Abstract

We show that tenth-grade teacher expectations affect students' likelihood of college completion. Our approach leverages a unique feature of a nationally representative dataset: two teachers provided their educational expectations for each student. Identification exploits teacher disagreements about the same student, an idea we formalize using a measurement error model. We estimate an elasticity of college completion with respect to teachers' expectations of 0.12. On average, teachers are overly optimistic, though white teachers are less so with black students. More accurate beliefs are counterproductive if there are returns to optimism or sociodemographic gaps in optimism. We find evidence of both.

I. Introduction

AT least since Becker (1964) cast schooling as an investment in human capital, economists have sought to understand the factors that drive variation in educational outcomes. Sociodemographic gaps in educational attainment have received particular attention, as education facilitates upward economic and social mobility (Bailey & Dynarski, 2011) and increased earnings (Card, 1999). Attainment gaps are especially concerning if they reflect suboptimal investments in human capital by underrepresented or historically disadvantaged groups.

Teacher expectations constitute one potentially important, but relatively understudied, educational input that might contribute to sociodemographic gaps in educational attainment. Despite pervasive views that teacher expectations matter, however, it is difficult to credibly identify their causal effects on student outcomes (Brophy, 1983; Jussim & Harber, 2005; Ferguson, 2003). The reason is that observed correlations between teacher expectations and student outcomes could reflect either accurate forecasts or a causal relationship.1 In the first case, expectations do not drive student outcomes, but instead reflect the same factors that drive educational attainment. In the second case, a causal impact arises if incorrect (i.e., biased) teacher expectations create self-fulfilling prophecies in which investments made in or by students are altered, thereby leading to outcomes that resemble teachers' initially incorrect beliefs (Loury, 2009; Glover, Pallais, & Pariente, 2017).

In this paper, we provide evidence that teacher expectations affect educational attainment. To identify causal effects, we exploit a unique feature of a nationally representative longitudinal data set: two teachers provide their educational expectations for each student.2 When teachers disagree about a particular student, which they frequently do, they provide within-student, within-semester variation in expectations, which we argue below is conditionally random. We leverage this variation to identify the impact of expectations on educational attainment. Intuitively, our analysis uses one teacher's expectation to control for unobserved factors that both teachers use to form expectations and that affect student educational attainment.3 Our approach addresses the fundamental endogeneity problem that arises if teacher expectations reflect omitted variables that also drive education outcomes (Gregory & Huang, 2013; Boser, Wilhelm, & Hanna, 2014).

Our study begins by documenting several interesting patterns in the teacher expectations data. First, we find that teacher expectations predict student outcomes, though teachers appear to be optimistic. Second, teacher expectations respond in expected ways to information that would presumably affect college going, such as family income, standardized test scores, and ninth-grade GPA. Third, teachers frequently disagree about how far a given student will go in school, which is key to our identification strategy.

Next, we provide evidence that teacher expectations causally affect student outcomes. Our main analyses use OLS regressions and condition on both teachers' expectations. Using this strategy, our estimates provide consistent and compelling evidence of a causal impact of teachers' expectations on the likelihood of college completion. The estimates suggest that the elasticity of the likelihood of college completion with respect to teachers' expectations is about 0.12. We are not the first to provide evidence that teachers' biased beliefs can become self-fulfilling prophecies. In an earlier contribution, Rosenthal and Jacobson (1968) report effects of informing teachers that some randomly selected students are high aptitude. These students eventually perform better on tests, which provides some evidence for the view that teacher expectations matter in the sense that randomly assigned biases can become self-fulfilling prophecies. Our findings using a longitudinal data set show that these so-called Pygmalion effects can have long-run impacts on educational attainment and that nonexperimentally induced teacher expectations can generate self-fulfilling prophecies.4

A possible concern is that high teacher expectations not only raise the probability of a college degree, but also induce some students to enroll in postsecondary education without completing a degree. Given relatively low labor market returns to “some college,” this could suggest a downside to high expectations for some students. Instead, we find that the high expectations raise college graduation rates not only by lowering the likelihood of attaining a high school diploma or less, but also by reducing the probability that students begin some postsecondary education without completing a four-year degree. In other words, high expectations reduce the probability of several levels of educational investment with low returns. We also find that the positive impacts of high expectations extend to other longer-run outcomes, including higher rates of employment and lower usage of public benefits.

We also examine potential mechanisms explaining why teacher expectations matter. One possibility is that expectations do not improve teaching or learning, but instead lead teachers to ease students' pathway to college, for example, by writing stronger college recommendation letters. Alternatively, teachers might directly impart expectations to students or do so indirectly by modifying how they teach (Burgess & Greaves, 2013; Ferguson, 2003; Mechtenberg, 2009). We show that high expectations translate to positive changes in twelth-grade GPA and to slightly longer amounts of time spent on homework, perhaps reflecting increases in student effort. High teacher expectations also raise students' own expectations in the twelth grade about their educational attainment.5 This suggests that teacher expectations matter in part because they shift students' own views about their educational prospects.

Motivated by our main findings, we develop an econometric model to jointly estimate the production of teacher expectations and student outcomes. Drawing on lessons from the measurement error literature (Hu & Schennach, 2008), the model formalizes two key ideas.6 First, the same factors jointly generate teacher expectations and student educational attainment. These include observables like parental income, along with variables that teachers might observe, but the econometrician does not, such as student ambitions or household-level shocks (e.g., a parent's health problems). These variables are captured by a latent factor, θi, the distribution of which we estimate and which governs the objective probability of school completion absent the impact of teacher expectations. Second, the model formalizes the idea that teacher disagreements about the same student, treated in the model as measurement error around θi, can be used to identify causal estimates of the impact of teacher expectations on student outcomes.

Estimates of the measurement error model corroborate our initial findings on the impact of teacher expectations. Moreover, we use the model to recover the distribution of teacher biases, defined as the difference between teacher expectations and expected educational attainment absent bias. Recovering the distribution of bias is in part motivated by earlier findings that white teachers' expectations are systematically lower than black teachers' expectations for black students (Gershenson, Holt, & Papageorge, 2016). Yet it is unclear whether white teachers are too pessimistic, black teachers are optimistic, or both. We find that teachers are on average too optimistic: their expectations are higher than the student's objective probability of college completion captured by θi and other model parameters.7 However, there are racial differences. White teachers are systematically less optimistic about black students. An overlooked nuance is therefore that white teachers are less biased about black students in that their expectations are closer to the objective probability. However, since higher expectations lead to better outcomes, this “accuracy” amounts to a selective lack of optimism that harms black students. The model also reveals that most racial differences in expectations arise not because white teachers have exceedingly low expectations for promising black students. Rather, differences are subtle. They emerge for students with moderate levels of θi, which maps to an objective probability of college completion between 30% and 90%. In such cases, white students are given the benefit of the doubt in that teachers report that they expect college completion, while for black students with a similar θi, the teacher expects something less, such as “some college with no degree.”

In summary, our findings show that expectations matter and that systematic differences in expectations put black students at a disadvantage. Shifting the production of expectations so that black students faced the same expectations (the same benefit of the doubt as white students with similar θi) would help to close achievement gaps. The next question is how to narrow expectations gaps across groups. One possibility is to hire more black teachers since they are more likely to have higher expectations for black students than are white teachers, but doing so is not a costless endeavor and may not be feasible, especially in the short run. A second possibility is rooted in the idea that racial expectations differences tend to be relatively small and subtle. Teachers may therefore be unaware of their own biases, a phenomenon known as implicit bias. Emerging scholarship provides mounting evidence that relatively low-cost interventions (e.g., simply informing teachers of their biases) can reduce implicit biases and change teacher behaviors in ways that help traditionally underserved or disadvantaged students (Okonofua, Paunesku, & Walton, 2016; Alesina et al., 2018). We return to a discussion of both policies in section V.

Apart from its links to growing literature on implicit biases, our paper relates more generally to literature on the consequences of bias, including self-fulfilling prophecies. For example, Glover et al. (2017) study how managers' negative implicit biases against certain demographic groups can lead to lower performance for these groups. Another set of studies examines biases in beliefs and human capital investments and their consequences for investment decisions (Cunha, Elo, & Culhane, 2013; Wiswall & Zafar, 2015). Regarding teacher biases, Lavy and Sand (2015) identify primary school teachers in Israel who have “pro-boy” grading bias by comparing students' scores on blind and nonblind exams.8 Most similar to us, Jones and Hill (2018) provide evidence that teacher expectations improve student test scores. Their findings use different data (from North Carolina) and a different identification strategy and thus complement ours. A key difference is that we examine older students and longer-run outcomes such as college completion and labor supply.

Our study contributes to a large body of research examining how teachers and schools affect student outcomes. A number of studies have shown that teachers are important inputs in the education production function (Chetty, Friedman, & Rockoff, 2014b; Hanushek & Rivkin, 2010). Other studies have recognized that same-race teachers are more effective, especially for racial minorities (Fairlie, Hoffmann, & Oreopoulos, 2014; Dee, 2004; Gershenson et al., 2018). However, it remains unclear what specific behaviors and characteristics make teachers effective (Staiger & Rockoff, 2010). Our study suggests one possible mechanism: teachers' expectations. Moreover, while our main results leverage within- school, within-classroom differences in expectations, our findings are consistent with the idea that schools can improve performance by creating a “culture of high expectations” (Dobbie & Fryer, 2013; Fryer, 2014). This may be particularly salient for relatively disadvantaged students who rarely interact with college-educated adults outside school settings (Jussim & Harber, 2005; Lareau, 2011; Lareau & Weininger, 2008).

