USINGMAIMONIDES’RULETOESTIMATETHEEFFECT...

USING MAIMONIDESrsquo RULE TO ESTIMATE THE EFFECTOF CLASS SIZE ON SCHOLASTIC ACHIEVEMENT

JOSHUA D ANGRIST AND VICTOR LAVY

The twelfth century rabbinic scholar Maimonides proposed a maximum classsize of 40 This same maximum induces a nonlinear and nonmonotonic relation-ship between grade enrollment and class size in Israeli public schools todayMaimonidesrsquo rule of 40 is used here to construct instrumental variables estimatesof effects of class size on test scores The resulting identication strategy can beviewed as an application of Donald Campbellrsquos regression-discontinuity design tothe class-size question The estimates show that reducing class size induces asignicant and substantial increase in test scores for fourth and fth gradersalthough not for third graders

When asked about their views on class size in surveysparents and teachers generally report that they prefer smallerclasses This may be because those involved with teaching believethat smaller classes promote student learning or simply becausesmaller classes offer a more pleasant environment for the pupilsand teachers who are in them [Mueller Chase and Walden 1988]Social scientists and school administrators also have a long-standing interest in the class-size question Class size is oftenthought to be easier to manipulate than other school inputs and itis a variable at the heart of policy debates on school quality andthe allocation of school resources in many countries (see egRobinson [1990] for the United States OFSTED [1995] for theUnited Kingdom and Moshel-Ravid [1995] for Israel)

This broad interest in the consequences of changing class sizenotwithstanding causal effects of class size on pupil achievementhave proved very difficult to measure Even though the level ofeducational inputs differs substantially both between and withinschools these differences are often associated with factors such asremedial training or studentsrsquo socioeconomic background Possi-bly for this reason much of the research on the relationship

This work was funded by grant 96-001151 from the US-Israel BinationalScience Foundation We thank Nora Cohen and the staff at the Chief Scientistrsquosoffice in the Israel Ministry of Education and Yigal Duchan and Soa Mintz in theOffice for Information Technology at the Ministry for help with data Thanks alsogo to hardworking research assistants Phillip Ellis and Jonathan Guryan Wehave beneted from the helpful comments of Michael Boozer Guido Imbens AlanKrueger Aaron Yelowitz seminar participants at Harvard University Universityof Pennsylvania Princeton University and the June 1996 lsquolsquoEconometrics in TelAvivrsquorsquo workshop and from the editor and two referees The authors bear soleresponsibility for the content of this paper

r 1999 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics May 1999

533

between class size and achievement is inconclusive In widelycited meta-analyses of class-size research Glass and Smith [1979]and Glass Cahen Smith and Filby [1982] conclude that smallerclasses raise childrenrsquos test scores Card and Krueger [1992a1992b] also found that lower pupil-teacher ratios in school areassociated with higher adult earnings while randomized trials inTennessee and Ontario provide evidence for benecial effects ofrandomly assigned reductions in class size [Finn and Achilles1990 Wright Shapson Eason and Fitzgerald 1977] But resultsfrom the Glass et al meta-analyses have been questioned [Slavin1989] and Hanushekrsquos [1986 1996] surveys of research on theeffects of school inputs including pupil-teacher ratios report arange of ndings Recently Card and Kruegerrsquos studies of theschool qualityearnings link have also been challenged [HeckmanLayne-Farrar and Todd 1995]

Although recent years have seen renewed interest in theclass-size question academic interest in this topic is not only amodern phenomenon the choice of class size has been of concernto scholars and teachers for hundreds of years One of the earliestreferences on this topic is the Babylonian Talmud completedaround the beginning of the sixth century which discusses rulesfor the determination of class size and pupil-teacher ratios in biblestudy The great twelfth century Rabbinic scholar Maimonidesinterprets the Talmudrsquos discussion of class size as follows lsquolsquoTwenty-ve children may be put in charge of one teacher If the number inthe class exceeds twenty-ve but is not more than forty he shouldhave an assistant to help with the instruction If there are morethan forty two teachers must be appointedrsquorsquo [Hyamson 1937 p58b]1 Interestingly while Maimonidesrsquo maximum of 40 studentswas partly derived by interpreting the Talmud this rule leads tosmaller classes than the Talmudic rule which allows a maximumsize of 492

1 This is from Chapter II of lsquolsquoLaws Concerning the Study of Torahrsquorsquo in Book Iof Maimonidesrsquo Mishneh Torah The same chapter discusses compulsory schoolattendance (at public expense from the age of six or seven for boys) the penalty fornonenforcement of compulsory attendance laws (excommunication of the entiretown) hours of instruction (long) holidays (few) use of corporal punishment(limited) qualications for teaching positions (strict) competition between schoolsfor students (permitted desirable) and busing school students between towns toschools of higher quality (permitted only if the towns are not separated by a river)

2 The Talmudic portion that Maimonides relied on is lsquolsquoThe number of pupilsassigned to each teacher is twenty-ve If there are fty we appoint two teachersIf there are forty we appoint an assistant at the expense of the townrsquorsquo (quote fromChapter II page 21a of the Baba Bathra English translation on page 214 ofEpstein [1976])

QUARTERLY JOURNAL OF ECONOMICS534

The importance of Maimonidesrsquo rule for our purposes is thatsince 1969 it has been used to determine the division of enroll-ment cohorts into classes in Israeli public schools The maximumof 40 is well-known to school teachers and principals and it iscirculated annually in a set of standing orders from the DirectorGeneral of the Education Ministry3 As we show below this rulegenerates a potentially exogenous source of variation in class sizethat can be used to estimate the effects of class size on thescholastic achievement of Israeli pupils To see how this variationcomes about note that according to Maimonidesrsquo rule class sizeincreases one-for-one with enrollment until 40 pupils are enrolledbut when 41 students are enrolled there will be a sharp drop inclass size to an average of 205 pupils Similarly when 80 pupilsare enrolled the average class size will again be 40 but when 81pupils are enrolled the average class size drops to 27

Maimonidesrsquo rule is not the only source of variation in Israeliclass sizes and average class size is generally smaller than whatwould be predicted by a strict application of this rule But Israeliclasses are large by United States standards and the ceiling of 40students per class is a real constraint faced by many schoolprincipals The median class size in our data is 31 pupils with 25percent of classes having more than 35 pupils and 10 percenthaving more than 38 pupils A regression of actual class size atmidyear on predicted class-size using beginning-of-the-year enroll-ment data and Maimonidesrsquo rule explains about half the variationin class size in each grade (in a population of about 2000 classesper grade)4

In this paper we use the class-size function induced byMaimonidesrsquo rule to construct instrumental variables estimates ofclass-size effects Although the class-size function and the instru-ments derived from it are themselves a function of the size ofenrollment cohorts these functions are nonlinear and nonmono-tonic We can therefore control for a wide range of smoothenrollment effects when using the rule as an instrument The

3 The original policy was laid out in a 1966 memo making the maximum of 40effective as of the 1969 school year [Israel Ministry of Education 1966] Mai-monidesrsquo discussion of class-size ceilings was noted in the press release announc-ing the legislation proposing a 30-pupil maximum [Israel Ministry of Education1994] The pre-1969 elementary school maximum was 50 or 55 depending ongrade [Israel Ministry of Education 1959]

4 A bivariate regression of class size on the mathematical expression ofMaimonidesrsquo rule has an R2 of 49 in the 1991 population of 2018 fth gradeclasses The corresponding R2 for 2049 fourth grade classes is 55 and thecorresponding R2 for 2049 third grade classes is 53

USING MAIMONIDESrsquo RULE 535

resulting evidence for a causal impact of class size on test scores isstrengthened by the fact that even when controlling for otherenrollment effects the up-and-down pattern in the class sizeenrollment size relationship induced by Maimonidesrsquo rule matchesa similar pattern in test scores Since it seems unlikely thatenrollment effects other than those working through class sizewould generate such a pattern Maimonidesrsquo rule provides anunusually credible source of exogenous variation for class-sizeresearch This sort of identication argument has a long history insocial science and can be viewed as an application of Campbellrsquos[1969] regression-discontinuity design for evaluation research tothe class size question5

The paper is organized as follows Following a description ofIsraeli test score data in Section I Section II presents a simplegraphical analysis Section III describes the statistical model thatis used for inference and briey outlines the connection withCampbell [1969] Section IV reports the main estimation resultsand Section V interprets some of the ndings Section VI con-cludes The results suggest that reductions in class size induce asignicant and substantial increase in math and reading achieve-ment for fth graders and a modest increase in reading achieve-ment for fourth graders On the other hand there is little evidenceof an association between class size and achievement of any kindfor third graders although this may be because the third gradetesting program was compromised

I DATA AND DESCRIPTIVE STATISTICS

The test score data used in this study come from a short-livednational testing program in Israeli elementary schools In June of1991 near the end of the school year all fourth and fth graderswere given achievement tests designed to measure mathematicsand (Hebrew) reading skills The tests are described and theresults summarized in a pamphlet from the National Center forEducation Feedback [1991] The scores used here consist of acomposite constructed from some of the basic and all of the moreadvanced questions in the test divided by the number of ques-tions in the composite score so that the score is scaled from 1ndash100

5 A recent application of regression-discontinuity ideas in economics is vander Klauww [1996] Other related papers are Akerhielm [1995] which usesenrollment as an instrument for class size and Hoxby [1996] which usespopulation to construct instruments for class size


This composite is commonly used in Israeli discussions of the testresults6 As part of the same program similar tests were given tothird graders in June 1992 The June 1992 tests are described inanother pamphlet [National Center for Education Feedback 1993]7

The achievement tests generated considerable public controversybecause of lower scores than anticipated especially in 1991 andbecause of large regional difference in outcomes After 1992 thenational testing program was abandoned

Our analysis began by linking average math and readingscores for each class with data on school characteristics and classsize from other sources The details of this link are described inthe Data Appendix Briey the linked data sets contain informa-tion on the population of schools covered by the Central Bureau ofStatistics [1991 1993] Censuses of Schools These are annualreports on all educational institutions at the beginning of theschool year (in September) based on reports from school authori-ties to the Israel Ministry of Education and supplemented byCentral Bureau of Statistics data collection as needed Informa-tion on beginning-of-the-year enrollment was taken directly fromthe computerized les underlying these reports and the classes inthe schools covered by the reports dene our study populationThe data on class size are from an administrative source andwere collected between March and June of the school year thatbegan in the previous September

The unit of observation in the linked data sets and for ourstatistical analysis is the class Although micro data on studentsare available for third graders in 1992 for comparability with the1991 data we aggregated the 1992 micro data up to the classlevel The linked class-level data sets include information onaverage test scores in each class the spring class size beginning-of-the-year enrollment in the school for each grade a town

