urquiola 2009
TRANSCRIPT
Class-Size Caps, Sorting, and the
Regression-Discontinuity DesignBy MIGUEL URQUIOLA AND ERIC VERHOOGEN
Presented by
OGWUIKE C. Obinna (Advocate)
AYEDEGUE T. Patric (prosecutor)
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.
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)
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)
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.
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.
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.
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?
KEY OBJECTIVE
This paper hopes to clarify the literature on the
effect of class size on student performance by
using a Regression Discontinuity Design.
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
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
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.
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.
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.
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()
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
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
Tq
/ln
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
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
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
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.
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.
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
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.
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.
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.
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!
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.
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
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