Class size

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© 2011 Taylor & Francis

Class size, class composition, and the distribution of student achievement

Ryan Bosworth*

Department of Applied Economics, Utah State University, Logan, Utah USA Taylor and FrancisCEDE_A_568698.sgm

(Received 29 March 2010; final version received 1 March 2011) 10.1080/09645292.2011.568698Education Economics0964-5292 (print)/1469-5782 (online)Article2011Taylor & [email protected]

Using richly detailed data on fourth- and fifth-grade students in the North Carolina public school system, I find evidence that students are assigned to classrooms in a non-random manner based on observable characteristics for a substantial portion of classrooms. Moreover, I find that this non-random assignment is statistically related to class size for a number of student characteristics and that failure to control for classroom composition can severely bias traditionally estimated class size effects. Teacher-fixed effects and classroom composition controls appear to be effective at addressing selection related to classroom composition. I find heterogeneity in class size effects by student characteristics – students who struggle in school appear to benefit more from class size reductions than students in the top of the achievement distribution. I find that smaller classes have smaller achievement gaps on average and that class size reductions may be relatively more effective at closing achievement gaps than raising average achievement; however, class size effects on both average achievement and achievement gaps are small.

Keywords: class size; education; achievement gap

JEL Classifications: I20; I21; I28

1. Introduction

Although class size reductions have been frequently suggested as a way to boost student achievement and close achievement gaps, research indicates that estimating the actual effects of class size changes can be difficult because these effects can depend on classroom composition (Boozer and Rouse 2001; Dobbelsteen et al. 2002; Lazear 2001), student type (Krueger and Whitmore 2001, 2002), student peers (Hanushek et al. 2003; Whitmore 2005), and teacher strategy (Betts and Shkolnik 1999). Research in this area indicates also that changes in class size can influence the distribution (e.g., standard deviation) of outcomes within a classroom (Bosworth and Caliendo 2007; Maasoumi, Millimet, and Rangaprasad 2005). Classroom composi- tion, in particular, has been a potential confounding factor for the researchers seeking to understand the effect of class size on student outcomes because students may be assigned to classrooms in a non-random fashion.

Although theory suggests that administrators have the incentive to sort students into classrooms in a non-random fashion (Lazear 2001), and parents may push to have their children assigned to particular teachers (Hui 2003; Lareau 1987; Sieber 1982),

*Email: [email protected]

Education Economics, 2014 Vol. 22, No. 2, 141 – 165, http://dx.doi.org/10.1080/09645292.2011.568698

142 R. Bosworth

many involved with public education object to sorting students based on ability, particularly at young ages (e.g., see, Hall, Prevatte, and Cunningham 1995). Hence, the degree to which students are non-randomly assigned to classrooms in public schools may vary substantially, and it is unclear whether or not this sorting is system- atically related to class size.1 The non-random assignment of students to classes has been frequently cited as a possible explanation for the ‘class size puzzle’ – the failure of the empirical literature to consistently find the expected relationship between class size and student achievement.

Using richly detailed data on fourth- and fifth-grade students in the North Carolina public school system, I attempt to answer three major questions regarding the produc- tion of education. First, are elementary school students systematically sorted into classes based on abilities or socio-demographic attributes in a way that is related to class size? Second, can the use of detailed student- and classroom-level data to control for the non-random assignment of students to classes improve the estimation of class size effects? Finally, how does class size influence the distribution of student achieve- ment within a classroom?

I find evidence that students are assigned to classrooms in a non-random manner for a substantial portion of classrooms. Moreover, I find that this non-random assign- ment is statistically related to class size for a number of student characteristics and that failure to control for classroom composition can severely bias traditionally estimated class size effects. I use three methods to control for the classroom composition selec- tion problem: (1) a detailed set of control variables describing classroom composition, (2) teacher-fixed effects models, and (3) sub-samples of classrooms free from direct evidence of non-random selection on observable characteristics. Using these methods, I find evidence that students who struggle in school benefit more from class size reductions than students at the top of the achievement distribution and that class size is positively related to the standard deviation of student achievement within a class- room. However, I find that the magnitude of the achievement effects related to class size I estimate is very small, suggesting that class size reduction policies are unlikely to produce substantial improvement in student achievement. Although the data I use allow for a very extensive set of controls for classroom composition, it should be noted that these estimates may still be influenced by omitted unobserved variables.

2. Related literature

The effect of class size on educational achievement has been the subject of numerous empirical studies over a very long time period. Although conventional wisdom suggests that educational achievement ought to be higher in smaller classes, this expectation has not been consistently confirmed by the extensive literature on the subject (see Lazear (2001), Akerhielm (1995), or Hoxby (2000) for summaries of the literature. Also see Urquiola (2006), Wossmann and West (2006), Burke (2003), Ding and Lehrer (2005), Krueger (2003), Becker and Powers (2001), and Lindahl (2005)). This disconnect between basic intuition and empirical results has spurred researchers to develop theoretical explanations (Bosworth and Caliendo 2007; Lazear 2001), design large-scale experiments (Krueger 1999), and undertake sophisticated econo- metric studies (see, e.g., Angrist and Lavy 1999; Bonesronning 2003, 2004; Hoxby 2000; Ma and Koenker 2006, among others).

Studies of class size effects may suffer from omitted variables’ bias if the researcher fails to control for class composition as composition is related to class size.

Education Economics 143

Dobbelsteen, Levin, and Oosterbeek (2002), for example, show that controlling for the number of pupils with similar IQ in the class can change class size estimates consid- erably – a result echoed in this study. This paper is also similar in spirit to Boozer and Rouse (2001), who control for the ‘ability’ of the class when addressing the effect of class size. The data used in this study allow for a more complete description of class- room composition than these previous studies.

Although rare, controlled experiments can provide exogenous variation in both class size and composition. Krueger (1999) reports the results obtained using data from a large-scale experiment in Tennessee and finds relatively large class size effects, although other researchers (Hanushek 1999) have criticized features of this study. Milesi and Gamoran (2006), Sims (2008), and Jepsen and Rivkin (2009) provide evidence that experimental class size results may not generalize to non- experimental settings.

The use of richly detailed data to control for classroom composition has the poten- tial to overcome the omitted variables’ problem while avoiding the potential for behavior changes due to the awareness of being observed (Hawthorne effects) in an experimental approach.

