Disrupting ‘Secondary’ Class Effects on Educational

Outcomes*

Aleksei Opacic

Harvard University

June 11, 2021

Abstract

The education system plays a crucial role as both a mediator and moderator of intergenerational

reproduction. While a large portion of the association between parental income and child outcomes op-

erates through educational attainment, the school and college system is a primary locus of intervention

for policy-makers wishing to increase rates of mobility. In this paper, I argue that a theoretical distinc-

tion from sociologists of education - namely, between primary and secondary class effects on educational

outcomes - is useful for constructing a set of realistic and informative policy interventions to promote

equality of life chances. Specifically, I make two contributions. First, I demonstrate how primary and

secondary effects can be understood within a targeted intervention framework, and distinguish between

two types of secondary intervention to neutralize ‘secondary’ class effects on educational outcomes. I

clarify how these interventions can be understood as a hypothetical field experiment. Second, I demon-

strate how, under certain identification and credibility assumptions, the corresponding interventional

effects can be identified with observational data, and propose a set of imputation and weighting estima-

tors that can be combined with machine-learning methods to estimate them. I demonstrate the utility of

distinguishing between these two types of intervention in understanding the sources of and possibilities

to disrupt intergenerational income persistence using the NLSY97 cohort.

1 Introduction

One of the strongest predictors of adult socio-economic attainment is your family income during child-

hood. On average, a 10 percentile point increase in parent income rank is associated with a 3.41 per-

centile increase in a child’s income rank in adulthood (Chetty et al., 2014, 2020). This association between

parental income and adult attainment - variously called intergenerational persistence or its complement,

*Direct all correspondence to Aleksei Opacic, Department of Sociology, Harvard University, 1737 Cambridge Street, CambridgeMA 02138; email: aopacic@g.harvard.edu. Many thanks to Xiang Zhou for incredibly helpful comments and advice.

1

intergenerational mobility - is important because it gives us some indication of the degree of opportu-

nity in society (Hout, 1988; Breen, 2004). When intergenerational persistence is high, individuals from

low income households tend to stay poor, and those from wealthier households tend to stay wealthy. By

contrast, when intergenerational persistence is low - that is, when mobility is high - individuals’ socio-

economic outcomes are to a greater degree independent of the socio-economic advantages or disadvan-

tages that characterized their upbringing.

For social scientists and policy-makers concerned with increasing rates of intergenerational mobility,

a natural question to ask is what sort of interventions might be most effective towards this goal. Provid-

ing an answer is a challenging theoretical and empirical task. Theoretically, it requires an understanding

of the causal factors and processes that shape intergenerational mobility: the complex interactions be-

tween the economic and non-economic resources of an individual and their family, on the one hand, and

broader social institutions such as schooling systems, colleges and the labour market, on the other. Such

an understanding might, for example, lead us to the education system as a crucial site of intergenerational

reproduction, given that a large portion of the association between parental and child income is mediated

through educational attainment (Bloome et al., 2018; Breen and Müller, 2020). At the same time, hypo-

thetical interventions that are truly informative must be tempered with an eye to the practical. It seems

more reasonable to consider an intervention to school resources than to alter class- or race-specific test

score distributions per se (Jackson and VanderWeele, 2018).

Counterfactual interventions also pose an empirical challenge to the researcher. We might conceptu-

alize the goal of suggesting effective interventions as evaluating a hypothetical field experiment whose

results we wish to know without actually undertaking the experiment. Of course, such an aim shifts

our research objective into the counterfactual realm. But while the tools of causal inference have enabled

us, with the due assumptions, to make inferences about counterfactuals in the present, there are some

important differences where we seek to make predictions about causal relationships under some social

setup that does not as yet exist - a scenario in which all potential outcomes are unobserved (see Jackson

and Arah, 2020).

The aim of this paper is to draw on insights from sociologists of education on two dimensions of

class-based educational inequalities to define two theoretically-motivated education-based interventions

designed to promote intergenerational mobility, and to produce a set of credible estimates for levels of

mobility we would observe under these interventions. In the following, I first argue that the distinction

between between class inequalities in educational outcomes produced from primary effects (class effects

on academic performance) and those produced from secondary effects (class effects on individuals’ prob-

ability of making an educational transition, net of performance) (Boudon, 1974; Breen and Goldthorpe,

2

1997) is especially informative from a policy standpoint. While it is difficult to imagine how we might

reduce performance effects given the years of cumulative and multidimensional (dis)advantage among

children from different socio-economic groups that impact on attainment, a more realistic form of inter-

vention might target resource and informational constraints at educational transition junctures. Next,

I argue that, despite the utility of the primary-secondary effects distinction from a policy perspective,

the original theory has a number of weaknesses from a theoretical and a causal identification perspec-

tive. I therefore offer a corrective in the context of interventional effects by distinguishing between two

types of ‘secondary interventions’: what I label ‘weak’ and ’strong’ secondary intervention, which can

be conceptualized from a field-experimental standpoint. Third, I show how, under some relatively weak

assumptions and by carefully delimiting the scope of the intervention, we can express these interventions

in terms of observational data. I further propose a set of imputation and weighting estimators that can be

used to estimate secondary interventions. Finally, I show empirically the utility of considering alternative

credible interventional effects by examining intergenerational income persistence of racial majority and

minority groups of the NLSY97 cohort.

2 Primary and secondary effects in an interventionist context

2.1 Secondary class effects are conducive to policy interventions

The education system plays a vital role in social reproduction. Children from high income families are

likely to attain higher levels of education than their socio-economically disadvantaged peers (Ziol-Guest

and Lee, 2016; Duncan et al., 2017). In turn, higher levels of education lead to higher returns in the

labour market (Autor et al., 2008; Baum et al., 2010). On the one hand, because of these twin processes,

a major portion of the association between parental and child income is mediated through educational

attainment (Blau and Duncan, 1967; Bloome et al., 2018). On the other hand, these facts point towards

the education system, and in particular, class-based inequalities in educational outcomes, as a key site of

policy intervention if we wish to break the link between social origin and social destination. Nevertheless,

to clearly articulate an effective and practical set of educational interventions designed to disrupt these

inequalities requires a theoretical understanding of the mechanisms producing this inequality.

Sociologists interested in understanding inequality in educational opportunity have typically drawn

a distinction between two distinct mechanisms underpinning such inequality, first articulated by Boudon

(1974). First, primary effects refer to the effects of social background on academic performance - the fact

that children from higher class backgrounds, on account of their superior economic, social, and cultural

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resources, tend to outperform their disadvantaged peers in standardized tests and public examinations.1

By contrast, secondary effects are the effects of class background on the decision an individual, in con-

junction with their parents, teachers and peers, makes about whether to transition to a higher level of

education, conditional on prior performance (for instance, whether or not to continue to university educa-

tion conditional on high school GPA). e.g. Breen and Goldthorpe, 1997; Jackson, 2013. Class inequalities

in educational outcomes can then be understood as being produced from these two sources. Indeed, a

salient finding from the empirical literature on primary and secondary effects is that, across a range of

countries and time periods, working-class children are less likely than their advantaged peers to tran-

sition to a higher level of education, even when they have the same attainment level as these advan-

taged peers (Jackson et al., 2007; Jackson, 2013; Morgan, 2012) This approach is formalized in the directed

acyclic graphs (DAGs) presented in Figure 1. Consider the top DAG (A), which illustrates the assumed

data-generation process in many applications of primary and secondary effects. Let A denote family

income, Z be a measure of high school GPA, and M be an indicator denoting whether an individual tran-

sitions to college. Family income or social class background, can then affect individuals’ choice decisions

at an educational juncture both indirectly through performance, A → Z → M as well as directly, net of

performance, A→ M.

One important payoff to making a distinction between these two types of effects is that it draws at-

tention to the distinct processes underlying different aspects of educational inequality, and thus different

policy solutions. In particular, primary effects on performance are understood as the consequence of

a complex interaction between the cultural, economic and social resources of children and their families

and the educational system: the superior economic resources at the disposal of better-off families, and the

higher cultural fluency of better educated parents and their familiarity with the schooling system, for ex-

ample. By contrast, class differences in educational decisions at transitions can be seen as stemming from

an evaluation of costs and benefits of the various educational options available - decisions conditioned

by class-specific resources and informational constraints – and on the ‘perceived probabilities of more or

less successful outcomes’ of those in different classes (Breen and Goldthorpe, 1997, p.276). For example,

education costs provide a one key source of discrepancy in attainment as they decrease the proportion

of working-class families whose resources will meet the costs of further education (in addition to indi-

rect earnings-forgone through pursuing an educational pathway) (Goldthorpe, 2007). While social policy

does target the mechanisms producing primary effects in the form of educational resources, the extent of

cumulative (dis)advantage that underpins primary effects is surely less amenable to policy intervention

1In this paper, I refer to social background broadly as material circumstances of upbringing, and use this term interchangeablywith class origin and family income. Although I operationalize social origin in terms of family income, the approach I outline is ofcourse generalizable to other domains of childhood material advantage.

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than the cost benefit calculus underpinning secondary effects. One could envisage such policy interven-

tions in the form of informational campaigns, financial incentives and support and types of class-based

affirmative action. In other words, we may be able to credibly infer what would gaps in adult income

between individuals from different parental income groups look like if individuals from less advantaged

backgrounds would ‘exploit’ their demonstrated academic ability as advantaged children at educational

transitions - if we were to ‘neutralize’ secondary class effects on educational outcomes.