Finally, our paper contributes to research on the importance of subjective expectations. The idea that subjective beliefs (rather than objective probabilities) drive individual behavior is not new (Savage, 1954; Manski, 1993). However, despite mounting evidence that subjective expectations can affect important economic outcomes, they have only recently entered into economic analyses of decision making (Manski, 2004; Hurd, 2009).9 One reason is that it is difficult to assess whether beliefs have causal effects on outcomes absent experimentally induced exogenous variation. We examine the conditions under which multiple subjective expectations about a single latent objective probability can be used to identify biased expectations and causal impacts of expectations via self-fulfilling prophecies.

II. The ELS 2002 Data and Patterns in Teacher Expectations

A. The Data

Data come from the Educational Longitudinal Study of 2002 (ELS 2002), a nationally representative survey of the cohort of U.S. students who were in tenth grade in spring 2002. The ELS contains rich information on students' sociodemographic backgrounds and secondary and postsecondary schooling outcomes (including educational attainment through 2012, or within eight years of an on-time high school graduation). Students were sampled within schools, and school identifiers facilitate within-school (school fixed effects, FE) analyses. The data also contain a number of observed school and teacher characteristics, including teachers' experience, demographic background, credentials, and expectations and perceptions of specific students.

The main analytic sample is restricted to the 6,060 students for whom these variables are observed.10 Because there are two teacher expectations per student, the analytic sample contains 12,130 student-teacher pairs.11 Table 1 summarizes the students who compose the analytic sample. Column 1 does so for the full sample, and columns 2 to 5 do so separately by student race and sex. The outcome of interest, educational attainment, is summarized in three ways: percentage of students who earn a four-year college degree (or more), percentage who fail to complete high school, and average years of schooling. About 45% of students in the sample completed a four-year degree, though whites and women were significantly more likely to do so than blacks and men, respectively.12 This is consistent with demographic gaps in educational attainment observed in other data sets (Bailey & Dynarski, 2011; Bound & Turner, 2011; Cameron & Heckman, 2001). The racial gaps in educational attainment are particularly stark, as whites were about 20 percentage points (69%) more likely to complete college than blacks, while blacks were twice as likely as whites to fail to complete high school. Racial differences in educational attainment also appear in figure 1a, which provides a histogram for educational attainment categories for the full sample, and then separately for blacks, whites, males, and females.
Figure 1.

Teacher Expectations and Student Outcomes

Panel 1a is a histogram of the percentage of the subsample of students who fall in the given educational attainment category. “HS” is high school. “Graduate degree” includes masters, PhD, and professional degrees. Panel 1b shows the percentage of students who complete a four-year college degree by ELA and math teacher expectations.

Figure 1.

Teacher Expectations and Student Outcomes

Panel 1a is a histogram of the percentage of the subsample of students who fall in the given educational attainment category. “HS” is high school. “Graduate degree” includes masters, PhD, and professional degrees. Panel 1b shows the percentage of students who complete a four-year college degree by ELA and math teacher expectations.

Table 1.
Analytic Sample Means—Students
AllWhiteBlackMaleFemale
(1)(2)(3)(4)(5)
Educational attainment      
Completed College or more 0.45 0.49 0.29 0.43 0.47 
Completed less than high school 0.01 0.01 0.02 0.01 0.01 
Education completed, years 14.67 14.83 14.08 14.51 14.81 
 (2.06) (2.06) (1.84) (2.05) (2.07) 
Teacher expectations      
Expect college or more, English 0.64 0.67 0.48 0.60 0.67 
Expect less than high school, English 0.01 0.01 0.03 0.02 0.01 
ELA teacher expected years 15.65 15.78 14.86 15.48 15.80 
 (2.23) (2.14) (2.21) (2.29) (2.16) 
Expect college or more, math 0.63 0.66 0.44 0.61 0.65 
Expect less than high school, Math 0.01 0.01 0.03 0.01 0.01 
Math teacher expected years 15.51 15.65 14.66 15.43 15.59 
 (2.09) (1.99) (2.07) (2.16) (2.03) 
Teacher expectations disagree 0.21 0.20 0.25 0.21 0.21 
Math teacher has higher expectation 0.10 0.10 0.11 0.10 0.10 
Academic background      
Reading assessment 52.82 54.67 46.71 52.39 53.21 
 (9.83) (9.26) (8.99) (10.20) (9.47) 
Math assessment 53.01 54.71 45.77 54.00 52.12 
 (9.67) (8.78) (8.88) (10.13) (9.15) 
Ninth-grade GPA 2.92 3.02 2.44 2.82 3.01 
 (0.78) (0.73) (0.76) (0.78) (0.77) 
Demographics and Socioeconomic Status      
Household Income less than 20,000 0.11 0.06 0.26 0.09 0.13 
Household Income more than 100,000 0.18 0.21 0.08 0.19 0.17 
Mother high school diploma or less 0.34 0.29 0.39 0.32 0.35 
Mother has a bachelor's or more 0.31 0.34 0.23 0.33 0.29 
Teacher      
ELA teacher nonwhite 0.10 0.05 0.26 0.10 0.10 
Math teacher nonwhite 0.11 0.06 0.21 0.11 0.11 
ELA teacher black 0.04 0.02 0.20 0.04 0.04 
Math teacher black 0.04 0.02 0.16 0.03 0.04 
Observations 6,060 3,970 610 2,870 3,190 
AllWhiteBlackMaleFemale
(1)(2)(3)(4)(5)
Educational attainment      
Completed College or more 0.45 0.49 0.29 0.43 0.47 
Completed less than high school 0.01 0.01 0.02 0.01 0.01 
Education completed, years 14.67 14.83 14.08 14.51 14.81 
 (2.06) (2.06) (1.84) (2.05) (2.07) 
Teacher expectations      
Expect college or more, English 0.64 0.67 0.48 0.60 0.67 
Expect less than high school, English 0.01 0.01 0.03 0.02 0.01 
ELA teacher expected years 15.65 15.78 14.86 15.48 15.80 
 (2.23) (2.14) (2.21) (2.29) (2.16) 
Expect college or more, math 0.63 0.66 0.44 0.61 0.65 
Expect less than high school, Math 0.01 0.01 0.03 0.01 0.01 
Math teacher expected years 15.51 15.65 14.66 15.43 15.59 
 (2.09) (1.99) (2.07) (2.16) (2.03) 
Teacher expectations disagree 0.21 0.20 0.25 0.21 0.21 
Math teacher has higher expectation 0.10 0.10 0.11 0.10 0.10 
Academic background      
Reading assessment 52.82 54.67 46.71 52.39 53.21 
 (9.83) (9.26) (8.99) (10.20) (9.47) 
Math assessment 53.01 54.71 45.77 54.00 52.12 
 (9.67) (8.78) (8.88) (10.13) (9.15) 
Ninth-grade GPA 2.92 3.02 2.44 2.82 3.01 
 (0.78) (0.73) (0.76) (0.78) (0.77) 
Demographics and Socioeconomic Status      
Household Income less than 20,000 0.11 0.06 0.26 0.09 0.13 
Household Income more than 100,000 0.18 0.21 0.08 0.19 0.17 
Mother high school diploma or less 0.34 0.29 0.39 0.32 0.35 
Mother has a bachelor's or more 0.31 0.34 0.23 0.33 0.29 
Teacher      
ELA teacher nonwhite 0.10 0.05 0.26 0.10 0.10 
Math teacher nonwhite 0.11 0.06 0.21 0.11 0.11 
ELA teacher black 0.04 0.02 0.20 0.04 0.04 
Math teacher black 0.04 0.02 0.16 0.03 0.04 
Observations 6,060 3,970 610 2,870 3,190 

This table presents means of variables where students are the unit of analysis. Standard deviations for nonbinary variables are reported in parentheses. Ninth-grade GPAs are on a 4.0 scale. Math and reading assessment scores are on a 0–100 scale. All sample sizes are rounded to the nearest 10 in accordance with NCES regulations for restricted data.

The key teacher expectation variable is based on teachers' responses to the following question: “How far do you think [STUDENT] will go in school?” Teachers answered this question by selecting one of seven mutually exclusive categories.13 In most of our subsequent analysis, we exploit a unique feature of the ELS 2002's design: two teachers, one math and one English language arts (ELA), provided their subjective expectations and perceptions of each student. Teachers' expectations are summarized in the next section of table 1. Overall, about 64% of teachers expected the student to complete a four-year college degree. This suggests that teachers, on average, are too optimistic about students' college success, as only 45% of students complete a four-year degree. This overoptimism is apparent in each demographic group, though teachers' expectations for black students are significantly lower than for white students, as are expectations for male students relative to females. This points to an interesting feature in the data that foreshadows our results: teacher expectations for black students are not necessarily low relative to observed outcomes. Rather, they are less inflated relative to observed outcomes compared to expectations for white students. Still, observed racial and sex gaps in expectations are consistent with the patterns in actual educational attainment already described, suggesting that teachers' expectations are informative. However, while math and ELA teachers' expectations are similar on average, ELA teachers' expectations tend to be slightly higher, particularly among black students. This shows that teachers occasionally disagree about how far a particular student will go in school. Specifically, teachers disagree on slightly more than 20% of students, with math teachers having higher expectations in slightly less than half of those cases. Below, we investigate the sources of teacher disagreements and consider how such disagreements can be leveraged to identify the impact of expectations on student outcomes.

