ESSAYS ON THE ECONOMICS OF SPECIAL EDUCATION
By
Hassan Allen Enayati

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics - Doctor of Philosophy
2014

ABSTRACT
ESSAYS ON THE ECONOMICS OF SPECIAL EDUCATION
By
Hassan Allen Enayati
This dissertation is separated into three independent chapters. The first chapter examines the e↵ect of federal regulations at reducing racial disproportionality in special education
identification. Since the 1960s, educators and policymakers have been concerned with the
overrepresentation of Black students in special education compared to their White counterparts. In 2004, Congress reauthorized the Individuals with Disabilities Education Act
(IDEA) and included in it financial incentives for school districts to reduce their level of
racial disproportionality in special education programs. This chapter presents evidence of
the e↵ect of the penalties on the Black-White gap in special education representation in
Michigan. Using a di↵erence-in-di↵erences approach, I exploit the variation by race in the
marginal cost of an additional special education student for districts close to the policy
threshold. The results indicate that the IDEA policy change induced sanctioned districts to
lower their relative disproportionality measure by 41 percent, which was achieved by reducing
the proportion of all Black students receiving special education services.
The second chapter further investigates disparities in special education placement across
multiple dimensions. Factors leading to the placement of students into special education
programs have become a central discussion point in special education policy over the past
decade. This essay examines the roles that race and socio-economic status play on special
education identification. Using data describing Michigan school districts between 2002 and
2010, this chapter analyzes several features of special education: placement, amount of

services provided, and type of disabilities identified. I find evidence that socio-economic
status and relative race influence participation rates.
The third and final chapter explores the role of reduced class size on special education
placement. In the 1980s, Tennessee implemented the Project STAR experiment, which randomly assigned elementary-school students to di↵erent sized classrooms from kindergarten
through third grade. Using this randomization of students to classrooms, I estimate that
reduced class size net of its e↵ect on test scores increased special education participation by
2.8 percentage points. Evidence of increased special education identification supports the
hypothesis that smaller class sizes lead to increased teacher understanding of students’ needs
for services.

To Samantha.

iv

ACKNOWLEDGMENTS

I am forever indebted to my advisor, Todd Elder. As a teacher and mentor, he helped
me develop as an economist during my graduate studies at Michigan State University. His
generosity of time and thoughtful comments were profoundly important in the completion
of this dissertation. I also wish to thank Steven Haider for working tirelessly with me over
the years. His detailed comments and suggestions shaped not only this dissertation but also
my abilities as an economist. The guidance and feedback from my other advisors, Scott
Imberman and Gary Troia, were invaluable.
My time at Michigan State University granted me the opportunity to work with and learn
from faculty members and fellow graduate students. Some of the faculty that assisted in
my development as an economist include Mike Conlin, Ronald Fisher, Robert Floden, Leslie
Papke, Barbara Schneider, Gary Solon, and Je↵ Wooldridge. My friends and fellow graduate
students not only informed this project with their comments but also made the undertaking
more enjoyable. I am grateful to Otavio Bartalotti, Michael Bates, Quentin Brummet, Paul
Burkander, Steven Dieterle, Daniel Litwok, Michelle Maxfield, Gina Schmitt, Brian Stacy,
and Paul Thompson.
The successful completion of my graduate studies and this dissertation would not have
been possible without the love and support of my family. I thank my parents, Massoud
and Terri, for fostering my passion for learning and am grateful for the encouragement of
my sister, Lauren, and stepfather, Rodney. Finally, I dedicate this dissertation to my wife,
Samantha, for her patience, support, and love at every moment of this process.
The work reported here was supported in part by a Pre-Doctoral Training Grant from
the Institute of Education Sciences, U.S. Department of Education (Award # R305B090011)
v

to Michigan State University. The opinions expressed here are those of the author and do
not represent the views of the Institute or the U.S. Department of Education. All remaining
errors are my own.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

Chapter 1 The Impact of Disproportionality Regulations on Identification
into Special Education Programs . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Identification Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Di↵erence-in-Di↵erences Estimates . . . . . . . . . . . . . . . . . . .
1.5.2 Di↵erential E↵ects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.3 Impact of Regulations on Other Outcomes . . . . . . . . . . . . . . .
1.5.4 Validity of Di↵erence-in-Di↵erences Approach . . . . . . . . . . . . .
1.5.5 Did IDEA Reduce Inappropriate Disproportionality? . . . . . . . . .
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
6
9
12
12
14
17
19
22
23

Chapter 2 Disparities in Special Education Identification: Evidence from
Michigan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Special Education Identification and Services Delivered . . . . . . . .
2.3.2 Di↵erential Participation Patterns by Race . . . . . . . . . . . . . . .
2.3.3 Regression Framework . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
29
34
34
40
42
44

Chapter 3 Class Size and Special Education Placement:
nessee’s Project STAR . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Project STAR . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Identification Strategy . . . . . . . . . . . . . . . . . . .
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

46
46
49
52
54
60

vii

Revisiting Ten. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A Tables for “The Impact of Disproportionality Regulations on Identification into Special Education Programs” . . . . . . . . . . . . . . .
Appendix B Tables for “Disparities in Special Education Identification: Evidence
from Michigan” . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix C Tables for “Class Size and Special Education Placement: Revisiting
Tennessee’s Project STAR” . . . . . . . . . . . . . . . . . . . . . . .
Appendix D Figures for “The Impact of Disproportionality Regulations on Identification into Special Education Programs” . . . . . . . . . . . . . .
Appendix E Figures for “Disparities in Special Education Identification: Evidence
from Michigan” . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix F Figures for “Class Size and Special Education Placement: Revisiting
Tennessee’s Project STAR” . . . . . . . . . . . . . . . . . . . . . . .
Appendix G Appendices for “The Impact of Disproportionality Regulations on
Identification into Special Education Programs” . . . . . . . . . . .
Appendix H Appendices for “Disparities in Special Education Identification: Evidence from Michigan” . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix I Appendices for “Class Size and Special Education Placement: Revisiting Tennessee’s Project STAR” . . . . . . . . . . . . . . . . . . . .

62
63
72
74
78
81
102
104
113
118

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

viii

LIST OF TABLES

Table A.1

Characteristics of Districts By Verified Ratio . . . . . . . . . . . . .

63

Table A.2

DD E↵ect of IDEA Sanctions on Racial Disproportionality . . . . .

64

Table A.3

Di↵erential E↵ects of IDEA Sanctions on Racial Disproportionality .

65

Table A.4

DD E↵ect of IDEA Sanctions on Other Outcomes . . . . . . . . . .

66

Table A.5

Disproportionality in Free and Reduced-Price Meal Eligibility . . . .

67

Table A.6

Characteristics of Ineligible Districts By Verified Ratio . . . . . . . .

68

Table A.7

DDD E↵ect of IDEA Sanctions on Racial Disproportionality . . . .

69

Table A.8

Potential Causes of Disproportionality . . . . . . . . . . . . . . . . .

71

Table B.1

Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Table B.2

Disparities in Special Education Identification . . . . . . . . . . . .

73

Table C.1

Means of Selected Characteristics . . . . . . . . . . . . . . . . . . .

74

Table C.2

E↵ect of Small Class Size on Fourth Grade Special Education
Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Table C.3

Test Score Means by Class Size and Special Education Status . . . .

76

Table C.4

E↵ect of Small Class Size and Special Education Status on Academic
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Table G.1

Role of Control Variable Set on DD Impact of Penalties . . . . . . . 107

Table I.1

Transition Matrix of Special Education Status by Grade . . . . . . . 118

Table I.2

Characteristics of Students by Grade-Specific Special Education
Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

ix

LIST OF FIGURES

Figure D.1

Impact of Penalties on the Black Verified Ratio . . . . . . . . . . . .

78

Figure D.2

Impact of Penalties on the Black Verified Ratio for Ineligible
Districts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Figure D.3

Risk by Race Over White Composition . . . . . . . . . . . . . . . .

80

Figure E.1

Special Education Participation Over Homestead Taxable Values . .

81

Figure E.2

Disability Categories Over Homestead Taxable Values . . . . . . . .

82

Figure E.3

Special Education Participation Over Median Household Income . .

83

Figure E.4

Disability Categories Over Median Household Income . . . . . . . .

84

Figure E.5

Special Education Participation Over the Percent of Students Eligible
Free and Reduced-Price Meals . . . . . . . . . . . . . . . . . . . . .

85

Disability Categories Over the Percent of Students Eligible Free and
Reduced-Price Meals . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Figure E.7

Special Education Participation Over District Budget . . . . . . . .

87

Figure E.8

Disability Categories Over District Budget . . . . . . . . . . . . . .

88

Figure E.9

Special Education Participation Over District Urbanicity . . . . . .

89

Figure E.10 Disability Categories Over District Urbanicity . . . . . . . . . . . .

90

Figure E.11 Special Education Participation Over the Share of Eighth Graders
That Met or Exceeded Math Standards . . . . . . . . . . . . . . . .

91

Figure E.12 Disability Categories Over the Share of Eighth Graders That Met or
Exceeded Math Standards . . . . . . . . . . . . . . . . . . . . . . .

92

Figure E.13 Special Education Participation Over White Composition . . . . . .

93

Figure E.14 Disability Categories Over White Composition . . . . . . . . . . . .

94

Figure E.15 Risk of Special Education Over White Composition . . . . . . . . .

95

Figure E.6

x

Figure E.16 Risk of Special Education Over Homestead Taxable Values . . . . .

96

Figure E.17 Risk of Special Education Over Median Household Income . . . . .

97

Figure E.18 Risk of Special Education Over the Percent of Students Eligible Free
and Reduced-Price Meals . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure E.19 Risk of Special Education Over District Budget . . . . . . . . . . . .

99

Figure E.20 Risk of Special Education Over District Urbanicity . . . . . . . . . . 100
Figure E.21 Risk of Special Education Over the Share of Eighth Graders That
Met or Exceeded Math Standards . . . . . . . . . . . . . . . . . . . 101
Figure F.1

Distributions of Initial Z-Scores . . . . . . . . . . . . . . . . . . . . 102

Figure F.2

Distributions of Fourth Grade Z-Scores . . . . . . . . . . . . . . . . 103

Figure G.1

Impact of Threat on the Black Verified Ratio . . . . . . . . . . . . . 109

Figure G.2

Impact of Threat on the Black Verified Ratio for Ineligible Districts

Figure G.3

Risk by Race Over Relative Income Gap . . . . . . . . . . . . . . . 111

Figure G.4

Relative Income Gap Over White Composition . . . . . . . . . . . . 112

Figure H.1

Comparison Relationships With White Composition . . . . . . . . . 113

Figure H.2

Scatter Plot of Special Education Intensity and General Education
Participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure H.3

Special Education Participation Over Black Composition . . . . . . 115

Figure H.4

Disability Categories Over Black Composition . . . . . . . . . . . . 116

Figure H.5

Risk of Special Education Over Black Composition . . . . . . . . . . 117

xi

110

Chapter 1
The Impact of Disproportionality Regulations
on Identification into Special Education Programs

1.1

Introduction

Despite considerable attention from educators and policymakers, minority children consistently represent a disproportionate share of students in special education programs. The
relative di↵erence in identification rates, i.e., special education placement rates, between
Black and White students has been at the center of the discussion since the 1960s (Albrecht
et al., 2012; Donovan and Cross, 2002; Dunn, 1968; Patton, 1998). In a 2008 paper, racial
disproportionality in special education was described as “among the most longstanding and
intransigent issues” in education (Skiba et al., 2008). It is commonly argued that these
di↵erences in placement rates stem from the inappropriate identification of students, which
could harm their educational outcomes (Hosp and Reschly, 2003).
Congress reauthorized the Individuals with Disabilities Education Act (IDEA) in 2004,
including in it components designed to reduce disproportionality that were in accordance
with the belief that inappropriate placement drives representation di↵erences. Congress
emphasized the priority it gave to reducing racial over/underrepresentation by making it
one of three monitored target areas of the act. Specifically, the policy required states to
collect and analyze enrollment data to determine racial representation levels. If a school
district’s level of disproportionality was found to be excessive and reoccurring, the district

1

was labeled as having “significant disproportionality” and mandated to reserve fifteen percent
of its federal IDEA funds, originally intended for special education programs, for Early
Intervention Services targeted toward students at higher risk of receiving special education
services.1
This paper evaluates the e↵ect of financial sanctions associated with the IDEA 2004
on Black overrepresentation in Michigan special education programs. I use a di↵erence-indi↵erences approach at the district level to exploit the variation in Black disproportionate
representation before and after the policy. Additional analysis examines whether the change
in relative disproportionality was driven by reductions in Black representation or increases
in White representation. Finally, I compare districts evaluated by the policy to districts
ineligible to participate.
I find that the financial sanctions were successful in reducing Black overrepresentation.
A significant disproportionality label lowered relative Black representation by 41 percent in
following years. Moreover, this decline in Black overrepresentation was achieved through
a reduction in the proportion of Black students receiving special education services. The
examination of analogous changes for ineligible districts, a third di↵erencing group, serves
as a robustness check of the main estimates and supports the common trends assumption
required for the unbiasedness of the estimates. I then show evidence consistent with inappropriate identification practices: the racial composition of one’s peers has a significant
relationship on one’s special education status. These results suggest that the reductions
in disproportionality induced by the policy led to positive well-being e↵ects for Michigan
students in special education programs.2
1 The term school district replaces the term local education agency (LEA), which is the language used
in the IDEA. In Michigan, an LEA refers to traditional school districts, charter schools, state-operated
education facilities, and intermediate school districts.
2 This conclusion focuses on the e↵ect of the policy on special education students but does not speak to

2

1.2

Background

If one were to assume that all children, regardless of race, are equally likely to have a condition requiring special education services, it is clear why the stark levels of disproportionate
representation have been a focal point for researchers and policymakers.3 Despite the fact
that the disparity was identified in the 1960s, the overrepresentation of Black students has
persisted in special education programs. In 1975, Black students represented 15 percent of
the student body but 38 percent of all students labeled as cognitively impaired. In 1991, 16
percent of all students were Black while 35 percent of those identified as cognitively impaired
were Black (Patton, 1998). By 2004, Black students represent 17 percent of the total student
population and 33 percent of students with a cognitive impairment placement.4
Within the last decade, several studies have expanded the understanding of disproportionality. Evidence indicates disproportionality begins prior to formal schooling and is exhibited in lower skill levels as well as behaviors less conducive to the traditional school setting
(Farkas, 2003; Morgan et al., 2012; Morrier and Gallagher, 2010). Even after controlling
for economic and demographic variables, Black students are disproportionately likely to be
labeled as cognitively impaired (Oswald et al., 1999). Finally, prior academic achievement
strongly predicts special education identification and also attenuates the socio-economic estimates (Hibel et al., 2010; Hosp and Reschly, 2004).
the e↵ect on general education students.
3 In Michigan, a student with a disability is a student, within certain age limits, who was designated by a
team of professionals as having at least one impairment that necessitates special education services. After the
district has identified a student as being a candidate for services, the student is given several assessments, so
an individual education program team can discuss the findings and determine whether the student is eligible
for services. If the student’s parent or guardian accepts the o↵er for services, the student may then begin
receiving special education services. The declassification out of special education follows a similar structure
of recommendation, evaluation, and ending with the determination of the multidisciplinary team.
4 The 2004 numbers were based on the author’s calculations using the National Center for Education
Statistics’ Common Core of Data and the Data Accountability Center’s IDEA Data.

3

The appropriateness of racial disproportionality in special education has been discussed
through two channels: inappropriate and appropriate causes. The arguments that point to
inappropriate sources highlight the subjectivity in the identification process, inherent bias in
the tests, discrimination, and socio-cultural di↵erences between teachers and students (Dunn,
1968; Patton, 1998). Alternatively, disproportionality may be due to appropriate reasons.
Donovan and Cross (2002) argue that certain minority groups are more likely to be exposed
to poverty and environmental toxins, such as lead paint, which in turn increase the need
for special education services. Hence, racial disproportionality in poverty could cause racial
disproportionality in special education programs. Under the premise that disproportionality
in poverty is causing disproportionality in special education, the overrepresentation in special
education of the racial group most subjected to poverty and poor living conditions would be
based on the students’ need for additional services and would thus be an appropriate form
of disproportionality in special education.
If participation in special education programs created strictly positive e↵ects, then the
overrepresentation of Black students seen since the 1960s would not be a concern. However,
Hosp and Reschly (2003) summarize two avenues that would lead to deleterious consequences
of special education programs. First, the actual label of special education may lower the
teacher’s expectations for the student. Second, the physical act of separating the special
education students from the general education classroom could lead to lower levels of access
to the curriculum. Therefore, if disproportionality was due to inappropriate identification
practices, students incorrectly placed into special education programs would be unnecessarily
exposed to these negative impacts.
This paper studies the e↵ect of the financial regulations associated with the 2004 reauthorization of the IDEA in Michigan. The IDEA was originally passed in 1975 as an extension
4

of the Rehabilitation Act of 1973 with the goal of ensuring a free and appropriate public education to all students. The reauthorization passed in 2004, became e↵ective during
the 2005-06 academic year, and began sanctioning in Michigan during the Fall of 2007. It
marked a clear change towards outcomes-based enforcement similar to No Child Left Behind
(Ramanathan, 2008). In an attempt to lower disproportionality, Congress created financial
penalties for districts found to have significant disproportionality, i.e., repeated instances of
high levels of representational disparities.5 The consequence for a significant disproportionality label is the reservation of 15 percent of federal IDEA flow-through funds for one year.
In total, the Federal share of special education revenue represents 15 percent of all special
education revenue since 2007.6 These reserved funds must be used for Early Intervention
Services targeted at the overrepresented racial group.7
In addition to the reallocation of the IDEA funds, districts with significant disproportionality face additional incentives to comply with the reduction of disproportionality. Penalized
districts bare the time and resource cost of a state audit into the district’s identification
policies and practices. The IDEA reauthorization also required public reporting of the significant disproportionality label and the findings of the state audit, which could lead to
public relations issues and pressure from the community.
In Michigan, significant disproportionality labels are determined by eligibility conditions
and relative representation measures. For a district to be eligible for representation calculations within a particular racial group, it must include at least 10 members of the racial
subgroup in the special education program, 30 or more students aggregated across all races
5 While Congress outlined the structure for the sanctions, it gave states discretion over the formula, upper
threshold, and repetition pattern.
6 The estimate of the federal share of total special education revenue was based on the author’s calculations
using the baseline analytic sample of Michigan districts.
7 Early Intervention Services are designed to provide young students on the margin of a special education
label with additional support in an attempt to prevent the need for more extensive special education services.

