return to schooling

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Economettica Vo l. 69, No. (Se ptem ber, 20011, 1127-1160

E ST IM A T IN G T H E R E T U R N T O S C H O O L IN G : P RO G R E S SO N S O M E P E R S I ST E N T E C O N O M E T R I C P R O B L E M S

This paper reviews a set of recent studies that have attempted to measure the causaleffect of education on labor market earnings by using institutional features of the supplyside of the education system as exogenous determin ants of schooling outcom es. simpletheoretical model that highlights the role of comparative advantage in the optimalschooling decision is presented and used to motivate an extended discussion of economet-ric issues, including the properties of ordinary least squares and instrumental variablesestimators. A review of studies that have used compulsoiy schooling laws, differences inthe accessibility of schools, and similar features as instrumental variables for completededucation, reveals that the resulting estimates of the return to schooling are typically asbig or bigger than the corresponding ordin aly least squares estimates. On e interpretationof this finding is that marginal returns to education among the low-education subgroupstypically affected by supply-side innovations tend to be relatively high, reflecting their highmarginal costs of schooling, rather than low ability that limits their return to education.

KEYWORDS:eturns to education, ability bias, random coefficients

OVERTHE PAST DECADE there has been a resurgence of interest in the study ofthe causal links between education and labor market success. Part of thisrenewed in terest stem s from a rise in the "return" to education , especially in theU.S. labor market, and a search for the causes of the growing disparitiesbetween more and less-educated workers (Katz and Autor (1999)). Part isattributable to the revival of interest in the determinants of economic growth,and a new focus on the role of hum an capital in the develop men t process (Tope1(1999)). Finally, many countries are experiencing rapid growth in their secondaryand post-secondary school enrollment rates, leading to a concern about therelative costs and benefits of higher ed ucation for those wh o were no t previouslyreceiving it.

In addition to the stimulus provided by these key substantive issues, interestin the joint structure of education and earnings has been heightened by thebelief that som e progress has bee n made-and m ore may follow-in th e verydifficult task of uncovering the causal effect of education in labor marketoutcomes. The basic idea underlying this new thrust of research is that institu-tional features of the education system can be used to form credible instrumen-tal variables for individual schooling outcomes that can cut through the GordianKnot of endogenous schooling and unobserved ability. The use of supply-sidevariables to help resolve identification problems on the demand side of theI Fisher-Schultz Lecture delivered to the Eu rop ean M eeting of the Econom etric Society, Septem -ber, 1998. I am grateful to Joshua Angrist, Michael Boozer, Ken Chay, Andrew Hildreth, Alan

Krueger, and a co-editor for comments o n earlier versions of this material, and to Jam es Powell for


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1128 DAVID CARDeducation market is a natural outgrowth of standard econometric practice.Nevertheless, th e idea attrac ted very little atten tion in the first wave of microe-conom etric studies of education and earnings in the 1960's and 1970's. Inde ed, itis one of the few methodological issues that is not discussed thoroughly inGriliches' (1977) landmark survey of the first-wave literature.In this paper, I present a survey and partial synthesis of the recent literaturethat has used supply-side featu res of th e education system to help identify thecausal effect of education. In interpreting this literature I believe it is helpful towork from a theoretical an d econ om etric viewpoint that explicitly recognizes thepossibility that returns to education may vary across the population, dependingon such characteristics as family background and ability. This perspective helpsto reconcile various findings in the literature, and also provides a usefulframework for generating new hypotheses and insights about the connectionbetween education and earnings.The paper begins with the presentation of a simple theoretical model ofendogenous schooling. This model is then used to motivate an extended discus-sion of various econometric issues. Finally, I present a selective review of therecent literature on estimating the economic returns to education, drawing onstudies of the U S and other developed economies, as well as a handful ofstudies of developing econ omies.

