follows that, notwithstanding the interpretive complications that the role of preferences and talents may entail, there are good normative reasons to be concerned that the gross origin-by-destination association may be increasing in the United States.

The foregoing takes for granted a social commitment to equalizing opportunity and simply suggests that, insofar as such a commitment exists, the association parameters coming out of a mobility table provide one important measure of the extent to which that commitment has been realized. It is especially important to monitor this commitment given its role in justifying inequality. In the United States, there is still a widespread belief that opportunities to get ahead are openly available (rather than dependent on birth), a presumption that plays a central role in legitimating the soaring inequality of our times. We look to social science to establish whether, consistent with such an account, inequality is indeed the outcome of a fair and open contest rather than some “rigged competition.”

Countervailing Effects on Mobility

In explaining trends in social mobility, a host of factors may of course be relevant, but two stand out as especially important. The “education hypothesis” focuses on the effects of mass education in reducing the intergenerational transmission of class (e.g., Hout 1998; Pfeffer and Hertel 2015). The “income hypothesis” suggests, to the contrary, that the rise in income inequality amounts to an unprecedented infusion of additional resources among the higher reaches of the class structure, an infusion that will work to increase the amount of reproduction. We review each of these competing hypotheses in turn.

The simplest form of the education hypothesis posits a decline in the effects of class origins on educational attainment. If a college education becomes increasingly open to children from all classes, then we can expect a corresponding decline in the association in a mobility table. Although some evidence for this mechanism has been found in several countries (e.g., Breen and Jonsson 2007; Breen et al. 2009; cf. Shavit and Blossfeld 1993; Shavit, Yaish, and Bar-Haim 2007), there is little, if any, evidence of a substantial reduction in the effects of class origins in the U.S. case (e.g., Bailey and Dynarski 2011; Chetty et al. 2014; Duncan et al. 2014; Pfeffer and Hertel 2015; also see Gamoran 2001; Hout, Raftery, and Bell 1993; Morgan and Kim 2006; Roksa et al. 2007).

The main way, therefore, in which mass education has instead exerted its effects is through a simple compositional effect. In a classic paper, Hout (1988) showed that the association between origin and destination withers away among college graduates, the implication being that educational upgrading shifts the population toward a low-association regime (see also Breen 2004; Vallet 2004). This line of reasoning led Beller and Hout (2006, 28) to conclude that the rising share of U.S. men with college degrees is “one major reason for the declining correlation between fathers’ and sons’ occupations.” In a recent U.S. study, Pfeffer and Hertel (2015) argued that class mobility increased gradually over the twentieth and early twenty-first centuries, an increase that is almost entirely attributable to a compositional effect of this sort (see Breen [2010a] for a similar argument for England).

The evidence for the compositional argument may be strong, but Torche (2011) has recently shown that it takes on a somewhat weaker form within the population of advanced degree holders. For those holding such degrees, Torche finds that the intergenerational association is no longer zero, although for our purposes the more important point is that the association within the advanced degree population is still weaker than that prevailing among those without college. The latter result means that the overall compositional effect should grow weaker as an increasing share of the college population holds an advanced degree.

It follows that, all else equal, the association between origins and destinations should follow the trend in the proportion of the population with a college degree.¹ In Figure 1, we have graphed this proportion between 1950 and 2010 (for those aged 25–60), a graph that will likely surprise few. As it reveals, there is a rapid increase between 1950 and 1980 in the college-educated share of men, whereas thereafter this share continues to grow but at a much slower pace. Among women, the share with college barely changes between 1950 and 1960, then increases at a slower pace than the men’s share over the following decade, and ultimately takes on a fast and steady growth rate between 1970 and 2010.

Figure 1

Proportion of 25 to 60 Year-Olds with a Bachelor’s Degree or More, 1950–2010

We next consider the income hypothesis. Although it is newer than the education hypothesis, it is also more frequently featured in contemporary speculation about mobility trends (Andrews and Leigh 2009; Corak 2013; Krueger 2012; OECD 2010; see also Burtless and Jencks 2003). The relationship between income inequality and class mobility nonetheless remains rather poorly developed in the literature. Insofar as it has been discussed, the main argument is that rising inequality provides privileged families with more resources that can then be lavished on their children, resources that raise their chances of securing desirable class positions for themselves (e.g., Pollak et al. 2011).² By this logic, inequality of condition and of opportunity are now understood as varying together, even though scholars have typically been at pains to stress that they are analytically distinct.³

How might parents in privileged classes use their newfound income? The available evidence (see Kornrich and Furstenberg 2013; Putnam 2015) suggests that they will increase the human, cultural, and social capital of their children via high-quality childcare and preschool, educational toys and books, after-school training and test preparation, science-related summer camps, elite preparatory schools, prestigious college degrees, a “finishing school” vacation in Europe, and stipends or allowances that free them from the need to work during high school and college. As the takeoff plays out, privileged parents can also more readily afford privileged residential neighborhoods, with accordingly improved access to high-quality public schools, neighborhood amenities that assist in human-capital formation (e.g., libraries), and peers that can provide all manner of career advantages (Durlauf 1996; cf. Mayer 2001).⁴

The responses to the takeoff that we have discussed to this point take place when children are still living with their parents (e.g., paying for after-school classes) or are pursuing their college degree (e.g., paying for tuition). Although we suspect that the takeoff will have its strongest effects on adults who were exposed to it as children, we certainly cannot rule out the possibility that it also affects the opportunities of adult children. We can imagine that well-off parents may decide (1) to finance, via loans or gifts, a late-adult professional degree; or (2) to provide in-kind or direct economic support when their adult children are unemployed, support that then allows their children to maintain a high reservation wage (rather than settle quickly for a lesser position). In some cases, parents might also help their adult children pursue entrepreneurial opportunities by providing start-up resources, physical space, or implicit insurance in case of failure. If the takeoff indeed affects adult opportunities in this way, its effects will register without the prolonged lag that arises when it operates exclusively on children (who must then age into the labor force).

This line of reasoning implies that, all else equal, the trend in social fluidity should follow the trend in income inequality, with a lag that depends on the extent to which these “adult mechanisms” are operating. The well-known trend in family income inequality is shown in Figure 2 (see also Atkinson, Piketty, and Saez 2011; Grusky and Cumberworth 2013). It bears noting that this trend takes a rather different form than that pertaining to college completion (Figure 1). Even if the takeoff in income inequality has the effects we have outlined, this means that the gross association between origins and destinations will not necessarily increase in recent decades (i.e., “rigidification”). We may instead find that the takeoff simply slows down the education-generated decline in that association. It is precisely this ambiguity about the outcome of the contest between education and income inequality that motivates our analyses. Although it is unlikely that none of the takeoff-induced processes outlined above are operative, what remains unclear is whether they are strong enough to undermine the largely countervailing effects of the expansion of mass education. It is also unclear how other forces affecting trend may play out. In stressing the special role of education and inequality, we do not mean to rule out the further complicating effects of yet other forces, such as the rise of cohabitation, blended families, and other more complicated family forms.⁵

Figure 2

Family Income Inequality, 1947–2009

Previous Research on Trend

When concerns about rigidification have previously surfaced in U.S. history, they invariably have proven overblown. The evidence from other countries and earlier time periods indicates that mobility regimes change only slowly, and, insofar as a trend in social fluidity has been teased out, it is typically in the direction of increasing equality of opportunity (e.g., Breen et al. 2009; Breen 2010a; Pfeffer and Hertel 2015; Vallet 2001). Is there any reason to believe that there is now a trend in the opposite direction?

This question has been addressed most frequently within the literature on income and earnings mobility that uses the intergenerational elasticity (IGE) to assess trend. In fact, there has been a minor resurgence of such analyses, a development that has been partly motivated by concerns about the effects of rising income inequality (e.g., Aaronson and Mazumder 2008; Bloome and Western 2011; Hertz 2007; Lee and Solon 2009; Mayer and Lopoo 2004, 2005; Mazumder 2012). These studies of income and earnings mobility are of immense interest, but they have produced very mixed evidence, making it difficult to draw any clear conclusion on trend. As Lee and Solon (2009) pointed out in a recent review, available estimates of trend in economic mobility are highly imprecise, mainly because the typically used surveys (e.g., Panel Study of Income Dynamics) are based on small samples. Also, IGEs are sensitive to cross-generation changes in the variances of the earnings or income distributions, meaning that they cannot of course be interpreted as measures of association.⁶ Although Chetty et al. (2014) have exploited the full population of tax records and applied a measure that is not sensitive to the variance (i.e., rank-rank correlation), it is difficult to reach any definitive conclusions from their results because they were obliged to rely on proxies for economic standing for the youngest cohorts (i.e., the probability of attending college).⁷ Moreover, whatever the trend in economic mobility might be, there is of course no reason to assume that it will mimic the trend in class mobility.

