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Attendance, Completion, and Heterogeneous Returns to College: A Causal Mediation Approach

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Title: Attendance, Completion, and Heterogeneous Returns to College: A Causal Mediation Approach
Language: English
Authors: Xiang Zhou (ORCID 0000-0001-8634-7360)
Source: Sociological Methods & Research. 2024 53(3):1136-1166.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 31
Publication Date: 2024
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
High Schools
Secondary Education
Descriptors: Outcomes of Education, Educational Benefits, Educational Status Comparison, Disadvantaged Youth, High School Graduates, College Attendance, College Graduates, Reentry Students, Social Science Research, Cost Effectiveness, Advantaged, Comparative Testing
DOI: 10.1177/00491241221113876
ISSN: 0049-1241
1552-8294
Abstract: A growing body of social science research investigates whether the economic payoff to a college education is heterogeneous -- in particular, whether disadvantaged youth can benefit more from attending and completing college relative to their more advantaged peers. Scholars, however, have employed different analytical strategies and reported mixed findings. To shed light on this literature, I propose a causal mediation approach to conceptualizing, evaluating, and unpacking the causal effects of college on earnings. By decomposing the total effect of attending a four-year college into several direct and indirect components, this approach not only clarifies the mechanisms through which college attendance boosts earnings, but illuminates the ways in which the postsecondary system may be "both an equalizer and a stratifier." The total effect of college attendance, its direct and indirect components, and their heterogeneity across different subpopulations are all identified under the assumption of sequential ignorability. I introduce a debiased machine learning (DML) method for estimating all quantities of interest, along with a set of bias formulas for sensitivity analysis. I illustrate the proposed framework and methodology using data from the National Longitudinal Survey of Youth, 1997 cohort.
Abstractor: As Provided
Entry Date: 2024
Accession Number: EJ1434929
Database: ERIC
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  Value: <anid>AN0178879688;som01aug.24;2024Aug09.05:11;v2.2.500</anid> <title id="AN0178879688-1">Attendance, Completion, and Heterogeneous Returns to College: A Causal Mediation Approach </title> <p>A growing body of social science research investigates whether the economic payoff to a college education is heterogeneous — in particular, whether disadvantaged youth can benefit more from attending and completing college relative to their more advantaged peers. Scholars, however, have employed different analytical strategies and reported mixed findings. To shed light on this literature, I propose a causal mediation approach to conceptualizing, evaluating, and unpacking the causal effects of college on earnings. By decomposing the total effect of attending a four-year college into several direct and indirect components, this approach not only clarifies the mechanisms through which college attendance boosts earnings, but illuminates the ways in which the postsecondary system may be both an equalizer and a stratifier. The total effect of college attendance, its direct and indirect components, and their heterogeneity across different subpopulations are all identified under the assumption of sequential ignorability. I introduce a debiased machine learning (DML) method for estimating all quantities of interest, along with a set of bias formulas for sensitivity analysis. I illustrate the proposed framework and methodology using data from the National Longitudinal Survey of Youth, 1997 cohort.</p> <p>Keywords: returns to college; heterogeneity; causal inference; causal mediation; debiased machine learning</p> <hd id="AN0178879688-2">Introduction</hd> <p>Education is long perceived as a ticket to the American dream, a pathway to economic success regardless of a person's circumstances of birth. Back in 1848, Horace Mann portrayed education as "a great equalizer of the conditions of men" ([<reflink idref="bib22" id="ref1">22</reflink>]). In his 2020 presidential campaign, Joe Biden envisioned a plan for higher education so that it serves as a gateway to economic opportunity for everyone, "regardless of their parents' income or the color of their skin."[<reflink idref="bib5" id="ref2">5</reflink>] Biden's emphasis on the role of higher education in social mobility is echoed by public opinion — the vast majority of Americans believe that nowadays a college education is "necessary to get ahead" ([<reflink idref="bib14" id="ref3">14</reflink>]).</p> <p>Echoing the public and political discourse on the role of higher education in equalizing opportunities, a growing body of social science research has investigated whether the economic payoff to a college education is heterogeneous — in particular, whether disadvantaged youth can benefit more from attending and completing college relative to their more advantaged peers. If so, it would be apt for us to characterize higher education as an "equalizer," in which case inducing more youth into college would potentially reduce inequality and improve intergenerational mobility.</p> <p>This body of research, however, has yielded mixed findings ([<reflink idref="bib16" id="ref4">16</reflink>]). On the one hand, several studies suggest that the economic payoff to a college education may be greater for students from disadvantaged backgrounds than for their more advantaged peers (e.g., [<reflink idref="bib5" id="ref5">5</reflink>]; [<reflink idref="bib1" id="ref6">1</reflink>]; [<reflink idref="bib24" id="ref7">24</reflink>]; [<reflink idref="bib3" id="ref8">3</reflink>]; [<reflink idref="bib52" id="ref9">52</reflink>]; [<reflink idref="bib13" id="ref10">13</reflink>]). These studies have variously measured (dis) advantage using race/ethnicity, parental income, or the propensity score, i.e., the probability of attending or completing a four-year college given an array of observed pre-college characteristics. In particular, [<reflink idref="bib3" id="ref11">3</reflink>]) find that young people with the lowest propensity scores — typically students from minority and low-income backgrounds — appear to benefit the most from a bachelor's degree (henceforth BA degree), a pattern they call "negative selection." On the other hand, economic studies that pay close attention to unobserved sorting into college suggest a theory of "positive selection," i.e., individuals self-select into college on the basis of their anticipated payoffs to attending college, and those most likely to attend college reap the highest economic returns from it ([<reflink idref="bib41" id="ref12">41</reflink>]; [<reflink idref="bib6" id="ref13">6</reflink>]; but see [<reflink idref="bib51" id="ref14">51</reflink>] for a reanalysis and reinterpretation of Carneiro et al.'s data). More recently, by modeling the earnings return to college as a flexible function of the propensity score, scholars have reported a more nuanced, U-shaped pattern of college effects, especially among men ([<reflink idref="bib7" id="ref15">7</reflink>]; see also [<reflink idref="bib50" id="ref16">50</reflink>]). In addition, a related strand of research on intergenerational income mobility suggests that once selection processes are adjusted for, the association between parental income and child income is about as strong among college graduates as among non-graduates, a finding that casts doubt on the equalizing potential of a college degree ([<reflink idref="bib46" id="ref17">46</reflink>]; [<reflink idref="bib12" id="ref18">12</reflink>]; but see [<reflink idref="bib19" id="ref19">19</reflink>]).</p> <p>While it is beyond the scope of this paper to fully reconcile the seemingly incongruent findings on heterogeneous college effects, I highlight an important distinction that has so far received insufficient attention in this body of research, namely, the distinction between attending college and completing a BA degree. In fact, almost all previous research on the economic payoff to higher education has treated college as a dichotomous variable, that is, whether a young adult with a high-school diploma or equivalent has attended (e.g., [<reflink idref="bib6" id="ref20">6</reflink>]; [<reflink idref="bib52" id="ref21">52</reflink>]), or graduated from (e.g., [<reflink idref="bib3" id="ref22">3</reflink>]; [<reflink idref="bib7" id="ref23">7</reflink>]), a four-year college by a certain age. Such a dichotomous approach has several limitations. First, it fails to distinguish the "direct effect" of college attendance (short of a BA degree) from its "continuation value," i.e., its effect on earnings via the possibility it creates for attaining higher levels of education, particularly a BA degree ([<reflink idref="bib15" id="ref24">15</reflink>]). This distinction is consequential because patterns of effect heterogeneity may differ sharply between the direct effect of college attendance and its continuation value. In particular, whereas the direct effect of college attendance may be equalizing, i.e., being larger among more disadvantaged students ([<reflink idref="bib13" id="ref25">13</reflink>]), its continuation value may be stratifying, i.e., favoring students from more advantaged backgrounds. The latter is plausible because minority and low-income college-goers are much less likely to complete college relative to their white and more affluent peers ([<reflink idref="bib2" id="ref26">2</reflink>]; [<reflink idref="bib10" id="ref27">10</reflink>]). In this case, a dichotomous approach based on either attendance or completion would obscure the opposing patterns of effect heterogeneity associated with different stages of the educational pipeline.</p> <p>Second, studies that focus on the effect of a BA degree on earnings often conflate high-school graduates and college dropouts under the umbrella of "non-graduates," and compare college graduates with non-graduates that are similar on a set of pre-college characteristics. This practice may lead to bias because it adjusts only for selection into college, but not <emph>selection out of college</emph>. A college dropout and a college graduate who share the same pre-college characteristics may differ substantially in their postsecondary characteristics, such as college quality, college GPA, and field of study. To the extent that these postsecondary characteristics affect both the chance of college completion and earnings, they are confounders of their causal relationship, which, if not adjusted for, will lead to biased estimates. Moreover, treating high-school graduates and college dropouts as a whole may engender spurious patterns of effect heterogeneity. For example, if we aim to examine heterogeneous effects of a college degree across students with different income backgrounds, high-income non-graduates may be more likely than low-income non-graduates to have attended college in the first place. Thus, if college experience per se (short of a BA degree) boosts earnings — for example, through its effects on human capital, social capital, and career-related information (see [<reflink idref="bib13" id="ref28">13</reflink>] for a detailed discussion) — the estimated effect of a BA degree among high-income youth might be smaller than that among low-income youth simply because the comparison group for high-income college graduates is, on average, more likely to have enjoyed the benefits of a college experience.</p> <p>To overcome the limitations of the dichotomous approach, I introduce a causal mediation framework for studying the effects of higher education on earnings and their heterogeneity across individuals with different backgrounds. Specifically, by treating BA completion as a mediator that transmits the effect of college attendance on earnings (see Figure 1), the proposed framework enables us to decompose the average total effect of attending a four-year college into four distinct components: (i) the direct effect of college attendance (short of a BA degree) on earnings, (ii) the probability of BA completion given college attendance, (iii) the net effect of BA completion on earnings, and (iv) a residual component reflecting the covariance between BA completion and its net effect on earnings. Each of these components may follow a distinct pattern of effect heterogeneity. For example, the direct effect of college attendance (i) and the net effect of BA completion (iii) may both follow a pattern of negative selection ([<reflink idref="bib3" id="ref29">3</reflink>]), but the opposite is likely true for the probability of BA completion given college attendance (ii). Thus, the proposed decomposition not only clarifies the mechanisms through which college attendance boosts earnings, but, more importantly, illuminates the ways in which the postsecondary system may be <emph>both an equalizer and a stratifier</emph>.</p> <p>Graph: Figure 1. Direct and indirect effects of college on earnings. Note: Factors that may confound the relationships between college attendance, BA completion, and earnings are omitted.</p> <p>When we observe a rich set of individual-, family-, and contextual-level characteristics that may affect a person's selection into and out of college, it is reasonable to entertain the assumption of sequential ignorability ([<reflink idref="bib29" id="ref30">29</reflink>]), which, in our context, means that (a) given observed pre-college characteristics, no unobserved confounding exists for the effect of college attendance on BA completion and earnings, and (b) among college goers, given observed pre-college and postsecondary characteristics, no unobserved confounding exists for the effect of BA completion on earnings. I show that under sequential ignorability, the total effect of college attendance, its direct and indirect components, and their heterogeneity across different subpopulations are all identified.</p> <p>Despite the identification result, given the large number of pre-college and postsecondary characteristics we will likely need to adjust for, estimation methods based purely on parametric models may suffer from model uncertainty and large biases due to model misspecification (e.g., [<reflink idref="bib44" id="ref31">44</reflink>]). To minimize model dependency while preserving statistical efficiency, I introduce a debiased machine learning (DML; [<reflink idref="bib8" id="ref32">8</reflink>]; [<reflink idref="bib33" id="ref33">33</reflink>]; [<reflink idref="bib47" id="ref34">47</reflink>]) method for estimating all quantities of interest. Through the use of flexible machine learning methods, carefully constructed estimating equations, and sample splitting, the DML estimators are not only robust to model misspecification but also immune to the regularization and overfitting biases that often afflict machine learning estimators of statistical parameters. I illustrate the proposed framework and DML method using data from the National Longitudinal Survey of Youth, 1997 cohort (NLSY97).