Despite the important insight that the distinction between primary and secondary effects can bring

to interventional effects, there are several limitations to the traditional definition of these effects that

need to be addressed. Consider next the more elaborated causal DAG (B) below, which features two

additional vectors representing an unobserved vector ~U denoting cultural resources as well as genetic

inheritance, which affects all other sets of variables in the DAG, and a vector of ‘intermediate’ variables

~X on the causal path from family income to performance (such as educational expectations, school type

and neighborhood quality). Note that our outcome is now adult income than educational transition (in

keeping with this paper’s motivating example of disrupting intergenerational mobility associations).2

If we are to understand the causal structure as more complex than the top DAG presumes, then a

causal interpretations of primary and secondary effects quickly becomes far less straightforward. First,

in this more elaborated DAG, disparities in high school GPA, college attendance and adult income across

family income groups arise in several ways. First, unobserved variables ~U such as parental ability affect

both parental income A as well as intermediate variables ~X such as child ability. Crucially, this is a non-

mediating path - it does not capture the effect of family income on ~X. In addition, there are also forward

paths emanating from family income: for instance, family income might affect the type of neighborhood

a child grows up in or the quality of school they attend.3 If we do not observe all of the components in

~U, then we will be unable to identify the causal effect of family income or class. Second, in addition to

the two causal pathways elaborated in (A) we have an additional causal path A → ~X → M. As Morgan

(2012) writes, this causal path cannot be considered a mechanistic elaboration of Boudon’s conception

of secondary effects understood as simply class-specific cost benefit analyses. Instead, they are a ‘sepa-

rate component of the net association between class and college entry that is best attributed to a broad

structural interpretation’ (p. 33). Thus, while if we modeled the data using the naive DAG in Panel A,

this path will be absorbed into the secondary effect, yet these variables are not explicit components of the

choice process that is thought to generate the causal secondary effects suggested by Boudon.

2Note in addition that, while ~X is multivariate, we are agnostic about the causal relationships among its constitutive variables.We assume that A and ~X precede Z which precedes M which precedes Y, though we are also agnostic about the temporal orderingof A and X.

3In the formal language of DAGs, we now have 4 backdoor paths from A to M: A ← U → M, A ← ~U → ~X → M, A ← U →Z → M, A← U → ~X → Z → M, and A← U → Z ← ~X → M.

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Figure 1: Two DAGs showing alternative models of class inequalities in educational attainment. Thefirst corresponds to the simple model of primary and secondary effects as described in the literature(e.g. Jackson et al., 2007; Jackson, 2013). The second corresponds to the more elaborated (and realistic)setting where we have two additional vectors: pre-family income unobserved confounders ~U, as well asa set of intermediate variables ~X, which encompasses a range of aspects of social disadvantage duringupbringing. Note that ~U has forward paths to every vertex in the DAG.

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2.2 We can define two secondary interventions through a field-experimental ideal

How then might we use primary and secondary mechanisms to inform targeted interventions? Under a

targeted intervention, we might want to ask what gaps in adult income, Y, would look like if we ‘neu-

tralized’ secondary effects in the sense of imposing some distribution among children from different

backgrounds. I define two types of intervention, which I illustrate graphically in Figure 2:

1. First ‘weak’ secondary interventions can be understood as those effects that capture class-differentiated

choice decisions at a juncture operating net of the effect of class origin on intermediate variables ~X.

In other words, such an intervention would not disrupt the path from family income to college

attendance through ~X; only directly net of ~X.

2. By contrast, ‘strong’ secondary effects refer to interventions that would neutralize the path from

parental income to college attendance net of high school performance that operates both through

intermediate variables ~X as well as through other pathways. The distinction compared with weak

secondary interventions therefore entails blocking an additional pathway from family income to

college attendance.We can further subdivide this strong form of intervention into two forms. The

first refers to cases where the strong intervention is applied to both high and low income groups,

while the second captures instances where the strong intervention is applied only to low income

groups. The latter quantity corresponds to an intervention targeted solely to alter the proportion of

lower income children attending college while not altering admissions policies or behavior for high

income children, and as such represents a form of class-based affirmative action (AA).4 I refer to the

former uniformally-applied intervention as the ‘uniform-strong’ intervention, and the latter as the

‘AA-strong’ intervention.

The advantages of this approach are twofold. First, when we consider A as simply a descriptive marker

that informs a population disparity, then we do not need to identify the ‘effect’ of income at all.5 It there-

fore conveniently sidesteps identification issues of A on M present in the original primary-secondary

effects framework; many of the backdoor paths considered by Morgan are simply not relevant to inter-

ventional effects. To recall, disparities in high school GPA, college attendance and adult income across

family income groups arise in several ways. First, unobserved variables ~U such as parental ability af-

fect both parental income A as well as intermediate variables ~X such as child ability. Crucially, this4This latter quantity is closely aligned with Proposition 4 in Jackson and Vanderweele (2018), where the interventional distri-

bution f (m|a) is applied only to individuals from the disadvantaged social group.5I adopt notation from the mediation literature concerned with effect identification of A to ease comparison, but A is simply

indicative of a collection of individuals, and is not of direct causal interest

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is a non-mediating path - it does not capture the effect of family income on ~X. In addition, there are

also forward paths emanating from family income: for instance, family income might affect the type of

neighborhood a child grows up in or the quality of school they attend. As a result, the interventiouns I

consider deactivate both the forward path(s) from A to M (net of high school GPA, and, in the case of

the strong intervention, additionally, net of intermediate variables), and the backdoor paths from A to M.

Specifically, the strong intervention which equalizes college attendance within high-school GPA groups

deactivates the forward paths A → M and A → ~X → M as well as the backdoor paths A ← ~U → M

and A← ~U → ~X → M, while the weak intervention deactivates only one forward path A→ M and one

backdoor path A← ~U → M .

Second, we have approached the issue of defining an intervention to secondary class effects on ed-

ucational transitions by creating two distinct interventions. Clearly, defining a secondary intervention

solely as an effort to neutralize the ‘direct effect’ of family income net of performance begs the question

of how to treat variables on the (backdoor) pathway from A to M, college transition. As we can see in

the DAG, there are two pathways from family income to college attendance (which may be backdoor

paths through unobserved variable ~U the pathway that operates through intermediate variables such as

neighborhood type, and the pathway that operates directly, net of these intermediate variables). This

observation then naturally leads us to consider two different types of secondary intervention on what the

traditional literature labels the ‘direct’ pathway from class background to college, net of performance.

To define strong and weak secondary effects formally, I follow Lundberg (2020) in defining my esti-

mand in terms of a hypothetical field experiment. Consider the set-up where a and a∗ are two levels of

family income we wish to compare, i.e. A ∈ {a, a∗}with a∗ representing low parental income groups. For

instance, if we dichotomized family income we may have a = 1 and a∗ = 0. Imagine drawing a sample

S from a population P of interest, and then intervening to assign each individual in the sample a level

of education M = m ; the value Y(m) then denotes the potential value of adult earnings that individual

i would take under that that level of education. For instance, 1na

∑ni:Ai=a Yi(m) is the average outcome

of units in the sample S when each individual with A = a has been exposed to m level of education.

Additionally, let P(M|u) denote the cumulative distribution function (CDF) of M (college attendance)

among those with a particular set of characteristics u. In addition letM|u denote a random draw from

this distribution.

Turning first to the weak secondary intervention, let P(M|X, Z, a) denote the cumulative distribution

function (CDF) of M (college attendance) among those with high school GPA (Z) grade Z and interme-

diate variables value X among those with family income level A = a, andM|X,Z,a, a random draw from

this distribution. Note that X and Z are random variables, whereas A is fixed to A = a, indicating that

8

Figure 2: A ‘weak’ secondary intervention removes the association between family income A and itsconfounders ~U on college attendance, conditional on high school GPA Z and on intermediate variables~X such as school type, neighborhood of origin, and peer expectations. A ‘strong’ secondary interventionremoves the association between family income and its confounders on college attendance, conditional onhigh school GPA, but unconditional on intermediate variables. family income A purely as a demographicmarker, the DAG accomodates unobserved cultural and genetic confounders ~U of the effect of familyincome on each set of variables in the model. Note that ~U has forward paths to every vertex in theDAG. Moreover, for expositional simplicity I illustrate the case where intermediate variables ~X are ‘post’-family income, though the flexibility of treating family income as a demographic marker means that myframework is agnostic about whether ~X occur before or after family income. The only chronologicalrequirement is that A and ~X occur before Z, which in turn occurs before M, which occurs before Y.

9

the intervention randomly assigns college attendance M among individuals from all income backgrounds

such that it follows the same marginal distribution as is observed in the group A = a, within Z and X

groups as currently observed. Therefore, the quantity

θa,a1 , E[Y(M|X,Z,a)|A = a1]

reflects the expected mean over repeated samples S where we impose the CDF P(M|X, Z, a) among

all individuals with family income level A = a1.6

While this formula is general in that it denotes the interventional quantity E[Y(M|X,Z,a)|A = a1] for

any combination of a, a1 ∈ {a∗, a}, for weak secondary interventions we are interested only in the case

where a ≥ a1; i.e. θaa∗ and θaa . For instance, θa,a∗ = E[Y(M|X,Z,a)|A = a∗] captures the expected outcome

of individuals from low income backgrounds if we intervened to send these individuals to college at the

same rate as individuals from high income backgrounds with the same high school GPA (Z) score and

value of intermediate variables X.7

Turning next to the strong secondary intervention, M|Z,a denotes a random draw from P(M|Z, a) -

the distribution of M (college attendance) among those with high school GPA Z among those with family

income level a1, such that

ψa,a1 , E[Y(M|Z,a)|A = a1]

reflects the expected mean over repeated samples S where we impose the distributionP(M|Z, a)

among all individuals with family income level A = a1, and

ψaa∗ , E[Y(M|Z,a)|A = a∗]

captures the expected mean over repeated samples of individuals from low income backgrounds if

we intervened to send these individuals to college at the same rate as individuals from high income

backgrounds with the same high school GPA (Z) score (i.e. if we imposed the distribution of college

attendance among high income children with a particular GPA on low income children, conditional on

GPA).8 Note in addition that the contrasts

6We could equally define this quantity in terms of a population-level expectation, rather than as a sample average, i.e.θa,a∗ , E[Y(M|x,z,a)|A = a∗]. Identification proofs for both are identical, but the version I present in the main text is usefulfor interpretation purposes, as I will clarify in the next section.