It is accordingly useful to consider the relatively few studies that have directly examined trends in occupational or class mobility during the post-takeoff period or some part of it (Beller 2009; Beller and Hout 2006; Pollak et al. 2011; Pfeffer and Hertel 2015). Notably, Pollak et al. (2011) have analyzed trends in social fluidity between 1962 and 2006, using data from several national surveys. Because they relied on highly disaggregate classifications, the power of their models was limited, and it proved difficult for them to reach any definitive conclusion about trend. In an earlier analysis, Beller (2009) sought mainly to demonstrate that the mother’s class matters in intergenerational mobility, but she also provided, perhaps for the first time, some evidence of decreasing social fluidity in the United States during our period of interest. Using social class measures for both parents, Beller found that the intergenerational association between 1994 and 2006 was substantially larger for men from the 1965 to 1979 birth cohorts than for men in two older cohorts. The results for women, however, are less clear.⁸ As we discuss in more detail below, Beller’s important finding relies on strong assumptions regarding the age profile of class membership, assumptions that we will avoid in our own analyses. Finally, Pfeffer and Hertel (2015) analyzed class mobility for cohorts of men born in six periods, covering 1883–1921 to 1970–1982. Although their main descriptive finding is a gradual and small increase in social class mobility over time, the point estimates also imply a reduction in social fluidity between the last two cohort groups in their study. This reduction is, however, very small and not statistically significant.⁹

Because there is a shortage of direct evidence on mobility trends, concerns about rigidification have been fueled mainly by indirect evidence that speaks to some of the mechanisms through which the income hypothesis may be operating. For instance, Reardon (2011) documents a growing performance gap in math and reading achievement between high-income and low-income children, while Kornrich and Furstenberg (2013) show a marked increase in the amount of money that parents in the top income decile spend on such goods and services as high-quality daycare and babysitting, private schooling, books and tutoring, and college tuition and fees (also see Putnam 2015). Although spending on children also increased at the bottom of the income distribution, this increase is far smaller in both absolute and relative terms (also see Kaushal, Magnuson, and Waldfogel 2011). As best as we can tell, upper-class parents are indeed spending ever more on the “reproduction project,” and it is at least plausible that this spending is achieving its intended effect (cf. Mayer 1997).

The latter indirect evidence is suggestive, but it cannot of course substitute for direct measurements of trend in social mobility. The small number of available direct measurements (especially Beller 2009) also reveal some hints of declining mobility, but these studies should clearly be supplemented with an analysis that directly focuses on the takeoff. We turn to that analysis after introducing an amendment to the simple income hypothesis featured to this point.

The Top Income Hypothesis

The income-inequality hypothesis implies a proportional “stretching out” of the interclass gaps in family income that should make all types of exchange less common (see Pollak et al. 2011). By this interpretation, one expects an across-the-board increase in the association between class origins and destinations of the sort that might be teased out, for example, with a conventional association model (e.g., Xie 1992). We will label this formulation as the “simple income hypothesis.”

It is perhaps more plausible, however, that the effects of the takeoff will register principally within the upper regions of the class structure. This modification of the income hypothesis is attractive because the professional-managerial class has been far and away the biggest beneficiary of the inequality takeoff. Whereas average family income rose within this class by a full 31.8 percent between 1979 and 2010 (i.e., from $80,490 to $106,120), it rose within the other classes by just 4.9 percent over that same period (i.e., from $56,890 to $59,690). As Figure 3 shows, the income gap between the professional-managerial class and all other classes is now very large, whereas the income gaps among the remaining classes are not much different from what prevailed in 1979.¹⁰ The implication is simple: if rising income inequality is indeed driving changes in mobility, we would expect its effects to register almost exclusively in the divide between the professional-managerial class and all other classes.¹¹ We refer to this account as the “top income hypothesis.”¹²

Figure 3

Mean Family Income by Class of Household Head, 1949–2010

The mechanical distributional effect just discussed is not the only rationale for the top income hypothesis. There is, after all, a long tradition of scholarship suggesting that professional-managerial culture and institutions are tailor-made for reproductive purposes (e.g., Bourdieu and Passeron 1977). Within this literature, the professional-managerial class is not just represented as especially oriented toward and anxious about reproduction (if only for loss aversion reasons), but also especially skilled in realizing its agenda by choosing the right neighborhoods, buying high-quality preschool, purchasing after-school training, and otherwise engaging in “concerted cultivation” (Lareau 2003). The top income hypothesis thus suggests that, by virtue of increasing the resources at the disposal of professionals and managers, the takeoff works to potentiate their natural reproductive tendencies.

The methodological implication of this hypothesis is that the takeoff should register most prominently in the odds ratio pertaining to the advantage of the professional-managerial class relative to all other classes. The simple income hypothesis directs us, alternatively, to average across all odds ratios, an approach that will blunt our capacity to detect change insofar as the top income hypothesis is indeed on the mark. We will carry out tests aimed at discriminating between these two hypotheses.

Data, Measurement, and Design

We analyze data drawn from the 1972 to 2010 General Social Surveys (GSS; see Davis, Smith, and Marsden 2010). Because the annual and biannual samples of the GSS are too small to conduct analyses by gender and age group, we are obliged to pool the data into four periods, one for each decade (i.e., 1972–1980, 1982–1990, 1991–2000, and 2002–2010). In Table 1, the decade-specific sample sizes and descriptive statistics are provided, with the right panel reserved for the “two-parent” variable that is available only for the last two decades (see below for details on the construction of this variable).

Table 1

Sample Descriptives (Unweighted Proportions)

Variables	1970s	1980s	1990s	2000s	Variables	1990s	2000s
Female	.53	.55	.55	.54	Two-parent class ^a
					Professional/manager, professional/manager	.07	.09
Age 25–40	.51	.57	.52	.45	Professional/manager, routine white collar	.05	.05
Age 32–50	.51	.54	.61	.56	Professional/manager, self-employed	.02	.02
Age 42–60	.46	.40	.45	.52	Professional/manager, skilled/supervisor	.02	.03
					Professional/manager, unskilled	.05	.06
Respondent’s class					Professional/manager, farm	.01	.00
Professional/ manager	.28	.34	.39	.43	Professional/manager, nonresident parent	.03	.04
Routine white collar	.20	.17	.14	.12	Professional/manager, out of labor force	.08	.07
Self- employed	.06	.07	.07	.06	Routine white collar, routine white collar	.01	.01
Skilled/ supervisor	.12	.11	.11	.10	Routine white collar, self-employed	.02	.01
Unskilled	.35	.31	.29	.29	Routine white collar, skilled/supervisor	.02	.03
					Routine white collar, unskilled	.03	.03
Father’s class					Routine white collar, farm	.00	.00
Professional/ manager	.14	.19	.23	.25	Routine white collar, nonresident parent	.02	.03
Routine white collar	.04	.04	.05	.03	Routine white collar, out of labor force	.02	.01
Self- employed	.12	.12	.10	.10	Self-employed, self-employed	.01	.01
Skilled/ supervisor	.17	.16	.16	.16	Self-employed, skilled/supervisor	.01	.01
Unskilled	.26	.24	.22	.23	Self-employed, unskilled	.02	.02
Farm	.16	.11	.08	.06	Self-employed, farm	.00	.00
Nonresident parent	.12	.13	.15	.18	Self-employed, nonresident parent	.01	.01
					Self-employed, out of labor force	.04	.04
					Skilled/supervisor, skilled/ supervisor	.01	.01
					Skilled/supervisor, unskilled	.06	.07
					Skilled/supervisor, farm	.00	.00
					Skilled/supervisor, nonresident parent	.01	.02
					Skilled/supervisor, out of labor force	.06	.04
					Unskilled, unskilled	.09	.08
					Unskilled, farm	.02	.01
					Unskilled, nonresident parent	.07	.07
					Unskilled, out of labor force	.07	.06
					Farm, farm	.00	.00
					Farm, nonresident parent	.01	.00
					Farm, out of labor force	.04	.03
					Nonresident parent, out of labor force	.03	.02
Sample size	6,854	7,902	9,221	7,422	Sample size	7,138	7,278

Mother’s class is only available starting in 1994.

We have coded age into overlapping categories (25–40, 32–50, 42–60) because the data are very sparse. This decision to resort to overlapping categories entails no methodological complication given that our key comparisons are across periods but within age groups. For those who prefer nonoverlapping categories, it is merely a matter of confining attention to our youngest and oldest age groups. In Appendix A, Figure A1 offers a graphical representation of the respondents in each of the age groups by listing their birth cohort, exact age, years observed, and the years in which they were 16 to 18 years old.

The GSS provides information on father’s occupation, mother’s occupation, respondent’s occupation (at the time of the survey), sex, age, and other variables that aid in coding social class (e.g., employment status). Unfortunately, the GSS does not include a quantitative measure of parental income, thus making it impossible to carry out a direct individual-level test of the income hypothesis. Although the GSS does include a subjective measure of relative income, a measure of this sort is unusable for our purposes because changes in inequality may leave rank in the income distribution unaffected.

For the sake of comparability with most research in this area, we use an approximation to the Erikson, Goldthorpe, and Portacarero (EGP) class scheme (see, e.g., Erikson and Goldthorpe 1992).¹³ This scheme includes the following class categories: (1) professionals and managers (EGP I/II), (2) routine white-collar workers (EGP IIIa), (3) self-employed (nonfarm) workers with and without employees (EGP IVab), (4) skilled manual workers and supervisors of manual workers (EGP V/VI), (5) unskilled manual and nonmanual workers (EGP VIIa/IIIb), and (6) self-employed farmers and farm laborers (EGP IVc/VIIb).¹⁴

We have made small adjustments to the EGP categories to prevent various types of selective processes from affecting our assessments of trend. Rather than excluding respondents who, at age 16, had one or more absent parents, we have added a new category to the EGP scheme, labeled “nonresident parent.” Likewise, rather than excluding mothers who are out of the labor force, we have again treated “out of the labor force” as a distinct class category. The resulting expanded scheme appears in Table 2.

