The missing "one-offs": the hidden supply of high-achieving, low-income students.

Hoxby, Caroline ; Avery, Christopher

ABSTRACT We show that the vast majority of low-income high achievers do not apply to any selective college. This is despite the fact that selective institutions typically cost them less, owing to generous financial aid, than the two-year and nonselective four-year institutions to which they actually apply. Moreover, low-income high achievers have no reason to believe they will fail at selective institutions since those who do apply are admitted and graduate at high rates. We demonstrate that low-income high achievers' application behavior differs greatly from that of their high-income counterparts with similar achievement. The latter generally follow experts' advice to apply to several "peer," a few "reach," and a couple of "safety" colleges. We separate low-income high achievers into those whose application behavior is similar to that of their high-income counterparts ("achievement-typical") and those who apply to no selective institutions ("income-typical"). We show that income-typical students are not more disadvantaged than the achievement-typical students. However, in contrast to the achievement-typical students, income-typical students come from districts too small to support selective public high schools, are not in a critical mass of fellow high achievers, and are unlikely to encounter a teacher who attended a selective college. We demonstrate that widely used policies--college admissions recruiting, campus visits, college mentoring programs--are likely to be ineffective with income-typical students. We suggest that effective policies must depend less on geographic concentration of high achievers.

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In this study we show that a large number--probably the vast majority--of very high-achieving students from low-income families do not apply to a selective college or university. (1) This is in contrast to students with the same test scores and grades who come from high-income backgrounds: they are overwhelmingly likely to apply to a college whose median student has achievement much like their own. This gap is puzzling because we find that the subset of high-achieving, low-income students who do apply to selective institutions are just as likely to enroll and progress toward a degree at the same pace as high-income students with equivalent test scores and grades. Added to the puzzle is the fact that very selective institutions not only offer students much richer instructional, extracurricular, and other resources, but also offer high-achieving, low-income students so much financial aid that these students would often pay less to attend a selective institution than the far less selective or nonselective postsecondary institutions that most of them do attend.

We attempt to unravel this puzzle by characterizing low-income, very high achieving students in the U.S. using a rich array of data, including individual-level data on every student who takes one of the two college assessments, the ACT and the SAT. We divide the low-income, very high-achieving students into those who apply similarly to their high-income counterparts ("achievement-typical" behavior) and those who apply in a very dissimilar manner ("income-typical" behavior). We do this because we are interested in why some low-income high achievers appear to base their college-going on their achievement, whereas others base it on their income. We find that income-typical students are fairly isolated from other high achievers, both in terms of geography and in terms of the high schools they attend. In fact, their lack of concentration is such that many traditional strategies for informing high-achieving students about college--for instance, college admissions staff visiting high schools, or after-school programs that provide mentoring--would be prohibitively expensive. We also show that income-typical students have a negligible probability of meeting a teacher, high school counselor, or schoolmate from an older cohort who attended a selective college.

In contrast, we show that achievement-typical students are highly concentrated. Some of these low-income students attend a small number of "feeder" high schools that contain a critical mass of high achievers. Some feeder schools admit students on the basis of an exam or previous grades; others are magnet schools; still others contain a subpopulation of low-income students in a student body that is generally affluent. Since these high schools are nearly all located in the largest school districts of very large metropolitan areas (not even in medium-size metropolitan areas), their students are far from representative of high-achieving, low-income students in general. Moreover, we show evidence that suggests that these schools may be "tapped out"--that their students are already so intensively recruited by selective colleges that further recruitment may merely shift students among similar, selective colleges, and not cause students to change their college-going behavior in more fundamental ways.

The evidence that we present is descriptive, not causal. This is an important distinction. For instance, we cannot assert that a high-achieving, low-income student would act like an achievement-typical student rather than an income-typical student if he or she were moved to a large metropolitan area with a high school that practices selective admission. Moreover, we do not assert that income-typical students would have higher welfare if they applied to college in the same way that achievement-typical and high-income high achievers do. We leave such causal tests for related studies in which we conduct randomized, controlled interventions. Nevertheless, our descriptive evidence makes three important contributions. First, it documents that the number of low-income high achievers is much greater than college admissions staff generally believe. Since admissions staff see only the students who apply, they very reasonably underestimate the number who exist. Second, our evidence suggests hypotheses for why so many low-income high achievers apply to colleges in a manner that may not be in their best interest, and that is certainly different from what similarly high-achieving, high-income students do. Most of our hypotheses are related to the idea that income-typical students--despite being intelligent, literate, and on colleges' search lists (that is, the lists to which selective colleges mail brochures)--lack information or encouragement that achievement-typical students have because they are part of local, critical masses of high achievers. Third, our descriptive evidence allows us to explain why some traditional interventions are unlikely to change the situation and allows us to identify other interventions that could plausibly do so.

