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    Economics o f Education Review, Vol. 14, No. 3, pp. 229-241, 1995Jt~De~smon Elsevier Science LtdPrinted in Great Britain0272-7757/95 $9.50+0.000272--7757(95)00004--6

    D o e s C l a s s S i z e M a t t e r ?KAREN AKERHIELM

    Mathtech, Inc., 202 Carnegie Center, Suite 111, Princeton, NJ 08540, U.S.A.

    Abstract--This paper analyzes pupil-specific public school data unavailable in previous studies andapplies instrumental variable econometric methods to account for nomrandom allocation of students todifferent class sizes and the endogeneity of the class size variable. By using better data and improvedstatistical techniques, this paper shows that there are returns to investing in smaller classes for certainstudents and it provides some evidence on why past literature has produced such inconsistent findingson the effects of class size. [ J E L 121]

    I. INTRODUCTION 2. OVERVIEW OF THE LITERATURE

    WHETHER OR NOT to reduce class size is one of the While there is not consistent empirica l evidence onmost contentious aspects of the debate on educational the link between class size and student achievement,reform, mainly because past research has failed to some studies have shown that small class size isshow any consistent relationship between class size important for certain types of students, such as low-and student performance. Of a review of 112 studies achieving students, elementary school students, andon class size, only 23 found a statistically significant students from low socio-economic backgroundsrelationship between class size and student achieve- (Summers and Wolfe, 1977; ERS, 1986; Dolan andment and only fourteen of these exhibited a negative Schmidt, 1987).relation; the other nine studies showed that as class Other studies have analyzed the link between classsize increased, students performed better on tests, not size and future earnings , bypassing the immediateworse. 'There is little apparent merit for schools to effects on student achievement. One recent studypursue their ubiquitous quest for lowered class sizes', found that males who went to schools with low pupi]-(Hanushek, 1986, p. 1167). teacher ratios ended up with higher wages and that

    This paper contributes to the debate on class size the closing over time of the class size gap betweenby analyzing pupil-specific public school data blacks and whites explains about one-fifth of the clos-unavailable in previous studies and by applying ure in the black-white wage gap between 1960 andeconometric methods that account for non-random 1980 (Card and Kreuger, 1992).allocation of students to different class sizes. By using There are two major reasons why past research hasbetter data and improved statistical techniques, this revealed such inconsistent results. First, due to miss-paper shows that there are returns to inves ting in ing data on a student 's actual class size, past researchsmaller classes for certain students and it provides often used the pupil-t eacher ratio for an entire schoolsome evidence on why the past literature has pro- as the indicator of class size (Coleman e t a l . , 1966:duced such inconsistent findings. Coleman e t a l . , 1982; Chubb and Moe, 1990; Card

    [Manuscript received 16 March 1994; revision accepted for publication 6 December 1994.]22 9

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    2 3 0 E c o n o m i c s o f E d u c a t io n R e v i e wa n d K r e u g e r, 1 9 9 2 a ll u se d a g g r e g a te r a ti o s) . T o t h e 3 . T H E M O D E Lexten t tha t an aggrega te ra t io d i f fe r s f rom the c l a sss i ze tha t a s tuden t was ac tua l ly expose d to , measure - The spec i f i ca t ion of the educa t iona l p rocess usedment e r ror ex i s ts , b i a s ing the coe f f i c i en t o f t he c l a ss in th is pape r is based on the theore t ica l f rame wo rk ofs i ze va r i ab le toward ze ro . an educa t iona l p roduc t ion func t ion mo de l in which

    In add it ion , pup i l - t eache r ra t ios a re de f ined a s the the ach ievemen t (o r ou tpu t ) o f s tuden t i i n sub jec t jnum ber o f s tuden t s in the school d iv ided by the num- (a s measu red by t e s t scores) i s r e l a ted to va r ious fam-ber o f fu l l - time t eache rs fo r an en t i re school and ma y i ly backgro und , com mu ni ty , and school re sourcehave no th ing to do w i th ac tua l c l a ss s ize. Th i s ra t io inpu t s (Hanushek , 1979; L ink and Mul l igan , 1991) .o f t en inc ludes gu idance counse lors , p r inc ipa l s , and The mo de l can thus be wr i t t en a s :spec ia l educa t ion t eache rs in the count o f t eache rs ,and thus the lo we r the ra t io, the higher the no n-teac h- Y~j = et + 13~X~+ 132Wij + 133Zij + ij (1 )ing staff , regard less of actual c lass size . Mo reov er ,even i f the ra t io on ly inc ludes ac tua l t eache rs , scho ol swith the sam e pup i l - tea che r ra t io m ay hav e signif i- wh ere Yij i s the test score for chi ld i in subject j , X~i s a vec tor con ta in ing fami ly and communi ty cha rac -can t ly d i f fe ren t c l a ss s i zes depending on the ave ragenum ber o f hours o f t each ing requ i red (Bow les and t e r is t i cs fo r ch i ld i, Wi j i s cha rac te r i s ti c s o f ch i ld i ' sLevin , 1968). teacher in subject j , Zi j i s chi ld i ' s c lass size in subjectThe second prob lem i s tha t, a s oppo sed to us ing j , and % i s an unmeasured e r ror com pon ent whichinc ludes inpu t s such a s inna te ab i l i t y and mot iva t ion .expe r imenta l (o r randomized) da ta co l l ec t ed a s pa r to f a spec i f ic c l a ss s i ze eva lua tion , pas t s tud ies have The e r ror t e rm can a l so be thou ght o f a s ' unobs e rvedof t en re l ied on da ta tha t were pa r t o f a l a rge r na t iona l t e s t -t ak ing ab i l i t y ' .su rvey implem ented fo r o the r purposes . As a re sul t, The ques t ion th i s pape r a sks i s, wha t i s t he e f fec t Ofc la ss s i ze on a s tude n t ' s ach ievem ent , ho ld ing va r iousprev ious re sea rch has used d a ta in which s tuden t a llo -ca t ion to d i f fe ren t c l a ss s izes ma y no t be a rando m fami ly , comm uni ty , and teache r cha rac te ri s t ic s con-s t an t? Class s i ze may a f fec t ach ievement i f s tuden tprocess .If a scho ol has a del iberate pol ic y to assign diff icul t part ic ipat ion and sat isfact ion, student a t tendanc e,or l e ss ab le s tuden t s to sma l l e r c l a sses , t hen any pos i - t each ing prac t i ces and /or the amo unt o f ind iv id-ua l i zed a t t en t ion rece ived by the s tuden t va r i e s fo rt ive e f fec t o f sma l l c l a ss s i ze on s tuden t pe r form ance d i f fe ren t c l a ss s i zes (Smi th and Glass , 1980; Cahenmay be d i sgu i sed because such s tuden t s may t end to e t a l . , 1983).score lower on t es ts . L ikewise , i f a school has a po l - How ever , s tuden t s ma y no t be a l loca ted rando mlyicy o f a ss ign ing advan ced s tuden t s to the sma l l e r t o d i f fe ren t c l a ss s izes. Suppose a school has a de l ib -c l a sses , t hen a pos i t ive re l a t ion be tween sma l l c l a ss e ra t e po l i cy to a ss ign l e ss ab le s tuden t s to sma l l e rs i ze and s tuden t pe r form ance may be ove rs t a t ed . In c l a sses . In such a si tuat ion , t he a ssumpt ion o f Co vthese examples , c la ss s ize (and the a l loca t ion of s tu - (Z ,~ ) = 0 i s v io l a t ed when us ing OL S because thedent s to va r ious c la ss s i zes) is a cho ice va r i ab le and re l a tionsh ip be tween Z (c l a ss s i ze ) and ~ (unobse rv edthus the e s t ima t ion of the c l a ss s i ze - s tuden t ach ieve - test taking abi l i ty) i s posi t ive . As a resul t , 133 wil l bement re la t ionsh ip us ing ord ina ry l eas t squa res (OLS) upwa rd ly b ia sed , f a l se ly sugges t ing tha t a s c l a ss s i zewil l pro vide biased coeff ic ients, increases, stud ents ' cog ni t ive skil ls increase as wel l.