Table 2

Expanded Class Schemas

Full class schema, Respondent

Professionals/Managers

Routine white collar

Self-Employed

Skilled/Supervisor

Unskilled

Full class schema, Father

Professionals/Managers

Routine white collar

Self-Employed

Skilled/Supervisor

Unskilled

Farm

Nonresident parent

Full class schema, Mother (available from 1994 on)

Professionals/Managers

Routine white collar

Self-Employed

Skilled/Supervisor

Unskilled

Farm

Nonresident parent

Out of labor force

NOTE: For respondents not living with their father at age 16, the occupation of a father substitute (i.e., stepfather, male relative, or other male adult with whom respondent was living) was used when available. If the respondent was not living with any male adult, the origin was coded as “nonresident father.” The same rules were used to construct the mothers’ expanded class. In models using the combined father-mother class, the scheme represents class combinations without taking into account which class pertains to which parent. The combinations “Farm, Self-Employed” and “Nonresident parent, Nonresident parent” were dropped to avoid zero marginal totals.

The results that Beller (2009) reports make it clear that, whenever possible, it is important to use information from both parents simultaneously. Although it is conventional to simply take the “dominant occupation” and characterize the family’s class situation in terms of it, we think it is important to more fully exploit the information that mixed-class families convey. At the same time, because our samples are too small to use the full cross-classification of father’s and mother’s categories, we have instead developed a symmetric scheme that aggregates across each possible type of a mixed-class family. The family with a professional father and self-employed mother is, for example, coded into the same category as a family with a professional mother and self-employed father. This approach, which yields a total of thirty-four class categories (see Table 1), allows us to take seriously Beller’s (2009) injunction that mothers matter while also recognizing that, given the GSS data, sample-size constraints are binding. We will refer to the resulting classification as our “two-parent” scheme.

The top income hypothesis implies that the strongest evidence of change will be found in the odds ratios pertaining to the professional-managerial class. This hypothesis can be tested by creating subtables that isolate the odds ratios that it identifies as most and least affected by the takeoff. We thus proceed by creating four types of mobility tables: (1) “full class” tables that use all categories of the class schemes of Table 2; (2) residual “NPM tables” that are formed by excluding all origin and destination categories pertaining to the professional-managerial class;¹⁵ (3) extended professional-managerial tables (“PM-1 tables”) that collapse the routine white collar, self-employed, skilled/supervisor, unskilled, and farm categories into a single non-PM category and retain all other categories (including “nonresident” and “out of labor force”);¹⁶ and (4) reduced professional-managerial tables (“PM-2 tables”) that combine the non-PM category in the PM-1 tables with all other categories (“nonresident,” “out of labor force”) and thus take a simple 2 × 2 form.¹⁷

If the top income hypothesis is on the mark, the trend should be strongest in those tables that isolate the odds ratios pertaining to the professional-managerial class (i.e., the PM-1 and PM-2 tables). By contrast, the “full class” tables provide a blunter tool with which to detect trend, as they also incorporate odds ratios that, under the top income hypothesis, are not expected to change very much. Finally, the top income hypothesis implies that the NPM tables should reveal very little evidence of a takeoff-induced trend, as they do not contain the very odds ratios (i.e., those pertaining to the professional-managerial class) that are most affected by rising inequality. If, on the other hand, the simple income hypothesis is on the mark, the odd ratios in the full, NPM, and PM tables should change at roughly the same pace.¹⁸

We restrict the analysis to male and female respondents between 25 and 60 years old (inclusive) who have nonmissing responses on age, gender, social class, and parental class (for either the father-only or two-class schemes). For each of the above types of mobility table, we have constructed separate cross-classifications for all combinations of gender, age group, and period. This yields a total of twenty-four mobility tables for our analyses based on father’s class and a total of twelve mobility tables for our analyses based on the two-parent scheme. We have only twelve tables in the latter case because a measure of mother’s occupation is only available within the GSS for the last two time periods (i.e., starting in 1994).

As noted above, our analyses will be based on age-specific tables for each period, an approach that privileges age and period over cohort. It is of course more common to apply a cohort approach when studying trends in social fluidity (e.g., Breen and Jonsson 2007). When Beller (2009), for example, analyzed trends in social fluidity, she adopted a cohort approach that involved applying controls for age and its square in her models. These controls are needed to statistically adjust for the widely different ages at which members from different cohorts are observed. There are, however, two important drawbacks to her approach: (1) the validity of the estimates depends on getting the class-age profile right (see Hertz [2007] for a related point in the context of economic mobility) and (2) the biasing effects of changes in the process of selection into the labor force could be substantial (especially of course for women).

How, then, do we proceed? Unlike Beller (2009), we rely on cross-period comparisons of respondents falling within the same age group, with a particular focus on the 25 to 40 age group. This group is of special interest because, for the latest period, virtually all of its members were exposed to the takeoff while still children. By contrast, the vast majority of those 25 to 40 years old in the 1970s and 1980s became 18 before 1980, which means that they grew up before the takeoff in income inequality unfolded (see Figure A1). The 1990s is an intermediary period because only about half of those in the 25 to 40 age group during the 1990s grew up during the takeoff. Even when privileged children did experience the takeoff before becoming adults, they were nonetheless often teenagers when it commenced, which means that their parents typically did not receive their extra infusion of money in time to make early human capital investments (such as purchasing high-quality preschool). Moreover, this group experienced the initial part of the takeoff, when better-off parents were not yet receiving as much additional income as they later did. The upshot is that, within our young age group, the effects of the takeoff might start to operate in the 1980s (through “adult mechanisms” exclusively), should begin to show up with some force in the 1990s, and will register far more prominently in the 2000s.

We have to bear in mind, however, that two main forces are likely affecting the trend, not just exposure to the takeoff (i.e., the “income hypothesis”) but also to educational expansion (i.e., the “education hypothesis”). Although we could have proceeded by disaggregating our mobility tables by education (e.g., Breen 2010a), it is inadvisable to do so with the full-class tables, given how small our samples are. The fallback solution is to at least examine the relevant age-specific time series on college education (see Figure 4). As Figure 4 indicates, both women and men experienced a sharp upward trend in the proportion with college degrees, although the upward trend in the youngest cohort is much more pronounced for women than for men (until the last decade).

Figure 4

Percentage with a Bachelor’s Degree or More

The foregoing commentary can be formalized in the six hypotheses listed in Table 3. The first hypothesis expresses the relatively wide range of trend lines that the countervailing forces of educational expansion and rising inequality might produce. If the effects of the takeoff are exceedingly strong, they may more than offset the effects of educational expansion, thus producing an accelerating rigidification of the sort featured in the direst predictions. If, however, the effects of the takeoff are comparatively weak, we will only find a slowdown in the rate of decline in the origin-destination association. It follows that the association might (1) increase at an accelerating rate between the 1970s and the 2000s, (2) decrease at a decreasing rate between the 1970s and 2000s, or (3) stop decreasing and start increasing as the inequality takeoff unfolds. These three possibilities all imply that, between the 1970s and 2000s, the trend in association is convex towards the time axis. We therefore label it the “convexity hypothesis.”

Table 3

Hypotheses about Trend in Origin-Destination Association for the Youngest Age Group (Ages 25–40)

Name	Description	Results
1. Convexity	The origin-destination association is convex toward the time axis	Supported
2. Accelerating change	The origin-destination association either (1) increases more or falls less between the 1990s and the 2000s than in prior periods or (2) falls during the decades up to the 1990s but increases after the 1990s	Supported
3. Initial gender-specific change	The decline in association between the 1970s and 1980s is more prominent for women than for men	Supported
4. Recent gender-neutral change	The trend in association is the same for women and men between the 1990s and 2000s	Point estimates are inconsistent with hypothesis, but null hypothesis of equality of change is not rejected
5. Simple income	The trends are similar across the full class, NPM, and PM tables	Not supported
6. Top income	The trends are more prominent within the PM tables	Supported

NOTE: FC = full class; NPM = non-professionals/managers; PM = professionals/managers.

The second hypothesis follows from our argument that the 2000s is the home ground of the income hypothesis. In this most recent decade, the youngest age group was exposed to the takeoff from a very young age and in its more advanced and extreme stages, meaning that the intergenerational association will either increase more sharply or decline less sharply than in prior periods. We have thus labeled it the “accelerating change hypothesis.”

The third hypothesis notes that, because the educational expansion was markedly more prominent for women than men between the 1970s and 1980s (see Figure 4), the decline in the association parameter should likewise be more prominent. This hypothesis assumes that, between the 1970s and 1980s, it was too early for the takeoff to have had any significant effects and that the education expansion was accordingly the main force behind change. The fourth hypothesis states that, because men and women experienced roughly the same educational expansion after the 1990s, we should not expect any major gender difference in the trend line for the association parameters. These two hypotheses thus refer to “gender-specific” change in the early decades and “gender-neutral” change in the more recent periods.

The final two hypotheses in Table 3 pertain to the locus of the takeoff’s effects. The top-income hypothesis assumes that the takeoff mainly affects the advantage accruing to the professional-managerial class, while the simple income hypothesis assumes that the takeoff affects exchange among all classes. We adjudicate between these two hypotheses by determining whether the trends pertaining to the professional-managerial odds ratios are especially prominent.