5

in the special education program, and more than 100 total students not of the targeted race.
Once eligibility is confirmed for a particular race, the state calculates what is called the
verified ratio. The verified ratio can take on six di↵erent forms depending on the size of the
district and its racial subgroups.8 One common component across the formulae is the risk
of special education, e.g., the Black risk of special education measures the number of Black
students in special education divided by the total number of Black students.9 The simplest
form of the verified ratio divides the risk of one racial group by the risk of all other racial
groups. Thus, a Black verified ratio of 2.7 means a Black student is 2.7 times more likely
to receive special education services than a non-Black student. Finally, a district obtains a
significant disproportionality label if its verified ratio for a particular race exceeds 3.0 for
two consecutive years.

1.3

Data

While previous work studied the existence and causes of disproportionate representation,
this paper is the first to quantitatively analyze the impact of significant disproportionality
regulations under the IDEA 2004 on special education identification rates. To conduct this
analysis, I use three data sources: the Michigan Compliance Information System (MI-CIS)
Special Education Operating LEA Summary Report, the MI-CIS Special Education StateISD Summary Report, and the Michigan Center for Performance and Information (CEPI)
data reports. The MI-CIS data provide special education head counts by race while the CEPI
data contain information on total enrollment by race and percent of students receiving free
8 The six forms are designed to create comparability across the district and subgroup size distributions.
The appendix provides a detailed discussion of the verified ratio calculation procedures.
9 The Black risk is defined as the number of Black special education students divided by the total number
Number of Black Sp Ed Students
of Black students enrolled in the district times 100: Black Risk =
⇤ 100.
Total Number of Black Students

6

and reduced-price meals. The full panel data set covers 9,112 district-year observations for
851 traditional school districts and charter schools between the Fall terms of 2001 through
2011.10,11
The LEA Reports provide information regarding the district in which a student received
her special education services. Three of the formulae for the final verified ratio used by the
state employ this district of service data while the remaining three use district of residence
data as a way to account for school of choice induced mobility. Because the state uses the
lower of the verified ratios calculated with either the district of service or residence data,
the verified ratios used in this paper are larger than or equal to those used by the state.
As a result, my estimation strategy will treat some districts as a↵ected by the policy when
they were in fact not under pressure from the regulations. Assuming inaccurately labeled
districts do not respond to the policy more than the districts actually facing the sanctions,
the estimates of the e↵ect of the IDEA 2004 will be attenuated toward zero.
To analyze the e↵ect of disproportionality regulations, I restrict the full sample to districtyear observations with eligible verified ratios, at least three consecutive years of prior eligibility, and non-missing lagged sanctions variables. The analytic sample includes 1,249
district-year observations for 183 districts from 2004 to 2011.12
10 For simplicity, I will continue to refer to both traditional school districts and charter schools as school
districts; however, the baseline analytic sample includes only traditional school districts.
11 To protect the identity of individual students, the LEA Reports censor cells with low numbers of students.
Prior to 2011, cells with less than 6 students were reported as missing while the cuto↵ was reduced to 5
students in 2011. This type of censoring does not a↵ect the main results as districts need at least 10 students
of a particular race in the special education program alone to qualify for disproportionality calculations. One
of the validity checks of the baseline results extends the analysis to districts not eligible for calculations due
to low student counts. In this case, I am only able to determine the special education head counts by race
if the district had at least 6 total students of that race, which leaves 380 district-year observations in my
sample period with unaccounted total special education enrollment out of the analysis. Note that I am able
to determine the special education head count by race for districts with at least 6 total students of that race
by using within race risk estimates in the LEA Reports through 2010.
12 Charter schools are excluded from the base sample. When they are included in later analysis, 36 charter
schools add 150 district-year observations. Of the 6,098 non-charter school district-year observations, 68
percent were dropped from the analytic sample because they failed to meet the disproportionality eligibility

7

Table A.1 presents district characteristics broken up by policy period and disproportionality status, i.e., a Black verified ratio above 3. Columns 2 and 4 show that Black
overrepresentation existed in Michigan during the sample period. Prior to the 2007 policy
change, a Black student was approximately 1.3 times as likely to receive a special education
label compared to her non-Black counterpart when averaging across the state. Disproportionate districts during that same time identified Black students as needing special education
services 4.3 times more often than their non-Black counterparts. Looking across the columns,
the overall risk of special education remained relatively constant at 14 percent, which means
that disproportionately Black districts do not also tend to have higher identification rates
compared to the state as a whole. Separating the two main components of the Black verified
ratio, Black and White risk, reveals that the overrepresentation is due to drastically larger
shares of the total Black population in special education but not significantly lower shares
of White students. Prior to the policy, the Black risk of 55.41 percent is 2.96 times larger
than the average Black risk of 18.75 percent.
Comparing the total student populations across these groups shows that disproportionate
districts had between 20 to 30 percent fewer students than the state average. Additionally,
disproportionate districts have nearly 50 percent lower rates of free and reduced-price meal
eligibility. The Black share of the total student population in overrepresented districts was
roughly a tenth of the state average over the entire sample while the White share was nearly
a third larger than the state average.13 The stark di↵erences in the White composition across
district types suggests that students are more likely to be placed into a special education
program the more dissimilar they appear to their peers, a relationship further investigated
requirements, i.e., they were not subject to the policy in that year.
13 White (Black) composition is calculated as the total number of White (Black) students in a district
divided by the total student population.

8

in Section 1.5.5.
The final three rows of Table A.1 attempt to describe the amount of services delivered to
the average special education student.14 The total special education expenditures and special
education revenue from Federal sources, which is included in the total expenditure value, are
higher in disproportionate districts.15 Additionally, special education student-teacher ratios
are 30 percent lower for districts with overrepresentation.

1.4

Identification Strategy

To examine the e↵ects of the IDEA, I use a di↵erence-in-di↵erences approach, demonstrated
in Figure D.1. The figure plots policy period means of the Black verified ratio for two
types of districts: ones in penalty phases and ones in neither threat nor penalty phases. A
district is in a penalty (threat) phase when it has two (one) consecutive years with a Black
verified ratio above 3. Provided reducing disproportionality is within the scope of districts’
abilities, financial sanctions associated with high levels of overrepresentation should cause
districts to lower their levels of disproportionality. Figure D.1 shows that districts in a
penalty phase prior to the existence of the disproportionality sanctions had average verified
ratios of 5.57 while districts in a penalty phase during the policy period had lower mean
Black verified ratios of 3.47: a statistically significant reduction of 38 percent. Making the
same comparison for districts in neither threat nor penalty phases, we see an insignificant
increase in the verified ratio. The crude di↵erence-in-di↵erences estimate of the e↵ect of
the significant disproportionality sanctions on the Black verified ratio is a reduction by 2.16
14 These summarizations of the special education services delivered do not account for variation in student
need across districts.
15 The expenditure and revenue variables are per special education pupil and expressed as 2000 inflationadjusted dollars.

9

mean points. The significant reduction in disproportionality for overrepresented districts and
insignificant increase for non-overrepresented districts suggests that the financial regulations
of the IDEA 2004 were successful at lowering the disproportionality levels in the targeted
districts.
A regression-based di↵erence-in-di↵erences approach o↵ers the advantage of estimating
the e↵ect of the policy while controlling for district observable characteristics and systematic
di↵erences between the treatment and control groups. The parameters of the policy determined the classification of the di↵erencing groups. Because a significant disproportionality
label requires two consecutive years of overrepresentation, the treatment group consists of
districts with cases of disproportionality within the last two years. Whereas, the control
group does not have a history of disproportionality within the past two years. Given that
it takes two consecutive years of excessive overrepresentation to be penalized, it is possible
that districts respond not only to the penalties themselves but also to the threat of penalty,
i.e., exceeding the threshold for only one year. Thus, the empirical method must account for
the di↵erential e↵ects of both threat and also penalty phases across the policy periods.
Looking only at districts eligible for calculations by the state, I compare the verified
ratios of districts in either threat or penalty phases before and after the policy to districts
not in any danger of penalty before and after the policy. I implement this approach using
the following pooled OLS regression:

Yrdt = 1 + Zdt 2 + 3 IDEAt + 4 T hreatrdt + 5 (IDEAt ⇤ T hreatrdt )

(1.1)

+ 6 P enaltyrd,t 1 + 7 P enaltyrdt + 8 (IDEAt ⇤ P enaltyrdt ) + ✏rdt .
Yrdt represents a relative measure of the probability of receiving special education services

10

for race r in district d at time t. The outcome variables, Y , used in this study are the Black
verified ratio, Black risk, and White risk. The vector Zdt contains a cubic in years since 2000
and control variables such as percent of students eligible for free and reduced-price meals
and a quadratic of the White composition. IDEAt equals one for the post-policy years, i.e.,
2007-2011. The indicator T hreatrdt equals one when the verified ratio exceeded the cuto↵ of
three in the previous year. P enaltyrdt is an indicator variable equal to one if the district had
two consecutive years with a verified ratio over three.16 Thus whenever P enaltyrdt equals
one, T hreatrdt must also equal one. The inclusion of P enaltyrd,t 1 allows the model to
di↵erentiate between districts in penalty once from districts in penalty for consecutive years.
The use of the threat term means that both 7 and 8 are interpreted as the additional
e↵ect of being in the penalty phase. The interaction terms, (IDEAt ⇤ T hreatrdt ) and
(IDEAt ⇤ P enaltyrdt ), are indicator variables that discern districts in threat or penalty
phases between the pre and post-policy periods. In this specification, 5 identifies the e↵ect
of the threat of penalty while 8 isolates the additional e↵ect of the penalty itself. Given
this specification, the total e↵ect of the policy equals the sum of 5 and 8 . Lastly, standard
errors are clustered at the district level, making them robust to heteroskedasticity and serial
correlation (Bertrand et al., 2004). The benefit of the di↵erence-in-di↵erences specification
is that it compares district-years over the disproportionality threshold to those below the
threshold before and after the policy and thus yields easily interpreted results of the policy.
16 Since the penalty and threat variables are functions of the lagged dependent variable, the inclusion of
district fixed e↵ects would violate the exogeneity assumption of the fixed e↵ects model (Wooldridge, 2010).

11

1.5
1.5.1

Results
Di↵erence-in-Di↵erences Estimates

Table A.2 reports the estimates of the impact of the financial regulations of the IDEA
2004 on Black overrepresentation, based on Equation 1.1. Negative interaction terms would
show that the policy caused districts with excessively high levels of overrepresentation to
reduce their disproportionality compared to similar districts prior to the policy introduction.
Column 1 presents the estimated e↵ect on the Black verified ratio, the disproportionality
measure of interest to the policy. The total e↵ect of the disproportionality regulations of the
IDEA 2004 on the Black verified ratio is -2.31, which means that districts in penalty phases
after the policy had an average verified ratio 2.31 points lower than their similar pre-policy
counterparts. This decline yields an average verified ratio much closer to the significant
disproportionality threshold at 3.26. The estimate of the total e↵ect represents a 41 percent
reduction in the pre-policy mean for in-penalty districts.
The magnitude of the policy’s e↵ect is influenced by the fact that the districts targeted by
the regulations have roughly 2 percent Black composition, resulting in larger relative changes
in the Black risk and verified ratio for every reduction in Black special education head count.
Putting the impact of the policy on students in context, penalized districts during the postpolicy period provided special education services to 701 Black students and 8,742 non-Black
students while enrolling a total of 1,901 Black students and 80,096 non-Black students.17
While the total e↵ect of the policy represents a large reduction in percentage terms, it is
clear that a small number of Black students were directly a↵ected by the policy.
Looking separately at the two components of the total policy e↵ect for the Black verified
17 The head count totals were calculated using district enrollment counts from the first year of penalty.

12

ratio, I find that the additional e↵ect of the penalty, not the e↵ect of the threat, is driving the
significantly negative total e↵ect. The post 2007 penalty estimate of -2.92 reveals that the
sanctions induced a decline in disproportionality. Contrary to the hypothesis that districts
in threat of sanctions during the policy period would lower their levels of disproportionality
before actually being placed in a penalty phase, I find evidence that these districts had an
insignificant, positive estimate of 0.61.18 In sum, column 1 of Table A.2 shows that districts
do not respond to the threat of sanctions but do reduce their levels of Black disproportionality
once financial penalties are exacted.
Since the Black verified ratio compares the Black risk of special education, i.e., the
proportion of Black students in special education out of the total number of Black students,
to the non-Black risk, the verified ratio could be reduced by either decreasing the Black risk
or increasing the non-Black risk. Column 2 of Table A.2 shows a similar pattern of e↵ects
as Column 1. The post-policy threat term is still positive and largely insignificant while
the estimate for the additional e↵ect of the penalties remains both negative and significant
at -34.4. Overall, these estimates indicate that the policy reduced the proportion of Black
students receiving special education services by 29.6 percentage points, a 44 percent reduction
in the pre-policy mean among districts in a penalty phase. Given the racial distribution
in Michigan and the fact that districts with high Black overrepresentation have a White
composition of 89 percent, looking at the e↵ect of the policy on the White risk should
closely approximate the e↵ect on the non-Black risk, which is the group of students most
likely to see an increase in the verified ratio when the district is in a penalty phase for Black
18 A LexisNexis Academic search for documents and articles discussing disproportionality was performed
for each district that the State found to have significant disproportionality. The search produced zero results
in news sources. The lack of evidence of the communities discussing the districts’ threat or penalty statuses
supports the evidence of financial penalties driving the reductions in disproportionality.

13

overrepresentation.19 The estimated total e↵ect of the policy on White risk is a statistically
insignificant reduction by 1.83.
Columns 4 to 6 expand the first three columns by including charter schools into the analysis. While the regulations of the IDEA 2004 treat both traditional public school districts
and charter schools the same, the baseline analysis excludes charter schools for two reasons.
First, charter schools may be able to respond di↵erently to the policy than the average school
district because charter schools have total and special education enrollment counts 88 and 91
percent smaller than districts, respectively. Secondly, selection over the disability categories
serviced between school districts and charter schools could lead to di↵erent impacts of the
IDEA sanctions. Despite these concerns, the inclusion of the charter schools does not drastically alter the estimates or their interpretations. Thus, Table A.2 provides evidence that the
financial regulations of the IDEA 2004 caused districts with significant disproportionality to
lower their levels of Black overrepresentation, with this reduction achieved by curtailing the
Black risk of special education.

1.5.2

Di↵erential E↵ects

Because the districts are held accountable to disproportionality regulations each year, it
is possible that the policy has di↵erential e↵ects between districts in their first penalty
phase and districts with consecutive penalty phases.20 The empirical specification used in
Equation 1.1 does not distinguish separate policy e↵ects between these two types of districts.
I estimate these di↵erential e↵ects by partitioning the penalty variable into two mutually
19 The censored nature of the special education data prevent me from accurately estimating the risks
separately for members of the American Indian, Asian, Hispanic, and Multiracial subgroups.
20 By first penalty, I mean first occurrence in a sequence of penalties and not the first penalty ever seen in
a district.

14

exclusive variables based on the lag of the penalty variable: first penalty and consecutive
penalty. If a district is currently in a penalty phase but was not in a penalty phase during
the previous year, the first penalty indicator variable is set to one. Alternatively, when both
current and lagged periods are in a penalty phase, the consecutive penalty indicator has
a value of one. Replacing the penalty variable in Equation 1.1 with the first penalty and
consecutive penalty variables allows me to estimate the total e↵ects for districts in their first
penalty phase separately from districts in multiple penalty phases.
Table A.3 presents the results from this di↵erential e↵ects estimation. Similar to Table A.2, the total e↵ects are listed at the top of the table while the bottom of the table
consists of the additional e↵ect of each phase. Looking across the top section of Table A.3
shows that there is a di↵erence in the impact of the sanctions depending on the district’s
pattern of disproportionality, i.e., the e↵ect is larger for districts in their first penalty phase.
The estimated total impact on Black verified ratio for districts in their first penalty phase
is -2.57, and the estimate for districts with consecutive penalty phases is -2.05. These total e↵ects of the policy fail to be significantly di↵erent from one another with a p-value of
0.77 and represent a 46 and 37 percent decrease in the Black verified ratio, respectively. The
lower section of column 1 highlights the fact that the entire negative additional impact of the
policy comes from districts entering their first penalty phase, as documented by the -3.18 estimate on the post-policy penalty interaction. Moreover, districts beyond their first penalty
phase have a positive, though insignificant, additional e↵ect of the IDEA 2004 sanctions.
This pattern implies that the policy was e↵ective at reducing the levels of disproportionality
in districts first entering into a penalty phase but ine↵ective at continuing to lower Black
overrepresentation in districts with consecutive penalty phases.
Given these findings and the lack of evidence of an e↵ect of the threat phase from Ta15

ble A.2, it appears that the combination of financial sanctions, state audits, and reputation
costs initiated in the first year of penalty drive the overall impact of the policy. While it is
feasible that the reallocation of funds into Early Intervention Services would assist districts
both in more appropriately identifying special education students and also in preventing the
need for placement into special education programs, there are two reasons it is unlikely that
this reallocation of funds led to these results. First, the policy has a strong e↵ect in the first
year but not in subsequent years, which would imply both immediate and short-lived benefits
of the Early Intervention Services program. Secondly, districts had the opportunity to reallocate 15 percent of the IDEA funds towards Early Intervention Services regardless of their
penalty status, so the districts could have achieved these results without being sanctioned.
Columns 2 and 3 of Table A.3 describe how the policy a↵ected the two key components
of the Black verified ratio, Black and White risks. Similar to the baseline di↵erence-indi↵erences results, I find strong negative e↵ects on Black risk but insignificant estimates on
White risk. Again, the policy had the greatest impact among districts in their first penalty
phase with an estimated 37 percent reduction from pre-policy levels. Among districts who
remained in the penalty phase for multiple periods, the policy induced them to lower their
Black risk rates by 16 percent. The di↵erential total e↵ects on Black risk are statistically
di↵erent from each other with a p-value of 0.02. The bottom section of column 2 shows a 21
percent increase in Black risk as the additional e↵ect of the sanctions on districts remaining
in the penalty phase. Columns 1 and 2 demonstrate the e↵ectiveness of the policy weakened
for districts continually in penalty phases.
Incorporating eligible charter schools into the di↵erential e↵ects analysis yields results
similar to those in columns 1 through 3. The estimates on Black verified ratio are slightly attenuated, and the total e↵ect for first penalty exposure is imprecisely estimated. Comparing
16

the total e↵ect estimates for Black and White risks shows that although the impact of first
penalty on Black risk was 13 percent larger than the estimate without charter schools, the
e↵ect on White risk was 113 percent larger. Table A.3 also presents the first indication of
statistically significant evidence of an e↵ect of the policy on White risk. The additional e↵ect
of consecutive penalties on the sample including charter schools has an estimate of 4.76 while
both di↵erential total e↵ects are still statistically insignificant. The weakly positive impact
of the policy on districts in consecutive penalty phases would point to districts perversely
lowering their disproportionality levels by increasing the denominator of the Black verified
ratio, as opposed to reducing Black overrepresentation. The consistently large and significant e↵ects on Black risk, however, provide overwhelming evidence that districts lowered
disproportionality by reducing Black risk.
Table A.3 confirms the findings of the baseline di↵erence-in-di↵erences analysis. The
financial penalties of the 2004 IDEA induced districts and charter schools to lower their
levels of disproportionality by reducing their levels of Black risk. Moreover, the e↵ects of the
policy were strongest for districts in their first penalty phase and then slightly attenuated
for districts in consecutive penalty periods.

