1) The study examines how household choices of schools and school choices of class size affect regression discontinuity estimates of the effect of class size on student outcomes in Chile. 2) The authors develop an economic model showing that class size is an inverted-U shaped function of household income, which biases cross-sectional estimates, and that schools stack students at multiples of the class size cap, invalidating regression discontinuity estimates. 3) Using data from Chile, the authors find evidence supporting the predictions of the model, suggesting previous regression discontinuity studies may have overestimated the effects of class size on student performance.

1. Class-Size Caps, Sorting, and the
Regression-Discontinuity Design
By MIGUEL URQUIOLA AND ERIC VERHOOGEN
Presented by
OGWUIKE C. Obinna (Advocate)
AYEDEGUE T. Patric (prosecutor)

2. KEYWORDS
 CLASS-SIZE CAP: this is the highest number of students required
to make up a class size. This study takes 45 students for its
Class-Size Cap.
 SORTING: this is synonymous to classifying i.e. grouping and/or
strategic selection.
DISCONTINUITY: Some sort of arbitrary jump/change thanks to a
quirk in law or nature. We’re interested in the ones that make
very similar people get very dissimilar results.

3. DISCONTINUITY EXAMPLE
School Class Size
Maimonides’ Rule--No more than 40 kids in a class
in Israel.
40 kids in school means 40 kids per class. 41 kids
means two classes with 20 and 21.
(Angrist & Lavy, QJE 1999)

4. EXAMPLE (2)
 Union Elections
If employers want to unionize, NLRB holds
election. 50% means the employer doesn’t
have to recognize the union, and 50% + 1
means the employer is required to “bargain in
good faith” with the union.
(DiNardo & Lee, QJE 2004)

5. REGRESSION DISCONTINUITY
Run a regression based on a situation where
you’ve got a discontinuity.
Treat above-the-cutoff and below-the-cutoff like
the treatment and control groups from a
randomization.

6. RDD (2)
 Many times, random assignment is not possible e.g:
 Universal take-ups
 Non-excludable intervention
 Treatment already assigned
 When randomization is not feasible i.e. how can we
measure implementation features of a program to
measure its impact?
 The answer is QUASI-EXPERIMENTS; Regrssion
Discontinuity Design is a good example.

7. MOTIVATION
 Eric A. Hanushek (1995; 2003); says that class size has no systematic effect
on student achievement in either developed or developing countries.
 Alan B. Krueger (2003), Michael R. Kremer (1995); countered that this
conclusion is based largely on cross-sectional evidence and subject to
multiple potential sources of bias. They requested for further analyses using
experimental and quasi-experimental designs
 Joshua D. Angrist and Victor Lavy (1999); exploits the discontinuous
relationship between enrollment and class size that results from class-size
caps though using regression-discontinuity (RD) design.
Thus, Miguel Urquiola and Eric Verhoogen (2009) decided to examine how
schools’ choices of class size and households’ choices of schools affect
regression-discontinuity-based estimates of the effect of class size on
student outcomes.

8. RESEARCH QUESTIONS
What is the effect of Class Size on the
performance of the Students?
What is the relationship between
household income and quality of
education?

9. KEY OBJECTIVE
 This paper hopes to clarify the literature on the
effect of class size on student performance by
using a Regression Discontinuity Design.

10. DATA
Three types of schools in Chile’s primary school
system
Public/Municipal: funded per student, can’t turn
students away, max class size 45, typically low quality
Private subsidized/Voucher: same per student funding
from gov’t, same class size cap, but can select students
Private unsubsidized: no gov’t funding
40-58% of primary schools in Chile are private
Most private schools are for-profit & can charge
tuition

11. DATA (2)
Administrative information on schools’ grade-
specific enrollments and number of classrooms
Standardized testing data
Math and language performance
Student characteristics such as household income
and parental schooling

12. DATA (3)
 Public or municipal schools are run by roughly 300 municipalities
which receive a per-student “voucher” payment from the central
government. These schools cannot turn away students unless
demand exceeds capacity, and are limited to a maximum class
size of 45.2
In most municipalities, they are the suppliers of last resort.

13. DATA (4)
 Private subsidized or voucher schools are independent, and since
1981 have received exactly the same per-student subsidy as
municipal schools.3 They are also constrained to a maximum
class size of 45, but, unlike public schools, have wide latitude
regarding student selection.

14. DATA (5)
 Private unsubsidized schools are independent, do not
accept vouchers, receive no other explicit subsidies, and
are not bound by the class-size cap.
N.B: Parents can use the per-student voucher in any public or private voucher
school that is willing to accept their children.

15. MODEL
 Model parents’ demand for education in a standard discrete-
choice framework with quality differentiation (eg, BLP 1995)
 Model unsubsidized and voucher schools as profit maximizers
subject to the relevant constraints
 Don’t allow for entry, exit or sector switching
 Schools are heterogeneous in productivity parameter 
 Continuum of schools with density fu() or fv()

16. DEMAND
U(p(),q (x(),n(); ); ) = q ( x(),n(); ) − p() + ε
• U(p, q; ) = q – p + 
q = school quality, p = tuition
 = random match-specific utility; i.i.d. double exponential distribution
 = marginal willingness to pay (function of income)
• Derive:
s(p,q;  ) = Probability household  chooses school (p,q)
D(p,q) = Expected demand for school (p,q)
• Monopolistic competition
• Combines horizontal and vertical differentiation