</p> <hd id="AN0178879688-3">Unpacking Heterogeneous College Effects</hd> <p></p> <hd id="AN0178879688-4">A Causal Decomposition</hd> <p>We consider completion of a BA degree as an intermediate variable, i.e., a mediator, that transmits the effect of college attendance on earnings. Thus, the total effect of attending a four-year college on earnings can be decomposed into a direct effect of college attendance (short of a BA degree) and an indirect effect that operates through BA completion. The latter component is sometimes referred to as the "continuation value" of college attendance (e.g., [<reflink idref="bib15" id="ref35">15</reflink>]), and it is governed by a person's likelihood of BA completion given college attendance as well as the net effect of BA completion on earnings. Specifically, for individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> , let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>i</mi></msub></math> </ephtml> denote a binary indicator of attending a four-year college, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub></math> </ephtml> a binary indicator of BA completion, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math> </ephtml> labor market earnings. In addition, using the potential-outcomes notation ([<reflink idref="bib31" id="ref36">31</reflink>]), let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math> </ephtml> denote individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> 's potential status of BA completion if her college attendance status was set to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> </ephtml> , and let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></math> </ephtml> denote individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> 's potential earnings if her college attendance status was set to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> </ephtml> and BA completion status set to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> </ephtml> . The total effect (TE) of college attendance on earnings can then be expressed as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd /><mtd><mstyle displaystyle="true" scriptlevel="0"><msub><mrow><mi>T</mi><mi>E</mi></mrow><mi>i</mi></msub><mo>=</mo><msub><mi>Y</mi><mi>i</mi></msub><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mn>1</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mrow><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mstyle></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><msub><mi>Y</mi><mi>i</mi></msub><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mn>1</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mrow><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="1em" /><mtext>(because \;{M_{i}(0) =0}) </mtext></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><munder><mrow><munder><mrow><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder></mrow><mtext>direct effect of college attendance </mtext></munder><mo>+</mo><munder><mrow><munder><mrow><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><munder><mrow><munder><mrow><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mrow><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mrow><mo>⏟</mo></munder></mrow><mrow><mrow><mrow><mrow><mrow><mi mathvariant="normal">neteffectofBAcompletion</mi></mrow></mrow></mrow></mrow></mrow></munder></mrow><mo>⏟</mo></munder></mrow><mrow><mrow><mrow><mrow><mrow><mi mathvariant="normal">indirecteffectviaBAcompletion</mi></mrow></mrow></mrow></mrow></mrow></munder><mo>.</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>Thus, for individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> , the total effect of college attendance is governed by three components: the direct effect of college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> ), whether the person would complete a BA degree given college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> ), and the net effect of BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> ). The product of the latter two components constitutes the indirect effect of college via BA completion.</p> <p>Since for each individual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> , only one of the three potential outcomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> is observed, neither the direct effect of college attendance nor the net effect of BA completion can be computed at the individual level. We thus focus on the population- and group-level means of these effects. First, taking the expectation of equation (<reflink idref="bib1" id="ref37">1</reflink>) yields a population-level decomposition: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mtext>TE</mtext><mi>i</mi></msub><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></mrow></munder></mtd><mtd><mo>=</mo><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></mrow></munder><mo>+</mo><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi>π</mi><mtext>comp</mtext></msub></mrow></munder><mo>⋅</mo><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></mrow></munder></mtd></mtr><mtr><mtd /><mtd><mspace width="1em" /><mo>+</mo><munder><mrow><munder><mrow><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">[</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></mrow></munder></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo>+</mo><munder><mrow><munder><mrow><msub><mi>π</mi><mtext>comp</mtext></msub><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo>+</mo><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></mrow><mo>⏟</mo></munder></mrow><mrow><msub><mi mathvariant="normal">Δ</mi><mtext>ind</mtext></msub></mrow></munder><mo>.</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>Here, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> represents the average total effect of college on earnings, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> represents the average direct effect of college attendance on earnings, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>ind</mtext></msub></math> </ephtml> represents the average indirect effect via BA completion. The indirect effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>ind</mtext></msub></math> </ephtml> equals <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo>+</mo><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> represents the probability of BA completion if a person attended college, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> represents the average net effect of BA completion on earnings, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> is a component reflecting the covariance between BA completion and its net effect on earnings. Intuitively, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> is positive if those who would complete a BA degree given college attendance (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></math> </ephtml> ) can benefit more from a BA degree (i.e., larger <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> ) than those who would not complete a BA degree given college attendance (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> ), and negative if the opposite is true. According to the positive selection thesis ([<reflink idref="bib41" id="ref38">41</reflink>]; [<reflink idref="bib6" id="ref39">6</reflink>]), a positive <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> may arise if college goers possess knowledge about their individual-specific payoffs to a BA degree and decide whether to pursue a BA degree on the basis of their anticipated payoffs. A positive <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> may also arise for structural (rather than individual) reasons, for example, if the financial and cognitive resources of middle- and upper-class students allow them to both complete college at a higher rate and reap higher economic returns from a BA degree relative to their less advantaged peers.</p> <p>To see how each of the above components varies across individuals with different backgrounds, we can evaluate the conditional expectation of equation (<reflink idref="bib1" id="ref40">1</reflink>) given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> , some indicator of pre-college advantage. Analogous to the population-level decomposition (<reflink idref="bib2" id="ref41">2</reflink>), we have</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mrow><mi mathvariant="normal">TE</mi></mrow><mi>i</mi></msub><mo fence="false" stretchy="false">|</mo><msub><mrow><mi mathvariant="normal">S</mi></mrow><mi>i</mi></msub><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></munder></mtd><mtd><mo>=</mo><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></munder><mo>+</mo><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi>π</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></munder></mtd></mtr><mtr><mtd /><mtd><mo>⋅</mo><munder><mrow><munder><mrow><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></munder></mtd></mtr><mtr><mtd /><mtd><mspace width="1em" /><mo>+</mo><munder><mrow><munder><mrow><mrow><mi mathvariant="normal">Cov</mi></mrow><mo stretchy="false">[</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mrow><mo>:</mo><mspace width=".1em" /><mo>=</mo><mspace width=".1em" /><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></munder></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>π</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> represent the same components in equation (<reflink idref="bib2" id="ref42">2</reflink>) among individuals with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi></math> </ephtml> .</p> <p>The group-level decomposition (<reflink idref="bib3" id="ref43">3</reflink>) enables us to quantify the equalizing and stratifying roles of higher education. Specifically, the negative selection thesis ([<reflink idref="bib3" id="ref44">3</reflink>]) suggests that the direct effect of college attendance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> and the net effect of BA completion <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> may be particularly large among individuals from disadvantaged backgrounds, contributing to the equalizing role of higher education. On the other hand, ample empirical evidence indicates that college graduation rates are much higher among students from more advantaged backgrounds relative to their less privileged peers (e.g., [<reflink idref="bib2" id="ref45">2</reflink>]). Thus the component <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> is likely an increasing function of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> , contributing to the stratifying role of higher education. Furthermore, as noted earlier, the positive selection thesis suggests that college students may possess knowledge about their idiosyncratic payoffs to a BA degree and act on it. If such a pattern of self-selection is present and if it is stronger among more advantaged youth than among less advantaged youth (e.g., due to unequal access to information about their idiosyncratic returns to a BA degree or unequal capacities to act on such information), then the within-group covariance component <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> may also be an increasing function of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> , contributing to the stratifying role of higher education. Given these competing forces, an expansion in college enrollment would have the potential to reduce inequality if the equalizing roles of college (e.g., those associated with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> ) outweigh its stratifying roles (e.g., those associated with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> ).</p> <hd id="AN0178879688-5">Identification</hd> <p>Since the average total effect and its direct and indirect components all depend on potential outcomes, they cannot be directly estimated from data. We first need to identify these quantities — i.e., write them as functions of observed data only — under appropriate assumptions. In particular, the quantities of interest outlined in the previous section are all identified under the assumption of sequential ignorability ([<reflink idref="bib29" id="ref46">29</reflink>]), which, simply speaking, means that given observed covariates, no unobserved confounding exists for the causal relationships among college attendance, BA completion, and earnings. Specifically, if we use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> to denote a set of observed pre-college characteristics that may confound the causal effects of college attendance and BA completion on earnings, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> to denote a set of observed postsecondary characteristics (e.g., college GPA) that may additionally confound the causal effect of BA completion on earnings, the sequential ignorability assumption states that (a) conditional on pre-college characteristics <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , college attendance is independent of both potential earnings and potential college completion status (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊥</mo><mo>⊥</mo><mi>A</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi></math> </ephtml> ), and (b) conditional on pre-college characteristics <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and postsecondary characteristics <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , BA completion is independent of potential earnings among college goers (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊥</mo><mo>⊥</mo><mi>M</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>Z</mi></math> </ephtml> ). Figure 2 contains a directed acyclic graph (DAG) relating college attendance, BA completion, and earnings to potential pre-college and postsecondary confounders of their relationships. Here, the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> vectors are assumed to capture a broad range of potential confounders of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> relationships, such as socioeconomic background, cognitive and noncognitive skills, motivation, personality traits, and social capital. Under this DAG, sequential ignorability will hold if all of these potential confounders are observed and accurately measured. In practice, however, some of these confounders (e.g., motivation) are likely unobserved or imperfectly measured. Thus, in most (if not all) empirical studies, we should view sequential ignorability as a working assumption and conduct a sensitivity analysis (see below) to assess the direction and magnitude of potential bias due to unobserved confounding.