    Exper imen ta l s tud ies in Ind iana and Tennessee , i n S imi la rly , i f a school has a sys t emat i c po l i cy to a ss ignwhich s tuden t s were rand om ly ass igned to d i f fe ren t advanc ed s tuden t s to the sma l l e r c l a sses , t hen a nega -c l a ss s i zes and fo l lowed ove r t ime , foun d s ign if i can t , t i ve re la t ionsh ip be tween Z and ~ w i ll dow nwa rd lypos i t ive e f fec t s o f sma l l c l a sses on e l ementa ry scho ol b i a s the OL S re la t ionsh ip be tween c l a ss s i ze and s tud-s tuden t ach ievement (McGiver in e t a l . , 1989; W ord en t pe r formanc e , imply ing a s t ronger assoc ia t ione t a l . , 1990) . By show ing the impo r tance of sma l l be tween sma l l c l a sses and s tuden t ach ievem ent thanc lasses when the re ex i s t s a rando m a l loca tion of s tu - i s ac tua l ly the case .den t s to d i f fe ren t c l a ss s i zes, t he re su l ts f rom exper - The so lu tion en ta i ls i ns t rument ing c l a ss s i ze, us ingimenta l s tud ies sugges t t ha t a non- ra ndo m a l loca t ion a two-s t age l east squa res (2SLS ) approach , based onma y mas k the t rue re l a tionsh ip be tween c la ss s i ze and exog enou s schoo l - l eve l va r iab le s a s the ins t rument ss tuden t ach ievement , fo r c l a ss s ize . Thus , s tuden t ach ievem ent i s e s t ima ted

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    D o e s C l a s s S i z e M a t t e r ? 2 3 1condi t i ona l on the p rocess o f non- ran dom a l loca t ion Tab le 1 a l so show s tha t l owe r ab i l it y s tuden t s ( a sof s tuden t s t o d i f fe ren t c l a ss s i ze s a s fo l l ows: de f ined l a t er i n t h is pape r ) have sma l l e r c l a sse s on

    ave rage . When t - t e s t s were cons t ruc t ed tha t comparedY~j = ot + 13tXi + 132Wij + 1337.ij + % (2) the diff eren ces in clas s size m ean s am on g eac h pa ir

    o f subgroup o f s tuden t s , fo r a l l poss ib l e compar i sons(excep t fo r u rban non-whi t e and u rban whi t e sc i encewhere Z i s endo geno us and thus mu s t be p red i c ted s tuden ts ) , t he nu l l hypo thes i s o f s imi l a r c la ss s i ze

    us ing exogen ous va r i ab le s t ha t a re r e l a t ed to Z and means fo r t he two subgroups was re j ect ed .no t wi th e .

    The ma jor p rob lem wi th t h i s mode l spec i f i ca t i on i su s i n g d a t a f r o m o n e y e a r o n l y ( in t h is c a s e t h e e i g h th O L S R e g r e s s i o n R e s u l t sg rade ) wh ich ma y no t p rov ide a su f f i ci en t con t ro l fo r Tab le 2 d i sp l ays t he re su l t s o f runn ing the OLSregress ion fo r a l l four sub jec t s , us ing the mode lini ti al ab i l it y . How ever , t he cor re l a t i on be tween desc r ibed by equa t ion (1 ). Th i s t ab l e a l so show s theinna t e ab il i ti e s and c l a ss s i ze , a f t e r con t ro l l i ng fo r means and s t anda rd dev ia t ions o f each o f t he va r i -f a m i l y b a c k g r o u n d f a c t o r s s u c h a s i n c o m e a n d e d u -cat ion, i s l ike ly to be smal l (Ha nush ek, 1979) . ables used. In a l l fou r subjects , the pu pi l ' s ac tual c lasss i ze has a pos i t i ve i n f luence on ach ievement ( a s mea -

    sured by t e s t score s ) whi l e fo r t h ree o f t he fou r s t~b-4 . N E L S D A T A S E T jects (a l l except sc ience) , the effec t i s s igni f icant (a t

    The da t a used in t h is ana lys i s a re f rom the U .S . t he f i ve pe rcen t l eve l fo r ma th and Engl i sh and a t t heDep a r tment o f Edu ca t ion ' s Na t iona l Educa t ion t en pe rcen t l eve l fo r h i s to ry ) . Ove ra l l , u s ing OL SLong i tud ina l S tudy (NE LS) o f 1988. In t h is na t ion- show s tha t t he re la t i onsh ip be tween c l a ss s i ze anda l ly repre sen ta ti ve sample , ove r 24 ,000 e igh th g rade rs s tuden t ach i eveme nt , a f t e r con t ro l l i ng fo r f ami ly ,f rom pub l i c and p r iva t e schoo l s t h rough out t he U.S . com mu ni ty , and teache r i npu t s , i s pos i t i ve and s ig -were su rvey ed in 1988 and t e s t ed in fo ur sub jec t s n i f ican t , sugges t ing tha t a s c l a ss s i ze i nc reases , s t ud-(ma them a t i c s , Eng l i sh , sc i ence , and h i s to ry ). On e pa r - en t pe r fo rman ce inc reases a s we l l.en t and tw o t eache rs o f each s tuden t ( r epre sen t ingt w o o f t h e f o u r s u b je c ts ) w e r e a ls o s u r v e y e d a s w a s S p l i n e T e c h n i q u e san admin i s t ra to r f rom each schoo l . On ave rage , each Sp l ines were used to de t e rmine i f t he re a re ce r t a inof t he 1 ,052 pa r t i c ipa t ing schoo l s was repre sen ted by ranges fo r wh ich changes i n c l a ss s ize m ay no t have24 s tuden t s and f ive t eache rs . The cogn i t i ve t e s ts con - an im pac t on t e s t score s . Rev iew s o f t he c l a ss s i zet a ined a to t a l o f 116 i t ems to be com ple t ed in 85 l it erature (ERS , 1986 ; Gla ss and Smi th , 1979) haveminu te s , o f t en conc lud ed tha t chang es i n c l a ss s i ze s have no

