EC 508 Boston University Unconditional Cash Transfers Worksheet
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EXPERIMENTAL ESTIMATES OF EDUCATION
PRODUCTION FUNCTIONS*
ALAN B. KRUEGER
This paper analyzes data on 11,600 students and their teachers who were
randomly assigned to different size classes from kindergarten through third grade.
Statistical methods are used to adjust for nonrandom attrition and transitions
between classes. The main conclusions are (1) on average, performance on
standardized tests increases by four percentile points the rst year students
attend small classes; (2) the test score advantage of students in small classes
expands by about one percentile point per year in subsequent years; (3) teacher
aides and measured teacher characteristics have little effect; (4) class size has a
larger effect for minority students and those on free lunch; (5) Hawthorne effects
were unlikely.
I. INTRODUCTION
The large literature on the effect of school resources on
student achievement generally nds ambiguous, conicting, and
weak results. Even quantitative summaries of the literature tend
to reach conicting conclusions. For example, based on the fact
that most estimates of the effect of school inputs on student
achievement are statistically insignicant, Hanushek [1986] concludes, There appears to be no strong or systematic relationship
between school expenditures and student performance. By contrast, Hedges et al. [1994] conduct a meta-analysis of (a subset of )
the studies enumerated by Hanushek and conclude, the data are
more consistent with a pattern that includes at least some
positive relation between dollars spent on education and output,
than with a pattern of no effects or negative effects.
Much of the uncertainty in the literature derives from the fact
* I thank Helen Bain, a founder and principal director of Project STAR, for
providing me with the data used in this study, Jayne Zaharias, DeWayne Fulton,
and Van Cain for answering several questions regarding the data, and Jessica
Baraka, Aaron Saiger, and Diane Whitmore for providing outstanding research
assistance. The STAR data have been collected and maintained by the Center of
Excellence for Research in Basic Skills at Tennessee State University. The STAR
data are available from www.telalink.net/, heros. Helpful comments on my
research were provided by Charles Achilles, Jessica Baraka, Ronald Ehrenberg,
William Evans, Jeremy Finn, John Folger, Victor Fuchs, Joseph Hotz, Lawrence
Katz, Cecilia Rouse, James P. Smith, two referees , and seminar participants at the
Milken Institute, Massachusetts Institute of Technology, National Bureau of
Economic Research, Princeton University, Vanderbilt University, University of
California at Los Angeles, the Kennedy School (Harvard University), the London
School of Economics, Stockholm University, the Econometric Society, World Bank,
and Society of Labor Economists. Financial support was provided by the National
Institute of Childhood Health and Development.
r 1999 by the President and Fellows of Harvard College and the Massachusetts Institute of
Technology.
The Quarterly Journal of Economics, May 1999
497
498
QUARTERLY JOURNAL OF ECONOMICS
that the appropriate specicationincluding the functional form,
level of aggregation, relevant control variables, and identicationof the education production function is uncertain.1 Some
specications do consistently yield signicant effects, however.
Notably, estimates that use cross-state variation in school resources typically nd positive effects of school resources, whereas
studies that use within-state data are more likely to nd insignicant or wrong-signed estimates (see Hanushek [1996]).2 Many of
these specication issues arise because of the possibility of
omitted variables, either at the student, class, school, or state
level. Moreover, functional form issues are driven in part by
concern for omitted variables, as researchers often specify education production functions in terms of test-score changes to difference out omitted characteristics that might be correlated with
school resources (although such differencing could introduce
greater problems if the omitted characteristics affect the trajectory of student performance). A classical experiment, in which
students are randomly assigned to classes with different resources, would help overcome many of these specication issues
and provide guidance for observational studies.