7Intuitively, setting M to a particular distribution is equivalent to assigning each individual a random draw from that distri-bution. Note in addition that, compared with other causal estimands, it is only meaningful in general to talk about interventionaleffects in the average, since each individual’s Y(M|x,z,a) depends on the random quantityM|x,z,a

8Note that we could also write the weak and strong interventions, respectively, as θa,a1 , ES [yS ,a1 (M|x,z,a)] and ψa,a1 ,ES [yS ,a1 (M|x,a)], preserving the notation used in Lundberg (2020), where the expectation is defined over hypothetical repeated

10

(a) E[Y(M|Z,a)|A = a]−E[Y(M|Z,a)|A = a∗]

(b) E[Y|A = a]−E[Y(M|Z,a)|A = a∗]

represent, respectively, (a) the expected disparity over repeated samples in income between high and

low parental income groups after the strong intervention is applied to both high and low income groups

(the ‘uniform-strong’ intervention), and (b) the expected disparity over repeated samples in income be-

tween high and low parental income groups after the strong intervention is applied only to low income

groups (the ‘AA-strong’ intervention). The latter quantity corresponds to an intervention targeted solely

to alter the proportion of lower income children attending college while not altering admissions policies

or behavior for high income children, and as such represents a form of class-based affirmative action

(AA).9

Importantly, these two types of secondary intervention map onto different types of policy as might be

delivered in practice. Figure 3 summarizes this mapping. First, weak secondary interventions can be con-

sidered the weaker form of intervention as they only block pathways from family income to attendance

that operate net of structural aspects of socio-economic upbringing - i.e. pathways which are thought

to capture the cost-benefit calculus aspect of educational transitions. Thus, an intervention of this sort

would pertain to policy aimed directly to alter the cost-benefit calculus of individuals from different class

backgrounds - for instance, to informational resources targeted at low income children, or needs-based

grants or financial incentives to apply or enroll in college. By contrast, strong secondary interventions are

the more radical since they are interventions that would not be sensitive to - or whose efficacy would not

be shaped by - other aspects of disadvantaged students’ upbringing environment. In other words, since

they block the composite path from family income to attendance through both the cost-benefit calculus

and structural aspects of upbringing (but not the path through GPA performance), they capture a world in

which college admission depends only on high school performance and no other class-contingent factors.

An intervention of this sort might refer to targeted university admissions or quotas for a representative

admission of individuals from different class origins, within GPA groups. Two points are of note here.

First, both interventions operate at the level of college enrolment rather than of attainment, which make

them distinct from the concept of ‘controlled mobility’ introduced in Zhou (2019) (see Section 2.4 for

samples. I opt for population level quantities in the main text for notational simplicity, but I seek to make inferences only about alimited subsample, as I specify in the next section.

9This latter quantity is closely aligned with Proposition 4 in Jackson and Vanderweele (2018), where the interventional distri-bution f (m|a) is applied only to individuals from the disadvantaged social group.

11

further discussion).10 Second, neither of these interventions alter the association between family back-

ground and intermediate variables or high school GPA. The intervention is solely with respect to college

attendance conditional on high school GPA (strong intervention), or conditional on both high school GPA

and intermediate variables (weak intervention).

Of course, policy interventions to equalize transition rates to higher education for those within the

same GPA bracket are likely to be unsatisfactory for reducing class educational inequalities in general

insofar as low income students are constrained by their lower average GPA scores, as well to the extent

that class inequalities in adult income persist among college graduands. I explore these issues more

thoroughly in Section 6.

10Moreover, because Zhou’s (2019) intervention concerns BA completion, rather than college enrolment, it hinges on both ensur-ing access and ensuring completion among students. It could therefore be considered a composite secondary-primary intervention.

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Figure 3: Educational interventions to reduce class-based inequalities in adult earnings operate at dif-ferent levels. Most broadly, they can be designed to alter either individuals’ educational performanceor individuals’ probabilities of transitioning to the subsequent stage of education (e.g. college), net ofperformance. These correspond, respectively, to primary- and secondary-based interventions, only thelatter of which I consider in this paper. Secondary interventions can then be further divided into twotypes: (a) those that only block the direct pathway from parent income to college attendance net of bothperformance and intermediate variables such as school, neighborhood and other family characteristics(a weak secondary intervention), and (b) those that block the composite path from parent income to col-lege attendance comprising both the direct effect in (a) and the path through intermediate variables. Theimportant takeaway from this distinction is that weak and strong secondary interventions capture twodifferent forms of policy intervention: whether we intervene to alter individuals’ cost-benefit calculus forinstance through an informational or grant-based approach (weak secondary), or through interveningto specify a particular admissions policy (strong secondary). Whether this admissions policy applies toeveryone or just to disadvantaged children underpins the final distinction between ‘uniform-strong’ and‘AA-strong’ interventions.

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2.3 We can approximate the ideal field experiment using observational data

Conceptualizing these interventions as hypothetical experiments is useful because it helps clarify our

research goal, and is unbounded in scope. We may want to ask what mobility would look like under

an intervention to the whole population of high-school goers. Of course, in practice, since we cannot

change admissions policies twice over across colleges, approximating these counterfactual interventions

using observational data is the most feasible way to estimate these estimands. As I demonstrate below,

a key advantage of the interventions that I propose is that they are identified under relatively weak

assumptions - namely, no unobserved confounding of the effect of college attendance M on adult earnings

Y, conditional on all antecedent variables.11 This assumption would be met in the DAG in Figure 2, if ~U

does not affect M and Y.

Despite the advantages of the estimation approach I propose, there are two validity-related difficulties

when we shift from the theoretical field experimental estimand at the population level, to estimating this

estimand with observational data.

The first regards causal identification. While a clear advantage of the approach I propose compared

with the traditional primary-secondary effects literature is the ability to sidestep identification of family

income on on educational transitions/adult income (see Section 2.2), this framework still requires iden-

tification of the effect of college attendance M on adult earnings Y (see formal identification proofs in

Appendix A). In practice, this requires no unobserved confounding of the effect of college attendance

M on adult earnings Y, conditional on all antecedent variables.12 This assumption would be met in the

DAG in Figure 2, but would be violated if there existed unobserved confounders of the M− Y relation-

ship. If we fail to meet this assumption then we would fail to approximate the ideal field experiment

using observational data.

The second regards external validity. In practice, a key limitation of estimating the counterfactual

disparities I suggest with observational data concerns the size of the population about which we are

able to credibly make a counterfactual claim. This trade-off between making a claim with credible yet

broad scope is succinctly laid out in Lundberg (2020, p. 8-12) in the context of ‘gap-closing estimands’,

which closely parallel the interventions I propose here. This is a particular issue if the effect of college

11Again, since A is used in purely a descriptive sense as a demographic marker, we do not require the assumptions of nounobserved treatment-mediator or treatment-outcome confounding. Note in addition that we are agnostic about the causal orderingof A andM1. A can be seen as either causally prior to or causally post any of the component variables ofM1 (e.g. family incomedetermines the type of school a child attends, but parental education level or neighborhood type affects income (e.g. Wodtke et al2011)). Either of these causal relationships may be true, but weak secondary interventions are identified in all cases.

12Again, since A is used in purely a descriptive sense as a demographic marker (of family income), we do not require anyadditional identification assumptions in the model. Note in addition that we are agnostic about the causal ordering of A and X: Acan be seen as either causally prior to or causally post any of the component variables of X (e.g. family income determines the typeof school a child attends, but parental education level or neighborhood type affects income e.g. Wodtke et al., 2011. Either of thesecausal relationships may be true in reality, but weak and strong secondary interventions are identified in all cases.

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attendance on adult earnings is partly a function of the proportion of individuals in the population who

receive a college degree. In this case, then we can only use observational data to estimate the effect of

college attendance when the observed proportion of individuals obtain a degree. Yet, the interventions I

consider in this piece will alter the proportion of individuals attending college since they serve to make

individuals from particularly lower class backgrounds less constrained in their college options. Claims

about a counterfactual world in which more people attend college are however at best speculative.13 We

might expect, for example, labour market returns to college to be very different in this world than in

reality, where fewer than 40% of young adults enroll in postsecondary education.14

There are at least two options for addressing this issue. The first simply concerns interpretation. Infer-

ence about the counterfactual world where we undertake such a set of interventions to college attendance

will be strongest when we interpret it as a local claim about hypothetical mobility rates in a small fraction

of the population, rather about the whole population. As Lundberg (2020) notes, this latter claim is often

relevant from the perspective of policy-makers, who cannot intervene on the whole population at once

(p.12). Moreover, secondary interventions can readily be conceptualized at the school-level: one could

imagine admissions policies or outreach programs changing at a set of schools, in which case the coun-

terfactual mobility rates I estimate would apply to the subpopulations attending these particular those

schools.