    e f fec t on s tuden t ach i evement i n t he mid- ranges o f5 . E S T I M A T I O N R E S U L T S c l a ss s i ze ( i.e ., 2 0 t o 3 0 s t ud e n ts ) . S i n c e t h e re h a v e

    been seve ra l de f in i ti ons o f t he 'm id- ran ge ' o f c l a ssAl l e s t ima t ion and t e s t ing w as cond uc ted fo r t he s i ze i n t he l it erature, t h i s sec t ion exam ines two poss-

    four d i f fe ren t sub jec t s t o see i f c l a ss s i ze e f fec t s d i f fe r i b l e scena r ios a s show n in Tab le 3 .by sub jec t ma t t e r and to take advan tage o f NE LS In bo th scena r ios , t h ree c l a ss s i ze ' sp l i ne ' va r i ab l e sc l a ss room-spec i f i c da t a . we re i nc luded in t he regre ss ion to repre sen t t he l ow,

    As Tab le 1 i ll us tra te s , t he ave rage c l a ss s ize and midd le and h igh ranges o f c l a ss s i ze . Whi l e ac tua lthe pup i l - t eache r r a t i o fo r a l l four sub jec ts , i s l a rge r c l a ss s ize was t he va r i ab l e tha t r epre sen ted the sma l lfo r non-whi t e s tuden t s t han fo r whi t e s tuden t s (no t r ange o f c l a ss s i ze, two new va r i ab l e s were c rea t edcon t ro l l i ng f o r o the r cha rac te r i s ti c s ) . Th i s i s i ncon- t o r epre sen t t he midd le and l a rge ranges . ~ As a r e su l ts i s ten t wi th recen t r e sea rch tha t found s imi l a r pu p i l - o f t h is spec i fi ca t ion , t he coe f f i c i en t o f t he sma l l - rangeteache r r a t i os fo r whi t e s and b l acks (Booze r e t a l . , variable (b~p~i,ej ) represents the effec t o f ea ch1992; Ca rd and Kreuge r , 1992). Urban non -whi t e s add i t iona l s tuden t in the sma l l r ange o f c l a ss s izehad s ign i f i can tly l a rge r c l a sse s than u rban whi t e s fo r whi l e the coe f fi c i en t s added toge the r f rom the sma l l -a l l sub ject s excep t sc i ence . Ove ra l l , t he ave rage range and the mid- rang e ( b s p l l n e l + b s p l i n e 2 ) r e p r e s e n tpup i l - t eache r r a t io i s abou t 18 o r 19 , cons i s t en t ly and the e f fec t o f an add i t iona l s tuden t i n t he mid- range .subs t an t ia l l y l ower t han ave rage c l a ss s ize . L ikew ise b s p J i n e I + bspline2 + bspline3 equa l s t he e f fec t on

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    23 2 Economics of Education ReviewTable 1. Class size by type of public school student

    (N=4741) (N=4973) (N=4645) (N=4850)Type of Average Av er ag e Av er ag e Averagepublic school math English sc ience historystudent class size class size class size class sizeNon-white 25.33 25.05 25.36 26.09White 23.71 23.95 24.57 24.53t-Test for differences 8.16 6.15 4.46 8.14in meansUrban 25.70 25.36 25.60 26.65Non-urban 23.73 23.97 24.55 24.49t-Test for differences 9.72 7.37 6.00 10.89in meansUrban non-white 26.t9 25.87 25.52 27.06Urban white 25.02 24.72 25.67 26.22t-Test for differences 3.33 3.50 0.48 2.40in meansNon-lower ability 24.78 24.94 25.05 25.25Lower ability 21.39 21.01 23.17 23.15t-Test for differences 13.19 16.35 7.57 7.83in means

    (N=4741) (N=4973) (N=4645) (N=4850)Type of Average Av er ag e Ave ra ge Averagepublic school PIT Ratio P/T Ratio PtT Ratio P/I" RatiostudentNon-white 19.18 18.63 18.53 18.73White 17.45 17.67 17.79 17.73t-Test for differences 9.67 5.77 4.14 6.08in meansUrban non-white 19.55 18.88 18.01 19.14Urban white 18.21 18.37 17.74 18.21t-Test for differences 5.13 1.95 1.09 3.55in meansNote: The t-tests are based on the assumption of separate variances. Assuming pooledvariances yielded "t-tests of similar magnitudes. P/T = Pupil/Teacher.

    achievement of one additional student in the large Finally, the null hypothesis to be tested is:range of class size.

    Table 4 displays the spline results. For clar ity 's Ho "bspline2=bspline3= 0.sake, the coefficients from the other variables areomitted from the table (there were no sign changesor significant differences of magnitude of the coef- If Ho is accepted, this implies that the slope of theficients of the other variables as compared to Table class size-achievement relation is constant and can2). As shown at the bottom of the table, the effect of be represented by the class size variable itself withoutthe mid-range is negative for both scenarios for his- the additional spline variables. In all eight cases (fourtory and science, while the positive coefficients in the subjects across two scenarios), the null hypothesislower ranges for math and English achievement was rejected, implying that the effect of class size isswamp the negative coefficients in the mid-range to not independent of the level of class size and thatyield an overall positive effect. 2 there are nonl inearities in the effects of class size.

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    D o e s C l a s s S iz e M a t t e r ? 233Table 2, OLS regression results by subject area

    Math subject (N = 4741) English subject (N = 4973)F=74 R2=0 .18 F=71 R2=0.17Dependent variable = Std. err. of est. = 7.82 Std. err. of est. = 4.26Test score (number right):' Mean of dep. var. = 22.64 Mean of dep. var. = 12.95Std. Std.Var iab le Coeff . t -S ta t Me an dev . Coeff . t -S ta t Me an dev .