The final two panels of table 1 summarize students' academic and socioeconomic characteristics. A comparison of columns 2 and 3 shows that white students have significantly higher test scores, GPAs, and household incomes than black students, as well as better educated mothers, all of which is consistent with longstanding racial disparities in academic performance and socioeconomic status (Fryer, 2010). Another notable difference by student race is in their assigned teacher's race: black students are four to five times as likely as white students to be assigned a black teacher, which is due to nonwhite teachers being more likely to teach in majority nonwhite schools (Hanushek, Kain, & Rivkin, 2004; Jackson, 2009). Nonetheless, the majority of students, white and black, have white teachers. Columns 4 and 5 of table 1 show that girls have higher GPAs and perform better on reading assessments than boys, while boys perform better on math assessments.

In our analytic sample, 11% of teachers are nonwhite, and nonwhite teachers are evenly represented across subjects and sex.14 White teachers tend to have more experience than black teachers. Moreover, black teachers are significantly more likely to teach black students than are teachers from other racial backgrounds. Looking further into racial differences between teachers, white teachers, compared to black teachers, are more likely to be male, experienced, and hold teaching certificates, and these differences are statistically significant.

In this section and in most subsequent analyses, we focus on the college-completion margin because recent research explicitly notes that individuals with some college, but less than a four-year degree, have socioeconomic trajectories that closely resemble those of high school graduates (Lundberg, Pollak, & Stearns, 2016). This choice is also due to the striking patterns observed in figure 1a: blacks are significantly more likely than whites to complete only some college. This suggests that college completion, relative to college entrance, is an important margin to consider in the analysis of racial attainment gaps. Thus, we define students' educational attainment and teachers' educational expectations for the student in the same way: the student outcome of interest in the primary analyses is an indicator for “student completed a four-year college degree or more” (as of 2012, eight years removed from an on-time high school graduation) and the independent variables of interest are indicators for “teacher expects a four-year college degree or more.”

B. Key Patterns in Teacher Expectation Data

Teacher expectations are predictive.

Figure 1b plots the percentage of students who complete a four-year college degree for each category of teacher expectations, separately for math and ELA teachers. Higher expectations are associated with a higher probability of college completion. Interestingly, however, teacher forecasts are subject to error. For example, of students for whom ELA teachers expect some college (but not college completion), roughly 15% go on to obtain a four-year degree. Forecast errors tend to be in the opposite direction, however: fewer than 60% of students whose math or ELA teachers expect a four-year degree actually obtain one. This pattern extends to students for whom teachers expect a graduate degree, who obtain at least a four-year degree roughly 80% and 85% of the time, respectively. In other words, though teacher expectations are predictive of student outcomes, on average teachers overestimate educational attainment, which is consistent with patterns in table 1.

The production of teacher expectations.

Understanding the determinants of teacher expectations is a precursor to credibly identifying the impact of those expectations on student outcomes. However, previous analyses of the association between teacher expectations and student outcomes generally pay short shrift to the formation of teacher expectations. Thus, one contribution of this study is a systematic analysis of the teacher expectation production function. We show that factors that would presumably affect educational attainment also produce teacher expectations. We do so by estimating equations of the form
Tij=Xiβj+νij,j{M,E}.
(1)
The usual suspects are predictive of teacher expectations (see table 2). Columns 1 to 3 show that higher income, being white, and higher GPA are associated with higher teacher expectations. When we evaluate these factors jointly in column 4, we find higher expectations for Asians and lower expectations for Hispanics. Interestingly, if we adjust for parental income and GPA, black students do not face lower expectations. Similar patterns are found for math teacher expectations, but expectations for black students are lower even after we have controlled for ninth-grade GPA and household income.
Table 2.
Teacher Expectations Production Function
ELA Teacher ExpectationsMath Teacher Expectations
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
HH income 20K–35K 0.07**   0.02 0.02 0.10***   0.05** 0.05** 
 (0.03)   (0.02) (0.02) (0.03)   (0.02) (0.02) 
HH income 35K–75K 0.19***   0.09*** 0.09*** 0.19***   0.08*** 0.08*** 
 (0.02)   (0.02) (0.02) (0.02)   (0.02) (0.02) 
HH income 75K–100K 0.25***   0.11*** 0.11*** 0.27***   0.13*** 0.13*** 
 (0.03)   (0.02) (0.02) (0.03)   (0.02) (0.02) 
HH income > 100K 0.29***   0.12*** 0.12*** 0.30***   0.13*** 0.13*** 
 (0.03)   (0.02) (0.02) (0.03)   (0.02) (0.02) 
Student is American Indian  −0.21**  -0.04 -0.04  −0.25**  -0.08 -0.08 
  (0.10)  (0.07) (0.07)  (0.10)  (0.10) (0.10) 
Student is Asian  0.11***  0.05** 0.05**  0.12***  0.05** 0.05** 
  (0.03)  (0.02) (0.02)  (0.03)  (0.02) (0.02) 
Student is black  −0.17***  -0.01 -0.00  −0.21***  −0.05** −0.05** 
  (0.03)  (0.02) (0.02)  (0.03)  (0.02) (0.02) 
Student is Hispanic  −0.17***  −0.04* −0.04*  −0.16***  -0.03 -0.03 
  (0.03)  (0.02) (0.02)  (0.02)  (0.02) (0.02) 
Student is multiple race  -0.05  0.00 -0.00  −0.07**  -0.02 -0.02 
  (0.03)  (0.03) (0.03)  (0.03)  (0.03) (0.03) 
GPA for all ninth-grade courses   0.35*** 0.33*** 0.33***   0.35*** 0.34*** 0.34*** 
   (0.01) (0.01) (0.01)   (0.01) (0.01) (0.01) 
School FE Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 
Teacher characteristics No No No No Yes No No No No Yes 
Observations 6,060 6,060 6,060 6,060 6,060 6,060 6,060 6,060 6,060 6,060 
R2 0.28 0.27 0.49 0.50 0.50 0.28 0.28 0.50 0.50 0.51 
Adjusted R2 0.19 0.18 0.43 0.44 0.44 0.20 0.19 0.44 0.44 0.44 
ELA Teacher ExpectationsMath Teacher Expectations
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
HH income 20K–35K 0.07**   0.02 0.02 0.10***   0.05** 0.05** 
 (0.03)   (0.02) (0.02) (0.03)   (0.02) (0.02) 
HH income 35K–75K 0.19***   0.09*** 0.09*** 0.19***   0.08*** 0.08*** 
 (0.02)   (0.02) (0.02) (0.02)   (0.02) (0.02) 
HH income 75K–100K 0.25***   0.11*** 0.11*** 0.27***   0.13*** 0.13*** 
 (0.03)   (0.02) (0.02) (0.03)   (0.02) (0.02) 
HH income > 100K 0.29***   0.12*** 0.12*** 0.30***   0.13*** 0.13*** 
 (0.03)   (0.02) (0.02) (0.03)   (0.02) (0.02) 
Student is American Indian  −0.21**  -0.04 -0.04  −0.25**  -0.08 -0.08 
  (0.10)  (0.07) (0.07)  (0.10)  (0.10) (0.10) 
Student is Asian  0.11***  0.05** 0.05**  0.12***  0.05** 0.05** 
  (0.03)  (0.02) (0.02)  (0.03)  (0.02) (0.02) 
Student is black  −0.17***  -0.01 -0.00  −0.21***  −0.05** −0.05** 
  (0.03)  (0.02) (0.02)  (0.03)  (0.02) (0.02) 
Student is Hispanic  −0.17***  −0.04* −0.04*  −0.16***  -0.03 -0.03 
  (0.03)  (0.02) (0.02)  (0.02)  (0.02) (0.02) 
Student is multiple race  -0.05  0.00 -0.00  −0.07**  -0.02 -0.02 
  (0.03)  (0.03) (0.03)  (0.03)  (0.03) (0.03) 
GPA for all ninth-grade courses   0.35*** 0.33*** 0.33***   0.35*** 0.34*** 0.34*** 
   (0.01) (0.01) (0.01)   (0.01) (0.01) (0.01) 
School FE Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 
Teacher characteristics No No No No Yes No No No No Yes 
Observations 6,060 6,060 6,060 6,060 6,060 6,060 6,060 6,060 6,060 6,060 
R2 0.28 0.27 0.49 0.50 0.50 0.28 0.28 0.50 0.50 0.51 
Adjusted R2 0.19 0.18 0.43 0.44 0.44 0.20 0.19 0.44 0.44 0.44 

*p<0.10, **p<0.05, and ***p<0.01. The dependent variable is a binary indicator equal to 1 if the teacher expects the student to complete a four-year college degree or more and 0 otherwise. Parentheses contain standard errors that are robust to clustering at the school level. Estimates are from OLS regressions of equation (1). “Teacher characteristics” controls include teacher race and gender dummies, years of experience, and whether the teacher majored in the subject he or she teaches. School FE refers to school fixed effects.

These estimates highlight how the correlation between teacher expectations and student outcomes may reflect how teachers respond to information about students that could affect their educational attainment. If we regress educational attainment on one teacher's expectation, the estimated coefficient is unlikely to represent a causal effect, as omitting income would lead to an upwardly biased estimate since income presumably drives educational attainment but is also associated with higher expectations. More generally, the production function estimates show that the same factors that drive teacher expectations are also likely to drive educational attainment. Omitting such factors thus leads to biased coefficients. Moreover, there are likely to be other factors that teachers observe and that we do not observe that also affect teacher expectations and student outcomes, which would lead to omitted variables bias despite adjusting for observable student characteristics.