</p> <p>Graph: Figure 2. Hypothesized causal relationships in a direct acyclic graph.</p> <p>Equation (<reflink idref="bib2" id="ref47">2</reflink>) implies that to identify the total effect of college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> ) and its various components ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> ), it suffices to identify the following expected potential outcomes: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> . Here I omit the subscript <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> for conciseness. Under sequential ignorability, these quantities are identified via Robins's ([<reflink idref="bib28" id="ref48">28</reflink>]; [<reflink idref="bib29" id="ref49">29</reflink>]) g-formula: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>M</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo>∫</mo><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>z</mi><mo>,</mo><mi>M</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo>∫</mo><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>z</mi><mo>,</mo><mi>M</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>z</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> </ephtml> denotes the cumulative distribution function of a random variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> . It is easy to see that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> is identified by equation (<reflink idref="bib4" id="ref50">4</reflink>), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> identified by equation (<reflink idref="bib6" id="ref51">6</reflink>) minus equation (<reflink idref="bib5" id="ref52">5</reflink>), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> identified by equation (<reflink idref="bib7" id="ref53">7</reflink>) minus equation (<reflink idref="bib5" id="ref54">5</reflink>), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> identified by equation (<reflink idref="bib8" id="ref55">8</reflink>) minus equation (<reflink idref="bib7" id="ref56">7</reflink>), and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> identified by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub><mo>−</mo><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo>−</mo><msub><mi>π</mi><mtext>comp</mtext></msub><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> . Components of the group-level decomposition are identified analogously, except that all quantities involved in equations (<reflink idref="bib4" id="ref57">4</reflink>)–(<reflink idref="bib8" id="ref58">8</reflink>) should now be conditioned on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi></math> </ephtml> .</p> <p>The sequential ignorability assumption is weaker than the ignorability assumption previously invoked for studying the effect of a college degree on earnings. For example, [<reflink idref="bib3" id="ref59">3</reflink>]) used a dichotomous approach that directly compares college graduates with non-graduates that are similar on a set of pre-college characteristics. This approach implicitly assumes that conditional on pre-college characteristics <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , BA completion status is independent of potential earnings under completion and non-completion (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊥</mo><mo>⊥</mo><mi>M</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi></math> </ephtml> ). This assumption is stronger than sequential ignorability because it rules out (a) a direct effect of college attendance on earnings (the arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> in Figure 2) <emph>and</emph> (b) postsecondary characteristics that may confound the effect of BA completion on earnings (the noncausal path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo stretchy="false">←</mo><mi>Z</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> in Figure 2). By contrast, sequential ignorability allows for both (a) and (b).</p> <p>The sequential ignorability assumption is also weaker than the ignorability assumption required for identifying the natural direct and indirect effects (NDE and NIE) in a generic causal mediation analysis ([<reflink idref="bib39" id="ref60">39</reflink>]; [<reflink idref="bib17" id="ref61">17</reflink>]). The latter requires a conditional independence relationship between the so-called cross-world counterfactuals given pretreatment confounders <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , namely, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⊥</mo><mo>⊥</mo><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>a</mi><mo>*</mo></msup><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><mi>X</mi></math> </ephtml> , which rules out post-treatment confounding of the mediator-outcome relationship. By contrast, all components of our effect decomposition are identified under sequential ignorability, which allows for observed post-treatment confounders <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> . To understand this result, note that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> correspond to a controlled direct effect (CDE; [<reflink idref="bib26" id="ref62">26</reflink>]; [<reflink idref="bib30" id="ref63">30</reflink>]) and a controlled mediator effect (CME; [<reflink idref="bib45" id="ref64">45</reflink>]), both of which are identified under sequential ignorability. Interestingly, in our context, because <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> is also the NDE, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>ind</mtext></msub><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> is the NIE. Hence, equation (<reflink idref="bib2" id="ref65">2</reflink>) is a more fine-grained decomposition of the ATE than the two-component decomposition routinely considered in causal mediation analysis; yet, due to the constraint <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> , all components are identified under sequential ignorability.</p> <p>Nonetheless, sequential ignorability is still a strong and unverifiable assumption, which can be violated whenever unobserved confounders exist for any of the causal relationships involved. For example, in my empirical illustration below, some of the potential confounders depicted in Figure 2 such as motivation are not directly measured. This is a common scenario in observational studies of college effects, or for that matter, in observational studies in general. Thus, in practice, it is prudent to view sequential ignorability as a working assumption and report the sensitivity of estimated causal effects to potential violations of sequential ignorability (e.g., [<reflink idref="bib4" id="ref66">4</reflink>]). Later in this section, I outline a bias factor approach for performing sensitivity analysis in our context.</p> <hd id="AN0178879688-6">Estimation</hd> <p>Equations (<reflink idref="bib4" id="ref67">4</reflink>)–(<reflink idref="bib8" id="ref68">8</reflink>) and their group-level counterparts can be estimated via a variety of methods, such as g-computation ([<reflink idref="bib28" id="ref69">28</reflink>], [<reflink idref="bib29" id="ref70">29</reflink>]), sequential g-estimation ([<reflink idref="bib40" id="ref71">40</reflink>]; [<reflink idref="bib18" id="ref72">18</reflink>]), regression-with-residuals ([<reflink idref="bib48" id="ref73">48</reflink>]; [<reflink idref="bib42" id="ref74">42</reflink>]), inverse probability weighting (IPW; [<reflink idref="bib36" id="ref75">36</reflink>]), and residual balancing ([<reflink idref="bib49" id="ref76">49</reflink>]) (see [<reflink idref="bib47" id="ref77">47</reflink>] for an overview of various estimation methods). Yet, all of these methods rely on the correct specification of at least two parametric models about <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> , or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> (implicitly or explicitly). Given the large number of pre-college covariates and postsecondary characteristics we are likely to encounter in practice, estimators based purely on parametric models may suffer large biases due to model misspecification. To minimize model dependency, I now introduce a debiased machine learning (DML; [<reflink idref="bib8" id="ref78">8</reflink>]; [<reflink idref="bib33" id="ref79">33</reflink>]; [<reflink idref="bib47" id="ref80">47</reflink>]) method for estimating quantities (<reflink idref="bib4" id="ref81">4</reflink>)-(<reflink idref="bib8" id="ref82">8</reflink>) and their group-level counterparts.</p> <p>In our context, the DML approach is characterized by three key elements: a sample-splitting technique called cross-fitting, the construction of a "Neyman-orthogonal signal" for each of the target parameters in equations (<reflink idref="bib4" id="ref83">4</reflink>)–(<reflink idref="bib8" id="ref84">8</reflink>), and, when estimating the group-level decomposition (<reflink idref="bib3" id="ref85">3</reflink>), a linear model of the Neyman-orthogonal signal on our measure of pre-college advantage <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> . Specifically, it involves the following steps:</p> <p></p> <ulist> <item> Randomly partition the analytical sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">I</mi></mrow></math> </ephtml> into <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math> </ephtml> equal-sized subsamples: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">I</mi></mrow><mn>1</mn></msub><mo>,</mo><msub><mrow><mi mathvariant="script">I</mi></mrow><mn>2</mn></msub><mo>...</mo><msub><mrow><mi mathvariant="script">I</mi></mrow><mi>J</mi></msub></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math> </ephtml> is recommended to be a small number such as 5 ([<reflink idref="bib8" id="ref86">8</reflink>]);</item> <p></p> <item> For each subsample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">I</mi></mrow><mrow><mspace width=".1em" /><mi>j</mi></mrow></msub></math> </ephtml> ,</item> <p></p> <item> Use the observations in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">I</mi></mrow><mi mathvariant="normal">∖</mi><msub><mrow><mi mathvariant="script">I</mi></mrow><mrow><mspace width=".1em" /><mi>j</mi></mrow></msub></math> </ephtml> (i.e., all observations but those in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">I</mi></mrow><mrow><mspace width=".1em" /><mi>j</mi></mrow></msub></math> </ephtml> ) to fit a flexible machine learning model for each of the following "nuisance functions":[<reflink idref="bib6" id="ref87">6</reflink>] <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo form="prefix" movablelimits="true">Pr</mo><mo stretchy="false">[</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo form="prefix" movablelimits="true">Pr</mo><mo stretchy="false">[</mo><mi>M</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo form="prefix" movablelimits="true">Pr</mo><mo stretchy="false">[</mo><mi>M</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>a</mi><mo stretchy="false">]</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>z</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mrow><mi>Z</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn></mrow></msub><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>Z</mi><mo>,</mo><mi>m</mi><mo stretchy="false">]</mo></math> </ephtml> ;</item> <p></p> <item> For each observation in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">I</mi></mrow><mrow><mspace width=".1em" /><mi>j</mi></mrow></msub></math> </ephtml> , use estimates of the above models to construct a set of "Neyman-orthogonal signals," one for each potential outcome: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>M</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> .</item> <p></p> <item> In the full sample, use the above signals for potential outcomes to construct the corresponding signals for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> . For example, the signal for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , the signal for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>−</mo><msup><mi>Y</mi><mo>*</mo></msup><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , and so on. The sample averages of these signals constitute the DML estimates of the corresponding quantities, and the covariance component is estimated by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>cov</mtext></msub><mo>=</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>tot</mtext></msub><mo>−</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>att</mtext></msub><mo>−</mo><msub><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub></math> </ephtml> .</item> <p></p> <item> To assess effect heterogeneity by pre-college advantage (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> ), fit a linear model of the corresponding signal (constructed in step 3) on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> . The heterogeneity of the covariance component by pre-college advantage is estimated by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>tot</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> .</item> </ulist> <p>In step 2(b), the Neyman-orthogonal signals are plug-in estimates of the recentered efficient influence functions for the expectations of the corresponding potential outcomes ([<reflink idref="bib33" id="ref88">33</reflink>]). Their analytical expressions are given in Supplementary Material A. These signals satisfy several interesting properties, which, when combined with cross-fitting, yield estimators that are not only robust to model misspecification but also consistent and asymptotically normal under mild conditions.