    Constant 8.83 10.86 6.80 15.54Family income ($10,000's) 1.00 9.27 3.71 2.98 0.59 10.18 3.65 2.83Family income squared -0.0 4 -5.5 3 -0.0 2 -6.7 3Family composition 1.06 4.16 0.65 0.48 0.41 3.03 0.64 0.48(1 = married)Parent educ. (1 =beyond 2.31 9.21 0.61 0.49 1.25 9.49 0.60 0.49H.S.)Sex of student (I = male) 0.60 2.63 0.49 0.50 -0 .8 5 - 7.0 1 0.49 0.50Race of student (1 = white) 2.55 9.02 0.72 0.45 1.39 9.50 0.69 0.46Preschool (1 = attended) 0.89 3.67 0.50 0.50 0.46 3.53 0.50 0.50% of single parent families in -2. 09 -3.21 0.29 0.18 -1 .66 -4 .95 0.30 0.19schoolUrban (1 = Yes) -0. 81 -2 .5 6 0.23 0.42 -0. 01 -0 .0 6 0.23 0.42.Race of teacher (1 = White)" 1.90 5.08 0.87 0.34 0.85 4.20 0.88 0.32Sex of teacher (1 = Male)" 0.40 1.61 0.44 0.50 -0 .3 7 - 2. 61 0.25 0.43Years of experience* 0.32 4.87 14.62 7.31 0.06 1.52 15.07 6.92Years of experience squared* -0.01 -3. 85 -0.001 -0 .90Class size" 0.12 6.25 24.17 6.10 0.08 7.71 24.29 5.68Science subject (N = 4645) History subject (N = 4850)F=5 0 R2=0.13 F=5 5 Rz=o.14Dependent variable = Std. err. of est. = 4.20 Std. err. of est. = 5.02Test score (number right)" Mean of dep. var. = 13.82 Mean of dep. var. = 19.44

    Std. Std.Var iab le Coeff . t -S ta t Me an dev . Coeff . t -S ta t Me an dev .Constant 8.52 16.59 12 .9 0 24.66Family income ($10,000 's) 0.47 8.08 3.89 3.10 0.60 8.94 3.95 3.29Family income squared -0.0 2 -5.33 -0.0 2 -5. 82Family composition 0.31 2.25 0.64 0.48 0.48 2.92 0.64 0.48(1 = married)Parent educ. ( I = beyond 1.22 9.01 0.62 04 9 1.59 9.86 0.62 0.49H.S.)Sex of student (I = male) 0.61 4.97 0.48 0.50 0.39 2.67 0.49 0.50Race of student (1 = white) 1.35 8.90 0.74 0.44 1.17 6.50 0.74 0.44Preschool (1 = attended) 0.60 4.53 0.51 0.50 0.77 4.97 0.52 0.50% of single parent families in -1 .31 -3 .4 6 0.28 0.17 0.55 1.20 0.28 0.17schoolUrban (1 = Yes) 0.04 0.27 0.21 0.41 -0 .6 6 -3 .3 3 0.21 0.40Race of teacher (1 = white)" 1.06 3.41 0.96 0.20 0.97 3.71 0.91 0.29Sex of teacher (1 = male)" 0.19 1.46 0.57 0.50 -0 .11 -0 .71 0.61 0.49Years of experience" 0.10 2.79 13.85 7.74 0.06 1.50 15.29 7.54Years of experience squared" -0.0 03 -2. 48 -0.001 -0 .92Class size" 0.002 0.13 24.78 5.22 0.03 1.91 24.93 5.73* = These variables are subject specific. For example, in the math student regress ion, the race, sex and experience of thestudent' s math teacher and the studen t's math class size are the relevant variables. Likewise, the dependent variable is thestudent' s math test score.

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    2 3 4 Economics o f Education Revie wT a b l e 3 . D e f in i ng t h e c l as s s i z e s p li n e v a r ia b le s D i s t r i b u t i o n o f S t u d e n t s b y A b i l i t y i n t o C l a s s

    S i z e sV a r ia b le T h e p o s i t i v e O L S i m p a c t o f c l a s s s i z e o n t e s tSpl ines Scenar io 1 Scenar io 2 def in i t ion s c o r e s , a n d t h e p o s i t i v e e f f e c t s o f t h e l o w e r r a n g e s o f

    S p l in e 1 0 - 1 5 0 - 2 0 Ac tu a l c la ss c l a s s s i z e a s i l l u s t r a t e d b y th e s p l in e r e s u l t s , c o u ld b e( small- range) S tudents S tudents s ize a r e f l ec t ion tha t s tud ents w ho a re do ing po or l y (orI f c l. w h o a r e t r a c k e d i n t o s p e c i a l e d u c a t i o n c l a s s e s ) a r e

    s iz e > 1 5 ( 20 ) , p l a c e d i n t o v e r y s m a l l c l a s s e s i n o r d e r t o r e c e i v e s p e -v a r = c i a l a t t e n t i o n . T o e x p l o r e t h i s l i n e o f r e a s o n i n g , t h eS p l in e 2 1 6 - 3 0 2 1 - 3 0 CI . s iz e- 15 (2 0 ); f o l lo w in g v a r i a b le wa s a n a l y z e d : a b i l i ty l e v e l o f th e(mid-range) Students Students else zero c l a s s ( h i g h , m i d d l e , o r l o w ) f o r a l l f o u r s u b j e c t s . T h eI f c l . s ize >30 ,v a r = a b i l i t y l e v e l o f t h e s t u d e n t ' s c l a s s i s f r o m a q u e s t i o n

    S p l in e 3 C1. s i ze - 3 0; e l s e d i r e c te d to th e t e a c h e r a n d i s e x p r e s s e d a s a v a r i a b le( la rge- range) 31+ Students 31+ Students ze ro w he re on e eq ua ls be ing in a low er ab i l i ty c lass and

    z e r o e q u a l s o t h e r w i s e . A b o u t 1 4 t o 1 8 p e r c e n t o f t h es t u d e n t s a r e c h a r a c t e r i z e d a s b e i n g i n l o w e r a b i l i t yc l a s s e s , a s d e f i n e d b y t h e t e a c h e r. F i g u r e s 1 t h r o u g h

    T a b l e 4. Resu l ts f rom OLS regress ions with sp l ines1. Regression results fo r class size variablesMath English History Science(N = 4741) (N = 4973) (N = 4850) (N = 4645)

    Coeff. t-Stat. Coeff. t-Stat. Coeff. t-Stat. Coeff. t-Stat.Scenario 10-1 5 Students 0.44 4.27 0.31 5.11 0.40 4.78 0.32 3.671 6 -3 0 S tu de nt s - 0 . 3 8 - 3 . 2 6 - 0 . 2 5 - 3 . 6 1 - 0 . 4 2 - 4 . 5 4 - 0 . 3 3 - 3 . 5 931+ Students 0 .16 1 .52 -0 .0 6 -0 .9 2 0 .08 1 .28 -0 .03 -0 .3 8Scenario 20-2 0 Students 0.24 4.55 0.20 6.53 0.17 4.21 0.17 4.382 1 - 3 0 S tu de nt s - 0 . 2 2 - 2 . 7 1 - 0 . 1 7 - 3 . 6 9 - 0 . 2 1 - 3 . 8 7 - 0 . 2 3 - 4 . 4 131+ Students 0.22 1.84 -0 .0 0 -0 .0 3 0.12 1.75 0.05 0.692. Joint test o f b(mid) = b(large) = 0Scenario 1 F-test 5.46 8.51 10.33 7.21Scenario 2 F-test 3.82 8.81 7.52 10.49Cr i t ica l va lue a t 5% s ign if icance = 33. Impact of an additional student in class on achievement in each of thethree ranges Formula Math English History ScienceScenario 10-1 5 Students b(small) 0.44 0.31 0.40 0.3216-30 Students b ( small) + b(mid) 0 .06 0 .06 -0 .0 2 -0 . 0131+ Students b(sma ll) + b(m id) 0.22 0.00 0.06 -0 .0 4+b( la rge)Scenario 20-2 0 Students b(small) 0.24 0.20 0.17 0.1721-3 0 Students b ( small) +b(mid) 0 .02 0 .03 -0 .0 4 -0 .0 631+ Students b(sm all) + b(mid ) 0.24 0.03 0.08 -0. 01+b( la rge)