1.5.3

Impact of Regulations on Other Outcomes

Extending the comparison of means evaluated in Table A.1, I investigate the impact of the
IDEA regulations on outcomes not directly targeted by the policy. The existence of an
e↵ect could suggest important changes between the treatment and control groups across
time that would compromise the di↵erence-in-di↵erences strategy. Table A.4 applies the
empirical specification of Equation 1.1 on the following outcome variables: overall risk, total
enrollment, Black composition, White composition, percent of students eligible for free and
17

reduced-priced meals, and special education student-teacher ratio.21
Table A.4 reports no e↵ect of the policy on any of the independent variables. The overall
risk of special education placement could have been a↵ected either positively or negatively
as an unintended consequence of the sanctions; however, there exists no evidence of either a
total e↵ect or an additional e↵ect of the policy. Changes in the size, racial shares, or poverty
status of the total student population would represent compositional changes in the control
and treatment groups across time. I find no evidence of these types of compositional changes.
Despite the penalties reallocating Federal funds away from special education services and
also mandating greater Early Intervention Services, Table A.4 shows no impact on the special
education student-teacher ratio. The only consistently significant e↵ect across the columns
of the table is the threat estimate, which was expected given the di↵erences in means for the
disproportionate districts compared to all districts observed in Table A.1.
In addition to the above outcomes, it is possible that the type of student receiving special education services changed in response to the reauthorization of the IDEA. To examine
this issue and drawing upon prior evidence of overrepresentation of impoverished students
in special education identification, I test for socioeconomic disproportionality in special education programs (Dunn, 1968). To implement this test, I analyze the relative rate at which
special education students are eligible for free and reduced-price meals compared to general
education students.22
Over the sample period, special education students had similar eligibility rates as to the
21 The quadratic in White composition as a control variable was dropped from the regressions for Black and
White composition. Likewise, percent of students eligible for free and reduced-priced meals was eliminated
as a control in its own regression.
22 Due to data limitations, special education student eligibility in free and reduced-price meals is available
during the 2005-06 through the 2011-12 academic years. Similar to other variables collected from the LEA
Reports, special education student eligibility in free and reduced-price meals also su↵ers from censored
observations. Due to the data being censored when the headcount fell below 6, the following analysis relies
upon 642 observations matching the baseline di↵erence-in-di↵erences sample.

18

total population of students, 44.8 percent compared to 44.3 percent, respectively. Calculating a verified ratio by free and reduced-price meal eligibility comparing special education
students to general education students reveals that special education students were overrepresented in eligibility status by 28 percent. Table A.5 presents a crude di↵erence-in-di↵erences
estimate of an insignificant reduction of 0.74 mean points for the impact of the penalties
of the IDEA on socioeconomic overrepresentation. Additionally, Table A.5 shows that data
availability and limitations force for the analysis of socioeconomic disproportionality to rely
upon only 2 pre-policy instances of penalty phases. Finally, I applied the regression specification outlined in Equation 1.1 to the socioeconomic disproportionality measure; this analysis
resulted in an estimated penalty e↵ect of -1.71 (with a p-value of 0.36) and threat e↵ect of
0.24 (with a p-value of 0.05). Thus, I find evidence of overrepresentation of students with
lower socioeconomic status in special education programs but no e↵ect of the regulations
at reducing this disproportionality or changing the composition, outside of race, of students
participating in special education services.

1.5.4

Validity of Di↵erence-in-Di↵erences Approach

The validity of the di↵erence-in-di↵erences results from Table A.2 depends upon the assumption of a common time trend. This assumption requires that without treatment, the treated
and control groups would have followed common paths. If this assumption does not hold,
the main results are biased.
To address this concern, I incorporate a third di↵erencing group: districts ineligible for
disproportionality calculations. Districts were classified as ineligible under Black special education identification if any of the following enrollment requirements were true: fewer than
10 Black students receiving special education services, fewer than 30 students of any race
19

in the special education program, or fewer than or equal to 100 non-Black students in the
total student body. Due to a change in the availability of data in 2011, I am only able to
identify the ineligible districts through 2010, which leads to an additional 201 districts with
705 district-year observations entering into the analysis.23,24 The inclusion of the ineligible
districts addresses the common trends assumption. If the eligible treated districts exhibited
a di↵erent trend than the eligible control districts, by lowering their levels of Black overrepresentation over this time period for reasons other than the IDEA, we would expect to
see a similar reduction in disproportionate representation for ineligible districts with Black
overrepresentation.
Figure D.2 illustrates the policy period means of the Black verified ratio for ineligible
districts by penalty phase. Calculation ineligible districts in penalty phases increased their
pre-policy average Black verified ratio of 3.54 by a statistically insignificant 0.11 mean points.
Likewise, the ineligible districts in neither threat nor penalty phases saw an increase in their
average Black verified ratio from 1.29 to 1.51. These di↵erences suggest that the policy had
a statistically insignificant reduction of -0.11 mean points on ineligible districts in penalty.
In the di↵erence-in-di↵erences approach, I compared districts with a recent history of disproportionality to districts without that history before and after the policy. The di↵erencein-di↵erence-in-di↵erences approach extends this analysis by including ineligible districts
23 For an ineligible district to avoid censored special education head counts for a given race, it must have at
least six students of that race in either special or general education. Including the ineligible charter schools
leads to the addition of 290 units with 1,007 observations.
24 I also drop all district-year observations from the 2011-2012 academic year for the eligible districts.

20

with and without histories of disproportionality using the following specification:

Yrdt = ✓1 + Zdt ✓2 + ✓3Erdt + ✓4 IDEAt + ✓5 (Erdt ⇤ IDEAt ) + ✓5 T hreatrdt
+✓6 (Erdt ⇤ T hreatrdt ) + ✓7 (IDEAt ⇤ T hreatrdt ) + ✓8 (Erdt ⇤ IDEAt ⇤ T hreatrdt )
+✓9 P enaltyrd,t 1 + ✓10 P enaltyrdt + ✓11 (Erdt ⇤ P enaltyrdt ) + ✓12 (IDEAt ⇤ P enaltyrdt )
+✓13 (Erdt ⇤ IDEAt ⇤ P enaltyrdt ) + rdt .

(1.2)

Erdt is an indicator variable equaling one when the district is eligible for the verified ratio calculation.25 The inclusion of the eligibility term creates six new estimators and di↵erentiates
the e↵ect of the policy between the district eligibility types.
Table A.7 presents the key estimates from Equation 1.2 and strongly supports the results
from Table A.2. Including the third di↵erencing group yields a similar estimated policy e↵ect
of -2.17 where again the impact of the penalty, P ost2007 ⇤ Elig ⇤ P enalty, itself – not the
threat, P ost2007 ⇤ Elig ⇤ T hreat – is driving the result. In accordance with the hypothesis
that ineligible districts with disproportionality would have no incentive to reduce their levels
of overrepresentation, the estimates of P ost2007⇤P enalty and P ost2007⇤T hreat are close to
zero but imprecisely estimated. Similar to the di↵erence-in-di↵erences estimates, columns 2
and 3 indicate that the reduction in overall Black disproportionality was achieved by reducing
the Black risk and not by increasing the White risk. Finally, the inclusion of the smaller
charter schools in columns 4 through 6 only mildly dampens the estimated e↵ect of the policy
on the Black verified ratio and Black risk. Using the di↵erence-in-di↵erence-in-di↵erences
approach, I calculate that the penalties associated with the IDEA 2004 caused the Black
verified ratio to drop by 39 percent and the Black risk to drop by 52 percent, which compare
25 There are no cases of a district switching eligibility status while in either a threat or penalty phase.

21

closely to the di↵erence-in-di↵erences total e↵ects of 41 and 44 percent, respectively. Since
the di↵erence-in-di↵erence-in-di↵erences approach served as a validity check of the baseline
di↵erence-in-di↵erences identification strategy, the similarity of the estimates across models
supports the use and reliability of the di↵erence-in-di↵erences results.

1.5.5

Did IDEA Reduce Inappropriate Disproportionality?

Figure D.3 further investigates the role of total White student composition on special education risk. Looking across the White composition distribution, I find that the risk for Black
students increases as the percent of White students increases. Likewise, the special education
risk for White students is largest in districts with the lowest shares of White students and
decreases as the percent of White students grows. Figure D.3 demonstrates that for both
White and Black students, the risk of special education increases the less the student looks
like her peers, which is evidence supporting the existence of inappropriate identification.
This graphic, however, does not account for the possibility that poverty, or even relative
resources, across races drives the levels of Black overrepresentation.26 I address this alternative cause of disproportionality by assessing the role of White composition on risk of special
education placement controlling for available financial resources and other possible factors
using the following OLS specification:

Rrd = 0 + 1 Fd + 2 Wd + 3 Incwd + 4 Incbd + ⌘rdt .

(1.3)

Rrdt represents the median risk of special education placement for race r in district d for
26 I looked at the potential role of teacher gender on risk of special education and found that it magnifies (attenuates) the estimate of White student composition on Black (White) risk. Due to limited data
availability on teacher racial shares, the variable was excluded from the following regression.

22

years 2004 through 2009.27 Fd is the median percent of students eligible for free and reducedprice meals in the district. Wd equals the median White composition of the district. The
median White and Black incomes, in logs, within the district are represented by Incwd and
Incbd , respectively.
Table A.8 reports the estimates of the factors in special education placement, based on
Equation 1.3. Even after accounting for the impact of financial resources for White and Black
households, the district’s White composition plays both an economically and statistically
significant role on the risk of special education placement with an estimated e↵ect ranging
between 9 and 11 percent. While Figure D.3 and Table A.8 do not provide conclusive causal
evidence of inappropriate identification in Michigan special education placement, they do
point to inappropriate identification as at least a partial reason for the disproportionality
seen in the data.

1.6

Conclusion

The inappropriate identification of students into special education programs in the United
States remains a legitimate concern. As Hosp and Reschly (2003) describe, placement into a
special education program may harm a student by simply lowering the teacher’s expectation
for the student or limiting the curriculum and peer exposure the student experiences. In
other words, inappropriate placement may jeopardize a student’s achievement. Coupled with
the fact that Black students are historically overrepresented in special education programs,
it is apparent why Congress chose to include disproportionality regulations in the 2004
reauthorization of the IDEA.
27 Median values during this time period were used to match the available income data from the 2009
5-year American Community Survey.

23

This study presented the first quantitative estimates of the e↵ect of the financial sanctions
associated with the IDEA 2004 using Michigan enrollment data. Unlike prior work, which
analyzed endogenous variation in disproportionality, I was able to exploit the exogenous
variation induced by significant disproportionality sanctions. The policy’s requirement of
two consecutive years of excessive disproportionality allowed for the analysis of the impact
of the threat of sanctions separately from the impact of the sanctions. After confirming
the existence of Black overrepresentation in Michigan during the sample period, the study
showed that the financial sanctions themselves, not their threat, lowered disproportionality
levels by 41 percent, The reduction in disproportionality e↵ectively positioned the penalized
districts at the threshold for the verified ratio. The lack of an estimated impact of the
policy for districts in the threat phase shows that districts failed to lower their levels of
Black overrepresentation before incurring penalties associated with the IDEA 2004, which
further strengthens the impact sanctions had on district behavior. The results for the total
e↵ect of the IDEA 2004 show not only that the policy reduced disproportionality among
overrepresented districts but also that lowering disproportionality is within the scope of
school districts, a contested issue by itself (Byrnes, 2011).
Evaluating the potential success of the regulations hinges crucially on which groups are
a↵ected by lowering Black disproportionality. The baseline approach found that a 44 percent
drop in the risk of special education for Black students created the reduction in overall Black
disproportionality. Since the policy’s sanctions were based on a disproportionality measure
comparing the risk of special education placement among Black students to the risk among
non-Black students, districts with high levels of Black overrepresentation could have avoided
penalties by solely increasing the risk of the non-Black student population. Extending the
baseline analysis to the risk of special education for White students, the racial majority in
24

Michigan, failed to reveal this perverse behavior, as estimates were insignificantly negative.
To address the concern of di↵erential e↵ects of the policy, the baseline specification was
modified to account for the e↵ect of the first penalty separately from the e↵ect of consecutive
penalty phases. The results echo the original findings in that sanctioned districts reduced
their levels of disproportionality by lowering the risk of special education for Black students.
Moreover, this analysis also revealed that the policy had its strongest e↵ect on districts in
their first penalty phase.
As a means to verify the validity of the assumptions needed by the di↵erence-in-di↵erences
strategy, a third di↵erencing group of ineligible school districts was added to the study. The
addition of this group provided evidence of the common time trends assumption. The results
of the di↵erence-in-di↵erence-in-di↵erences specification upheld the earlier estimated total
e↵ects of the policy and thereby support the di↵erence-in-di↵erences approach.
This paper demonstrates policy-induced reductions in Black overrepresentation and sheds
light on the student well-being e↵ects of disproportionality regulations. If disproportionality
existed due to inappropriate identification, perhaps due to discrimination, then the policy
successfully lowered the unnecessary exposure to special education for some students, thereby
increasing student well-being for those students who would have otherwise participated in
special education programming. Alternatively, under the assumption that disproportionality
exists due to justifiable reasons, e.g., racial disproportionality in poverty rates, the IDEA
2004 may have in fact harmed students by creating artificial costs to placement and thus
preventing students from receiving the services they need. By showing that the racial composition of one’s peers correlates with one’s own risk of special education after accounting
for the relative income gap between races, I argue that this relationship provides evidence of
inappropriate identification partially leading to Black overrepresentation in Michigan. The
25

indications of inappropriate identification combined with the policy’s requirement of Early
Intervention Services for students of the overrepresented race leads me to conclude that the
disproportionality regulations of the IDEA 2004 increased student well-being for students
otherwise likely to participate in special education programs.

26

Chapter 2
Disparities in Special Education Identification:
Evidence from Michigan

2.1

Introduction

Special education programs provided by the Michigan Department of Education served nearly
218,000 of Michigan’s 1.6 million students in 2011. Those students received individualized
instruction and assistance for disabilities including vision, hearing, cognitive, and emotional
impairments. The Michigan Office of Special Education provides those services in an e↵ort
to guarantee a free and appropriate public education.
Beyond its size, Michigan’s special education program is costly. In 2010, Michigan received 389 million dollars in federal funds for special education services and contributed over
964 million dollars of its own funds for special education. The instruction of these special
education students required over 19 thousand full-time equivalent (FTE) special education
instructors and aides. Overall, the Michigan special education program educates over thirteen percent of the students in the state while employing thousands of teachers and using
over 1.3 billion dollars.
Given the value of the services provided as well as the cost, it is important to understand the determinants of special education placement. This paper addresses this need by
examining disparities in special education participation, intensity, and services across racial,
socio-economic, and school-level dimensions. To what degree does special education identification vary by family and community resources? How does the composition of students

27

in a district correspond with the level and type of special education services provided in
the district? To analyze the socio-economic dimension, I focus on three measures of family
and community resources: the share of students eligible for free and reduced-price meal programs, the taxable values of homes within the district, and the median household income
within the district. The potential disparities in special education across racial dimensions
are investigated by focusing on the share of White students in the district. Finally, I analyze
the role of district characteristics by focusing on district financial budget, urbanicity, and
standardized test score performance.
The motivation to examine disparities in special education stems from a large literature
documenting known di↵erences in participation rates. Beginning in 1960s, researchers and
policymakers have discussed the racial and socio-economic disproportionalities in special education programs (Dunn, 1968).1 Between 1975 and 2004, Black students have maintained
nearly double their expected participation rate in the cognitively impaired category conditional on their share of the student population (Patton, 1998).2 More recently, researchers
have shown that conditional on demographic and economic variables, the overrepresentation
of Black students labeled as cognitively impaired persists; however, the inclusion of prior
academic achievement diminishes those findings (Hibel et al., 2010; Hosp and Reschly, 2004;
Oswald et al., 1999). Evidence supports the claim that it is within the scope of districts
1 While medical health outcomes are a separate entity from special education outcomes, it is important
to note that racial disparities also exist in youth health outcomes. Research has documented that Black
students are less likely to receive mental health services compared to their White counterparts. Additionally,
the relationship between health services receipt and education switches sign across races: an increase in
education is associated with a decrease (increase) in service receipt for Black (White) young adults (Broman,
2012). In a similar vein to this paper, a recent study demonstrated that Black young adults who attended
schools with higher White student composition report poorer health outcomes than Black young adults from
lower White composition schools (Goosby and Walsemann, 2012).
2 The 2004 estimates are from the author’s calculations using the National Center for Education Statistics’
Common Core of Data and the Data Accountability Center’s IDEA Data (National Center for Education
Statistics, U.S. Department of Education, 2013).

28

to a↵ect their levels of disproportionality: districts have been shown to respond to financial
incentives to lower the overrepresentation of Black students (Enayati, 2014). Focusing specifically on Michigan, variation in special education services may be the result of the four-part
system of funding special education programs (Jalilevand and Conlin, 2014).
The remainder of this paper will proceed as follows. Section 2 outlines the data sources
used in this analysis. Section 3 provides both a graphical illustration of dimensions of special
education disparities and also a regression framework to quantify the associations between
the dimensions and the special education measures. Finally, I provide concluding remarks
in Section 4.

2.2

Data

This paper relies on six di↵erent data sources to generate variables and trends of value to its
objective. The data are broken down into four categories: special education specific data,
general Michigan school data, district characteristics data, and finance data. The special
education head count data come from the Michigan Compliance Information System (MICIS). MI-CIS provides district-level reports on special education variables for the academic
years 2002-2003 through 2010-2011. The advantage of the MI-CIS is that they provide
special education head counts by race and disability category, separately.3 The MI-CIS reports detail head counts for thirteen disability categories: cognitive impairment, emotional
impairment, hearing impairment, visual impairment, physical impairment, speech and lan3 In an attempt to protect the identity of individual students, the LEA Reports censor disability category
and race cells with less than six students of that disability category or race (across special and general
education) in the district, respectively. This censoring introduces missing values into the data set. The
concern over the reliability of measures based on counts from districts with less than six relevant students
gives perspective to the value of these cells.