Evidence for the Young Age Group

We proceed by estimating simple association models that condition on a common pattern of odds ratios but allow the strength of these odds ratios to vary across periods (Xie 1992; Erikson and Goldthorpe 1992). The resulting “unidiff model” may be represented as follows:

\log (F_{i j k}) = λ + λ_{i}^{O} + λ_{j}^{D} + λ_{k}^{P} + λ_{i k}^{O P} + λ_{j k}^{D P} + ψ_{i j} ϕ_{k},

where $F_{i j k}$ is the expected frequency, $λ$ is the intercept, $λ_{i}^{O}$ is the marginal effect for the ith origin class, $λ_{j}^{D}$ is the marginal effect for the jth destination class, $λ_{k}^{P}$ is the marginal effect for the kth period, $λ_{i k}^{O P}$ captures changes across periods in the origin marginal effects, $λ_{j k}^{D P}$ captures changes across periods in the destination marginal effects, $ψ_{i j}$ refers to the interactions between origin and destination class, and $ϕ_{k}$ indexes the extent to which those interactions grow weaker or stronger across periods. In most cases, we report the estimate for $ϕ_{k} / ϕ_{1990 s}$ , as the 1990s is a convenient contrast category in assessing many of the hypotheses discussed above. We instead report $ϕ_{k} / ϕ_{1970 s}$ whenever doing so suits the hypothesis we seek to examine. For purposes of comparison, we will also fit (1) the model of conditional independence (which drops $ψ_{i j} ϕ_{k}$ ) and (2) the model of constant association (which drops $ϕ_{k}$ ). We will carry out one-tail tests of our hypotheses, as appropriate, within the PM-2 tables (see Table 3). The derivation for these tests can be found in Appendix B.

The results from our analysis are presented in Tables 4a to 4g, Tables 5 and 6, and Figures 5a to 5f. The fit statistics for the models are available in Tables 4a to 4g; the estimates of $ϕ_{k} / ϕ_{1990 s}$ or $ϕ_{k} / ϕ_{1970 s}$ are available in Table 5; one-sided tests of change between the 1990s and 2000s are available in Table 6; and line graphs of estimates of $ϕ_{k} / ϕ_{1990 s}$ are available in Figures 5a to 5f. Because the results and fit statistics for the PM-1 and PM-2 models are very similar, we will only discuss the results for the PM-2 models.

Table 4A

Results of Fitting Log-Linear Models to Father × Respondent Full-Class Mobility Tables

Four Periods (1970s, 1980s, 1990s, 2000s)								Two Periods (1990s, 2000s)
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	7,333	96	829.7	.00	−24.7	14		CI	3,670	48	444.9	0	50.9	14.5
CA	7,333	72	88.5	.09	−552.3	3.7		CA	3,670	24	28.3	.25	−168.7	3
Unidiff	7,333	69	85.2	.09	−528.9	3.5	.352	Unidiff	3,670	23	26.9	.26	−161.8	2.9	.242
Women 25–40								Women 25–40
CI	8,855	96	720.70	.00	−151.8	11.4		CI	4,479	48	346.9	0	−56.7	11.2
CA	8,855	72	88.80	.09	−565.6	3.5		CA	4,479	24	36	.05	−165.8	3.1
Unidiff	8,855	69	67.00	.55	−560.1	2.7	.000	Unidiff	4,479	23	31.4	.11	−162	2.9	.032
Men 32–50								Men 32–50
CI	8,046	96	968.1	.00	104.8	14.4		CI	4,445	48	559.9	0	156.7	15
CA	8,046	72	82.9	.18	−564.5	3.6		CA	4,445	24	15	.92	−186.6	2.1
Unidiff	8,046	69	81.0	.15	−539.5	3.5	.593	Unidiff	4,445	23	14.8	.9	−178.4	2	.631
Women 32–50								Women 32–50
CI	9,491	96	776.40	.00	−102.8	11.4		CI	5,310	48	385.8	0	−25.9	10.4
CA	9,491	72	125.40	.00	−534.0	4.2		CA	5,310	24	44.2	.01	−161.7	3.2
Unidiff	9,491	69	104.30	.00	−527.6	3.8	.000	Unidiff	5,310	23	43.7	.01	−153.6	3.1	.497
Men 42–60								Men 42–60
CI	6,565	96	860.3	.00	16.6	14.7		CI	3,623	48	449.4	0	56	14.8
CA	6,565	72	100.8	.01	−532.0	4.5		CA	3,623	24	30.4	.17	−166.3	3.5
Unidiff	6,565	69	94.8	.02	−511.7	4.2	.109	Unidiff	3,623	23	29.3	.17	−159.2	3.4	.306
Women 42–60								Women 42–60
CI	7,736	96	685.90	.00	−173.6	12.2		CI	4,346	48	293.4	0	−108.7	10.5
CA	7,736	72	136.40	.00	−508.2	4.7		CA	4,346	24	28.6	.24	−172.5	2.7
Unidiff	7,736	69	113.10	.00	−504.7	4.1	.000	Unidiff	4,346	23	28.6	.19	−164.1	2.7	.975

NOTE: CI = conditional independence model; CA = constant association model; BIC = Bayesian information criterion; DI = Dissimilarity Index. “P diff” is the p-value from a chi-square test of the difference in G² between the constant association and unidiff models.

Table 4B

Results of Fitting Log-Linear Models to Father × Respondent NPM Mobility Tables

Four Periods (1970s, 1980s, 1990s, 2000s)								Two Periods (1990s, 2000s)
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	3,947	60	273.3	.00	−223.5	9.6		CI	1,884	30	135.3	0	−90.9	9.4
CA	3,947	45	54.2	.16	−318.4	3.7		CA	1,884	15	16.3	.37	−96.9	2.6
Unidiff	3,947	42	53.5	.11	−294.3	3.6	.870	Unidiff	1,884	14	16.2	.3	−89.3	2.6	.960
Women 25–40								Women 25–40
CI	4,668	60	147.60	.00	−359.3	6.5		CI	2,117	30	60.4	0	−169.3	5.9
CA	4,668	45	46.10	.43	−334.1	3.1		CA	2,117	15	18.4	.24	−96.4	3.2
Unidiff	4,668	42	44.30	.37	−310.5	3	.617	Unidiff	2,117	14	18.3	.19	−88.9	3	.701
Men 32–50								Men 32–50
CI	4,368	60	311.4	.00	−191.5	9.6		CI	2,240	30	158.8	0	−72.6	9.5
CA	4,368	45	38.5	.74	−338.7	3.5		CA	2,240	15	11.2	.73	−104.5	2.8
Unidiff	4,368	42	37.2	.68	−314.8	3.3	.730	Unidiff	2,240	14	10.4	.73	−97.6	2.5	.352
Women 32–50								Women 32–50
CI	5,121	60	194.70	.00	−317.8	6.9		CI	2,504	30	68.6	0	−166.2	5.1
CA	5,121	45	61.50	.05	−322.9	4		CA	2,504	15	12.7	.62	−104.7	2.4
Unidiff	5,121	42	52.60	.13	−306.1	3.6	.031	Unidiff	2,504	14	12.5	.56	−97	2.4	.671
Men 42–60								Men 42–60
CI	3,788	60	334.6	.00	−159.7	10.7		CI	1,891	30	172.7	0	−53.6	10.9
CA	3,788	45	59.1	.08	−311.7	4.5		CA	1,891	15	17	.32	−96.2	3.5
Unidiff	3,788	42	58.2	.05	−287.8	4.4	.839	Unidiff	1,891	14	16	.31	−89.6	3.6	.330
Women 42–60								Women 42–60
CI	4,450	60	234.70	.00	−269.3	8.3		CI	2,096	30	71.6	0	−157.8	5.4
CA	4,450	45	78.80	.00	−299.2	4.8		CA	2,096	15	13.2	.59	−101.6	2.7
Unidiff	4,450	42	64.20	.02	−288.7	3.4	.002	Unidiff	2,096	14	13.1	.52	−94	2.7	.824

NOTE: See Table 4A for definitions.