</p> <p>Below, I use the estimand <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , i.e., average potential earnings under non-college-attendance, to illustrate the logic of the DML method. As noted above, under sequential ignorability, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> equals <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> (equation 5). To simplify exposition, let us denote this quantity by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> </ephtml> . Its Neyman-orthogonal signal can be written as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mi>A</mi></mrow><mrow><mn>1</mn><mo>−</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><mi>Y</mi><mo>−</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mrow><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>Z</mi><mo>,</mo><mi>M</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></math> </ephtml> denotes observed data, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width=".1em" /><mo>:</mo><mo>=</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></math> </ephtml> is the conditional mean of earnings given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width=".1em" /><mo>:</mo><mo>=</mo><mo form="prefix" movablelimits="true">Pr</mo><mo stretchy="false">[</mo><mi>A</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><mi>X</mi><mo stretchy="false">]</mo></math> </ephtml> is the propensity score of attending college given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> denote the empirical estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></math> </ephtml> , respectively. In equation (<reflink idref="bib9" id="ref89">9</reflink>), we use the notation " <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math> </ephtml> " to highlight that the signal depends on both the observed data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> </ephtml> and the estimated models <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> . This quantity is useful because <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>θ</mi></math> </ephtml> , suggesting that we can estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> </ephtml> by first estimating the outcome and propensity score models <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math> </ephtml> and then taking a sample mean of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub><mo>=</mo><msub><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mi>n</mi></msub><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mi>n</mi></msub><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo><mspace width=".1em" /><mo>:</mo><mo>=</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><munder><mo>∑</mo><mi>i</mi></munder><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></math> </ephtml> denotes the operation of computing a sample mean. In this sense, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><msub><mi>O</mi><mi>i</mi></msub><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math> </ephtml> can be interpreted as the "contribution" of observation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> to the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub></math> </ephtml> . Second, the conditional mean of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>s</mi></math> </ephtml> is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>A</mi><mo>=</mo><mn>0</mn><mo stretchy="false">]</mo><mi>d</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo fence="false" stretchy="false">|</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , which, under sequential ignorability, equals <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><mi>S</mi><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></math> </ephtml> . Thus, we can estimate the latter by averaging <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math> </ephtml> among members of group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> . When <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> is continuous, however, we cannot estimate such conditional means nonparametrically. Thus, in step 4 of the above procedure, we fit a linear model of the signal <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> , which can be seen as a first-order approximation of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><mi>S</mi><mo>=</mo><mi>s</mi><mo stretchy="false">]</mo></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> is continuous but is equivalent to taking a group-specific average of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></math> </ephtml> when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> is discrete.</p> <p>To understand the property of the estimator <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub></math> </ephtml> , it is best to consider the following decomposition of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub><mo>−</mo><mi>θ</mi></math> </ephtml> ) ([<reflink idref="bib20" id="ref90">20</reflink>]), <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo><msqrt><mi>n</mi></msqrt><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo></mrow><msub><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mi>n</mi></msub><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mrow><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><munder><mrow><munder><mrow><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><msub><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mi>n</mi></msub><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mi>A</mi></munder><mo>+</mo><munder><mrow><munder><mrow><msqrt><mi>n</mi></msqrt><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo>−</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mi>B</mi></munder></mtd></mtr><mtr><mtd /><mtd><mspace width="1em" /><mo>+</mo><munder><mrow><munder><mrow><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><msub><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mi>n</mi></msub><mo>−</mo><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo>−</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder></mrow><mi>C</mi></munder><mo>,</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mi>g</mi><mo>=</mo><mo>∫</mo><mi>g</mi><mi>d</mi><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow></math> </ephtml> denotes the expectation of a function <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> of observed data under the true distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is treated as fixed. In equation (<reflink idref="bib11" id="ref91">11</reflink>), term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> has a mean of zero and variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> . By the central limit theorem, it converges to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mtext>Var</mtext><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> </ephtml> . Thus, by Slutsky's theorem, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></math> </ephtml> will also converge to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mtext>Var</mtext><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></math> </ephtml> if terms <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> </ephtml> are asymptotically negligible, i.e., if they converge to zero in probability. The latter condition can also be written as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><msub><mi>o</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><msub><mi>o</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>First, it can be shown that term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> is in the order of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><msub><mi>O</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>μ</mi><mo fence="false" stretchy="false">‖</mo><mo>⋅</mo><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>π</mi><mo fence="false" stretchy="false">‖</mo><mo stretchy="false">)</mo></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">‖</mo><mo>⋅</mo><mo fence="false" stretchy="false">‖</mo></math> </ephtml> denotes the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml> -norm of a function with respect to probability measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">P</mi></mrow></mrow></math> </ephtml> . The multiplicative structure of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>μ</mi><mo fence="false" stretchy="false">‖</mo><mo>⋅</mo><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>π</mi><mo fence="false" stretchy="false">‖</mo></math> </ephtml> facilitates the use of machine learning methods to estimate the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math> </ephtml> functions. To see this connection, note that due to the data-driven nature of machine learning algorithms, they generally do not provide <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt></math> </ephtml> -consistent estimates of the underlying functions, such as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math> </ephtml> . However, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt></math> </ephtml> -consistency is not required of either <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>μ</mi><mo fence="false" stretchy="false">‖</mo><mo>⋅</mo><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>π</mi><mo fence="false" stretchy="false">‖</mo></math> </ephtml> to converge to zero. In fact, provided that both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> are consistent at a faster-than- <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup></math> </ephtml> rate, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>μ</mi><mo fence="false" stretchy="false">‖</mo><mo>⋅</mo><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>π</mi><mo fence="false" stretchy="false">‖</mo></math> </ephtml> will be <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><msub><mi>o</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo stretchy="false">)</mo><msub><mi>o</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>o</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> , rendering term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> asymptotically negligible. This condition, unlike the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt></math> </ephtml> -consistency required for the nuisance functions in conventional estimators such as IPW, is achievable for many machine learning methods such as Lasso ([<reflink idref="bib8" id="ref92">8</reflink>]). Second, it can be shown that when cross-fitting is used, term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> </ephtml> is in the order of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>O</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mo fence="false" stretchy="false">‖</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo><mo>−</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">‖</mo><mo stretchy="false">)</mo></math> </ephtml> , which will also be asymptotically negligible if both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow></math> </ephtml> are consistent.</p> <p>In sum, when cross-fitting is used in combination with estimating equation (<reflink idref="bib10" id="ref93">10</reflink>), the resulting estimator will be consistent and asymptotically normal provided that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>n</mi></msqrt><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>μ</mi><mo fence="false" stretchy="false">‖</mo><mo>⋅</mo><mo fence="false" stretchy="false">‖</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo>−</mo><mi>π</mi><mo fence="false" stretchy="false">‖</mo><mo>=</mo><msub><mi>o</mi><mrow><mspace width=".1em" /><mi>p</mi></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> , a condition achievable even when the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math> </ephtml> functions are estimated with flexible machine learning methods. This property of the DML approach makes it highly attractive in our context, in which the rich sets of background characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ) and postsecondary characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> ) (see the next section) make it unrealistic for us to correctly specify parametric models for college attendance and college completion, which would be required to justify conventional methods such as IPW. Since the asymptotic variance of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>θ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>DML</mtext></msub></math> </ephtml> is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mo stretchy="false">[</mo><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mi>μ</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , we can construct a plug-in estimate of the standard error as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mrow><mover><mrow><mi>V</mi><mi>a</mi><mi>r</mi></mrow><mo>^</mo></mover></mrow><mrow><mo>[</mo><mrow><mi>φ</mi><mo stretchy="false">(</mo><mi>O</mi><mo>;</mo><mrow><mover><mi>μ</mi><mo stretchy="false">^</mo></mover></mrow><mo>,</mo><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></msqrt><mo>/</mo><mi>n</mi></math> </ephtml> .</p> <p>Although the above reasoning is for the estimand <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">E</mi></mrow></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> </ephtml> , the same logic applies to our DML estimators of the other expected potential outcomes (i.e., equations 4, 6-8) and the causal effects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ([<reflink idref="bib47" id="ref94">47</reflink>]). Their standard errors can all be estimated through the empirical variances of the corresponding Neyman-orthogonal signals. Estimates of the group-level causal effects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> are given by the predicted values of the linear models in Step 4, and their standard errors can be estimated through the robust ("sandwich") estimator of the corresponding regression coefficients ([<reflink idref="bib33" id="ref95">33</reflink>]). The covariance components <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , as noted above, are estimated using the plug-in estimators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>cov</mtext></msub><mo>=</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>tot</mtext></msub><mo>−</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>att</mtext></msub><mo>−</mo><msub><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>tot</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> . Their standard errors can be estimated through the empirical variances of their influence functions, which are detailed in Supplementary Material B.</p> <hd id="AN0178879688-7">A Bias Factor Approach to Sensitivity Analysis</hd> <p>For a generic causal mediation analysis, [<reflink idref="bib38" id="ref96">38</reflink>]) and [<reflink idref="bib37" id="ref97">37</reflink>]) introduced a bias factor approach for assessing the sensitivity of estimated total, direct, and indirect effects to unobserved confounding. In our context, this approach can be adapted to derive a set of bias formulas for the total effect of college ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> ), the direct effect of attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> ), and the net effect of completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ).</p> <p>First, let us consider the total effect of college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> ), which may be confounded by unobserved individual characteristics that affect both college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ) and earnings ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ). For analytical tractability, we consider a binary unobserved confounder <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> , say a personality trait that predisposes a person to prefer cognitive tasks over noncognitive tasks, that affects both college attendance and earnings. Under some simplifying assumptions regarding the homogeneity of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> relationships, the bias for the estimated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> is given by (see Supplementary Material C) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>bias</mtext><mo stretchy="false">[</mo><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub><mo stretchy="false">]</mo><mo>=</mo><msub><mi>α</mi><mtext>tot</mtext></msub><msub><mi>β</mi><mtext>tot</mtext></msub><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub></math> </ephtml> denotes the difference in the prevalence of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> between high school graduates ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>0</mn></math> </ephtml> ) and college goers ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn></math> </ephtml> ) given pre-college covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>tot</mtext></msub></math> </ephtml> denotes the average difference in earnings between those with and without <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> given college attendance status <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> and pre-college covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> .</p> <p>Second, unobserved confounders may exist for the causal effect of BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> ) and earnings ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ). In this case, while the total effect of college attendance may still be unbiased, the direct effect of college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> ) and the net effect of BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ) can be over- or under-estimated. To explore the direction and magnitude of potential bias, let us again consider a binary unobserved confounder <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> , say availability of a supportive social network, that affects both BA completion and earnings but may itself be affected by college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ). Under some simplifying assumptions regarding the homogeneity of the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> - <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> relationships, the biases for the estimated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> are given by (see Supplementary Material C) <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>bias</mtext><mo stretchy="false">[</mo><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">]</mo><mo>=</mo><mo>−</mo><msub><mi>π</mi><mtext>comp</mtext></msub><msub><mi>α</mi><mtext>comp</mtext></msub><msub><mi>β</mi><mtext>net</mtext></msub><mo>,</mo></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!bias</mtext><mo stretchy="false">[</mo><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">]</mo><mo>=</mo><msub><mi>α</mi><mtext>comp</mtext></msub><msub><mi>β</mi><mtext>net</mtext></msub><mo>,</mo></math> </ephtml> </p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> is the probability of BA completion given college attendance (see equation 2), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>comp</mtext></msub></math> </ephtml> denotes the difference in the prevalence of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> between college dropouts ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>M</mi><mo>=</mo><mn>0</mn></math> </ephtml> ) and college graduates ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>M</mi><mo>=</mo><mn>1</mn></math> </ephtml> ) given both pre-college and postsecondary characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> ), and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>net</mtext></msub></math> </ephtml> denotes the net difference in earnings between those with and without the unobserved characteristic <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> .</p> <p>The above formulas can also be used to assess the sensitivity of group-level causal effects <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> . In this case, the sensitivity parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>tot</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>comp</mtext></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>net</mtext></msub></math> </ephtml> are group-specific, i.e., depending on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>s</mi></math> </ephtml> . It is clear that if these sensitivity parameters are identical between individuals with different values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> </ephtml> , estimated patterns of effect heterogeneity will be unaffected. In other words, our estimates of effect heterogeneity will be biased only if there are group differences in these sensitivity parameters. For example, if we found that low-propensity college goers benefit more from completing college than high-propensity college goers, potential bias in this finding would be <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mtext>comp</mtext><mtext>low \;propensity</mtext></msubsup><msubsup><mi>β</mi><mtext>net</mtext><mtext>low \;propensity</mtext></msubsup><mo>−</mo><msubsup><mi>α</mi><mtext>comp</mtext><mtext>high \;propensity</mtext></msubsup><msubsup><mi>β</mi><mtext>net</mtext><mtext>high \;propensity</mtext></msubsup></math> </ephtml> . In the next section, we illustrate this approach by applying it to our estimates from the NLSY97 data.</p> <hd id="AN0178879688-8">Empirical Illustration</hd> <p></p> <hd id="AN0178879688-9">Data, Measures, and Implementation</hd> <p>Below I illustrate the proposed methods using data from the National Longitudinal Survey of Youth, 1997 cohort (NLSY97).[<reflink idref="bib7" id="ref98">7</reflink>] The NLSY97 began with a nationally representative sample of 8,984 men and women at ages 12–17 in 1997. These individuals were interviewed annually through 2011 and biennially thereafter. I limit my analytical sample to respondents who had completed at least a high-school diploma or GED by age 22 and had valid earnings information at ages 30-33, the oldest ages for which data for the youngest respondents in NLSY97 are available ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>576</mn></math> </ephtml> ).</p> <p>I construct five sets of variables, each corresponding to a node in Figure 2: college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ), BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> ), earnings ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ), pre-college characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ), and postsecondary characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> ). Specifically, college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ) denotes whether the respondent had attended a four-year college by age 22, and BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> ) denotes whether the respondent had received a BA degree by age 29. A respondent is coded as a <emph>college goer</emph> (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn></math> </ephtml> ) if she had either attended a four-year college by age 22 or received a BA degree by age 29, and as a <emph>high school graduate</emph> otherwise (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>0</mn></math> </ephtml> ). Among college goers, a respondent is coded as a <emph>college graduate</emph> (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mn>1</mn></math> </ephtml> ) if she had received a BA degree by age 29, and as a <emph>college dropout/stopout</emph> (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mn>0</mn></math> </ephtml> ) otherwise. Earnings denote the natural logarithm of the respondent's average annual earnings at ages 30–33 (inflation-adjusted to 2019 dollars). To accommodate respondents with zero earnings (due to unemployment, labor force nonparticipation, and incarceration), I add a small constant (<reflink idref="bib1" id="ref99">1</reflink>,000 dollars) to the respondent's average annual earnings before taking the log transformation. To assess the robustness of my findings to this measurement choice, I have conducted parallel analyses using the percentile rank of earnings as the outcome. The results are similar to those reported below (see Supplementary Material D).</p> <p>Guided by the DAG in Figure 2, I include in the vector of pre-college characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ) several groups of variables: (a) basic demographic variables (gender, race, ethnicity, age in 1997); (b) socioeconomic background (parental education, parental income, parental assets, co-residence with both biological parents, presence of a paternal figure, rural residence, southern residence); (c) cognitive and noncognitive skills (percentile score on the Armed Services Vocational Aptitude Battery test, high school GPA, an index of substance use [ranging from 0 to 3], an index of delinquency [ranging from 0 to 10], whether the respondent had any children by age 18); and (d) peer and school-level characteristics (college expectation among peers, 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 using mother's years of schooling; when mother's years of schooling is unavailable, it is measured using father's years of schooling. Parental income is measured as the average annual parental income from 1997 to 2001. Both parental income and parental assets are inflation-adjusted to 2019 dollars.</p> <p>Similarly, following the DAG in Figure 2, I include in the vector of postsecondary characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> ) several variables pertaining to college quality as well as the respondent's field of study, college GPA, and the amounts of loans that the student has taken to finance college. In each survey wave of the NLSY97, respondents were asked to report, if any, the names of the colleges in which they were currently or most recently enrolled. Since some respondents attended more than one college, I focus on the college in which the respondent had been enrolled for the longest time by age 29. The college characteristics include: (a) college type, which is a trichotomous variable denoting whether the college is a public institution, a private not-for-profit institution, or a for-profit institution; (b) college selectivity, operationalized as three dummy variables denoting whether the college is one of the "most competitive," "highly competitive," and "very competitive" colleges in Barron's Profile of American Colleges 2000; (c) graduation rate, operationalized as the percentage of students graduating within six years of enrollment measured in 2002; and (d) "upward mobility rate," measured as the percentage of students who reach the top quintile of the income distribution among those with parents in the bottom quintile of the income distribution. Data on graduation rates and upward mobility rates come from the Department of Education's Integrated Postsecondary Education Data System (IPEDS) and the Opportunity Insights project ([<reflink idref="bib9" id="ref100">9</reflink>]), respectively. In each survey wave, respondents who were currently or recently enrolled in college were also asked to report their major field of study. I use a dummy variable to denote whether the field of study in which the respondent had majored for the longest time by age 29 is a STEM field. College GPA is measured as the respondent's cumulative GPA from the Post-Secondary Transcript Study (PSTRAN). Finally, I include two variables representing the total amounts of loans that the respondent had taken from family and friends and from other sources (including the federal government) to pay for college by age 29. Previous studies suggest that educational debt affects both the likelihood of college completion (e.g., [<reflink idref="bib11" id="ref101">11</reflink>]) and labor market outcomes (e.g., [<reflink idref="bib25" id="ref102">25</reflink>]). In my analytical sample, some components of the pre-college characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ) and postsecondary characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> ) contain a small fraction of missing values. They are handled by multivariate imputation via chained equations, with ten imputed data sets. The standard errors of our parameter estimates are adjusted using Rubin's ([<reflink idref="bib32" id="ref103">32</reflink>]) method.</p> <p>After constructing the analytical sample, I apply the DML algorithm to implement the decompositions (<reflink idref="bib2" id="ref104">2</reflink>) and (<reflink idref="bib3" id="ref105">3</reflink>). To examine effect heterogeneity by pre-college advantage, I compare individuals with different estimated propensity scores of attending college.[<reflink idref="bib8" id="ref106">8</reflink>] Previous research has advocated the use of the propensity score as a summary index of pre-college advantage in socioeconomic and academic resources ([<reflink idref="bib3" id="ref107">3</reflink>]; [<reflink idref="bib43" id="ref108">43</reflink>]). Thus, heterogeneous returns to college between individuals with lower and higher propensity scores signify the equalizing versus stratifying roles of college. Given that recent research has reported U-shaped patterns of effect heterogeneity by the propensity score ([<reflink idref="bib50" id="ref109">50</reflink>]; [<reflink idref="bib7" id="ref110">7</reflink>]), I discretize the estimated propensity score into its quintiles and report quintile-specific estimates of all quantities of interest. Following [<reflink idref="bib8" id="ref111">8</reflink>]), I use five-fold cross-fitting, meaning that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo>=</mo><mn>5</mn></math> </ephtml> . All nuisance functions, including the propensity score of college attendance, are estimated using a super learner ([<reflink idref="bib35" id="ref112">35</reflink>]) composed of Lasso and random forest.[<reflink idref="bib9" id="ref113">9</reflink>] The NLSY sampling weights are used in the estimation of all nuisance functions and target parameters.