Teacher disagreements.

A consistent pattern in the data is that teachers frequently disagree about a particular student's educational prospects, which we leverage in our identification strategy. The transition matrices reported in table S5 in appendix A document the frequency of such disagreements.15 Overall, teachers disagree for about 40% of students. Aggregating all possible responses into a binary measure of “expects a four-year college degree or more” still leads to disagreements over about 25% of students. The modal disagreement is over whether students who enter college will earn a four-year degree rather than more substantial disagreements. This suggests that disagreements are often subtle and might hinge on arbitrary factors that do not directly affect student outcomes. This is key to the identification strategy discussed below.

III. Main Results

In this section, we provide arguably causal evidence that higher teacher expectations lead to higher educational attainment. Our main identification strategy leverages teacher disagreements, as disagreements generate within-student, within-semester variation in teacher expectations. To the degree that these disagreements are conditionally random, we can use this variation to estimate causal effects. We first present the main results for educational attainment and then discuss additional outcomes and possible mechanisms.

A. Evidence That Teacher Expectations Matter

Table 3 presents OLS estimates of linear regressions of the form
yi=γETEi+γMTMi+Xiβ+εi,
(2)
where y denotes student outcomes, which are coded as binary indicators of college completion in the baseline model, T's denote teacher expectations, and i indexes students.16
Table 3.
OLS Estimates of Effect of Expectations on Educational Attainment
All StudentsWhiteBlack
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)
ELA teacher expectations 0.48***  0.31*** 0.30*** 0.24*** 0.15*** 0.14*** 0.17*** 0.16*** 0.14*** 0.17** 
 (0.01)  (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.02) (0.08) 
Math teacher expectations  0.48*** 0.31*** 0.31*** 0.25*** 0.16*** 0.13*** 0.15*** 0.13*** 0.14*** 0.11 
  (0.01) (0.02) (0.02) (0.01) (0.02) (0.02) (0.03) (0.02) (0.02) (0.07) 
Teacher controls No No No Yes Yes Yes Yes Yes Yes Yes Yes 
Student SES No No No No Yes Yes Yes Yes Yes Yes Yes 
Ninth grade GPA No No No No No Yes Yes Yes Yes Yes Yes 
School FE No No No No No No Yes No Yes Yes Yes 
Teacher dyad FE No No No No No No No Yes No No No 
Observations 6,060 6,060 6,060 6,060 6,060 6,060 6,060 3,600 3,600 3,970 610 
R2 0.22 0.22 0.28 0.30 0.34 0.37 0.45 0.59 0.46 0.48 0.65 
Adjusted R2 0.22 0.22 0.28 0.29 0.34 0.37 0.38 0.18 0.37 0.39 0.31 
All StudentsWhiteBlack
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)
ELA teacher expectations 0.48***  0.31*** 0.30*** 0.24*** 0.15*** 0.14*** 0.17*** 0.16*** 0.14*** 0.17** 
 (0.01)  (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) (0.02) (0.02) (0.08) 
Math teacher expectations  0.48*** 0.31*** 0.31*** 0.25*** 0.16*** 0.13*** 0.15*** 0.13*** 0.14*** 0.11 
  (0.01) (0.02) (0.02) (0.01) (0.02) (0.02) (0.03) (0.02) (0.02) (0.07) 
Teacher controls No No No Yes Yes Yes Yes Yes Yes Yes Yes 
Student SES No No No No Yes Yes Yes Yes Yes Yes Yes 
Ninth grade GPA No No No No No Yes Yes Yes Yes Yes Yes 
School FE No No No No No No Yes No Yes Yes Yes 
Teacher dyad FE No No No No No No No Yes No No No 
Observations 6,060 6,060 6,060 6,060 6,060 6,060 6,060 3,600 3,600 3,970 610 
R2 0.22 0.22 0.28 0.30 0.34 0.37 0.45 0.59 0.46 0.48 0.65 
Adjusted R2 0.22 0.22 0.28 0.29 0.34 0.37 0.38 0.18 0.37 0.39 0.31 

*p<0.10, **p<0.05, and ***p<0.01. The dependent variable is a binary indicator equal to 1 if the student completed a four-year college degree or more, and 0 otherwise. Parentheses contain standard errors that are robust to clustering at the school level. Estimates are from OLS regressions of equation (2). Student socioeconomic status (SES) controls include indicators for household income and mother's educational attainment, as well as indicators for student race, sex, and if a language other than English is spoken at home. “Teacher controls” include teacher race and gender dummies, years of experience, and whether the teacher majored in the subject he or she teaches. “School FE” refers to school fixed effects and Teacher dyad FE refers to Math–ELA teacher pair fixed effects. Estimates in column (9) are from a school fixed-effects model, but estimated on the subsample of students for whom teacher dyad fixed effects are identified.

Either γE or γM can be restricted to equal 0, where E and M index ELA and math teachers, respectively. The vector X includes a progressively richer set of statistical controls, up to and including school or teacher fixed effects (FE). Standard errors are clustered by school, as teachers and students are nested in schools (Angrist & Pischke, 2008).

Columns 1 and 2 of table 3 report simple bivariate regressions of y on the ELA and math teachers' expectations, respectively. The point estimates are nearly identical, positive, and strongly statistically significant. Of course, these positive correlations cannot be given causal interpretations because there are many omitted factors that jointly predict student outcomes and teachers' expectations (e.g., household income). In subsequent columns of table 3, we attempt to reduce this omitted-variables bias by explicitly controlling for such factors. In column 3, we simultaneously condition on both teachers' expectations. Interestingly, though both estimates of γj decrease in magnitude, they remain nearly identical to one another and both remain individually statistically significant. The decline in magnitude suggests that one teacher's expectation can be viewed as a proxy for factors that both teachers observe and could generate a correlation between expectations and outcomes. That both teachers' expectations remain individually significant indicates substantial within-student variation in teacher expectations (i.e., teachers frequently disagree).

It is possible that teacher disagreements are not fully random if we fail to condition on additional information. Therefore, we would expect expectations to become less predictive of outcomes once we control for factors that potentially affect both. Thus, subsequent columns of table 3 continue to add covariates to the model, which lead to a similar pattern in the estimated γj: the estimated effects of expectations decrease in magnitude but remain positive, similar in size to one another, and individually statistically significant. The largest drop in the size of the coefficient occurs when we adjust for ninth-grade GPA, which suggests that teacher expectations, in particular disagreements, might exhibit different patterns depending on a student's earlier grades.17 We return to this point when discussing threats to identification in appendix B (tables S11– S14 and figure S1). One consequence is that we control for ninth-grade GPA in our subsequent analyses.18

Our preferred model, which conditions on students' sociodemographic background, past academic performance, and school FE, is reported in column 7. These estimates suggest that conditional on the other teacher's expectation and a rich set of observed student characteristics including sex, race, household income, mother's educational attainment, ninth-grade GPA, and performance on math and ELA standardized tests, the average marginal effect of changing a teacher's expectation that a student will complete college from 0 to 1 increases the student's likelihood of earning a college degree by about 15 percentage points.

Column 8 shows that the preferred point estimates are robust to controlling for teacher FE. Specifically, this model controls for the ELA–math teacher dyad FE. That is, we compare students who had the same pair of math and ELA teachers.19 Two caveats to this analysis are of note. First, this approach can only be applied to the subsample of math–ELA teacher dyads who taught multiple students in the ELS 2002 analytic sample. To verify that the teacher-FE results are not driven by this necessary sample restriction, in column 9 we estimate the preferred school-FE specification using the restricted teacher-FE sample and see that the point estimates are similar. Second, the ELS 2002 does not provide actual teacher identifiers, so we create teacher identifiers using a probabilistic matching process, which is necessarily prone to measurement error. This procedure makes within-school matches based on teachers' race, sex, subject, educational attainment, experience, and college majors and minors. The algorithm is likely to perform well given the relatively large number of observable teacher characteristics and the fact that the sample is limited to teachers of tenth graders; still, the possibility remains that teachers with identical observable profiles are incorrectly coded as being the same teacher. For these reasons, we take the school-FE estimates in column 7 as the preferred baseline estimates, though it is reassuring that the teacher-FE estimates are remarkably similar. Finally, columns 10 and 11 show that the point estimates are similar in magnitude for white and black students, though the black sample estimates are less precise, likely due to the smaller sample size.

To interpret the preferred point estimate of 0.14 in column 7, consider that this reflects a change in expectation from 0% chance of completing a college degree to 100% chance of completing the degree. Such a drastic change in expectations is unlikely to be of policy interest and likely to be an out-of-sample change. Rather, the policy-relevant change in teachers' expectations is in the range of a 10 or 20 percentage point increase in the probability that a teacher places on a student's completing college, which corresponds with the unconditional black-white gap in expectations shown in table 1. The corresponding marginal effects of these changes on the likelihood that the student graduates from college are about 1.4 and 2.8 percentage points, respectively. From the base college completion rate of 45%, these represent modest but nontrivial increases in the graduation rate of 3.1 to 6.2%. These effect sizes are remarkably similar to those found in other evaluations of primary school inputs' impacts on post secondary outcomes. For example, Dynarski, Hyman, and Schanzenbach (2013) find that assignment to small classes in primary school increased college graduation rates by 1.6 percentage points. Similarly, Chetty et al. (2014b) find that a 1 SD increase in teacher effectiveness increases the probability that a student attends at least four years of college between the ages of 18 and 22 by about 3.2%.20 Still, even with these rich controls and conditioning on the other teacher's expectation, the threat of omitted-variables bias remains. We discuss 2SLS estimates of equation (2) in appendix B (table S14).