The second is to insist on making a global, population-level claim (i.e. about counterfactual mobility

rates in the whole population) by explicitly changing the estimand into one that preserves the marginal

distribution of college attendance at its observed value. An estimand that changes not the prevalence

of college-goers but rather that simply changes who goes to college - i.e. that simply shuffles college ad-

missions among individuals from different socio-economic backgrounds - can be endowed with a global

rather than local interpretation, and is more credibly estimated using observed data. We can easily ex-

tend the estimation framework I develop here to numerically derive a set of weighting constraints that

preserve the marginal distribution of college attendance. In future analyses, I plan to compare both ap-

proaches - thus exploring counterfactual mobility rates under (a) an intervention to a small sample of the

population (equal to the size of my survey sample) and (b) an intervention to the entire population where

the number of college attendees does not change.

In Appendix A, I offer formal identification proofs for the strong and weak interventions. Under

the assumption of no unobserved confounding of the effect of college attendance on earnings, the weak

13Jackson and Arah (2020) refer to these extrapolation issues as violations of the assumption that the intervention is systempreserving: intervening on the treatment shifts its values but does not change the statistical relationships that define the system.Formally, this general equilibrium threat can be understood as a violation of the Stable Unit Treatment Value Assumption (SUTVA)- the assumption that yi(~m) = yi(m), i.e. that the potential outcome of unit i under an intervention to send that individual to collegeis independent of the college attendance statuses of other individuals in the population.

14From the NLSY97 cohort, I calculate that approximately 36% of individuals enrol in a 2- or 4-year college by the age of 25.

15

secondary intervention is then non-parametrically identified as follows:

θaa∗ =∫∫

E[Y|a∗, x, z, m]dP(m|x, z, a)dP(x, z|a∗) (1)

This identification formula for the weak secondary intervention is closely related to the generalized

mediation functional in mediation analysis in the case of no pre-treatment confounders (see Zhou, 2020).

Note additionally that we can factorize E[Y|A = a] as follows:

E[Y|A = a]

=∫∫

E[Y|a, x, z, m]dP(m|x, z, a)dP(x, z|a)

= E[Y(M|x,z,a)|A = a]

, θaa

In other words, the expectation of adult income under the weak intervention among individuals from

high parent backgrounds is equal to their average observed outcome. For the strong secondary interven-

tion, in terms of observed data, we can write ψaa∗ ≡ E[Y(M|x,a)|A = a∗] as

ψaa∗ =∫∫∫

E[Y|a∗, x, z, m]dP(m|x, a)dP(z|x, a∗)dP(x|a∗) (2)

Note the key distinction between this quantity and the identification result for the weak intervention

considered above is simply that the PMF for college attendance in the strong secondary intervention lacks

intermediate variables in the conditioning set, reflecting the fact that this intervention breaks the pathway

from family income to college attendance via intermediate variables, while the weak intervention does

not. Note additionally the relationship between parental income and ~X (intermediate variables) and Z

(high school GPA) is preserved in both types of intervention.15

2.4 Comparison with existing approaches: controlled mobility, conditional equal-

ization and the randomized analog of the mediation formula

Strong and weak secondary interventions are closely related to three alternative estimands recently con-

sidered in sociology and epidemiological literature, all of which fall into a general category of ‘interven-

15Additionally, in the case of the weak secondary intervention, the interventional estimand among high income children reducesto the average observed outcome among high income children: ψaaa = E[Y(a)]. By contrast, in the case of the strong secondaryintervention, only the ‘AA-strong’ interventional estimand reduces to the average observed outcome among high income children,while the ‘uniform-strong’ intervention does not.

16

tional effects’. In general, interventional effects refer to counterfactual quantities in which the natural

distribution of a mediator under one exposure condition is replaced by that mediator’s distribution from

the other exposure condition.

To demonstrate the intimate connections between these, note that we can use the notation introduced

in Section 2.2 to write any stochastic assignment rule that randomly allocates individuals to a level of

educational attainment M as a function of observed characteristic. Generally, then , we have the following

formula:

τstoch.a,a∗ (m) = E[Y(M|u︸︷︷︸

∗

)|A = a]−E[Y(M|u︸︷︷︸∗

)|A = a∗] (3)

where E[Y(M|U)|A = a] denotes the average potential outcome in category A = a under an interven-

tion to equalize the distribution of M across social groups conditional on their value of U. U may be the

empty set, in which the CDF is simply the marginal distribution of M. Alternatively, it may be a vector of

covariates, which means that we are definining an interventional distribution conditional on some set of

covariates U, which can be defined in relation to the specific intervention we consider. Note that U may

consist of both random and fixed elements: random elements denote those characteristics over which we

define an intervention, but on which we do not intervene directly; fixed elements denote the Generally,

this quantity is identifiable so long as E[Y(m)] is identifiable, in which case we marginalize E[Y(m)] over

P(m|u), which denotes the cumulative distribution function (CDF) of M|u.

First, consider the case when P(m|u) = 1 and when u is the empty set. In this case, the interven-

tion is ‘deterministic’: each unit is deterministically assigned a value of the intervention, rather than

stochastically according to some assignment rule. This estimand is equivalent to that of ‘controlled mo-

bility’ proposed in Zhou (2019), when M is defined as an indicator for BA attainment, which captures

the remaining disparity in adult income by parental income groups after an intervention to set college

attainment to M = 1 for all individuals.

Consider next the alternative estimand Lundberg (2020) labels ‘conditional equalization’, which ’as-

signs a treatment to each unit according to the conditional distribution within the covariate stratum’ Z in

which that unit is observed, but independently of the category A within that stratum. Conditional equal-

ization can be written in terms of the above formulas, withMm|z and dP(m|z), respectively, replacing the

underbraced components of the equations.

Conditional interventions are further distinct from another estimand considered in the interventional

effects literature: randomized interventional effects (VanderWeele et al., 2014), where the underbraced

components of the equations are replaced by Mm|a. This intervention coincides with proposition 4 in

17

Jackson and Vanderweele (2018), though in the case where we have no pre-treatment confounders. Intu-

itively, such an approach would shift the focus away from eliminating ‘direct’ paths from family income

to college attendance net of high school performance, towards eliminating disparities in college atten-

dance entirely, regardless of whether those disparities arise through high school performance or through

alternative pathways. In contrast with the secondary interventions considered earlier, this estimand is

the quantity obtained under randomly assigning college attendance among low income students such

that they follow the same marginal distribution as among all those from high-income backgrounds, but

not conditional on high school performance (strong secondary interventions) or high school performance

and other class-based aspects of upbringing (weak secondary interventions).

Each of these approaches comes with its relative advantages and disadvantages compared with the

secondary interventions I propose here. First, to be sure, an intervention to send everyone to college

attainment is more radical than strong or weak secondary interventions, but arguably, interventions are

most informative when identificiable from observed data. An intervention to everyone’s college status

would likely drastically change returns to college; further; estimating this quantity is highly susceptible

to positivity extrapolations - when we impute a set of counterfactual outcomes for types of individuals

not or rarely observed in the treatment condition - in disparity estimands where we manipulate evey-

rone’s treatmnet condition to M = m. Do we really know how enrolling high school dropouts in higher

education would impact their earnings when we observe near to no such cases in our datasets? The

approach I suggest in this paper sidesteps this issue by considering interventions conditional on GPA.

Further, conditional equalization may not be entirely satisfactory as an intervention. Consider the

motivating setup where A is an indicator variable denoting family income, Z, high school GPA, M an

indicator of college attendance, and Y, earnings. While conditional equalization uses the distribution

f (m|z) for both family income groups, this distribution reflects both the ‘net effect’ of academic perfor-

mance on college attendance (Z → M) and the confounding effect of family income (Z ← A → M). In

other words, conditional equalization represents an intervention that does not fully purge the interven-

tion of the lingering effect of family background. Finally, randomized interventions, unlike secondary

interventions, do not constrain low income students to a college attendance distribution that depends

on their (low) GPA performance, and therefore block not only the path from family income to college

attendance net of GPA but, in addition, the path from family income to college attendance via GPA. The

advantage of this approach is that represents an even more radical intervention than the strong secondary

intervention considered above, and it neutralizes both secondary and primary class effects on attainment.

The disadvantage is that it is difficult to envisage such an intervention in practice, as it would require the

college admissions process to disregard any signal of academic ability (as proxied through high school

18

GPA). It is therefore perhaps a less realistic or at any rate foreseeable intervention compared with strong

and weak forms of secondary intervention.

3 Estimating weak and strong interventional effects

3.1 Estimation by imputation and weighting

The equations in the previous section suggest that both weak and strong secondary interventions can be

estimated via numerical integration or a Monte Carlo simulation approach (e.g. Imai et al., 2010).16 A

limitation of this approach however is that estimates of these conditional density or probability functions

tend to be noisy if any of the mediators are continuous or multivariate. Fortunately, we can rewrite both

of the integrals that identify weak and strong interventions, respectively, in terms of probability functions

that are more amenable to estimation.

Consider first weak secondary interventions. Without loss of generality, consider the case where fam-

ily income takes two levels. We note that the integral θaa∗ =∫∫

E[Y|a∗, x, z, m]dP(m|x, z, a)dP(x, z|a∗) can

be expressed as a series of iterated expectations of observed/imputed outcomes:

θaa∗ = Ex,z|a∗Em|a,x,zE[Y|a∗, x, z, m]

Since this approach does not require modeling of any of the mediators, it is especially advantageous

when one or more mediators is multivariate or continuous. This equation then suggests the following

regression-imputation procedure (see Zhou and Yamamoto, 2020):

1. Since θa,a ≡ E[Y(M|x,a)|A = a] = E[Y|A = a] by construction, and because A is treated as a

demographic marker with no ‘pre-treatment’ covariates, θa,a can be estimated by the average of the

observed outcome Y among units with A = a.