1 A M O D EL O F EN D O G EN O U S SCH O O LIN GMost of the conceptual issues underlying the interpretation of recent studiesof the return to education can be illustrated in the framework of a simple modelthat builds on Becker (1967). In such a model individuals face a marketopportunity locus that gives the level of earnings associated with alternativeschooling choices, and reach an optimal schooling decision by balancing thebenefits of higher schooling (which are reaped over the lifecycle) against the

costs (which are born early on). Traditionally, it is assumed that individuals seekto maximize the discounted present value of earnings, net of schooling costs(see, e.g., Willis (1986)). This is appropriate if people can borrow or lend at afixed interest rate, and if they are indifferent between attending school orworking during their late teens and early twenties. More generally, however,different individuals may have different aptitudes and tastes for schoolingrelative to work, and this variation may lead to differences in the optimal levelof schooling across individuals.Assume that individuals have an infinite planning horizon that starts at themin imu m school-leaving age ( t 0) an d tha t they accrue a flow of utility inperiod t that dep ends on consumption c( t) in period t and on whether they a rein school (and working part time) o r ou t of school and working full time. Utility


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1129ETU R N TO SC H O O LIN Grelative disutility of school versus work for th e t th year of schooling.' Finally,assume that individuals discount future utility flows at a subjective discount ratep and make a once-for-all decision on when to leave ~ c h o o l . ~ifecycle utility,conditional on schooling an d a given consum ption profile is

Le t y (S, t) de not e real earnings at age t of an individual who has com pletedyears of post-compulsory schooling (with t o . ~Assume that individualswho are in school at time t work part time and earn p(t), and pay tuition costsof T( t). Moreover, assume that t he individual can borrow o r lend freely at afixed interest rate R. Under these conditions the intertemporal budget con-straint is

An individual's optimal schooling choice and optimal consumption pathmaximize

The derivative of this expression with respect to is

where

'1n principle 4 ( S ) can be negative (if schooling is preferre d to work) or positive. For simplicity Iam treating ho urs of work both during and after the completion of school as exogenous.3 ~ a r dnd Lemieux (2000) examine school-leaving behavior of young men and wom en in theNational Longitudinal Survey of Youth and find that about one quarter of those who leave schoolreturn at some point in the future. However, mo re than half of the returners comple te one semesteror less of additional schooling. Angrist and Newey (1991) study the earnings changes associated witheducation increments acquired after young men first en ter t he labor market on a full-time basis.4 ~ h earnings function y (S ,t) may reflect productivity a nd/ or signaling effects or higherschooling. As not ed below, some recent stu dies identify th e causal effect of education by com parin gschooling and earnings differences across cohorts or other groups. In the presence of signaling


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1130 DAVID CARDrepre sent s the marginal benefit of th e S t h unit of schooling (expressed in periodS dollars), and

M C ( S ) = y ( S , S ) y ( S ) T (S ) l / ~ e - ( ~ ~ ) ' + ( ~ )represents the marginal cost of the Sth unit of schooling (also in period Sdollars). Notice that if + (S ) 0, th en M C (S ) is ind epe nde nt of preferences anddepen ds only on the net opportunity cost of schooling (y(S, S ) -p (S )) plustuition costs (T(S)). Otherwise, MC(S) also includes a term capturing therelative disutility of school versus work.Assuming that MC(S) rises faster than MB(S), a necessary and sufficientcondition for an optimal schooling choice is that M C (S ) MB (S). T o proceed ,assume that log earnings are additively separable in education and years ofpost-schooling experience (Mincer (1974)). Then the earnings function can bewritten as y(S , t) f( S )h (t S ) (with h(0) I) , and the marginal benefit of theS th unit of schooling is

M B (S ) f ' ( ~ ) / ~ h ( r ) e ~ d r~ ' ( s ) H ( R ) ,where H ( R ) is a decreasing fun ction of the interest ra te. In particular, ifearnings are fixed after th e com pletion of schooling (i.e., h (t ) for all t ) th enH ( R ) 1 /R . M ore generally, if earnings follow a concave lifecycle profile, thenH ( R ) 1 /( R -g ), where g is the constant growth rate that is equivalent tothe lifecycle profile (i.e. /reg' e - R ' d ~ / , h(r)epR'dr) .Under separability, the marginal costs and marginal benefits of additionalschooling are equated when

The left-hand side of this expression is the proportional increase in earnings(per year) associated with the Sth unit of schooling. The right-hand side is theannuitized m arginal cost of th e S th un it of schooling, expressed as a fraction offoregone earnings. Ignoring tuition costs and earnings while in school, and anydisutility of schooling relative to work, and assuming that earnings are fixed overthe lifecycle, this expression reduces to th e well-known condition '(S) /f(S ) R(see, e.g., Willis (1986)), which implies that individuals invest in schooling untilthe marginal return is equal to the interest rate.T o consider a mo re general case, assume that u(c (t)) log c(t) . Th en the firstorder conditions for an optimal consumption profile, together with the lifecyclebudget constraint, imply that