6 In 1990 the Israel Ministry of Education created a testing center headed bythe chief scientist in the ministry to develop and run a cognitive testing program inprimary schools The resulting curriculum-based exams were pretested in the fallof 1990 The math tests included computational geometry and problem-solvingquestions The reading tests included questions evaluating grammar skills andreading comprehension The fourth grade tests included 45 math questions and 57reading questions The fth grade tests included 48 math questions and 60 readingquestions Among these fteen questions are considered basic for the purposes ofthe score composite and the remainder more advanced

7 The 1992 exams included 40 math questions of which 20 were consideredbasic The math composite score includes ten of the basic questions plus twenty ofthe more advanced questions The reading exams included 44 questions of which20 were considered basic The reading composite includes ten of the basic readingquestions plus all of the more advanced questions


identier and a school-level index of studentsrsquo socioeconomicstatus that we call percent disadvantaged (PD)8 Also included arevariables identifying the ethnic character (JewishArab) andreligious affiliation (religioussecular) of schools

Except for higher education schools in Israel are segregatedalong ethnic (JewishArab) lines Within the Jewish public schoolsystem there are also separate administrative divisions andcurricula for secular and religious schools This study is limited topupils in the Jewish public school system including both secularand religious schools These groups account for the vast majorityof school children in Israel We exclude students in Arab schoolsbecause they were not given reading tests in 1991 and because noPD index was computed or published for Arab schools until 1994The PD index is a key control variable in our analysis because it iscorrelated with both enrollment size and test scores Also ex-cluded are students in independent religious schools which areassociated with ultra-orthodox Jewish groups and have a curricu-lum that differs considerably from that in public schools

The average elementary school class in our data has about 30pupils and there are about 78 pupils per grade This can be seenin Panel A of Table I which reports descriptive statistics includ-ing quantiles for the population of over 2000 classes in Jewishpublic schools in each grade (about 62000 pupils) Ten percent ofclasses have more than 37 pupils and 10 percent have fewer than22 pupils The distribution of test scores also shown in the tablerefers to the distribution of average scores in each class Per-pupilstatistics ie class statistics weighted by class size are reportedin Appendix 1 The average score distributions for fourth and fthgrade classes are similar but mean scores are markedly higherand the standard deviations of scores lower for third graders Webelieve the difference across grades is generated by a systematictest preparation effort on the part of teachers and school officialsin 1992 in light of the political fallout resulting from what werefelt to be were disappointing test results in 1991

8 The PD index is discussed by Algrabi [1975] and is used by the Ministry ofEducation to allocate supplementary hours of instruction and other schoolresources It is a function of pupilsrsquo fathersrsquo education and continent of birth andfamily size The index is recorded as the fraction of students in the school who comefrom what is dened (using index characteristics) to be a disadvantaged back-ground


TABLE IUNWEIGHTED DESCRIPTIVE STATISTICS

Variable Mean SD

Quantiles

010 025 050 075 090

A Full sample5th grade (2019 classes 1002 schools tested in 1991)

Class size 299 65 21 26 31 35 38Enrollment 777 388 31 50 72 100 128Percent disadvantaged 141 135 2 4 10 20 35Reading size 273 66 19 23 28 32 36Math size 277 66 19 23 28 33 36Average verbal 744 77 642 699 754 798 833Average math 673 96 548 611 678 741 794

4th grade (2049 classes 1013 schools tested in 1991)


3rd grade (2111 classes 1011 schools tested in 1992)


B 1 2 5 Discontinuity sample (enrollment 36ndash45 76ndash85 116ndash124)

5th grade 4th grade 3rd grade

Mean SD Mean SD Mean SD

(471 classes224 schools)

(415 classes195 schools


Class size 308 74 311 72 306 74Enrollment 764 295 785 300 757 282Percent disadvantaged 136 132 129 123 145 146Reading size 281 73 283 77 246 62Math size 285 74 287 77 248 63Average verbal 745 82 725 78 862 63Average math 670 102 687 91 842 70

Variable denitions are as follows Class size 5 number of students in class in the spring Enrollment 5September grade enrollment Percent disadvantaged 5 percent of students in the school from lsquolsquodisadvantagedbackgroundsrsquorsquo Reading size 5 number of students who took the reading test Math size 5 number of studentswho took the math test Average verbal 5 average composite reading score in the class Average math 5average composite math score in the class


A The Discontinuity Sample

Maimonidesrsquo rule can be used to identify the effects of classsize because the rule induces a discontinuity in the relationshipbetween enrollment and class size at enrollment multiples of 40Since this discontinuity is the source of identifying informationsome of the analysis that follows is restricted to schools withenrollments in a range close to the points of discontinuity9 PanelB of Table I shows descriptive statistics for one such lsquolsquodiscontinu-ity samplersquorsquo dened to include only schools with enrollments inthe set of intervals [3645] [7685][116125] Slightly fewer thanone-quarter of classes come from schools with enrollments in thisrange Average class size is a bit larger in this 6 5 discontinuitysample than in the overall sample But the average characteris-tics of classes in the discontinuity sample including test scoresand the PD index are otherwise remarkably similar to those forthe full sample

II GRAPHICAL ANALYSIS

The class-size function derived from Maimonidesrsquo rule can bestated formally as follows Let es denote beginning-of-the-yearenrollment in school s in a given grade and let fsc denote the classsize assigned to class c in school s for that grade Assuming thatcohorts are divided into classes of equal size we have

(1) fsc 5 es [int ((es 2 1)40) 1 1]

where for any positive number n the function int (n) is thelargest integer less than or equal to n Equation (1) captures thefact that Maimonidesrsquo rule allows enrollment cohorts of 1ndash40 to begrouped in a single class but enrollment cohorts of 41ndash80 are splitinto two classes of average size 205ndash40 enrollment cohorts of81ndash120 are split into three classes of average size 27ndash40 and soon

Although fsc is xed within schools in practice enrollmentcohorts are not necessarily divided into classes of equal size Inschools with two classes per grade for example only about

9 We thank a referee (Caroline M Hoxby) for suggesting an analysis in thissubsample Hahn Todd and van der Klaauw [1997] explore a related nonparamet-ric approach to regression-discontinuity estimation


one-quarter of the classes are of equal size On the other handeven though the actual relationship between class size andenrollment size involves many factors in Israel it clearly has a lotto do with fsc This can be seen in Figures Ia and Ib which plot theaverage class size by enrollment size for fth and fourth gradepupils along with the class-size function The dashed horizontal

FIGURE IClass Size in 1991 by Initial Enrollment Count Actual Average Size and as

Predicted by Maimonidesrsquo Rule


lines in the gures mark the class sizes where the class-sizefunction has corners The gures show that at enrollment levelsthat are not integer multiples of 40 class size increases approxi-mately linearly with enrollment size But average class size dropssharply at integer multiples of 40 ie at the corners of the classsize function

The gures show that average class size never reaches 40when enrollment is less than 120 even though the class sizefunction predicts a class size of 40 when enrollment is either 4080 120 etc This is because schools can sometimes afford to addextra classes before reaching the maximum class size For ex-ample schools may receive funds to support more classes if theyhave a high PD index [Lavy 1995] These funds represent adeliberate attempt to offset the effects of socioeconomic back-ground and can also be used to add hours of instruction andteachers to those schools where the PD index is high On the otherhand manipulation of class size by parents is limited by the factthat Israeli pupils must attend a neighborhood school Overowclasses caused by large enrollments and Maimonidesrsquo rule areconducted in school libraries and other temporary classrooms ifneed be10 Of course parents can circumvent Maimonidesrsquo rule bymoving to another school district Unlike in the United Stateshowever very few Israeli children attend private schools

It is also noteworthy that average class sizes do not drop asmuch at the corners of the class size function as fsc predicts This isbecause the beginning-of-the-year enrollment data are not neces-sarily the same as enrollment at the time the class-size data werecollected (for example if enrollment has fallen then an initiallylarge cohort will not necessarily have been split) and because afew classes are reported to include more than 40 pupils11 In spiteof this reduction in predictive power for midyear class size itseems more attractive to predict class size using beginning-of-the-year measures of enrollment since early measures are less likelythan contemporaneous measures to have been affected by thebehavior of parents or school officials

10 Exceptions can be made in response to written requests but pupils aregenerally required to attend school in their lsquolsquolocal registration arearsquorsquo whichtypically includes only one religious and one secular school Moreover lsquolsquoPrincipalsmay not refuse to register a pupil in their schoolrsquos registration area and may notregister a pupil who does not live in the arearsquorsquo [Israel Ministry of Education 1980Part B6a]

11 The empirical analysis is restricted to schools with at least 5 pupilsreported enrolled in the relevant grade and to classes with less than 45 pupils


In addition to exhibiting a strong association with averageclass size the class-size function is also correlated with theaverage test scores of fourth and fth graders (although not thirdgraders) This can be seen in Figures IIa and IIb which plotaverage reading test scores and average values of fsc by enrollmentsize in enrollment intervals of ten Figure IIa plots the scores of

FIGURE IIAverage Reading Scores by Enrollment Count and the Corresponding Average

Class Size Predicted by Maimonidesrsquo Rule


fth graders and Figure IIb plots the scores of fourth graders12

The gures show that test scores are generally higher in schoolswith larger enrollments and therefore larger predicted classsizes Most importantly however average scores by enrollmentsize can be seen to exhibit an up-and-down pattern that is at leastin part the mirror image of the class-size function

The overall positive correlation between scores and enroll-ment is partly attributable to that fact that larger schools in Israelare more likely to be located in relatively prosperous big citieswhile smaller schools are more likely to be located in relativelypoor lsquolsquodevelopment townsrsquorsquo outside of major urban centers In factenrollment size and the PD index measuring the proportion ofstudents who come from a disadvantaged background are highlynegatively correlated

After controlling for this lsquolsquotrend associationrsquorsquo between testscores and enrollment size and between test scores and PD thereis a negative association between fsc and scores This can be seenin Figures IIIa and IIIb which plot residuals from regressions ofaverage reading scores and the average of fsc on average enroll-ment and PD index for each interval Again the x-axis isenrollment size Although the approximate mirror-image relation-ship between detrended average scores and detrended fsc is clearlynot deterministic this pattern is evident for the reading scores ofpupils in both grades and as shown in Figure IIIc for the mathscores of fth graders In a regression of detrended average scoreson detrended average fsc the slopes are roughly 2 22 for fthgradersrsquo reading scores and 2 11 for fourth gradersrsquo readingscores Thus the estimates for fth graders imply that a reductionin predicted class size of ten students is associated with a 22 pointincrease in average reading scores a little more than one-quarterof a standard deviation in the distribution of class averages

III MEASUREMENT FRAMEWORK

The gures suggest a clear link between the variation in classsize induced by Maimonidesrsquo rule and pupil achievement but they