2. Estimate E[Y|a∗, x, z, m] by fitting a model for the outcome conditional on A, M, X and Z, and

obtain predicted values for all units at A = a∗ and their observed values of X and Z.

3. Fit a model for the imputed outcomes obtained in (1), E[Y|a∗, x, z, m], conditional on A, X and Z,

and obtain predicted values for all units at A = a and their observed values of X and Z.

4. Estimate θaa∗ by averaging the fitted values of Em|a,x,zE[Y|a∗, x, z, m] among all units with A = a∗.

16Specifically, this would involve fitting models for the observed outcome and conditional densities or probabilities of the media-tors f (m3|a∗m2), f (m2|a, m1) and f (m1|a), simulating counterfactual values of the mediator and of the outcome given the simulatedmediator values, and computing the integral using the simulated values.

19

Standard errors can then be obtained by bootstrapping the entire procedure. Steps 1-4 can easily be gen-

eralized to instances where A is multivariate, in which case the procedure is analogous but the contrasts

a and a∗ may refer to arbitrary levels of family income.17 Further, because the iterated expectations are

agnostic about functional form, we can use any model, including flexible machine learning methods,

to estimate the outcome models. This approach can help reduce model dependence, especially in cases

where the mediators are high-dimensional and when non-linearities and interactions are likely to exist. In

order to improve convergence rates of the machine-learning estimators employed, and for semiparamet-

ric/asymptotic efficiency, we can alternatively use a debiased-machine learning approach to estimate this

intervention (Chernozhukov et al., 2017). This approach is characterised by two components: first, the

use of a Neyman orthogonal estimating equation which makes estimates of targeted parameter ’locally

robust’ to estimates of the nuisance function; second, the use of a K-fold cross-fitting algorithm. In my

main analyses, I employ this debiased machine-learning approach for the weak secondary intervention,

which I detail further in Appendix B.

Consider next the strong secondary intervention. Again, like the weak secondary intervention, it

would be difficult to estimate the identifying equation directly when any of the mediators are continuous

or multivariate. Further, the integral expression for the strong intervention cannot be expressed as a series

of iterated iterations that would enable us to pursue a regression-imputation approach. Fortunately,

however, using Bayes’ rule, we can rewrite Equation 1 as a function of odds ratios of M:

∫∫∫E[Y|a∗, x, z, m]dP(m|a, z)dP(z|x, a∗)dP(x|a∗) = E

[Y

f (M|a, X)

f (M|a∗, X, Z)|a∗]

A proof is shown in Appendix A. Since M is an indicator for college attendance, the density function

f (M|u) becomes a more estimable probability mass function, Pr(M = m|u). This equation therefore sug-

gests the following weighting-based estimator for the strong secondary interventions among low income

children

ψaa∗ =1n ∑

i

[Yi

f (Mi|a, Xi)

f (Mi|a∗, Zi, Xi)|a∗]

and

ψaa =1n ∑

i

[Yi

f (Mi|a, Xi)

f (Mi|a, Zi, Xi)|a]

for high-income children when applying the ‘uniform-strong’ intervention. This method can be seen

as an extension of the weighting-based estimators proposed in Vanderweele et al. (2014) Approach 3, to

17Note that when A is continuous, an additional step to fit a further model as a function of A would be required.

20

estimate the ‘randomized interventional analog of the natural (in)direct effect, and suggests the following

procedure:

1. Estimate a model (1) for M conditional on A and X; for individuals with observed level of family

income A = a∗, obtain predicted values at A = a.

2. Estimate a model (2) for M conditional on A, Z and X; for individuals with observed level of family

income A = a∗, obtain predicted values at A = a∗.

3. For individuals with observed level of family income A = a∗:

(a) Obtain predicted values from model (1) at A = a;

(b) Obtain predicted values from model (2) at A = a∗;

(c) A weighted average of observed Yi among individuals with observed level of family income

A = a∗, where the weights are the ratio of the predicted probabilities obtained in steps (a) and

(b) constitutes an estimate of ψaa∗ .

4. For individuals with observed level of family income A = a:

For the ‘uniform-strong’ intervention:

(a) Obtain predicted values from model (1) at A = a;

(b) obtain predicted values from model (2) at A = a;

(c) A weighted average of observed Yi among individuals with observed level of family income

A = a, where the weights are the ratio of the predicted probabilities obtained in steps (a) and

(b) constitutes an estimate of ψaa.

For the ‘AA-strong’ intervention:

(a) Calculate E[Y|A = a] as the simple average among individuals from high income backgrounds.

Standard errors can then be obtained by bootstrapping the entire procedure. Again, the procedure can

easily be generalized to instances where A is multivariate, in which case the procedure is analogous but

the contrasts a and a∗ may refer to arbitrary levels of family income. In addition, because the estimator

has been derived without any parametric assumptions, flexible machine-learning methods can also be

used to fit the propensity score models for M.

Further, because the identification formulae are agnostic about functional form, we can use any model,

including flexible machine learning methods, to estimate the propensity score models for the mediators.

21

This approach can help reduce model dependence, especially in cases where the ‘intermediate variables’

~X are high-dimensional and when non-linearities and interactions exist. For all models in the analyses

that follow, I use a super learner consisting of Lasso and random forest.

3.2 General equilibrium threats: weighting constraints

As I detal in Section 2.3, one key limitation of the estimands I propose is that they involve changing the

proportion of individuals attending college. This is problematic insofar as the effect of college attendance

on adult earnings is partly a function of the proportion of individuals in the population who receive a

college degree. Claims about a counterfactual world in which more people attend college are at best spec-

ulative. Two options I consider are (a) adapting our interpretation of the estimates we obtain as simply

hypothetical mobility rates in a small fraction of the population (see Lundberg, 2020) , and (b) changing

the estimand into one that preserves the marginal distribution of college attendance at its observed value.

Since in practice, we can only use observational data to estimate the effect of college attendance when the

observed proportion of individuals obtain a degree, an estimand that altershanges not the prevalence of

degree-holders but rather that simply changes who holds a degree can be endowed with a global rather

than local interpretation.

We can easily extend the estimation framework I develop above to numerically derive a set of weight-

ing constraints that preserve the marginal distribution of college attendance in future analyses, I plan to

compare both approaches (exploring counterfactual mobility rates under both an intervention to a small

sample of the population, and alternatively as an intervention to the entire population where the number

of college attendees does not change. To this end, we can therefore additionally impose the constraint on

the quantities obtained via my proposed estimation proceudres that ensures P(M = m) is preserved by

selecting the weight wi such that

∫Pr(M = m|a, x, z)widP(x, z) = Pr(M = m)

for the weak secondary intervention, and

∫Pr(M = m|a, z)widP(z) = Pr(M = m)

for the strong secondary intervention. Using the empirical distribution as estimates of P(x, z) and

P(z), the resultant weights can therefore be calculated numerically as wi = Pr(M=m)Pr(Mi=m|ai ,xi ,zi)

and wi =

Pr(M=m)Pr(Mi=m|ai ,zi)

, for the weak and strong interventions, respectively.

REDO

22

These formulas seem incorrect. I was talking about a weighted average of Pr(M=1|a,x,z) across

different values of a as a "composite intervention" such that the marginal distribution is unchanged.

So the weight should be a function of a: w(a) and the intervention is a summation of Pr(M=1|a,x,z)w(a)

over a, subject the constraint that w(a) sum to 1.

4 Data

To estimate weak and strong secondary interventions, I draw primarily on the National Longitudinal

Survey of Youth 1997 (NLSY97), which began with a nationally representative sample of men and women

at ages 12 to 18 in 1997. The population amenable to the interventions I consider are all students who

completed a high school diploma or GED (‘high school graduates’). I exclude students who dropped out

of high school since they would be ineligible for college entry, and thus for the interventions I propose.

I measure parental income (A) as the average family income reported in the five earliest survey waves

(1997 to 2001), and adult income by averaging respondent annual earnings between ages 30 and 33. Both

variables are adjusted for inflation to 2019 dollars using the personal consumption expenditures index

(PCE). I treat respondent’s annual earnings as the sum of their self-reported wage and salary income and

income from farms and businesses. Although total family income arguably captures a more complete

picture of economic (dis)advantage in adulthood, focusing on individual income enables a more focused

analysis for two reasons. First, because family income is function of extra-labour market processes such

as assortative mating in addition to processes in the labour market, the counterfactual disparities I es-

timate would capture the effect of college on labour market and marital outcomes, which may differ

and even counteract each other (Zhou, 2019) Second, for more global interpretations of the counterfac-

tual disparities, family-level measures of adult income would require the stricter assumption that neither

labour market nor marital market outcomes are a function of the proportion of individuals attending col-

lege, which is harder to maintain than solely the first component. As is common practice in the income

mobility literature (Chetty et al., 2014, 2020; Bloome et al., 2018; Zhou, 2019), I transform both parental

and respondent/adult earnings into their percentile ranks. This enables me to capture ‘relative’ rates of

income mobility, which consider the intergenerational persistence of income net of overall changes in

the marginal distribution of income over time and thus more directly measures equality of opportunity

(Torche, 2015; Bukodi and Goldthorpe, 2018). I calculate the income ranks with respect to the population

of high-school completers, and adjust them using the NLSY97 sampling weights.

Compared with the NLSY79, which could also in theory be used to estimate the interventions I con-

sider, the NLSY97 facilitates estimation of interventional effects that are likely to be closer to those we

23

would obserce if we undertook the ideal field experiment, since it traces the educational and labor mar-

ket experiences of a younger cohort. Nevertheless, such an approach necessarily comes with a trade-off,

since it only enables us to measure adult earnings in these cohort members’ early thirties. As has been

noted in the mobility literature, measures of adult income at younger ages are likely to act as poor prox-

ies for permanent adult income, and thus misrepresent true rates of mobility (e.g. Blanden, 2013; Bloome

et al., 2018). I therefore plan to re-estimate the interventional effects I consider using the NLSY79, with

measures of adult income at later ages.