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1131ETURN TO S HOOLINGequa l to tuition costs (i.e., T(S ) = y (S )) . Th en an optimal schooling choicesatisfies the condition

Relative to the baseline case that ignores preferences for school versus workand post-schooling earnings growth, this expression introduces two additionalconsiderations to the determination of optimal schooling. First, the interest ratemust b e adjusted t o reflect lifecycle earnings growth. Second, marginal cost h asto account for the relative disutil ity of attending the St h year of ~ c h o o l i n g .~Inspection of equation (1) suggests that individual heterogeneity in theoptimal schooling choice can arise from one of two sources: differences in theeconomic benefits of schooling, represented by heterogeneity in the marginalreturn to schooling f '(S )/f( S); o r differences in the marginal costs of schooling,repre sen ted by heterogeneity in d( S). A simple specification of these het ero-geneity com ponen ts is

where b, and r, are random variables with m eans b and ? and second momentsa;, q and a, and 12, and k , ar e non negative constants. hese assump tionsimply that the optimal schooling choice is linear in the individual-specificheterogeneity terms:where k 12, 12, is assumed to be strictly positive.At the equilibrium level of schooling described by equation (4) individual i'smarginal return to schooling is

This model gives rise to a nondegenerate distribution of marginal returns acrossth e popu lation u nless o ne of two conditions is satisfied: k , 0 and ? for allA m or e complex expression arises if par t tim e earnings while in school do not fully offset tuition.

For example, if tuition costs and part time earnings are constant T t ) T ; t ) = p , i t can beshown that

If tuition costs are small relative to lifetime earnings, the term in square brackets is close to 1implying

6 ~ o t ehat if individuals are indifferent between school and work, then k , 0 and I. R g . nthis case variation in r arises because individuals face different interest rates, or differential growth


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1132 D VID C RDi (a situation to which Becker (1967) refe rred as equality of opportunity ); or12, = 0 and b,=b for all i (a case to which Becker refe rred as equality ofability ).7 The average marginal return to education is =E [ P ] =E [ b , lz,S,]= b k ,S . This is the expected increase in average log earnings if a randomsample of the population acquired an additional unit of education. As will bediscussed in m or e deta il below, is not necessarily the relevant marginal retu rnfor evaluating any particular schooling intervention. Nevertheless, it forms auseful benchmark against which to compare the probability limit of variousestimators of the retu rn to schooling.For the labor market as a whole the distribution of marginal returns toschooling is endogenous: a greater supply of highly-educated workers willpresumably lower b , and might also affect other characteristics of the distribu-tion of bi.' From the perspective of a cohort of young adults deciding on theireducation, however, the distribution of returns to education is arguably exoge-nous. therefore prefer to interpret equation (4) as a partial equilibriumdescription of the relative education choices of a cohort, given the institutionalenvironment and economic conditions that prevailed during their late teens andearly twenties. Differences across cohorts in these background factors will leadto further variation in the distribution of marginal returns to education in thepopulation as a whole.

2. ECONOMETRIC ISSUES R ISED BY ENDOGENOUS SCHOOLINGA. O LS Estimates of the Return to Schooling

T o understand th e implications of the preceding m odel for observed schoolingand earnings outcomes, note that equation (2) implies a m odel for log earningsof the formlog yi = ai b ,S i k l S i

where a is a person-specific constant of integrat i~n.~his is a somewhat moregeneral version of the semi-logarithmic functional form adopted in Mincer(1974) and hundreds of subsequent studies. In particular, individual heterogene-ity is allowed to affect both the intercept of the earnings equation (via i andthe slope of the earnings-schooling relation (via b,) . It is convenient to rewritethis equation as

7 ~ h e r es also an uninteresting case in which both and I are degenerate .%ee Freeman (1986) and Willis (1986) for some discussion of the general equilibrium implica-tions of optimal schooling models.