B. Additional Results

Other outcomes.

A possible downside of high expectations is an increase in the number of students who enroll in college but do not obtain a college degree. This could occur if expectations encourage ill-prepared students to attempt college. Given relatively low returns to “some college,” high expectations could lead to a waste of students' time and money.

To investigate this possibility, we estimate a multinomial logit model (MNL) with three mutually exclusive outcomes: high school graduation or less, college enrollment without a degree, and completion of a college degree. Average partial effects (APE) are reported in table S15 in appendix B. Similar to estimates presented in table 3, specifications include increasingly rich sets of controls as we move from the left to the right.21 Consistent with earlier results, we find that higher teacher expectations increase the probability of college completion by 13 percentage points. However, we find declines of about 6 percentage points in both the probability of completing high school and of enrolling in but failing to complete college.22 This suggests that the group of students being induced into enrolling in, but failing to complete, college is small.

We also consider the impact of teacher expectations on longer-run outcomes, including employment, marital status, and measures of financial well-being (e.g., home ownership and use of public benefits). These variables are measured twelve years after the baseline survey. Results are presented in table S16 in appendix B. For each outcome, we use the preferred specification corresponding to column 7 of table 3, which conditions on teacher controls, student SES, ninth-grade GPA, and school FE. Coefficient estimates tend to be relatively noisy, and some are only marginally statistically significant. However, they are of the expected sign and provide evidence that the positive impacts of teacher expectations on educational attainment extend to associated longer-run socioeconomic outcomes. For example, high ELA teacher expectations lead to a 5 percentage point increase in the probability of being employed (either full or part time) and a 7 percentage point drop in using public benefits. High expectations also lead to lower probabilities of being married and having children, which suggests that high expectations may lead some individuals to postpone starting a family in order to invest more in their education. Given the impact of education on life cycle outcomes, evidence provided here corroborates our result that high expectations raise educational attainment. These results also underscore concerns about low expectations, which can harm students for years to come.

Mechanisms.

Having shown that higher teacher expectations raise educational attainment—and may have additional impacts on later outcomes—we now turn to a discussion of mechanisms that could explain how. One possibility is that high expectations have no direct impact on student behavior or learning but function solely through changes in how teachers perceive students. This could affect a student's chances of successfully completing college if, for example, teachers write stellar recommendation letters for college. Alternatively, teachers with high expectations might modify how they interact with a student or how they allocate their time and effort, which could affect student learning more concretely. Yet another possibility is that teachers' expectations shift students' own expectations about their ultimate educational attainment, which can translate to shifts in their own behavior.

It is difficult to identify exact mechanisms because we do not observe teacher-student interactions. However, the ELS contains some twelth-grade outcomes that shed some light on why tenth-grade teacher expectations matter for longer-run outcomes. Specifically, we examine twelth-grade GPA, time spent on homework, and student expectations. We estimate versions of equation (2) for these twelth-grade outcomes and sometimes augment the model to control for their lagged (tenth-grade) values. It is unclear whether the lags should be included, because on the one hand, tenth-grade values might be influenced by teacher expectations and including these lagged values may create bias by “overcontrolling.”23 On the other hand, excluding the lags may exacerbate omitted variables bias if these tenth-grade characteristics create teacher disagreements. We prefer the lag-score specifications reported in even-numbered columns of table S17 in appendix B, as these estimates are more conservative and capture the growth in the intermediate outcome attributable to teacher expectations. Still, it is reassuring that the estimates with and without lags are similar.

We first examine twelth-grade GPA. Columns 1 and 2 provide evidence that teacher expectations lead to a higher GPA. For math teachers, the coefficient is 0.11, and for ELA teachers it is 0.16, where mean GPA is 3.04. This change could reflect better student performance due to changes in teacher or student effort decisions. It could also reflect easier grading (or easier classes), which could facilitate a student's path to college if a higher GPA increases the set of colleges to which a student is accepted.

Is there more direct evidence of changes in student behavior? While teacher effort and time allocations are not observed, students' time investments are. In particular, we examine how many hours students spend on homework. We find that higher teacher expectations in the tenth grade lead to increases in time spent doing homework in the twelth grade of roughly one-third to one-half of an hour. Scaling these coefficients to reflect a more reasonable 10% to 20% change in expectations suggests that a 20% change in the math teacher's expectations would lead to a rise of about 7 minutes per week spent on homework. While very modest in size, this result provides some evidence of actual changes in student behavior, which could explain higher grades and is inconsistent with the idea that GPA merely reflects easier grading or easier classes.

To examine these shifts a bit further, we conclude by asking if high teacher expectations affect students' own expectations about their future. This would suggest that teacher expectations matter in part through their impact on how students view their educational pathways and futures. We find strong evidence that high tenth-grade teacher expectations shift students' expectations upward. For example, adjusting for tenth-grade expectations, high tenth-grade teacher expectations lead to a rise of 8 to 10 percentage points in the probability that a student believes he or she will attain a college degree.

It is difficult to identify how the factors examined here interact, in part because they are likely jointly determined and mutually reinforcing. For example, if a teacher allocates more time to a student due to high expectations, a possible response is that the student puts forth more effort and thus earns a higher GPA, leading to higher expectations. Alternatively, a teacher with high expectations could grade more easily, which might lead a student to have higher expectations and thus put forth more effort. Still, the results in this section suggest that high expectations do lead to observable changes in student behaviors, performance in school and to a broader shift in students' own expectations about their future. Together, these mechanisms shed light on how teacher expectations can become self-fulfilling prophecies and buttress a causal interpretation of the main results.

IV. A Joint Model of Expectations and Outcomes

In this section, we develop and estimate a joint model of teacher expectations and student outcomes. The model formalizes the idea that teacher disagreements can be used to identify causal estimates of the impact of teacher expectations on student outcomes. The model posits an unobserved latent factor θi that maps to an objective probability, absent teacher expectations, that students complete a college degree. Teacher expectations are treated as measurements of this latent factor. Teacher bias is treated like forecast error, defined as the difference between expectations and what a student would achieve absent bias.

The model serves two key purposes. First, it provides a different approach to estimate the impact of teacher expectations on student outcomes, one that explicitly incorporates the idea that the same set of factors—summarized by θi and some of which are unobserved by the econometrician—jointly determine teacher expectations and student college degree completion. Second, by recovering θi, we can compute teacher bias or forecast error, defined as the difference between teacher expectations and θi. We can thus use the estimated model to examine the distribution of biases for different teacher-student pairs. In particular, we examine bias for different teacher and student race pairs. For example, white teachers have lower average expectations than do black teachers for the same black student. Recovering bias allows us to assess whether in such cases, black teachers are too optimistic or white teachers are too pessimistic (or both). We now introduce a simple linear version of the model to illustrate its mechanisms and interpretations.

A. Theoretical Model

Let yi be the outcome of interest and Tji, j{E,M}, be the variables measuring teacher j's expectations about student i's outcome. Let the true model of educational attainment be
yi=c+θi+bEiγE+bMiγM+εYi,
(3)
where bji=bji(Tji,θi) represents teacher j's bias for student i and is a function of teacher j's expectation and the latent factor, and εYi is a mean-zero educational achievement shock. The parameters of interest are the coefficients γj that map these biases to outcomes. Similar to Cunha, Heckman, and Schennach (2010), we assign an economic interpretation to θi. This is not a student fixed effect and should not be interpreted as a measure of student ability or skill. Rather, it is a latent variable that captures heterogeneity in the objective probability that a student observed in the tenth grade will eventually graduate from college.24 That is, c+θi gives the expected probability that student i will graduate from college in the absence of teacher biases (bEi=bMi=0). The same latent variable will be used in the production function of teacher expectations to capture how teachers observe many of the factors that determine this objective probability. θi thus includes variables observed by the econometrician (e.g., parental income), along with variables that are not observed by the econometrician but are observed by teachers and jointly affect teacher expectations and student outcomes, such as a student's ambition, motivation, or career plans. Finally, including biases in teachers' expectations in the education production function is an innovation of the current study that formally allows for self-fulfilling prophecies.
We initially assume that teacher expectation production functions are defined as follows:
TEi=cE+φEθi+εEi,
(4)
TMi=cM+φMθi+εMi.
(5)
Using the education production function, equation (3), along with the teacher expectations equations, we define bias as the difference between teacher expectations and the objective college completion probability, which we define as expected yi. Here, the expectation is conditional on teachers assuming no impact of their bias (or that their bias is equal to 0), which we can relax in a way we discuss below. Formally, bias is defined as
bji=Tji-E[Yi|θi,bEi=0,bMi=0]=Tji-c-θi=(cj-c)+(φj-1)θi+εji.
(6)

This expression shows that to generate bias, teacher expectations can deviate from the objective college completion probability in three ways. First, the mean of expectations could be systematically different, captured by the difference between cj and c. Second, teachers may have different beliefs about the role of the latent factor, captured by φj. Notice that if φj>0, the magnitude of bias rises with θi. If φj<0, it falls for students with a higher objective likelihood of college completion. Third, there is random forecast error, which we assume is independent of the disturbance term in the education production function.25

We highlight two features of the baseline linear model. First, the model as written implicitly assumes that the impact of expectations is the same as the impact of bias. To see this, substitute the definition of bias in equation (6) into equation (3) and rearrange terms. The education production function can be rewritten as
yi=(1-γE-γM)c+(1-γE-γM)θi+TEiγE+TMiγM+εYi.
(7)

That is, while replacing teacher biases with teacher expectations would change the interpretation of some of the parameters of the model, the impact of teacher expectations and of teacher bias are both governed by γj.