Our previous work (Avery, Hoxby, and others 2006) was perhaps the first to identify the phenomena described in this paper, but there is now a small literature on the topic of "undermatching." We especially note the work of William Bowen, Matthew Chingos, and Michael McPherson (2009), Eleanor Dillon and Jeffrey Smith (2012), and Amanda Pallais (2009). Relative to those studies, our study's strengths are its comprehensiveness (we analyze the entire population of high-achieving students, not a sample); our complete characterization of each U.S. high school, including its history of sending its students to college; our ability to map students to their exact high schools and neighborhoods (this allows us to investigate exactly what they experience); and our use of accurate administrative data to identify students' aptitude, application behavior, college enrollment, and on-time degree completion. The sheer comprehensiveness and accuracy of our data are what allow us to test key hypotheses about why some high-achieving, low-income students are income-typical and others are achievement-typical. Our data also allow us to assess which interventions might plausibly (and cost-effectively) alter such behavior.

The paper is organized as follows. In the next section we present some background on college policies directed toward low-income high achievers. In section II we describe our data sources. In section III we present a descriptive portrait of very high-achieving U.S. students--their family incomes, parents' education, race, ethnicity, and geography. In section IV we show that high-achieving students' college application behavior differs greatly by family income. We also show that, conditional on applying to a college, students' enrollment, college grades, and degree receipt do not differ by family income (among students with similar incoming qualifications). In section V we divide low-income high achievers into achievement-typical and income-typical groups. We then compare factors that might affect the college application behavior of these groups. In section VI we consider several interventions commonly directed toward low-income high achievers, and we demonstrate that they are likely to be cost-prohibitive for income-typical students. To drive the point home, we contrast colleges' difficulty in identifying low-income high achievers with their ease in identifying top athletes. In section VII we conclude by discussing which hypotheses we have eliminated and which still need testing, and we speculate on the sort of interventions that could plausibly test whether income-typical students' welfare would be greater if they were better informed.

I. Background on College Policies Directed toward Low-Income High Achievers

Many students from low-income families have poor college outcomes: they do not attend college, they drop out before attaining a degree, they earn so few credits each term that they cannot graduate even in 1.5 times the "correct" time to degree, or they attend institutions with such poor resources that even when they do graduate, they earn much less than the median college graduate. These poor college outcomes are often attributed to low-income students being less academically prepared for college and less able to pay for college. These are certainly valid concerns. As we show later, high-income students (those from families in the top income quartile) are in fact much more likely to be high achievers at the end of high school than are low-income students. Nevertheless, some low-income students are very high achievers: at the end of high school, they have grades and college assessment scores that put them in the top 10 percent of students who take one of the ACT or SAT college assessment exams or, equivalently, the top 4 percent of all U.S. secondary school students.

High-achieving, low-income students are considered very desirable by selective colleges, private and public, which are eager to make their student bodies socioeconomically diverse without enrolling students who are unprepared for their demanding curricula. The ultimate evidence of colleges' eagerness is their financial aid policies, which, as we shall show, are very generous toward such students. However, we have also observed this eagerness personally among hundreds of college leaders and their admissions staff. Many spend considerable amounts on recruiting the low-income students who do apply and on (not always successful) programs designed to increase their numbers of low-income applicants. There are many reasons for selective institutions to prefer socioeconomic diversity. These include, to name just a few, a deep respect for merit regardless of need; the fact that students whose lives were transformed by highly aided college education tend to be the most generous donors if they do become rich; a belief that a diverse student body makes instruction and research more productive; and pressure from society.

In recent years, selective schools' aid for low-income high achievers has become so generous that such students' out-of-pocket costs of attendance are zero at the nation's most competitive schools, and small at other very selective schools. Figure 1 shows the distribution of annual income in 2008 for families with a child in the 12th grade--a good indicator for a family having a child of college-going age in the next year. The 20th percentile of this distribution was $35,185. Table 1 shows the out-of-pocket costs (including loans) such a student would have experienced in the 2009-10 school year at a variety of selective and nonselective institutions. The table is organized based on institutions' selectivity as classified by Barron's Profiles of American Colleges: most competitive, very competitive, competitive 4-year institutions, nonselective 4-year institutions, and (nonselective by definition) community colleges and other 2-year institutions. Table 1 also shows the colleges' comprehensive cost for a student who needs no financial aid (the "sticker price") and their instructional expenditure per student. What the table reveals is that a low-income student who can gain admission to one of the most selective colleges in the U.S. can expect to pay less to attend a very selective college with maximum instructional expenditure than to attend a nonselective 4-year college or 2-year institution. In short, the table demonstrates the strong financial commitment that selective colleges have made toward becoming affordable to low-income students. (2)