29

guage impairment, early childhood developmental delay, specific learning disability, severe
multiple impairment, Autism spectrum disorder, traumatic brain injury, deaf-blindness, and
other health impairment.4 The special education FTE data come from the Michigan Department of Treasury’s Tax Policy Analysis division.5,6 The ratio of a district’s special education
FTE and its special education head count measures the average amount of special education
time each student with a special education label receives. This metric provides a sense of
the intensity, in terms of time, of special education services provided to special education
students.
The general education enrollment and student count data come from the Michigan Center
for Performance and Information (CEPI). CEPI o↵ers multiple data sets at the district level.
This paper uses CEPI data to determine measures of total enrollment, total enrollment by
race, special education enrollment, share of eighth grade students who met or exceeded the
standardized math test, and percent of students eligible for free and reduced-price meals.7
Among the students present on the days the MEAP is administered, the population of exam
takers includes general education students as well as the special education students whose
individualized education plans hold them, with the addition of test accommodations, to the
same standards as the general education students. The combination of the special education
and CEPI data creates the special education participation rates used in the analysis.
The urbanicity measure was obtained from the National Center for Education Statis4 In this paper, I aggregate hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness into a single category called Physical. This aggregation does not suggest equivalence among the needs of students with these disabilities; rather, these variables
were grouped together to assist in the analysis due to their small share of all special education students.
5 The Michigan Department of Treasury obtained the FTE values from the Department of Education.
6 The author thanks Michael Conlin for sharing these data.
7 Proficiency is determined by the student’s performance on the eighth grade math section of the MEAP
exam. The Michigan Department of Education assembles a panel to determine the cut scores required for a
student to met or exceed proficiency standards (MDE, 2008).

30

tics’ Elementary/Secondary Information System’s urban-centric locale variable. Using 2000
Census geography and the district’s address, the urbanicity measure describes a district’s
proximity to populous areas by assigning it into one of twelve categories. The categories
are a function of two sets of labels, area type and size. For the purposes of this essay, the
size dimension was eliminated, which reduces the categorizations to four types: city, suburb,
town, and rural.
Finally, the financial data were obtained from the Michigan Department of Education
Bulletin 1014, which houses district-level data on revenue and expenditure categories, and
the American Community Survey. Homestead taxes are determined by the taxable value for
primary residences but do not include businesses, rentals, and secondary homes. Homestead
taxable values serve as a good proxy for the value of the homes occupied by the students of
that district and act as a finer measure of family resources than the binary variable of free
and reduced-price meals. To allow for comparability across districts, taxable values were
adjusted to measure per pupil dollars.8 The per pupil adjustment is by the total number of
pupils in the district. District budgets are defined as the operating income’s percent of total
revenue.9 This measure of budget attempts to capture the short-run financial stress faced
by districts. The median household income variable comes from the American Community
Survey’s 2009 five-year measure, which draws upon data collected between 2004 and 2009.
Given that the district-level household income measure requires five years of collection, each
district has one median income value replicated for all years where the district is observed
in the panel.
The sample of districts used in this paper pulls information from the five sources detailed
8 Homestead taxable values were converted into 2010 U.S. dollars using the Consumers Price Index.
9 District budget equals the di↵erence between total revenue and total expenditures divided by total
revenue times one hundred.

31

above. In total, I have 4,198 district-year observations covering 505 traditional public school
districts between the 2002-03 and 2010-11 academic years. The analysis of this sample relies
upon figures generated using kernel-weighted local polynomial regressions. The advantage of
this type of scatterplot smoothing procedure is that it allows me to avoid making functional
form assumptions about the relationship between variables.10
Table B.1 presents the summary statistics for the sample of districts. On average, districts
identified 13.6 percent of their students for special education programs. The average student
in a district spent 3.6 percent of their time in school receiving special education services.
The special education intensity measure of 0.26 means that the average special education
student spent a quarter of her day receiving special education services.
A courser measure of intensity is represented by the share of special education students
that spend at least 80 percent of their school day in the general education classroom. While
the special education FTE variable is a continuous measure within the unit interval, the
share of students in general education classrooms for at least 80 percent of the day is a
function of a dichotomous variable. Additionally, this new measure is only available for the
2011-12 academic year, which is outside of the sample used in this paper. As a check of the
data quality of the FTE measure, I examined the correlation between the 2010 district-year
observations of special education intensity and the 2011 district general education value,
which is -0.65.11 This analysis supports the use of the FTE measure within this paper.
A summary of demographic characteristics shows that the average district is located in
communities with an average homestead taxable value of 130 thousand dollars per pupil
10 It is important to note that this style of analysis is meant to provide a description of the correlation
between special education participation and several dimensions. It does not allow the author or reader to
draw causal claims.
11 Figure H.2 presents a scatter plot of the relationship between special education intensity and the share
of students in general education classrooms for at least 80 percent of the day.

32

and had a median household income of 50 thousand dollars. Districts report 38 percent of
their students eligible for free and reduced-price meals. As indicated by the budget measure,
districts’ total revenue exceed their total expenditures by six-tenths of one percent of their
total revenue. Considering district racial composition, 86 percent of districts’ total student
body identify as White. The urbanicity measure reveals that the majority of districts are
categorized as either rural or suburban, 49 and 26 percent, respectively. The final district
demographic is the percent of students who met or exceeded the eighth-grade math standard.
Table B.1 shows that the average district had over 68 percent of its eighth graders score as
proficient or above.
Finally, the risk of special education measures describe the likelihood that a student of a
particular race participates in special education programs.12,13 For example, the White risk
measure of 14.2 means that, for the average district, one-seventh of all its White students are
identified into special education programs. The risks for both White and Hispanic students
are close to the average participation rate of 13.6 percent. On the other hand, Black students
are more likely to receive a special education label - as they have a risk of 19.4 percent. This
overrepresentation of Black students in special education programs is initial evidence of
disparities in special education placement by race, which will be examined further in the
following section.
12 The lower number of observations for the risk of special education variables is due to the censored special
education by race data from the MI-CIS reports.
13 The risk of special education is not reported for American Indians, Asians, or Multiple Races students,
but their enrollment counts are included in the calculations.

33

2.3

Findings

2.3.1

Special Education Identification and Services Delivered

Variation in family resources corresponds with disparities in special education placement.
One potential explanation for the disproportionate representation of Black students in special education is that race acts as a proxy for poverty which induces appropriate disproportionality (Donovan and Cross, 2002; Dunn, 1968; Patton, 1998).14 Figure E.1 plots the
relationship between special education participation and a proxy for family resources, the
natural log of homestead taxable values per pupil. All three measures of participation decline over the taxable values distribution. The 5 percent drop in special education head
counts per total pupil count from 14.1 percent to 13.3 percent indicates that districts with
higher taxable values tend to identify more students into special education programs. The
54 percent drop in participation rates by FTE, from 6.1 percent to 2.8 percent, shows that
high taxable value districts provide less special education services to the average student
from the total student population compared to districts in communities with lower taxable
values.15
The special education intensity measure focuses on the amount of time a special education
student receives special education services. Confirming findings by Jalilevand and Conlin
(2014), Figure E.1 shows that, on average, special education students from districts in the
highest taxable value per pupil communities spend 22.1 percent of their time in special
14 Here, the racial disproportionality in special education participation is driven by the racial disproportionality in poverty. This type of disproportionality is considered “appropriate” because exposure to poverty
has been shown to increase the need for special education services. The alternative to appropriate disproportionality is inappropriate disproportionality, which is often explained by discrimination.
15 In this paper, the amount of services received, when discussing special education intensity and FTE
measures, refers specifically to the amount of time spent receiving those services.

34

education programs; whereas, special education students attending districts in communities
from the far left tail of the distribution spend only 35.6 percent of their time at school
receiving special education services.16 This di↵erence represents a 38 percent reduction in
the amount of services received.
Jalilevand and Conlin (2014) assert that the negative relationship between special education intensity and homestead taxable values is due to di↵erences in the disability categories
served in districts across the taxable value distribution. Figure E.2 considers this claim by
examining the identification patterns by disability category. Figure E.2 shows that the least
wealthy communities have 14.6 percent of their special education students identified with
a cognitively impaired label while the wealthiest communities have only 6.6 percent. Likewise, moving from the left to the right tail of the taxable value distribution corresponds to
113 percent, 81 percent, and 19 percent growth in other health impairments, Autism spectrum disorder, and speech and language impairments, respectively. A relative comparison
of disability category intensities reconciles the drop in intensity seen in Figure E.1 with the
results of Figure E.2. The average intensity measure for students with Autism is 0.89 while
cognitively impaired students have an average intensity of 0.83 (Michigan Department of
Education: Office of Special Education, 2008).17 Thus, the drop in overall intensity is due
to the larger decline in the share of students with a cognitively impaired label.
While the shares of students with cognitive impairments and Autism spectrum disorder
move in opposite directions as the wealth of the community increases, it is unclear whether
these patterns are driven by di↵erent types of students across districts or di↵erences in how
16 The taxable value measure is adjusted to represent the taxable values per total student in the district.
The special education intensity measure reports the share of a special education student’s day that is spent
receiving special education services.
17 Within disability category intensities were calculated by the author using the tables provided in the 2008
MDE OSE Report.

35

districts are labeling students into special education programs. Focusing on the features
which may lead to di↵erences in the way districts label students, two elements are examined: the extent to which the categories have overlapping criteria and the cost of servicing
categories.
Overlapping criteria present the greatest concern in this context for the categories described as “soft” due to the subjective judgments factored into the determination of a special
education label (Harry and Anderson, 1994; Hibel et al., 2010).18 In their 2010 paper, Hibel
et al. argue that, among the soft categories, it is necessarily true that the normative standards used in the labeling process are based on the sample of peers within the school. Thus,
we would observe di↵erences in who is identified under various categories when we look across
districts. Focusing on the malleability, as a function of the Michigan guidelines for special
education identification, of cognitive impairment and Autism spectrum disorder reveals that
the categories have one overlapping condition, low academic performance. However, Autism
spectrum disorder does not demand an academic deficiency while a cognitive impairment
label requires below average academic performance, i.e., two standard deviations below the
mean on intellectual assessments and standardized math and reading scores in the bottom
six percentiles. Additionally, Autism spectrum disorder has been described as a “cognitive style” not a “cognitive deficit,” as evidenced by local cognitive strengths, e.g., savant
skills with a specific topic, alongside “weakened [global] coherence” (Happ´e, 1999; Happ´e
and Frith, 2006). While it is true that cognitive impairment and Autism spectrum disorder
share common criteria, the Michigan Department of Education clearly describes cognitive
impairment as lying in the cognitive domain whereas Autism spectrum disorder manifests
18 Soft categories include cognitive impairment, Autism spectrum disorder, specific learning disability, as
well as speech and language (Harry and Anderson, 1994; Hibel et al., 2010).

36

primarily through impairments with social interaction and communicative behavior.
The second element that may contribute to the way in which districts label students is
the related cost of providing the services. The average costs per special education pupil for
programs and services varies greatly by category (Michigan Department of Education: Office of Special Education, 2008). Among students in the general education setting receiving
services only, the per pupil cost ranged from $2,569 for speech services to $6,304 for learning
disability services. Special education programs have higher individual costs and a wider
distribution with early childhood programs costing only $17,687 per pupil and visual impairment programs costing $76,877. The expense of the program may by a key factor in the
observed pattern in the relative shares of students with cognitive impairments and Autism
spectrum disorder. The cost of cognitive impairment programs ranges between $19,737 and
$35,645 per pupil for mild to severe labels, respectively. Alternatively, the average cost of
an Autistic impairment program is $52,762. One important factor in the cost of the programs is the maximum student-to-sta↵ ratio requirements: 2.5:1 for Autistic impairment and
between 3.75:1 and 15:1 for cognitive impairment.19 Thus, districts in economically disadvantaged communities may not be able to a↵ord the additional expense of providing Autistic
impairment services, which may lead to the observed opposite trends seen in Figure E.2.
Figures E.3 and E.4 illustrate the role of household income on special education disparities. Median household income within a district provides another measure of the socioeco19 The student-to-staffing requirements are complex functions of both the maximum number of students
allowed in a single classroom program and also a additional number of students that would require an
additional aide (Michigan Department of Education: Office of Special Education, 2008). If a district o↵ers
just one classroom program for Autistic impairment, the maximum number of students is five. If the district
o↵ers multiple programs for Autistic impairment, the average student-to-sta↵ ratio can not exceed five. In
both of the prior two cases, an aide is required if there are more than two students in the program. On
the other hand, the ratios for cognitive impairment programs depends upon the type of impairment: mild,
moderate, or severe. In high schools, the maximum number of students with a mild cognitive impairment
label allowed in a program with just one teacher is fifteen. The cap of fifteen students with a severe cognitive
impairment requires one teacher and three aides.

37

nomic status of families within the community. As with homestead taxable values, both the
participation rate and intensity decline over the income distribution, by 18 and 46 percent
respectively. While the percent of special education students with a cognitively impaired
label fell by 8 percentage points over the homestead value distribution, it dropped by 10.6
percentage points moving across the income distribution. When using a di↵erent measure
of family and community resources, this paper continues to demonstrate large disparities in
special education participation, intensity, and category.
The percent of students eligible for free and reduced-price meals serves as another proxy
for family resources. Figure E.5 shows that participation via head count and FTE is increasing but that the FTE rate is growing faster, illustrated by the rise in intensity. Districts with
less disadvantaged students have the lowest intensity as well as participation by either head
count or FTE. Again, I find that as districts serve more disadvantaged students they tend to
also have higher shares of students with cognitive impairments and lower shares of students
with other health impairments, Autism spectrum disorder, and speech and language impairments. The sharp drops seen at the far right tails in Figures E.5 and E.6 are supported
by far fewer observations than the rest of the distribution, as shown by the wide confidence
intervals and the fact that the 95th percentile of the free and reduced-price meal distribution
has a value of 69 percent. Given the correlation of -0.49, the similar results between free and
reduced-price meal eligibility and the examination of disparities along homestead taxable
values provides evidence of the robustness of the findings along the family and community
resources dimension.
The next set of dimensions considered in this analysis are the characteristics of the districts themselves. The district budget measure used in this paper reports total revenues in
excess of total expenditures as a percent of total revenues. While there is variation in the
38

budget variable, as evidenced by the standard deviation of 3.14, Figures E.7 and E.8 demonstrate no relationship between the distribution of budgets and special education disparities.
Participation rates drop by only 1.1 percentage points, and moreover have maximum and
minimum values with overlapping 95 percent confidence intervals. Overall, the graphic evidence does not support the existence of a relationship between district budgets and special
education disparities.
Another potentially impactful characteristic of districts is the urbanicity. Figure E.9
shows participation is fairly flat over the four categories of urbanicity but the FTE measure
drops more substantially, as seen by the 31 percent drop in special education intensity. Although the urbanicity and community resource measures, i.e., homestead taxable values and
median household incomes, portray similar patterns in intensity, the urbanicity measure is
not simply another proxy for community resources. Calculating the average median household incomes for each urban locale, I find a non-monotonic relationship: 46 thousand dollars
in cites, 57 thousand dollars in suburbs, 47 thousand dollars in towns, and 48 thousand dollars in rural locations. While Figures E.1 and E.3 would suggest that cities, towns, and rural
locations have similar special education intensities with a lower value for suburbs, Figure E.9
illustrates a monotonic decline in intensity as proximity to populous areas decreases. Similar
to the above findings, Figure E.10 shows that lower intensity districts have relatively higher
shares of students with specific learning disabilities and other health impairments.
Figures E.11 and E.12 illustrate the relationship between special education disparities
and district academic performance. The results follow the same patterns as the disparities
seen over the community resource dimensions. Namely, large declines in both participation
and intensity, reductions of 34 and 60 percent, respectively. Also, the districts with the
highest intensity have the highest share of students with a cognitive impairment label and
39

the lowest shares of students with Autism, speech and language impairments, and other
health impairments.
The final dimension considered in this analysis of disparities in overall special education
participation is the racial composition of the district. Here, I examine the relationship
between special education status and the share of the total student body that is White.
Figure E.13 demonstrates that having a higher White student composition corresponds with
both lower proportions of students participating in special education programs and also
lower intensity of services. The share of students with a special education label falls by 23
percentage points while the intensity of services drops by 48 percentage points. As seen
in earlier graphics, Figure E.14 shows that higher shares of cognitively and emotionally
impaired students correlates with higher overall intensity. Matching well with the literature
documenting racial disproportionality, Figures E.13 and E.14 illustrate the observation that
the districts with the fewest White students, which in Michigan corresponds with high Black
student composition (correlation of -0.88), have the highest special education participation
rates, highest intensity, and largest shares of cognitively impaired students (Donovan and
Cross, 2002; Oswald et al., 1999).

2.3.2

Di↵erential Participation Patterns by Race

This section of the analysis dives deeper into racial disparities in special education by examining within race participation rates, i.e., risks of special education. Figure E.15 extends the
analysis of Figures E.13 and E.14 by plotting the risks of special education for White, Black,
and Hispanic students over the White composition distribution. The White risk is highest
in districts with the fewest White students and lowest in the districts with the most White
students: dropping by 29 percentage points. Likewise, Figure E.15 illustrates that as the
40

White composition of a district increases, the Black risk steadily increases by 6.0 percentage
points, which represents growth of 40 percent. As in Enayati (2014), Figure E.15 portrays
a system where the more dissimilar a student looks to her peers, the more likely she is to
participate in special education programs.
Comparing the socio-economic dimensions in Figures E.16 through E.18 reveals similar
patterns. Hispanic participation rates are relatively flat across all three dimensions. Black
students are more likely to participate in special education programs if they are enrolled in
less disadvantaged districts. On the other hand, the White risk increases as districts become
more disadvantaged. These patterns could be explained by the relative relationship between
each of these dimensions and White composition. The percent of free and reduced-price meal
eligible students drops by 44 percentage points over the White composition distribution while
taxable values increase by 0.81 log points.20
The role of school characteristics on racial disparities in special education placement are
examined in Figures E.19 through E.21. Similar to the earlier results, separating the risk of
special education participation by race over the budget measure does not reveal any strong
patterns of disparities. On the other hand, racial risk by urbanicity and academic performance reveal stronger results. White risk drops by 16 and 75 percent over the urbanicity
and performance measures, respectively. Over the same two dimensions, Black risk increases
by 19 and 27 percent, respectively. The patterns observed in the socio-economic and school
characteristic measures are strikingly similar, which then suggests that some variables may
be acting as proxies for other impactful dimensions. In the following section, I use a regression framework analysis, which allows me to examine the correlations between participation
and the socio-economic and school characteristics dimensions while controlling for the White
20 See Appendix Figure H.1 for the relevant analysis.

41

composition.

2.3.3

Regression Framework

Figures E.1 through E.21 illustrate the piecemeal relationships between special education
participation rates and racial, socio-economic, as well as school characteristic measures. To
provide a sense of the relative correlation of each disparity dimension on placement rates
while also accounting for the other dimensions, I consider the following linear probability
model:

Yit = 0 + 1 HT Vit + 2 F ARMit + 2 Wit + 3 Bit + 4 Mit + Ci 5 + t + ✏ij ,

(2.1)

where Yit represents the special education participation measure of interest for district i
in year t.21 Special education participation measures include special education rates by
head count and FTE, intensity, and risk of special education by race. To facilitate the
interpretation of the table, I convert all special education participation measures to percent
values, laying between zero and one hundred. HT Vit is the log of per pupil homestead taxable
values. F ARMit and Wit refer to the percent students in the district that are eligible for
free and reduced-price meals and the White student composition of the district, respectively.
Bit and Mit represent the budget and math test scores, respectively. School characteristics
are contained within Ci : log of median household income and urbanicity measures. The t
term accounts for year fixed e↵ects. Standard errors are made robust by clustering at the
district level.
Columns 1 through 3 of Table B.2 report the estimates of the disparities in special
21 These estimates describe the associations between the dependent and independent variables and do not
support causal claims.