Table 4C

Results of Fitting Log-Linear Models to Father × Respondent PM-1 Mobility Tables

Four Periods (1970s, 1980s, 1990s, 2000s)								Two Periods (1990s, 2000s)
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	7,333	8	392.9	.00	321.7	9.5		CI	3,670	4	246.6	0	213.8	11.2
CA	7,333	6	6.2	.40	−47.2	1.1		CA	3,670	2	2.8	.25	−13.6	1.1
Unidiff	7,333	3	1.6	.66	−25.1	0.4	.205	Unidiff	3,670	1	0.5	.49	−7.7	0.4	.128
Women 25–40								Women 25–40
CI	8,855	8	332.50	.00	259.8	7.7		CI	4,479	4	186.6	0	152.9	8.6
CA	8,855	6	13.70	.03	−40.8	1.6		CA	4,479	2	6.4	.04	−10.5	1.5
Unidiff	8,855	3	2.80	.42	−24.5	0.6	.012	Unidiff	4,479	1	0.7	.39	−7.7	0.5	.018
Men 32–50								Men 32–50
CI	8,046	8	460.0	.00	388.1	9.5		CI	4,445	4	301.2	0	267.6	11.2
CA	8,046	6	12.4	.05	−41.5	1.4		CA	4,445	2	1.8	.41	−15	0.7
Unidiff	8,046	3	9.5	.02	−17.5	0.8	.402	Unidiff	4,445	1	0.6	.45	−7.8	0.4	.276
Women 32–50								Women 32–50
CI	9,491	8	310.90	.00	237.7	6.9		CI	5,310	4	207.6	0	173.3	8.1
CA	9,491	6	9.10	.17	−45.8	1		CA	5,310	2	1	.62	−16.2	0.5
Unidiff	9,491	3	6.20	.10	−21.2	0.8	.411	Unidiff	5,310	1	0.6	.45	−8	0.4	.526
Men 42–60								Men 42–60
CI	6,565	8	305.0	.00	234.7	7.8		CI	3,623	4	161.7	0	129	8.6
CA	6,565	6	10.9	.09	−41.9	1.3		CA	3,623	2	0.8	.66	−15.6	0.7
Unidiff	6,565	3	3.8	.29	−22.6	0.6	.069	Unidiff	3,623	1	0.4	.55	−7.8	0.3	.497
Women 42–60								Women 42–60
CI	7,736	8	222.70	.00	151.1	5.9		CI	4,346	4	139.7	0	106.2	6.8
CA	7,736	6	13.40	.04	−40.3	1.4		CA	4,346	2	2.2	.34	−14.6	0.7
Unidiff	7,736	3	12.70	.01	−14.1	1.2	.875	Unidiff	4,346	1	1.9	.17	−6.5	0.8	.603

NOTE: See Table 4A for definitions.

Table 4D

Results of Fitting Log-Linear Models to Father × Respondent PM-2 Mobility Tables

Four Periods (1970s, 1980s, 1990s, 2000s)								Two Periods (1990s, 2000s)
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	7,333	4	390.7	0	355.1	9.5		CI	3,670	2	244.7	0	228.3	11.2
CA	7,333	3	4.4	.22	−22.3	0.8		CA	3,670	1	2.4	.12	−5.8	1.1
Unidiff	7,333	0	0	n/a	0	0	.222	Unidiff	3,670	0	0	n/a	0	0	.121
Women 25–40								Women 25–40
CI	8,855	4	327.7	0	291.3	7.7		CI	4,479	2	185.2	0	168.4	8.6
CA	8,855	3	11.9	.01	−15.4	1.4		CA	4,479	1	6	.01	−2.4	1.5
Unidiff	8,855	0	0	n/a	0	0	.008	Unidiff	4,479	0	0	n/a	0	0	.014
Men 32–50								Men 32–50
CI	8,046	4	450.4	0	414.4	9.4		CI	4,445	2	299.6	0	282.8	11.2
CA	8,046	3	2.8	.42	−24.2	0.7		CA	4,445	1	1.2	.27	−7.2	0.7
Unidiff	8,046	0	0	n/a	0	0	.420	Unidiff	4,445	0	0	n/a	0	0	.272
Women 32–50								Women 32–50
CI	9,491	4	302.9	0	266.3	6.9		CI	5,310	2	200.9	0	183.7	8.1
CA	9,491	3	2.7	.44	−24.8	0.5		CA	5,310	1	0.3	.57	−8.3	0.3
Unidiff	9,491	0	0	n/a	0	0	.441	Unidiff	5,310	0	0	n/a	0	0	.568
Men 42–60								Men 42–60
CI	6,565	4	301.2	0	266.1	7.8		CI	3,623	2	160.9	0	144.5	8.6
CA	6,565	3	7.1	.07	−19.3	1.1		CA	3,623	1	0.4	.51	−7.8	0.4
Unidiff	6,565	0	0	n/a	0	0	.069	Unidiff	3,623	0	0	n/a	0	0	.508
Women 42–60								Women 42–60
CI	7,736	4	207.3	0	171.5	5.8		CI	4,346	2	126.1	0	109.3	6.8
CA	7,736	3	1.3	.73	−25.6	0.4		CA	4,346	1	0	.86	−8.3	0.1
Unidiff	7,736	0	0	n/a	0	0	.728	Unidiff	4,346	0	0	n/a	0	0	.861

NOTE: See Table 4A for definitions.

Table 4E

Results of Fitting Log-Linear Models to Two-Parent × Respondent Full-Class and NPM Mobility Tables (1990s, 2000s)

Full-Class								NPM
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	3,125	256	721.9	.00	−1338.2	19.6		CI	1,365	144	247.7	.00	−791.8	14.5
CA	3,125	128	166.1	.01	−863.9	7.3		CA	1,365	72	87.5	.10	−432.3	7.3
Unidiff	3,125	127	163.5	.02	−858.5	7.4	.108	Unidiff	1,365	71	87.6	.09	−425.0	7.2	1.000
Women 25–40								Women 25–40
CI	3,796	256	667.10	.00	−1442.8	16.8		CI	1,504	144	216.10	.00	−837.4	12
CA	3,796	128	147.30	.12	−907.7	6.4		CA	1,504	72	82.70	.18	−444.1	7.6
Unidiff	3,796	127	145.00	.13	−901.7	6.1	.130	Unidiff	1,504	71	79.20	.24	−440.2	6.6	.062
Men 32–50								Men 32–50
CI	3,836	256	818.3	.00	−1294.3	18.9		CI	1,719	144	296.0	.00	−776.8	14.3
CA	3,836	128	150.7	.08	−905.6	6.5		CA	1,719	72	85.9	.13	−450.5	7.5
Unidiff	3,836	127	150.8	.07	−897.3	6.5	1.000	Unidiff	1,719	71	83.6	.14	−445.3	7.1	.133
Women 32–50								Women 32–50
CI	4,549	256	779.10	.00	−1377.1	16.5		CI	1,894	144	234.90	.00	−851.8	11.3
CA	4,549	128	152.00	.07	−926.1	6.1		CA	1,894	72	73.70	.42	−469.6	5.6
Unidiff	4,549	127	151.80	.07	−917.8	6.1	.723	Unidiff	1,894	71	73.60	.39	−462.2	5.6	.776
Men 42–60								Men 42–60
CI	3,205	256	683.2	.00	−1383.4	18.7		CI	1,522	144	278.3	.00	−776.9	15
CA	3,205	128	122.8	.61	−910.5	6.1		CA	1,522	72	62.9	.77	−464.7	6.5
Unidiff	3,205	127	122.6	.59	−902.6	6.1	.727	Unidiff	1,522	71	61.8	.77	−458.5	6.3	.288
Women 42–60								Women 42–60
CI	3,831	256	639.30	.00	−1472.9	15.4		CI	1,677	144	231.60	.00	−837.6	11
CA	3,831	128	134.50	.33	−921.6	5.6		CA	1,677	72	79.90	.25	−454.7	6.4
Unidiff	3,831	127	134.60	.31	−913.3	5.6	1.000	Unidiff	1,677	71	78.20	.26	−449.0	6.3	.191

NOTE: See Table 4A for definitions.

Table 4F

Results of Fitting Log-Linear Models to Two-Parent × Respondent PM-1 and PM-2 Mobility Tables (1990s, 2000s)

PM-1								PM-2
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	3,129	14	266.3	.00	153.7	12.8		CI	3,129	2	229.2	0	213.1	12.8
CA	3,129	7	6.0	.55	−50.4	1.5		CA	3,129	1	2	.16	−6.1	1.2
Unidiff	3,129	6	3.5	.74	−44.8	1.2	.117	Unidiff	3,129	0	0	n/a	0	0	.161
Women 25–40								Women 25–40
CI	3,800	14	201.40	.00	86.0	9.7		CI	3,800	2	158	0	141.5	9.7
CA	3,800	7	12.90	.07	−44.8	2.3		CA	3,800	1	4.3	.04	−4	1.6
Unidiff	3,800	6	9.10	.17	−40.3	1.9	.052	Unidiff	3,800	0	0	n/a	0	0	.039
Men 32–50								Men 32–50
CI	3,839	14	291.6	.00	176.1	12.1		CI	3,839	2	256.2	0	239.7	12.1
CA	3,839	7	7.1	.42	−50.7	1.5		CA	3,839	1	3.1	.08	−5.2	1.3
Unidiff	3,839	6	4.1	.66	−45.4	1.0	.087	Unidiff	3,839	0	0	n/a	0	0	.079
Women 32–50								Women 32–50
CI	4,556	14	302.30	.00	184.4	10.2		CI	4,556	2	218.9	0	202	10.2
CA	4,556	7	17.40	.02	−41.6	2.7		CA	4,556	1	0.5	.48	−7.9	0.5
Unidiff	4,556	6	17.30	.01	−33.2	2.7	.830	Unidiff	4,556	0	0	n/a	0	0	.479
Men 42–60								Men 42–60
CI	3,208	14	159.4	.00	46.4	9.4		CI	3,208	2	139.5	0	123.4	9.4
CA	3,208	7	4.6	.71	−51.9	1.2		CA	3,208	1	0.1	.72	−7.9	0.3
Unidiff	3,208	6	4.5	.61	−43.9	1.3	.826	Unidiff	3,208	0	0	n/a	0	0	.715
Women 42–60								Women 42–60
CI	3,837	14	221.30	.00	105.8	9.1		CI	3,837	2	157.7	0	141.2	9.1
CA	3,837	7	3.50	.84	−54.3	1.3		CA	3,837	1	0.3	.57	−7.9	0.4
Unidiff	3,837	6	3.00	.81	−46.5	1.1	.491	Unidiff	3,837	0	0	n/a	0	0	.569

NOTE: See Table 4A for definitions.