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    D o e s C l a s s S i z e M a t t e r ? 2 3 54 d e p i c t c l a s s s i z e d i s t r i b u t i o n s b y a b i l i ty f o r a l l f o u r a c h i e v e m e n t a n d t o n e g a t i v e a n d i n s i g n i f i c a n t f o rs u b j e c t s . A s s h o w n , l o w e r a b i l i t y s t u d e n t s a r e d i s p r o - m a t h a n d E n g l i s h a c h i e v e m e n t , s u g g e s t i n g r e t u r n s t op o r t i o n a t e l y r e p r e s e n t e d i n t h e lo w e r r a n g e s o f c la s s i n v e s t i n g i n s m a l l e r c l a s s e s f o r so m e s t u d e n ts .s i z e ? O v e r a l l , c l a s s s i z e e f f e c t s a r e s i g n i f i c a n t f o r s c i e n c e

    a n d h i s t o r y b u t s m a l l , a s c o m p a r e d t o f a m i l y b a c k -I n s t r u m e n t a l Variable Approach g r o u n d e f f e c t s . F o r e x a m p l e , a o n e u n i t d e c r e a s e i nT h e e v i d e n c e p r e s e n t e d a b o v e s u g g e s t s th a t l o w e r a s t u d e n t ' s s c i e n c e c l a s s s i z e w i l l i n c r e a s e t h a t s t u d -a b i l i t y s t u d e n t s a r e n o t r a n d o m l y a l l o c a t e d t o d i f f e r e n t e n t ' s s c i e n c e te s t s c o r e b y . 08 p o i n t s ( b a s e d o n a 2 5c l a s s s i z e s a n d a r e d i s p r o p o r t i o n a t e l y f o u n d i n t h e p o i n t t e s t w i t h a m e a n o f 1 3. 82 a n d a s t a n d a r d d e v i -s m a l l e r c l a s s e s . O n e w a y t o c o r r e c t f o r t h e c o r r e la t i o n a t i o n o f 4 .5 ) . L i k e w i s e , d e c r e a s i n g a s t u d e n t ' s h i s to r yb e t w e e n c l a s s s i z e a n d u n o b s e r v e d t e s t - t a k i n g a b i l i t y c l a s s s i z e b y o n e p e r s o n w o u l d o n l y i n c r e a s e t h e s t u d -w o u l d e n t a i l a n i n s tr u m e n t a l v a r i a b l e p r o c e d u r e th a t e n t ' s t e s t s c o r e b y 0 .0 7 p o i n t s ( b a s e d o n a 3 0 p o i n tp r e d i c t s c l a ss s i z e to a c c o u n t f o r t h e e n d o g e n e i t y o f t e s t w i t h a m e a n o f 1 9 .4 4 a n d a s t a n d a r d d e v i a t i o n o fc l a s s s i z e . B a s e d o n a t h e o r e t i c a l k n o w l e d g e o f h o w 5 .4 ) . D e c r e a s i n g c l a s s s i z e f r o m t h e c u r r e n t a v e r a g es c h o o l s w o r k , t h i s p a p e r u s e s th e a v e r a g e c l a s s s i z e o f 2 5 t o 1 5 w o u l d i m p r o v e th e s tu d e n t ' s s c i e n c e a n df o r a g i v e n s u b j e c t i n t h e s t u d e n t ' s s c h o o l a n d t h e h i s t o r y t e s t s c o r e s b y 0 .8 a n d 0 . 7 o f a p o i n t , r e s p e c t -e i g h t h g r a d e s t u d e n t e n r o l l m e n t i n t h e s c h o o l a s i v e l y . T h u s , t h e i n c r e m e n t a l b e n e f i t s m a y n o t s u r p a s si n s t r u m e n t s f o r c l a s s s i z e. t h e i n c r e m e n t a l c o s t s o f d e c r e a s i n g c l a s s s i z e . 4T h e a v e r a g e c l a s s s i z e v a r i a b l e i s e x o g e n o u s t o a n F i n a l l y , a s t h e s p l in e r e s u l t s s u g g e s t e d a b o v e , t h ei n d i v i d u a l s tu d e n t b u t i s c o r r e l a t e d s t r o n g l y w i t h a n e f f e c t o f c l a s s s i z e , e v e n a f t e r c o r r e c t i n g f o r n o n - r a n -i n d i v i d u a l s t u d e n t ' s a c t u a l c l a s s s i z e . T h i s v a r i a b l e i s d o m a l l o c a t i o n o f s t u d e n ts , m a y n o t b e i n d e p e n d e n tc o n s t r u c t e d b y ta k i n g t h e a v e r a g e c l a s s s i z e o f a ll t h e o f t h e l e v e l o f c l a s s s iz e . H o w e v e r , i t i s p r o b l e m a t i cs t u d e n t s i n t h e s c h o o l i n a g i v e n s u b j e c t t h a t t o c r e a t e s p l i n e c o v a r i a t e s w h i c h i n c l u d e a n e n d o g e n -r e s p o n d e d t o t h e N E L S s u r v e y . B y a g g r e g a t i n g th e o u s c o m p o n e n t . O n e a p p r o a c h m a y b e t o r e c r e a t e t h ec l a s s s i z e v a r i a b l e , t h e w i t h i n - s c h o o l n o n - r a n d o m s p l i n e s a s d e f i n e d a b o v e , b u t u s i n g t h e a v e r a g e c l a s ss t u d e n t a l l o c a t i o n p r o b l e m i s b y p a s s e d w h i l e a t th e s i z e v a r i a b l e in s t e a d o f a c tu a l c l a s s s i z e . T h e r e s u l t ss a m e t i m e a v o i d i n g t h e p o t e n t i a l p r o b l e m s o f u s i n g a o f t h i s s p e c i f i c a t i o n y i e l d e d a n e g a t i v e a n d s i g n i f i c a n tp u p i l - t e a c h e r r a t i o , w h i c h m a y n o t h a v e a n y r e l a t i o n e f f e c t o f c l a ss s iz e i n t h e f o l l o w i n g c a t e g o r i e s : m a t ht o a c t u a l c l a s s s i z e i n a s c h o o l , c l a s s e s i n t h e s m a l l - r a n g e o f b o t h s c e n a r i o s ( s c e n a r i o