12 Intervals of ten were used to construct the gures instead of thesingle-value intervals in Figures Ia and Ib because the test score data have moreidiosyncratic variation than the class-size data The enrollment axes in the guresrecord interval midpoints Averages were computed for schools with enrollmentsbetween 9 and 190 This accounts for over 98 percent of classes The last interval(165 on the x-axis) includes enrollments from 160ndash190


do not provide a framework for formal statistical inferenceAlthough the micro data for fourth and fth graders are un-available a model for individual pupilsrsquo test scores is used todescribe the causal relationships to be estimated For the ith

FIGURE IIIAverage Test (ReadingMath) Scores and Predicted Class Size by Enrollment

Residuals from Regressions on Percent Disadvantaged and Enrollment


student in class c and school s we can write

(2) yisc 5 X8s b 1 nsc a 1 microc 1 h s 1 e isc

where yisc is pupil irsquos score Xs is a vector of school characteristicssometimes including functions of enrollment and nsc is the size ofclass c in school s The term microc is an iid random class componentand the term h s is an iid random school component Theremaining error component e isc is specic to pupils The rst twoerror components are introduced to parameterize possible within-school and within-class correlation in scores The class-size coeffi-cient a is the parameter of primary interest

Our interpretation of equation (2) is that it describes theaverage potential outcomes of students under alternative assign-ments of nsc controlling for any effects of Xs Although equation (2)is linear with constant coefficients this is not necessary forestimates of a to have a valid causal interpretation For exampleif nsc were randomly assigned conditional on Xs then a would be aweighted average response along the length of the individualcausal response functions connecting class size and pupil scores(see Angrist and Imbens [1995] and Section V below) Since nsc isnot randomly assigned in practice it is likely to be correlated withpotential outcomes (in this case the error components in (2))Thus OLS estimates of (2) do not have a causal interpretationalthough instrumental variables estimates still might The causalinterpretation of instrumental variables estimates turns onwhether it is reasonable to assume that after controlling for Xsthe only reason for any association between instruments and testscores is the association between instruments and class size Wediscuss this assumption further below

Equation (2) is cast at the individual level because it is pupilswho are affected by class size In practice however the literatureon class size often treats the class as the unit of analysis and notthe pupil Examples of class-level analyses of data from random-ized experiments are Finn and Achilles [1990] and Wright et al[1977] Since class size is naturally xed within classes andstudent test scores are correlated within classes little is lost instatistical precision from this aggregation Moreover as notedabove we have no option other than a class-level analysis forfourth and fth graders because the micro-level data are unavail-able To make the analyses from different years comparable wealso aggregated the 1992 data on third graders to the class level


Grouping equation (1) the class-level estimating equations havethe form

(3) ysc 5 X 8s b 1 nsc a 1 h s 1 [microc 1 e sc]

where overbars denote averages The term [microc 1 e sc] is the class-level error term while the random school component h s capturescorrelation between class averages within schools13

Efficient regression estimators with grouped data reweightthe data to make the grouped residuals homoskedastic In thiscase however simply weighting by class size does not make theresiduals in (3) homoskedastic because of the random-effects errorstructure Moreover without assuming that the behavioral rela-tionship of interest is truly linear with constant coefficientsstatistical theory provides little guidance as to the choice ofweighting scheme [Deaton 1995 Pfefferman and Smith 1985] Wetherefore report conventional ordinary least squares (OLS) andinstrumental variables estimates of (3) along with standarderrors corrected for intraschool correlation using the formulas inMoulton [1986] Allowing for a heteroskedastic grouped errorterm has little impact on inferences so that the grouped errors aretreated as homoskedastic Correction for the correlation of classaverages within schools leads to 10ndash15 percent larger standarderrors than the usual formulas

A Instrumental Variables and Regression-Discontinuity Designs

The approach taken here exploits the fact that the regressorof interest (class size) is partly determined by a known discontinu-ous function of an observed covariate (enrollment) In a seminaldiscussion of nonexperimental methods in evaluation researchCampbell [1969] considered a similar problem how to identify thecausal effect of a treatment that is assigned as a deterministicfunction of an observed covariate that is also related to theoutcomes of interest14 Campbell used the example of estimatingthe effect of National Merit scholarships on applicantsrsquo later

13 Finn and Achilles [1990] also used a model with random school effects inan analysis of class-level averages to analyze data from the Tennessee ProjectSTAR (StudentTeacher Achievement Ratio) experiment

14 Goldberger [1972] discusses this in the context of compensatory educationprograms See also Thistlewaithe and Campbell [1960] and Campbell and Stanley[1963]


academic achievement when the scholarships are awarded on thebasis of past achievement He argued that if the assignmentmechanism used to award scholarships is discontinuous egthere is a threshold value of past achievement that determineswhether an award is made then one can control for any smoothfunction of past achievement and still estimate the effect of theaward at the point of discontinuity This is done by matchingdiscontinuities or nonlinearities in the relationship betweenoutcomes and past achievement to discontinuities or nonlineari-ties in the relationship between awards and past achievement

The graphs discussed in the previous section can be seen asapplying Campbellrsquos [1969] suggestion to the class-size question(see especially Campbellrsquos Figures 12ndash14) The up-and-downpattern in the conditional expectation of test scores given enroll-ment is interpreted as reecting the causal effect of changes inclass size that are induced by changes in enrollment Thisinterpretation is plausible because the class-size function isknown to share this pattern while it seems likely that any othermechanism linking enrollment and test scores will be muchsmoother

Campbell [1969] argued that when the rule relating covari-ates to treatment is not deterministic something he called alsquolsquofuzzy regression-discontinuityrsquorsquo the regression-discontinuitymethod breaks down Although later discussions of regression-discontinuity methods reversed this negative position (eg Cookand Campbell [1979] Trochim [1984]) the connection between theuse of fuzzy regression discontinuity and instrumental variablesmethods was not made explicit until van der Klauuwrsquos [1996]study of the effects of nancial aid awards The class-size problemalso provides an example of how a fuzzy regression discontinuitycan be analyzed in an instrumental variables framework In thiscase instrumental variables estimates of equation (3) use discon-tinuities or nonlinearities in the relationship between enrollmentand class size (captured by fsc) to identify the causal effect of classsize at the same time that any other relationship betweenenrollment and test scores is controlled by including smoothfunctions of enrollment in the vector of covariates In practice thisincludes linear polynomial and piecewise linear functions of es15

15 van der Klaauw [1996] exploits a fuzzy regression discontinuity bysubstituting a nonparametric estimate of the conditional expectation of treatmentfor the endogenous regressor (nancial aid) A similar approach is discussed bySpiegelman [1976] and Trochim [1984] This lsquolsquoplug-inrsquorsquo method is not literally the


The identifying assumptions that lay behind this approachcan be expressed formally by introducing some notation for thelsquolsquorst-stagersquorsquo relationship of interest

(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc

where p 0 and p 1 are parameters and as before Xs is a vector ofschool-level covariates that includes functions of enrollment esand measures of pupil socioeconomic status The error term j sc isdened as the residual from the population regression of nsc on Xs

and the instrument fsc This residual captures other factors thatare correlated with enrollment These factors are probably alsorelated to pupil achievement which is why OLS estimates of (3)do not have a causal interpretation Since fsc is a deterministicfunction of es and es is almost certainly related to pupil test scoresfor reasons other than effects of changing class size the keyidentifying assumption that underlies estimation using fsc as aninstrument is that any other effects of es on test scores areadequately controlled by the terms in X 8s b in (3) and lsquolsquopartialledoutrsquorsquo of the instrument by the term X 8s p 0 in equation (4)

To assess the plausibility of this assumption it helps toconsider why es is related to test scores in the rst place Onereason already noted is that in Israel socioeconomic status isinversely related to local population density Also better schoolsmight face increased demand if parents selectively choose dis-tricts on the basis of school quality On the other hand more-educated parents might try to avoid large-enrollment schools theyperceive to be overcrowded Any of these effects seem likely to besmooth however whereas the variation in test scores withenrollment has a rough up-and-down pattern that mirrors Mai-monidesrsquo rule Nevertheless it remains an untestable identifyingassumption that nonclass-size effects on test scores do not dependon enrollment except through the smooth functions included in XsFor this reason we experiment with a wide range of alternativespecications for the relationship of interest

A nal identifying assumption is that parents do not selec-tively exploit Maimonidesrsquo rule so as to place their children inschools with small classes Selective manipulation could occur ifmore-educated parents successfully place children in schools withgrade enrollments of 41ndash45 knowing that this will lead to smaller

same as instrumental variables unless a linear regression is used to construct therst-stage tted values


classes in a particular grade In practice however there is no wayto know whether a predicted enrollment of 41 will not decline to 38by the time school starts obviating the need for two small classesin the relevant grade And even if there was a way to predict thisaccurately we noted earlier that parents are not free to transferchildren from one elementary school to another except by movingOf course parents who discover they got a bad draw in thelsquolsquoenrollment lotteryrsquorsquo (eg enrollment of 38 instead of 41) mightthen elect to pull their kids out of the public school systementirely Private elementary schooling is rare in Israel outside ofthe ultra-orthodox community Nevertheless for this reason wedene fsc as a function of September enrollment and not enroll-ment at the time testing was done even though the latter is morehighly correlated with class size

IV ESTIMATION RESULTS

A OLS Estimates for 1991

OLS estimates with no control variables show a strongpositive correlation between class size and achievement Control-ling for PD however the positive association largely disappearsand in some cases becomes negative These ndings can be seenin Table II which reports coefficients from regressions of the mathand reading scores of fourth and fth graders on class size the PDindex and enrollment size In a regression of the average readingscores of fth graders on class size alone the class-size effect is aprecisely estimated 221 but when the PD index is added as acontrol variable the estimated class-size effect falls to 2 031 witha standard error of 022 The addition of PD also eliminates mostof the positive association between class size and math scores

Lavy [1995] previously observed that the positive associationbetween class size and test scores in Israel is largely accounted forby the association between larger classes and higher PD amongpupils The importance of family background in the United Stateswas also a key point in the Coleman [1966] report on educationoutcomes and has been emphasized more recently in the meta-analysis by Hedges Laine and Greenwald [1994] However notethat controlling for PD in the Israeli data does not completelyeliminate the positive association between class size and mathscores Also the negative OLS estimates of effects of class size onreading scores are small and at best marginally signicant One


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probable reason for these ndings is that selection bias in therelationship between test scores and class size is generated withinschools as well as between schools For example school principalsmay group children who are having trouble with their schoolworkinto smaller classes In addition to eliminating bias due todifferences between schools our instrumental variables strategyhas the potential to eliminate bias from nonrandom selectionwithin schools

B Reduced-Form and Instrumental Variables Estimates for 1991

The reduced-form relationship between predicted class size( fsc) and actual class size reported in Table III for a variety ofspecications shows that higher predicted class sizes are associ-ated with larger classes and lower test scores The top panel ofTable III reports the results of regressions on fsc with controls forPD only and with controls for both PD and enrollment size Theeffect of fsc on class size ranges from 54 to 77 and is very preciselyestimated The negative association between fsc and test scores isstrongest for fth graders but there is a precisely estimatednegative association between fourth grade reading scores and fsc

as well It is also noteworthy that the reduced-form relationshipsbetween fsc and reading scores in both grades are largely insensi-tive to the inclusion of a control for enrollment size On the otherhand there is no evidence of a relationship between math scoresand predicted class size for fourth graders