Second, our definition of bias assumes that teachers, when forming expectations, do not know that their expectations can directly affect student outcomes. We can relax this assumption by allowing teachers to know that the impact of their own expectations is γj and to form their expectations accordingly. This model is presented in appendix C. As long as the impact of teacher expectations is not too large, even if teachers are aware of the impact of their expectations, expectations (and bias) do not approach infinity, that is, teachers cannot generate arbitrarily high outcomes solely through high expectations.

B. Empirical Implementation

Instead of estimating the linear model described above, we make three changes to accommodate the data. First, we impose a probit functional form on the education and teacher expectation production functions to address the binary nature of the dependent variables. Second, we allow for differences by student and teacher race in the education production function parameters along with differences by student and teacher race in the production of teacher expectations. This allows racial mismatch between teachers and students to affect whether and to what degree teachers are biased. Third, we assume that the latent factor θi is normally distributed:
θiN(0,σθ2).
(8)
Analogous equations for equations (3), (4), and (6) are thus given by
Pr(yi=1)=Φ(c+θi+Giβ+bEiγE+bMiγM),
(9)
Pr(Tji=1)=Φ(cj+φjθi+Giβj+Dji×[cj,D+φj,Dθi+Giβj,D]),
(10)
bjiTji-Φ(c+θi+Giβ),
(11)

where Φ is the standard normal CDF and ninth-grade GPA (Gi) is added as another control following our findings in section III. The indicator Dji takes the value of 1 if student i faces an other-race subject-j teacher, and 0 otherwise. This captures how teacher-student racial mismatch can change how teachers form expectations for a given student with a singular objective probability of college completion. We present a more detailed discussion of the empirical model in appendix C. There, we also discuss how use of the probit specification requires means additional measurement equations for identification.

C. Estimates

Panel A of table 4 reports parameter estimates of the education production function defined by equation (9). Panel B reports estimates of the teacher expectation production functions defined by equation (10).26 Column 1 in panel A of table 4 reports parameter estimates for white students, and the estimated γj suggest that teacher expectations have positive, statistically significant effects on the probability that white students complete a four-year degree.27 The estimated β is positive and statistically significant, indicating that students with higher ninth-grade GPAs are significantly more likely to earn a four-year college degree than their counterparts with lower GPAs. This result is intuitive and provides a useful check of the model, since GPA is a known proxy for academic ability that predicts college completion (Bound & Turner, 2011).

Table 4.
Measurement Error Model Estimates
A. Education Production Function
WhitesBlacks
(1)(2)
γE 0.52*** 0.50***   
 (0.06) (0.16)   
γM 0.55*** 0.23   
 (0.06) (0.16)   
β 0.50*** 0.27**   
 (0.05) (0.11)   
c −0.46*** −0.83***   
 (0.05) (0.14)   
σθ 0.51*** 0.80***   
 (0.05) (0.14)   
APE     
bE 0.18*** 0.14***   
 (0.02) (0.05)   
bM 0.20*** 0.07   
 (0.02) (0.04)   
Elasticities     
bE 0.12*** 0.18***   
 (0.02) (0.06)   
bM 0.13*** 0.08   
 (0.02) (0.05)   
N 3,970 610   
B. Teacher Expectation Production Function 
 Whites Blacks 
 ELA Math ELA Math 
 (1) (2) (3) (4) 
c 0.58*** 0.56*** 0.47** 0.53*** 
 (0.03) (0.03) (0.19) (0.19) 
cD -0.09 0.23 -0.26 −0.53*** 
 (0.13) (0.15) (0.21) (0.2) 
φ 1.47*** 1.68*** 0.94*** 1.38** 
 (0.18) (0.2) (0.32) (0.55) 
φD -0.45 0.00 -0.21 -0.52 
 (0.45) (0.39) (0.32) (0.51) 
β 0.55*** 0.5*** 0.44** 0.14 
 (0.04) (0.04) (0.18) (0.21) 
βD 0.23 0.16 0.05 0.31 
 (0.19) (0.14) (0.2) (0.23) 
APE     
D -0.03 0.06 −0.10* −0.27*** 
 (0.04) (0.04) (0.06) (0.07) 
N 3970 610 
A. Education Production Function
WhitesBlacks
(1)(2)
γE 0.52*** 0.50***   
 (0.06) (0.16)   
γM 0.55*** 0.23   
 (0.06) (0.16)   
β 0.50*** 0.27**   
 (0.05) (0.11)   
c −0.46*** −0.83***   
 (0.05) (0.14)   
σθ 0.51*** 0.80***   
 (0.05) (0.14)   
APE     
bE 0.18*** 0.14***   
 (0.02) (0.05)   
bM 0.20*** 0.07   
 (0.02) (0.04)   
Elasticities     
bE 0.12*** 0.18***   
 (0.02) (0.06)   
bM 0.13*** 0.08   
 (0.02) (0.05)   
N 3,970 610   
B. Teacher Expectation Production Function 
 Whites Blacks 
 ELA Math ELA Math 
 (1) (2) (3) (4) 
c 0.58*** 0.56*** 0.47** 0.53*** 
 (0.03) (0.03) (0.19) (0.19) 
cD -0.09 0.23 -0.26 −0.53*** 
 (0.13) (0.15) (0.21) (0.2) 
φ 1.47*** 1.68*** 0.94*** 1.38** 
 (0.18) (0.2) (0.32) (0.55) 
φD -0.45 0.00 -0.21 -0.52 
 (0.45) (0.39) (0.32) (0.51) 
β 0.55*** 0.5*** 0.44** 0.14 
 (0.04) (0.04) (0.18) (0.21) 
βD 0.23 0.16 0.05 0.31 
 (0.19) (0.14) (0.2) (0.23) 
APE     
D -0.03 0.06 −0.10* −0.27*** 
 (0.04) (0.04) (0.06) (0.07) 
N 3970 610 

*p<0.10, **p<0.05, and ***p<0.01. Panel A reports parameter estimates of the education production function, equation (9). The dependent variable is a binary indicator equal to 1 if the student completed a four-year college degree or more, and 0 otherwise. Parameters related to ELA teacher expectations are marked with a subscript E, and parameters related to math teachers are denoted with a subscript M. Panel B reports parameter estimates of the teacher expectations production function, equation (10). Standard errors (in parentheses) are computed by constructing the Hessian of the likelihood function using the outer product measure. Standard errors for the average partial effects (APE) and elasticities are calculated using the delta method.

The magnitudes of these probit coefficients cannot be directly interpreted, so we also report the APE of teachers' expectations, the main independent variables of interest, on the likelihood of earning a four-year degree.28 The APEs indicate that for white students, on average, the impact of either teacher changing from not expecting to expecting a college degree is about a 20 percentage point increase in the likelihood of the student's completing a four-year degree. These estimates are remarkably similar to the APEs shown in table 3 of 0.13 and 0.14 for math teachers and ELA teachers, respectively. The similarity between these two approaches lends additional credence to the interpretation of these estimates as causal effects of teacher expectations on students' long-run educational attainment. These effects, moreover, translate into statistically significant elasticities of college completion with respect to biases of about 0.12 to 0.13. Column 2 of table 4 reports parameter estimates for black students, and the estimated γj once again suggest that teacher expectations have positive effects on educational attainment. However, only the ELA teacher's expectation is statistically significant at traditional confidence levels, and this coefficient is larger for ELA than for math teachers, again consistent with the OLS estimates found in table 3.29 The estimated β is again positive and statistically significant, though smaller in magnitude than for white students.

The variance and mean of θi, via the probit function, determine the objective probability (absent teacher bias and along with GPA) that a student will complete college. Consistent with realized educational outcomes, a comparison of columns 1 and 2 shows that the distribution of θi for black students is centered to the left of that for white students and exhibits greater variance. This means that upon reaching the tenth grade, black students are already disadvantaged relative to their white counterparts in terms of college completion probability. Again, this does not reflect their ability, but instead captures racial disparities in the multitude of investments over the life cycle, including factors such as school quality, neighborhood effects, and early childhood environments and resources. Our model is designed to separate the objective probability (which teachers use to form their expectations) from the impact of teacher expectations via self-fulfilling prophecies.

Panel B of table 4 reports the parameter estimates of the teacher expectation production functions. The first two columns report the parameter estimates for white students' ELA and math teachers, respectively. The production of teacher expectations for white students is broadly similar across subjects: the other-race teacher indicators are both statistically insignificant, as are their corresponding APE, which is consistent with the lack of a racial-mismatch effect on teachers' expectations for white students. Also, intuitively, teachers' expectations are increasing in both θi and ninth-grade GPA.