This paper provides an econometric analysis of the only
large-scale randomized experiment on class size ever conducted in
the United States, the Tennessee Student/Teacher Achievement
Ratio experiment, known as Project STAR. Project STAR was a
longitudinal study in which kindergarten students and their
teachers were randomly assigned to one of three groups beginning
in the 19851986 school year: small classes (1317 students per
teacher), regular-size classes (2225 students), and regular/aide
classes (2225 students) which also included a full-time teachers
aide. After their initial assignment, the design called for students
to remain in the same class type for four years. Some 60007000
students were involved in the project each year. Over all four
years, the sample included 11,600 students from 80 schools. Each
school was required to have at least one of each class-size type,
and random assignment took place within schools. The students
1. There is also debate over what should be the appropriate measure of school
outputs (see Card and Krueger [1996]). Whereas education researchers tend to
analyze standardized test scores, economists tend to focus on students educational
attainment and subsequent earnings.
2. Hanushek attributes this difference to omitted state-level variables that
bias the state-level studies, although it is possible that endogenous resource
decisions within states (e.g., assignment of weaker students to smaller classes as
required by compensatory education) bias the within-state micro-data estimates,
and that the interstate estimates are unbiased.
EXPERIMENTAL ESTIMATES
499
were given a battery of standardized tests at the end of each
school year. In a review article Mosteller [1995] described Project
STAR as a controlled experiment which is one of the most
important educational investigations ever carried out and illustrates the kind and magnitude of research needed in the eld of
education to strengthen schools.
The STAR data have been examined extensively by an
internal team of researchers. This analysis has found that students in small classes tended to perform better than students in
larger classes, while students in classes with a teacher aide
typically did not perform differently than students in regular-size
classes without an aide (see Word et al. [1990], Finn and Achilles
[1990], and Folger and Breda [1989]). Past research primarily
consists of comparisons of means between the assignment groups,
and analysis of variance at the class level. In this research, little
attention has been paid to potential threats to the validity of the
experiment or to the longitudinal structure of the data.
As in any experiment, there were deviations from the ideal
experimental design in the actual implementation of Project
STAR. First, students in regular-size classes were randomly
assigned again between classes with and without full-time aides
at the beginning of rst grade, while students in small classes
continued on in small classes, often with the same set of classmates.3 Re-randomization was done to placate parents of children
in regular classes who complained about their childrens initial
assignment. Because analysis of data for kindergartners did not
indicate a signicant effect of a teacher aide on achievement in
regular-size classes, it was felt that this procedure would create
few problems. But if the constancy of ones classmates inuences
achievement, then the experimental comparison after kindergarten is compromised by the re-randomization.
A second limitation of the experiment is that approximately
10 percent of students switched between small and regular classes
between grades, primarily because of behavioral problems or
parental complaints. These nonrandom transitions could also
compromise the experimental results. Furthermore, because some
students and their families naturally relocate during the school
year, actual class size varied more than intended in small classes
(11 to 20) and in regular classes (15 to 30). Finally, as in most
3. If a school had more than one small class, students could be moved between
small classes.
500
QUARTERLY JOURNAL OF ECONOMICS
longitudinal studies of schooling, sample attrition was common
half of students who were present in kindergarten were missing in
at least one subsequent year. And some students may have
nonrandomly switched to another public school or enrolled in
private school upon learning their class-type assignments. These
limitations of the experiment have not been adequately addressed
in previous work.
This paper has three related goals. First, to probe the
sensitivity of the experimental estimates to aws in the experimental design. Second, to use the experiment to identify an appropriate specication of the education production function to estimate
with nonexperimental data. Third, to use the experimental results to interpret estimates from the literature based on observational data. The conclusion makes a rough attempt to compare the
benets and costs of reducing class size from 22 to 15 students.