While class size and composition may, indeed, be endogenously determined prior to instruction beginning, from the perspective of teacher and student, once the school year begins, class size and composition are largely determined. Simply put, the prob- lem of endogenous determination of class size and composition can be viewed also as an omitted variables’ problem that can, in principle, be overcome with richly detailed information on class composition.

Researchers have recognized also a link between class size and achievement gaps. For example, recent theoretical work by Bosworth and Caliendo (2007) suggests that the benefits from a smaller class size should be most pronounced for groups that typically struggle in school. Substantial empirical evidence exists in support of this idea. See, for example, Krueger (1999), Maasoumi, Millimet, and Rangaprasad (2005), Krueger and Whitmore (2001), Hanushek, Kain, and Rivkin (2002), Blatchford et al. (2003), Bonesronning (2003), Cooper and Cohn (1997), and the references therein.

3. Data and empirical strategy

The data used for this project have been provided by the North Carolina Education Research Data Center (NCERDC).2 The data center was established in 2000–2001 to provide researchers with access to a large store of data at the North Carolina Depart- ment of Public Instruction and other agencies. The NCERDC is housed in the Center for Child and Family Policy at Duke University and contains district-, school-, teacher-, classroom-, and student-level information. Of particular use in the current study is information on the demographic and exceptionality characteristics of each student and classroom. The NCERDC contains information on students for race– gender categories (e.g., Hispanic female) and exceptionality categories (e.g., academ- ically gifted, learning disabled). This detailed information makes it possible to describe the composition of each classroom in detail using a wide array of variables. Coupled with End-of-Grade (EOG) test results in reading and mathematics, the data allow the estimation of relationships between test outcomes, student characteristics, classroom characteristics, and interaction effects while controlling for school and teacher effects. Table 1 provides summary statistics for key variables.3

144 R. Bosworth

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Education Economics 145

In this study, I focus on outcomes for fourth- and fifth-grade students in the North Carolina public school system. The advantage of focusing on this age group is that students are old enough that academic progress is easily measurable using standard- ized tests, but instruction generally takes place with one teacher working with the same group of students.

3.1. Tests for within-school sorting

The fact that students are not always assigned to classrooms in a random manner has been well-established by Clotfelter, Ladd, and Vigdor (2006b) and Rothstein (2010) and using the same source data employed in this study. To illustrate this, I similarly test whether or not each fourth- or fifth-grade classroom in the North Carolina public school system appears to be a random draw from the cohort of students entering each grade in a particular school and χ2 test statistics for 13 different student characteristics (about 300,000 separate tests).4 For each classroom, I can compute the number of students with and without a given characteristic and use the proportion of the incom- ing cohort in each school-grade that has the same characteristic to compute the expected number of students in each classroom with that characteristic. Intuitively, χ2 tests can ‘flag’ classrooms that are unlikely to be the result of a random draw from the incoming cohort.5 I compute these tests not only to show that systematic sorting is apparent in the data, but also for the purpose of creating subsamples of the data free from direct evidence of systematic sorting for use in the empirical analysis below. Table 2 provides a summary of the proportions of classrooms with statistical evidence of systematic sorting by various characteristics.

Table 2. Percentage of classrooms with statistically significant evidence of non-random assignment by observable characteristics (95 confidence level test).a

Fourth grade Fifth grade

2001 2002 2003 2004 2001 2002 2003 2004

Socio-demographic characteristics Black 4.7 4.2 4.3 3.8 4.4 5.1 4.8 4.7 Hispanic 4.1 3.9 4.1 4.4 3.7 4.0 4.5 4.5 Other race/ethnicity 3.9 3.4 3.3 4.1 3.8 4.4 4.2 4.3 White 12.2 12.0 11.7 12.1 11.8 11.9 12.9 11.7 Female 0.2 0.4 0.4 0.4 0.2 0.3 0.6 0.3 Free/reduced lunch 8.4 8.3 7.4 7.7 8.8 8.1 8.8 8.3 Parent high education 17.9 17.8 19.2 17.4 16.9 17.1 18.5 18.6

Exceptionality characteristics Academically gifted 8.7 8.6 8.5 8.7 9.8 9.0 9.4 9.5 Learning disabled 7.1 6.5 5.7 5.2 7.6 7.3 6.2 6.9

Previous year score Top 25% math 6.0 5.3 5.8 5.6 6.3 6.0 6.1 6.9 Top 25% reading 4.7 5.1 5.3 4.9 5.8 5.9 6.0 5.8 Bottom 25% math 5.3 5.3 4.7 5.6 5.4 4.9 6.3 6.2 Bottom 25% reading 4.7 5.2 5.3 5.4 5.8 5.6 5.4 6.3

Observations (classrooms) 4213 4305 4237 4354 4113 4281 4334 4267

aRelative to expected concentrations based on school-grade cohort.

146 R. Bosworth

Many classrooms exhibit evidence of non-random assignment. This proportion is quite large for some characteristics, especially income (as measured by eligibility for free and reduced-lunch programs), parental education level, and the exceptionality characteristics: ‘academically gifted’ and ‘learning disabled.‘6 I also find that about 12% of fourth- and fifth-grade classrooms show evidence of a non-random proportion of White students. Although non-random assignment of White students necessarily indicates non-random assignment of students of other races, I find that no more than 4.7% of classrooms exhibit statistically significant evidence of the non-random assignment of Black students, Hispanic students, or students of other racial or ethnic background. Any systematic sorting by racial characteristics appears to be related to a White/non-White pattern rather than (for example) a White/Hispanic or White/Black pattern.

The most striking patterns in Table 2 are that 16.9–19.2% of classrooms (depend- ing on year and grade) contain an unusual proportion of students with a parent who has more than a high school-level education and that there is a statistically significant evidence of non-random assignment by gender in only 0.2–0.6% of classrooms. The number of classrooms with evidence of non-random assignment by gender is so low as to suggest that teachers and/or administrators are actually ‘unsorting’ students by gender – it appears that schools actively take steps to avoid unusual proportions of males and females in classrooms. The high proportion of classrooms with an unusual proportion of students with highly educated parents is consistent with Clotfelter, Ladd, and Vigdor’s (2006b) description of ‘parent-push’ sorting.