[FIGURE 1 OMITTED]

In related work (Avery, Hoxby, and others 2006), we analyze Harvard University's introduction of zero costs for students with annual family incomes of $40,000 and below starting in 2005. (Harvard is a relevant option for the students we analyze in this paper.) Harvard's policy was

quickly imitated or outdone by the institutions with which it most competes: Yale, Princeton, Stanford, and some others. All such institutions subsequently raised the bar on what they considered to be a low enough income to merit zero costs, to the point where even students from families with income above the U.S. median can often attend such institutions for free. Although less well endowed institutions followed suit to a lesser extent (usually by setting the bar for zero costs at a lower family income than the aforementioned institutions did), the result was very low costs for low-income students at selective institutions, as table 1 shows.

In our other work we show that Harvard's policy change had very little effect--at least, very little immediate effect--on the income composition of its entering class. We estimate that it increased the number of low-income students by approximately 20, in a class of more than 1,600 (Avery, Hoxby, and others 2006, table 1, top row). Interestingly, this very modest effect was not a surprise to many college admissions staff. They explained that there was a small pool of low-income high achievers who were already "fully tapped," so that additional aid and recruiting could do little except shift them among institutions that were fairly similar. Put another way, they believed that the overall pool of high-achieving, low-income students was inelastic. Many felt that they had already tried every means open to them for recruiting low-income students: guaranteeing need-blind admission, (3) disproportionately visiting high schools with large numbers of free-lunch-eligible students, (4) sending special letters to high achievers who live in high-poverty ZIP codes, (5) maintaining strong relationships with guidance counselors who reliably direct low-income applicants to them, (6) coordinating with or even running college mentoring programs for low-income students, (7) paying a third-party organization for a guaranteed minimum number of low-income enrollees, (8) sponsoring campus visits for students from local high schools known to serve low-income families, and personally contacting students whose essays suggest that they might be disadvantaged. Although the admissions staff believed that they might succeed in diversifying their student bodies by poaching from other selective schools or lowering their admissions standards for low-income students, they did not expect additional aid together with more of the same recruiting methods to affect matters much. (9) (Note that the methods we use in this paper to identify low-income students are not available to college admissions staff.) (10)

In this paper, we show that--viewed one way--the admissions staff are correct. The pool of high-achieving, low-income students who apply to selective colleges is small: for every high-achieving, low-income student who applies, there are from 8 to 15 high-achieving, high-income students who apply. Viewed another way, however, the admissions staff are too pessimistic: the vast majority of high-achieving, low-income students do not apply to any selective college. There are, in fact, only about 2 high-achieving, high-income students for every high-achieving, low-income student in the population. The problem is that most high-achieving, low-income students do not apply to any selective college, so they are invisible to admissions staff. Moreover, we will show that they are unlikely to come to the attention of admissions staff through traditional recruiting channels.

II. Data Sources and Identifying High-Achieving, Low-Income Students

We attempt to identify the vast majority of U.S. students who are very high achieving. Specifically, we are interested in students who are well prepared for college and who would be very likely to be admitted to the majority of selective institutions (if they applied). Thus, as mentioned above, we choose students whose college assessment scores place them in the top 10 percent of test takers based on either the SAT I (combined math and verbal) or the ACT (comprehensive). (11) Since only about 40 percent of U.S. secondary school students take a college assessment, these students are in the top 4 percent of U.S. students. We include in our target group only those students who self-report a grade point average of A- or higher in high school. In practice, this criterion for inclusion hardly matters once we condition on having test scores in the top 10 percent. (12)

Our key data come from the College Board and ACT, both of which supplied us with student-level data on everyone in the high school graduating class of 2008 who took either the ACT or the SAT I. (13) Apart from students' test score history, these data sets contain students' high school identifiers, self-reported grades, race and ethnicity, and sex. Validation exercises have shown that students self-report their grades quite accurately to the College Board and ACT (with just a hint of upward bias), probably because students perceive the organizations as playing a semiofficial role in the college application process (Freeberg 1988). The data also contain answers to numerous questions about students' high school activities and their plans for college.