II. BACKGROUND
ON
PROJECT STAR
AND
DATA
A. Design and Implementation
Project STAR was funded by the Tennessee legislature, at a
total cost of approximately $12 million over four years.4 The
Tennessee legislature required that the study include students in
inner-city, suburban, urban, and rural schools. The research was
designed and carried out by a team of researchers at Tennessee
State University, Memphis State University, the University of
Tennessee, and Vanderbilt University. To be eligible to participate
in the experiment, a public school was required to sign up for four
years and be large enough to accommodate at least three classes
per grade, so within each school students could be assigned to a
small class (1317), regular class (2225 students), or regular plus
a full-time aide class.5 The statewide pupil-teacher ratio in
kindergarten in 19851986 was 22.3, so students assigned to
regular classes fared about as well as the average student in the
state [Word et al. 1990]. Schools with more than 67 students per
grade had more than three classes. One limitation of the comparison between regular and regular/aide classes is that in grades 13
each regular class had the services of a part-time aide 2533
4. This section draws heavily from Word et al. [1990] and Folger [1989].
5. Participating schools had an average per-pupil expenditure in 19861987
of $2724, compared with the statewide average of $2561.
EXPERIMENTAL ESTIMATES
501
percent of the time on average, so the variability in aide services
was restricted.6
The cohort of students who entered kindergarten in the
19851986 school year participated in the experiment through
third grade. Any student who entered a participating school in a
relevant grade was added to the experiment, and participating
students who repeated a grade, skipped a grade, or left the school
exited the sample. Entering students were randomly assigned to
one of the three types of classes (small, regular, or regular/aide) in
the summer before they began kindergarten.7 Students were
typically notied of their initial class assignment very close to the
beginning of the school year. Students in regular classes and in
regular/aide classes were randomly reassigned between these two
types of classes at the end of kindergarten, while students initially
in small classes continued on in small classes. Notice, however,
that results from the kindergarten year are uncontaminated by
this feature of the experiment.
Because kindergarten attendance was not mandatory in
Tennessee at the time of the study, many new students entered
the program in rst grade. Additionally, students were added to
the sample over time because they repeated a grade or because
their families moved to a school zone that included a participating
school. In all, some 2200 new students entered the project in rst
grade and were randomly assigned to the three types of classes.
Another 1600 and 1200 new students entered the experiment in
the second and third grades, respectively. Newly entering students were randomly assigned to class types, although the uneven
availability of slots in small and regular classes often led to an
unbalanced allocation of new students across class types.
A total of 11,600 children were involved in the experiment
over all four years. After third grade, the experiment ended, and
all students were assigned to regular-size classes. Although data
have been collected on students through ninth grade, the present
study only has access to data covering grades K3. Data were
6. The reason that regular classes often had a teacher aide is that the ethic
underlying the study was that students in the control group (i.e., regular classes)
would not be prevented from receiving resources that they ordinarily would
receive.
7. The procedure for randomly assigning students was as follows. Each school
prepared an alphabetized enrollment list. Algorithms were centrally prepared
which assigned every kth student to a class type; the algorithm was tailored to the
number of enrolled students. A random starting point was used by each school to
apply the algorithm. The schools were audited to ensure that they followed
procedures for random assignment.
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QUARTERLY JOURNAL OF ECONOMICS
collected on students each fall and spring during the experiment.
Class type is based on the class attended in the fall. All students
who attended a STAR class in either the fall or spring are included
in the database.
Unfortunately, the STAR data set does not contain students
original class type assignments resulting from the randomization
procedure; only the class types that students actually were
enrolled in each year are available. It is possible that some
students were switched from their randomly assigned class to
another class before school started or early in the fall. To
determine the frequency of such switches, we obtained and
(double) entered data on the initial random assignments from the
actual enrollment sheets that were compiled in the summer prior
to the start of kindergarten for 1581 students from 18 participating STAR schools.8 It turns out that only 0.3 percent of students in
the experiment were not enrolled in the class type to which they
were randomly assigned in kindergarten. Moreover, only one
student in this sample who was assigned a regular or regular/aide
class enrolled in a small class. Consequently, in the analysis
below, we will refer to the class type in which students are
enrolled during the rst year they enter the experiment as their
initial random assignment.