Table 4G

Results of Fitting Log-Linear Models to Father × Respondent Mobility Tables (1970s, 1980s)

Full-Class								PM-2
Model	N	df	G ²	p	BIC	DI	P diff	Model	N	df	G ²	p	BIC	DI	P diff
Men 25–40								Men 25–40
CI	3,663	48	383.8	0	−10	13.6		CI	3,663	2	144.4	0	128	7.6
CA	3,663	24	29.8	.19	−167.1	3.1		CA	3,663	1	0.1	.8	−8.1	0.2
Unidiff	3,663	23	28.7	.19	−160	3	.292	Unidiff	3,663	0	0	n/a	0	0	.803
Women 25–40								Women 25–40
CI	4,376	48	373.9	0	−28.5	11.6		CI	4,376	2	142.4	0	125.7	6.7
CA	4,376	24	30.6	.17	−170.6	3		CA	4,376	1	5.8	.02	−2.6	1.4
Unidiff	4,376	23	16.7	.82	−176.1	1.8	.000	Unidiff	4,376	0	0	n/a	0	0	.016
Men 32–50								Men 32–50
CI	3,601	48	409	0	15.9	13.6		CI	3,601	2	152.3	0	135.9	7.3
CA	3,601	24	40.5	.02	−156	4.1		CA	3,601	1	1.4	.24	−6.8	0.7
Unidiff	3,601	23	39.6	.02	−148.7	4	.343	Unidiff	3,601	0	0	n/a	0	0	.238
Women 32–50								Women 32–50
CI	4,181	48	389.4	0	−10.9	12.6		CI	4,181	2	102.8	0	86.1	5.3
CA	4,181	24	34	.09	−166.1	3.3		CA	4,181	1	2.1	.15	−6.2	0.8
Unidiff	4,181	23	22.9	.46	−168.8	2.5	.001	Unidiff	4,181	0	0	n/a	0	0	.148
Men 42–60								Men 42–60
CI	2,942	48	409	0	25.6	14.7		CI	2,942	2	139.9	0	124	6.8
CA	2,942	24	19.7	.71	−171.9	3.1		CA	2,942	1	0.5	.49	−7.5	0.4
Unidiff	2,942	23	19.1	.7	−164.6	2.9	.413	Unidiff	2,942	0	0	n/a	0	0	.486
Women 42–60								Women 42–60
CI	3,390	48	387.6	0	−2.6	14.2		CI	3,390	2	81.7	0	65.4	4.6
CA	3,390	24	33.2	.1	−161.9	3.5		CA	3,390	1	0	.93	−8.1	0
Unidiff	3,390	23	28.5	.2	−158.5	3.2	.030	Unidiff	3,390	0	0	n/a	0	0	.933

NOTE: See Table 4A for definitions.

Table 5

Unidiff Parameter Phi

	Sons				Daughters
	1970s	1980s	1990s	2000s	1970s	1980s	1990s	2000s
Respondent ages 25–40
Father, all periods, full-class	1.05	0.94	1.00	1.14	1.68	1.08	1.00	1.29
Father, all periods, NPM	1.06	0.91	1.00	1.03	1.27	1.12	1.00	0.76
Father, all periods, PM-1	0.91	0.95	1.00	1.22	1.46	0.99	1.00	1.43
Father, all periods, PM-2	0.92	0.96	1.00	1.22	1.46	0.97	1.00	1.45
Father, 1970s-1980s, full-class	1.00	0.89			1.00	0.65
Father, 1970s-1980s, NPM	1.00	0.85			1.00	0.88
Father, 1970s-1980s, PM-1	1.00	1.05			1.00	0.68
Father, 1970s-1980s, PM-2	1.00	1.04			1.00	0.67
Father, 1990s-2000s, full-class			1.00	1.13			1.00	1.31
Father, 1990s-2000s, NPM			1.00	0.99			1.00	0.85
Father, 1990s-2000s, PM-1			1.00	1.22			1.00	1.43
Father, 1990s-2000s, PM-2			1.00	1.22			1.00	1.45
Two-parent, 1990s-2000s, full-class			1.00	1.19			1.00	1.19
Two-parent, 1990s-2000s, NPM			1.00	0.93			1.00	0.52
Two-parent, 1990s-2000s, PM-1			1.00	1.22			1.00	1.35
Two-parent, 1990s-2000s, PM-2			1.00	1.21			1.00	1.40
Respondent ages 32–50
Father, all periods, full-class	1.05	0.91	1.00	1.02	1.64	1.09	1.00	1.08
Father, all periods, NPM	0.89	0.91	1.00	0.83	1.88	1.15	1.00	1.01
Father, all periods, PM-1	1.15	0.93	1.00	1.14	1.14	0.86	1.00	1.09
Father, all periods, PM-2	1.15	0.94	1.00	1.14	1.16	0.87	1.00	1.09
Father, 1970s-1980s, full-class	1.00	0.90			1.00	0.68
Father, 1970s-1980s, NPM	1.00	1.01			1.00	0.65
Father, 1970s-1980s, PM-1	1.00	0.84			1.00	0.76
Father, 1970s-1980s, PM-2	1.00	0.82			1.00	0.75
Father, 1990s-2000s, full-class			1.00	1.04			1.00	1.09
Father, 1990s-2000s, NPM			1.00	0.85			1.00	1.13
Father, 1990s-2000s, PM-1			1.00	1.14			1.00	1.09
Father, 1990s-2000s, PM-2			1.00	1.14			1.00	1.09
Two-parent, 1990s-2000s, full-class			1.00	1.01			1.00	0.98
Two-parent, 1990s-2000s, NPM			1.00	0.76			1.00	0.99
Two-parent, 1990s-2000s, PM-1			1.00	1.24			1.00	1.03
Two-parent, 1990s-2000s, PM-2			1.00	1.25			1.00	1.10
Respondent ages 42–60
Father, all periods, full-class	1.16	1.10	1.00	0.89	1.70	1.35	1.00	0.98
Father, all periods, NPM	0.99	0.95	1.00	0.86	2.12	1.76	1.00	0.83
Father, all periods, PM-1	1.36	1.21	1.00	0.90	1.12	1.17	1.00	1.06
Father, all periods, PM-2	1.36	1.21	1.00	0.90	1.18	1.20	1.00	1.03
Father, 1970s-1980s, full-class	1.00	0.91			1.00	0.77
Father, 1970s-1980s, NPM	1.00	0.93			1.00	0.82
Father, 1970s-1980s, PM-1	1.00	0.90			1.00	1.00
Father, 1970s-1980s, PM-2	1.00	0.89			1.00	1.02
Father, 1990s-2000s, full-class			1.00	0.90			1.00	1.00
Father, 1990s-2000s, NPM			1.00	0.84			1.00	0.94
Father, 1990s-2000s, PM-1			1.00	0.90			1.00	1.10
Father, 1990s-2000s, PM-2			1.00	0.90			1.00	1.03
Two-parent, 1990s-2000s, full-class			1.00	1.04			1.00	0.98
Two-parent, 1990s-2000s, NPM			1.00	1.22			1.00	0.69
Two-parent, 1990s-2000s, PM-1			1.00	0.96			1.00	1.11
Two-parent, 1990s-2000s, PM-2			1.00	1.06			1.00	1.10

Table 6

One-Sided Tests for Changes in Log Odds Ratio in PM-2 Models

	Men						Women
	Father				Two-Parent		Father				Two-Parent
	1990s–2000s		1970s–1980s		1990s–2000s		1990s–2000s		1970s–1980s		1990s–2000s
	Percent Change	p-value	Percent Change	p-value	Percent Change	p-value	Percent Change	p-value	Percent Change	p-value	Percent Change	p-value
Age 25–40	22%	0.061	4%	0.598	21%	0.081	45%	0.007	−34%	0.008	40%	0.020
Age 32–50	14%	0.136	−18%	0.119	25%	0.040	9%	0.284	−24%	0.074	10%	0.239
Age 42–60	−10%	0.746	−11%	0.243	7%	0.358	3%	0.430	2%	0.533	10%	0.284

NOTE: “Percent change” refers to the change in the log odds ratio in the PM-2 models (from the 1990s to the 2000s or from the 1970s to the 1980s). The null hypothesis for the 1990s–2000s one-sided tests is that the 1990s–2000s change in the log odds ratio is smaller than or equal to zero. The null hypothesis for the 1970s–1980s tests is that the 1970s–1980s change in the log odds ratio is greater than or equal to zero. See Appendix B for description and derivation of the test.