In addition to parent and child income, I construct three sets of variables. First, I measure college

attendance as a binary variable denoting whether an individual has ever enrolled in a 4-year college (with

the reference category indicating completion of high school studies). I set the cutoff of college attendance

at age 25, and only treat individuals who began their postsecondary education as 4-year college enrollees

(“four-year beginners”). Categorizing community-college transfers as high-school graduates enables a

more focused analysis: individuals who transition from 2- to 4-year colleges are qualitatively distinct

from those who begin at 4-year colleges Ciocca Eller and DiPrete, 2018, since including transfer students

in the analysis would necessitate some measure of associates degree (AA) performance which would be

considered in college admissions processes. Second, I measure high school GPA Z as a credit-weighted

average of GPA from high school classes. I use NLSY97 transcript data rather than self-reported GPA and

curricular measures to measure academic performance, which increases confidence in the validity of our

estimates.

Third, I construct a set of background and school characteristic variables as part of the intermediate

variables ~X in my model. To recall, these variables are important for two reasons: first, they are neces-

sary in order to plausibly estimate the effect of college attendance on earnings, which is only identified

if I appropriately adjust for all confounding of the M− Y relationship. Second, these variables delineate

a particular pathway through which family income is associated with college attendance, and as such

play an important role in the interventions themselves. In particular, the weak secondary intervention I

propose concerns equalizing college attendance within subgroups defined by both high school GPA and

social-structural aspects of (dis)advantage that affect college attendance. While in practice it is difficult to

measure all of the components of this set of variables, the NLSY97 is advantageous as it contains a wealth

of information on individuals’ family and social background, making identification of the effect of college

attendance on earnings more plausible. I include a range of variables that include demographic charac-

teristics (age in 1997, gender and race), non-economic aspects of family background (parental education,

whether the respondent lived with both biological parents, presence of a father figure, and southern or

rural residence), ability, (percentile score on the ASVAB test, a measure of substance use and of delin-

24

quency, whether the respondent had any children by age 18, and peer and school-level characteristics

(peer college expectations, an indicator for whether the individual attended a private high school, and

three dummy variables denoting whether the respondent ever had property stolen at school, was ever

threatened at school, and was ever in a fight at school). In particular, parental education is measured us-

ing mother’s years of schooling or, if missing, father’s years of schooling. Because I treat family income

as purely a descriptive marker, we need not worry about whether such variables are strictly descendants

of family income as opposed to pre-treatment. I restrict the sample to respondents with non-missing

information on all variables of interest, yielding an analytic sample of N = 3, 737 individuals.18

5 Results

5.1 Sample characteristics

To get a sense of the extent of inequalities by parental income groups, Table 1 presents descriptive statis-

tics of my analytic sample. For expositional simplicity, in Table 1 I dichotomize parental income by me-

dian parental income. These dichotomized statistics provide a picture consistent with inequalities across

all parental income strata, as shown in Figures 9 and 10 in the Appendix, which show the fitted values

from regressing each covariate as a spline function of parental income rank. Indeed, many of the vari-

ables I consider have an almost linear relationship with parental income, indicating how socio-economic

advantage persists throughout the income distribution, and not just between the top and bottom halves.

A comparison between low and high income groups reveals considerable differences by class back-

ground in background characteristics, and educational and labour market outcomes. On all measured

social background characteristics, poor children are substantially disadvantaged compared with children

from high income backgrounds. Poor children are more likely to have had school experiences disrupted

by violence or threatening behaviour than their socio-economically advantaged peers, and less likely to

be surrounded by school peers expecting college education or to have attended a private high school. Re-

garding the family unit, poor children are far less likely to have been living with their biological parents,

with both parents, and with parents who have a high level of education (measured in years). The average

poor child also lived in a family with far fewer parental assets.

Stark differences for children of different parental income brackets persist throughout young adult-

hood. The average credit-weighted GPA of a high school graduate from the bottom 50% of family earn-

18I adopt this approach because of the sheer computational time required to boostrap on multiple imputed datasets. However,MI increases the sample size by approximatley 1.7. After submission of this project, I will therefore either boostrap on the im-puted datasets or try to develop an alternative variance estimation strategy for the strong secondary intervention (for the weakintervention, I can use the EIFs proposed in Zhou (2020) for inference, as I currently do on the non-bootstrapped sample.

25

ings is 2.65, compared with 2.99 for individuals in the top 50% of family earnings. Poorer children are

also less likely to have progressed beyond only high school education, and, if they do ‘make it’ to college,

are less likely to have attended attended a four-year college (32% as opposed to 60% for high income

children), and are half as likely as their more advantaged peers to have attended a four-year college with-

out transferring (20% vs 42%). Class origin inequalities also leave their mark on adult socio-economic

outcomes. The gap in annual earnings between individuals from a high and low income background is

on average 12.1 percentiles; the corresponding figure for hourly wages is similar, at 13.6 percentiles. In

terms of real dollars, these differences translate to approximately $16,000 and $6, respectively. In keeping

with my focus on intergenerational income persistence, I focus on these percentile outcomes (specifically,

of earnings) as my primary dependent variable in the analyses that follow. Thie backdrop of intense in-

equalities by parental income motivates an exploration of what might happen to class-based inequalities

under a series counterfactual secondary interventions

5.2 Weak and strong secondary interventions

The left panel of Figure 4 presents current income gaps in adult percentile earnings rank by parental in-

come quintiles. Extant class-baed inequalities in adult socio-economic are stark: individuals coming from

the lowest parental income quintile attain, on average adult earnings in the 37th percentile, while those

who were brought up in highest parental income quintile on average take home earnings in the 58th per-

centile. In other words, coming from a high rather than low income household in childhood increases

expected earnings by 21 percentiles. Using the estimation strategy outlined above, I next estimate aver-

age earnings under a weak secondary intervention. The figure shows that under such an intervention,

this gap in expected income between individuals from the highest and lowest parental income brackets

would diminish to 17 percentiles. Interestingly, a reduction of adult earnings disparities is unique for the

gap between the first and fifth parental income quantiles; the gap between the 2nd, 3rd and 4th parental

quantiles, on the one hand, and the 5th parental quantile, remain largely unchanged under the weak sec-

ondary intervention. This interesting finding could point to the fact that the mechanism of disadvantage

that weak secondary interventions neutralize - namely, the cost-benefit calculus at an educational tran-

sition juncture - is especially relevant to individuals from a low-income background. These individuals’

decisions, one might suspect, are especially sensitive to the cost of higher education in contrast to the

economic payoff to immediately entering the job market, and it is perhaps these individuals who have

minimal access to information about the long-term payoffs to college and financial aid available.

Next, I estimate the counterfatual expected earnings among different parental income quintiles under

the strong secondary intervention. As Figure 5 demonstrates, the strong secondary intervention raises

26

Table 1: Conditional means in educational and labour market outcomes, as well as background charac-teristics, by family income (dichotomized).

27

Figure 4: Expected adult income ranking under (a) no intervention and (b) a weak secondary interven-tion. Standard errors for the weak intervention are constructed using the empirical analog of estimatedinfluence functions

expected earnings for all parental income groups, and not just for the bottom quintile, as was the case

with the weak secondary intervention. For example, average earnings among the bottom four parental

income quintiles would be expected to increase by 10.8, 10.2, 9 and 7.5 percentage points, respectively,

under such an intervention. The blue and green point estimates for the 5th parental income quintile

group correspond, respectively, to this group’s expected earnings under (a) the ‘uniform-strong’ and (b)

‘AA-strong’ interventions. To recall, these estimands refer to an intervention that (a) applies the strong

intervention to both high and low income groups, and (b) applies the strong intervention only to low in-

come groups. What we see is that the ‘uniform-strong’ also serves to increase the average earnings of

individuals from the highest parental income quintile, albeit by a smaller amount (6 percentage points)

than corresponding increases for lower income quintiles. Therefore, although the strong secondary inter-

vention compresses the gaps in adult income by parental background (gap in expected earnings between

1st and 5th quintiles is 16.0 percentage points) margianlly than the weak intervention (that same gap is

17.2 percentage points) and far more than the gap under no intervention (a gap of 21.0 precentage points),

the equalizing potential of the strong intervention is somewhat attenuated by the fact it promotes the ex-

pected earnings of children from high income backgrounds. Instead, therefore, we could consider (b)

the ‘AA-strong’ intervention, which does not alter the distribution of high income children attending

college (i.e. preserves this distribution at its current level). This quantity can be seen as correspond-

28

Figure 5: Expected adult income ranking under (a) no intervention and (b) a strong secondary interven-tion. Blue and green point estimates for the 5th parental income quintile group correspond to this group’sexpected earnings under the ‘uniform-strong’ and ‘AA-strong’ interventions, respectively. Standard er-rors for the strong intervention are obtained via the nonparametric boostrap (500 replications).

ing to an intervention targeted solely to alter the proportion of lower income children attending college

while not altering admissions policies or behavior for high income children, and as such represents a

form of class-based affirmative action. As the green point estimate and confidence interval show, under

this alternative type of secondary intervention, individuals from the 4th parental income quintile would

marginally overtake the most advantaged children in terms of expected earnings. Under this ‘AA-strong’

intervention, the gap in average adult earnings by parent income quintile would be reduced from 16.0

percentage points to 10.2 percentage points.