A limitation of the experiment is that baseline test score
information on the students is not available, so one cannot
examine whether the treatment and control groups looked
similar on this measure before the experiment began. Nonetheless, if the students were successfully randomly assigned between
class types, one would expect those assigned to small- and
regular-size classes to look similar along other measurable dimensions at base line. Tables I and II provide some evidence on the
differences among students assigned to the three class types.
Table I disaggregates the data into waves, based upon the
grade the students entered the program, because this was the rst
time the students were randomly assigned to a class type. The
sample consists of all students who were enrolled in a STAR class
when the fall or spring data were collected. Sample means by class
type for several variables are presented. As one would expect,
students assigned to small classes had fewer students in their
class than those in regular classes, on average. There are small
8. I thank Jayne Zaharias for providing the enrollment sheets. The sample I
analyze excludes twins; schools were allowed to assign twins to the same class if
that was the schools ordinary practice.
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EXPERIMENTAL ESTIMATES
COMPARISON
OF
TABLE I
MEAN CHARACTERISTICS OF TREATMENTS
UNADJUSTED DATA
AND
CONTROLS :
A. Students who entered STAR in kindergartenb
Variable
1.
2.
3.
4.
5.
6.
Free lunchc
White/Asian
Age in 1985
Attrition rated
Class size in kindergarten
Percentile score in kindergarten
Small
Regular
Regular/Aide
Joint
P-Valuea
.47
.68
5.44
.49
15.1
54.7
.48
.67
5.43
.52
22.4
49.9
.50
.66
5.42
.53
22.8
50.0
.09
.26
.32
.02
.00
.00
B. Students who entered STAR in rst grade
1.
2.
3.
4.
5.
6.
Free lunch
White/Asian
Age in 1985
Attrition rate
Class size in rst grade
Percentile score in rst grade
.59
.62
5.78
.53
15.9
49.2
.62
.56
5.86
.51
22.7
42.6
.61
.64
5.88
.47
23.5
47.7
.52
.00
.03
.07
.00
.00
C. Students who entered STAR in second grade
1.
2.
3.
4.
5.
6.
Free lunch
White/Asian
Age in 1985
Attrition rate
Class size in third grade
Percentile score in second grade
.66
.53
5.94
.37
15.5
46.4
.63
.54
6.00
.34
23.7
45.3
.66
.44
6.03
.35
23.6
41.7
.60
.00
.66
.58
.01
.01
D. Students who entered STAR in third grade
1.
2.
3.
4.
5.
6.
Free lunch
White/Asian
Age in 1985
Attrition rate
Class size in third grade
Percentile score in third grade
.60
.66
5.95
NA
16.0
47.6
.64
.57
5.92
NA
24.1
44.2
.69
.55
5.99
NA
24.4
41.3
.04
.00
.39
NA
.01
.01
a. p-value is for F-test of equality of all three groups.
b. Sample size in panel A ranges from 6299 to 6324, in panel B ranges from 2240 to 2314, in panel C ranges
from 1585 to 1679, and in panel D ranges from 1202 to 1283.
c. Free lunch pertains to the fraction receiving a free lunch in the rst year they are observed in the
sample (i.e., in kindergarten for panel A; in rst grade in panel B; etc.) Percentile score pertains to the average
percentile score on the three Stanford Achievement Tests the students took in the rst year they are observed
in the sample.
d. Attrition rate is the fraction that ever exits the sample prior to completing third grade, even if they
return to the sample in a subsequent year. Attrition rate is unavailable in third grade.
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QUARTERLY JOURNAL OF ECONOMICS
P-VALUES
FOR
TESTS
OF
TABLE II
WITHIN-SCHOOL DIFFERENCES
AND REGULAR/AIDE CLASSES
Variable
1.
2.
3.
4.
5.
6.
