CLASS-SIZE CAPS, SORTING, AND THE

REGRESSION-DISCONTINUITY DESIGN

BY MIGUEL URQUIOLA AND ERIC VERHOOGEN

PRESENTED BY

OGWUIKE CLINTON OBINNA (ADVOCATE)

AYEDEGUE TAYE 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)

MORE EXAMPLE

• 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.

FURTHER ON RDD

• 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

• Contentious literature on whether class size matters• To develop a viable a model in which:• Households sort into schools of different quality levels• Schools can choose their quality level and class size

• Explore and to an extent evaluate liberalized Chilean education market• Forcast the Implications of model for reference purposes:• Class-size is an inverted-U function of hh income (this will bias cross-

sectional estimates)• Stacking occurs at class size cap (this will invalidate regression

discontinuity estimates)

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

MORE ON DATA

• 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

MORE ON DATA

• public or municipal schools are run by roughly 300 municipalities which receive a perstudent “voucher” payment from the central government. These schools cannot turn awaystudents unless demand exceeds capacity, and are limited to a maximum class size of 45.2In most municipalities, they are the suppliers of last resort.

MORE ON DATA

• private subsidized or voucher schools are independent, and since 1981 have receivedexactly the same per-student subsidy as municipal schools.3 They are also constrained to amaximum class size of 45, but, unlike public schools, have wide latitude regarding studentselection.

MORE ON DATA

• private unsubsidized schools are independent, do not accept vouchers, receive no otherexplicit 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 willingto accept their children.

SUMMARY OF 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 hh 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

nxTq/

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

IMPLICATION OF THE MODEL

• 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

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.

MORE ON CRITIQUE

• 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 Sizeon 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.

MORE ON BIBLOGRAPHY• 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 anUrban 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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