5.3 Heterogeneity by racial groups

I next examine heterogeneity in secondary interventional effects by racial/ethnic groups. To this end, I

repeat the analysis conducted in the previous section for racial-ethnic subgroups. The dependent vari-

able of interest - annual earnings - is again ranked in relation to the distribution of all individuals who

at least complete high school, regardless of racial group. Figures 6, 7 and 8 show expected adult earn-

ings by parental income quintile groups under a series of secondary interventions for whites, Hispanics

and Blacks, respectively. First, there are presently considerable inequalities in adult attainment by racial

group, even among individuals from the same parental quintile. For example, while whites and Hispan-

29

ics in the bottom income quintile have average earnings in the 39-40th percentile, average earnings for

Blacks in the equivalent group are 8 percentage points lower. Similarly, while whites and Hispanics in

the top income quintile have average earnings in the 57-60th percentiles, Blacks in the same group have

average earnings in only the 47th percentile. Consequently, the gap in average earnings between the bot-

tom and top parent income groups differs across race: this gap is 18.4 percentage points for whites, 20.6

percentage points for Hispanics and 15.6 percentage points for Blacks. The smaller gap for Blacks reflects

the lower and more compressed income distribution among this group.

Turning now to the weak form of secondary intervention, the estimated increase in expected earn-

ings among whites and Hispanics is similar, at approximately 2-4 percentage points for children in the

bottom three parental quintiles. By contrast, Black individuals are not estimated to benefit from such

an intervention; the expected increase in earnings among this group under a weak intervention is sub-

stantively insignificant. Consequently, gaps in annual earnings between the lowest and highest parental

income groups would be expected to decrease by 1.9 percentage points for whites, 4.0 percentage points

for Hispanics, and 0.9 precentage points for Blacks.Under the strong intervention, the story is quite dif-

ferent. Specficially, for whites, a strong secondary intervention would see expected earnings increase by

6-8 percentiles for the bottom three parental quintiles. The result is even more stark for Hispanics in the

equivalent quintile groups: Hispanics in the bottom three parental quintiles would be expected to earn

between 7 and 12 percentiles more under this form of intervention. Unlike in the case of weak secondary

interventions, Blacks also stand to benefit highly from the strong form of secondary intervention - more

so than whites but slightly less than Hispanics. Black individuals in the bottom parental quintile would

see average earnings increase by 10.6 percentage points, while expected earnings for Blacks in the second

and third quintile groups increase by between 6 and 7.5 percentage points under such an intervention.

Overall, a ‘uniform-strong’ secondary intervention would see the gap in adult percentile earnings be-

tween the lowest and highest quintile groups drop by 3.6 percentage points for whites, 5.8 percentage

points for Hispanics, but –.2 percentage points for Blacks. The fact that the ‘uniform-strong’ intervention

does not reduce the overall gap between the first and fifth quintiles for Black individuals reflects the low

relatively earnings of high parental income Black groups under no intervention, and that Blacks from

all income groups stand to benefit from this intervention. However, under the ‘AA-strong’ intervention

(which applies the strong secondary intervention only to low-income groups), the gap between low and

high income Black students would essentially be reduced to 0.

More generally, how might we make sense of the fact that the equalizing effect of the strong secondary

intervention is more prominent for Blacks than whites, while the weak form of secondary intervention is

slightly more effective for whites? One possible explanation might be that (a) the pathway from family in-

30

come to college attendance net of both high school GPA and intermediate variables constitutes a weaker

effect for Blacks than whites, while (b) the composite path from family income to college attendance

through both this direct path and through intermediate variables constitutes a stronger effect for Blacks

than whites (see Figure 2 for a comparison of these two pathways). Specifically, (a) might correspond

to the fact that the class-based cost-benefit calculus (Boudon, 1974; Breen and Goldthorpe, 1997) might

interact with one’s racial group. Perhaps ethnic minority students expect larger benefits from higher ed-

ucation, for instance because they expect levels of ethnic discrimination in the labor market to be reduced

if they attain higher qualifications. Alternatively, given the significant obstacles that Black parents would

have faced to make it to a similar class position as that of their white counterparts, we might expect Black

individuals with a particular income level to be more positively selected on attributes salient for their

children’s educational decision-making than whites. On the other hand, (b) might result from the fact

that Black children, on average, suffer from a broader set of structural disadvantages as compared with

their white peers, even within the same income level - such as growing up in an impoverished neigh-

bourhood, being surrounded by lower-income peers and attending lower-performing high schools (e.g.

Wodtke et al., 2011; Sampson, 2012). Therefore, a strong secondary intervention, which intervenes to

block the composite path from family income to college attendance via (i) a class-based cost-benefit cal-

culus and (ii) structural factors encapsulated in the set of intermediate variables, blocks a set of pathways

that are presently more constraining for Blacks than for whites.

31

Figure 6: Gaps in white adults’ income ranking under a series of stochastic educational interventions.Blue and green point estimates for the 5th parental income quintile group correspond to this group’s ex-pected earnings under the ‘uniform-strong’ and ‘AA-strong’ interventions, respectively. Standard errorsfor the strong intervention are obtained via the nonparametric boostrap (500 replications).

32

Figure 7: Gaps in Hispanic adults’ income ranking under a series of stochastic educational interventions.Blue and green point estimates for the 5th parental income quintile group correspond to this group’s ex-pected earnings under the ‘uniform-strong’ and ‘AA-strong’ interventions, respectively. Standard errorsfor the strong intervention are obtained via the nonparametric boostrap (500 replications).

33

Figure 8: Gaps in Black adults’ income ranking under a series of stochastic educational interventions.Blue and green point estimates for the 5th parental income quintile group correspond to this group’s ex-pected earnings under the ‘uniform-strong’ and ‘AA-strong’ interventions, respectively. Standard errorsfor the strong intervention are obtained via the nonparametric boostrap (500 replications).

34

6 Conclusion and future steps

The aim of this paper has been to define and estimate a series of stochastic interventional effects that can

usefully inform us about the levels of mobility we would likely observe under a series of interventions

to the schooling/college system. Its starting point was the recognition that a long-appreciated distinc-

tion among sociologists of education between two dimensions of class-based educational inequalities is

useful from the perspective of designing policy. Theoretically, I have sought to improve upon the orig-

inal primary-secondary effects theory by re-considering identification issues in the context of mobility

interventions. My proposal is to consider two types of ‘secondary interventions’: what I label ‘weak’

and ’strong’ secondary intervention, which can be conceptualized from a field-experimental standpoint.

Empirically, I have shown how these estimands can be both identified from observational data under the

corresponding assumptions, and have proposed a regression-imputation and weighting strategy that can

be used for estimation in practice. The flexibility of the estimation procedures I propose is such that we

can use any flexible, including machine-learning model, to implement them. Using the NLSY97 as a case

study, I have shown the insights that these distinct interventional effects can bring for understanding

policy correctives to inequalities of opportunity.

I’ve been constrained by time in terms of the empirical analyses I have been able to undertake, and

finish the piece by simply listing those which I would like to explore further in the immediate future. The

first, most mundanely, is a replication of the main analyses using different measurement specifications,

as I allude to in the data section. This would include analyzing interventional effects at different college

attendance cutoffs, measuring adult income as total family, rather than solely individual, income, and

analyzing the effects using the NLSY79 to capture adult earnings at a later stage in the life-cycle. This

might be more relevant if I pursue this as a substantive rather than as a methods piece, however.

Second, as I highlight in Section 2.3, there are two possibilities in terms of estimating the interven-

tional effects with observational data. The first, which I have pursued here, is to simply interpret the

interventional estimates as ‘local’. The second, which involves applying the weighting procedure I detail

in Section 3.2, involves an additional estimation step that preserves the marginal distribution of college

attendees under the intervention. This second approach therefore enables a more ‘global’ interpretation

to the resulting estimates. I would like to provide comparison analyses for these two approaches.

The third regards potential a theoretical concern regarding the anticipatory component of choice.

One potential criticism of my classification of strong and weak interventions is that some of the vari-

ables I include in the vector ~X of intermediate variables (such as peer expectations), might alternatively

be considered indicators of anticipatory decision-making that is part of the secondary class effect. By

35

marginalizing out college attendance conditional on these variables, the weak intervention would artif-

ically downwardly bias the inequality-reducing capability of the weak intervention. I would therefore

like to re-estimate the weak and strong interventions considering this alternative classification of a subset

of the intermediate variables.

Next, I plan to undertake a formal decomposition of the ‘limitations’ of the interventions I propose.

As I have noted at a number of points throughout this paper, policy interventions to equalize transition

rates to higher education for those within the same GPA bracket are likely to be unsatisfactory for re-

ducing class educational inequalities in general insofar as low income students are constrained by their

lower average GPA scores, as well to the extent that class inequalities in adult income persist among

college graduands. This latter aspect of the persistent effect of social background on attainment, even

among college goers, might be particularly concerning as a source of inequality of opporunity when

within-educational group inequality by parent background has increased in recent years (Bloome et al.,

2018). One could therefore seek to perform a formal decomposition of the sources of lingering inequality

on both a pre- and post-intervention sample. For instance, one could pursue the following decomposi-

tion:

{E[Y|a]−∫

E[Y|a∗, x, z, m]dP(m|x, a, z)dP(z|a, x)dP(x|a))︸ ︷︷ ︸residual inequality in X,Z,M groups

}

+{∫

E[Y|a∗, x, z, m]dP(m|x, a, z)dP(z|a, x)dP(x|a)−∫

E[Y|a∗, x, z, m]dP(m|x, a, z)dP(z|a, x)dP(x|a∗)︸ ︷︷ ︸residual inequality in X (intermediate variables)

}

+{∫

E[Y|a∗, x, z, m]dP(m|x, a, z)dP(z|a, x)dP(x|a∗)−∫

E[Y|a∗, x, z, m]dP(m|x, a, z)dP(z|a∗, x)dP(x|a∗)︸ ︷︷ ︸residual inequality in GPA

}

Such a decomposition could also point towards fruitful research avenues to explore. For instance,

if it turns out that residual inequality in GPA is a large driver of the remaining discrepancy in income

outcomes, then this could act as motivation for a more radical set of interventions additionally seek to

eradicate the path from parent income to college attendance via high school GPA.