Importantly, the College Board and ACT data contain a full list of the colleges to which students have sent their test scores. Except in rare circumstances, a student cannot complete an application to a selective college without having the College Board or ACT send his or her verified test scores to the college. Thus, score sending is

necessary but not sufficient for a completed application. Put another way, score sending may exaggerate but cannot understate the set of selective colleges to which a student applies. Past studies have found that score sending corresponds closely with actual applications to selective colleges (Card and Krueger 2005, Avery and Hoxby 2004). Students who are admitted under an Early Decision or Early Action program often do not apply to colleges other than the one that admitted them early. However, such students typically send scores to all of the schools to which they would have applied had the Early school not admitted them (Avery, Glickman, Hoxby, Metrick 2013). Thus, it is somewhat better for our purposes to observe score sending than actual applications: score sending more accurately reveals the set of selective colleges to which the student would have applied. Note, however, that as most 2-year colleges and some nonselective colleges do not require verified ACT or SAT I scores, we do not assume that a student who sends no scores is applying to no postsecondary institutions. Rather, that student is applying to no selective institution.

For some of our analyses, we need to know where students actually enrolled and whether they are on track to attain a degree on time (June 2012 for baccalaureate degrees for the class of 2008). We therefore match students to their records at the National Student Clearinghouse, which tracks enrollment and degree receipt. We match all low-income high achievers and a 25 percent random sample of high-income high achievers. We do not match all students for reasons of cost.

The addresses in the data are geocoded for us at the Census block level, the smallest level of Census geography (22 households on average). We match each student to a rich description of his or her neighborhood. The neighborhood's racial composition, sex composition, age composition, and population density are matched at the block level. Numerous sociodemographic variables are matched at the block group level (556 households on average): several moments of the family income distribution, adults' educational attainment, employment, the occupational distribution, several moments of the house value distribution, and so on. We also merge in income data from the Internal Revenue Service (IRS) at the ZIP code level.

In addition to these data on the graduating class of 2008, we have parallel data for previous cohorts of students who took an ACT or a College Board test. (We have one previous cohort for the ACT and more than 10 previous cohorts for the College Board tests.) We use the previous cohort data in a few ways that will become clear below.

We create a profile of every high school, public and private, in the U.S., using administrative data on enrollment, graduates, basic school characteristics, and sociodemographics. The sources are the Common Core of Data (United States Department of Education 200%) and the Private School Survey (United States Department of Education 2009b). By summarizing our previous cohort data at the high school level, we also create profiles for each school of their students' usual test scores, application behavior, and college plans. For instance, we know how many students from the high school typically apply to each selective college or to any given group of selective colleges. Finally, we add high schools' test scores, at the subgroup level, for each state's statewide test mandated by the No Child Left Behind Act of 2001. These scores are all standardized to have a zero mean and a standard deviation of 1.

We estimate a student's family income rather than rely on the student's self-reported family income. We do this for a few reasons. First, both the College Board's and the ACT's family income questions provide a series of somewhat wide income "bins" as potential answers. Second, although the College Board's questionnaire appears to elicit unbiased self-reports of family income, students make substantial unsystematic mistakes when their data are compared to their verified data used in financial aid calculations (the CSS Profile data). Third, about 62 percent of students simply do not answer the College Board's family income question. Fourth, although the ACT's questionnaire elicits a high response rate, its question refers to the fact that colleges offer more generous financial aid to students with lower family incomes. This framing apparently induces students to underestimate their family incomes: we find that students often report family incomes that are lower than the 10th percentile of family income in their Census block group.

We predict students' family income using all the data we have on previous cohorts of College Board students, matched to their CSS Profile records (data used by financial aid officers to compute grants and loans). That is, using previous cohorts, we regress accurate administrative data on family income using all of our Census variables, the IRS income variables, the high school profile variables, and the student's own race and ethnicity. In practice, the income variables from the Census have the most explanatory power. Our goal is simply to maximize explanatory power, and many of the variables we include are somewhat multicollinear. We choose predicted income cutoffs to minimize Type I error (false positives) in declaring a student to be low-income. Specifically, we choose cutoffs such that, in previous cohorts, only 8 percent of students who are not actually in the bottom quartile of the income distribution are predicted to be low-income. We recognize that by minimizing Type I error, we expand Type II error, but it is less worrisome for our exercise if we mistakenly classify a low-income student as middle-income than if we do the reverse. This is because we wish to characterize the college-going behavior of students who are low-income. Since we also find that there are more high-achieving, low-income students than college admissions staff typically believe, we make decisions that will understate rather than overstate the low-income, high-achieving population.