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R E T U R N TO S HOOLING 1133where ai cri - a o has mean 0.Equations 4) and ( 5 ) together describe atwo-equation system for schooling and earnings in terms of the underlyingrandom variables a i , b i , and vi.Ignoring other covariates (or assuming these have already been conditionedout) it is straightforward to derive the implications of this model for conven-tional ordinary least squares ( O L S ) estimates of the return to schooling. Toproceed, consider the linear projections of ai and ( b i b ) on observed school-ing:

where represents the me an of schooling and E [ S i u i ]E [ S i u , ]0.Substitut-ing these expressions into (5 ) , the earnings function can be w ritten a slog yi constant ( b A, 4 , s ) ~ ~ i k , ) ~ : u , u i S i .( 4 ,

In g eneral t he orthogonality of u, and S , doe s not imply that E [ u , S ] 0: thus,the residual component viSi may be correlated with schooling. However, if thethird central m om ents of the joint distributio n of bi and ri are all zero, then u,S,will be uncorrelated with Si.10 Moreover, in this case E [ ( S , S 3 ] 0 implyingthat the linear projection of S? on S , has slope 2 3 . Under this assumption theprobability limit of the ordinary least squares ( O L S ) regression coefficient b,,,from a regression of log earnings on schooling is( 7 ) plim b,,, b A, q0S 2 3 x ( 4 , i k , )

More generally equation ( 7 ) includes an additional term that depends on thethird moments of b, and vi.ll'O~l te rna t ive l~ ,f the orthogonality condition E [ S , u , ] 0 is replaced by the assumption E[u i IS , ]0 (i.e., that the conditional expectation of bi is linear in S , ) , then E [ u , S ; l E { E [ u , S ? Sill 0 .In the general case the linear projection of S? on Si has slope

Z S E [ s , / v a r [ ~ ~ ] ,an d) ~ ]Taking these expressions into consideration, equation (7) includes another term:


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1134 DAVID CARDEquation 7) generalizes the conventional analysis of ability bias in therelationship betwee n schooling and earnings.12 Sup pose that ther e is no h etero -geneity in the marginal benefits of schooling (i.e., bi =b and that log earningsare linear in schooling (i.e. 12, = 0). Then (7) implies that

plim b,,, b = Awhich is the standard expression for the asymptotic bias in the estimated returnto schooling that arises by applying the om itted variables form ula to a n earning smo del with a constant schooling coefficient b. According to the model presentedhere, this bias arises through the correlation between the ability component a,and the marginal cost of schooling r,. If marginal costs are lower for people whowould tend to earn more at any level of schooling, then a < 0, implying thatA > 0.If both the intercept and slope of the earnings function vary across individu-als, then th e situation is a little m or e complex. Since people w ith a higher retu rnto education have an incentive to acquire more schooling, a cross-sectionalregression of earning s o n schooling is likely to yield an upw ard-biased estim ateof the average marginal return to schooling, even ignoring variation in theintercepts of the earnings function. The magnitude of this endogeneity orcomparative advantage bias d epen ds on th e relative importance of bi in deter-mining the overall variance of schooling outcomes. Specifically, the fraction ofthe variance of schooling attributable to differences in the slope of theearnings-schooling relation (as opposed to differences in tastes or access tofunds) can be defined as

Assuming th at a (i.e., that the marginal benefits of schooling ar e no higherfor people with higher marginal costs of schooling), f is bounded between 0 and1.The projection coefficient defined in equation (6b) is , = cov[bi, Si]/var[S i]=k Thus, the endogeneity bias component in the O LS estimator is $oS=k fwhich is prop ortional to Even ignoring the traditio nal ability bias term A,,b,,, is ther efor e an upward-biased estimate of with a larger bias the m oreimportant are the comparative advantage incentives that lead individuals withhigher returns to schooling to acquire m ore schooling.The analysis so far has ignored any problems arising from the mismeasure-m ent of schooling. Griliches (1977) argued t hat m easu rem ent e rror s in schoolingwould lead to downward bias in the OLS estimate of the effect of schooling onearnings th at could partially offset any upward ability biases. To se e this, assum etha t observed schooling (s:) differs from true schooling (S ,) by an additive

2Throughout this paper I use the term bias to refer to the difference between the probability