Figure 5A

Unidiff Parameter Phi, Ages 25–40, Father-Only Tables (Reference Period: 1990s)

Figure 5B

Unidiff Parameter Phi, Ages 25–40, Two-Parent Tables (Reference Period: 1990s)

Figure 5C

Unidiff Parameter Phi, Ages 32–50, Father-Only Tables (Reference Period: 1990s)

Figure 5D

Unidiff Parameter Phi, Ages 32–50, Two-Parent Tables (Reference Period: 1990s)

Figure 5E

Unidiff Parameter Phi, Ages 42–60, Father-Only Tables (Reference Period: 1990s)

Figure 5F

Unidiff Parameter Phi, Ages 42–60, Two-Parent Tables (Reference Period: 1990s)

We begin by evaluating our first two hypotheses. As may be recalled, these two hypotheses express the key argument that educational expansion drives the trend at the beginning of the time series, whereas the effects of rising income inequality increasingly express themselves at the end of the time series. This formulation leads to the hypothesis that the trend in association is convex to the time axis (i.e., hypothesis 1) and that the association-increasing effects of the takeoff become increasingly prominent at the end of the time series (i.e., hypothesis 2). Is there any evidence in support of these hypotheses?

There indeed is. This conclusion is evident by first examining the trend in phi within the father-only tables (see Figure 5a). In all cases, the point estimates reveal a curve that is convex to the time axis, a result that is consistent with the presumption that the effects of the takeoff are revealing themselves increasingly over time. For sons, the phi parameter in the full-class table drops from 1.05 in the 1970s to 0.94 in the 1980s, but then increases from 1.00 in the 1990s to 1.14 in the 2000s (Table 5, ages 25–40, row 1). In the PM-2 table, phi is roughly constant in the early decades (ranging from 0.92 in the 1970s to 1.00 in the 1990s), but then increases to 1.22 in the 2000s (Table 5, ages 25–40, row 4). The curve appears to be yet more convex for daughters. In the full-class table, phi drops from 1.68 in the 1970s to 1.08 and 1.00 in the 1980s and 1990s, but then goes back up to 1.29 in the 2000s (Table 5, ages 25–40, row 1). This “U-turn” assumes a near-perfect form in the corresponding PM-2 table. As Figure 5a and Table 5 reveal, phi drops from 1.46 in the 1970s to 0.97 and 1.00 in the 1980s and 1990s, but then restores to 1.45 in the 2000s (Table 5, ages 25–40, row 4). We are thus seeing convex curves within the father-only tables that are consistent with our first and second hypotheses.

Are these changes significant? Although the null hypothesis of no change in the association can only be rejected at conventional levels of significance in the models for daughters (see Tables 4a and 4d, left panels), that particular test is blunt and imperfect for the purposes of assessing whether the curve is convex.¹⁹ Moreover, when we carry out one-sided tests in the PM-2 tables, we find that the null hypothesis of no change between the 1990s and 2000s is rejected at p = .061 for sons and p = .007 for daughters (see Table 6). When we turn to the two-parent tables, we of course cannot estimate the full curve, as data are only available for the last two decades. The rate of change over these two decades is, however, roughly the same as shows up within our full father-only tables. These results, which appear in Figure 5b, reveal a nontrivial increase in the point estimates from the 1990s to the 2000s. For sons, the estimate of phi for the full-class model increases 19 percent from the 1990s to the 2000s (Table 5, ages 25–40, row 13), an increase that is significant in our two-sided test at p = .108 (Table 4e, left panel). Within the PM-2 table, the corresponding estimate of phi increases 21 percent over this period (Table 5, ages 25–40, row 16), an increase that is significant at p = .081 in the one-sided test (Table 6). For daughters, the estimate of phi increases 19 percent in the full-class table (Table 5, ages 25–40, row 13), a change that is only significant at p = .130 in our two-sided test (Table 4e, left panel). By contrast, the estimate increases by 40 percent in the PM-2 table (Table 5, ages 25–40, row 16), with the corresponding one-sided test significant at p = .020 (Table 6).

It is clear by now that the recent change is usually larger and more frequently significant (at conventional levels) in analyses pertaining to the professional-managerial odds ratios (i.e., PM-2 tables). This result, which is directly consistent with the top-income hypothesis, shows up even more clearly when attention turns to the NPM tables (which exclude the professional-managerial categories). As Figure 5a and 5b reveal, all evidence of an increase in association disappears in the NPM tables, a result that implies that the takeoff is mainly working to increase professional-managerial reproduction. Within the NPM tables for sons, phi increases by 3 percent between the 1990s and 2000s in the father-only analyses (Table 5, ages 25–40, row 1), while it decreases by 7 percent over this period in the two-parent analyses (Table 5, ages 25–40, row 14). Within the NPM tables for daughters, phi decreases by 24 percent between the 1990s and 2000s in the father-only analyses (Table 5, ages 25–40, row 1), while it decreases by 48 percent over this period in the two-parent analyses (Table 5, ages 25–40, row 14). These results provide strong support for the top-income hypothesis.

The only remaining hypotheses of interest pertain to the gender interactions. We have argued that, because the early expansion of education played out especially prominently for women, they would likely experience a correspondingly prominent initial decline in association. This gender interaction should, however, wither away in the more recent decade, when the rate of expansion became roughly the same for men and women. Are the data consistent with these expectations?

We do indeed find evidence of an initial decline in association that is more prominent for women than for men. When the left and right panels of Figure 5a are compared, we see that the association for sons changes only barely between the 1970s and 1980s, whereas the association for daughters drops substantially, just as their dramatic increase in college education would imply. These declines are significant for women (Table 5, ages 25–40, rows 5 and 8; Table 4g; Table 6). Moreover, when we test the null hypothesis that the log odds ratio is changing equally for women and men (in the PM-2 tables), that test is rejected with p = .058.

We care more, however, about trends in the recent period, when the takeoff in income inequality is in play. During this period, the educational expansion is no more prominent for women than for men (see Figure 4), which leads us to believe that the effect of the takeoff should register equivalently for each gender. The point estimates, which we have already presented, instead suggest a rather more prominent uptick in association for daughters (compare the left and right panels of Figure 5a). Although the point estimates are thus inconsistent with our expectations, we also cannot reject the claim that the trend is the same across genders. That is, when we test the null hypothesis that the log odds ratio is changing equally for women and men (in the PM-2 tables), that hypothesis cannot be rejected in either the father-only tables (p = .403) or the two-parent tables (p = .479).²⁰ The most appropriate conclusion given these results is to withhold judgment on this hypothesis.

Evidence from Older Age Groups

The foregoing evidence from the young age group is consistent with the argument that the takeoff in inequality has increased the intergenerational association. For sons and daughters alike, we find evidence that phi changes along a convex curve, a pattern that becomes even more pronounced for the tables pertaining to the professional-managerial odds ratios. The latter result is consistent with the top income hypothesis.

We can provide further evidence on the top income hypothesis by comparing these results to those that obtain for the two older age groups. As Figure 4 shows, education is again expanding within both of these older age groups, although that expansion slows down in the last decade for women and disappears for men. The education effect, by itself, should therefore lead to a decline within these older groups in the intergenerational association (albeit not for men in the last decade). In the oldest age group, one would not expect any substantial countervailing effect of the takeoff, given that these respondents mainly grew up before the takeoff was in play (even when they were observed in the 2000s). If any effects of the takeoff are to be found for this group, they would therefore mainly reflect the growing capacity of aging parents to assist their adult children, a type of takeoff effect that is likely to be comparatively minor. The same is not the case, however, for the intermediary group of 32- to 50-year-olds. Within the 2000s sample, approximately two-thirds of this age group became 18 after 1980, meaning that those with privileged origins could have benefited, as children, from the infusion of cash that their parents were beginning to receive. It follows that we should find some evidence of a takeoff effect, at least in the 2000s, within this intermediary age group.

The results are largely consistent with these expectations. If we consider first the oldest age group (i.e., 42–60 years old), the estimate for phi declines from 1.16 to 0.89 for the full father-by-son tables and from 1.70 to 0.98 for the full father-by-daughter tables (Table 5, ages 42–60, row 1). The test statistics in Table 4a reveal that this change is significant at p = .109 for sons and p = .000 for daughters. The corresponding PM-2 tables show a steep decline for sons but not for daughters, whereas the point estimates for the two-parent tables are highly variable, with some suggesting a decline, others suggesting stability, and yet others suggesting an increase. The null hypothesis of no change in association cannot be rejected at any conventional significance level for the latter two-parent tables (see Tables 4e and 4f). Taken together, we thus see some evidence of an overall decline in association, just as the education hypothesis would suggest. As anticipated, the origin-destination association does not appear to have increased in the 2000s, presumably because this age group is too old to have experienced the takeoff as a child.

If the middle age group (i.e., 32–50 years old) is considered next, here we expect to find some evidence of the takeoff’s effects in the 2000s, as members of this age group did frequently experience it while still a child. We may expect, therefore, results that are midway between those of the youngest age group and those of the oldest age group. This is indeed what we largely find. If we consider, for example, the change in the full-class association between the 1990s and the 2000s, we see a 14 percent increase for young sons, a 2 percent increase for middle-aged sons, and an 11 percent decrease for older sons. The corresponding results for daughters are a 29 percent increase (young), 8 percent increase (middle-aged), and 2 percent decrease (old). The middle-age group likewise assumes an intermediary position for the PM-2 association within father-only tables (between the 1990s and the 2000s). In this case, we see a 22 percent increase for young sons, a 14 percent increase for middle-aged sons, and a 10 percent decrease for old sons. The coefficients for daughters similarly reveal a 45 percent increase (young), 9 percent increase (middle-aged), and 3 percent increase (old). These results are consistent with the view that the takeoff’s effects are attenuated within the intermediary age group because a smaller proportion of that group experienced the takeoff as a child.