More generally, it is not important for our exercise that our measure of income be precise. What matters for our exercise is that the students we analyze are, in fact, capable of gaining admission at selective colleges--at which time the college's financial aid policies will be implemented. We are confident that the students we analyze are capable of being admitted because we are using the same score data and similar grade data to what the colleges themselves use. Also, we show later that we can accurately predict the colleges at which students enroll, conditioning on the colleges to which they applied. We would not be able to make such accurate predictions if we lacked important achievement and other data that colleges use in their admissions processes.

Hereafter, we describe as low-income any student whose estimated family income is at or below the cutoff for the bottom quartile of the 2008 distribution of incomes among families who had a child in his or her senior year of high school: $41,472. (14) We describe as high-income any student whose estimated family income is at or above the cutoff for the top quartile of the same distribution: $120,776. See figure 1 for other percentiles.

III. A Portrait of High-Achieving Students in the O.S.

Who and where are the high-achieving students in the United States? In this section, we briefly characterize them, leaving more detailed analysis of the low-income, high-achieving group for later.

Figure 2 shows that 34 percent of high achievers have estimated family income in the top quartile and 27 percent have estimated family income in the third quartile. That is, high-income families are overrepresented in the high-achieving population. However, 22 percent and 17 percent of high achievers have estimated family incomes in, respectively, the second and bottom quartiles. We estimate that there are at least 25,000 and probably about 35,000 low-income high achievers in each cohort in the United States. (15)

Table 2 shows that among high achievers, those who are from higher-income families do have slightly higher college assessment scores, but the difference is small. The average low-income high achiever scores at the 94.1th percentile. The average high-income high achiever scores at the 95.7th percentile.

Data on the parental education of high achievers are unfortunately very incomplete, because ACT takers are not asked to report their parents' education, and 52 percent of SAT I takers fail to answer the question about their parents' education. Moreover, SAT I takers are apparently less likely to report their parents' education when it is low. We base this assessment on the observation that parents' education is more likely to be missing for students who live in Census block groups with low adult education. For what they are worth, however, the data on the parents' education are shown in figure 3. (16) More precisely, we show the greater of the father's reported educational attainment and the mother's reported educational attainment. Of students who report their parents' education, 50.7 percent say that at least one parent has a graduate degree, 27.9 percent say that at least one parent has a baccalaureate degree, and another 6 percent cite "some graduate school" (but no degree); 11.6 percent claim that at least one parent has an associate's degree or "some college or trade school" (but no degree), and only 3.8 percent report neither parent having more than a high school diploma. Perhaps the most interesting thing about the parents' education data is that they seem to indicate that high achievers are reluctant to report that they have poorly educated parents. This is in contrast to the family income data from the same College Board questionnaire. Many students did not answer the income question, but those who did answered it in an unbiased (albeit fairly inaccurate) way.

Figure 4 displays information on high achievers' race and ethnicity, which 98 percent of students voluntarily report on the ACT or the College Board questionnaire. Of all high achievers, 75.8 percent say that they are white non-Hispanic, and another 15.0 percent say that they are Asian. The remaining 9.2 percent of high achievers are associated with an underrepresented minority, (17) either Hispanic (4.7 percent), black non-Hispanic (1.5 percent), Native American (0.4 percent), or mixed race/ethnicity (2.6 percent). If we focus on low-income high achievers only (figure 5), we see that 15.4 percent are underrepresented minorities. Interestingly, the entire increase in this share comes out of the percentage who are white. Asians make up 15.2 percent of low-income high achievers, almost identical to their share of all high achievers.

A key takeaway from figure 5 is that a student's being an underrepresented minority is not a good proxy for his or her being low-income. Thus, if a college wants its student body to exhibit income diversity commensurate with the income diversity among high achievers, it cannot possibly attain this goal simply by recruiting students who are underrepresented minorities. If admissions staff do most of their outreach to low-income students by visiting schools that are largely Hispanic and black, the staff should realize that this strategy may lead to a student body that is diverse on specific racial and ethnic dimensions but that is not diverse in terms of family income.