42

education identification using Equation 2.1. Looking across these three columns reveals that
only a subset of the analyzed regressors have a statistically significant association with the
dependent variable after conditioning for the full set of independent variables. Notably,
the special education measures considered in columns 1 through 3 do not have a strong
relationship with either the budget or median income measures but do have statistically
significant associations with taxable homestead values, free and reduced-price meal eligibility,
racial composition, academic performance, and urbanicity. For example, column 1 indicates
that increasing the homestead taxable value by one percent corresponds to a 0.008 percentage
point increase in the special education participation rate. Likewise, increasing the share of
students who at least met the math standards by 1 percentage point is associated with a 0.6
percentage point reduction in participation. The intensity measure in column 3 shows similar
results to column 1 with the exceptions that the impact of homestead taxable values now
dominates over the impact of free lunch. Additionally, the White compositon of a district is
a significant predictor of intensity. A benefit of the regression framework over the graphical
analysis is that we are able to observe that both the homestead taxable values and free and
reduced-price meal eligibility are both significant predictors of participation rates even after
controlling for the other, with a one percentage point increase in free lunch eligbility linked
to a 0.05 percentage point increase in participation.
The disparities in racial risk levels are considered in columns 4 through 6. The evidence
suggests that community resources, as measured by homestead taxable values, are positively
associated with White and Black risks of special education. Alternatively, academic performance and urbanicity are now only significant in one of the two main risk categories.22
22 As evidenced by the graphical analysis, the variation in Hispanic risk is poorly captured by the dimensions
examined in this anaylsis.

43

Finally, the White composition of the school district has estimates of opposite sign between
White and Black risks, i.e., increasing the White composition by one percentage point corresponds to a 0.1 percentage point decrease in White risk but a 0.08 percentage point increase
in Black risk. Thus, the regression framework supports the eariler conclusion that looking
dissimilar to one’s peers increases the probability of special education participation.

2.4

Conclusion

Special education programs in Michigan provide services to one in every seven public school
students. The average additional cost of educating a special education student in 2007
was $11,551 compared to the average K-12 student cost of $9,177 (MDE, 2007). Given the
prevalence and cost of the program, it is crucial for educators and policymakers to understand
the current patterns and disparities in special education participation.
This paper first addressed disparities in overall special education identification and services by exploring the role of family and community resources on special education participation. Evidence consistently indicates that districts serving more economically advantaged
families and communities provide less intense special education services, which appears to
be driven by lower participation rates. Specifically, the share of a special education student’s
day spent receiving special education services is the lowest in the most advantaged districts.
The reasons for this pattern should be investigated further. It could be due to variation
in the need for special education services, as in poverty leading to higher special education
participation, variation in the involvement of and advocacy by parents in the identification
process, or as a consequence of the special education funding system in Michigan (Donovan
and Cross, 2002; Dunn, 1968; Jalilevand and Conlin, 2014; Patton, 1998).

44

Disparities associated with school district characteristics were the next dimension examined in this essay. While the district’s budget is not correlated with the special education
disparities investigated in this study, both district academic performance and urbanicity are
strong predictors of special education participation and intensity. There exists a monotonic
reduction in intensity levels as the proximity to an urban locale decreases. Evidence strongly
supports a negative relationship between district-averaged academic performance and both
special education participation and intensity. Moreover, the urbanicity and performance
relationships persist after controlling for the influence of community financial resources.
Finally, this paper demonstrated a strong correlation between special education placement and district racial composition. Special education intensity drops as the White composition of a district increases. Moreover, I support earlier suggestive evidence of inappropriate
identification practices by showing that a student’s probability of being in a special education program increases as she looks, in terms of race, more dissimilar to her peers (Enayati,
2014). Hosp and Reschly (2003) document that a special education label may harm some
students by either lowering the teacher’s expectations for the student or reducing the access
to the general education curriculum. Given special education costs as well as the potential
for negative side e↵ects, the disparities of who is placed into special education programs and
why they are identified needs to be further understood so as to assure the most beneficial
learning environment for all students.

45

Chapter 3
Class Size and Special Education Placement:
Revisiting Tennessee’s Project STAR

3.1

Introduction

In 2011, federal grants allocated to states for special education programs totaled over 12
billion dollars, representing 35 percent of all federal funds distributed to elementary and secondary programs. These funds provided services to 13 percent of all K-12 students (National
Center for Education Statistics, U.S. Department of Education, 2013). A central tenet of the
Individuals with Disabilities Education Act, the federal act under which special education
funds are distributed to states, is access to free and appropriate public education, which
among other criteria is designed to meet the individual needs of the student and result in
educational benefits. Special education programs are large in scale, costly to fund, and provide critical services to disadvantaged students. It is thus essential to understand the factors
that influence the identification of students with disabilities.
Early research in the field documented the roles of race and socio-economic status in
determining special education placement by demonstrating racially disproportionate representation of Black students with a cognitively impaired label (Dunn, 1968). Between 1975
and 1991, Black students represented roughly double the expected share of students with
a cognitively impaired label given their share in the total population (Patton, 1998).1 In
addition to the overrepresentation of Black students in special education programs, male
1 This expectation is under the assumption that all children, regardless of race, have the same propensity
for conditions requiring special education services.

46

students have a history of out numbering female students with an odds ratio of at least 2:1
since 1976 (Coutinho and Oswald, 2005; Oswald et al., 2003).
More recently we have gained a richer understanding of how school-level characteristics
influence special education participation rates. Hibel (2010) found schools with higher minority populations corresponded with a lower likelihood of a special education label while
increases in socio-economic status were associated with higher participation rates. A potential explanation for these findings is that wealthier schools have more resources available for
special education programs and thus o↵er services to more students. Recent work in health
economics has shown that both the absolute and relative age of a student within his or her
school e↵ects the likelihood of a determinant of mental health diagnoses (Elder, 2010; Elder
and Lubotsky, 2009; Evans et al., 2010). Overall, a large body of literature has documented
the significance of student-specific and school-specific characteristics on the probability of
participating in a special education program.
A current method of special education identification is the Responsiveness to Intervention
(RTI) approach (Fuchs et al., 2003). One of primary goals of RTI is to accurately identify
students with disabilities (Fuchs and Fuchs, 2009). With the support of the professional
community, Congress included provisions in the 2004 reauthorization of the Individuals with
Disabilities Education Act that permit states to use strategies similar to those outlined
under the RTI framework (Zirkel and Krohn, 2008). RTI methods follow a tiered approach
of delivering research-based instructional strategies to students (Canter et al., 2008). A
crucial component in multiple tiers of RTI is increased instructional intensity, which can be
achieved by decreasing class size (Mellard et al., 2010).2 Zahorik (1999) demonstrated that
2 Mellard et al. (2010) outline ten variables to increase instructional intensity; one of which being reduced
class size.

47

smaller class sizes leads to increased individualized instruction through increased knowledge
of students as well as more instructional time as result of fewer behavioral problems.
While reduced class size is a way to increase instructional intensity, it is important to
understand its total impact on special education identification. The total e↵ect of class size
on special education participation is an empirical question. Smaller class sizes should lead to
increases in individualized instruction compared to regular-sized classrooms (Zahorik, 1999).
The e↵ect of this extra instruction is ambiguous. It could allow students on the margin
of joining a special education program the additional support needed to avoid a formal
placement.3 On the other hand, it is also possible that the higher levels of individualized
instruction allow the teacher to more accurately understand the needs of her students and
hence increase placement.
This paper aims to enrich the understanding of reduced class size’s role on special education identification by using Tennessee’s Project STAR (Student/Teacher Achievement
Ratio) experiment. The randomization of Project STAR o↵ers a prime opportunity to test
the e↵ect of class size on special education placement. It allows me to exploit the exogenous
variation in class size induced by the random assignment of students to classrooms to address
the concern of omitted variables bias.
Evidence from this paper supports the hypothesis that the primary impact of small classes
on identification is to a↵ord teachers the time to more fully understand their students’ needs
for special education services. Exploiting the variation in special education participation
induced by class size reductions, I estimate the total e↵ect of small class size as an imprecise
increase in the special education placement rate of 1.8 percentage points. When the model
3 Special education need and programs vary significantly by disability category and intensity. While this
paper discusses special education participation as a binary outcome, it represents a spectrum of services
o↵ered to students. With this in mind, the margin along which I expect class size to a↵ect special education
participation is directly a↵ected by a student’s ability to catch up to grade-level expectations.

48

controls for prior academic performance, I find that small classes increased the probability of
special education placement by 2.8 percentage points. This increase represents a 25 percent
rise in the odds of placement rates among fourth-grade students.
The paper is organized as follows. In Section 2, I briefly describe the experimental design
and discuss the sample of students. Section 3 outlines the identification strategy. Section
4 reports the results of the analysis of the role of reduced class size on special education
participation. Section 5 concludes.

3.2

Project STAR

Project STAR was a large-scale randomized experiment conducted in seventy-nine schools
throughout Tennessee. It randomly assigned students to one of three class types based on
size: small with a goal of thirteen to seventeen students, regular with a goal of twenty-two
to twenty-five students, and regular with a teacher’s aide. The assignment of teachers to
classrooms was also randomized. Notably, all randomization took place within the school,
so the following analysis focuses on within-school variation.
Project STAR followed one cohort of students as they moved from kindergarten to third
grade between 1985 and 1989. In total, the experiment included observations on 11,600
students. At the time of the experiment, kindergarten was not required, so only 55 percent
of all students participated in the first year of the experiment. Students enrolling in Project
STAR schools after kindergarten were also randomly assigned to one of the three class size
types. Students entering in first, second, and third grades represent 20 percent, 14 percent,
and 11 percent, respectively.
While the experimental stage of Project STAR ended in the third grade, students were

49

tracked through high school. Of particular importance to this paper, fourth-grade teachers
were asked to complete a questionnaire, the Fourth-Grade Student Participation Questionnaire (FGSPQ), for a random subset of the former Project STAR students in their class.
Included in this survey was a question asking if the student attends special education classes
outside of the classroom. While measures of special education participation in kindergarten
and first grade are available, Finn et al. (2007) caution against the use of these variables
due to concerns with the distributions of the measures. Special education participation rates
are low overall and even drop from 3.2 percent in kindergarten to 1.2 percent in first grade.
Additionally, the dynamics of participation are concerning with only 6.6 percent of students
observed in a special education program in first grade given they participated in kindergarten. Of those same kindergarten special education students, 29.3 percent were observed
in fourth-grade special education programs.4 Given the data quality concerns, I focus on the
fourth-grade special education measure in this paper.
The full FGSPQ sample consisted of 2,207 students. The analytic sample used in this
paper restricted the observations to 2,025 students from each of the seventy-nine schools.5
One hundred seventy-three observations are dropped due to missing special education status,
gender, or free lunch status information. An additional nine observations are dropped to
create a dichotomy in race between White and Black groups.6
Two well documented threats to the experimental design were the moving of students
across class types and the addition of new experiment participants in the later grades.
4 Appendix Tables I.1 and I.2 document the kindergarten and first-grade special education measures in
greater detail. Of the first-grade special education students who were enrolled in kindergarten, only 22.2
percent were also in a kindergarten special education program. First-grade special education students also
have lower small class assignment and higher free lunch status than kindergarten or fourth grade.
5 School representation is based on initial school placement in the first year of Project STAR participation.
This sample of students entered the experiment in either kindergarten or first grade.
6 The nine dropped observations consisted of four Asian students, four Hispanic students, and one student
with an “Other” race status.

50

Krueger (1999) finds that roughly 10 percent of students were shu✏ed across class types
in a nonrandom manner. As a way to address this concern, I only use the initial class assignment.7 By using only the classroom characteristics from a student’s initial year in Project
STAR, I maintain the exogeneity of classroom assignment created by the experiment’s initial
randomization.
Table C.1 reports the student characteristics for the analytic sample. Of the 2,025 students with a complete FGSPQ, 11.5 percent of the students participated in a special education program. Using data from the National Center for Education Statistics, I calculate
that 12.5 percent of all students in Tennessee were in special education programs during the
1990-91 academic year. The comparability of these percentages gives credence to the external validity, at least within Tennessee and during this time period, of the results. Comparing
the participation rates across class types shows that small and regular rooms have the same
rate of 11 percent while rooms with aides slightly higher rates of 13 percent.
Student demographics are also shown in Table C.1. The sample is evenly mixed between
males and females. Black students constitute 23 percent of the observations, which is 13
percentage points lower than the full Project STAR sample.8 Disadvantaged students, as
measured by free lunch status, represent 41 percent of the sample. All three of the demographic measures are evenly represented in the three classroom types.
Table C.1 also details the actual number of students in the classroom. This number could
deviate from the expected number due to students moving between room types after the
initial assignment. Columns 2 through 4 demonstrate that the initial assignment appears to
7 This approach can not address the potential heterogenous e↵ects of created by the length exposure to
the studied classroom characteristics.
8 The di↵erence in the racial composition between the samples is due to the fact that inner-city Memphis
schools did not participate in the Tennessee Comprehensive Assessment Program in 1989, which precluded
them from participating in the FGSPQ.

51

correspond appropriately to the number of actual classmates, which is evidence of compliance
with the randomization in the initial year for this analytic subsample.
Finally, Table C.1 reports the averaged performance of the math and reading scores of
the Stanford Achievement Test (SAT-9) in a student’s initial year in the project as well as
in fourth grade. Following Schanzenbach (2006), separate z-scores for math and reading
exams are calculated by standardizing by the mean and standard deviation of the scores
from the full sample of Project STAR, non-small class students. I then averaged the two
z-scores to produce a single performance measure.9 Column 1 shows that in both the initial
year and also in fourth grade, the FGSPQ subsample performed better on the SAT-9 than
the full sample, which is observed by the positive, non-zero means among regular and aide
classrooms. Students from small classrooms scored 0.50 standard deviations higher on the
SAT-9 compared to the full-sample, non-small classroom students and roughly 0.3 standard
deviations higher than their FGSPQ non-small class peers. While these numbers do not
account for within-school fixed e↵ects, they do suggest that the impact of small classes on
test scores was large in the FGSPQ sample, a relationship explored in further detail in
Table C.4.

3.3

Identification Strategy

The contribution of this paper to the literature examining special education placement stems
from the randomization of students to classrooms under Project STAR. In his 1999 paper,
9 If one of the math or reading scores was missing, I used the non-missing score as the single performance
measure.

52

Krueger outlines the following model:

Yij = ↵1 Sij + ↵2 Fij + ⇠ij ,

(3.1)

where Yij , Sij , and Fij refer to achievement level, school characteristics, and family background for student i from school j, respectively. He argues that the omitted variables bias
due to excluded relevant variables, such as innate ability, is remedied by Project STAR’s randomization. The random assignment of students to various class sizes means that class size
is independent of the omitted variables. Thus, Project STAR created exogenous variation
in class size.
The randomization under Project STAR allows me to estimate the impact of class size
reductions on special education placement by comparing average participation rates across
class types. Because the randomization occurred within schools and only for the initial
assignment, the analysis must control for school-by-entry-wave fixed e↵ects. Following the
framework developed by Krueger and Whitmore (2001), I first estimate the impact of classroom characteristics on special education placement using the following fixed-e↵ects linear
probability model:

Yis = 0 + 1 SM ALLis + 2 AIDEis + 3 Xis + ↵sw + ✏is ,

(3.2)

where Yis is a fourth-grade special education status indicator variable for student i who
initially entered Project STAR in school s. SM ALLis and AIDEis are indicator variables
equal to one if the student was originally assigned to a small class and a regular class with
an aide, respectively. Given the linear probability model specification, 1 is interpreted as

53

the change in the probability of participating in fourth-grade special education induced by
an initial small class assignment. Xis is a vector of covariates including gender, race, and
free lunch status. To control for school-by-entry-wave e↵ects, ↵sw is included as a fixed
e↵ect, where w represents the wave the student joined the experiment, i.e., kindergarten,
first grade, second grade, or third grade. Finally, standard errors are clustered at the schoolby-entry-wave level to account for heterogeneity.

3.4

Results

Table C.2 reports the estimates of small class size on special education placement using
Equation 3.2. Column 1 presents the estimate of small class size from a regression with no
controls beyond school-by-entry-wave fixed e↵ects. The coefficient indicates a statistically
insignificant increase of 1.8 percentage points in the likelihood of fourth-grade special education placement, which represents 16 percent increase over the base participation rate of 11
percent. The positive direction of the estimate suggests that the hypothesis that increased
individualized instruction led to teachers being able to more accurately identify students who
need special education services. Column 2 builds upon the first column by incorporating the
indicator for assignment to a regular class with an aide. Given that the addition of an aide
to a classroom increases the instructor-to-student ratio, one might expect the introduction
of an aide to produce similar results as small class assignment. Like much of the literature
examining the impact of class size on test scores, I find a statistically and economically insignificant estimate of 0.01 on assignment to a room with an aide (Krueger, 1999; Krueger
and Whitmore, 2001). In an attempt to increase precision, column 3 includes individual
demographic variables; however, no additional precision is observed. The similarity in the

54

small class estimates with no demographic controls supports the exogeneity of class size
assignment. Across columns 1 through 3, the estimated e↵ect of small class assignment on
special education participation holds steady between 0.18 and 0.19.
Table C.2 extends Equation 3.2 by incorporating the z-score from the initial year a
student was in the experiment. The inclusion of this variable allows me to compare the
likelihood of special education placement across class assignments after controlling for the
academic performance of the student. Because special education services are provided to
students with conditions limiting their academic progress, test score measures are strongly
associated with special education participation, e.g., the estimate of fourth-grade z-score on
contemporaneous special education placement is -0.08 with a standard error of 0.01. The
inclusion of the academic performance measure does, however, change the interpretation of
1

to the impact of reduced class size net of the e↵ect of small class assignment on test

scores. The initial z-score is used to most closely capture a student’s baseline achievement.
Column 4 reports a significant decrease in the likelihood of special education placement by
3.5 percentage points associated with a one standard deviation increase in test performance.
The inclusion of the initial z-score also results in a statistically significant estimate of 0.028
on small class assignment. Thus, assignment to a small class net of its role on test scores
increases special education participation by 2.8 percentage points, which represents a 25
percent rise over the regular class rate. Controlling for a measure of academic performance
caused the point estimate for small class assignment to increase from 0.019 to 0.028. Given
the well documented success of assignment to small classrooms at increasing test scores,
the jump in the estimates reveals that the impact of reduced class size on test score was
dampening the estimate of small class assignment on special education placement (Krueger,
1999; Krueger and Whitmore, 2001). Again, I find a positive relationship between small class
55

assignment and participation rates, which points to an increase in teacher understanding of
student needs.
The partially functional relationship between standardized test scores and special education participation implies that if increased individualized instruction leads to increases
in special education placement rates, the marginal students entering special education programs will likely have higher academic achievement.10 Figures F.1 and F.2 illustrate the
distributions of the z-scores for the initial and fourth-grades scores, respectively, by class
type and special education status. Examining the initial z-score, I find that for each class
type the special education distribution is shifted to the left of the general education distribution. Moreover, there is evidence that both the general education and special education
students from small classes out performed the general education students from regular classrooms. If the mechanism behind the increased likelihood of special education placement
is teachers developing a deeper understanding of student need for services, I would expect
to see a thicker right-tail of the special education distribution from small classes than the
distribution from regular classes. Figure F.1 displays evidence of a higher density of special
education students from small classes coming from the right tail of the z-score distribution.
Figure F.2 illustrates the z-score distributions by the time the students are in fourth grade.
As expected, I find that the distribution of general education students from small classes
is farther to the right than general education students from regular classes. Now, however,
special education students from all class types have a similar modal z-score. Despite this
alignment of the special education distributions, the larger density of high scorers from the
small classes persists.
10 Special education programs provide additional services to students with disabilities impeding their academic progress. While not all students in special education programs will have low test scores, it has been
shown that the average score for a special education student is below the average score for a general education
student (Hibel et al., 2010).