3.2. Education production functions

Standard practice in the economics of education literature has been to estimate an ‘education production function’ that relates achievement to class size and a number of other covariates.7 Often a lack of sufficiently disaggregated data has led to research that relates average (classroom, school, or cohort) achievement to class size and a set of control variables. Using the detailed student level data provided by the NCERDC, I model student achievement as follows: Let Aijt represent the achievement level of student i in classroom j at time t:

where the vector Xijt may include, depending on the specification, various measures of student, classroom, teacher, school, or district characteristics. The vector Xijt may also include school- or teacher-fixed effects. εijt may be thought of as an iid error term. Notably, α2 is written α2i in Equation (2) indicating one of the major contributions of this study: estimating differential class size effects for different student types by allowing the parameter α2 to vary systematically with student characteristics.

Equation (2) may be specified alternatively as a first difference model (which assumes the coefficient α1 equals 1) where the dependent variable is the difference between this year’s score and the previous year’s score, or lagged achievement may be omitted entirely (which assumes α1 = 0). EOG tests are administered beginning in third grade in mathematics and reading, allowing the inclusion of the previous year’s test score as a control variable or the estimation of a first difference model. I use the lagged model specification in Equation (2) as the primary basis for analysis and conduct a sensitivity analysis using alternative specifications.

A Aijt ijt i ijt= + + + +−α α α α ε0 1 1 2 3 1class size Xijt ijt ( )

Education Economics 147

Another contribution of this study is an examination of the empirical relationship between class size and the standard deviation of classroom achievement. In Equation (3) let SDjt represent the standard deviation of achievement in classroom j at time t.

Xjt represents, as before, a vector of observed classroom (or teacher) and school char- acteristics.

4. Empirical results

This section of the paper is organized around the three key questions posed in Section 1. Section 4.1 investigates the relationship between class size and classroom compo- sition. Section 4.2 summarizes the results of models of the effect of class size and class composition on student achievement. Section 4.3 provides a sensitivity analysis of alternative specifications. Section 4.4 describes heterogeneity in class size effects. Section 5 concludes.

4.1. Class size and class composition

Although the χ2 tests results summarized in Table 2 suggest that students are some- times non-randomly assigned to classes based on certain characteristics, they cannot tell the direction of the sorting or if it is related to variation in class size. If estimates of class size effects can be explained by non-random assignment of children to class- rooms, we would expect evidence that class size is systematically related to classroom composition.

The data provided by the NCERDC allow the calculation of a comprehensive set of classroom composition variables. These variables include the percentage of the class that is female, Black, Hispanic, other (non-White) ethnicity, or on free or reduced lunch. Also included are the proportion of the classroom that has a parent with more than a high school-level education, the proportion classified as an ‘academically gifted,’ and the proportion classified as ‘learning disabled.’ Finally, I calculate the proportions of the class that scored in the top or bottom 25% of the distribution in reading or mathematics on the previous year’s EOG test. To address the idea that heterogeneity itself may influence student outcomes, I calculate two measures of classroom heterogeneity for use as control variables. These variables, labeled ‘race/ gender heterogeneity’ and ‘exceptionality heterogeneity,’ are allowed to vary from 0 to 1 where 0 indicates a perfectly homogenous classroom and 1 indicates a perfectly heterogeneous classroom.8

To investigate the relationship between class size and classroom composition, I regress classroom size on variables describing the classroom composition. I estimated these models using both school-fixed effects models and teacher-fixed effects models. Furthermore, I estimate both the school- and the teacher-fixed effects models using two different subsamples, labeled ‘Sub-Sample 1’ and ‘Sub-Sample 2.’ Sub-Sample 1 excludes all classrooms with any direct evidence of non-random assignment (i.e., if any one of the 13 separate χ2 -tests for that classroom suggests evidence of non- random assignment, that classroom is excluded from the sample). Sub-Sample 2 uses a stronger exclusion criterion: if any classroom shows evidence of non-random assign- ment, the entire school-grade is excluded. While this stronger criterion excludes a

SD class size Xjt jt jt jt= + + +β β β ε0 1 3 2( )

148 R. Bosworth

large portion of the sample, it also excludes classrooms that may not show statistically significant evidence of sorting but are clearly affected by non-random assignment given that they are assigned from the same cohort as classrooms with direct evidence of sorting.9

The school-fixed effects models include controls for teacher gender, race, and experience level. The teacher-fixed effects models include controls for school district attributes: state-level per-pupil expenditure, federal-level per-pupil expenditure, and local-level per-pupil expenditure as well as the average daily membership (students) in the district. All models contain year fixed effects. To save space the coefficients on these control variables are not reported. All regression results reported in this study (unless otherwise indicated) use White’s heteroskedasticity robust standard errors.

The results in Table 3 suggest that classroom composition is statistically signifi- cantly related to class size. In particular, I find that Black students and students on free or reduced lunch tend to be assigned to smaller classes, as are students categorized as academically gifted or learning disabled. Students with highly educated parents or who scored in the top 25% of their cohort in reading tend to be assigned to larger classes. In general, these effects are robust to different specifications and persist in models that use only Sub-Sample 1 or 2.10 The fact that these effects persist even in the sub-samples underscores the need to control for classroom composition when estimating class size effects.

The statistics presented in Tables 2 and 3 provide evidence that non-random assignment exists, and this non-random assignment is related to class size. With the exception of students labeled ‘academically gifted,’ groups that typically struggle in school are more likely to be in small classes, while students with highly educated parents are more likely to be in larger classes – a situation that would lead to tradition- ally estimated class size effects that are biased toward zero or positive values.

4.2. Class size effects – achievement

Class size is generally considered endogenous because the choices of teachers, admin- istrators, and parents that influence test scores also influence the size and composition of classes. These choices manifest themselves as variation in class size and class composition. Noting that these inputs are determined prior to instruction beginning, the problem of endogenous class size can be thought of as an omitted variables’ prob- lem. The statistical models reported in this section can provide consistent estimates of class size effects if all factors that influence learning, other than class size, can be appropriately controlled for.

One advantage of the NCERDC data is the extreme detail that allows the construc- tion of the most extensive set of classroom composition control variables of any study to date. However, even the extensive set of control variables and the fixed effects tech- niques employed here cannot guarantee that effects from unobserved student (or teacher) attributes do not influence the estimates of class size effects reported.