The lower half of the table reports estimates from the samespecication using only classes in the 1 52 5 discontinuity sam-ple Although here the estimates are less precise the pattern issimilar to that in the full sample With or without enrollmentcontrols there is strong evidence of a negative association be-tween reading scores and predicted class size for fth gradersWith enrollment controls there is a signicant negative associa-tion between predicted class size and the math scores of fthgraders For fourth graders the association between predictedclass size and reading scores in the discontinuity sample isnegative and close in magnitude to that in the full samplealthough not signicantly different from zero On the other handthe effects of predicted class size for fth graders are larger(though not signicantly different) in the discontinuity samplethan in the full sample

Instrumental variables estimates for fth graders are re-ported in Table IV These results correspond to the reduced-form


TAB

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specications reported in Table III as well as other specicationsThe instrumental variables estimate of the effect of class size onthe reading scores of fth graders in a model without any controlsfor enrollment size is 2 16 with a standard error of 04 Theestimates (standard errors) from models including linear andquadratic controls for enrollment size reported in columns (2)ndash(3) range from 2 26 (08) to 2 28 (07) Without enrollmentcontrols the instrumental variables estimate for fth grade mathscores is virtually zero But in models with linear and quadraticenrollment controls the instrumental variables estimates for themath scores of fth graders are similar to the estimates in thecorresponding models for reading scores For example the esti-mated class-size effect on math scores from a model with linearcontrols reported in column (8) is 2 23

A major concern in assessing the internal validity of esti-mates based on a regression discontinuity design is whethercontrols for effects of the variable that generates the discontinuityare adequate Therefore in addition to reporting results frommodels with linear and quadratic controls for enrollment we alsoreport results from a model that includes a continuous piecewiselinear trend with slopes identical to the slope of fsc on the linearsegments For example the slope in the range [4180] is 12 Sovariability around the piecewise linear trend is generated solelyby the jumps in Maimonidesrsquo rule at the points of discontinuityThe trend is dened on the interval [0160] as follows

es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]

The idea behind the piecewise linear model is that once the trendeffects of the covariate generating the discontinuity are com-pletely controlled there should be no need to hold any othercovariates xed Results from models with the piecewise lineartrend are reported in columns (4) and (10) of Table V forspecications that include no controls other than this trend As inthe other specications these results show a negative associationbetween class size and test scores although the effects are smallerand less precisely estimated than in models with parametriccontrols for enrollment effects and controls for PD Adding PD to


TAB

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the piecewise linear specication generates larger estimates forfth graders and smaller estimates for fourth graders

Other columns in Table IV report estimates using classes inthe 1 5 2 5 discontinuity sample These specications correspondto the reduced-form specications reported in Table III Here toothe purpose of the analysis is to emphasize the variability in classsize generated by jumps in class size at the points of discontinuityMost of these estimates while less precise are substantiallylarger than those for the full sample In three out of four casesthey are signicantly different from zero in spite of the reducedsample size

The instrumental variables estimates for fourth gradersreported in Table V also show a robust and in some casesstatistically signicant negative association between class sizeand reading achievement although the effects for fourth gradersare smaller than the effects for fth graders The estimate(standard error) in a model without enrollment controls is 2 11(04) and with a linear enrollment control the estimate is 2 13(06) The estimate from a model including quadratic enrollmentcontrols is not signicantly different from zero although it is stillnegative Dropping PD and adding a piecewise linear enrollmentcontrol leads to an estimate of about 2 15 (08) Estimates for thereading scores of fourth graders in the 1 52 5 discontinuitysample are similar to those for the full sample but not signi-cantly different from zero Estimates of effects on fourth gradersrsquomath scores are much weaker than the corresponding estimatesfor reading scores none of the estimates is signicantly differentfrom zero and the fourth grade math estimates in the discontinu-ity sample are positive17

C Additional Results for 1991

Results for a number of additional specications are reportedin Tables VI and VII The estimates in Table VI use only classesclose to the point of discontinuity18 As before the 1 5 2 5 disconti-nuity sample is limited to classes in schools where grade enroll-ment is in the set [3645][7685][116125] similarly a 1 32 3discontinuity sample includes classes in schools where grade

17 Using enrollment at the time tests were taken to construct the Mai-monidesrsquo rule instrument (instead of September enrollment) estimates of effectson fourth grade math scores are signicantly different from zero although stillonly about two-thirds as large as the corresponding fourth-grade verbal estimates

18 Variations on the full-sample models are reported in our working paper[Angrist and Lavy 1997]


TAB

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enrollment is in the set [3843][7883][118123] Unlike theestimates with parametric enrollment controls reported in TablesIV and V the results in Table VI are from models where control forenrollment effects consists solely of two dummies indicating eachof the rst two of segments in the discontinuity samples Soestimates in the 1 5 2 5 discontinuity sample are from models thatinclude the dummy variables d1sc 5 1[36 es 45] and d2sc 51[76 es 85] but conditional on being in any one of the threesegments in the discontinuity sample there is no control forenrollment effects The idea here is that if the discontinuitysample is narrow enough fsc is a valid instrument withoutcontrolling for enrollment effects

Another difference between the results in Table VI and earlierresults is that instead of using fsc itself as an instrument a set ofthree dummy variable instruments is used where the instru-ments indicate enrollments in the upper half of each the threesegments that make up the discontinuity samples For examplein the 1 5 2 5 discontinuity sample the instruments are

z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]

Since predicted class size is less than 32 when any of the zjsc 5 1and is more than 32 otherwise (in the discontinuity samples) thisinstrument set is generated by the dummy zsc 1[ fsc 32] fullyinteracted with a variable for enrollment segment This is equiva-lent to using zsc as instrument but allowing the reduced-formeffect of zsc on class size to vary by segment About half of classes inthe 6 5 discontinuity sample have zsc 5 1

In models with no exogenous covariates use of any single zjsc

as an instrument with data from segment j generates a Waldestimate for the effect of class size based on comparisons ofaverage test scores by the values of zsc in schools with enrollmentsin segment j Use of the three variables z1sc z2sc z3sc as instru-ments while controlling for segment effects produces a linearcombination of the three Wald estimates for each segment [An-grist 1991] This setup captures the quasi-experimental spirit ofidentication using Maimonidesrsquo rule because the resulting esti-mator is constructed from simple comparisons of means

Instrumental variables estimates of effects on fth gradereading and math scores using binary instruments in 6 5 and 6 3discontinuity samples are all negative Some of the estimates are


signicantly different from zero and most are larger than esti-mates in the full sample although also with much larger standarderrors For example the estimate (standard error) from a modelwith no covariates other than segment dummies in the 1 32 3discontinuity sample is 2 45 (24) Estimates for the readingscores of fourth graders are also negative and marginally signi-cant in the 1 32 3 discontinuity sample when the model ex-cludes PD

The second set of additional estimates reported in Table VIIconsists of results from models where the effect of class size on testscores is interacted with PD This specication is used to seewhether the benets of smaller classes vary with pupil back-ground The instruments in this case are fsc and PDfsc Toincrease precision estimates of models pooling fourth and fthgraders were also computed These models include a dummy forfourth graders The estimates by grade generate negative interac-tion terms although the interaction terms are signicant for fthgraders only Pooled estimates without interaction terms re-ported in columns (5) and (7) lie between the previously reportedgrade-specic estimates and are signicant for both test scoresPooled estimates with interaction terms reported in columns (6)and (8) of the table generate negative main effects and signicantnegative interaction terms for both test scores although the maineffect for math scores is not signicantly different from zeroOverall the estimates strongly suggest that the benets of smallclasses are larger in schools where there is a high proportion ofpupils who come from a disadvantaged background Similarndings regarding pupil backgroundclass size interactions werereported by Summers and Wolfe [1977] in a study of Philadelphiasixth graders

D Results for 1992 (Third Graders)

The OLS estimates for third graders reported in columns (2)and (6) of Table VIII show essentially no relationship betweenclass size and test scores Reduced-form effects of fsc on third gradeclass size reported in column (1) are much the same as the effectsof fsc on fourth and fth grade class size But estimates from aregression of third grade test scores on fsc PD and enrollmentsize reported in columns (3) and (7) offer little evidence of arelationship between fsc and scores Finally while the instrumen-tal variables estimates for third graders reported in columns (4)(5) (8) and (9) are all negative they are smaller than the


TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD


estimates for fourth and fth graders None of the instrumentalvariables estimates are precise enough to be statistically distin-guishable from zero19

One possible explanation for the weak ndings for thirdgraders is that the effects of class size may be cumulative Sinceenrollment cohorts tend to progress through elementary schooltogether fth graders who happen to be in enrollment cohortsthat generate small class sizes may have been grouped into smallclasses in earlier grades Years of experience in small classes maybe required before any benets are detectable This sort ofcumulative effect would also explain why the effects for fourthgraders are smaller than those for fth graders It is worth notinghowever that Krueger [1999] found no evidence of cumulativeeffects in his reanalysis of the STAR data

A more likely explanation for the absence of effects on thirdgraders is the fact that testing conditions were very different in1992 when a variety of (noneducational) activities were directed

19 Results using pupil data are similar after the standard errors arecorrected for intraclass correlation

TABLE VIIIESTIMATES FOR THIRD GRADERS

Classsize Reading comprehension Math

(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV

Mean score 863 841(sd) (61) (68)Regressors

Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)

Percent disad- 2 044 2 176 2 175 2 177 2 177 2 110 2 112 2 112 2 110vantaged (009) (011) (011) (012) (012) (013) (013) (014) (013)

Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)

Enrollment 004 2 023squared100 (007) (008)

fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056

The unit of observation is the average score in the class Standard errors are reported in parenthesesStandard errors were corrected for within-school correlation between classes There are 2111 third gradeclassess The RF column heading denotes reduced-form estimates


toward increasing test scores and reducing the variation in scoresacross schools The official report of the 1992 test results [NationalCenter for Education Feedback 1993] highlights major differencesbetween the 1992 and 1991 waves of the testing program Forexample regular class teachers (as well as an outside examproctor) were present when tests were taken in 1992 but not in 199120

As noted at the beginning of the paper the descriptivestatistics in Table I show higher scores and lower variance in1992 Additional evidence that testing conditions changed is thefact that the average score was over 90 percent correct in 25percent of 1992 classes (over 500 classes) In 1991 this was true ofno more than eight classes in any grade or subject Also our 1992estimation results show much smaller effects of PD on test scoresthan were observed in 1991 Since the distribution of the PD indexover towns and cities was essentially unchanged this nding isconsistent with the hypothesis that test preparation reduced theinformation about pupil abilities contained in the scores