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Figure 6 is a choropleth map showing the number of high-achieving students in each county of the United States. Counties are an imperfect unit of observation because some are large in land area and some are small. Nevertheless, they are the most consistent political units in the United States. (18) The darker is the county's coloring, the more high-achieving students it contains. What the map demonstrates is that critical masses of high-achieving students are most likely to be found in the urban counties in southern New England (Massachusetts, Connecticut, Rhode Island), the Mid-Atlantic (New York, New Jersey, eastern Pennsylvania), southern Florida, and coastal California from the Bay Area to San Diego. The other critical masses are more scattered, but a person familiar with U.S. geography can pick out Chicago (especially), Houston, Dallas-Fort Worth, Atlanta, and some smaller cities. In short, if one's goal were to visit every county where one could gather at least 100 high achievers, one could concentrate entirely on a limited number of cities on the East and West Coasts and a few cities in between.

[FIGURE 7 OMITTED]

Some part of the above statement is due to the fact that high-income, highly educated parents are somewhat concentrated in the aforementioned areas, and such parents, as we have shown, are somewhat more likely to have high-achieving children. However, some part of the above statement is due purely to population density. That is, even if children in all counties were equally likely to be high-achieving, there would still be critical masses of them in densely populated counties, and vice versa. The choropleth map in figure 7 illustrates the role of population density by showing the number of high-achieving students per 17-year-old in each county. The darker a county is, the higher is its decile on this relative measure. The map makes it clear that this relative measure is far less concentrated than the absolute measure that favors densely populated counties. In fact, one can see a belt of counties that tend to produce high achievers running from Minnesota and the Dakotas south through Missouri and Kansas. A good number of counties in Appalachia, Indiana, and the West outside of coastal California also tend to produce high achievers. In short, if one's goal were to meet a nationally representative sample of high achievers, one's trip could not be concentrated on a limited number of counties on the coasts and a few cities in between.

IV. College Applications, Enrollment, and Degree Receipt among High-Achieving Students in the U.S.

In this section, we analyze the college application choices, enrollment decisions, and on-time degree receipt of high-achieving students in the United States, paying attention to how low-income students' behavior does or does not differ from that of high-income students. Because colleges in the United States are so varied and large in number, we characterize them by the college assessment score of their median student, expressed as a percentile of the national college assessment test score distribution. This statistic, although admittedly insufficient to describe colleges fully, has important qualities. First, it is probably the single best, simple indicator of selectivity--much better than a college's admissions rate, for instance (Avery, Glickman, Hoxby, and Metrick, 2013). Second, when an expert college counselor advises students on how to choose a portfolio of schools to which to apply, he or she usually tells students to apply to a few schools that are a "reach," four or more schools that are "peer" or "match," and one or more schools that are "safe." Similar advice is widely available on the Internet sites of college advising organizations with a strong reputation, including the College Board and the ACT. Expert college counselors use schools' median test scores to define "reach" schools (typically, those whose median score is more than 5 percentiles above the student's own), "peer" schools (typically, those where the school's median score is within 5 percentiles of the student's own), and "safety" schools (typically, those whose median score is 5 to 15 percentiles below the student's own). (19) Naturally, the exact cutoffs for these categories vary from expert to expert, and high-achieving students are often advised to apply to their state's public flagship university, even if it falls below the safety zone. (20) High-achieving students are generally advised to apply to at least eight schools.

IV. A. College Application Behavior: A Graphical Analysis

In this subsection, we provide graphical evidence of what students' application portfolios look like. This presentation is somewhat informal but useful for fixing ideas and defining categories before we move to the formal econometric analysis in the next subsection. In what follows, an "application" is defined as sending a test score to a college. (21)

Figure 8 is a histogram of the application portfolios of high-income students. It is important to understand how this and subsequent figures are constructed. On the horizontal axis is the difference between the applied-to college's median test score and the student's own score, in percentiles. Thus, if an application is located at zero, the student is applying to a peer school whose median student has exactly the same score. An application at, say, +8 is a reach, and an application at, say, -13 is a safety. Since nonselective colleges do not require their students to take college assessments (and thus do not report a median student score), an application to a nonselective school is placed at -94, which is zero minus the average percentile score of high-achieving students in the data. It is not obvious where to place applications to nonselective schools, but -94 has the advantage that such applications cannot be mistaken for applications to a school that is selective but that sets a very low bar.

Each student is given a weight of 1 in the histogram, and this weight is split evenly over that student's applications. This is to ensure that the histogram does not overrepresent the behavior of students who apply to more schools, since, after all, each student will enroll at just one (initially at least). Thus, if a student puts all of his or her eggs in one basket and applies to a single +8 school, that student's full weight of 1 will show up in the +8 bar. If a student applies to one +8 school, one +6 school, one +4 school, and so on down to one -8 school, one 9th of that student's weight will show up in each of the relevant bars. Note that each bar is 2 percentiles wide.