Table 4 presents the results of models with standardized student EOG test scores for reading and mathematics in fourth and fifth grade as the dependent variables.11 In this table, I report the results of models that estimate traditional scalar class size effects, first with school and then with teacher-fixed effects. The school-fixed effects models rely on within-school variation in class size to identify class size effects. These models can provide consistent estimates of class size effects if students are randomly assigned to classes within-school grades or if other control variables (such as class

Education Economics 149

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150 R. Bosworth

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Education Economics 151

composition controls) adequately account for all other factors that influence test scores. The teacher-fixed effects models rely on within-teacher variation in class size across years to identify class size effects. These models can provide consistent esti- mates of class size effects if the variation in a given teacher’s class size across years is due to random influences such as population variation. As with the models of class size, the school-fixed effects models include controls for teacher gender, race, and experience level, the teacher fixed effects models include controls for school district attributes, and all models contain year fixed effects. To save space the coefficients on these control variables are not reported. All regression results use White’s heteroske- dasticity robust standard errors unless otherwise indicated.12

In the first two columns in Table 4, we see that class size effects estimated with school-fixed effects and no classroom composition controls are positive and statisti- cally significant for both mathematics and reading EOG scores – the commonly encountered and counter-intuitive result known as the class size puzzle. However, the addition of a comprehensive set of composition control variables results in intuitively appealing and statistically significantly negative class size effects, suggesting that models that fail to control for classroom composition may suffer from omitted vari- ables’ bias (Boozer and Rouse 2001; Dobbelsteen Levin, and Oosterbeek 2002; Lavy and Schlosser 2007; Lavy et al. 2008; Wossmann and West 2006). I also find that class size effects estimated with school-fixed effects and classroom composition controls are extremely similar to those estimated with teacher-fixed effects without composition controls. Moreover, the addition of classroom composition controls to teacher-fixed effects models has no substantive effect on estimates of class size effects – providing suggestive evidence that classroom composition controls and teacher- fixed effects both may be effective at addressing the selection problem (Milesi and Gamoran 2006). Teacher-fixed effects models are especially appealing given evidence that student selection problems may be driven by a teacher–student matching process (Clotfelter, Ladd, and Vigdor 2006b). The fact that teacher-fixed effects models provide similar estimates to models with classroom composition controls also suggests that omitted teacher characteristics may be just as important as a source of bias as omitted classroom characteristics.

While the class size effects I estimate are statistically significant, they are very small in magnitude. For example, estimates of class size effects using a teacher- fixed effects model and classroom composition controls suggest that a one-student decrease in class size is associated with a 0.0052 standard deviation increase in mathematics tests scores. For reading, the estimate is 0.0032. These estimates are roughly 1/10th the size estimated by Krueger (1999) using data from the Tennessee STAR experiment, although Krueger’s estimates represent an upper bound of published estimates. The class size effects I estimate are consistent with those found by, for example, Wossmann and West (2006), Pong and Pallas (2001), Hoxby (2000) and Cooper and Cohn (1997), and others who find smaller (or zero) class size effects. Maasoumi, Millimet, and Rangaprasad (2005) utilize non-parametric tests for stochastic dominance and find little causal effect of class size reductions for class sizes above 20 students. Rivkin, Hanushek, and kain (2005) suggest that class size effects are small relative to teacher quality effects. The small class size effects estimated here are, of course, consistent with omitted variables’ bias from unob- served sources. I cannot rule out the possibility that these estimates are biased toward zero, even with the very extensive set of composition control variables used in these models.

152 R. Bosworth

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Education Economics 153

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154 R. Bosworth

4.3. Class size effects – sensitivity analysis

I investigate several alternative estimation procedures designed to assess the robust- ness of these estimates of class size effects.13 First, I employ a bootstrapping procedure to provide alternative estimates of the standard errors associated with class size effects that are free from assumptions about the distribution of the error term.14 Second, I esti- mate first difference models of student achievement. As noted above, these models use the difference between the student’s current year (standardized) EOG test score and the previous year’s (standardized) EOG test score as the dependent variable. First difference models, in this context, assume that the parameter α1 in Equation (1) is equal to 1. Third, I estimate models that omit the previous year’s test score from the model entirely, assuming that the parameter α1 in Equation (1) is equal to 0. Finally, I use the previously constructed sub-samples and restrict the sample to those class- rooms or school grades with no statistically significant evidence of non-random assignment. Because classrooms that sort students on characteristics observable to the researcher may also be more likely to sort students on characteristics unobservable to the researcher, we might expect estimates of class size effects from this sub-sample to differ from those in Table 4. I also estimate models that use school-by-year fixed effects rather than school-fixed effects and year fixed effects. I estimate the school- by-year fixed effects models using the both the full sample and the sub-samples. The results of these alternative estimation procedures are reported in Table 5. I find that bootstrapped standard errors are extremely close to White’s heteroskedasticity robust standard errors and, importantly, that models that include classroom composition controls or teacher-fixed effects are quite robust to alternative specifications. However, class size effects estimated with school-fixed effects and without composi- tion controls appear sensitive to specification choice. All specifications that include classroom composition controls or teacher-fixed effects provide class size estimates that are negative, statistically significant, and of similar magnitude. Interestingly, the results from the subsamples are extremely close the results from the full sample in all cases. These results suggest that the inclusion of teacher-fixed effect or classroom composition controls can reduce the influence of omitted variables’ bias in estimates of class size effects.

4.4. Class size effects – heterogeneity

Theoretical and empirical work in the economics of education suggests that teachers may respond to changes in class size by altering teaching strategies in a way that bene- fits some students more than others. For example, Betts and Shkolnik (1999) examine a sample of over 2000 classrooms and find that ‘when class size is reduced teachers shift time away from group instruction and toward individual instruction.’15

The results reported in Table 6 (full sample) and Table 7 (sub-samples) suggest that class size changes may affect different types of students differently. In addition to the classroom composition controls used previously, these models contain interaction terms between class size and 10 different dummy variables reflecting student socio- demographic characteristics: female, Black, Hispanic, other ethnicity, free/reduced lunch, parent high education, academic gifted, learning disabled, and dummy vari- ables indicating that the student was in the top or bottom 25% of the distribution in the corresponding subject the previous year. I also include intercept dummy variables for these characteristics.