</p> <hd id="AN0178879688-10">Results</hd> <p>Table 1 reports estimates of the average total effect (ATE) and its direct and indirect components (i.e. equation 2). The first column shows that the estimated ATE of attending a four-year college on log earnings is 0.39, implying a 47.7% earnings premium ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mn>0.39</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0.477</mn></math> </ephtml> ). The next two columns indicate that the bulk of the ATE is indirect, i.e., through the possibility of completing a BA degree. Without completing a BA degree, the average direct effect of college attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> ) is estimated at 0.14, or a 15% earnings premium ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mn>0.14</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0.15</mn></math> </ephtml> ) relative to high school graduates. The last three columns show estimates of the three components that compose the indirect effect via BA completion: the probability of BA completion given attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> ), the net effect of BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ), and the covariance between BA completion and its net effect on earnings ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> ). Among them, the covariance component is very small; thus the indirect effect ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>ind</mtext></msub><mo>=</mo><msub><mi>π</mi><mtext>comp</mtext></msub><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub><mo>+</mo><msub><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math> </ephtml> ) is largely determined by the product of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.57</mn><mo>*</mo><mn>0.47</mn><mo>=</mo><mn>0.27</mn></math> </ephtml> ). In particular, the estimated net effect of BA completion implies an earning premium of 60% ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mn>0.47</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0.60</mn></math> </ephtml> ) for BA holders compared with college dropouts/stopouts. The sum of the estimated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> is 0.61, which can be interpreted as the <emph>joint effect</emph> of attending and completing a four-year college on earnings. In other words, the earnings premium associated with attending and completing a four-year college as opposed to not attending college is about <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mn>84</mn><mtext>%</mtext></math> </ephtml> ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mn>0.61</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0.84</mn></math> </ephtml> ).</p> <p>Graph</p> <p>Table 1. Decomposition of the Average Total Effect (ATE) of College Attendance on Log Earnings.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /></colgroup><thead><tr><th align="left">Total Effect (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math></p>)</th><th align="left">Direct Effect (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math></p>)</th><th align="left">Indirect Effect (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>ind</mtext></msub></math></p>)</th><th align="left">Completion Prob. (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>π</mi><mtext>comp</mtext></msub></math></p>)</th><th align="left">Completion Effect (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math></p>)</th><th align="left">Covariance Term (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>cov</mtext></msub></math></p>)</th></tr></thead><tbody><tr><td>0.39 (0.05)</td><td>0.14 (0.06)</td><td>0.25 (0.04)</td><td>0.57 (0.01)</td><td>0.47 (0.07)</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.02 (0.02)</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note:</emph> Numbers in parentheses are estimates of standard errors, which are constructed using the empirical variances of the corresponding influence functions and adjusted for multiple imputation via Rubin's ([<reflink idref="bib32" id="ref114">32</reflink>]) method.</p> <p>Figure 3 shows estimates of the total effect and its various components in each of the propensity score quintiles. We find suggestions of nonlinearity in several components, such as the total and direct effects of attendance, although estimation uncertainty prevents us from reaching a definitive conclusion. However, several patterns are discernible for the lowest-propensity individuals, i.e., those in the first quintile. On the one hand, their estimated direct effect of attendance is particularly large (0.45), much larger than those for the other quintiles, whose direct effect estimates are all relatively small and statistically indistinguishable from zero. On the other hand, their estimated indirect effect via BA completion is exceptionally small; in fact, it is negative. This finding is counterintuitive if we construe the indirect effect as reflecting the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo stretchy="false">→</mo><mi>M</mi><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> in Figure 2. Since both the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> (i.e., the probability of BA completion given attendance) and the effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> (the net effect of BA completion) are positive, how can the indirect effect of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> via <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> </ephtml> be negative? This is due to the (estimated) covariance component for the lowest-propensity group ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> ), which is not only negative but larger in absolute value than <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> , rendering the indirect effect estimate ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>ind</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mover><mi>π</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>comp</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mover><mi mathvariant="normal">Δ</mi><mo stretchy="false">^</mo></mover></mrow><mtext>cov</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></math> </ephtml> ) negative. Substantively, the negative covariance means that among the lowest-propensity individuals, those who would benefit more from completing college are less likely to complete college given attendance, a pattern we might call "negative selection among the least advantaged." As a result of their particularly large direct effect and exceptionally small indirect effect, the total effect of college among the lowest-propensity individuals appears comparable to that for their more advantaged peers (e.g., those in the fourth and fifth quintiles). Clearly, without the effect decomposition, the sharp and countervailing patterns of effect heterogeneity between the lowest-propensity individuals and their more advantaged peers would be obscured.</p> <p>Graph: Figure 3. Estimates of the total effect and its components by propensity score quintile. Note: Line ranges represent 95% confidence intervals.</p> <hd id="AN0178879688-11">Sensitivity Analyses</hd> <p>Due to data limitations, some of the theoretical constructs depicted in Figure 2, such as motivation, personality traits, and social capital, are not directly captured in the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> vectors. Below, I illustrate how the bias factor approach to sensitivity analysis described earlier can be employed to assess the direction and magnitude of potential biases due to such unobserved confounders. In particular, let us consider the total effect of college ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> ), the direct effect of attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> ), and the net effect of completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ) for individuals in the lowest and highest propensity score quintiles. First, to the extent that an unobserved confounder <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> (e.g., a personality trait that predisposes a person to prefer cognitive tasks over noncognitive tasks) affects both college attendance and earnings, the bias for our total effect estimate is given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub><msub><mi>β</mi><mtext>tot</mtext></msub></math> </ephtml> (equation 12). Given the symmetry of the bias formula, let us consider only cases where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi></math> </ephtml> is positively associated with log earnings, i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>tot</mtext></msub><mo>></mo><mn>0</mn></math> </ephtml> , while leaving the sign of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub></math> </ephtml> unconstrained. Columns 3-4 of Table 2 report the bias-adjusted estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> for the lowest- and highest-propensity individuals across a range of potential values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>tot</mtext></msub></math> </ephtml> . Given that an unobserved characteristic that boosts earnings is likely also positively associated with college attendance, we may focus on the lower part of Table 2, where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>tot</mtext></msub></math> </ephtml> are both positive. In this case, although our estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> will be upwardly biased, they are quite robust to unobserved confounding for both groups. For example, even if the unobserved characteristic increases log earnings by 0.3 (given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ) and its prevalence differs by as much as 30 percentage points between high school graduates and college goers (given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> </ephtml> ), the bias-adjusted estimates of the total effect are still sizable — 0.29 and 0.38 for the least and the most advantaged groups, respectively.</p> <p>Second, if an unobserved confounder exists for the effect of BA completion on earnings (e.g., social capital accumulated during college), the biases for our estimates of the direct effect of attendance and the net effect of completion are given by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msub><mi>π</mi><mtext>comp</mtext></msub><msub><mi>α</mi><mtext>comp</mtext></msub><msub><mi>β</mi><mtext>net</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>comp</mtext></msub><msub><mi>β</mi><mtext>net</mtext></msub></math> </ephtml> (equations 13 and 14), respectively. Columns 5-8 of Table 2 report the bias-adjusted estimates of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> across a range of potential values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>comp</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>net</mtext></msub></math> </ephtml> . When assessing <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtext>bias</mtext><mo stretchy="false">[</mo><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub><mo stretchy="false">]</mo></math> </ephtml> for lowest- and highest-propensity individuals, I replace <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>π</mi><mtext>comp</mtext></msub></math> </ephtml> with its DML estimate for the corresponding group. Given the symmetry of these formulas, let us consider only cases where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>net</mtext></msub><mo>></mo><mn>0</mn></math> </ephtml> . Since an unobserved characteristic that boosts earnings is likely positively associated with BA completion, it is reasonable to assume that <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>comp</mtext></msub></math> </ephtml> is also positive. Thus, we may focus on the lower panels of Columns 5-8, which suggest that the direct effect of attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> ) is likely underestimated and the net effect of BA completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ) is likely overestimated. In general, our estimates of the BA completion effect are fairly robust for both the lowest- and highest-propensity groups. The estimated direct effect of attendance, on the other hand, is much more robust for the least advantaged youth than for the most advantaged youth.</p> <p>Graph</p> <p>Table 2. Sensitivity Results for the Total Effect of College ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> ), the Direct Effect of Attendance ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> ), and the Net Effect of BA Completion ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> ) for Individuals in the Lowest and Highest Propensity Score (PS) Quintiles.</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /><col align="center" /></colgroup><thead><tr><th align="left" colspan="2">Sensitivity Parameters</th><th align="left" colspan="2">Total Effect (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math></p>)</th><th align="left" colspan="2">Direct Effect of Attendance (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math></p>)</th><th align="left" colspan="2">Net Effect of Completion (<p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math></p>)</th></tr><tr><th align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">α</mi></math></p></th><th align="left"><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">β</mi></math></p></th><th align="left">1st PS Quintile</th><th align="left">5th PS Quintile</th><th align="left">1st PS Quintile</th><th align="left">5th PS Quintile</th><th align="left">1st PS Quintile</th><th align="left">5th PS Quintile</th></tr></thead><tbody><tr><td>0</td><td>0</td><td>0.38</td><td>0.47</td><td>0.45</td><td>0.10</td><td>0.37</td><td>0.42</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.3</td><td>0.1</td><td>0.41</td><td>0.50</td><td>0.44</td><td>0.08</td><td>0.40</td><td>0.45</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.3</td><td>0.2</td><td>0.44</td><td>0.53</td><td>0.43</td><td>0.05</td><td>0.43</td><td>0.48</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.3</td><td>0.3</td><td>0.47</td><td>0.56</td><td>0.42</td><td>0.03</td><td>0.46</td><td>0.51</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.2</td><td>0.1</td><td>0.40</td><td>0.49</td><td>0.44</td><td>0.09</td><td>0.39</td><td>0.44</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.2</td><td>0.2</td><td>0.42</td><td>0.51</td><td>0.44</td><td>0.07</td><td>0.41</td><td>0.46</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.2</td><td>0.3</td><td>0.44</td><td>0.53</td><td>0.43</td><td>0.05</td><td>0.43</td><td>0.48</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.1</td><td>0.1</td><td>0.39</td><td>0.48</td><td>0.45</td><td>0.10</td><td>0.38</td><td>0.43</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.1</td><td>0.2</td><td>0.40</td><td>0.49</td><td>0.44</td><td>0.09</td><td>0.39</td><td>0.44</td></tr><tr><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo xmlns="">−</mo></math></p>0.1</td><td>0.3</td><td>0.41</td><td>0.50</td><td>0.44</td><td>0.08</td><td>0.40</td><td>0.45</td></tr><tr><td>0.1</td><td>0.1</td><td>0.37</td><td>0.46</td><td>0.45</td><td>0.11</td><td>0.36</td><td>0.41</td></tr><tr><td>0.1</td><td>0.2</td><td>0.36</td><td>0.45</td><td>0.46</td><td>0.12</td><td>0.35</td><td>0.40</td></tr><tr><td>0.1</td><td>0.3</td><td>0.35</td><td>0.44</td><td>0.46</td><td>0.13</td><td>0.34</td><td>0.39</td></tr><tr><td>0.2</td><td>0.1</td><td>0.36</td><td>0.45</td><td>0.46</td><td>0.12</td><td>0.35</td><td>0.40</td></tr><tr><td>0.2</td><td>0.2</td><td>0.34</td><td>0.43</td><td>0.46</td><td>0.14</td><td>0.33</td><td>0.38</td></tr><tr><td>0.2</td><td>0.3</td><td>0.32</td><td>0.41</td><td>0.47</td><td>0.16</td><td>0.31</td><td>0.36</td></tr><tr><td>0.3</td><td>0.1</td><td>0.35</td><td>0.44</td><td>0.46</td><td>0.13</td><td>0.34</td><td>0.39</td></tr><tr><td>0.3</td><td>0.2</td><td>0.32</td><td>0.41</td><td>0.47</td><td>0.16</td><td>0.31</td><td>0.36</td></tr><tr><td>0.3</td><td>0.3</td><td>0.29</td><td>0.38</td><td>0.48</td><td>0.18</td><td>0.28</td><td>0.33</td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note:</emph> The sensitivity parameters <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> refer to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>tot</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>tot</mtext></msub></math> </ephtml> for the total effect of college and to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mtext>comp</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mtext>net</mtext></msub></math> </ephtml> for the direct effect of attendance and the net effect of BA completion.