    A s c h o o l ' s e i g h t h g r a d e e n r o l l m e n t i s a l s o e x o g e n - 1 e f f e c t = - 0 . 4 7 , s c e n a r i o 2 e f f e c t = - 0 . 3 4 ) , s c i e n c eo u s to an i n d i v i d u a l s t u d e n t b u t m a y id e n t i f y w h i c h a n d h i s t o r y c l a s s e s in t h e m i d - r a n g e s o f s c e n a r i o 2s c h o o l s a r e m o r e l i k e l y t o a l l o c a t e t h e i r s t u d e n t s i n a ( s c i e n c e e f f e c t = - 0 . 1 4 ; h i s t o r y e f f e c t = - 0 . 1 4 ) .n o n - r a n d o m w a y . L a r g e r s c h o o l s m a y h a v e t h e s c a l e T h e s e r e s u l t s u p h o l d t h e p r e v i o u s f i n d i n g s o f :a n d r e s o u r c e s t o o f f e r s p e c i a l i z e d s m a l l c l a s s e s f o r ( 1 ) n o n l i n e a r i t i e s i n c l a s s s i z e e f f e c t s , a n d ( 2 ) r e t u r n sl o w e r a b i l i t y s t u d e n t s; it m a y n o t b e c o s t - e f f e c t i v e f o r t o i n v e s t i n g i n s m a l l e r s c i e n c e a n d h i s t o r y c l a s s e s .s m a l l e r s c h o o l s t o o f f e r t h e s e s e r v i c e s . O v e r a l l ,s c ho o l s iz e i s a n i m p o r ta n t p r e d ic t o r o f a s c h o o l 's 6 . E X T E N S I O N S O F T H E M O D E Lc u r r i c u l u m a n d o f t h e p r e v a l e n c e o f t r a c k i n g s t u d e n t sb y a b i l i t y ( M o n k a n d H a i l e r , 1 9 9 3 ; M o n k , 1 9 87 ; I n o r d e r t o e x a m i n e c l a s s s i z e e f f e c t s m o r e c l o s e l y ,O a k e s , 1 9 8 5 ; P o w e l l et a l . , 1 9 85 ). t h e l o w - a c h i e v i n g s t u d e n t s t h a t a r e g r o u p e d i n t h e

    T a b l e 5 s h o w s t h e r e s u l t s o f r e g r e s s i n g c la s s s i z e s m a l l c la s s e s w e r e d r o p p e d f o r a n a l y s i s ( d e f i n e d a so n a l l o f t h e e x o g e n o u s v a r i a b l e s i n t h e s y s t e m a s l o w - a b i l i t y s t u d e n t s i n c l a s s e s f e w e r t h a n 1 8w e l l a s t h e t w o - s t a g e l e a s t s q u a r e s ( 2 S L S ) e s ti m a t i o n s t u d e n ts ) . W h e n u s i n g O L S o n th i s s u b s a m p l e o f s t u -w i t h th e p r e d i c t e d v a l u e o f c l a s s s i z e e n t e r e d in t o t h e d e n t s , t h e e f f e c t o f c l a s s s i z e w a s n e g a t i v e a n d s i g -a c h i e v e m e n t e q u a t i o n , a s d e s c r i b e d i n e q u a t i o n ( 2 ) . n i f i ca n t o n l y f o r s c i e n c e a n d h i s t o r y a c h i e v e m e n t , j u s tT h e s e a r e c o n t r a s t e d w i t h th e O L S r e g r e s s i o n r e s u l t s a s w a s f o u n d fr o m u s i n g t h e 2 S L S t e c h n i q u e o n t h ef r o m T a b l e 2 . f u l l s a m p l e o f s t u d e n t s, b u t w i t h e v e n l o w e r m a g n i -

    T h e 2 S L S r e s u l t s s h o w t h a t o n c e t h e e n d o g e n e i t y t u d e s (t h e c o e f f i c i e n t s o f th e c la s s s i z e v a r i a b l e w e r eo f c l a s s s i z e i s t a k e n i n t o a c c o u n t , t h e e f f e c t o f c l a s s - 0 . 0 4 a n d - 0 . 0 3 f o r s c i e n c e a n d h i s t o r y a c h i e v e -s i z e o n s tu d e n t a c h i e v e m e n t c h a n g e s f r o m p o s i t i v e t o m e n t , r e s p e c t i v e l y ) . T h e s i m i l a r i t y o f t h e s e f i n d i n g sn e g a t i v e a n d s i g n i f i c a n t f o r s c i e n c e a n d h i s t o r y s u g g e s t s t h a t t h e 2 S L S t e c h n i q u e a p p r o p r i a t e l y c o n -

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    Fig ure l , Dis t r ibut ion of s tudents by ab i l i ty to math c lass F ig ure 3. Dis t r ibut ion of s tudents by abi l ity to science c lasssizes, sizes.

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    I B u m d o n M o ~ n g A ~ m l e o f Fm J r C h i ~ ~ m )( h ~ d o n M o~InO A ~ ~ F mf C ~ S ~ w lFig ure 2. Dis t r ibut ion of s tudents by abi l i ty to Engl i sh c lasss izes . F ig ure 4. Dis t r ibut ion of s tudents by abi l i ty to history c lasssizes.

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    Does Class Size Matter? 2 3 9Ta ble 6. Interactions o f class size and student characteristics

    Coefficients (and t-stats.) of interaction termMath English Science HistoryInteraction term (N = 4741) (N = 4973) (N = 4645) (N = 4850)Student race* c lass s ize -0 .2 0 0.06 -0.0 1 -0 .08( - 3 . 3 2 ) ( 1. 92 ) ( - 0 . 2 2 ) ( - 2 . 0 4 )Student sex* class size 0.07 0.01 -0 .0 2 0.01(! .23) (0 .20) (-0 .59 ) (0 .28)Family income* c lass s ize -0 .0 1 -0.0 03 0.01 -0.0 1( - 1 .47) ( -0 . 57 ) (1 .34) ( - 1 .49)Urban school* class size 0.14 -0 .0 5 0.03 0.16( 1.96) ( - 1.13) (0.64) (3.57)It is the average predicted class size variable that is used to create these interaction variables.Interactions were also tried with the average class size variable, leading to similar results .t-Statistics of the interaction v ariables a re in parentheses.