Education Economics 155

The results using the full sample (Table 6) and the sub-samples (Table 7) are in general very similar. The negative and statistically significant coefficients on the class size interaction terms suggest that female students and students on free/reduced lunch may benefit from reduced class sizes more than other groups. In contrast, students with more highly educated parents or the top (bottom) 25% of the distribution from the previous year appear to benefit less from class size reductions (although these effects are not generally statistically significant in the sub-samples).

Although I do not find robust evidence that Black students do relatively better in smaller classes (as in Krueger and Whitmore 2002), and I do find that females benefit relatively more from smaller classes (in contrast to Whitmore 2005), the results of this study broadly align with theoretical and empirical work suggesting that the benefits

Table 5. Sensitivity analysis of class size effects.a

Math Reading Math Reading

Class composition controls N N Y Y

School-fixed effects: Bootstrapped standard errors 0.0009 0.0024 −0.0037 −0.0026

(3.36)*** (9.27)*** (14.37)*** (9.61)*** First difference −0.0031 −0.0023 −0.0036 −0.0025

(11.83)*** (8.01)*** (13.46)*** (8.46)*** Last-year score omitted 0.0192 0.0200 −0.0040 −0.0031

(45.28)*** (46.72)*** (9.93)*** (7.53)*** Sub-Sample 1 (classroom) 0.0002 0.0019 −0.0044 −0.0031

(0.75) (5.48)*** (13.26)*** (8.62)*** Sub-Sample 2 (school-grade) 0.0004 0.0008 −0.0031 −0.0033

(0.772) (1.611) (−6.530)*** (−6.513)***

Teacher-Fixed effects:

Bootstrapped standard errors −0.0045 −0.0025 −0.0052 −0.0032 (11.21)*** (6.16)*** (13.13)*** (8.09)***

First difference −0.0056 −0.0040 −0.0051 −0.0032 (14.60)*** (9.21)*** (13.30)*** (7.22)***

Last-Year score omitted −0.0004 0.0021 −0.0057 −0.0034 (0.61) (3.40)*** (9.36)*** (5.42)***

Sub-Sample 1 (classroom) −0.0052 −0.0028 −0.0058 −0.0033 (9.61)*** (4.57)*** (10.74)*** (5.42)***

Sub-Sample 2 (school-grade) −0.0057 −0.0035 −0.0063 −0.0040 (−6.641)*** (−3.678)*** (−7.358)*** (−4.174)***

School-by-year fixed effects Full sample 0.0012 0.0026 −0.0034 −0.0026

(4.85)*** (9.43)*** (13.13)*** (8.98)*** Sub-Sample 1 (classroom) 0.0007 0.0023 −0.0042 −0.0031

(2.19)** (6.16)*** (12.09)*** (8.17)*** Sub-Sample 2 (school-grade) 0.0013 0.0010 −0.0022 −0.0033

(2.635)*** (1.841)* (−4.447)*** (−6.083)***

*, **, and ***, statistically significant t-ratios at the 10%, 5% and 1% levels, respectively. at-ratios in parentheses.

156 R. Bosworth

from smaller class sizes should be most pronounced for students who typically strug- gle in school (Blatchford et al. 2003; Bonesronning 2003; Bosworth and Caliendo 2007; Cooper and Cohn 1997; Hanushek, Kain, and Rivkin 2002; Krueger 1999; Krueger and Whitmore 2001). The results are also broadly consistent with the idea that teachers may alter teaching strategies in a strategic response to class size changes and that class size reductions do not benefit all students equally. However, it should be emphasized that these heterogeneous class size effects, while statistically significant, are extremely small in magnitude, suggesting that the benefits of trying to tailor class size to different groups would not be worth the cost in terms of policy complexity.

If teachers care about the distribution of outcomes in addition to the class average, class size effects may be diluted if teachers choose to spend some of the extra time per student using instruction methods designed to tighten the distribution of outcomes.

Table 6. Heterogeneity in class size effects.a

School-fixed effects Teacher-fixed effects

Math Reading Math Reading

Class size −0.0041 −0.0025 −0.0056 −0.0025 (−6.786)*** (−3.790)*** (−9.152)*** (−3.549)***

Class size* 1(Female) −0.0012 −0.0010 −0.0010 −0.0008

(−2.459)** (−1.973)** (−2.465)** (−1.716)* 1(Black) −0.0002 0.0009 −0.0007 −0.0003

(−0.254) (1.284) (−1.163) (−0.443) 1(Hispanic) −0.0011 0.0007 −0.0006 0.0003

(−0.956) (0.519) (−0.615) (0.287) 1(Other ethnicity) −0.0005 −0.0003 0.0019 0.0007

(−0.377) (−0.225) (1.790)* (0.561) 1(Free/reduced lunch) −0.0018 −0.0011 −0.0009 −0.0011

(−3.042)*** (−1.735)* (−1.732)* (−1.789)* 1(Parent high education) 0.0023 0.0016 0.0017 0.0011

(4.206)*** (2.698)*** (3.346)*** (2.005)** 1(Academic gifted) 0.0015 −0.0006 0.0004 −0.0011

(1.840)* (−0.821) (0.584) (−1.501) 1(Learning disabled) 0.0005 −0.0008 0.0005 0.0000

(0.526) (−0.702) (0.591) (0.044) 1(Top 25% previous year) 0.0016 0.0004 0.0018 0.0005

(2.255)** (0.630) (2.997)*** (0.747) 1(Bottom 25% previous year) 0.0004 −0.0014 0.0008 −0.0009

(0.581) (−1.885)* (1.513) (−1.334) Observations 494,433 491,311 589,049 585,386 R-squared 0.73 0.68 0.77 0.70

*, **, and ***, statistically significant t-ratios at the 10%, 5% and 1% levels, respectively. at-ratios in parentheses. Coefficients not reported: dummy variables for student type, last-year score, class composition controls, teacher controls, District Controls, school fixed effects, teacher fixed effects, year fixed effects, grade fixed effects, constant. Dependent variable is standardized EOG scores for fourth-and fifth-grade students, years 2001–2004.

Education Economics 157

T ab

le 7

. H

et er

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158 R. Bosworth

T ab

le 7

. (C

o n ti

n u ed

).

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am p le

1 :

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sr o o m

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.