Conclusions

We have provided the first comprehensive evidence on the claim that the takeoff in income inequality is reducing class mobility in the United States. This hypothesis is difficult to test because the takeoff, although now some 40 years old, is not likely to register its full effects on mobility except among relatively recent entrants to the labor force. Because of this difficulty, some scholars have attempted to make inferences about trend on the basis of indirect evidence, such as (1) the association between class origins and academic performance (Reardon 2011), (2) the distribution of parental spending on children (Kornrich and Furstenberg 2013), or (3) the association between income inequality and mobility within a cross-section of countries (Krueger 2012).

We have instead proceeded by searching for trend within GSS data among those age groups and social classes that are most likely to evince trend. Although the data are sparse and caution is clearly warranted, our key conclusion is that the GSS data do indeed reveal a recent increase in the intergenerational association among young and even middle-age adults. We cannot of course establish the mechanisms by which this rigidification has been generated, but the factual status of this result is independent of whether the takeoff in income inequality produced it. It is nonetheless relevant that the observed trend is largely consistent with predictions coming out of our two-factor model featuring the effects of (1) the expansion of mass education and (2) the takeoff in income inequality. Just as this model implies, the association curve has the expected convex shape in the two younger age groups, and the change in association also appears to be accelerating in the most recent decade (presumably as the full effects of the takeoff register). These results suggest that the takeoff may be implicated, but of course we can hardly deliver a direct test without a direct measure of family income. In the ideal design, the mobility tables would be disaggregated by both family income and respondent’s education, thus allowing the two-factor model to be properly evaluated.

The main predictions coming out of our two-factor model were on the mark save in one respect. For the most recent period, the model does not imply any differences in trend across genders, yet our point estimates did nonetheless reveal some differences (entailing a larger change for women). This seeming anomaly may reflect nothing more than sampling variability given that we could not reject the null hypothesis that the fall in social fluidity is the same for men and women. It is also possible, however, that there are real differences in trend for men and women. These differences might have arisen because daughters born into the professional-managerial class have been especially able to exploit the new infusion of resources that their parents received. We of course cannot rule out the alternative possibility that the gender interaction results from the operation of mechanisms completely unrelated to the takeoff.

The increase in association uncovered by our models takes the form of rising professional-managerial reproduction. We have thus rejected the simple income hypothesis in favor of what we have dubbed the “top income hypothesis.” There are two reasons why one might expect the takeoff to principally affect reproduction in the professional-managerial class. Most obviously, this class was a main beneficiary of the takeoff, whereas the income gaps between other classes remained roughly the same. This mechanical effect may also combine powerfully with the special anxiety about reproduction that professional-managerial parents are often presumed to have. Because this class cares about reproduction, once additional resources become available it will likely commit them to reproductive ends, such as buying into high-end neighborhoods with good schools and advantageous networks. The takeoff has in this sense allowed the professional-managerial class to more reliably realize its strong interest in reproduction.

Footnotes

Appendix A

Figure A1 provides detailed information on the three age groups in our analysis (25–40, 32–50, and 42–60 years old). For cohorts born between 1911 and 1985, the figure shows (1) the year in which the children were ages 16 to 18 and (2) the survey year or years from which they were drawn. The horizontal line shows the approximate time the takeoff in inequality started. It thus allows us to identify (1) the proportion of children in each age group and period that were between 16 and 18 years old after the takeoff started and (2) the number of post-takeoff years that the respondents experienced as a child.

Appendix B

An important advantage of the PM-2 tables is that, when only two periods are compared, they allow us to conduct one-sided or one-tailed tests. If “P diff” denotes the p-value of the chi-square test of the difference in G² between the constant association and unidiff models, the p-value of the one-sided tests can be computed as follows:

\begin{matrix} H_{0} & p - v a l u e \\ \frac{ϕ_{2000 s}}{ϕ_{1990 s}} \leq 1 & {\begin{matrix} \frac{P d i f f}{2} & i f \frac{ϕ_{2000 s}}{ϕ_{1990 s}} \geq 1 \\ 1 - \frac{P d i f f}{2} & i f \frac{ϕ_{2000 s}}{ϕ_{1990 s}} \leq 1 \end{matrix} \\ \frac{ϕ_{2000 s}}{ϕ_{1990 s}} \geq 1 & {\begin{matrix} 1 - \frac{P d i f f}{2} & i f \frac{ϕ_{2000 s}}{ϕ_{1990 s}} \geq 1 \\ \frac{P d i f f}{2} & i f \frac{ϕ_{2000 s}}{ϕ_{1990 s}} \leq 1 . \end{matrix} \end{matrix}

These one-sided tests can be derived by observing that, given a PM-2 table with only two periods (e.g., 1990s, 2000s), we can drop $ψ_{p m, p m} ϕ_{1990 s}$ + $ψ_{p m, p m} ϕ_{2000 s}$ from the unidiff model and substitute instead $η_{p m, p m}$ + $η_{p m, p m}^{2000}$ to represent association. The resulting model is a re-parameterization of the unidiff model in which $η_{p m, p m} = ψ_{p m, p m} ϕ_{1990 s}$ and $η_{p m, p m}^{2000} = ψ_{p m, p m} (ϕ_{2000 s} - ϕ_{1990 s})$ . Indeed, the 1990s log odds ratio is represented in these two models as $ψ_{p m, p m} ϕ_{1990 s}$ and $η_{p m, p m}$ , respectively, while the 2000s log odds ratio is represented as $ψ_{p m, p m} ϕ_{2000 s}$ and $η_{p m, p m} + η_{p m, p m}^{2000}$ , respectively. Hence we have:

η_{p m, p m} = ψ_{p m, p m} ϕ_{1990 s}

η_{p m, p m} + η_{p m, p m}^{2000} = ψ_{p m, p m} ϕ_{2000 s}

ψ_{p m, p m} ϕ_{1990 s} + η_{p m, p m}^{2000} = ψ_{p m, p m} ϕ_{2000 s}

η_{p m, p m}^{2000} = ψ_{p m, p m} (ϕ_{2000 s} - ϕ_{1990 s}) .

As long as $ψ_{p m, p m} > 0$ (an assumption without which the unidiff model cannot be estimated), the chi-square test of the difference in G² between the constant association and unidiff models is both a test of the null hypothesis that $ϕ_{2000 s} = ϕ_{1990 s}$ and a test of the null hypothesis that $η_{p m, p m}^{2000} =$ . Because the square of a variable with a standard normal distribution follows the chi-squared distribution with 1 degree of freedom, we can compute the p-value of one-sided tests as follows:

\begin{matrix} H_{0} & p - v a l u e \\ (ϕ_{2000 s} - ϕ_{1990 s}) \leq 0 & 1 - Φ (s i g n (ϕ_{2000 s} - ϕ_{1990 s}) {(x^{2})}^{1 / 2}) \\ (ϕ_{2000 s} - ϕ_{1990 s}) \geq 0 & Φ (s i g n (ϕ_{2000 s} - ϕ_{1990 s}) {(x^{2})}^{1 / 2}) \end{matrix}

where $x^{2}$ is the chi-square statistic from the difference in G² between the constant association and unidiff models, $Φ (z)$ is the cumulative distribution function of the standard normal distribution, and sign(y) is the sign function. The latter function returns a positive sign if its argument is positive and a negative sign otherwise.

These formulas are identical to the ones provided above (which can be computed more directly). We have the following equalities:

Φ ({(x^{2})}^{\frac{1}{2}}) = 1 - \frac{F (x^{2}, 1)}{2}

Φ (- {(x^{2})}^{\frac{1}{2}}) = \frac{F (x^{2}, 1)}{2},

where $F (z, 1)$ is the cumulative distribution function of the chi-square distribution with 1 degree of freedom. Therefore, for the null that $(ϕ_{2000 s} - ϕ_{1990 s}) \leq 0$ , if $s i g n (ϕ_{2000 s} - ϕ_{1990 s}) > 0$ , we have:

1 - Φ ({(x^{2})}^{\frac{1}{2}}) = 1 - 1 + \frac{F (x^{2}, 1)}{2} = \frac{F (x^{2}, 1)}{2} = \frac{P d i f f}{2} .

If $s i g n (ϕ_{2000 s} - ϕ_{1990 s}) < 0$ , we have:

1 - Φ (- {(x^{2})}^{\frac{1}{2}}) = 1 - \frac{F (x^{2}, 1)}{2} = 1 - \frac{P d i f f}{2} .

These are the formulas used above for that null hypothesis. The formulas for the null that ( $s i g n (ϕ_{2000 s} - ϕ_{1990 s}) \geq 0$ ) follow immediately.

Notes

Pablo A. Mitnik is a research associate at the Stanford Center on Poverty and Inequality. He conducts research on intergenerational social and economic mobility, low-wage work, intragenerational mobility policies, and quantitative methods.

Erin Cumberworth is a doctoral candidate in the Department of Sociology at Stanford University. Her research focuses on poverty, inequality, social mobility, and public policies related to poverty and inequality.

David B. Grusky is Barbara Kimball Browning Professor in the School of Humanities and Sciences, Director of the Stanford Center on Poverty and Inequality, and coeditor of Pathways Magazine. His recent books include Social Stratification (2014), Occupy the Future (2013), The New Gilded Age (2012), The Great Recession (2011), The Inequality Reader (2011), and The Inequality Puzzle (2010).

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