V CHARACTERIZATIONS AND COMPARISONS

A Characterizing Affected Groups

This section is concerned with the external validity of ourndings First we ask whether the classes affected by Mai-monidesrsquo rule are representative of all classes in Israeli gradeschools The problem of characterizing the group affected by abinary instrument is discussed by Angrist and Imbens [1995] forthe case of a multinomial treatment Here the treatment is classsize and the instrument can be taken to be Z 5 1 2 zsc which wasused to construct the discontinuity-sample estimates reported inTable VI (The normalization is reversed so Z 5 1 means biggerclasses) In what follows we drop subscripts indexing observa-tions and use uppercase variables to denote random variableswith the same distribution as for a randomly chosen class

Suppose that a set of pupils would have average test score Yj

when grouped into classes of size j where j can take on values1ndash40 Yj is a potential outcome that is we imagine that for each

20 Preparation for the 1992 tests is described in the report as follows [page3] lsquolsquoDuring the past year there was an intense and purposeful remedial effort onthe part of the elementary school division [in the Ministry of Education] in a largenumber of schools with high failure rates in 1991 Similarly in light of last yearrsquosscores [in 1991] and because of the anticipated new tests [in 1992] there was anintensive remedial effort on the part of schools district supervisors counselorsand othersrsquorsquo


set of pupils all of the elements of Y1 Y40 are well-denedeven though only one of thesemdashcorresponding to the pupilsrsquo actualclass sizemdashis ever observed The average causal effect of increas-ing class size by one is E[Yj 2 Yj 2 1] for each j [ [240] We couldlearn about these average effects in an experiment where sets ofpupils are randomly grouped into classes of different sizes (as wasdone in Tennessee) Similarly let NZ be the potential class sizethat would be realized if the binary instrument Z were randomlyassigned In practice an experiment that accomplishes this wouldhave to manipulate enrollment perhaps by randomly sendingsmall groups of pupils to different schools in the discontinuitysample The difference in means E[N Z 5 1] 2 E[N Z 5 0] 5E[N1 2 N0] is the average causal effect of Z on class size in such anexperiment

Although the empirical work is motivated by a model wherepotential outcomes vary linearly with class size according to aregression function that is the same for all classes this is almostcertainly not an accurate description of the causal effect ofchanging class size Angrist and Imbens [1995] discuss theinterpretation of linear IV estimators in models where the under-lying causal response function is both heterogeneous and nonlin-ear The main result is that if Z is independent of potentialoutcomes and other technical conditions are satised then theWald estimator using Z as an instrument can be written in termsof potential outcomes as

(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]

where the summation is from j 5 2 to j 5 40Formula (5) suggests a two-part answer to the question lsquolsquowho

is affected by the instrumentrsquorsquo First the range of variationinduced by the instrument consists only of values j where theprobability that Z causes class size to go from less than j pupils toat least j pupils P[N1 $ j N0 ] is positive The magnitude ofP[N1 $ j N0 ] is also of interest because a particular class size jis more important if this is large Second for a given j theprobability of being in the affected group (ie of having P[N1 $j N0] 0) may vary with the characteristics of schools or pupils


in the class For any observable characteristic denoted by W wecan ask how P[N1 $ j N0 W ] varies with W

Assuming that Z is independent of N0 and N1 (ie theinstrument is lsquolsquoas good as randomly assignedrsquorsquo) the size of theaffected group at class size j is just the difference in cumulativedistribution functions (CDF) of class size with the instrumentswitched off and on The CDFs of class size by values of Z areplotted in Figures IVa and IVb for fth and fourth grade classes inthe 1 5 2 5 discontinuity sample The gap between the two CDFsis largest for class sizes between 22 and 36 with especially largegaps in the 28ndash35 range Classes of this size are not unusual inIsrael where the median size is 31 but this is larger than istypical for the United States

By denition the group most affected by the instrument Zattends schools with enrollments close to points of discontinuity Acomparison of descriptive statistics for the 1 52 5 discontinuitysample and the full sample suggests that there is nothingparticularly special about attending school with grade enroll-ments in a range close to the point of discontinuity On the otherhand conditional on attending schools with enrollments in thisrange classes affected by the rule might still be special in someway In practice we can only look for unusual rst-stage relation-ships based on observed characteristics like PD and school sizeThe question of how the P[N1 $ j N0 W ] vary with an observedcharacteristic W can be addressed by noting that (again given theassumptions in Angrist and Imbens [1995])

(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]

which is simply the rst-stage relationship between Z and Nevaluated at W21

One clear and unsurprising pattern in the right-hand side of(6) is variation by school size Controlling for PD and segmenteffects classes in the discontinuity sample for fth graders have107 more pupils if Z 5 1 on the rst enrollment segment 44 morepupils if Z 5 1 on the second enrollment segment and 11 morepupils if Z 5 1 on the third enrollment segment So estimatesusing Maimonidesrsquo rule are driven primarily by smaller schoolsIn fact this can be seen clearly in Figures Ia and Ib which show

21 This expression is derived using the facts that P[N1 $ j N0 W ] is adifference in CDFs and that the integral of one minus the CDF of a positiverandom variable equals the mean


sharper drops in class size at enrollments of 40 than at 80 andsharper drops at enrollments of 80 than at 120 Conditional onenrollment there are also differences in the impact of Mai-monidesrsquo rule by PD Doubling PD at the mean is estimated toreduce the impact of Z on class size by 23 pupils On balance

FIGURE IVCDFs of Class Size in the 6 5 Discontinuity Subsample

Separately by Value of a Binary Instrument Based on Maimonidesrsquo Rule


therefore the analysis of affected groups indicates that theestimates presented here are affected disproportionately by smallerschools and by schools with fewer disadvantaged pupils thanaverage although the variation in impact along this seconddimension is much more modest than the rst

B Comparisons

The literature on class size and scholastic achievement oftenreports a summary statistic known as lsquolsquoeffect sizersquorsquo This is the testscore change that would result from a given change in class sizedivided by the standard deviation of the scores Finn and Achilles[1990] discuss two versions of this one using the standarddeviation of test scores among pupils and one using the standarddeviation of class means Since the overall variance is naturallylarger than the between-class variance measures based on therst standard deviation are always smaller than measures basedon the second The only measure that can be used here is thesecond because we do not have the micro test score data for fourthand fth graders However note that since class size is aclass-level intervention it seems reasonable to measure impactsrelative to the distribution of average scores

The Tennessee STAR experiment described by Finn andAchilles [1990 Table 5] yielded effect sizes of about 13 s ndash27samong pupils and about 32 s to 66s in the distribution of classmeans We can compare our results with the Tennessee experi-ment by calculating effect size for a reduction in class size of eightpupils as was done (on average) in the Tennessee experimentMultiplying this times the instrumental variables estimate forreading scores from column (2) in the table for fth graders (anestimate of 2 275 in a model with enrollment controls) gives aneffect size of about 29 s ( 5 22 points) in the distribution of classmeans The effect size is probably about 18 s among pupils22

Thus our estimates of effect size for fth graders are at the lowend of the range of those found in the Tennessee experiment Theeffect sizes based on estimates for fourth grade reading scores areonly about half as large as those for fth graders equal to roughly13s in the distribution of class means

22 This calculation is based on the ratio of between-class to total variation inthe third grade micro data


Another way to make this comparison is to use Kruegerrsquos[1999 Table VIII] instrumental variables estimates of per-pupileffects of reductions in class size for the STAR data (ie IVestimates of the coefficient on class size using the experimentalrandom assignment as an instrument for actual class size)Converting Kruegerrsquos IV estimates into standard deviation unitsusing the standard deviation of class average percentile scores (anestimate of 77 with a standard deviation of about sixteen pointsfor the Stanford Achievement Test) gives a per-pupil effect size ofabout 048 The corresponding gure for Israeli fth graders(reading scores) is 036 using estimates from the full sample and071 using estimates from the discontinuity sample Estimates forIsraeli fourth graders are much smaller about 017ndash019 Thus inper-pupil terms as well most of the estimates reported here are atthe low end of the range found in the STAR experiment Whilethese results may seem undramatic even apparently small effectsizes can translate into large movements through the scoredistribution [Mosteller 1995] For example the gap between thequartiles and the median reading score for the class averages ofIsraeli fth graders is less than two-thirds of a standard deviation

We can also compare the results reported here with theinstrumental variables estimates reported by Akerhielm [1995]Boozer and Rouse [1995] and Hoxby [1996] In a study usingdistrict-level population as an instrument for class size in a paneldata set for Connecticut school districts Hoxby [1996] nds noevidence of a relationship between class size and test scoresHoxby also reports results from a specication using a predictedclass size variable constructed by dividing the population intogroups close to twenty as well as population size itself as aninstrument This specication uses an instrument similar to theone used here but it does not control directly for population orenrollment effects Rather Hoxbyrsquos approach uses panel datamodels with district-specic intercepts and trends

Using grade enrollment and school-level average class size asinstruments Akerhielm [1995 page 235] nds statistically signi-cant effects on science and history achievement on the order of15s (in the pupil score distribution) for a ten-pupil reduction ineighth grade class size Akierhlemrsquos estimates may be affected bya possible secular association between enrollment and test scoresthat is not caused by changes in class size Using the sameNational Education Longitudinal Study (NELS) data set Boozerand Rouse [1995] report instrumental variables estimates on the


order of 29s in equations that control for base-year test scoresThe Boozer and Rouse instruments are indicators of state maxi-mum sizes for special education classes Both the Akerhielm andBoozer and Rouse ndings are similar to those reported here forfourth and fth graders

VI CONCLUSIONS

This paper presents a variety of OLS and instrumentalvariables estimates of the effect of class size on the reading andmath scores of elementary school children in Israel The rawpositive correlation between achievement and class size is clearlyan artifact of the association between smaller classes and theproportion of pupils from disadvantaged backgrounds Instrumen-tal variables estimates constructed by using functions of Mai-monidesrsquo rule as instruments for class size while controlling forenrollment and pupil background consistently show a negativeassociation between larger classes and student achievementThese effects are largest for the math and reading scores of fthgraders with smaller effects for the reading scores of fourthgraders Results for the math scores of fourth graders are notsignicant though pooled estimates for fourth and fth gradersare signicant and precise on both tests

Even though the effects reported here are mostly smallerthan those reported in the Tennessee STAR experiment they maynevertheless represent important gains relative to the distribu-tion of Israeli test scores The Israeli Parliament recently begandebating a bill that would lower the maximum legal class size to30 Using the cohort size distributions in our data we estimatethat the new law would reduce average elementary-school classsizes from 31 to about 25 and reduce the upper quartile from 35 to27 These reductions will clearly be expensive to implementrequiring something like 600 additional classes per grade But thendings reported here imply that the resulting change in Mai-monidesrsquo rule could have an impact equivalent to moving twodeciles in the 1991 distribution of class averages