Education Economics 159

Because a shrinking class size acts like an expanding budget constraint, teachers may spend the extra time per student ‘purchasing’ a higher class average or a tighter distribution or some mix of the two. This effect, predicted by theory (Bosworth and Caliendo 2007; Brown and Saks 1987) and empirically documented (Betts and Shkolnik 1999; Gamoran 1989; Maasoumi, Millimet, and Rangaprasad 2005), suggests that when class sizes shrink the associated effect on average, achievement may be positive, zero, or even negative; however, zero or small class size effects should be accompanied by a tighter distribution of achievement.

Table 8 presents the results of models that relate the (standardized) standard devi- ation of EOG math and reading test scores to class size and other classroom charac- teristics.16 As might be expected from the results in Tables 6 and 7, these models show a statistically significantly positive relationship between the standard deviation of classroom achievement and class size for math and reading for both school- and teacher-fixed effects models. This positive relationship persists even after controlling for classroom composition and when using teacher-fixed effects.

The effects of class size on the standard deviation of achievement are small in absolute terms. However, compared to class size effects on achievement (in terms of the effect on the standardized dependent variable), the effect of class size on the standard deviation of achievement is larger than the effect of class size on average achievement. The largest estimated class size effect from Table 4 suggests that a one-student decrease in class size is associated with a 0.0052 standard deviation increase in tests scores. The smallest estimated effect of class size on the standard deviation of achievement from Table 8 suggests that a one-student decrease in class size is associated with a 0.0109 standard deviation increase in classroom standard deviation. These estimates suggest that, in a distributional sense, the effect of class size on the standard deviation of achievement is at least twice as large as the effect on achievement.

One implication of the above findings is that policies designed to reduced class sizes may be more effective at closing or reducing achievement gaps than raising aver- age achievement (cites here). However, both effects are still very small in magnitude, and changes in class size policy may be ineffective tools to use to either close achieve- ment gaps or raise average attainment.

5. Conclusion

Using detailed student- and classroom-level data on fourth- and fifth-grade students in North Carolina public schools, I find evidence of a non-trivial amount of non-random assignment of students to classes and that this non-random assignment is related to class size for some student types.

Class size effects estimated with school-fixed effects that do not control for class- room composition are (counter-intuitively) positive and statistically significant. School-fixed effects models that control for classroom composition result in negative and statistically significant estimated class size effects, suggesting omitted variables’ bias in models that fail to control for classroom composition. Models with teacher- fixed effects provide negative estimates of class size effects whether or not classroom compositions controls are included, suggesting that classroom composition and teacher attributes are related. Models that do not control for classroom composition or teacher characteristics are more sensitive to the choice of estimation procedure. I find no substantial difference between class size effects estimated using a sub-sample of

160 R. Bosworth

T ab

le 8

. C

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Education Economics 161

T ab

le 8

. (C

o n ti

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a t -r

at io

s in

p ar

en th

es es

. C o

ef fi

ci en

ts n

o t

re p

o rt

ed :

te ac

h er

c o n tr

o ls

, d is

tr ic

t co

n tr

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ea ch

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ea r

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ra d e

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o n st

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en d

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ta n

d ar

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ev ia

ti o n o

f E

O G

t es

t sc

o re

.

162 R. Bosworth

classrooms with no evidence of non-random assignment and class size effects estimated using the full sample.

I find some statistically significant evidence of heterogeneity in class size effects by student type. I find evidence that female students and low-income students benefit more than the average student from class size reductions. However, the magnitude of these estimated relative gains is extremely small.

I estimate the relationship between class size and the standard deviation of student achievement within a classroom to be positive on average, even after controlling for classroom composition. Teachers may use part of the additional time per student that results from smaller class sizes to pursue a tighter distribution of achievement. I find that class size reductions appear to both raise average attainment and help close achievement gaps and that class size reductions may be relatively more effective at closing achievement gaps than raising average attainment. However, I estimate the effect of class size on both average achievement and achievement gaps to be very small in magnitude.

Acknowledgements I am grateful for helpful comments from Roger von Haefen, Laura Taylor, Trudy Cameron, two anonymous referees, and seminar participants at North Carolina State University and Utah State University. Any errors are mine.

Notes 1. Plausible mechanisms for this sorting include (1) administrators and teachers purposefully

assigning students that struggle to smaller classes in an effort to help close achievement gaps; (2) administrators may assign the best teachers to larger classes and/or a different mix of student types in an effort to maximize the sphere of influence of these teachers (Burns and Mason 1998, 2002; Gamoran 1989); and (3) parents who highly value education may pressure administrators to have their children assigned to the best teachers (Clotfelter, Ladd, and Vigdor 2006b; Hui 2003; Lareau 1987; Sieber 1982).The first two explanations would be consistent with the incentives provided by the No Child Left Behind Act. Clot- felter, Ladd, and Vigdor (2006b) offer persuasive empirical evidence that ‘more highly qualified teachers are matched with more advantaged students.’ (See also Clotfelter, Ladd, and Vigdor 2005, 2006a). Cohen-Zada and Reuven (2008) and Urquiola and Verhoogen (2009) show how non-random assignment can arise from the institutional features of the school. See also Burns and Mason (1998, 2002).

2. NCERDC data are not available to the general public, but academic researchers can apply for access. Detailed information on the data available from the NCERDC can be found on their website: http://www.pubpol.duke.edu/centers/child/ep/nceddatacenter/index.html.

3. For this study, I restrict the analysis to classrooms with at least 15 students. Classrooms smaller than 15 students generally contain high proportions of students categorized as having one or more learning disabilities.

4. I calculate these tests for 13 different student characteristics whereas Clotfelter, Ladd, and Vigdor (2006b) calculate them for six characteristics. I have also performed this analysis using (less-familiar) G-tests. Sokal and Rohlf (1994) suggest that G-tests may perform better in small samples, although the results I obtain using G-tests are very similar to those obtained using χ2 tests. See also Clotfelter, Ladd, and Vigdor (2006b).