The results for black students, reported in columns 3 and 4, are broadly similar. However, there is one notable difference: for black students, there are significant negative effects of student-teacher racial mismatch on teachers' expectations. This is consistent with estimates reported in Gershenson et al. (2016). Specifically, pooled estimates of student-FE LPMs in Gershenson et al. (2016) find that racial mismatch reduces the probability that teachers expect a black student will complete a college degree by 0.09. However, when allowing the effect to vary by subject, the racial mismatch effect is about twice as large for math teachers (0.15) as for ELA teachers (0.07). This pattern, and the effect sizes, are remarkably similar to those reported in columns 3 and 4 in panel B of table 4. That the measurement error model estimated here produces similar evidence regarding the impact of student-teacher racial mismatch on teachers' educational expectations for black students, despite using a demonstrably different econometric approach and estimation procedure, cross-validates the measurement error model and lends additional support to the causal interpretation of the estimated impact of teacher expectations on educational attainment.

D. The Distribution of Bias by Race

Thus far, the model confirms previous results suggesting that student-teacher racial mismatch reduces teachers' educational expectations for black students. However, the results in table 4 do not speak directly to long-debated questions about whether, to what extent, and in what direction teacher expectations are biased. The model developed in section IVB, and specifically equation (11), provides answers to these questions.

Figure 2 plots kernel density estimates of the distributions of the biases in teachers' expectations separately by student race, subject, and student-teacher race congruence.30 Panel A shows the distributions of ELA teachers' biases. For both same- and other-race ELA teachers of both white and black students, the average bias is positive. In other words, teachers are overly optimistic on average, which is consistent with patterns observed in the raw ELS data documented in table 1 and figure 1b. Also, for both same- and other-race ELA teachers, the average amount of bias is similar for both white and black students. However, the average positive bias (overoptimism) is slightly larger for black students when evaluated by a black teacher. This is consistent with evidence of smaller effects of student-teacher racial mismatch on ELA teachers' expectations for black students. The similarity in means is misleading, however, as it obfuscates more pronounced differences across the distribution. Specifically, there is more mass at zero bias for blacks than for whites, as many teachers accurately predict that black students will not complete college, and this is true for both same- and other-race teachers. There is similarly more mass at one (the upper bound of bias) for blacks than whites, which is due to both same- and other-race teachers being more likely to expect black students to complete college, even when the objective probability of their doing so is nil. White students, meanwhile, are more likely than blacks to receive positive bias in the range of about 0.1 to 0.7, which means that both same- and other-race teachers are more likely to give white students the benefit of the doubt and expect a four-year degree when their objective probability of completing college is in the 30% to 90% range.
Figure 2.

Distribution of Bias by Student Race

These figures show probability distribution functions of teacher bias for different teacher and student race pairs. Vertical lines represent mean bias. Panel a shows the distribution of bias for white and black students with same- and other-race ELA teachers. Panel b shows the analogous distributions of math teacher bias. Bias is defined in equation (11).

Figure 2.

Distribution of Bias by Student Race

These figures show probability distribution functions of teacher bias for different teacher and student race pairs. Vertical lines represent mean bias. Panel a shows the distribution of bias for white and black students with same- and other-race ELA teachers. Panel b shows the analogous distributions of math teacher bias. Bias is defined in equation (11).

Panel B of figure 2 similarly plots the distributions of math teachers' biases. Many of the qualitative patterns observed in panel A for ELA teachers are present here: biases are positive on average for all students, blacks are more likely than whites to receive 0 bias, and on average, black students receive more positive bias (overoptimism) than white students when evaluated by black teachers, while the opposite is true for white teachers' expectations. However, differences in the bias distributions of same- and other-race math teachers are more pronounced than the corresponding differences for ELA teachers. This is to be expected, given the result in panel B of table 4 that the effect of racial mismatch on expectations is significantly larger for math teachers than for ELA teachers. Indeed, these mean differences are driven by a notable increase in the frequency of objectively correct (zero-bias) expectations and a flattening of the right tail of the bias distribution for other-race teachers' expectations for black students. This raises a nuanced but important point: other-race math teachers' expectations for black students may be more accurate (less biased) than those made by black math teachers. However, this accuracy can propagate racial gaps in educational attainment because high expectations, even overly optimistic ones, have a positive impact on college completion. The results in figure 2 indicate that on average, all teachers are too optimistic about students' college-completion potential, but the degree of overoptimism is greater for black students assessed by black teachers relative to white teachers.

E. Bias and Racial Attainment Gaps

We have demonstrated racial differences in the production of bias along with the impact of expectations (including biased ones) on outcomes. However, we have yet to investigate how these two mechanisms interact to contribute to the racial gap in college completion. We begin to do so here by noting that the model distinguishes three types of racial differences that can influence racial gaps in educational attainment:

  1. Initial conditions, including ninth-grade GPA and the latent factor θi, which combine to identify the objective likelihood of college completion (net of the impact of bias) at the time tenth-grade teachers form expectations

  2. The mapping between initial conditions and teacher expectations governed by the parameters in equation (10), that is, racial disparities in the teacher expectations faced by students with the same θi and Gi

  3. The production function of student outcomes governed by parameters in equation (9)

Panels a and b of figure 3 illustrates how each of these factors contributes to racial disparities.31 Panel a plots the CDF of the probability that black and white students will obtain a four-year college degree. In panel b, we simulate the black-white college completion gap under the counterfactual in which blacks are assigned the same initial conditions as whites—the same distribution of θi and of Gi. Not surprisingly, this closes much of the attainment gap, as many of the differences in the distribution of educational attainment arise from factors occurring prior to the tenth grade. Still, even with the same initial conditions, black students do not face the same distribution of college completion as white students do. This means that some of the gap can be explained by how initial conditions map to expectations along with racial differences in how expectations produce outcomes.
Figure 3.

Decompositions and Counterfactual Expectations

Panel a plots the actual CDFs of Pr(Y=1) for black and white students who have white teachers. Panel b adds two sets of distributions under the counterfactuals in which (a) black students have the same θ and G as white students, and (b) black students further face the same teacher expectation production function Panels c and d show how teacher expectations change when black students face the same expectation production function from white ELA and math teachers, respectively, as white students do. Panels e and f compare white and black ELA and math teachers' expectations, respectively, for black students with given θ.

Figure 3.

Decompositions and Counterfactual Expectations

Panel a plots the actual CDFs of Pr(Y=1) for black and white students who have white teachers. Panel b adds two sets of distributions under the counterfactuals in which (a) black students have the same θ and G as white students, and (b) black students further face the same teacher expectation production function Panels c and d show how teacher expectations change when black students face the same expectation production function from white ELA and math teachers, respectively, as white students do. Panels e and f compare white and black ELA and math teachers' expectations, respectively, for black students with given θ.

One interesting feature of panel b of figure 3 is that black students with initial conditions suggesting a low probability of college completion might do better than their white counterparts if assigned the same initial conditions. The reason is that some black students with lower initial conditions may face higher positive bias. This can be seen in figure 2, where black students are more likely to face optimistic teachers. Nonetheless, toward the upper end of the distribution, whites outperform blacks despite having the same ninth-grade GPA and the same objective probability of completing college. Again, since θi does not represent innate ability, these results suggest that two students enter the tenth grade having the same objective probabilities (net of bias) of finishing college might experience different outcomes. This discrepancy is due to racial differences in the production and impact of biases, which thus exacerbates existing gaps.

To illustrate this point, the panel also shows what happens if black and white students not only have the same initial conditions but also the same mapping from initial conditions to teacher expectations. This has a small, additional impact on the gap. For individuals with relatively low or relatively high objective probabilities of college completion, the impact of the production of teacher bias is nearly 0. In fact, some black students in the lower tails are harmed if they face the same production of bias as white students. This is because black students with low θi tend to face higher expectations from white teachers. For black students in the middle of the distribution, however, facing the same mapping from initial conditions to teacher expectations as whites is helpful. This finding is consistent with the distributions of bias plotted in figure 2, which indicate that white students who begin with objective probabilities of college completion that are neither very high nor very low are more likely to be given the benefit of the doubt than are black students. Given that expectations matter, this can raise the attainment gap through self-fulfilling prophecies.

The remainder of the gap is closed when blacks counterfactually face the same education production function as whites, governed by the parameters in equation (9). Part of the education production function difference is due to differences in γj, particularly differences in math teachers' γs across races. Another difference is in β, which may reflect disparities in school quality.32 In general, figure 3 demonstrates that most of the attainment gap between blacks and whites arises from factors that occur prior to our observing them in the tenth grade, which is not surprising and underscores the importance of interventions in early childhood and primary school education. Still, initial conditions do not account for the entire gap, which is concerning since it means that teacher expectations can widen already alarming racial gaps. This is due to racial differences in the impact of bias on outcomes but also due to differences in the production of bias.

To further explore these patterns, in panels c to f of figure 3, we plot teachers' expectations for black students as a function of θi. In these panels, the solid lines show expectations for given student-teacher race pairs based on estimated model parameters. The dashed lines show counterfactual expectations using different assumptions about how expectations are formed. Panels c and d show how white ELA and math teachers' expectations change when we impose that for a given θi, black students face the same expectations normally reported about a white student. Among ELA teachers, there is an increase in expectations at all levels of θi due to such changes. At low levels of θi white teachers have similar expectations for black students and white students. However, at high levels of θi, black students benefit more from this change. For math, there is a muted decrease in expectations due to the positive effect of racial mismatch on math teachers' expectations. This is consistent with the bias distribution presented in figure 2. Panels e and f of figure 3 show how teachers' expectations for black students change when the expectation is formed by a black—rather than white—teacher. The plots show that across the distribution of θi, black students face higher expectations when paired with a black math or ELA teacher.