</p> <p>As noted earlier, if the sensitivity parameters are constant across the population, our findings of effect heterogeneity will be unchanged. However, the sensitivity parameters may differ between less and more advantaged individuals. Several processes may be at work. On the one hand, it is possible that low-propensity students who attend and complete college disproportionately possess some unobserved trait, such as motivation, that boosts both educational attainment and earnings, and that high-propensity youth who do not attend or complete college disproportionately face some unobserved barrier to educational attainment that also affects earnings. If so, biases due to the "imbalance" of unobserved confounders between treated and untreated individuals (i.e., the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> </ephtml> parameters in equations 12–14) will be larger for both the lowest- and highest-propensity youth than for their medium-propensity peers. On the other hand, unobserved traits such as motivation might have a greater effect on earnings among low-propensity youth than among high-propensity youth, whose advantaged socioeconomic backgrounds might dilute the influence of other factors. If so, biases due to the "impact" of unobserved confounders between treated and untreated individuals (i.e., the <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> parameters in equations 12–14) will be larger for the low-propensity individuals than for high-propensity individuals. Thus, compared with the first process, the second process is more likely to induce differential selection bias between the lowest- and highest-propensity groups. To be concrete, let us consider the direct effect of attendance, for which our estimated effect heterogeneity will be subject to a differential selection bias of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>π</mi><mtext>comp</mtext><mtext>5th \;quintile</mtext></msubsup><msubsup><mi>α</mi><mtext>comp</mtext><mtext>5th \;quintile</mtext></msubsup><msubsup><mi>β</mi><mtext>net</mtext><mtext>5th \;quintile</mtext></msubsup><mo>−</mo><msubsup><mi>π</mi><mtext>comp</mtext><mtext>1st quintile</mtext></msubsup><msubsup><mi>α</mi><mtext>comp</mtext><mtext>1st quintile</mtext></msubsup><msubsup><mi>β</mi><mtext>net</mtext><mtext>1st quintile</mtext></msubsup></math> </ephtml> . In this particular case, however, the differential selection bias would have to reach 0.35 to explain away the difference between the lowest- and highest-propensity individuals in their estimated <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> (0.45 versus 0.10). Considering the range of plausible values for our sensitivity parameters and the associated biases, it is highly unlikely that unobserved confounding plays a significant role in driving the observed effect heterogeneity in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>att</mtext></msub></math> </ephtml> . By contrast, our estimated differences in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>tot</mtext></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Δ</mi><mtext>comp</mtext></msub></math> </ephtml> between the first and fifth propensity quintiles are much smaller and consequently more sensitive to unobserved confounding.</p> <hd id="AN0178879688-12">Concluding Remarks</hd> <p>Higher education can be a double-edged sword in shaping inequality. It may serve as an equalizer if disadvantaged youth can benefit more from the experience of attending college and from obtaining a college degree than do their more advantaged peers. On the other hand, it reflects and reinforces preexisting inequalities. In the United States, minority and low-income students are much less likely than their white and more affluent peers to attend a four-year college, and, even when they do, they are less likely to graduate with a BA degree by their late twenties. In this paper, I have developed a potential-outcomes approach to conceptualizing, evaluating, and unpacking the causal effects of college on earnings. By decomposing the total effect of attending a four-year college into several direct and indirect components, this approach not only helps unveil the mechanisms through which college attendance boosts earnings, but illuminates and quantifies the equalizing and stratifying roles of college. Moreover, under the assumption of sequential ignorability, I have introduced a robust and efficient method for estimating all quantities of interest, along with a set of bias formulas for assessing the sensitivity of estimates to unobserved confounding.</p> <p>Applying the proposed framework and methodology to data from the NLSY97, I find evidence of both equalizing and stratifying roles of higher education. In particular, the estimated direct effect of college attendance is markedly larger among individuals from the lowest propensity score quintile than among their more advantaged peers. Yet, this equalizing effect is offset by the stratifying effects associated with unequal likelihoods of completing college (given attendance) and unequal covariances between BA completion and its net effect on earnings. The latter component is especially intriguing, as it reflects not an inequality in BA attainment or earnings returns per se, but <emph>an inequality in sorting</emph>: whereas more advantaged college-goers may be well informed about their idiosyncratic payoffs to a BA degree and poised to act on such information, their less advantaged peers may lack such information or the capacity to act on it, leading to a pattern of "negative selection among the least advantaged." As a result of these stratifying forces, the estimated total effect of attending a four-year college for the least advantaged youth is no larger than that for their more advantaged peers.</p> <p>Methodologically, the causal decomposition and the associated methods for estimation and sensitivity analysis constitute a new framework for analyzing the effects of higher education on earnings. Unlike the conventional practice of dichotomizing postsecondary attainment as either "college goers" versus "high school graduates" or "college graduates" versus "non-graduates," the new framework treats BA completion as a mediator that transmits the effect of college attendance on earnings. This approach not only maps more closely onto the sequential process by which people make educational transitions ([<reflink idref="bib23" id="ref115">23</reflink>]), but enables us, for the first time, to isolate the equalizing and stratifying roles of higher education. Moreover, it opens up new possibilities for future research on the nexus between education and earnings inequality. For example, while the present paper has focused on the effects of college attendance and BA completion, the methodological framework can be generalized to incorporate more educational transitions, such as high school attendance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math> </ephtml> high school graduation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math> </ephtml> college attendance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math> </ephtml> college graduation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math> </ephtml> postgraduate attendance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo></math> </ephtml> postgraduate degree, where the effect of each transition may show a distinct pattern of heterogeneity ([<reflink idref="bib34" id="ref116">34</reflink>]). Moreover, even within the journey from college enrollment to BA completion, the same approach can be applied to assess the roles of important milestones, such as persistence through the first year, and the extent to which they differ between more and less advantaged students. Future research can also adapt the proposed effect decomposition to unpack the economic payoff to attending a two-year college, which comprises not only a direct effect of attendance and an indirect effect via potential attainment of an AA degree, but also an indirect effect via potential transfer to a four-year institution and the associated prospect of attaining a BA degree. Given that two-year colleges currently enroll more than a third of all undergraduate students and that nearly half of all students completing a BA degree had some experience within a two-year institution ([<reflink idref="bib21" id="ref117">21</reflink>]), the relationships between two-year college attendance, eventual educational attainment, and earnings inequality constitute an important avenue for future research.</p> <hd id="AN0178879688-13">Supplemental Material</hd> <p>Graph: Supplemental material, sj-pdf-1-smr-10.1177_00491241221113876 for Attendance, Completion, and Heterogeneous Returns to College: A Causal Mediation Approach by Xiang Zhou in Sociological Methods & Research</p> <hd id="AN0178879688-14">Acknowledgements</hd> <p>The author thanks Paul Bauer, Derick Baum, Richard Breen, Aleksei Opacic, Ang Yu, and two anonymous reviewers for helpful comments on previous versions of this paper.</p> <ref id="AN0178879688-15"> <title> References </title> <blist> <bibl id="bib1" idref="ref6" type="bt">1</bibl> <bibtext> Attewell Paul, Lavin David, Domina Thurston, Levey Tania. 2007. Passing the Torch: Does Higher Education for the Disadvantaged Pay Off across the Generations? Russell Sage Foundation.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref26" type="bt">2</bibl> <bibtext> Bowen William G, Chingos Matthew M, McPherson Michael S. 2009. Crossing the Finish Line: Completing College at America's Public Universities. Princeton, NJ: Princeton University Press.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref8" type="bt">3</bibl> <bibtext> Brand Jennie E, Xie Yu. 2010. " Who Benefits Most from College? Evidence for Negative Selection in Heterogeneous Economic Returns to Higher Education." American Sociological Review. 75:273–302.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref50" type="bt">4</bibl> <bibtext> Breen Richard, Choi Seong-soo, Holm Anders. 2015. " Heterogeneous Causal Effects and Sample Selection Bias." Sociological Science. 2:351–69.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref2" type="bt">5</bibl> <bibtext> Card David. 1993. "Using Geographic Variation in College Proximity to Estimate the Return to Schooling".</bibtext> </blist> <blist> <bibl id="bib6" idref="ref13" type="bt">6</bibl> <bibtext> Carneiro Pedro, Heckman James J, Vytlacil Edward J. 2011. " Estimating Marginal Returns to Education." American Economic Review. 101:2754–81.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref15" type="bt">7</bibl> <bibtext> Cheng Siwei, Brand Jennie, Zhou Xiang, Xie Yu, Hout Michael. 2021. " Heterogeneous Returns to College over the Life Course." Science Advances. 7: eabg7641.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref32" type="bt">8</bibl> <bibtext> Chernozhukov Victor, Chetverikov Denis, Demirer Mert, Duflo Esther, Hansen Christian, Newey Whitney, Robins James. 2018. " Double/debiased Machine Learning for Treatment and Structural Parameters." The Econometrics Journal. 21:C1–68.</bibtext> </blist> <blist> <bibl id="bib9" idref="ref89" type="bt">9</bibl> <bibtext> Chetty Raj, Friedman John N, Saez Emmanuel, Turner Nicholas, Yagan Danny. 2020. " Income Segregation and Intergenerational Mobility Across Colleges in the United States." The Quarterly Journal of Economics. 135:1567–633.</bibtext> </blist> <blist> <bibtext> Ciocca Eller Christina, DiPrete A Thomas. 2018. " The Paradox of Persistence: Explaining the Black-White Gap in Bachelor's Degree Completion." American Sociological Review. 83:1171–214.</bibtext> </blist> <blist> <bibtext> Dwyer Rachel E, McCloud Laura, Hodson Randy. 2012. " Debt and Graduation from American Universities." Social Forces. 90:1133–55.</bibtext> </blist> <blist> <bibtext> Fiel Jeremy E. 2020. " Great Equalizer or Great Selector? Reconsidering Education as a Moderator of Intergenerational Transmissions." Sociology of Education. 93:353–71.</bibtext> </blist> <blist> <bibtext> Giani Matt S, Attewell Paul, Walling David. 2020. " The Value of an Incomplete Degree: Heterogeneity in the Labor Market Benefits of College Non-completion." The Journal of Higher Education. 91:514–39.</bibtext> </blist> <blist> <bibtext> Hanson Sandra L, Zogby John. 2010. " The Polls-Trends: Attitudes about the American Dream." Public Opinion Quarterly. 74:570–84.</bibtext> </blist> <blist> <bibtext> Heckman James J, Humphries John Eric, Veramendi Gregory. 2018. " Returns to Education: The Causal Effects of Education on Earnings, Health, and Smoking." Journal of Political Economy. 126:S197–S246.</bibtext> </blist> <blist> <bibtext> Hout Michael. 2012. " Social and Economic Returns to College Education in the United States." Annual Review of Sociology. 38:379–400.</bibtext> </blist> <blist> <bibtext> Imai Kosuke, Keele Luke, Yamamoto Teppei, et al. 2010. " Identification, Inference and Sensitivity Analysis for Causal Mediation Effects." Statistical Science. 25:51–71.</bibtext> </blist> <blist> <bibtext> Joffe Marshall M, Greene Tom. 2009. " Related Causal Frameworks for Surrogate Outcomes." Biometrics. 65:530–8.</bibtext> </blist> <blist> <bibtext> Karlson Kristian Bernt. 2019. " College as Equalizer? Testing the Selectivity Hypothesis." Social Science Research. 80:216–29.</bibtext> </blist> <blist> <bibtext> Kennedy Edward H. 2016. "Semiparametric Theory and Empirical Processes in Causal Inference." In Statistical Causal Inferences and Their Applications in Public Health Research, pp. 141–167. Springer.</bibtext> </blist> <blist> <bibtext> Ma Jennifer, Baum Sandy. 2016. " Trends in Community Colleges: Enrollment, Prices, Student Debt, and Completion." College Board Research Brief. 4:1–23.</bibtext> </blist> <blist> <bibtext> Mann Horace. 1957. " Twelfth Annual Report." In The Republic and the School: Horace Mann on the Education of Free Men, ed. Lawrence A. Cremin, pp. 79–112. New York: Teachers College Press.