5. Formally, χ2 tests are computed as: where Oi is the observed

count in a classroom in category i and Ei is the count expected under the null hypothesis. 6. It is worth noting that ‘academically gifted’ and ‘learning disabled’ are qualitative catego-

ries and the standards used to judge whether or not a student belongs in these categories are not necessarily uniform across, or even within, schools.

x O E Ei i i i

2 2= −∑[( ) / ]

Education Economics 163

7. The estimation of education production functions has a very long history. See Todd and Wolpin (2003) for an in-depth discussion of specification issues. See also, for example, Brown and Saks (1975), Babcock and Betts (2009), Belfield and Fielding (2001), and Fore- man-Peck and Foreman-Peck (2006). Barrow and Rouse (2005) provide an overview of methodologies in this area.

8. The exceptionality heterogeneity index for classroom j (EHj) is calculated as follows:

The race/gender index is calculated analogously using the following groups: Black females, Black males, Hispanic females, Hispanic males, White females, White males, other race/ethnicity females, and other race/ethnicity males.

9. These sub-samples are more restrictive than those used by Clotfelter, Ladd, and Vigdor (2006b) in their study of teacher effectiveness because I test for evidence of non-random assignment using 13 student characteristics rather than six.

10. School-fixed effects models suggest that classrooms with large concentrations of students categorized as ‘other ethnicity’ are larger, but teacher-fixed effects models suggest they are smaller. One possible explanation for this result may be the existence of several schools in North Carolina with large populations of Native American students.

11. The dependent variable is standardized by subtracting the mean EOG test score for that year and grade and dividing by the standard deviation of EOG test scores for that year and grade.

12. I have also estimated these models with standard errors clustered at the classroom level. These results are similar to those reported. Full results are available upon request.

13. Only the coefficients on the class size variable and the associated t-statistics from these models are reported in Table 5. Full results are available from the author upon request.

14. Although the inclusion of the previous year’s test score as a control variable is appealing on theoretical grounds, the inclusion of a lagged dependent variable may result in serial correlation in the errors. Although ordinary least squares (OLS) parameter estimates remain unbiased in the presence of serial correlation, the standard errors of those estimates may be calculated incorrectly. This bootstrapping procedure randomly draws, with replacement, individual student observations from the original dataset. When in a sample the size of the original dataset is obtained, OLS estimates of class size effects are obtained and recorded. By repeating this procedure, the characteristics of the resulting distribution of class size esti- mates can be used to construct standard errors. Given the large amount of computation time required to process the very large dataset and large number of fixed effects I use only 100 replications. Bootstrapping relies on the assumption that the estimation sample is represen- tative of the population. This assumption is very defensible here because the estimation sample is the population of fourth- and fifth-grade students in North Carolina public schools.

15. Sims (2008) finds that schools may also respond strategically to class size reductions by changing classroom composition. Bosworth and Caliendo (2007) provide theoretical evidence.

16. The unit of observation is the classroom standard deviation. These data are standardized by subtracting the mean classroom standard deviation for that year and grade and dividing by the standard deviation of classroom standard deviations for that year and grade.

References Akerhielm, K. 1995. Does class size matter. Economics of Education Review 14, no. 3: 229–

41. Angrist, J.D., and V. Lavy. 1999. Using Maimonides’ rule to estimate the effect of class size

on scholastic achievement. Quarterly Journal of Economics 114, no. 2: 533–75. Babcock, P., and J.R. Betts. 2009. Reduced-class distinctions: Effort, ability, and the educa-

tion production function. Journal of Urban Economics 65, no. 3: 314–22. Barrow, L., and C. Rouse. 2005. Causality, causality, causality: The view of education inputs

and outputs from economics. Working Paper No. 2005-15, Federal Reserve Bank of Chicago.

Becker, W.E., and J.R. Powers. 2001. Student performance, attrition, and class size given missing student data. Economics of Education Review 20, no. 4: 377–88.

EH AcademicGifted Non Exceptional Learning Disabledj j j j= − + − +1 2 2 2[(% ) (% ) (% ) ].

164 R. Bosworth

Belfield, C.R., and A. Fielding. 2001. Measuring the relationship between resources and outcomes in higher education in the UK. Economics of Education Review 20, no. 6: 589–602.

Betts, J.R., and J.L. Shkolnik. 1999. The behavioral effects of variations in class size: The case of math teachers. Educational Evaluation and Policy Analysis 21, no. 3: 193–213.

Blatchford, P., P. Bassett, H. Goldstein, and C. Martin. 2003. Are class size differences related to pupils’ educational progress and classroom processes? Findings from the insti- tute of education class size study of children aged 5–7 years. British Educational Research Journal 29, no. 5: 709–30.

Bonesronning, H. 2003. Class size effects on student achievement in Norway: Patterns and explanations. Southern Economic Journal 69, no. 4: 952–65.

Bonesronning, H. 2004. The determinants of parental effort in education production: Do parents respond to changes in class size? Economics of Education Review 23, no. 1: 1–9.

Boozer, M., and C. Rouse. 2001. Intraschool variation in class size: Patterns and implications. Journal of Urban Economics 50, no. 1: 163–89.

Bosworth, R., and F. Caliendo. 2007. Educational production and teacher preferences. Economics of Education Review 4: 487–500.

Brown, B.W., and D.H. Saks. 1975. The production and distribution of cognitive skills within schools. Journal of Political Economy 83, no. 3: 571–93.

Brown, B.W., and D.H. Saks. 1987. The microeconomics of the allocation of teacher’s time and student learning. Economics of Education Review 6, no. 4: 319–32.

Burke, W.M. 2003. The class size debate: Is small better? British Journal of Educational Studies 51, no. 4: 428–30.

Burns, R.B., and D.A. Mason. 1998. Class formation and composition in elementary schools. American Educational Research Journal 35, no. 4: 739–72.

Burns, R.B., and D.A. Mason. 2002. Class composition and student achievement in elemen- tary schools. American Educational Research Journal 39, no. 1: 207–33.

Clotfelter, C.T., H.F. Ladd, and J. Vigdor. 2005. Who teaches whom? Race and the distribu- tion of novice teachers. Economics of Education Review 24, no. 4: 377–92.

Clotfelter, C.T., H.F. Ladd, and J. Vigdor. 2006a. The academic achievement gap in grades 3 to 8. Working Paper 12207, National Bureau of Economic Research (NBER).

Clotfelter, C.T., H.F. Ladd, and J. Vigdor. 2006b. Teacher-Student matching and the assess- ment of teacher effectiveness. Journal of Human Resources 41, no. 4: 778–820.