[FIGURE 8 OMITTED]

Figure 8 shows that high-income students largely follow the advice of expert counselors. The bulk of their applications are made to peer schools. They apply to some reach schools as well, but they are mechanically limited in the extent to which they can do this: there are no reach schools for slightly more than half of the high-achieving students we study. (22) High-income high achievers also apply fairly frequently to safety schools. Although not shown in the figure, it is noteworthy that many such students apply to their state's flagship university. These schools vary greatly in selectivity, so that some such applications are in the safe range, but other applications to flagships appear far more safe than anyone would think necessary. For instance, an application by a high achiever to a flagship with a median score at the 50th percentile would end up at -40 to -50. Nevertheless, applying to these schools may be well-advised (see note 20).

[FIGURE 9 OMITTED]

The reader might be surprised to find that high-achieving, high-income students apply to some colleges that are nonselective on academic grounds. However, the schools in question are often specialty schools: music conservatories, art or design schools, drama or performing arts schools, cooking schools, and so on. Some of these are highly selective on nonacademic dimensions.

Figure 9 shows that unlike the high-income high achievers, few low-income high achievers follow the advice of expert counselors. More than 40 percent of the mass in the histogram loads on nonselective schools. (This is an underestimate because scores are not sent to some nonselective schools. If we included every nonselective enrollment as a nonselective application, the nonselective bar on the histogram would rise by 5.1 percentage points.) (23) Moreover, the nonselective colleges to which low-income students apply are rarely of the specialty type mentioned above. They are often local community colleges or local 4-year institutions with meager resources per student and low graduation rates. Much of the height of the nonselective bar is due to the fact that many low-income high achievers apply only to nonselective colleges, or to a nonselective college and a barely selective college.

Figure 10 overlays the histograms for low-income, middle-income, and high-income students who are high-achieving. It cuts off the portion of the histogram that shows nonselective colleges so as to focus on application choices among colleges that are selective to at least some degree. It will be observed that the behavior of the middle-income students (those from families in the two middle quartiles of the family income distribution) is about midway between that of their low- and high-income counterparts. Moreover, even within the subset of applications that are made to selective colleges, high-income students apply much more to peer colleges, and low-income students apply much more to colleges far below the safety level.

Figure 11 contains four panels. The top left-hand panel shows, for all high-achieving, low-income students, the histogram of the most selective college to which each student applied. The top right-hand panel shows the same histogram for high-achieving, high-income students. The bottom left-hand panel shows the histogram for the second most selective college to which a low-income student applied (or the most selective, for students who applied to a single college). The bottom right-hand panel shows the same histogram for high-income students. These histograms reveal that the vast majority of high-income high achievers' most selective applications fall within 10 percentiles of their test scores. Their second most selective applications are sent to less competitive, but not much less competitive schools: the vast majority fall between +10 and -15 percentiles. In contrast, low-income high achievers send their most selective applications to the entire range of colleges: nonselective and -60 to +10. Their second most selective applications are, again, to less competitive (but not necessarily much less competitive) schools. All of this suggests that there may be two distinguishable types of low-income high achievers: those who apply much as their high-income counterparts do, and those who apply in a manner that is very different.

[FIGURE 10 OMITTED]

In fact, 53 percent of low-income high achievers fit the profile we will hereafter describe as income-typical: they apply to no school whose median score is within 15 percentiles of their own, and they do apply to at least one nonselective college. At the other extreme, 8 percent of low-income high achievers apply in a manner that is similar to what is recommended and to what their high-income counterparts do: they apply to at least one peer college, at least one safety college with a median score not more than 15 percentiles lower than their own, and apply to no nonselective colleges. We hereafter designate such students as achievement-typical, noting that once a student fits the above criteria, he or she usually applies to several peer colleges, much as high-income students do.

[FIGURE 11 OMITTED]

The remaining 39 percent of low-income, high achievers use application strategies that an expert would probably regard as odd. For instance, we see some students apply to only a local nonselective college and one extremely selective and well-known college--Harvard, for instance. No expert would advise such a strategy because the probability of getting into an extremely selective, well-known college is low if a student applies to just one--even if the student's test scores and grades are typical of the college's students. Moreover, such a strategy reveals that the student is interested in extremely selective institutions yet is not applying to the other schools that are, for most purposes, indistinguishable from the one to which he or she applied. Another strategy that appears is a student applying to a single public college in his or her state that is selective but is much less selective than the state's flagship university. Although about half of these application choices could be motivated by distance from home, the other half cannot because the flagship university is nearer. Another strategy that falls into the idiosyncratic category is a student applying to a single private college outside his or her state that is selective, but much less selective and much poorer in resources than the student's private peer colleges would be. Such choices are odd because although the private peer colleges might offer fewer scholarships that are explicitly merit-based, they offer much more generous need-based aid, so that the student would pay less to attend and would enjoy substantially more resources. Furthermore, it is almost never sensible for a low-income student to apply to a single private, selective college: such a student can use competing aid offers to improve the aid package at his or her most preferred college.