    t r o i s f o r t h e p r o b l e m o f l o w e r a b i l i t y s t u d e n t s b e i n g e f f e c t o f c l a s s s i z e o n a c h i e v e m e n t c h a n g e s f r o md i s p r o p o r t i o n a t e l y f o u n d in t h e s m a l l e r c l a s s e s . W h e n p o s i t i v e t o n e g a t i v e f o r a l l f o u r s u b j e c t s a n d t o s i g -t h e s a m p l e w a s f u r t h e r s t r a ti f i e d i n t o v a r i o u s g r o u p s n i f i c a n t l y n e g a t i v e f o r s c i e n c e a n d h i s t o r y a c h i e v e -o f s t u d e n t s ( i . e . , b y a b i l i t y , g e n d e r , r a c e , e t c .) , t h e m e n t . T h i s s u g g e s t s t h a t i f s t u d e n t s a r e a s s i g n e d r a n -m a g n i t u d e o f c l a s s s i z e e f f e c t s r e m a i n e d r e l a t i v e l y d o m l y t o d i f f e r e n t c l a s s s i z e s , i n v e s t i n g i n s m a l l e rs m a l l f o r a l l s u b g r o u p s , c l a s s e s i n c e r t a i n s u b j e c t s m a y y i e l d p o s i t i v e r e t u r n s ,

    R e t u r n i n g t o t h e o r i g i n a l s a m p l e o f s tu d e n t s , a a s m e a s u r e d b y i n c r e a s e s i n s t u d e n t p e r f o r m a n c e .n u m b e r o f i n t e r a c t io n v a r i a b l e s w e r e c r e a t e d a s H o w e v e r , t h e s e e f f e c t s o n a c h i e v e m e n t a r e s m a l l i na n o t h e r m e t h o d t o i d e n t i f y w h e t h e r c la s s s i z e e f f e c t s s i z e a s c o m p a r e d t o f a m i l y b a c k g r o u n d e f f e c t s , a n dd i f f e r e d b y t y p e o f s t u d e n t. T h e f i t t e d v a l u e o f c l a s s t h e i n c r e m e n t a l b e n e f i t s o f d e c r e a s i n g c l a s s s iz e m a ys i z e ( a n d a l t e r n a t i v e l y , t h e a v e r a g e c l a s s s i z e v a r i a b l e ) n o t e x c e e d t h e a s s o c i a t e d c o s t s . F u r t h e r r e s e a r c h i sw a s m u l t i p l i e d b y t h e s t u d e n t ' s r a c e , s e x , f a m i l y n e e d e d t o i d e n t i f y o t h e r i n s t ru m e n t s , f o r c l a s s s i z e t h a ti n c o m e , a n d th e u r b a n i c it y o f th e s c h o o l t o c o n s t r u ct m a y m o r e a c c u r a t e l y g a u g e t h e m a g n i t u d e o f e f f ec t ,f o u r i n t e r a c ti o n v a r i a b l e s f o r e a c h o f t h e f o u r s u b j e c t a n d t o d e t e r m i n e w h i c h t y p e s o f s tu d e n t s b e n e f i t t h ea r e a s . A s s h o w n i n T a b l e 6 , n o n - w h i t e s tu d e n t s d i d m o s t f ro m s m a l l e r c l a s s s i z e s .b e t t e r i n s m a l l e r E n g l i s h c l a s s e s w h i l e w h i t e s t u d e n t s I n c lo s i n g , i n s t e a d o f r e l y i n g o n p a s t s tu d i e s t h a td i d b e t t e r i n s m a l l e r m a t h a n d h i s t o r y c l a s s e s . I n c l a i m t h r o w i n g m o n e y a t t h e s c h o o l s w i l l d o n o g o o d ,a d d i t i o n , n o n - u r b a n s t u d e n t s d i d b e t t e r i n s m a l l e r e d u c a t i o n a l p o l i c y m a k e r s s h o u l d q u e s t i o n h o w t h em a t h a n d h i s t o r y c l a s s e s . U n d e r n o s c e n a r i o d i d s e x s c h o o l r e s o u r c e v a r i a b l e s w e r e m e a s u r e d i n t h e s eo f s t u d e n t o r f a m i l y i n c o m e i n t e r a c t w i t h c l a s s s i z e s t u d i e s a n d s h o u l d f o c u s e i t h e r o n e x p e r i m e n t a li n a s i g n i f i c a n t m a n n e r . ( r a n d o m i z e d ) a s s e s s m e n t s o f s c h o o l r e s o u r c e s o r o n

    s t a t i s ti c a l t e c h n i q u e s t h a t p u r g e t h e a n a l y s i s o f t h e7 . C O N C L U S I O N S A N D P O L I C Y e n d o g e n ei t y o f s c ho o l r e so u r ce i np u ts . O n l y b y u s in g

    I M P L I C A T I O N S s ta t is t ic a ll y u n b i a se d m e a s u re s o f e d u c at io n a l i n p ut sc a n t h e d e b a t e o n e d u c a t i o n a l r e f o r m i n t h e U n i t e dS t a t e s b e i n f o r m a t i v e , r e l e v a n t a n d p r o d u c t i v e .

    T h i s p a p e r f o u n d t h a t p u b l i c s c h o o l s t u d e n t s a r en o t r a n d o m l y a l l o c a t e d t o d i f f e r e n t c l a s s s i z e s , w i t h Ac kn ow led gm ent s--I am gra teful to T . Paul Schul tz , Dun-the low er ab i l i ty s tuden t s d i s p ro por t ion a te ly found in can T homas, J im Akerh ie lm, Susan Pa rke r, and two anony-the s m a l le r c l a s s e s . W hen th i s i s t aken in to acco un t mous re fe rees o f th i s jou rna l fo r the ir comments and s ugges-us in g i ns tr um enta l va riab le regr ess ion m etho ds , the t ions . Any errors are sole ly a t tr ibutable to myself.

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    2 4 0 E c o n o m i c s o f E d u ca t i on R e v i e w