56

This graphical evidence supports the hypothesis that small classes allowed teachers to
better understand the needs of their students. Table C.3 reports the means of z-scores by
testing period, class type, and special education status.11 The first row of panel A shows
that general education students from small classes outscored their regular class peers by 0.28
standard deviations, the di↵erence between the small class mean of 0.52 and the regular class
mean of 0.24. The second row highlights that not only the general education students from
small classes but also the special education students from small classes, with an average zscore of 0.40, have higher test scores than the general education students from regular classes.
Overall, panel A demonstrates a small-to-regular test-score gap of 0.41 standard deviations
for special education students, which exceeded the same gap for general education students
by 0.13 standard deviations.
Panel B shows that by the fourth-grade special education students from small classes
are performing below the average among students from regular classes. The small-to-regular
di↵erential remained positive for both general and special education students; however, the
di↵erential for special education students dropped by 59 percent compared to the initial
z-score in panel A. The di↵erence in the fourth-grade di↵erentials reveals that the large lead
held by special education students from small classes had become a deficit compared to their
general education peers. Thus the test-score gap shrunk in both absolute and relative terms
for special education students compared to general education students.
Table C.4 utilizes the within school randomization of Project STAR to estimate the
impacts of both small class assignment and fourth-grade special education status on academic
performance.12 Panel B replicates the findings from Schanzenbach (2006), which are reported
11 In Table C.3, the regular classes with and without aides are combined into the single group.
12 The estimation follows the structure of Equation 3.2 where the dependent variable is the grade-specific
z-score. AIDEis is no longer included, which follows the broader Project STAR literature. Finally, the

57

in panel A.13 Panel C confirms the earlier indications from Tables C.1 and C.3 that the e↵ect
of reduced class size is substantially larger in this paper’s analytic sample compared to the full
Project STAR population. While the estimated e↵ect for the full population of small class
size on kindergarten test performance is an increase by 18 percent of a standard deviation,
the same estimate for the analytic sample is 10 percentage points of a standard deviation
higher at 0.28.
Panel D further illustrates that the di↵erential between panels B and C is driven by
the selection of students who participated in the fourth-grade survey in two ways. First,
traditionally worse performing schools from inner-city Memphis and rural areas did not
participate in fourth grade testing (Finn et al., 2007). Second, the large coefficients for
the interaction term indicate a di↵erential e↵ect of the reduced class size program for the
students in the FGSPQ sample. The estimates range between 14 percent and 20 percent
of a standard deviation and imply that the small class programs were more successful at
increasing test scores in the schools attended by students in the FGSPQ. The reason for this
di↵erence remains a question. Overall, the reduced class size program was highly e↵ective at
improving academic performance among the students in the FGSPQ, so one might expect this
sample of students to be prime candidates for exhibiting an e↵ect of small class assignment
on special education placement.
Panel E presents the estimates of interest in this line of inquiry and also elucidates
the relationship between reduced class size and special education participation. Looking at
both the fourth-grade z-score and earlier z-scores, I find strong positive coefficients for the
impact of small classes on academic performance, ranging between 19 and 29 percent of
special education indicator is now an independent variable and is interacted with the small class indicator.
13 I am unable to replicate the fourth-grade results due to di↵ering data sources for the fourth-grade sample.
Because this paper focuses on special education, the FGSPQ data best met the needs of the project.

58

a standard deviation. With the exception of kindergarten, fourth-grade special education
status has a statistically significant negative association with test scores. The interaction
term identifies any heterogeneous impact of reduced class size on academic performance by
special education status. The estimates are large, ranging between a reduction by 16 percent
of a standard deviation to an increase of 8 percent; however, each of the four estimates fails
to be significantly di↵erent from zero. Thus, I find no clear evidence of an additional impact,
either positive or negative, of special education status on reduced class size’s role on academic
performance.
If the primary e↵ect of reduced class size on participation was providing a stronger
learning environment and hence pushing the marginal special education student back into
general education status, I would have expected to find a negative estimate of reduced class
size on the probability of placement, i.e., assignment to a small class lowered the likelihood
of special education placement. This result is a consequence of the partially functional
relationship between special education status and test scores as well as the primary e↵ect
assumption as long the relative change in the special education threshold between small
and regular class types did not exceed the relative growth in test scores between small and
regular classes. Also under the primary e↵ect assumption, I would have anticipated a positive
interaction term for the estimate of small class size and participation on test scores because
the rightward shift of the small class test score distribution implies a thicker left tail for the
regular class students in special education. To the contrary, I find weak evidence of a positive
impact of reduced class size on special education placement rates, which coincides with the
hypothesis that increased individualized instruction led to increased teacher understanding of
student need. Additionally, the lack of a clear heterogenous impact, i.e., the small class and
special education interaction term, supports the evidence that factors outside of academic
59

performance influence the special education placement decision.

3.5

Conclusion

A longstanding and large literature has documented the role of individual-level and schoollevel characteristics on special education identification. Evidence of the overrepresentation
of Black students with cognitive impairment labels dates back to the 1960s. More recently,
researchers have discussed the high placement rates among males. Even newer evidence
supports that not only one’s own race but also the racial composition of one’s school a↵ects
the probability of special education participation (Enayati, 2014; Hibel et al., 2010).
Increased instructional intensity is a main feature of the RTI framework and can be
achieved through smaller class sizes. At the same time, lower student-to-teacher ratios
have been shown to increase student performance. These two e↵ects lead to a potentially
ambiguous impact of reduced class size on special education placement rates.
This paper aimed to determine the impact of reduced class size on special education identification. Examining the total e↵ect of reduced class sizes in the early grades of elementary
school, I find weak evidence of an increase in the fourth-grade special education participation
rate. Incorporating prior test scores into the model increases the estimate of small class size
and indicates that reduced class sizes correspond with increased identification rates. These
results as well as further analysis of the joint roles of reduced class size and special education
status on academic performance support the hypothesis that the primary e↵ect of class size
reduction on identifying students with disabilities is improved teacher understanding of student need. Linking these results to the RTI literature suggests that increased individualized
instruction gives a teacher the time necessary to identify student need for services in a way

60

that standardized-test-score deficiency might not.

61

APPENDICES

62

Appendix A

Tables for “The Impact of Disproportionality Regulations on Identification into Special Education Programs”

Table A.1: Characteristics of Districts By Verified Ratio
Pre 2007

Post 2007
(1)
(2)
(3)
(4)
All
Disproportionate
All
Disproportionate
Black Verified Ratio
1.30
4.27
1.35
4.02
(0.71)
(1.33)
(0.62)
(0.83)
Black Risk
18.75
55.41
18.57
52.53
(9.25)
(23.60)
(7.97)
(22.82)
White Risk
15.77
13.43
15.42
13.32
(5.38)
(5.61)
(5.53)
(6.96)
Overall Risk
14.53
13.96
14.22
14.04
(3.66)
(5.92)
(3.67)
(7.51)
Total Enrollment
6756.93
4889.50
5823.32
4877.94
(11352.09)
(1668.59)
(7596.17)
(2050.30)
Black Composition
20.69
1.99
20.84
2.43
(23.29)
(0.67)
(23.46)
(0.87)
White Composition
69.14
90.80
67.09
88.52
(24.55)
(3.81)
(25.25)
(5.94)
FARM
37.51
17.68
45.22
22.99
(23.38)
(12.10)
(22.95)
(19.45)
Special Education
8570.42
13332.32
8794.23
14652.14
Costs
(3365.98)
(7372.46)
(3475.50)
(6801.56)
Federal Special
991.74
1278.79
1303.34
1148.12
Education Revenue
(602.34)
(819.82)
(867.81)
(737.42)
Student-Teacher
20.25
15.42
24.26
16.27
Ratio
(7.38)
(4.43)
(34.04)
(5.55)
Observations
428
12
821
16
Threat Count
10
19
Penalty Count
4
15
Note: Standard deviations are reported in parentheses. A district is treated as disproportionate if
its current Black verified ratio exceeds 3.

63

Table A.2: DD E↵ect of IDEA Sanctions on Racial Disproportionality

Total E↵ect
Post 2007 * Penalty
Post 2007 * Threat
Penalty
Threat
Post 2007
Charters Included
Observations
R2
F

(1)
Black VR
-2.31⇤⇤⇤
(0.38)
-2.92⇤⇤⇤
(0.96)
0.61
(0.76)
2.40⇤⇤⇤
(0.77)
1.36⇤⇤⇤
(0.41)
-0.05
(0.04)
No
1249
0.57
49.62

(2)
Black Risk
-29.55⇤⇤
(13.94)
-34.41⇤⇤
(16.63)
4.85
(9.72)
23.68⇤⇤
(10.27)
23.38⇤⇤
(10.86)
-0.93
(0.77)
No
1249
0.35
9.26

(3)
White Risk
-1.83
(2.94)
-2.71
(3.76)
0.88
(2.31)
1.06
(2.62)
0.10
(1.78)
-0.14
(0.17)
No
1249
0.39
11.95

(4)
Black VR
-1.98⇤⇤⇤
(0.49)
-2.60⇤⇤
(1.03)
0.62
(0.76)
2.53⇤⇤⇤
(0.83)
1.36⇤⇤⇤
(0.41)
-0.06
(0.04)
Yes
1399
0.56
37.39

(5)
Black Risk
-31.29⇤⇤
(13.77)
-36.20⇤⇤
(16.73)
4.91
(9.74)
23.51⇤⇤
(10.45)
23.31⇤⇤
(10.85)
-0.82
(0.71)
Yes
1399
0.39
19.02

(6)
White Risk
-3.06
(3.40)
-4.09
(4.32)
1.03
(2.32)
0.64
(2.83)
0.07
(1.80)
-0.07
(0.24)
Yes
1399
0.37
9.84

Note: Each specification contains controls including a cubic in years since 2000, percent of students eligible for free
and reduced-price meals, and a quadratic of the White composition. Standard errors are reported in parentheses and
clustered at the district level. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

64

Table A.3: Di↵erential E↵ects of IDEA Sanctions on Racial Disproportionality

Total E↵ect:
First Penalty
Multiple Penalties
Post 2007 *
Consecutive Penalties
Post 2007 * Penalty
Post 2007 * Threat
Consecutive
Penalties
Penalty
Threat
Post 2007
Charters Included
Observations
R2

(1)
Black VR

(2)
Black Risk

(3)
White Risk

(4)
Black VR

(5)
Black Risk

(6)
White Risk

-2.57⇤⇤
(1.29)
-2.05⇤⇤⇤
(0.67)
0.52
(1.78)
-3.18⇤
(1.62)
0.61
(0.76)
-0.02
(1.15)
2.65⇤⇤
(1.32)
1.36⇤⇤⇤
(0.41)
-0.05
(0.05)
No
1249
0.57

-36.96⇤⇤⇤
(11.97)
-16.21⇤
(8.85)
20.76⇤⇤
(8.75)
-41.87⇤⇤⇤
(13.58)
4.91
(9.73)
-31.51⇤⇤⇤
(10.14)
40.38⇤⇤⇤
(10.01)
23.38⇤⇤
(10.91)
-1.15
(0.72)
No
1249
0.36

-2.38
(3.41)
0.20
(1.02)
2.58
(2.54)
-3.27
(3.59)
0.89
(2.32)
-6.03
(4.13)
3.92
(3.47)
0.11
(1.80)
-0.19
(0.16)
No
1249
0.39

-1.99
(1.28)
-1.91⇤⇤⇤
(0.60)
0.07
(1.63)
-2.60
(1.61)
0.62
(0.76)
-0.01
(1.15)
2.65⇤⇤
(1.32)
1.36⇤⇤⇤
(0.41)
-0.06
(0.05)
Yes
1399
0.56

-41.77⇤⇤⇤
(10.46)
-17.05⇤⇤
(8.33)
24.72⇤⇤⇤
(7.09)
-46.72⇤⇤⇤
(13.24)
4.95
(9.75)
-31.55⇤⇤⇤
(10.18)
40.36⇤⇤⇤
(10.02)
23.29⇤⇤
(10.89)
-0.98
(0.67)
Yes
1399
0.40

-5.07
(3.81)
-0.31
(1.24)
4.76⇤
(2.61)
-6.11
(4.35)
1.03
(2.33)
-6.16
(4.15)
3.90
(3.49)
0.07
(1.82)
-0.10
(0.24)
Yes
1399
0.37

Note: Each specification contains controls including a cubic in years since 2000, percent of students eligible for free
and reduced-price meals, and a quadratic of the White composition. Standard errors are reported in parentheses and
clustered at the district level. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

65

Table A.4: DD E↵ect of IDEA Sanctions on Other Outcomes

Total E↵ect
Post 2007 * Penalty
Post 2007 * Threat
Penalty
Threat
Post 2007
Charters Included
Observations
R2
F

Overall
Risk
-2.12
(3.16)
-3.31
(4.21)
1.19
(2.63)
1.07
(2.91)
0.56
(1.98)
-0.13
(0.15)
No
1249
0.17
8.41

Total
Enrollment
489.23
(1072.44)
777.00
(1085.37)
-287.77
(1359.06)
1177.00
(984.88)
-2779.20⇤⇤
(1404.85)
-249.45
(152.47)
No
1249
0.11
2.36

Black
Composition
1.29
(3.25)
3.96
(6.10)
-2.67
(3.99)
5.91
(5.34)
-7.87⇤⇤⇤
(2.94)
0.75
(0.60)
No
1249
0.45
17.77

White
Composition
0.62
(4.28)
-2.98
(8.01)
3.61
(4.95)
-10.25
(7.16)
10.08⇤⇤⇤
(3.82)
-0.78
(0.68)
No
1249
0.53
33.64

FARM
1.04
(4.33)
-2.88
(9.05)
3.92
(6.14)
-8.83
(7.79)
0.57
(4.24)
-1.31⇤
(0.67)
No
1249
0.56
105.53

Teacher
Ratio
2.52
(3.06)
4.15
(6.04)
-1.63
(5.84)
-2.70⇤⇤
(1.36)
-4.76⇤⇤⇤
(1.44)
2.22
(3.12)
No
1249
0.01
20.78

Note: The specifications for overall risk, total enrollment, and teacher ratio contain controls including a cubic
in years since 2000, percent of students eligible for free and reduced-price meals, and a quadratic of the White
composition. The racial composition specifications contain controls including a cubic in years since 2000 and
percent of students eligible for free and reduced-price meals. The FARM specification only includes controls for a
cubic in years since 2000 and a quadratic of the White composition. Standard errors are reported in parentheses
and clustered at the district level. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

66

Table A.5:
Disproportionality in Free and
Reduced-Price Meal Eligibility

Penalty = 0

Penalty = 1

Di↵erence

(1)
Pre 2007
1.34
(0.07)
[145]
5.00
(3.89)
[2]
3.66
(0.65)

(2)
Post 2007
1.21
(0.05)
[488]
4.13
(1.33)
[7]
2.92
(0.48)

(3)
Di↵erence
-0.13
(0.11)
-0.87
(3.09)
-0.74
(0.96)

Note: Standard errors are reported in parentheses, and
observations are in brackets. This table presents a crude
di↵erence-in-di↵erences analysis of the impact of financial
penalties on disproportionality in free and reduced-price
meal eligibility.

67

Table A.6: Characteristics of Ineligible Districts By Verified Ratio

Black VR
Black Risk
White Risk
Overall Risk
Total Enrollment
Black Composition
White Composition
FARM
Special Education
Costs
Federal Special
Education Revenue
Student-Teacher
Ratio
Observations
Threat Count
Penalty Count

Pre 2007
(1)
(2)
All
Disproportionate
1.40
3.85
(0.90)
(0.66)
18.57
46.45
(11.80)
(14.47)
14.38
12.11
(6.45)
(2.28)
13.63
12.30
(2.74)
(2.23)
2096.07
1588.78
(1181.31)
(862.30)
3.63
0.68
(14.43)
(0.25)
89.94
93.88
(15.30)
(4.75)
34.19
37.03
(16.22)
(16.19)
5349.15
5693.87
(1413.33)
(1800.74)
410.23
330.37
(586.33)
(487.71)
21.88
24.59
(7.11)
(10.25)
297
18
18
6

Post 2007
(3)
(4)
All
Disproportionate
1.65
4.21
(1.15)
(1.53)
20.93
50.28
(13.95)
(18.92)
14.22
12.50
(7.68)
(2.77)
13.37
12.78
(2.76)
(2.84)
1755.74
1386.51
(900.29)
(541.91)
3.32
0.92
(13.10)
(0.41)
89.64
91.68
(14.66)
(8.64)
41.86
40.61
(15.59)
(17.67)
5251.23
5065.87
(1515.71)
(1679.44)
534.42
669.91
(618.63)
(794.32)
25.29
24.85
(20.04)
(7.69)
404
41
30
9

Note: Standard deviations are reported in parentheses. A district is treated as disproportionate if
its current Black verified ratio exceeds 3.