Free lunch
White/Asian
Age
Attrition rate
Actual class size
Percentile score
AMONG
SMALL, REGULAR,
Grade entered STAR program
K
1
2
3
.46
.66
.38
.01
.00
.00
.29
.28
.12
.07
.00
.00
.58
.15
.48
.58
.00
.46
.18
.21
.40
NA
.00
.00
Each p-value is for an F-test of the null hypothesis that assignment to a small, regular, or regular/aide
class has no effect on the outcome variable in that grade, conditional on school of attendance.
All rows except 4 pertain to the rst grade in which the student entered the STAR program. The attrition
rate in row 4 measures whether the student ever left the sample after initially being observed.
differences in the fraction of students on free lunch, the racial mix,
and the average age of students in classes of different size,
although some of these differences are statistically signicant (see
rows 14).9 Because random assignment was only valid within
schools, these differences suggest the importance of controlling for
school effects as is done in Table II.
Table II presents p-values for joint F-tests of the differences
among small, regular, and regular/aide classes for the variables
presented in Table I. Unlike results reported in Table I, these
p-values are conditional on school effects. None of the three
background variables displays a statistically signicant association with class-type assignment at the 10 percent level, which
suggests that random assignment produced relatively similar
groups in each class size, on average. As an overall test of random
assignment, I regressed a dummy variable indicating assignment
to a small class on the three background measures in rows 13
and school dummies. For each wave, the student characteristics
had no more than a chance association with class-type assignment. Furthermore, if the same regression model is estimated for
a sample that pools all four entering waves of students together,
the three student characteristics are still insignicantly related to
assignment to a small class ( p-value 5 .58). Within schools, there
9. To be precise, the fraction on free lunch actually measures the fraction who
receive free or reduced-price lunch.
EXPERIMENTAL ESTIMATES
505
is no apparent evidence that initial assignment to class types was
correlated with student characteristics.
To check whether teacher assignment was independent of
observed teacher characteristics, I regressed each of three teacher
characteristics (experience, race, or education) on dummies indicating the class type the teachers were assigned to and school
dummies, and then performed an F-test of the hypothesis that the
class-type dummies jointly had no effect. These regressions were
calculated for each of the four grade levels, so there was a total of
twelve regressions. In each case, the p-value for the class-type
dummies exceeded .05.10 These results are as one would expect
with random assignment of teachers to the different class types.
There was a high rate of attrition from the project. Only half
the students who entered the project in kindergarten were
present for all grades K3. For the kindergarten cohort, students
in small classes were threefour percentage points more likely to
stay in the sample than those in regular-size classes. This pattern
was reversed among those who entered in rst grade, however.
Attrition could occur for several reasons, including students
moving to another school, students repeating a grade, and students being advanced a grade. Although I lack data on retention
rates for the early grades, Word et al. [1990] report that over the
four years of the project, 19.8 percent of students in small classes
were retained, while 27.4 percent of students in regular classes
were retained. This is consistent with the lower attrition rate of
students in small classes. Some of the analysis that follows makes
a crude attempt to adjust for possible nonrandom attrition.
It is virtually impossible to prescribe the exact number of
students in a class: families move in and out of a school district
during the course of a year; students become sick; and varying
numbers of students are enrolled in schools. As a result, in some
cases actual class size deviated from the intended ranges. Table
III reports the frequency distribution of class size for rst graders,
by assignment to small, regular, or regular/aide classes. Although
students assigned to small classes clearly were more likely to
attend classes with fewer students, there was considerable variability in class size within each class-type assignment, and some
overlap between the distributions.
10. In two cases the p-value was less than .10. Third grade teachers assigned
to small classes were less likely to have a masters degree or higher than were
teachers assigned regular-size classes, and rst grade teachers in small classes
had two more years of experience than those in regular-size classes (although less
experience than those in regular/aide classes).