    N O T E S1 . F o r e x a m p l e , f o r s c e n a r i o I , i f c la s s s i z e w a s g r e a t e r t h a n 1 5 s t u d en t s , t h e c r e a te d m i d - r a n g e v a r i a b l ewas se t equa l to ac tua l c lass s ize minus 15 ; i f c lass s ize was 15 or fewer s tuden ts , the c rea ted mid-range var iab le was se t equa l to ze ro . I f ac tua l c lass s ize was a l so g rea te r than 30 , the c rea ted la rge-range var iab le was se t equa l to ac tua l c lass s ize minus 30 ; i f no t , i t was se t to ze ro .2 . F o r e x a m p l e , i n i n t e r p r e t i n g t h e s p l i n e c o ef f i c ie n t s f o r s c i e n c e a c h i e v e m e n t in s c e n a r i o 2 , i n c r e a s i n g. s c ie n c e c l as s s i z e b y o n e i n a c l a s s o f 0 t o 2 0 s t u d e n t s w i l l a l s o i n c r e a s e s c ie n c e a c h i e v e m e n t b y 0 .1 7of a t es t po in t . However , i f c lass s ize increases by one in a c lass o f 21 to 30 s tuden ts , sc ience ach ieve-m e n t w i l l d r o p 0 .0 6 o f a t e s t p o i n t ( 0 .1 7 - 0 .2 3 ). L i k e w i s e , i n c r e a s i n g c l a s s s i z e b y o n e i n a c l a s s w i t hm o r e t h a n 3 0 s t u d e n t s w i l l d e c r e a se s t u d e n t a c h i e v e m e n t b y 0 .0 1 o f a t e s t p o i n t (0 .1 7 - 0 .2 3 + 0 .0 5 ) ,a l though th i s l a t t e r change i s no t s ta t i s t i ca l ly s ign i f ican t .3 . C o m p a r i n g t h e t w o d i s t r i b u t io n s o f s t u d e n t s f o r a l l f o u r s u b j e c ts y i e l d e d a P e a r s o n c h i - s q u a r e t e s ts t a t i s t i c t h a t t e s t s t h e n u l l h y p o t h e s i s t h a t t h e t w o d i s t r i b u t i o n s a r e r a n d o m s a m p l e s f r o m t h e s a m ep o p u l a t io n . T h i s c h i - s q u a r e t e s t s t a t is t ic ( w i t h d e g r e e s o f f r e e d o m e q u a l t o t h e n u m b e r o f d i f f e re n t c l a s ss i z e s m i n u s o n e ) e x c e e d e d t h e c r i t i c a l v a l u e o f 6 4 a t t h e 9 9 t h p e r c e n t i l e f o r a l l f o u r s u b j e c t s . T h u s ,t h e h y p o t h e s i s o f e q u a l c l a s s s i z e d i s t r i b u t i o n s f o r l o w e r a n d n o n - l o w e r a b i l i t y s t u d e n t s w a s r e j e c t e d( the ch i - square s ta t i s t i cs were : ma th - 477 , Eng l i sh - 497 , sc ience - 219 , h i s to ry - 309) .4 . A H a u s m a n s p e c i fi c a t io n te s t w a s e m p l o y e d t o c o m p a r e t h e O L S a n d 2 S L S e s t i m a t o r s a n d t o t e s tfo r the exogene i ty o f the c lass s ize var iab le . As seen in Tab le 5 , fo r a l l four sub jec t s , the ch i - squarec r i ti c a l v a l u e o f 2 5 ( f iv e p e r c e n t s i g n i f i ca n c e l e v e l w i t h 1 5 d e g r e e s o f f r e e d o m ) i s e x c e e d e d , r e j e c t in gt h e n u l l h y p o t h e s i s o f e q u a l i ty b e t w e e n t h e O L S a n d 2 S L S c o e f f ic i e n ts . T h a t i s , a l t h o u g h t h e c l a ss s i z ec o e f f ic i e n t s a r e s m a l l , t h e O L S a n d 2 S L S e s t i m a t e s c a n b e d i s t i n g u i s h e d f r o m o n e a n o t h e r o n s t a t i s ti c a lg r o u n d s . T h e s e r e s u l t s s u g g e s t th a t c l a s s s iz e i s n o t e x o g e n o u s a n d t h a t t h e O L S m o d e l m a y b e m i s s p e c i -fled.

    R E F E R E N C E S 'BOOZER, M.A . , KRUEGER, A.B. and WOLK1N, S . (1992) Race and school q ua l i ty s ince Brown v . Board o fEducation. Brookings Pa pers on Econom ic Activity: M icroecon omic s 1992. W a s h i n g t o n D C : B r o o k i n gsI n s t i tu t i o n , 2 6 9 - 3 3 8 .BOWLES, S . and LEVIN, H. M. (1968) Th e de te rm inan ts o f scho las t ic ach iev em ent - an appra i sa l o f some

    r e c e n t e v i d e n c e . The Journal o f Human Resources 3 , 3 -24 .CAHEN, L .S . , F ILBY, N. , MCCUTCt lEON, G. and KYLE, D.W . ' (1983 ) Class Size and Instruction: A FieldStudy. N e w Y o r k : L o n g m a n .CARD, D. an d KREUGER, A. (19 92) S chool qua l i ty an d b la ck-w hi te re la t ive ea rn ing s : A d i rec t assessmen t .The Quarterly Journal o f Eco nom ics 1 0 7 , 1 5 1 - 2 0 0 .CHUBB, J .E . an d MOE, T .M. (1990) Politics , M arkets & Ame rica's Schools . W a s h i n g t o n , D .C . : T h e B r o o k -ings Ins t i tu t ion .COLEMAN, J .S., HOFFER, T. an d KILOORE, S. (198 2) High School Achievement: Public, Catholic andPrivate Schools Compared. N e w Y o r k : B a s i c B o o k s .COLEMAN, J .S., CAMPBELL, E.Q., HOBSON, C.J. MCPARTLAND, J . , MOO D, A.M ., WEINFELD, F.D. andYORK, R.L . (1966) Equality of Educational Opportunity. W a s h i n g t o n , D .C .: N a t i o n a l C e n t e r fo r E d u -ca t ion S ta t i s t i cs .DOLAN, R.C. and SCHr~IDT, R .M. (1987) A ssess ing the im pac t o f expe ndi tu re on ach ievem ent : somem e t h o d o l o g i c a l a n d p o l i c y c o n s i d e r a t io n s . Econom ics o f Educat ion Rev iew 6 , 2 8 5 - 2 9 9 .ERS , I r~c . (1986) Class Size Research: A Relate d Cluster Analysis fo r Decision M aking. A r l i n g t o n , V A :ERS, Inc .GLASS, G.V. and SMITH, M.L. (1979) M eta-ana lys i s o f research on c lass s ize and ach ieveme nt . Edu-cational Evaluation and Polic y Analysis 1 , 2 -16 .HANUSHEK, E .A. (1986) The e cono mics o f sch ool ing : Produ c t ion and e f f ic iency in pub l ic schools . Journalo f Econom ic L i terature 2 4 , 1 1 4 1 - 1 1 7 7 .H A N U Sn E K, E .A . ( 1 9 7 9 ) C o n c e p t u a l a n d e m p i r i c a l i s s u e s i n t h e e s t i m a t i o n o f e d u c a t i o n a l p r o d u c t i o nfunc t ions . The Journal o f Human Resources 1 4 , 3 5 1 - 3 8 8 .LINK, C.R. an d MULLIGAN, J .G. (1991 ) Classm ates ' e f fec t s on b lack s tuden t ach ievem ent in pub l ic schoolc l a s s r o o m s . Econom ics o f Educat ion Rev iew 1 0 , 2 9 7 - 3 1 0 .McGIVERIN, J . , GILMAN, D. and TILLITSKI, C . (1989) . A Me ta-ana ly s i s o f the re la t ion sh ip be tw een c lasss i z e a n d a c h i e v e m e n t . The Elementary Schoo l Journal 90, 4 7 - 5 6 .

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