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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