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QUARTERLY JOURNAL OF ECONOMICS
DISTRIBUTION
OF
Actual
class size
in rst grade
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Average class size
CHILDREN
TABLE III
ACTUAL CLASS SIZES
GROUP IN FIRST GRADE
ACROSS
BY
RANDOM ASSIGNMENT
Assignment group in rst grade
Small
24
182
252
465
256
561
108
57
20
0
0
0
0
0
0
0
0
0
0
15.7
Regular
0
0
0
0
16
17
36
76
200
378
594
437
384
175
130
54
28
29
30
22.7
Aide
0
0
0
0
0
0
0
57
120
378
329
460
264
225
234
108
56
58
30
23.4
Actual class was determined by countin g the number of students in the data set with the same class
identication.
It is also virtually impossible to prevent some students from
switching between class types over time. Table IV shows a
transition matrix between class types for students who continued
from K1, 12, and 23 grades. If students remained in their same
class type over time, all the off-diagonal elements would be zero.
The re-randomization of students in regular classes in rst grade
is apparent in panel A. But in second and third grades, when
students were supposed to remain in their same type of class, 911
percent of students switched class-size types. Students were
moved between class types because of behavioral problems or, in
some cases, parental complaints. Obviously, if the movement
between class types was associated with student characteristics
(e.g., students with stronger academic backgrounds more likely to
move into small classes), these transitions would bias a simple
comparison of outcomes across class types.
507
EXPERIMENTAL ESTIMATES
TABLE IV
TRANSITIONS BETWEEN CLASS-SIZE IN ADJACENT GRADES
N UMBER OF STUDENTS IN EACH TYPE OF CLASS
A. Kindergarten to rst grade
First grade
Kindergarten
Small
Regular
Aide
All
Small
1292
126
122
1540
Regular
60
737
761
1558
Reg/aide
48
663
706
1417
All
1400
1526
1589
4515
B. First grade to second grade
Second grade
First grade
Small
Regular
Aide
All
Small
1435
152
40
1627
Regular
23
1498
115
1636
Reg/aide
24
202
1560
1786
All
1482
1852
1715
5049
C. Second grade to third grade
Third grade
Second grade
Small
Regular
Aide
All
Small
1564
167
40
1771
Regular
37
1485
76
1598
Reg/aide
35
152
1857
2044
All
1636
1804
1973
5413
To address this potential problem, and the variability of class
size for a given type of assignment, in some of the analysis that
follows initial random assignment is used as an instrumental
variable for actual class size.
B. Data and Standardized Tests
Students were tested at the end of March or beginning of
April of each year. The tests consisted of the Stanford Achievement Test (SAT), which measured achievement in reading, word
recognition, and math in grades K3, and the Tennessee Basic
Skills First (BSF) test, which measured achievement in reading
and math in grades 13. The tests were tailored to each grade
level. Because there are no natural units for the test results, I
scaled the test scores into percentile ranks. Specically, in each
grade level the regular and regular/aide students were pooled
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QUARTERLY JOURNAL OF ECONOMICS
together, and students were assigned percentile scores based on
their raw test scores, ranging from 0 (lowest score) to 100 (highest
score). A separate percentile distribution was generated for each
subject test (e.g., Math-SAT, Reading-SAT, Word-SAT, etc.). For
each test I then determined where in the distribution of the
regular-class students every student in the small classes would
fall, and the students in the small classes were assigned these
percentile scores. Finally, to summarize overall achievement, the
average of the three SAT percentile rankings was calculated.11 If
the performance of students in the small classes was distributed
in the same way as performance of students in the regular classes,
the average percentile score for students in the small classes
would be 50.
An examination of the correlations among the tests indicates
that the strongest correlations typically are between tests of the
same subject matter; for example, in second grade the SAT and
BSF reading tests have a correlation of .80. Tests of the same
subjects tend to have a higher correlation from one grade to the
next than tests of different subjects. The SAT and BSF tests are
also highly correlated with each other: the correlation between the
average SAT percentile and average BSF percentile is .79 in rst
grade and .85 in second grade. For most of the subsequent
analysis, the SAT exam is the primary focus of study because this
test has been used on a national level for a long period of time. The
main ndings are similar for the BSF test, however.