68

Table A.7: DDD E↵ect of IDEA Sanctions on Racial Disproportionality

Total E↵ect
Post 2007 * Eligible
* Penalty
Post 2007 * Eligible
* Threat
Post 2007 * Penalty
Post 2007 * Threat
Eligible * Penalty
Eligible * Threat
Penalty
Threat
Post 2007 * Eligible
Eligible
Post 2007
Charters Included
Observations
R2

(1)
Black VR
-2.17⇤⇤⇤
(0.75)
-2.81⇤⇤
(1.24)
0.63
(0.92)
0.02
(0.82)
-0.01
(0.52)
1.98⇤⇤
(0.80)
-0.19
(0.48)
0.44
(0.30)
1.56⇤⇤⇤
(0.25)
-0.18⇤⇤
(0.07)
0.21⇤⇤⇤
(0.08)
0.12
(0.08)
No
1789
0.37

(2)
Black Risk
-35.11⇤⇤
(14.41)
-41.12⇤⇤
(17.69)
6.01
(11.68)
8.97
(7.21)
-1.00
(6.49)
26.53⇤⇤
(11.28)
4.10
(11.77)
-3.32
(4.37)
19.39⇤⇤⇤
(3.77)
-1.65⇤
(0.98)
3.68⇤⇤⇤
(1.12)
0.58
(1.12)
No
1789
0.25

(3)
White Risk
-3.73
(3.04)
-4.22
(3.83)
0.49
(2.71)
1.82
(1.16)
0.47
(1.18)
3.05
(2.65)
1.73
(2.18)
-1.71⇤⇤
(0.86)
-1.31
(0.90)
0.02
(0.41)
-0.01
(0.94)
0.10
(0.33)
No
1789
0.34

69

(4)
Black VR
-1.92⇤⇤⇤
(0.69)
-2.16⇤
(1.18)
0.24
(0.91)
-0.40
(0.67)
0.39
(0.51)
1.68⇤⇤
(0.80)
0.24
(0.52)
0.75⇤⇤
(0.31)
1.16⇤⇤⇤
(0.30)
0.00
(0.16)
0.01
(0.18)
0.00
(0.11)
Yes
2152
0.14

(5)
(6)
Black Risk White Risk
-31.94⇤⇤
-5.19
(14.23)
(3.57)
-37.30⇤⇤
-8.89⇤
(17.65)
(5.00)
5.36
3.70
(11.41)
(3.71)
3.89
5.29⇤
(7.09)
(2.72)
-0.25
-2.49
(5.86)
(2.79)
25.01⇤⇤
5.84
(11.23)
(3.63)
6.24
-0.94
(11.65)
(3.21)
-1.58
-4.80⇤
(4.14)
(2.49)
⇤⇤⇤
17.60
1.50
(3.44)
(2.59)
-1.67⇤
-0.53
(0.86)
(0.44)
2.68⇤⇤⇤
-0.37
(0.97)
(0.92)
0.86
0.25
(0.99)
(0.49)
Yes
Yes
2152
2151
0.25
0.29
Continued on next page

F

(1)
Black VR
31.17

Table A.7 (cont’d)
(2)
(3)
Black Risk White Risk
11.61
11.28

(4)
Black VR
26.48

(5)
Black Risk
12.66

(6)
White Risk
10.57

Note: Each specification contains controls including a cubic in years since 2000, percent of students eligible for free
and reduced-price meals, and a quadratic of the White composition. Standard errors are reported in parentheses
and clustered at the district level. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

70
70

Table A.8: Potential Causes of Disproportionality

White Composition

(1)
Black Risk
0.10⇤⇤⇤
(0.02)

(2)
Black Risk
0.09⇤⇤⇤
(0.03)
4.72
(3.45)
-2.25
(2.15)

11.69⇤⇤⇤
(1.34)
168
0.09
20.94

-14.65
(30.31)
168
0.11
8.37

White Income
Black Income
FARM
Constant
Observations
R2
F

(3)
Black Risk
0.11⇤⇤
(0.05)
8.37
(6.19)
-2.20
(2.12)
0.07
(0.08)
-59.94
(54.74)
168
0.11
6.36

(4)
White Risk
-0.11⇤⇤⇤
(0.02)

(5)
White Risk
-0.09⇤⇤⇤
(0.02)
-2.80
(1.70)
-0.59
(0.77)

23.59⇤⇤⇤
(1.66)
168
0.30
30.90

58.80⇤⇤⇤
(14.47)
168
0.33
15.75

(6)
White Risk
-0.04⇤⇤
(0.02)
4.96⇤
(2.98)
-0.48
(0.77)
0.15⇤⇤⇤
(0.04)
-37.41
(29.39)
168
0.42
18.55

Note: Income measures (in 2009 inflation-adjusted dollars) come from the 2009 5-year American Community Survey
(ACS) and represent the median income, in logs, within a district by race. For comparability, risk, White composition,
and free and reduced-price meal measures are restricted to the same time period as the ACS data, i.e., 2004-2009, and the
median by district values are used. I additionally restrict the sample to the DD baseline sample without charter schools.
Due to the availability of district-level income data from the ACS and the time restriction, I lose 15 districts used in the
DD sample. Standard errors are reported in parentheses. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

71

Appendix B

Tables for “Disparities in Special Education Identification: Evidence from Michigan”

Table B.1: Summary Statistics

Special Education Participation (Headcount)
Special Education Participation (FTE)
Special Education Intensity
In General Education 80% of Day
Homestead Taxable Values Per Pupil
Median Household Income
Percent FARM
Budget
White Composition of Students
City
Suburb
Town
Rural
Met/Exceeded 8th Grade Math Standards
White Risk of Special Education
Black Risk of Special Education
Hispanic Risk of Special Education

Mean
.136
.036
.261
63.71
130042
49685
38.14
.058
85.83
.069
.256
.182
.493
68.43
14.22
19.35
13.74

SD
.032
.019
.092
14.41
50839
13049
18.20
3.14
18.10
.254
.437
.385
.500
14.97
5.14
11.40
8.65

N
4198
4198
4198
505
4198
505
4198
4198
4198
4198
4198
4198
4198
4198
4194
3016
3133

Note: Statistics drawn from a sample of 523 traditional public school districts from
the 2002-03 through the 2010-11 academic years. The taxable value and median
household income measures are per the total number of pupils in the district. All
dollar terms have been converted to 2010 U.S. dollars. Censored special education
by race data lead to the lower number of observations in the risk of special education
variables.

72

Table B.2: Disparities in Special Education Identification

Log Homestead
Taxable Values Per Pupil
Percent FARM
White Composition of
Students
Budget
Met/Exceeded 8th
Grade Math Standards
Log Median Income
City
Suburb
Town
Observations
R2

(1)
HC
0.813⇤
(0.414)
0.052⇤⇤⇤⇤
(0.014)
0.009
(0.010)
-0.011
(0.020)
-0.063⇤⇤⇤⇤
(0.009)
-0.035
(0.932)
0.668
(0.547)
0.698⇤⇤
(0.328)
0.490
(0.302)
4198
0.19

(2)
FTE
-0.393
(0.245)
0.016⇤⇤
(0.008)
-0.009
(0.008)
0.002
(0.012)
-0.035⇤⇤⇤⇤
(0.006)
-0.045
(0.481)
1.263⇤⇤⇤
(0.419)
0.767⇤⇤⇤⇤
(0.216)
0.610⇤⇤⇤⇤
(0.159)
4198
0.29

(3)
Intensity
-3.743⇤⇤⇤⇤
(0.969)
0.020
(0.031)
-0.053⇤
(0.031)
0.025
(0.053)
-0.096⇤⇤⇤⇤
(0.023)
-0.256
(2.118)
7.298⇤⇤⇤⇤
(1.790)
4.269⇤⇤⇤⇤
(0.968)
3.624⇤⇤⇤⇤
(0.796)
4198
0.28

(4)
White Risk
0.921⇤
(0.496)
0.026
(0.038)
-0.101⇤⇤
(0.050)
0.006
(0.027)
-0.089⇤⇤⇤⇤
(0.022)
-1.746
(2.060)
-1.442
(1.950)
-0.403
(0.911)
0.067
(0.533)
4194
0.29

(5)
Black Risk
2.827⇤⇤
(1.315)
0.035
(0.036)
0.076⇤⇤⇤
(0.024)
-0.098
(0.083)
-0.031
(0.033)
2.206
(2.288)
-0.725
(1.133)
-2.002⇤⇤
(0.991)
-0.804
(1.138)
3016
0.05

(6)
Hispanic Risk
0.425
(1.072)
-0.030
(0.034)
0.014
(0.030)
-0.042
(0.057)
-0.019
(0.026)
-4.577⇤⇤
(2.131)
-1.496
(1.226)
0.197
(0.833)
-1.121
(0.893)
3133
0.03

Note: Estimates from a linear probability model with year fixed e↵ects. Taxable values and median income values have been
converted to 2010 U.S. dollars. Again, censored special education by race data lead to the lower number of observations in
the risk of special education variables. Standard errors clustered at the district level. Significance levels: ⇤ p < 0.10, ⇤⇤ p <
0.05, ⇤⇤⇤ p < 0.01, ⇤⇤⇤⇤ p < 0.001.

73

Appendix C

Tables for “Class Size and Special Education Placement: Revisiting Tennessee’s
Project STAR”

Table C.1: Means of Selected Characteristics
(1)
All
Special Education 0.11
(0.32)
Female
0.50
(0.50)
Black
0.23
(0.42)
Free Lunch
0.41
(0.49)
Class Size
20.00
(4.02)
Initial Z-Score
0.31
(0.87)
th
4 Grade Z-Score 0.18
(0.90)
Observations
2025

(2)
Small
0.11
(0.31)
0.52
(0.50)
0.25
(0.43)
0.43
(0.50)
15.08
(1.56)
0.50
(0.90)
0.33
(0.90)
665

(3)
(4)
Aide Regular
0.13
0.11
(0.33) (0.31)
0.46
0.52
(0.50) (0.50)
0.21
0.23
(0.41) (0.42)
0.39
0.40
(0.49) (0.49)
22.60
22.20
(2.28) (2.26)
0.21
0.20
(0.82) (0.85)
0.08
0.13
(0.89) (0.90)
688
672

Note: Means with standard deviations in parentheses. Small,
aide, and regular refer to the three classroom types of Project
STAR. Free lunch and initial z-score are determined using data
from a student’s first year in the experiment.

74

Table C.2: E↵ect of Small Class Size on Fourth Grade Special
Education Placement

Small

(1)
0.018
(0.015)

(2)
0.018
(0.016)
0.001
(0.017)

(3)
0.019
(0.016)
0.000
(0.017)
-0.041⇤⇤⇤
(0.015)
0.029
(0.041)
0.087⇤⇤⇤
(0.018)

0.109⇤⇤⇤
(0.005)
2025
0.00

0.108⇤⇤⇤
(0.009)
2025
0.00

0.086⇤⇤⇤
(0.016)
2025
0.02

Aide
Female
Black
Free Lunch
Initial Z-Score
Constant
Observations
R2

(4)
0.028⇤
(0.016)
-0.001
(0.017)
-0.040⇤⇤⇤
(0.015)
0.020
(0.039)
0.077⇤⇤⇤
(0.018)
-0.035⇤⇤
(0.015)
0.100⇤⇤⇤
(0.016)
2025
0.03

Note: Estimates from a linear probability model with school-by-entrywave fixed e↵ects. Free lunch and initial z-score are determined using
data from a student’s first year in the experiment. Standard errors
clustered at the school-by-entry-wave level. Significance levels: ⇤ p <
0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

75

Table C.3: Test Score Means by Class Size and Special Education Status

Panel A - Initial Z-Score
General Education
Special Education
Di↵erence
Panel B - 4th Grade Z-Score
General Education
Special Education
Di↵erence

(1)
Regular

(2)
Small

(3)
Di↵erence

0.24
(0.02)
-0.01
(0.08)
-0.24
(0.07)

0.52
(0.04)
0.40
(0.12)
-0.11
(0.11)

0.28
(0.04)
0.41
(0.14)
0.13
(0.13)

0.18
(0.02)
-0.43
(0.09)
-0.61
(0.07)

0.40
(0.03)
-0.26
(0.14)
-0.66
(0.11)

0.22
(0.04)
0.17
(0.17)
-0.05
(0.13)

Note: Means with standard errors in parentheses.

76

Table C.4: E↵ect of Small Class Size and Special Education Status on Academic Performance
Kindergarten
Independent Var.
(1)
Panel A:

First Grade
(2)

Second Grade
(3)

Third Grade
(4)

Fourth Grade
(5)
0.035
(0.025)

Small

0.187⇤⇤⇤
(0.039)

0.189⇤⇤⇤
(0.035)

0.141⇤⇤⇤
(0.034)

0.152⇤⇤⇤
(0.030)

Small

0.183⇤⇤⇤
( 0.043)

0.191⇤⇤⇤
( 0.036)

0.131⇤⇤⇤
( 0.036)

0.149⇤⇤⇤
( 0.031)

Small

0.280⇤⇤⇤
( 0.056)

0.231⇤⇤⇤
( 0.052)

0.179⇤⇤⇤
( 0.057)

0.234⇤⇤⇤
( 0.046)

Small

0.095⇤⇤
( 0.042)
0.251⇤⇤⇤
( 0.026)
0.197⇤⇤⇤
( 0.050)

0.105⇤⇤
( 0.042)
0.269⇤⇤⇤
( 0.032)
0.145⇤⇤⇤
( 0.053)

0.053
( 0.037)
0.132⇤⇤⇤
( 0.030)
0.140⇤⇤⇤
( 0.054)

0.068⇤
( 0.040)
0.034
( 0.033)
0.170⇤⇤⇤
( 0.056)

0.290⇤⇤⇤
( 0.059)
-0.025
( 0.122)
-0.083
( 0.168)
5855
1542

0.228⇤⇤⇤
( 0.054)
-0.350⇤⇤⇤
( 0.091)
0.081
( 0.172)
6447
1996

0.185⇤⇤⇤
( 0.057)
-0.478⇤⇤⇤
( 0.102)
0.040
( 0.197)
5829
1903

0.255⇤⇤⇤
( 0.050)
-0.408⇤⇤⇤
( 0.115)
-0.106
( 0.185)
5929
1937

Panel B:

Panel C:

Panel D:

In FGSPQ
Small*In FGSPQ
Panel E:
Small
Sp Ed
Small*Sp Ed
Observations (B and D):
Observations (C and E):

0.195⇤⇤⇤
( 0.045)

0.223⇤⇤⇤
( 0.049)
-0.512⇤⇤⇤
( 0.107)
-0.156
( 0.196)
2025

Note: Estimates from a linear probability model with school-by-entry-wave fixed e↵ects. FGSPQ refers to the analytic sample
used in the paper. Sp Ed refers to fourth-grade special education status. Standard errors clustered at the school-by-entry-wave
level. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

77

Figures for “The Impact of Disproportionality Regulations on
Identification into Special Education Programs”
Figure D.1: Impact of Penalties on the Black Verified Ratio

2

Black VR
4

6

8

Sanctions Begin

0

Appendix D

2004

2005

2006

2007

2008

2009

2010

2011

Year
Policy Period Mean - No Penalty
Policy Period Mean - Penalty
Penalty Districts

Note: This figure plots the policy period means of the Black verified ratio for districts
in penalty phases and districts in neither threat nor penalty phases. Scatter points for
districts in neither threat or penalty phases were excluded for illustrative purposes. The
data have been restricted to the analytic sample used in the DD regressions, leaving
only traditional public school districts eligible for disproportionality calculations. For
interpretation of the references to color in this and all other figures, the reader is referred
to the electronic version of this dissertation.

78

Figure D.2: Impact of Penalties on the Black Verified Ratio for Ineligible
Districts

0

2

Black VR
4
6

8

10

Sanctions Begin

2004

2005

2006

2007
Year

2008

2009

2010

Policy Period Mean - No Penalty
Policy Period Mean - Penalty
Penalty Districts

Note: This figure plots the policy period means of the Black verified ratio for districts
in penalty phases and districts in neither threat nor penalty phases. Scatter points for
districts in neither threat or penalty phases were excluded for illustrative purposes. The
data have been restricted to the analytic sample used in the DDD regressions, leaving only
traditional public school districts ineligible for disproportionality calculations.

79

10

20

15

Risk
20

25

40
60
80
White Composition

30

100

Figure D.3: Risk by Race Over White Composition

0

2

4
6
Deciles of White Composition

Risk Black

Risk White

8

10

White Composition

Note: This graphic shows the relationship between the risk of special education placement
for Black and White students separately over the distribution of total White composition.
The sample has been restricted to the DD analytic sample without charter schools.

80

Figures for “Disparities in Special Education Identification:
Evidence from Michigan”

Special Education Intensity
.25
.3
.35

.4

.05
.1
.15
Special Education Per Pupil Enrollment

Figure E.1: Special Education Participation Over Homestead Taxable Values

0

.2

Appendix E

10.5

11
11.5
12
Log Homestead Taxable Values Per Pupil
Intensity

Headcount

12.5
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

81

20

25
30
35
40
Percent: Speech and Specific

45

Percent: Cognitive, Emotional, Autism,
Other, and Physical
0
5
10
15

Figure E.2: Disability Categories Over Homestead Taxable Values

10.5

11
11.5
12
Log Homestead Taxable Values Per Pupil
Cognitive
Physical

Speech
Emotional

Autism
Specific

12.5
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

82

0

.2

Special Education Intensity
.25
.3
.35

.4

.05
.1
.15
Special Education Per Pupil Enrollment

Figure E.3: Special Education Participation Over Median Household Income

10

10.5
11
Log Median Income
Intensity

Headcount

11.5
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

83

20

25
30
35
40
Percent: Speech and Specific

45

Percent: Cognitive, Emotional, Autism,
Other, and Physical
5
10
15
20

Figure E.4: Disability Categories Over Median Household Income

10

10.5
11
Log Median Income
Cognitive
Physical

Speech
Emotional

Autism
Specific

11.5
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

84

.05
.1
.15
.2
Special Education Per Pupil Enrollment
0

.2

Special Education Intensity
.3
.4

.5

Figure E.5: Special Education Participation Over the Percent of Students
Eligible Free and Reduced-Price Meals

0

20
Intensity

40
60
Percent FARM
Headcount

80

100
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

85

10

20
30
40
Percent: Speech and Specific

Percent: Cognitive, Emotional, Autism,
Other, and Physical
0
5
10
15
20
25

50

Figure E.6: Disability Categories Over the Percent of Students Eligible Free
and Reduced-Price Meals

0

20
Cognitive
Physical

40
60
Percent FARM
Speech
Emotional

80
Autism
Specific

100
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

86

.25

.04

Special Education Intensity
.26
.27
.28
.29

.06
.08
.1
.12
.14
Special Education Per Pupil Enrollment

.3

Figure E.7: Special Education Participation Over District Budget

-10

-5
Intensity

0
Headcount

5

10
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

87

20

30
40
Percent: Speech and Specific

Percent: Cognitive, Emotional, Autism,
Other, and Physical
2
4
6
8
10
12

50

Figure E.8: Disability Categories Over District Budget

-10

-5
Cognitive
Physical

0
Budget
Speech
Emotional

5
Autism
Specific

10
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

88

24

Special Education Intensity
26
28
30
32

4
6
8
10
12
14
Special Education Per Pupil Enrollment

34

Figure E.9: Special Education Participation Over District Urbanicity

City

Suburb

Town

Rural

Urbanicity
Intensity

Headcount

FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

89

25

30
35
40
Percent: Speech and Specific

45

Percent: Cognitive, Emotional, Autism,
Other, and Physical
4
6
8
10
12

Figure E.10: Disability Categories Over District Urbanicity

City

Suburb

Town

Rural

Urbanicity
Cognitive
Physical

Speech
Emotional

Autism
Specific

Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

90

0

.2

Special Education Intensity
.3
.4
.5

.05
.1
.15
.2
Special Education Per Pupil Enrollment

.6

Figure E.11: Special Education Participation Over the Share of Eighth
Graders That Met or Exceeded Math Standards