It is also worth considering whether results for Israel arelikely to be relevant for the United States or other developedcountries In addition to cultural and political differences Israelhas a lower standard of living and spends less on education perpupil than the United States and some OECD countries [Klinov1992 OECD 1993] And as noted above Israel also has largerclass sizes than the United States United Kingdom and Canada


So the results presented here may be showing evidence of amarginal return for reductions in class size over a range of sizesthat are not characteristic of most American schools On the otherhand while classes as large as those in Israel are not typical in theUnited States in 1991 the average eighth grade class size inCalifornia was 29 pupils not dramatically lower than the corre-sponding Israeli average of 3223

Finally our study serves to highlight an important methodo-logical point Hanushekrsquos [1995] widely cited survey of research onschool inputs in developing countries shows the same pattern ofweak effects reported in his surveys of results for the UnitedStates Like Hanushek an education survey in The Economist[1997] magazine recently interpreted the lack of an associationbetween education inputs and test scores as evidence that schoolresources have no causal effect on learning The ndings pre-sented here suggest that such conclusions are premature Obser-vational studies are often confounded by a failure to isolate acredible source of exogenous variation in school inputs Theregression-discontinuity research design overcomes problems ofconfounding by exploiting exogenous variation that originates inadministrative rules As in randomized trials like the STARexperiment when this sort of exogenous variation is used to studyclass size smaller classes appear benecial

DATA APPENDIX

A 1991 Data (Fifth and Fourth Graders)A computerized data le from the Central Bureau of Statis-

tics [1991] survey of schools includes 1027 Jewish public (secularand religious) schools with fth grade pupils in 2073 (nonspecialeducation) classes24 These data containing information collectedin September were given to us by the Central Bureau of Statis-tics Data on class size collected between March and Juneprovided by the Ministry of Education contained records for 2052of these classes with information on class size for 2029 of them

Data on average test scores came in two forms Ministry ofEducation programmers provided one le with information onaverage test scores and numbers of test takers for 1733 of the

23 These gures are from United States Department of Education [1996 p107] Utah with an average size of 30 had the largest classes in the United States

24 The relevant Central Bureau of Statistics [1991 p 67] report indicatesthat there were 1081 Jewish public elementary schools in 1990ndash1991 although notall of these have regular (nonspecial education) classes and not all have enrollmentin all grades


classes (about 85 percent) We also obtained a le that containedaverage test scores and numbers of test takers for each grade ineach school for 1978 of the classes Among the 296 classes missingclass-level average scores school-level averages were available forall but 5 Since there was never more than one class missing aclass-level score and we know the number of test takers in eachschool and in each class with nonmissing scores we were able toimpute the missing class-level average for all but the ve classesmissing both class-level and school-level averages Finally the PDindex and town ID were added to the linked and imputedclassschool data set from a separate Ministry of Education le onschools The PD index was available for every school in thedatabase

The construction of the fourth gradersrsquo data set follows that ofthe fth graders A computerized le from the Central Bureau ofStatistics [1993] survey of schools includes 1039 Jewish publicschools with fourth grade pupils in 2106 (nonspecial education)classes Data on class size provided by the Ministry of Educationcontained records for 2082 of these classes with information onclass size for 2059 classes

We were provided with class-level average scores in 1769 ofthe 2059 fourth grade classes and school-level averages in 2025 ofthe 2059 classes Among the 290 classes missing class-levelaverage scores school-level averages were available for all but 4Since there was never more than one class missing a class-levelscore and we know the number of test takers in each school and ineach class with nonmissing scores we were able to impute themissing class-level average for all but four of the classes missingboth class-level and school-level averages The PD index and townID were then added as with the fth graders

We checked the imputation of class-level averages from schoolaverages by comparing the school and class averages in schoolswith one class and by comparing the imputed and nonimputeddata School and class-level averages matched almost perfectly inschools with one class We were unable to detect any systematicdifferences between schools that were missing some class-leveldata and the schools that were not The empirical ndings are notsensitive to the exclusion of the imputed class-level averages

B 1992 Data (Third Graders)

Construction of the third graders data set differs from theconstruction of the fourth and fth graders data sets because we


were provided with micro data on the test scores of third gradepupils As with the fourth and fth graders we began with theCentral Bureau of Statistics [1993] survey of schools This in-cludes 1042 Jewish public schools with third grade pupils in 2193(nonspecial education) classes Data on class size provided by theMinistry of Education contained records with information onclass size for 2162 of these classes

We used micro data on the test scores of third graders tocompute average math and reading scores for each class Scoredata were available for 2144 of the 2162 classes with classsize information in the CBS survey of schools Finally we addedinformation on the PD index and town identities from a Ministryof Education le containing information on schools Therewas no information on the PD index for 34 of the 2144 classeswith data on size and test scores so that the third gradesample size is 2111 This is probably because new schoolswould not have had a PD index assigned at the time data inour school-level le were entered into the record-keepingsystem

APPENDIX 1 DESCRIPTIVE STATISTICS WEIGHTED BY CLASS SIZE

Variable Mean SD

Quantiles

010 025 050 075 090






MASSACHUSETTS INSTITUTE OF TECHNOLOGY AND NATIONAL BUREAU OF ECONOMIC

RESEARCH

HEBREW UNIVERSITY OF JERUSALEM

REFERENCES

Akerhielm Karen lsquolsquoDoes Class Size Matterrsquorsquo Economics of Education Review XIV(1995) 229ndash241

Algrabi M lsquolsquoIndicates to Characterize a Schoolrsquos Social Composition and a Methodfor Allocating Additional Funds Between Schoolsrsquorsquo (Hebrew) Megamot XXI(1975) 219ndash227

Angrist Joshua D lsquolsquoGrouped Data Estimation and Testing in Simple LaborSupply Modelsrsquorsquo Journal of Econometrics XLVII (1991) 243ndash266

Angrist Joshua D and Guido Imbens lsquolsquoAverage Causal Response in Models withVariable Treatment Intensityrsquorsquo Journal of the American Statistical Associa-tion XC (1995) 431ndash442

DESCRIPTIVE STATISTICS WEIGHTED BY CLASS SIZE

Variable Mean SD

Quantiles

010 025 050 075 090












Angrist Joshua D and Victor Lavy lsquolsquoUsing Maimonidesrsquo Rule to Estimate theEffect of Class Size on Scholastic Achievementrsquorsquo NBER Working Paper No5888 January 1997

Boozer Michael and Cecilia Rouse lsquolsquoIntraschool Variation in Class Size Patternsand Implicationsrsquorsquo NBER Working Paper No 5144 June 1995

Campbell Donald T lsquolsquoReforms as Experimentsrsquorsquo American Psychologist XXIV(1969) 409ndash429

Campbell Donald T and J C Stanley Experimental and Quasi-ExperimentalDesigns for Research (Chicago Rand-McNally 1963)

Card David and Alan Krueger lsquolsquoDoes School Quality Matter Returns toEducation and the Characteristics of Public Schools in the United StatesrsquorsquoJournal of Political Economy C (1992a) 1ndash40

Card David and Alan Krueger lsquolsquoSchool Quality and Black-White RelativeEarnings A Direct Assessmentrsquorsquo Quarterly Journal of Economics CVII(February 1992b) 151ndash200

Central Bureau of Statistics Educational Institutions Kindergartens SchoolsPost-Secondary and Higher Education 199091 Series of Education andCulture Statistics No 198 reprinted from the Supplement to the MonthlyBulletin of Statistics No 9 1991

Central Bureau of Statistics Educational Institutions Kindergartens SchoolsPost-Secondary and Higher Education 199192 in the Supplement to theMonthly Bulletin of Statistics No 11 November 1993 (Hebrew only)

Coleman J S et al Equality of Educational Opportunity (Washington DC U SGPO 1966)

Cook Thomas D and D T Campbell Quasi-Experimentation Design andAnalysis Issues for Field Settings (Boston Houghton-Mifflin Company 1979)

Deaton Angus lsquolsquoData and Econometric Tools for Development Analysisrsquorsquo Chapter33 in The Handbook of Development Economics Volume III J Behrman andT N Srinivasan eds (Amsterdam Elsevier Science BV 1995)

The Economist lsquolsquoEducation and the Wealth of Nationsrsquorsquo March 29 1997 15ndash16Epstein I Hebrew-English Translation of the Babylonian Talmud Baba Bathra

Volume I (London Soncino Press 1976)Finn Jeremy D and Charles M Achilles lsquolsquoAnswers and Questions about Class

Size A Statewide Experimentrsquorsquo American Educational Research JournalXXVII (Fall 1990) 557ndash577

Glass Gene V and M L Smith lsquolsquoMeta-Analysis of Research on Class Size andAchievementrsquorsquo Educational Evaluation and Policy Analysis I (1979) 2ndash16

Glass Gene V L S Cahen M L Smith and N N Filby School Class SizeResearch and Policy (Beverly Hills CA Sage 1982)

Goldberger Arthur S lsquolsquoSelection Bias in Evaluating Treatment Effects SomeFormal Illustrationsrsquorsquo University of Wisconsin Institute for Research onPoverty Discussion Paper 123ndash72 April 1972

Hahn Jinyong P Todd and W van der Klaauw lsquolsquoEstimation of Treatment Effectswith a Quasi-Experimental Regression-Discontinuity Designrsquorsquo University ofPennsylvania Department of Economics mimeo July 1997

Hanushek Eric lsquolsquoThe Economics of Schooling Production and Efficiency in PublicSchoolsrsquorsquoJournal of Economic Literature XXIV (September 1986) 1141ndash1177 lsquolsquoInterpreting Recent Research on Schooling in Developing Countriesrsquorsquo TheWorld Bank Research Observer X (August 1995) 227ndash246 lsquolsquoSchool Resources and Student Performancersquorsquo in Does Money Matter TheEffect of School Resources on Student Achievement and Adult Success GaryBurtless ed (Washington DC Brookings Institution 1996)

Heckman James J A Layne-Farrar and P Todd lsquolsquoDoes Measured School QualityReally Matter An Examination of the Earnings-Quality Relationshiprsquorsquo NBERWorking Paper No 5274 September 1995

Hedges Larry V R D Laine and R Greenwald lsquolsquoDoes Money Matter AMeta-Analysis of Studies of the Effects of Differential School Inputs onStudent Outcomesrsquorsquo Educational Researcher XXIII (1994) 5ndash14

Hoxby Caroline lsquolsquoThe Effects of Class Size and Composition on Student Achieve-ment New Evidence from Natural Population Variationrsquorsquo Harvard Depart-ment of Economics manuscript July 1996

Hyamson Moses Annotated English translation of Maimonidesrsquo Mishneh TorahBook I (The Book of Knowledge) (New York Jewish Theological Seminary1937)