V. Conclusion

Teacher expectations matter. Higher tenth-grade teacher expectations about students' educational outcomes lead to higher realized educational attainment. Our identification strategy leverages teacher disagreements. Our main reduced-form results use OLS regressions that include both teachers' expectations as right-hand-side variables. We also formalize our identification strategy using a model that treats teacher expectations as measurements, possibly with error, of a latent variable θi that captures the set of factors—some observed by teachers but not by the econometrician—that jointly generates teacher expectations and students' educational attainment. The estimated measurement error model corroborates our main OLS estimates of the impact of expectations on student outcomes. Moreover, we use the model to recover the distribution of teacher biases. Biases are defined as the difference between observed teacher expectations and θi. Measuring bias this way, we show that teachers are on average positively biased, but less so when white teachers are paired with black students, which puts black students at a disadvantage. Our findings also suggest that racial differences in expectations are often subtle and small. White students with moderate θi are more often given the benefit of the doubt by white teachers (i.e., teachers expect they will complete college) compared to black students with similar θi.

Our results open up a host of questions about policy. One conclusion is that diversifying the teaching workforce could be helpful. While this is no doubt true ceteris paribus, there is virtually no pipeline from which to draw new black teachers into the profession in the quantities needed to have a strong impact (Putman et al., 2016). Thus, it is important to consider other policies, such as interventions that moderate how white teachers form expectations and otherwise interact with their students. Here, there is some reason to be hopeful. If biases are subtle, small, and implicit (versus explicitly held negative views about certain groups rooted in racial animus), merely making teachers aware of their bias can affect how they treat students. For example, studying Italian teachers, Alesina et al. (2018) show that some implicitly biased teachers assign lower grades to immigrant students than to native students who perform equally well on blindly graded standardized tests. However, when made aware of their implicit bias, some teachers raise immigrant students' term grades. Other light touch interventions have shown promise in changing the behavior of teachers to better serve students from traditionally disadvantaged groups (Okonofua et al., 2016). If these types of findings extend to our context, which future research should examine, it would suggest that making teachers aware of expectations gaps across racial groups (in our case, the lack of optimism, which puts black students at a disadvantage) may lead to behavior changes that help to close achievement gaps.

Other extensions to our research could consider variation by gender, family income, or other factors in how teacher expectations diverge from objective probabilities. We could also use our framework for different populations or contexts. One possibility would be to examine the role of biased beliefs for students in earlier grades to assess whether they have stronger, or longer-lasting, effects. More generally, our approach could be used to assess the role of expectations in driving behavior in other contexts where individuals are tasked with making economic decisions under uncertainty. However, doing so using our framework would require data on multiple reports of expectations of a given outcome. Thus, the approach we develop here, along with our results, suggest that collecting multiple reports of subjective expectations could be used to identify causal effects of expectations on outcomes.

Notes

1

Both Jussim and Eccles (1992) and Jussim and Harber (2005) recognize how accuracy and self-fulfilling prophecies could create a correlation between expectations and outcomes.

2

Previous research has leveraged this feature to estimate the effect of student-teacher racial match on teachers' perceptions and expectations via student-fixed effects models (Dee, 2005; Gershenson, Holt, & Papageorge, 2016).

3

This approach is similar in spirit to comparing identical twins with different schooling levels to identify the returns to schooling (Ashenfelter & Krueger, 1994).

4

On negative bias and self-fulfilling prophecies, Rist (1970) provides a rather harrowing account of how subjective teacher perceptions, driven largely by social class, affected how both teachers and students behaved in the classroom. Eventually these behaviors produced student outcomes that corresponded to the teachers' initial and negative beliefs about students from lower social classes. Similarly, Loury (2009) develops an informal model in which taxi drivers incorrectly believe that black passengers are more likely than white passengers to rob them. This belief leads drivers to avoid black passengers. In response, black passengers with no criminal intent find other forms of transportation. This affects the composition of black passengers waiting for a cab so that in equilibrium, the original biases become true.

5

Fortin, Oreopoulos, and Phipps (2015) and Jacob and Wilder (2010) examine how students' expectations evolve over time and might explain demographic gaps in achievement.

6

This literature draws on the psychometric literature (see Goldberger, 1972, and Jöreskog & Goldberger, 1975), where an aim is to separate measurement error from an underlying latent factor (e.g., depression) captured imperfectly by a set of measurements.

7

As we explain when introducing the measurement error model, expected attainment absent bias is governed by a number of parameters, including the latent individual-specific term θi, returns to GPA, and mean educational attainment. The last two vary by racial group.

8

Terrier (2015) finds similar effects of bias in France.

9

Contributions to this line of work include the studies cited above along with numerous papers linking beliefs to economic behaviors such as voting (Chiang & Knight, 2011; DellaVigna & Kaplan, 2007; Gentzkow & Shapiro, 2006), risky sexual behavior (Delavande & Kohler, 2016), and financial decisions (Hudomiet, Kézdi, et al., 2011).

10

All sample sizes are rounded to the nearest ten in accordance with NCES regulations.

11

This does not mean that there are 12,130 different teachers because some students share one or both teachers. As best we can tell, the data set contains approximately 3,000 unique teachers. However, this total is likely mismeasured because the set lacks teacher identifiers. As explained below, we use a probabilistic matching procedure to determine which teacher observations come from the same person.

12

We document and discuss positive selection into the analytic sample in appendix A (tables S1--S3). Selection occurs on observables, which motivates which controls we use in our preferred specifications.

13

The exact question and categories are shown in appendix A.

14

Table S4 in appendix A summarizes the teachers represented in the analytic sample.

15

In our analysis, disagreements are used to identify the impact of expectations. Tables S6 and S7 in appendix A provide sample means for the students for whom teachers agree and disagree, respectively. Teacher expectations, educational attainment, math and ELA test scores, and ninth-grade GPA are lower for the group whose teachers' expectations disagree, though differences are more pronounced for white students. This means our estimates are more likely to be externally valid among disadvantaged whites and black students in general, the groups most at risk of not fulfilling their educational potential and for whom expectations may therefore be most salient.

16

We show robustness of estimates to the exclusion of students with very high or very low attainment and to alternative function form assumptions in appendix B (tables S8–S10).

17

This large drop does not depend on the order in which GPA and additional control variables are added, helping to alleviating concerns discussed in Gelbach (2016).

18

Ninth-grade GPA is predetermined in the sense that it is fixed before tenth-grade teachers form expectations about tenth-grade students. Moreover, it is determined prior to student-teacher classroom assignments in the tenth grade, which is important given that most sorting into classrooms is driven by past achievement (Chetty, Friedman, & Rockoff, 2014a).

19

We obtain similar point estimates if we instead condition on two-way teacher-specific FE (one for each subject's teacher) rather than on ELA–math teacher dyad FE. The difference between the teacher-FE strategies occurs when there are two math (or ELA) teachers in a given school in the ELS analytic sample.

20

Chetty et al. (2014b) do not observe college completion and instead use this as a proxy.

21

Ordered-logit models yield similar results. We omit school FE from these models to avoid the incidental parameters problem and computational issues in the MNL. We feel comfortable making this trade-off, as the results in table 3 are quite robust to adding school FE to a model that controls for these covariates.

22

In results available from us, we find that high teacher expectations also lower high school dropout rates.

23

Indeed, in results available on request from the corresponding author (N.P.), we find that teacher expectations affect tenth-grade GPA.

24

Our interpretation of θi as a factor that maps directly into the singular probability that an individual completes a four-year college degree means that it is sensible to be modeled as a singleton. If, as in Cunha et al. (2010), θi represented the skill(s) that facilitate college completion, it would make more sense to treat it as a multidimensional vector.

25

The simple linear model expressed above is identified using standard arguments from the measurement error literature (see Kotlarski, 1967).

26

Table S18 in appendix C reports “nuisance parameter” estimates from equation (S7). We also estimate a host of alternative specifications in tables S19 to S24 in appendix C.

27

We refer to effects of bias and teacher expectations interchangeably since there is a 1:1 relationship between these constructs, by definition, in equation (10).

28

Standard errors for the APE are computed via the delta method. The APE are evaluated at the mean value of θi, which is 0 by construction.

29

For black students, the math and ELA γj are not significantly different from one another.

30

Another way to illustrate these differences is using contour plots, which are presented in figure S2 in appendix C. These heat maps depict higher concentrations as brighter colors.

31

For each counterfactual simulation, this is done by drawing eGi,eEi,eMi, and θi 100,000 times using the distributional assumptions outlined in appendix C given our parameter estimates and simulating GPA, as well as ELA and math teacher expectations using equations (10) and (S7). The probability that black and white students will obtain a four-year college degree is then calculated using equation (9).

32

Indeed, if black students face white students' γj but different β, a small gap remains.

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Author notes

We gratefully acknowledge helpful comments from conference participants at the North American Meetings of the Econometric Society and the IZA Junior-Senior Labor Economics Symposium. Stephen B. Holt and Emma Kalish provided excellent research assistance. For helpful suggestions, we thank Tim Bartik, Barton Hamilton, Robert Moffitt, Robert Pollak, Yingyao Hu, Victor Ronda, Richard Spady, and seminar participants at Washington University in St. Louis, the W. E. Upjohn Institute, and Urban Institute. The usual caveats apply.

N.P. gratefully acknowledges that this research was supported in part by a grant from the American Educational Research Association (AERA). AERA receives funds for its AERA Grants Program from the National Science Foundation under NSF grant DRL-0941014. Opinions are our own and do not necessarily reflect those of the granting agencies.

A supplemental appendix is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/rest_a_00838.

Supplementary data