The average of the three SAT exams by class type is presented
in the last row of Table I. Figure I displays the kernel density of
the average test score distributions for students in small and
regular classes at each grade level.12 In all grades, the average
student in small classes performed better on this summary test
measure than did those in regular or regular/aide classes. There
does not seem to be a very strong or consistent effect of the teacher
aide, however. The rest of the paper probes the robustness of these
ndings.
11. Formally, denote the cumulative distribution of scores on test j (denoted
T j ) of students in the regular and regular/aide classes as F R (T j ) 5 prob [T jRi ,
T j ] 5 y j. For each student i in a small class, we then calculated F R (T jSi ) 5 yjSi.
Naturally, the distribution of y j for students in regular classes follows a uniform
distribution. We then calculated the average of the three (or two for BSF)
percentile rankings for each student. If one subtest score was missing, we took the
average of the two percentiles that were available; and if two were missing, we
used the percentile score corresponding to the only available test.
12. Note that because we have averaged over three percentile scores, the
distributions are not uniform for students assigned to regular classes.
EXPERIMENTAL ESTIMATES
509
FIGURE I
Distribution of Test Percentile Scores by Class Size and Grade
Observe also that the average test score of students in all
class types tends to be lower for those who entered the experiment
in higher grades. This pattern is likely to reect the fact that
kindergarten was optional and higher-achieving students were
more likely to attend kindergarten, as well as the tendency of
lower-achieving students to be retained and disproportionately
added to the sample at higher grade levels. Because of this feature
of the data, I control for the grade in which the student entered
Project STAR in some of the analysis below.
The Appendix presents means for several variables that are
available in the data set.
III. STATISTICAL MODELS
To see the advantage of a randomized experiment in estimating the effect of school resources on student achievement, consider
the following general model:
(1)
Yij 5
aSij 1
bFij 1
e ij,
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QUARTERLY JOURNAL OF ECONOMICS
where Yij is the achievement level of student i in school j, Sij is a
vector of school characteristics, Fij is a vector representing the
family background of the student, and e ij is a stochastic error
component. In principle, Sij and Fij include information cumulated
over the students life; for example, classroom size and teacher
qualications for each year the student attended school. The
entire history of family background variables and school resources
may contribute to students achievement in a given year. In
addition, childrens unobserved inherent ability may contribute to
their achievement. In any actual application we will generally
lack data on some relevant school, family, or student characteristics. These omitted variables will then appear in the error term. If
the omitted variables are correlated with the included variables,
then the estimated parameters will be biased.
If a school characteristic such as class size is determined by
random assignment, however, it will be independent of the
omitted variables. Thus, with random assignment, a simple
comparison of mean achievement between children in small and
large classes provides an unbiased estimate of the effect of class
size on achievement.
We begin analyzing the STAR data by estimating the following regression equation for students in each grade level:
(2)
Yics 5
b
0
1
b
1SMALLcs
1
b 2REG/Acs 1
b
3Xics
1
a
s
1
e
ics,
where Yics is the average percentile score on the SAT test of
student i in class c at school s, SMALLcs is a dummy variable
indicating whether the student was assigned to a small class that
year, REG/Acs is a dummy variable indicating whether the student
was assigned to a regular-size class with an aide that year, and Xics
is a vector of observed student and teacher covariates (e.g.,
gender). The independence between class-size assignment and
other variables is only valid within schools, because randomization was done within schools. Consequently, a separate dummy
variable is included for each school to absorb the school effects, a s.
The equation is estimated by ordinary least squares (OLS). In
calculating the standard errors, however, the error term e ics is
modeled in a components-of-va