0

20
40
60
80
Met/Exceeded 8th Grade Math Standards
Intensity

Headcount

100
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

91

50
10

20
30
40
Percent: Speech and Specific

Percent: Cognitive, Emotional, Autism,
Other, and Physical
0
10
20
30
40

Figure E.12: Disability Categories Over the Share of Eighth Graders That
Met or Exceeded Math Standards

0

20
40
60
80
Met/Exceeded 8th Grade Math Standards
Cognitive
Physical

Speech
Emotional

Autism
Specific

100
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

92

0

.25

Special Education Intensity
.3
.35
.4
.45

.5

.05
.1
.15
.2
Special Education Per Pupil Enrollment

Figure E.13: Special Education Participation Over White Composition

0

20

40
60
White Composition of Students

Intensity

Headcount

80

100
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

93

50
10

20
30
40
Percent: Speech and Specific

Percent: Cognitive, Emotional, Autism,
Other, and Physical
0
5
10
15
20
25

Figure E.14: Disability Categories Over White Composition

0

20

40
60
White Composition of Students

Cognitive
Physical

Speech
Emotional

80

Autism
Specific

100
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

94

10
15
20
25
Black and Hispanic Risk of Special Education

10

White Risk of Special Education
20
30
40
50

60

Figure E.15: Risk of Special Education Over White Composition

0

20

40
60
White Composition of Students
White

Black

80

100

Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

95

12
14
16
18
20
22
Black and Hispanic Risk of Special Education

10

White Risk of Special Education
15
20

25

Figure E.16: Risk of Special Education Over Homestead Taxable Values

10.5

11
11.5
12
Log Homestead Taxable Values Per Pupil
White

Black

12.5

Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

96

12
14
16
18
20
22
Black and Hispanic Risk of Special Education

10

White Risk of Special Education
20
30
40

50

Figure E.17: Risk of Special Education Over Median Household Income

10

10.5
11
Log Median Income
White

Black

11.5
Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

97

0

White Risk of Special Education
10
20
30
40

5
10
15
20
25
Black and Hispanic Risk of Special Education

Figure E.18: Risk of Special Education Over the Percent of Students Eligible
Free and Reduced-Price Meals

0

20

40
60
Percent FARM
White

Black

80

100

Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

98

10
15
20
25
Black and Hispanic Risk of Special Education

13

White Risk of Special Education
14
15
16
17

18

Figure E.19: Risk of Special Education Over District Budget

-10

-5
White

0
Black

5

10
Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

99

12
14
16
18
20
22
Black and Hispanic Risk of Special Education

14

White Risk of Special Education
15
16

17

Figure E.20: Risk of Special Education Over District Urbanicity

City

Suburb

Town

Rural

Urbanicity
White

Black

Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

100

0

White Risk of Special Education
20
40
60

80

10
15
20
25
Black and Hispanic Risk of Special Education

Figure E.21: Risk of Special Education Over the Share of Eighth Graders
That Met or Exceeded Math Standards

0

20
40
60
80
Met/Exceeded 8th Grade Math Standards
White

Black

100

Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

101

Figures for “Class Size and Special Education Placement: Revisiting Tennessee’s Project STAR”

.2

Density

.4

.6

Figure F.1: Distributions of Initial Z-Scores

0

Appendix F

-2

0

2

4

Initial Z-Score
Gen Ed: Regular
Sp Ed: Regular

Gen Ed: Small
Sp Ed: Small

Gen Ed: Aide
Sp Ed: Aide

Note: Kernel densities of initial z-scores by Project STAR class type and special education
status.

102

0

.2

Density

.4

.6

Figure F.2: Distributions of Fourth Grade Z-Scores

-4

-2
Gen Ed: Regular
Sp Ed: Regular

0
Fourth Grade Z-Score
Gen Ed: Small
Sp Ed: Small

2

4
Gen Ed: Aide
Sp Ed: Aide

Note: Kernel densities of fourth-grade z-scores by Project STAR class type and special
education status.

103

Appendix G

Appendices for “The Impact of Disproportionality Regulations on Identification
into Special Education Programs”

Disproportionate Representation and Significant Disproportionality
Disproportionality addresses the relative representation of students of a particular race receiving special education services, which the literature calls risk. Policy makers and educators
have long been concerned with this type of risk because it is believed that a child inappropriately assigned into special education bears the cost of the label without the high marginal
benefit of the services.14 The Office of Special Education and Westat provide states technical assistance in the calculation of various risk measures. There exist multiple methods to
calculate racial risk. The most basic measure attempts to identify the probability a student
from a given ethnic group will receive special education services. Risk, Rid , is calculated as
the number of students in a given race i receiving special education services divided by the
total number of students in that race enrolled in the school district d times one hundred.

Rid =

District d Sp Ed Enrl for race i
⇤ 100
District d Total Enrl for race i

Using the district-level risk, as defined above, states calculate three relative risk measures:
risk ratio, weighted risk ratio, and alternative risk ratio. Each of these measures has di↵erent
strengths and weaknesses, like comparability across districts and interpretation with small
student groups. The risk ratio, weighted risk ratio, and alternative risk ratio are defined,
14 It is worth noting that the risk concept used in this body of literature as well as the legislation only
controls for race. Recent papers, such as Coutinho and Oswald (2005), consider risk over gender.

104

respectively, as
RRid =

Rid
R id

(1 p )R
W RRid = P i id
pj Rjd
j6=i

ARRid =
where

Rid
R i

i means races not i, pi is the proportion of students in racial group i in the state,

and R i refers to the statewide risk of the comparison group

i. For example, a WRR

comparing Black students to all other students of 2.7 means that a Black student is 2.7
times more likely to receive special education services than all other student groups when
weighted by state level demographics, a case of overrepresentation.
With the above measures, the MDE calculates the district’s verified ratio, which is a
measure designed to both reduce the complications caused by low enrollment of a comparison
race and also allow for greater comparability across districts.15 The MDE calculates two
sets of the three ratios (W RRid , ARRid , and RRid ) using both the operating and resident
district data. The lower of the district’s relevant operating or resident district ratio is used
15 The MDE will only calculate the verified ratio when both of the following conditions hold. First, there
are 30 or more students in the LEA’s special education program. Second, the LEA’s total enrollment for all
other racial groups exceeds 100.

105

to calculate the verified ratio.
8
>
>
>
>
>
>W RRid ,
>
>
>
>
>
>
>
ARRid ,
>
>
>
>
<
V Rid = RR ,
id
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:

if District d Sum of Sp Ed Enrl for all races not i

10

if District d Sum of Sp Ed Enrl for all races not i < 10
if (there are 0 Black or White Sp Ed students) or
(there are less than 3 Black or White Sp Ed students and
W RRid

RRid

1)

106

Supplementary Tables for “The Impact of Disproportionality Regulations on Identification
into Special Education Programs”

Table G.1: Role of Control Variable Set on DD Impact of Penalties

Post 2007 * Penalty
Post 2007 * Threat
Penalty
Threat
Post 2007
Years
Years2
Years3

(1)
-2.92⇤⇤⇤
(0.96)
0.61
(0.76)
2.40⇤⇤⇤
(0.77)
1.36⇤⇤⇤
(0.41)
-0.05
(0.04)
-0.31⇤
(0.17)
0.04⇤
(0.02)
-0.00
(0.00)

(2)
-2.86⇤⇤⇤
(0.97)
0.56
(0.76)
2.41⇤⇤⇤
(0.77)
1.36⇤⇤⇤
(0.41)
0.11⇤⇤⇤
(0.03)

(3)
-2.91⇤⇤⇤
(0.96)
0.61
(0.76)
2.40⇤⇤⇤
(0.77)
1.36⇤⇤⇤
(0.41)
-0.02
(0.05)
-0.05
(0.05)
0.01⇤⇤
(0.00)

-0.01⇤⇤⇤
(0.00)

-0.00⇤⇤⇤
(0.00)

-0.01⇤⇤⇤
(0.00)

(4)
-2.92⇤⇤⇤
(0.96)
0.61
(0.76)
2.40⇤⇤⇤
(0.77)
1.36⇤⇤⇤
(0.41)
-0.07
(0.06)
-0.65
(1.03)
0.12
(0.22)
-0.01
(0.02)
0.00
(0.00)
-0.01⇤⇤⇤
(0.00)

-0.01

-0.01

-0.01

-0.01

Years4
FARM

(5)
-2.95⇤⇤⇤
(0.92)
0.64
(0.75)
2.28⇤⇤⇤
(0.72)
1.51⇤⇤⇤
(0.42)
-0.06
(0.05)
-0.52⇤⇤⇤
(0.18)
0.07⇤⇤⇤
(0.02)
-0.00⇤⇤⇤
(0.00)

(6)
-2.92⇤⇤⇤
(0.95)
0.62
(0.76)
2.35⇤⇤⇤
(0.76)
1.43⇤⇤⇤
(0.41)
-0.05
(0.05)
-0.34⇤
(0.17)
0.05⇤⇤
(0.02)
-0.00⇤
(0.00)

-0.01⇤⇤⇤
(0.00)

-0.01⇤⇤⇤
(0.00)

FARM2
White Composition

107

0.01⇤⇤⇤

(7)
-2.90⇤⇤⇤
(0.99)
0.59
(0.77)
2.45⇤⇤⇤
(0.80)
1.36⇤⇤⇤
(0.40)
-0.05
(0.04)
-0.16
(0.17)
0.02
(0.02)
-0.00
(0.00)

(8)
-2.90⇤⇤⇤
(0.94)
0.63
(0.75)
2.35⇤⇤⇤
(0.75)
1.35⇤⇤⇤
(0.41)
-0.05
(0.04)
-0.30⇤
(0.17)
0.04⇤
(0.02)
-0.00
(0.00)

(9)
-2.81⇤⇤⇤
(1.02)
0.62
(0.76)
2.64⇤⇤⇤
(0.78)
1.36⇤⇤⇤
(0.41)
-0.06
(0.05)
-0.32⇤
(0.17)
0.05⇤
(0.02)
-0.00⇤
(0.00)

-0.01⇤⇤⇤ -0.01⇤⇤⇤
(0.00)
(0.00)
0.00
(0.00)
-0.01
-0.00
-0.01
Continued on next page

White Composition2
Multi Penalty
Observations
R2
F

(1)
(0.00)
0.00⇤⇤⇤
(0.00)
0.49
(0.39)
1249
0.57
49.62

(2)
(0.00)
0.00⇤⇤⇤
(0.00)
0.49
(0.36)
1249
0.56
55.27

Table
(3)
(0.00)
0.00⇤⇤⇤
(0.00)
0.49
(0.38)
1249
0.57
53.64

G.1 (cont’d)
(4)
(5)
(0.00)
0.00⇤⇤⇤
(0.00)
0.49
0.48
(0.39)
(0.37)
1249
1249
0.57
0.52
47.81
56.08

(6)
(0.00)

0.50
(0.38)
1249
0.55
58.40

(7)
(0.00)
0.00⇤⇤⇤
(0.00)
0.54
(0.39)
1249
0.55
50.07

(8)
(0.01)
0.00⇤⇤
(0.00)
0.46
(0.38)
1249
0.57
49.80

(9)
(0.00)
0.00⇤⇤⇤
(0.00)

1249
0.57
31.28

Note: Each specification contains controls including a cubic in years since 2000, percent of students eligible for free and reducedprice meals, and a quadratic of the White composition. Standard errors are reported in parentheses and clustered at the district
level. Significance levels: ⇤ p < 0.10, ⇤⇤ p < 0.05, ⇤⇤⇤ p < 0.01.

108
108

Supplementary Figures for “The Impact of Disproportionality Regulations on Identification
into Special Education Programs”

Figure G.1: Impact of Threat on the Black Verified Ratio

0

2

Black VR
4

6

8

Sanctions Begin

2004

2005

2006

2007

2008

2009

2010

2011

Year
Policy Period Mean - No Threat
Policy Period Mean - Threat
Threat Districts

Note: This figure plots the policy period means of the Black verified ratio for districts in
threat but not penalty phases and districts in neither threat nor penalty phases. Scatter
points for districts in neither threat or penalty phases were excluded for illustrative purposes. The data have been restricted to the analytic sample used in the DD regressions,
leaving only traditional public school districts eligible for disproportionality calculations.

109

Figure G.2: Impact of Threat on the Black Verified Ratio for Ineligible Districts

0

2

Black VR
4
6

8

10

Sanctions Begin

2004

2005

2006

2007
Year

2008

2009

2010

Policy Period Mean - No Threat
Policy Period Mean - Threat
Threat Districts

Note: This figure plots the policy period means of the Black verified ratio for districts in
threat but not penalty phases and districts in neither threat nor penalty phases. Scatter
points for districts in neither threat or penalty phases were excluded for illustrative purposes. The data have been restricted to the analytic sample used in the DDD regressions,
leaving only traditional public school districts ineligible for disproportionality calculations.

110

10

-50

15

Risk

20

0
50
Relative White-Black Income Gap

25

100

Figure G.3: Risk by Race Over Relative Income Gap

0

2
4
6
8
Deciles of Relative White-Black Income Gap
Risk Black

Risk White

10

Income Gap

Note: This graphic shows the relationship between the risk of special education placement
for Black and White students separately over the distribution of the relative income gap.
The sample has been restricted to the DD analytic sample without charter schools.

111

0

20

40
60
80
White Composition

Relative White-Black Income Gap
10
20
30

40

100

Figure G.4: Relative Income Gap Over White Composition

0

2

4
6
Deciles of White Composition
Income Gap

8

10

Risk White

Note: This graphic shows the relationship between the relative income gap over the distribution of total White composition. The sample has been restricted to the DD analytic
sample without charter schools.

112

Appendix H

Appendices for “Disparities in Special Education Identification: Evidence from Michigan”

Supplementary Figures for “Disparities in Special Education Identification: Evidence from
Michigan”

30

10.8

40

11

Log $
11.2
11.4

50
60
Percent FARM

70

11.6

80

11.8

Figure H.1: Comparison Relationships With White Composition

0

20

40
60
White Composition
Taxable Values

80

100

FARM

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Taxable values are plotted over logged dollars.

113

0

Special Education Intensity
.2
.4
.6

.8

Figure H.2: Scatter Plot of Special Education Intensity and General Education Participation

20

40
60
80
Share in General Education at Least 80%

100

Note: Special education intensities are based on 2010 data while the share in general
education for at least 80 percent of the day are based on 2011 data.

114

0

.25

Special Education Intensity
.3
.35
.4
.45

.5

.05
.1
.15
.2
Special Education Per Pupil Enrollment

Figure H.3: Special Education Participation Over Black Composition

0

20
Intensity

40
60
Black Composition
Headcount

80

100
FTE

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. Special education per pupil enrollment measures participation rates by
headcount and FTE.

115

10

20
30
40
Percent: Speech and Specific

Percent: Cognitive, Emotional, Autism,
Other, and Physical
0
5
10
15
20
25

50

Figure H.4: Disability Categories Over Black Composition

0

20
Cognitive
Physical

40
60
Black Composition
Speech
Emotional

80
Autism
Specific

100
Other

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color. The Physical group refers to the aggregation of the following disability
categories: hearing impairment, visual impairment, physical impairment, severe multiple
impairment, traumatic brain injury, and deaf-blindness.

116

0

White Risk of Special Education
20
40

60

5
10
15
20
25
Black and Hispanic Risk of Special Education

Figure H.5: Risk of Special Education Over Black Composition

0

20

40
60
Black Composition
White

Black

80

100

Hispanic

Note: Broken lines represent the 95 percent confidence interval around the solid line of
the same color.

117

Appendix I

Appendices for “Class Size and Special Education Placement: Revisiting Tennessee’s
Project STAR”

Supplementary Tables for “Class Size and Special Education Placement: Revisiting Tennessee’s Project STAR”

K

Grade

1st

Grade

Table I.1: Transition Matrix of Special Education Status by
Grade

No
Yes
Missing
No
Yes
Missing

No
4363
113
2266
.
.
.

1st Grade
Yes Missing
28
1731
8
81
48
2963
.
.
.
.
.
.

4th Grade
No Yes Missing
1459 153
4510
29
12
161
421 99
4757
1903 256
4583
6
8
70
0
0
4775

Note: Cross tabulations of the counts of students by special education
status. Yes (No) refers to a student participating (not participating) in
special education programs in a specific grade. Missing means that the
cell containing special education status information had missing data in for
grade.

118

Table I.2: Characteristics of Students by Grade-Specific Special Education Status

Small
Aide
Female
Black
Free Lunch

Kindergarten
(1)
(2)
Sp Ed Gen Ed
0.36
0.30
(0.48) (0.46)
0.31
0.35
(0.46) (0.48)
0.38
0.49
(0.49) (0.50)
0.17
0.33
(0.38) (0.47)
0.57
0.48
(0.50) (0.50)

First Grade
(3)
(4)
Sp Ed Gen Ed
0.12
0.26
(0.33) (0.44)
0.49
0.36
(0.50) (0.48)
0.31
0.48
(0.47) (0.50)
0.19
0.33
(0.40) (0.47)
0.70
0.50
(0.46) (0.50)

Fourth Grade
(5)
(6)
Sp Ed Gen Ed
0.30
0.34
(0.46) (0.47)
0.36
0.33
(0.48) (0.47)
0.41
0.51
(0.49) (0.50)
0.16
0.24
(0.37) (0.43)
0.50
0.40
(0.50) (0.49)

-0.42
(1.06)
-0.21
(1.01)
-0.01
(1.30)

0.07
(0.94)
0.17
(0.97)
0.21
(0.89)

-0.27
(1.37)
-0.52
(1.19)
-0.68
(1.18)

0.17
(0.91)
0.06
(0.95)
0.10
(0.94)

0.35
(1.00)
0.08
(1.02)
-0.37
(1.16)

0.34
(0.87)
0.43
(0.83)
0.25
(0.83)

0.00
(0.00)
0.40
(0.49)
0.80
(0.40)
202

0.00
(0.00)
0.28
(0.45)
0.74
(0.44)
6122

0.57
(0.50)
0.00
(0.00)
0.83
(0.37)
84

0.34
(0.47)
0.00
(0.00)
0.68
(0.47)
6742

0.38
(0.49)
0.00
(0.00)
0.00
(0.00)
264

0.22
(0.41)
0.00
(0.00)
0.00
(0.00)
1909

Z-Score
Kindergarten
First Grade
Fourth Grade
Missing Special Education
Kindergarten
First Grade
Fourth Grade
Observations

Note: Means with standard deviations in parentheses. Sp Ed (Gen Ed) refers to special education
(general education) status in a given grade.

119

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