Israel Ministry of Education Standards for Compulsory Public EducationJerusalem 1959 (Hebrew) Memo from the Director General dated June 17 1966 (Hebrew) Director Generalrsquos Special Circular No 4 1980 (Hebrew) The Deputy Ministerrsquos Office press release dated July 24 1994 (Hebrew)

Klinov Ruth lsquolsquoPriorities in Public Resource Allocation for Educationrsquorsquo The Centerfor Social Policy Studies in Israel Jerusalem October 1992

Krueger Alan B lsquolsquoExperimental Estimates of Education Production FunctionsrsquorsquoQuarterly Journal of Economics CXIV (May 1999) 497ndash532

Lavy Victor lsquolsquoEndogenous School Resources and Cognitive Achievement in Pri-mary Schools in Israelrsquorsquo Hebrew University Falk Institute Discussion PaperNo 9503 1995

Loeb Susanna and John Bound lsquolsquoThe Effect of Measured School Inputs onAcademic Achievement Evidence from the 1920s 1930s and 1940s BirthCohortsrsquorsquo NBER Working Paper No 5331 November 1995

Moshel-Ravid Learning Teaching Education and Class Size A Review of theLiterature (Hebrew Pamphlet) (Jerusalem The Henrietta Sczold Institutethe National Institute for Research in the Behavioral Sciences 1995)

Mosteller Frederick lsquolsquoThe Tennessee Study of Class Size in the Early SchoolGradesrsquorsquo The Future of Children Critical Issues for Children and Youths V(SummerFall 1995) 113ndash127

Moulton Brent R lsquolsquoRandom Group Effects and the Precision of RegressionEstimatesrsquorsquo Journal of Econometrics XXXII (1986) 385ndash397

Mueller D C I Chase and J D Walden lsquolsquoEffects of Reduced Class Size inPrimary Classesrsquorsquo Educational Leadership XLV (1988) 48ndash50

National Center for Education Feedback The Results of Reading and MathematicsAchievement Tests Given to Fourth and Fifth Graders in June 1991 (Hebrewpamphlet) (Jerusalem Ministry of Education October 1991) Selected Analyses of the Results of the June 1992 Third Grade AchievementTests (Hebrew Pamphlet) (Jerusalem Ministry of Education March 1993)

OECD Centre for Educational Research and Innovation Education at a GlanceOECD Indicators (Paris Organization for Economic Cooperation and Develop-ment 1993)

OFSTED Class Size and the Quality of Education A Report from the Office of HerMajestyrsquos Chief Inspector of Schools (London Office for Standards in Educa-tion November 1995)

Pfefferman Daniel and T M F Smith lsquolsquoRegression Models for Grouped Popula-tions in Cross-Section Surveysrsquorsquo International Statistical Review LIII (1985)37ndash59

Robinson G E lsquolsquoSynthesis of Research on the Effects of Class Sizersquorsquo EducationalLeadership XLVII (1990) 80ndash90

Slavin E lsquolsquoClass Size and Student Achievement Small Effects of Small ClassesrsquorsquoEducational Psychology XXIV (1989) 99ndash110

Spiegelman C H lsquolsquoTwo Techniques for Establishing Treatment Effect in thePresence of Hidden Variables Adaptive Regression and a Solution of RiersolrsquosProblemrsquorsquo PhD thesis Northwestern University 1976

Summers Anita A and B Wolfe lsquolsquoDo Schools Make a Differencersquorsquo AmericanEconomic Review LXVII (September 1977) 639ndash652

Thistlewaithe D L and D T Campbell lsquolsquoRegression-Discontinuity Analysis AnAlternative to the Ex Post Facto Experimentrsquorsquo Journal of EducationalPsychology LI (1960) 309ndash317

Trochim William K Research Design for Program Evaluation The Regression-Discontinuity Approach (Beverly Hills CA Sage 1984)

U S Department of Education National Center for Education Statistics Educa-tion in States and Nations 2nd ed NCES-96-160 by Richard Phelps ThomasM Smith and Nabeel Alsalam (Washington DC U S GPO 1996)

van der Klaauw Wilbert lsquolsquoA Regression-Discontinuity Evaluation of the Effect ofFinancial Aid Offers on College Enrollmentrsquorsquo manuscript New York Univer-sity Economics Department December 1996

Wright Elizabeth N S M Shapson G Eason and J Fitzgerald Effects of ClassSize in the Junior Grades (Toronto Ontario Ministry of Education 1977)


between class size and achievement is inconclusive In widelycited meta-analyses of class-size research Glass and Smith [1979]and Glass Cahen Smith and Filby [1982] conclude that smallerclasses raise childrenrsquos test scores Card and Krueger [1992a1992b] also found that lower pupil-teacher ratios in school areassociated with higher adult earnings while randomized trials inTennessee and Ontario provide evidence for benecial effects ofrandomly assigned reductions in class size [Finn and Achilles1990 Wright Shapson Eason and Fitzgerald 1977] But resultsfrom the Glass et al meta-analyses have been questioned [Slavin1989] and Hanushekrsquos [1986 1996] surveys of research on theeffects of school inputs including pupil-teacher ratios report arange of ndings Recently Card and Kruegerrsquos studies of theschool qualityearnings link have also been challenged [HeckmanLayne-Farrar and Todd 1995]

Although recent years have seen renewed interest in theclass-size question academic interest in this topic is not only amodern phenomenon the choice of class size has been of concernto scholars and teachers for hundreds of years One of the earliestreferences on this topic is the Babylonian Talmud completedaround the beginning of the sixth century which discusses rulesfor the determination of class size and pupil-teacher ratios in biblestudy The great twelfth century Rabbinic scholar Maimonidesinterprets the Talmudrsquos discussion of class size as follows lsquolsquoTwenty-ve children may be put in charge of one teacher If the number inthe class exceeds twenty-ve but is not more than forty he shouldhave an assistant to help with the instruction If there are morethan forty two teachers must be appointedrsquorsquo [Hyamson 1937 p58b]1 Interestingly while Maimonidesrsquo maximum of 40 studentswas partly derived by interpreting the Talmud this rule leads tosmaller classes than the Talmudic rule which allows a maximumsize of 492

1 This is from Chapter II of lsquolsquoLaws Concerning the Study of Torahrsquorsquo in Book Iof Maimonidesrsquo Mishneh Torah The same chapter discusses compulsory schoolattendance (at public expense from the age of six or seven for boys) the penalty fornonenforcement of compulsory attendance laws (excommunication of the entiretown) hours of instruction (long) holidays (few) use of corporal punishment(limited) qualications for teaching positions (strict) competition between schoolsfor students (permitted desirable) and busing school students between towns toschools of higher quality (permitted only if the towns are not separated by a river)

2 The Talmudic portion that Maimonides relied on is lsquolsquoThe number of pupilsassigned to each teacher is twenty-ve If there are fty we appoint two teachersIf there are forty we appoint an assistant at the expense of the townrsquorsquo (quote fromChapter II page 21a of the Baba Bathra English translation on page 214 ofEpstein [1976])



























Variable Mean SD

Quantiles

010 025 050 075 090




















(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




















































































Variable Mean SD

Quantiles

010 025 050 075 090




















(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090














































































Variable Mean SD

Quantiles

010 025 050 075 090




















(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090








































































Variable Mean SD

Quantiles

010 025 050 075 090




















(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090

































































Variable Mean SD

Quantiles

010 025 050 075 090




















(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

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lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

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92)

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1)(2

36)

Per

cent

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372

23

692

369

24

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461

23

552

350

23

502

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24

35(0

14)

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13)

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nrol

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220

120

530

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79(0

09)

(026

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28)

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lmen

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00

052

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Pie

cew

ise

line

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040)

Roo

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SE

615

623

622

771

679

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834

840

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879

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6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

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aver

age

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ein

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clas

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rsar

ere

port

edin

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enth

eses

Sta

nda

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rors

wer

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rrec

ted

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onbe

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asse

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lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

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uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

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lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

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)(

034)

(034

)(0

17)

(017

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17)

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2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

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)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

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atio

nis

the

aver

age

scor

ein

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tan

dard

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rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

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rrec

ted

for

wit

hin

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oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

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clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




























































Variable Mean SD

Quantiles

010 025 050 075 090




















(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

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clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090































































(1) fsc 5 es [int ((es 2 1)40) 1 1]




















































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090










































































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090






































































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090
































































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




























































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




















































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

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ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090
















































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

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92)

(113

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131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

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372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

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13)

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7)(0

37)

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19)

(019

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nrol

lmen

t0

220

120

530

410

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79(0

09)

(026

)(0

28)

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37)

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lmen

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00

052

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Pie

cew

ise

line

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end

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193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

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1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

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clas

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tan

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rsar

ere

port

edin

par

enth

eses

Sta

nda

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rors

wer

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rrec

ted

for

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hin

-sch

oolc

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onbe

twee

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asse

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lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

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clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

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rrec

ted

for

wit

hin

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oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

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clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090










































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090

































































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




























































(4) nsc 5 X 8s p 0 1 fsc p 1 1 j sc













TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090

































































TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



























































TA

BL

EII

OL

SE

ST

IMA

TE

SF

OR

1991

5th

Gra

de4t

hG

rade

Rea

ding

com

preh

ensi

onM

ath

Rea

ding

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

743

673

725

699

(sd

)(8

1)

(99

)(8

0)

(88

)R

egre

ssor

sC

lass

size

221

20

312

025

322

076

019

014

12

053

20

402

210

550

09(0

31)

(02

6)(0

31)

(039

)(

036)

(04

4)(0

33)

(028

)(

033)

(036

)(0

33)

(039

)P

erce

ntdi

sadv

anta

ged

23

502

351

23

402

332

23

392

341

22

892

281

(01

2)(0

13)

(01

8)(

018)

(013

)(

014)

(016

)(0

16)

Enr

ollm

ent

20

020

172

004

014

(006

)(

009)

(00

7)(0

08)

Roo

tM

SE

754

610

610

936

832

830

794

665

665

866

782

781

R2

036

369

369

048

249

252

013

309

309

025

204

207

N2

019

201

82

049

204

9

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

s









TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090


































































TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



























































TAB

LE

III

RE

DU

CE

D-F

OR

ME

ST

IMA

TE

SF

OR

1991

5th

Gra

ders

4th

Gra

ders

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

Cla

sssi

zeR

eadi

ng

com

preh

ensi

onM

ath

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

A

Fu

llsa

mpl

e

Mea

ns29

974

467

330

372

568

9(s

d)

(65

)(7

7)

(96

)(6

3)

(80

)(8

8)

Reg

ress

ors

f sc

704

542

21

112

149

20

092

124

772

670

20

852

089

038

20

33(

022)

(02

7)(

028)

(03

5)(

039)

(04

9)(0

20)

(02

5)(

031)

(04

0)(

037)

(04

7)P

erce

nt

disa

dvan

tage

d2

076

20

532

360

23

552

354

23

382

054

20

392

340

23

402

292

22

82(

010)