17. QUALITY PRODUCTION TECHNOLOGY
Quality production technology:
 = school productivity,
T = technological maximum class size,
x is enrollment, n = # of classrooms,
x/n class size
Complementarity of  and x/n







nx
T
q
/
ln

18. SCHOOLS’ OPTIMIZATION PROBLEM
(p, n, x; ) = (p +  - c)x – nFc – Fs
p=tuition, n=# classrooms, x=enrollment, =per-student
subsidy, c=variable cost, Fc= classroom fixed cost, Fs =
school fixed cost
Constraints:
Enrollment cannot exceed demand: x  D(p,q)
Positive integer number of classrooms
Class size cap: x/n  45 (only applies to voucher schools)
The authors’ solve for the equilibrium

19. MODEL IMPLICATION
 TEST 1: There is a roughly inverted-U shaped relationship between
class size and average household income in equilibrium
 TEST 2: Schools will stack at enrollments there are multiples of 45,
implying discontinuous changes in average household income
with respect to enrollment

21. RESULT
Inverted-U shaped relationship found between
income and class size at voucher schools but not
unsubsidized schools
==> Cross-sectional regressions will
underestimate the effects of class size among
lower-income voucher schools and overstate it
among higher-income ones

22. Voucher schools stack at enrollments that are multiples of 45.
==>Average  of schools just at multiples of class size cap will be
strictly less than  of schools just above the multiple.
==>Since hh income is increasing in , this invalidates the
regression discontinuity design.

23.  The key prediction, borne out in data from Chile’s liberalized education
market, is that schools at the class-size cap adjust prices (or enrollments) to
avoid adding an additional classroom, which generates discontinuities in
the relationship between enrollment and household characteristics,
violating the assumptions underlying regression-discontinuity research
designs.

24. CONCLUSION
Authors develop a model of endogenous
household sorting and class size determination
They find that class-size is an inverted-U function
of household income (which biases cross-
sectional estimates)
They find that stacking occurs at class size cap
(which invalidates RD estimates)
Caveat: model only applicable if parents have
school choice and schools can adjust prices
and enrollment

25. ADDED VALUE
 An additional Literature to existing literatures on effect of Class Size on students’
outcomes.
 Contrary to several authors claim that class size (its reduction) impacts positively on the
student’s outcome as in Angrist and Lavy, 1999; Hoxby, 2000; Urquiola and Verhoogen,
2009 i.e. this paper demontrates an empirical evidence which shows that earlier studies
using RD estimates actually overestimates the effects of class size on students outcome.
 By using Public School system, they argue that the continuity assumptions underlying the
design are not like to be violated.

26. STRENGTH OF THE PAPER
 This paper is factful on Chile’s liberal educational system.
 It has contentious literature on whether class size matters.
 Creates and adopt a highly sophisticated model well caved for and into the
Chilean educational system.
 Adopts the Regression Discontinuity Design as a quasi-experimental approach.
 Continues and reshape previous works of Angrist and Lavy, 1999; Hoxby, 2000 on
Class Size.
 The paper implements the density discontinuity test suggested by McCrary (2008)
 The OLS and IV estimates passed Stock and Yogo (2005) f-statistics test in order to be
proven not weak.

27. CRITIQUE
1. Limited applicability of model.
2. Quality variable is not well-explained or defined. Is it
perceived quality? Or is it a measure of student
performance and outcomes?
3. If the latter, authors are assuming class size affects
quality, which seems circular.
4. Authors show that old methods don’t work, but they
don’t offer a new way to estimate effect.
5. Nevertheless, this paper does not clarify the literature
and point to a way forward.

28. CRITIQUE (2)
 Assumes that smaller class sizes improve school quality
and furthermore that this improvement will be larger at
higher quality schools. Writers are not thinking about the
quality that the parents pay for, not necessarily for the
quality of the output of the students – but it seems a bit
circular. The paper doesn’t actually address if class size
improves outcomes or not!

29. BIBLOGRAPHY
 Angrist, Joshua D., and Victor Lavy. 1999. “Using Maimonides’ Rule to
Estimate the Effect of Class Size
on Scholastic Achievement.” quarterly Journal of Economics, 114(2): 533–75.
 Asadullah, M. Niaz. 2005. “The Effect of Class Size on Student Achievement:
Evidence from Bangladesh.”
Applied Economics Letters, 12(4): 217–21.
 Banerjee, Abhijit V., Shawn Cole, Esther Duflo, and Leigh Linden. 2007.
“Remedying Education: Evidence from Two Randomized Experiments in
India.” quarterly Journal of Economics, 122(3): 1235–64.

30. BIBLOGRAPHY (2)
 Bartle, Robert G. 1976. The Elements of Real Analysis. 2nd ed. New York: John
Wiley & Sons.
 Bayer, Patrick J., Robert McMillan, and Kim Reuben. 2004. “An Equilibrium
Model of Sorting in an Urban Housing Market.” National Bureau of Economic
Research Working Paper 10865.
 Bressoux, Pascal, Francis Kramarz, and Corinne Prost. 2005. “Teachers’
Training, Class Size and Students’ Outcomes: Evidence from Third Grade
Classes in France.” Unpublished.
 Browning, Martin, and Eskil Heinesen. 2003. “Class Size, Teacher Hours and
Educational Attainment.”
Centre for Applied Microeconometrics Working Paper 2003–15

31. THANK YOU FOR LISTENING