Cohen-Zada, D., and E. Reuven. 2008. Maximum class size rules and the regression discontinuity design: The case of secondary public schools in Israel [Mimeo], Ben-Gurion University.

Cooper, S.T., and E. Cohn. 1997. Estimation of a frontier production function for the South Carolina educational process. Economics of Education Review 16, no. 3: 313–27.

Ding, W.L., and S.F. Lehrer. 2005. Class size and student achievement: Experimental esti- mates of who benefits and who loses from reductions. Working Paper No. 1046, Queen’s Economics Department.

Dobbelsteen, S., J. Levin, and H. Oosterbeek. 2002. The causal effect of class size on scholas- tic achievement: Distinguishing the pure class size effect from the effect of changes in class composition. Oxford Bulletin of Economics and Statistics 64, no. 1: 17–38.

Foreman-Peck, J., and L. Foreman-Peck. 2006. Should schools be smaller? The size- performance relationship for Welsh schools. Economics of Education Review 25, no. 2: 157–71.

Gamoran, A. 1989. Rank, performance, and mobility in elementary school grouping. Sociological Quarterly 30, no. 1: 109–23.

Hall, D., C. Prevatte, and P.M. Cunningham. 1995. Eliminating ability grouping and reducing failure in the primary grades. In No quick fix, ed. R. Allington and S. Walmsley, 137–58. Newark, DE: International Reading Association.

Hanushek, E.A. 1999. Some findings from an independent investigation of the Tennessee STAR experiment and from other investigations of class size effects. Educational Evalua- tion and Policy Analysis 21, no. 2: 143–63.

Hanushek, E.A., J.F. Kain, J.M. Markman, and S.G. Rivkin. 2003. Does peer ability affect student achievement? Journal of Applied Econometrics 18, no. 5: 527–44.

Hanushek, E.A., J. F. Kain, and S.G. Rivkin. 2002. Inferring program effects for special popu- lations: Does special education raise achievement for students with disabilities? Review of Economics and Statistics 84, no. 4: 584–99.

Education Economics 165

Hoxby, C.M. 2000. The effects of class size on student achievement: New evidence from population variation. Quarterly Journal of Economics 115, no. 4: 1239–85.

Hui, T.K. 2003. It’s teacher shopping season: Principals gently wield veto power over parents who request popular teachers. News and Observer, July 15, p. 1.

Jepsen, C., and S. Rivkin. 2009. Class size reduction and student achievement: The potential tradeoff between teacher quality and class size. Journal of Human Resources 44, no. 1: 223–50.

Krueger, A.B. 1999. Experimental estimates of education production functions. Quarterly Journal of Economics 114, no. 2: 497–532.

Krueger, A.B. 2003. Economic considerations and class size. Economic Journal 113, no. 485: F34–63.

Krueger, A.B., and D.M. Whitmore. 2001. The effect of attending a small class in the early grades on college-test taking and middle school test results: Evidence from Project STAR. Economic Journal 111, no. 468: 1–28.

Krueger, A.B., and D.M. Whitmore. 2002. Would smaller classes help close the black-white achievement gap? In Bridging the achievement gap, ed. J. Chub and T. Loveless, 11. Washington DC: Brookings Institute Press.

Lareau, A. 1987. Social class differences in family-school relationships: The importance of cultural capital. Sociology of Education 60, no. 2: 73–85.

Lavy, V., M.D. Paserman, and A. Schlosser. 2008. Inside the black of box of ability peer effects: Evidence from variation in low achievers in the classroom. Tel-Aviv: The Foerder Institute for Economic Research/The Sackler Institute for Economic Studies.

Lavy, V., and A. Schlosser. 2007. Mechanisms and impacts of gender peer effects at school. Working Paper No. W13292, NBER.

Lazear, E.P. 2001. Educational production. Quarterly Journal of Economics 116, no. 3: 777–803.

Lindahl, M. 2005. Home versus school learning: A new approach to estimating the effect of class size on achievement. Scandinavian Journal of Economics 107, no. 2: 375–94.

Ma, L.J., and R. Koenker. 2006. Quantile regression methods for recursive structural equation models. Journal of Econometrics 134, no. 2: 471–506.

Maasoumi, E., D.L. Millimet, and V. Rangaprasad. 2005. Class size and educational policy: Who benefits from smaller classes? Econometric Reviews 24, no. 4: 333–68.

Milesi, C., and A. Gamoran. 2006. Effects of class size and instruction on kindergarten achievement. Educational Evaluation and Policy Analysis 28, no. 4: 287–313.

Pong, S.L., and A. Pallas. 2001. Class size and eighth-grade math achievement in the United States and abroad. Educational Evaluation and Policy Analysis 23, no. 3: 251–73.

Rivkin, S.G., E.A. Hanushek, and J.F. kain. 2005. Teachers, schools, and academic achieve- ment. Econometrica 73, no. 2: 417–58.

Rothstein, J. 2010. Teacher quality in educational production: Tracking, decay, and student achievement. Quarterly Journal of Economics 125, no. 1: 175–214.

Sieber, R.T. 1982. The politics of middle class success in an inner-city public school. Boston University Journal of Education 30, no. 1: 30–47.

Sims, D. 2008. A strategic response to class size reduction: Combination classes and student achievement in California. Journal of Policy Analysis and Management 27, no. 3: 457–78.

Sokal, R.R., and F.J. Rohlf. 1994. Biometry: The principles and practice of statistics in biological research. New York: Freeman.

Todd, P.E., and K. Wolpin. 2003. On the specification and estimation of the production func- tion for cognitive achievement. Economic Journal 113, no. 485: F3–33.

Urquiola, M. 2006. Identifying class size effects in developing countries: Evidence from rural Bolivia. Review of Economics and Statistics 88, no. 1: 171–7.

Urquiola, M., and E. Verhoogen. 2009. Class-size caps, sorting, and the regression- discontinuity design. American Economic Review 99, no. 1: 179–215.

Whitmore, D. 2005. Resource and peer impacts on girls’ academic achievement: Evidence from a randomized experiment. American Economic Review 95, no. 2: 199–203.

Wossmann, L., and M. West. 2006. Class-size effects in school systems around the world: Evidence from between-grade variation in TIMSS. European Economic Review 50, no. 3: 695–736.

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