(00

9)(

012)

(01

3)(

017)

(01

8)(0

08)

(00

9)(

013)

(01

4)(

016)

(01

6)E

nrol

lmen

t0

430

100

310

270

010

19(

005)

(00

6)(

009)

(00

5)(

007)

(00

9)R

oot

MS

E4

564

386

076

078

338

284

204

136

646

647

837

81R

25

165

533

753

772

472

555

615

753

113

112

042

07N

201

92

019

201

82

049

204

92

049

B

Dis

cont

inui

tysa

mpl

e

Mea

ns30

874

567

031

172

568

7(s

d)

(74

)(8

2)

(10

2)(7

2)

(78

)(9

1)

Reg

ress

ors

f sc

481

346

21

972

202

20

892

154

625

503

20

612

075

059

012

(05

3)(

052)

(05

0)(

054)

(07

1)(

077)

(050

)(

053)

(05

6)(

063)

(07

2)(

080)

Per

cen

tdi

sadv

anta

ged

21

302

067

24

242

422

24

352

405

20

682

029

23

482

343

23

062

291

(02

9)(

028)

(02

7)(

029)

(03

9)(

042)

(029

)(

028)

(03

2)(

034)

(04

1)(

043)

Enr

ollm

ent

086

003

041

063

007

024

(01

5)(

015)

(02

2)(

014)

(01

7)(

022)

Roo

tM

SE

595

558

624

624

858

853

549

526

657

657

826

825

R2

360

437

421

421

296

305

428

475

299

299

178

182

N47

147

147

141

541

541

5

Th

efu

nct

ion

f sc

iseq

ual

toen

roll

men

t[i

nt(

(en

roll

men

t2

1)4

0)1

1]

Sta

nda

rder

rors

are

rep

orte

din

pare

nth

eses

S

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sT

he

un

itof

obse

rvat

ion

isth

eav

erag

esc

ore

inth

ecl

ass


TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



























































TAB

LE

IV2S

LS

ES

TIM

AT

ES

FO

R19

91(F

IFT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

744

745

673

670

(sd

)(7

7)

(82

)(9

6)

(10

2)R

egre

ssor

sC

lass

size

21

582

275

22

602

186

24

102

582

20

132

230

22

612

202

21

852

443

(040

)(0

66)

(081

)(

104)

(11

3)(1

81)

(056

)(0

92)

(113

)(

131)

(15

1)(2

36)

Per

cent

disa

dvan

tage

d2

372

23

692

369

24

772

461

23

552

350

23

502

459

24

35(0

14)

(014

)(0

13)

(03

7)(0

37)

(019

)(0

19)

(019

)(

049)

(049

)E

nrol

lmen

t0

220

120

530

410

620

79(0

09)

(026

)(0

28)

(012

)(0

37)

(036

)E

nrol

lmen

tsq

uare

d10

00

052

010

(011

)(0

16)

Pie

cew

ise

line

artr

end

136

193

(03

2)(

040)

Roo

tM

SE

615

623

622

771

679

715

834

840

842

949

879

910

N20

1919

6147

120

1819

6047

1

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scas

anin

stru

men

tfo

rcl

ass

size




es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090





























































es es [ [040]

20 1 (es2) es [ [4180]

(1003) 1 (es3) es [ [81120]

(1303) 1 (es4) es [ [121160]



TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



























































TAB

LE

V2S

LS

ES

TIM

AT

ES

FO

R19

91(F

OU

RT

HG

RA

DE

RS)

Rea

ding

com

preh

ensi

onM

ath

Ful

lsam

ple

12

5D

isco

ntin

uity

sam

ple

Ful

lsam

ple

12

5D

isco

nti

nui

tysa

mpl

e

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Mea

nsc

ore

725

725

673

687

(sd

)(8

0)

(78

)(9

6)

(91

)R

egre

ssor

sC

lass

size

21

102

133

20

742

147

20

982

150

049

20

502

033

20

980

950

23(0

40)

(059

)(0

67)

(08

4)(

090)

(128

)(0

48)

(070

)(0

81)

(09

2)(

114)

(160

)P

erce

ntdi

sadv

anta

ged

23

462

345

23

462

354

23

472

290

22

842

284

22

992

290

(014

)(0

14)

(014

)(

034)

(034

)(0

17)

(017

)(0

17)

(04

2)(0

43)

Enr

ollm

ent

005

20

400

172

020

007

023

(008

)(0

24)

(022

)(0

10)

(029

)(0

28)

Enr

ollm

ent

squa

red

100

021

006

(011

)(0

14)

Pie

cew

ise

line

artr

end

100

130

(02

6)(

028)

Roo

tM

SE

665

666

663

802

664

669

782

782

782

865

823

824

N20

4920

0141

520

4920

0141

5

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

ll2S

LS

esti

mat

esu

sef s

cas

anin

stru

men

tfo

rcl

ass

size










TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



































































TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



























































TAB

LE

VI

DU

MM

Y-I

NS

TR

UM

EN

TR

ES

UL

TS

FO

RD

ISC

ON

TIN

UIT

YS

AM

PL

ES

5th

grad

e4t

hgr

ade

Rea

din

gco

mpr

ehen

sion

Mat

hR

eadi

ng

com

preh

ensi

onM

ath

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

12

5S

ampl

e1

23

Sam

ple

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Reg

ress

ors

Cla

sssi

ze2

687

25

882

451

25

962

395

22

702

175

22

342

380

018

21

182

247

(197

)(1

98)

(23

6)(

254)

(25

4)(2

81)

(130

)(1

57)

(205

)(1

62)

(202

)(2

34)

Per

cent

dis-

24

642

452

24

332

416

23

502

372

22

912

323

adva

ntag

ed(0

39)

(045

)(

050)

(05

8)(0

34)

(043

)(0

43)

(055

)S

egm

ent

12

509

24

542

107

27

542

694

212

62

162

22

672

694

21

892

357

27

31(e

nrol

lmen

t(2

40)

(25

9(3

19)

(30

7)(3

34)

(38

0)(1

77)

(22

3)(2

90)

(22

1)(2

87)

(33

1)36

ndash45)

Seg

men

t2

21

642

218

22

962

157

22

172

289

21

522

216

23

832

115

22

502

396

(enr

ollm

ent

(14

1)(1

64)

(20

0)(1

83)

(21

4)(2

41)

(12

4)(1

59)

(21

0)(1

56)

(20

7)(2

39)

76ndash8

5)R

oot

MS

E7

467

248

679

419

1410

26

726

708

308

258

539

52N

471

302

471

302

415

265

415

265

Th

eta

ble

rep

orts

resu

lts

from

asa

mpl

eof

clas

ses

insc

hoo

lsw

ith

enro

llm

ent

clos

eto

poin

tsof

disc

onti

nu

ity

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dar

der

rors

are

repo

rted

inp

aren

thes

esS

tan

dard

erro

rsw

ere

corr

ecte

dfo

rw

ith

in-s

choo

lco

rrel

atio

nbe

twee

ncl

asse

sA

lles

tim

ates

use

1[f s

c

32]

and

inte

ract

ion

sw

ith

dum

mie

sfo

ren

roll

men

tse

gmen

tsas

inst

rum

ents

for

clas

ssi

zeS

ince

ther

ear

eth

ree

segm

ents

th

ere

are

thre

ein

stru

men

tsT

he

mod

els

incl

ud

edu

mm

ies

for

the

rs

ttw

ose

gmen

tsto

con

trol

for

segm

ent

mai

nef

fect

s




z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090





























































z1sc 5 1[41 es 45] z2sc 5 1[81 es 85]

z3sc 5 1[121 es 125]











TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090
































































TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

hgr

ade

Poo

led

esti

mat

es

Rea

ding

(1)

Mat

h(2

)R

eadi

ng

(3)

Mat

h(4

)

Rea

din

gM

ath

(5)

(6)

(7)

(8)

Reg

ress

ors

Cla

sssi

ze2

156

20

802

101

019

21

972

120

21

272

019

(074

)(1

04)

(067

)(

080)

(047

)(0

54)

(061

)(0

70)

Per

cent

disa

dvan

tage

d2

162

20

912

288

21

622

356

22

222

315

21

26(0

68)

(094

)(0

73)

(08

6)(0

12)

(056

)(0

15)

(071

)G

rade

42

193

21

891

521

57(1

58)

(160

)(1

93)

(194

)E

nrol

lmen

t0

180

360

040

180

130

100

290

26(0

09)

(012

)(0

08)

(01

0)(0

07)

(007

)(0

09)

(009

)In

tera

ctio

nC

lass

size

PD

20

082

010

20

022

005

20

052

007

(003

)(0

04)

(003

)(

003)

(002

)(0

03)

Roo

tM

SE

625

843

666

782

644

644

810

811

N20

1920

1820

4920

4940

6840

67

Th

eu

nit

ofob

serv

atio

nis

the

aver

age

scor

ein

the

clas

sS

tan

dard

erro

rsar

ere

port

edin

par

enth

eses

Sta

nda

rder

rors

wer

eco

rrec

ted

for

wit

hin

-sch

oolc

orre

lati

onbe

twee

ncl

asse

sA

lles

tim

ates

use

f scan

df s

cP

Das

inst

rum

ents

for

clas

ssi

zean

dcl

ass

size

PD








(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090



























































TA

BL

EV

IIP

OO

LE

DE

ST

IMA

TE

SA

ND

MO

DE

LS

WIT

HP

ER

CE

NT

DIS

AD

VA

NT

AG

ED

INT

ER

AC

TIO

NTE

RM

S

5th

grad

e4t

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(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090

































































(1)RF

(2)OLS

(3)RF

(4)IV

(5)IV

(6)OLS

(7)RF

(8)IV

(9)IV


Class size 2 020 2 052 2 040 023 2 005 2 068(027) (047) (055) (032) (056) (065)


Enrollment 019 0004 002 003 2 006 006 008 008 058(005) (005) (006) (006) (021) (006) (007) (008) (025)


fsc 691 2 036 2 003(025) (033) (038)

Root MSE 419 567 567 567 567 663 663 663 663R2 546 144 144 056 056














(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090






































































(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090





























































(5)E[Y Z 5 1] 2 E[Y Z 5 0]

E[N Z 5 1] 2 E[N Z 5 0]

5S E[Yj 2 Yj 2 1 N1 $ j $ N0]P[N1 $ j $ N0]

S P[N1 $ j $ N0]







(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090






























































(6) oj

P[N1 $ j N0 W ] 5 E[N WZ 5 1] 2 E[N WZ 5 0]










B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090
































































B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




























































B Comparisons











VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090
































































VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




























































VI CONCLUSIONS







DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090





























































DATA APPENDIX

















Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090





































































Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090






























































Variable Mean SD

Quantiles

010 025 050 075 090







RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090




























































RESEARCH


REFERENCES






Variable Mean SD

Quantiles

010 025 050 075 090

































































































































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