</bibtext> </blist> <blist> <bibtext> Mare Robert D. 1980. " Social Background and School Continuation Decisions." Journal of the American Statistical Association. 75:295–305.</bibtext> </blist> <blist> <bibtext> Maurin Eric, McNally Sandra. 2008. " Vive la Révolution! Long-Term Educational Returns of 1968 to the Angry Students." Journal of Labor Economics. 26:1–33.</bibtext> </blist> <blist> <bibtext> Minicozzi Alexandra. 2005. " The Short Term Effect of Educational Debt on Job Decisions." Economics of Education Review. 24:417–30.</bibtext> </blist> <blist> <bibtext> Pearl Judea. 2001. "Direct and Indirect Effects." In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, pp. 411–420. Morgan Kaufmann Publishers Inc.</bibtext> </blist> <blist> <bibtext> Polley E. C., van der Laan M SuperLearner. 2017. "Super Learner Prediction: R package, version 2.0-21".</bibtext> </blist> <blist> <bibtext> Robins James M. 1986. " A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period-Application to Control of the Healthy Worker Survivor Effect." Mathematical Modelling. 7:1393–512.</bibtext> </blist> <blist> <bibtext> Robins James M. 1997. "Causal Inference from Complex Longitudinal Data." In Latent Variable Modeling and Applications to Causality, pp. 69–117. Springer.</bibtext> </blist> <blist> <bibtext> Robins James M. 2003. "Semantics of Causal DAG Models and the Identification of Direct and Indirect effects." Highly Structured Stochastic Systems pp. 70–81.</bibtext> </blist> <blist> <bibtext> Rubin Donald B. 1974. " Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies." Journal of Educational Psychology. 66:688–701.</bibtext> </blist> <blist> <bibtext> Rubin Donald B. 1987. Multiple Imputation for Nonresponse in Surveys. Hoboken, NJ: John Wiley & Sons.</bibtext> </blist> <blist> <bibtext> Semenova Vira, Chernozhukov Victor. 2021. " Debiased Machine Learning of Conditional Average Treatment Effects and other Causal Functions." The Econometrics Journal. 24:264–89.</bibtext> </blist> <blist> <bibtext> Torche Florencia. 2011. " Is a College Degree Still the Great Equalizer? Intergenerational Mobility Across Levels of Schooling in the United States." American Journal of Sociology. 117:763–807.</bibtext> </blist> <blist> <bibtext> van der Laan Mark J, Polley Eric C, Hubbard Alan E. 2007. " Super Learner." Statistical Applications in Genetics and Molecular Biology. 6: Article 25.</bibtext> </blist> <blist> <bibtext> VanderWeele Tyler J. 2009. " Marginal Structural Models for the Estimation of Direct and Indirect Effects." Epidemiology (Cambridge, Mass.). 20:18–26.</bibtext> </blist> <blist> <bibtext> VanderWeele Tyler J. 2010. " Bias Formulas for Sensitivity Analysis for Direct and Indirect Effects." Epidemiology (Cambridge, Mass.). 21:540.</bibtext> </blist> <blist> <bibtext> VanderWeele Tyler J, Arah Onyebuchi A. 2011. " Bias Formulas for Sensitivity Analysis of Unmeasured Confounding for General Outcomes, Treatments, and Confounders." Epidemiology (Cambridge, Mass.). 22:42–52.</bibtext> </blist> <blist> <bibtext> VanderWeele Tyler J, Vansteelandt Stijn. 2009. " Conceptual Issues Concerning Mediation, Interventions and Composition." Statistics and its Interface. 2:457–68.</bibtext> </blist> <blist> <bibtext> Vansteelandt Stijn. 2009. " Estimating Direct Effects in Cohort and Case–control Studies." Epidemiology (Cambridge, Mass.). 20:851–60.</bibtext> </blist> <blist> <bibtext> Willis Robert J., Rosen Sherwin. 1979. " Education and Self-selection." Journal of Political Economy. 87:S7–36.</bibtext> </blist> <blist> <bibtext> Wodtke Geoffrey T, Zhou Xiang. 2020. " Effect Decomposition in the Presence of Treatment-induced Confounding: A Regression-with-residuals Approach." Epidemiology (Cambridge, Mass.). 31:369–75.</bibtext> </blist> <blist> <bibtext> Xie Yu, Brand Jennie, Jann Ben. 2012. " Estimating Heterogeneous Treatment Effects with Observational Data." Sociological Methodology. 42:314–47.</bibtext> </blist> <blist> <bibtext> Young Cristobal. 2009. " Model Uncertainty in Sociological Research: An Application to Religion and Economic Growth." American Sociological Review. 74:380–97.</bibtext> </blist> <blist> <bibtext> Zheng Cheng, Zhou Xiao-Hua. 2015. " Causal Mediation Analysis in the Multilevel Intervention and Multicomponent Mediator Case." Journal of the Royal Statistical Society: Series B: Statistical Methodology. 77: 581–615.</bibtext> </blist> <blist> <bibtext> Zhou Xiang. 2019. " Equalization or Selection? Reassessing the "Meritocratic Power" of a College Degree in Intergenerational Income Mobility." American Sociological Review. 84:459–85.</bibtext> </blist> <blist> <bibtext> Zhou Xiang. 2020. "Some Doubly and Multiply Robust Estimators of Controlled Direct Effects". arXiv preprint arXiv:2011.09569.</bibtext> </blist> <blist> <bibtext> Zhou Xiang, Wodtke Geoffrey T. 2019. " A Regression-with-residuals Method for Estimating Controlled Direct Effects." Political Analysis. 27:360–9.</bibtext> </blist> <blist> <bibtext> Zhou Xiang, Wodtke Geoffrey T. 2020. " Residual Balancing: A Method of Constructing Weights for Marginal Structural Models." Political Analysis. 28:487–506.</bibtext> </blist> <blist> <bibtext> Zhou Xiang, Xie Yu. 2016. " Propensity Score–based Methods Versus MTE-based Methods in Causal Inference: Identification, Estimation, and Application." Sociological Methods & Research. 45:3–40.</bibtext> </blist> <blist> <bibtext> Zhou Xiang, Xie Yu. 2020. " Heterogeneous Treatment Effects in the Presence of Self-selection: a Propensity Score Perspective." Sociological Methodology. 50:350–85.</bibtext> </blist> <blist> <bibtext> Zimmerman Seth D. 2014. " The Returns to College Admission for Academically Marginal Students." Journal of Labor Economics. 32:711–54.</bibtext> </blist> </ref> <ref id="AN0178879688-16"> <title> Footnotes </title> <blist> <bibtext> Replication materials are available in Open Science Framework: https://osf.io/psr3j/.</bibtext> </blist> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> Xiang Zhou https://orcid.org/0000-0001-8634-7360</bibtext> </blist> <blist> <bibtext> Supplemental material for this article is available online.</bibtext> </blist> <blist> <bibtext> https://joebiden.com/beyondhs/</bibtext> </blist> <blist> <bibtext> A nuisance function is a function that is not of our primary interest but necessary for constructing estimators of our target quantities (i.e., the components in equations (2) and (3)).</bibtext> </blist> <blist> <bibtext> Previous studies on the economic returns to college have often used data from the NLSY79 (e.g., [3]; [6]). I use the NLSY97 to illustrate the proposed methodology for two reasons. First, compared with the NLSY79, the NLSY97 traces the educational and labor market outcomes of a much younger cohort, making the results from my analyses more relevant to the experience of current and future cohorts of American youth. Second, compared with the NLSY79, the NLSY97 provides a richer set of postsecondary characteristics ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> </ephtml> ) that we can adjust for (e.g., college GPA) when estimating the causal effect of a BA degree, making the sequential ignorability assumption more plausible.</bibtext> </blist> <blist> <bibtext> In my analyses, the estimated propensity scores are treated as given. Thus, standard errors reported for the propensity-score-specific estimates of total, direct, and indirect effects should be viewed as approximate standard errors because they do not account for estimation uncertainty for the propensity score.</bibtext> </blist> <blist> <bibtext> A super learner is a weighted average of different machine learning methods designed to minimize prediction error. The algorithm is implemented in the R package SuperLearner ([27]).</bibtext> </blist> </ref> <aug> <p>By Xiang Zhou</p> <p>Reported by Author</p> </aug> <nolink nlid="nl1" bibid="bib22" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib14" firstref="ref3"></nolink> <nolink nlid="nl3" bibid="bib16" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib24" firstref="ref7"></nolink> <nolink nlid="nl5" bibid="bib52" firstref="ref9"></nolink> <nolink nlid="nl6" bibid="bib13" firstref="ref10"></nolink> <nolink nlid="nl7" bibid="bib41" firstref="ref12"></nolink> <nolink nlid="nl8" bibid="bib51" firstref="ref14"></nolink> <nolink nlid="nl9" bibid="bib50" firstref="ref16"></nolink> <nolink nlid="nl10" bibid="bib46" firstref="ref17"></nolink> <nolink nlid="nl11" bibid="bib12" firstref="ref18"></nolink> <nolink nlid="nl12" bibid="bib19" firstref="ref19"></nolink> <nolink nlid="nl13" bibid="bib15" firstref="ref24"></nolink> <nolink nlid="nl14" bibid="bib10" firstref="ref27"></nolink> <nolink nlid="nl15" bibid="bib29" firstref="ref30"></nolink> <nolink nlid="nl16" bibid="bib44" firstref="ref31"></nolink> <nolink nlid="nl17" bibid="bib33" firstref="ref33"></nolink> <nolink nlid="nl18" bibid="bib47" firstref="ref34"></nolink> <nolink nlid="nl19" bibid="bib31" firstref="ref36"></nolink> <nolink nlid="nl20" bibid="bib28" firstref="ref48"></nolink> <nolink nlid="nl21" bibid="bib39" firstref="ref60"></nolink> <nolink nlid="nl22" bibid="bib17" firstref="ref61"></nolink> <nolink nlid="nl23" bibid="bib26" firstref="ref62"></nolink> <nolink nlid="nl24" bibid="bib30" firstref="ref63"></nolink> <nolink nlid="nl25" bibid="bib45" firstref="ref64"></nolink> <nolink nlid="nl26" bibid="bib40" firstref="ref71"></nolink> <nolink nlid="nl27" bibid="bib18" firstref="ref72"></nolink> <nolink nlid="nl28" bibid="bib48" firstref="ref73"></nolink> <nolink nlid="nl29" bibid="bib42" firstref="ref74"></nolink> <nolink nlid="nl30" bibid="bib36" firstref="ref75"></nolink> <nolink nlid="nl31" bibid="bib49" firstref="ref76"></nolink> <nolink nlid="nl32" bibid="bib20" firstref="ref90"></nolink> <nolink nlid="nl33" bibid="bib11" firstref="ref91"></nolink> <nolink nlid="nl34" bibid="bib38" firstref="ref96"></nolink> <nolink nlid="nl35" bibid="bib37" firstref="ref97"></nolink> <nolink nlid="nl36" bibid="bib25" firstref="ref102"></nolink> <nolink nlid="nl37" bibid="bib32" firstref="ref103"></nolink> <nolink nlid="nl38" bibid="bib43" firstref="ref108"></nolink> <nolink nlid="nl39" bibid="bib35" firstref="ref112"></nolink> <nolink nlid="nl40" bibid="bib23" firstref="ref115"></nolink> <nolink nlid="nl41" bibid="bib34" firstref="ref116"></nolink> <nolink nlid="nl42" bibid="bib21" firstref="ref117"></nolink>
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  Data: Attendance, Completion, and Heterogeneous Returns to College: A Causal Mediation Approach
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  Data: English
– Name: Author
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  Data: <searchLink fieldCode="AR" term="%22Xiang+Zhou%22">Xiang Zhou</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-8634-7360">0000-0001-8634-7360</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2024 53(3):1136-1166.
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  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
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  Data: Y
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  Data: 31
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  Label: Publication Date
  Group: Date
  Data: 2024
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22High+Schools%22">High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink>
– Name: Subject
  Label: Descriptors
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  Data: <searchLink fieldCode="DE" term="%22Outcomes+of+Education%22">Outcomes of Education</searchLink><br /><searchLink fieldCode="DE" term="%22Educational+Benefits%22">Educational Benefits</searchLink><br /><searchLink fieldCode="DE" term="%22Educational+Status+Comparison%22">Educational Status Comparison</searchLink><br /><searchLink fieldCode="DE" term="%22Disadvantaged+Youth%22">Disadvantaged Youth</searchLink><br /><searchLink fieldCode="DE" term="%22High+School+Graduates%22">High School Graduates</searchLink><br /><searchLink fieldCode="DE" term="%22College+Attendance%22">College Attendance</searchLink><br /><searchLink fieldCode="DE" term="%22College+Graduates%22">College Graduates</searchLink><br /><searchLink fieldCode="DE" term="%22Reentry+Students%22">Reentry Students</searchLink><br /><searchLink fieldCode="DE" term="%22Social+Science+Research%22">Social Science Research</searchLink><br /><searchLink fieldCode="DE" term="%22Cost+Effectiveness%22">Cost Effectiveness</searchLink><br /><searchLink fieldCode="DE" term="%22Advantaged%22">Advantaged</searchLink><br /><searchLink fieldCode="DE" term="%22Comparative+Testing%22">Comparative Testing</searchLink>
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  Label: DOI
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  Data: 10.1177/00491241221113876
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  Data: 0049-1241<br />1552-8294
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  Data: A growing body of social science research investigates whether the economic payoff to a college education is heterogeneous -- in particular, whether disadvantaged youth can benefit more from attending and completing college relative to their more advantaged peers. Scholars, however, have employed different analytical strategies and reported mixed findings. To shed light on this literature, I propose a causal mediation approach to conceptualizing, evaluating, and unpacking the causal effects of college on earnings. By decomposing the total effect of attending a four-year college into several direct and indirect components, this approach not only clarifies the mechanisms through which college attendance boosts earnings, but illuminates the ways in which the postsecondary system may be "both an equalizer and a stratifier." The total effect of college attendance, its direct and indirect components, and their heterogeneity across different subpopulations are all identified under the assumption of sequential ignorability. I introduce a debiased machine learning (DML) method for estimating all quantities of interest, along with a set of bias formulas for sensitivity analysis. I illustrate the proposed framework and methodology using data from the National Longitudinal Survey of Youth, 1997 cohort.
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  Data: EJ1434929
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      – SubjectFull: Educational Benefits
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      – SubjectFull: Educational Status Comparison
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      – SubjectFull: Disadvantaged Youth
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      – SubjectFull: Comparative Testing
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      – TitleFull: Attendance, Completion, and Heterogeneous Returns to College: A Causal Mediation Approach
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