We have described a few salient strategies that appear among low-income high achievers who are neither achievement-typical nor income-typical. However, most of these students' portfolios do not evince any pattern that can be readily described. Thus, below we turn to an econometric analysis, in which we can simultaneously consider a large number of factors correlated with students' application choices.

IV. B. College Application Behavior: An Econometric Analysis

In this subsection we assess the factors that are associated with a student's choice of his or her application portfolio, using a conditional logit model in which a student can apply to all colleges in the United States but decides to apply only to some. This model is based on a random utility framework and assumes that the student prefers all colleges to which he or she applies over the colleges to which he or she does not apply. We do not assume anything about the student's preference ordering within the colleges to which he or she applied. (24) Each possible college matched with each student is an observation, and the dependent variable is a binary variable equal to 1 if the student submits an application to the college and zero otherwise.

The explanatory variables we consider are the difference between a school's median test score and the student's own test score if positive, the same difference if negative, (25) an indicator for the school's being nonselective, the distance between the student's home and the school, the square of this distance, an indicator for the school being the most proximate, an indicator for the school being public, an indicator for the school being in-state for the student, an indicator for the school being the flagship university of the student's state of residence, the sticker price of the college, the likely net cost of the college for the student, and the student-oriented resources per student at the college. We fully interact these explanatory variables with indicators for the student being low-income, high-income, or in between. Thus, we estimate separate coefficients for each income group. In the tables we do not show the coefficients for the middle-income group because they nearly always fall between those of the high- and low-income students, but the coefficients are available upon request.

Table 3 shows the results of this estimation. The coefficients are expressed as odds ratios so that a coefficient greater than 1 means that an increase in the covariate is associated with an increase in the probability that the student applies to the school, all other covariates held constant. Based on our graphical analysis, we expect to find very different coefficients for low- and high-income students, and we do. (26) Note that, although it is convenient to describe the coefficients as though they literally revealed preference, they should not be given such a strong interpretation or a causal interpretation. For instance, students might "disfavor" distance not because distance itself generates negative utility but because distant schools have, say, distinct cultures that the student dislikes.

We find that high-income students strongly favor reach colleges and disfavor safety colleges (those for which the score difference is negative). Per percentile of difference, this effect is much stronger on the reach side than on the safety side, but recall that high-achieving students can only reach a bit whereas they can apply to very safe schools. High-income students strongly dislike nonselective institutions. They also dislike higher net costs but (all else equal) like higher sticker prices. This is probably because higher sticker prices are associated with higher per-student resources, a characteristic they also like. High-income students dislike distance, but the quadratic term indicates that they dislike it only up to a point, after which they are fairly indifferent. They have a mild preference for in-state schools and their state's flagship university. They do not have a statistically significant preference for publicly controlled schools.

The low-income students exhibit several immediate contrasts. Such students strongly favor nonselective colleges. This was obvious in the graphical evidence. They do not disfavor schools whose median scores are lower than theirs. They slightly disfavor schools with higher sticker prices (recall that these were favored by high-income students) and do not have a preference for net costs that is statistically significantly different from zero. Low-income students do favor schools with higher expenditure per student, but not nearly as much as high-income students do. Distance is strongly disfavored for schools within 100 miles but, thereafter, low-income students are fairly indifferent to it. Low-income students favor in-state schools somewhat more than high-income students do, but low-income students do not exhibit a preference in favor of their state's flagship university. They slightly favor publicly controlled colleges.

Table 4 repeats the estimation but interacts the covariates with indicators for high-income students, middle-income students, low-income achievement-typical students, low-income income-typical students, and other low-income students. The estimated coefficients for achievement-typical students are fairly similar to those for high-income students. It is the income-typical students whose coefficients are strikingly different. Of course, these results are somewhat by design, given the way we categorized low-income students into achievement-typical and income-typical groups. However, the coefficients validate the categorization: achievement-typical students do pursue similar application strategies to high-income students. In the next section we assess which factors predict a student being achievement-typical and which predict a student being income-typical.