Finally, there are two additional components of the project I would love to consider more, especially

with your guidance! The first is the further substantive issue that the interventions I consider concern

college enrolment, rather than completion. Much work has shown vast class- and race-based inequalities

in college completion, and to the extent that many of the low-income individuals ‘sent’ to college under

36

the interventions I propose subsequently drop out (see Ciocca Eller and DiPrete, 2018), the strong and sec-

ondary policies could be deemed relatively inefficient. Future work could therefore consider a series of

dynamic interventions to college attendance and then completion. The second pertains to methodological

aspects of the project that would facilitate estimation. In particular, the bootstrapping procedure neces-

sary for inference of the strong secondary intervention is computationally inefficient; I wonder whether

it is possible to derive a consistent variance estimator for the strong intervention. Perhaps we could also

look into whether semiparametric estimation can be developed for the strong intervention, which would

additionally facilitate uncertainty estimation.

• end up 29 percentiles higher in the income distribution on average relative to children

• a. Check results under (a) alternative age cutoffs for college attendance (I currently use age 25) and

(b) different measures of college attendance (my treatment is 4 year college and I classify transfer

students as high school graduates). b. Replicate all analyses on the NLSY79 and ELS.

37

A Identification and Estimation of Equations 1 and 2

In this section, I offer formal identification proofs of the ‘weak’ and ‘strong’ secondary interventions I

propose, as well as proofs of the estimation strategies I undertake in the main text. Let A, X, Z, M and

Y be as in the main text. The following single independence assumption is sufficient to identify both

the weak and strong secondary interventions: Y(m) ⊥⊥ M|A, X, Z, i.e. there must be no unobserved

mediator-outcome confounding conditional on all antecedent variables.

First, for the ‘weak’ secondary intervention, E[Y(M|X,Z,a)|A = a∗], we have:

E[Y(M|X,Z,a)|A = a∗]

=∫

E[Y(M|x,z,a)|A = a∗, X = x, Z = z]dP(z|A = a∗, X = x)dP(x|A = a∗)

=∫

E[Y(m)|A = a∗, X = x, Z = z,M|x,z,a = m)]dP(M|x,z,a = m|A = a∗, X = x, Z = z)dP(z|A = a∗, X = x)dP(x|A = a∗)

=∫

E[Y(m)|A = a∗, X = x, Z = z, M = m]dP(m|X = x, Z = z, A = a)dP(z|A = a∗, X = x)dP(x|A = a∗)

=∫

E[Y|A = a∗, X = x, Z = z, M = m]dP(m|X = x, Z = z, A = a)dP(z|A = a∗, X = x)dP(x|A = a∗)

Note that this quantity can then be rewritten as follows:

∫∫E[Y|a∗, x, z, m]dP(m|x, z, a)dP(z|a∗, x)dP(x|a∗)

=∫

EM|x,z,aE[Y|a∗, x, z, M]dP(x, z|a∗)

= EX,Z|a∗EM|X,Z,aE[Y|a∗, X, Z, M]

Which provides the motivation for the regression-imputation approach proposed in the main text.

Second, for the ‘strong’ secondary intervention, E[Y(M|Z,a)|A = a∗], we have:

38

E[Y(M|Z,a)|A = a∗]

=∫

E[Y(M|z,a)|A = a∗, X = x, Z = z]dP(x, z|A = a∗)

=∫

E[Y(m)|A = a∗, X = x, Z = z,M|z,a = m]dP(M|z,a = m)dP(x, z|A = a∗)

=∫

E[Y(m)|A = a∗, X = x, Z = z]dP(m|Z = z, A = a)dP(x, z|A = a∗)

=∫

E[Y(m)|A = a∗, X = x, Z = z, M = m]dP(m|Z = z, A = a)dP(z|A = a∗, X = x)dP(x|A = a∗)

=∫

E[Y|A = a∗, X = x, Z = z, M = m]dP(m|Z = z, A = a)dP(z|X = x, A = a∗)dP(x|A = a∗)

To provide an estimator for the strong secondary intervention that avoids estimation of multiple den-

sities, which may be multivariate or continuous, we can rewrite the integral as follows:

∫∫∫E[Y|a∗, x, z, m]dP(m|a, z)dP(z|x, a∗)dP(x|a∗)

=∫∫∫∫

ydP(y, x, z, m|a)dP(z|a, x)dP(x|a)dP(a, x, z, m)

=∫∫∫∫

y f (y, x, z, m|a) f (m|a∗, z)f (m|a, x, z)

= E

[Y

f (M|a∗, Z)f (M|a, X, Z)

|a]

B Description of debiased machine-learning procedure used for weak

intervention

In the main text, I show how a regression imputation approach can be used to estimate the weak sec-

ondary intervention. This procedure is highly similar to that proposed in Zhou and Yamamoto (2020) for

estimation of path-specific effects (PSEs) in mediation analysis. In practice, however, to improve conver-

gence rates of the machine-learning estimators employed, and for semiparametric/asymptotic efficiency,

we can use a debiased-machine learning approach to estimate this intervention. This approach is char-

acterised by two components: first, the use of a Neyman orthogonal estimating equation which makes

estimates of targeted parameter ’locally robust’ to estimates of the nuisance function; second, the use of

39

a K-fold cross-fitting algorithm (Chernozhukov et al., 2017). Consider the set-up where a and a∗ are two

levels of family income we wish to compare, i.e. A ∈ {a, a∗} with a∗ representing low parental income

groups. Additionally, as in the main text, let A denote family income, Z be a measure of high school GPA,

M be an indicator denoting whether an individual transitions to college, and X a vector of intermediate

variables. To achieve semiparametric estimation, I use the following Neyman orthogonal signal based on

the efficient influence function (EIF) for the weak secondary intervention:

θ∗aa∗ =I(A = a∗)

π0 (a∗)π1 (a∗ | X)

π1 (a | X)

π2 (a | X, Z)π2 (a∗ | X, Z)

(Y− µ2 (X, Z))

+I (A = a)

π0 (a∗)π1 (a∗ | X)

π1 (a | X)(µ2 (X, Z)− µ1 (X))

+I (A = a∗)

π0 (a∗)(µ1 (X)− µ0)

+ µ0

where

µ2 (X, Z) , E[Y|a∗, X, Z, M]

µ1 (X) , EM|X,Z,aE[Y|a∗, X, Z, M]

µ0 , EX,Z|a∗EM|X,Z,aE[Y|a∗, X, Z, M]

π(a|U) , Pr(A = a|U)

This signal is identical to the estimator ψeif2a for the EIF of the generalized mediation functional (GMF)

proposed by Zhou (2020), although in the absence of pre-treatment confounders. For the second compo-

nent of the debiased-machine learning approach, ‘sample-splitting’, I adopt the following procedure:

1. Randomly split data into K folds: {S1, ...Sk}

(a) For each fold k, use the remaining (k − 1) folds (training sample) to fit a flexible machine-

learnng model for each of the following six nuisance functions: (i) E[Y|a∗, X, Z, M], (ii) EM|X,Z,aE[Y|a∗, X, Z, M],

(iii) EX,Z|a∗EM|X,Z,aE[Y|a∗, X, Z, M], (iv) Pr(A = a|x), (v) Pr(A = a|x), and (vi) Pr(A = a|x, z).

Additionally, π0(a∗) can be estimated as the simple proportion of individuals in the sample

with family income level A = a∗.

(b) For each fold k (estimation sample), calculate the signal θ∗aa∗ for each observation using the

equation given above.

2. Compute an estimate of the weak secondary effect by averaging the estimated influence functions

40

across all subsamples S1 through SK, for all units:

θaa∗ = n−1 ∑i

θ∗i,aa∗

Standard errors can be constructed using the sample variance of the estimated influence functions:19

Var(θaa∗) = E(θ∗i,aa∗ − ˆE(θ∗i,aa∗)]

2

As shown in the main text, E[Y|A = a] (i.e. the average observed outcome among individuals from

high parent backgrounds) factorizes into the expectation of their income under the weak intervention.

As such, we only require estimation for the lower income group. Note additionally that this debiased

machine-learning approach is valid for any paired contrasts of parent income, A ∈ {a, a∗}. In my main

analyses I bin parental income into quintiles, and thus extend this algorithm for each contrast {a∗ =

1; a = 5}, {a∗ = 2; a = 5}, . . . , {a∗ = 4; a = 5}.

19Since the sample estimation variance Var(EIF) coincides wth the semiparametric efficiency bound.

41

C Smoothed spline functions of variables used in analyses

42

Figure 9: Fitted conditional means of labour market and educational outcomes as a natural spline functionof parental income rank (with 3 degrees of freedom). Ribbons represent 95% confidence intervals. Agecutoff for attainment of all outcomes is set to 25.

43

Figure 10: Fitted conditional means of background characteristics as a funtion of parental income rank.Fitted values are obtained by a natural spline function with 3 degrees of freedom. Ribbons represent 95%confidence intervals. Age cutoff for attainment of all outcomes is set to 25.

44

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