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RETURN TO S HOOLING

error,

where ci has mean 0, variance aE2,nd is uncorrelated with e arnings. Assumingthat equation (7) describes the probability limit of an OLS estimator based ontrue schooling, the use of observed schooling will yield an OLS estimator with

where R = cov[s, , ~ ~ ] / v a r [ S P ]s the reliability of observed schooling (i.e., theslope of the linear projection of true schooling on observed schooling). If themeasurement errors are orthogonal to true schooling, then R =var[S , ] /{var[Si] aE2 and OLS using observed schooling will be downward-biasedrelative to the case of no measurement error. Rese arch in th e U.S. over the pastthree decades has concluded that the reliability of self-reported schooling is85-90 percent (Angrist and Krue ger (1999, Ta ble 9)), implying that the down-ward bias is o n the order of 10-15 percent-enough to offset a mod est upwardability bias.It is worth noting that equation (8) is valid regardless of whether themeasurement errors are orthogonal to true schooling or not. With correlatedmeasurement errors, however, the reliability of observed schooling may begreater or less than 1. M ore importantly, th e stan dard pro cedure for estimatingR based on the correlation between two alternative survey measures ofschooling, assumes that E [ S ic i] 0. Since schooling is measu red as a discretevariable with fixed upper and lower limits, this cannot be literally true: individu-als with the highest level of schooling cannot rep ort positive erro rs, while thosewith the lowest level cannot report negative errors. Mean-regressive measure-ment error is likely to lead conventional procedures to understate the actualrel iabil i ty of s~hooling. '~hus, the actual attenuation of an OLS estimate maybe less than 10 percent.

B . Instr~~mentalariables Estinzates of the Return to SchoolingSocial scientists have long recognized that the cross-sectional correlationbetween education and earnings may differ from the true causal effect ofeducation. A standard solution to th e problem of causal inference is the me thodof instrumental variables IV): a researcher posits the existence of an observ ablecovariate that affects schooling choices but is uncorrelated with (or evenindependent of) the ability factors a and b . Recently, much attention hasfocused on supply-side sources of variation in schooling, attributable to such1 3 ~ u p p o s ehat there are two equally noisy measures I and I of a tru e quantity with

x, = x el. Assume that the measurement errors are mean-regressive: E[e, 1x1 u ( x p , forj 1 , 2 , where p is the mean of I.Let u = e l E[e l and assume that uj 's are uncorrelated.


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36 DAVID CARD

features as the minimum school-leaving age, tuition costs, or the geographicproximity of schools. As in standard market settings, variables from the supplyside are an obvious source of identifying information for estimating demand-sideparameters.T o proceed , suppose tha t the m arginal cost com ponent r, is linearly related t oa set of observable variables 2,:

where v incorporates other unobserved taste and cost factors, and E [v iZ i]= 0.The optimal schooling choice is

where Zirr = b/k Zin- , /k and ti (b, b 77,)/12. If a , is the onlyindividual-specific component of ability, then equations (4 ) and 5 ) constitute astandard simultaneous equa tions system and the assum ption that Z i is uncorre-lated with ability ( E [ a i Z i ] 0) is sufficient to ensure that a n IV estimator basedon Zi will yield a consistent estimate of the average return to schooling b 14 Ifthere is heterogeneity in returns to education, however, somewhat strongerassumptions are needed for IV to yield a consistent estimate of the averagereturn to schooling, s ince the residual earnings component (bi b ) ~ , ay becorrelated with Z i even if Z i is orthogon al to bi.O ne sufficient condition is th at Z i is independe nt of individual abilities (a ,, b,)and the reduced form schooling residual 5 15In this case

Under independence the two conditional expectations in this expression areconstant for all values of implying tha t the coefficients from the second stageof a modified two-stage least squares system (in which actual schooling and itssquare are replaced by predicted schooling and its square) will be consistent forb and i k , , respectively. Thus the average marginal retu rn to schooling can beconsistently estimated by IV

I If earnings are a quadratic function of schooling, then Z , and its square can be used as


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RETURN TO SCHOOLING 1137A slightly weaker set of sufficient conditions for consistency was proposed byWooldridge (1997). In particular, suppose that the individual-specific hetero-geneity components are mean-independ ent of Z (i.e. , E[T , Z i] 0, E [ a i Z ,]

0 , and E[(b , Z i]O), and tha t two additional assum ptions a re satisfied:

Equ ation (9a) requires that the reduced form schooling residual in eq uation (4')be homoskedastic, while equation (9b) specifies that the conditional expectationof individual ability (b,) be linear in the schooling residual.16 under theseassumptions the conditional expectation of the residual earnings componentattributable to heterogeneity in returns is

implying that

Again, this condition ensures that the probability limits of the second-stagecoefficients of predicted education and its square in th e I V proc edure are b and;kl, respectively.17Unfortunately, the assumptions that Zi is independent of ability and thereduce d form schooling residual t i , o r the slightly weaker assumptions in (9a)and (9b), are likely to be violated when Z i is a variable representing exposure todifferent institutional structures o n the supply side of the educ ation system. T hereason is that the entire mapping between ability and schooling is likely to beaffected by a chang e in educationa l institutions, leading to a systematic correl a-tion between (b, b ) s i and Z i. T o illustrate this point in the context of themodel, consider IV estimation based on a schooling reform that leads to aproportional reduction in the marginal cost of schooling for students in aspecific set of schools (o r in a specific cohort). Assum e that th e joint distributionof abilities an d tastes (a ,, b,, r; is the same for individuals who attended thereform ed schools (indexed by Z i 1) and those who did not (indexed by Zi O),but that in the reformed schools the optimal school choice is given by

16 Equation (6b) above specifies the linear projection of b, on schooling as , Si E[Si]). If theconditional expectation is linear, it has the same coefficient as the linear projection.~ fk (i.e., earnings are a linear function of schooling) Wo oldridge shows that assump tions(9a) and (9b) can be replaced by E[(bi $1' Zi ] and E [ t l b, , Z ,] pl(b, -5 ) .2 Under these


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1138 DAVID CARDwhere 0 < 1. Let r = I: T,, and observe that among individuals who attendedthe unreformed schools,

whereas among individuals who attended reformed schools,-sj= ( b ? ) / k ( 1 8 ) r / k ( b , o T ,) /k

The reduced form schooling equation is thereforeS 1 = T O + Z i 7 T l t l ,

where t,= (1 Z ,) ti , Z j i , . Since the schooling reform lowers the effect ofcost differences in the optimal schooling decision, var[ Z i = 11ar[ Z j=01, violating indepe ndence and the homoskedasticity assump tion (9a). M oreover,unless a . = 0, cov[bi, ti Z i = 11f cov[b,, ti Z, = 01. Thus, assumption (9b) isunlikely to hold.Some evidence that changes in the institutional structure of the educationsystem affect the mapping between ability and schooling outcomes is presentedin Table I . This table reports th e coefficients of an I Q m easure from a series ofdescriptive regression models fit to the completed schooling of young men (age14-24 in 1966) in the N ational Long itudinal Study .ls A s shown in columns 1 and2, IQ is a strong predictor of education, explaining about 25 percent of thevariation in schooling outcomes among men in the sample. A one-standard

T A B L E IRELATIONSHIPETWEEN I Q A N D SCHOOLING

Pooled Near Not NealSample College College

(1) (2) (3) (4) 5 ) ( 6 )Coefficient of IQ 0.075 0.068 0.081 0.072 0.059 0.058(0.003) (0.003) (0.003) (0.004) (0.005) (0.006)Other Controls No Yes No Yes No YesR-squared 0.260 0.348 0.249 0.375 0.175 0.299Number of 2,061 2,061 1,460 1,460 601 601ObservationsNote Table reports coefficient of I Q in a lulear regression model for completed education 111 1976

Models in odd columns include no othe r controls. Models in even columns include both parents ed uca t~on.age and age-squared, indicators for race, fal~iily tructure at age 14. and region in 1966. Near Collegesubgroup are those rvhose county of residence in 1966 had a local 4-year college (public or private). Sanlpleincludes men in the NLS Young Men sample who were intenr~ewed n 1976 and who have va l~ d ducationdata for their parents and an IQ score obtained from their school records.

' 8 ~ c h o o l i n gs taken from the 1976 interview, when the men were 24-34 years old. IQ measures


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RETURN TO S HOOLING 1139deviation inc rease in I Q e.g. from 95 to 110) is associated with 1.1 additionalyears of schooling when other background factors are ignored, and 1.0 addedyears of schooling when age, race, parental education, region of residence, andfamily structure at age 14 are all taken into account.In th e mid-1960s about 30 percent of th e young men in the NLS sample livedin a county with n o local 4-year college either public o r private). As no ted inCa rd 1995), college proximity has a stron g effect on com pleted edu cation, evencontrolling for parental education, region, an d I Q . Assuming that the p resenceof a nearby college is unc orre lated with ability controlling for family bac k-ground factors), college proximity is a potential instrumental variable for school-ing. Even if ability is independent of college proximity, however, the remainingcolumns of Table I show that th e correlation between ed ucation an d ability asmeasured by IQ ) is different for me n who grew up n ear a college and those whodid not. Consistent with the idea that college proximity reduces the relativeimportance of cost factors in the schooling decision, the effect of IQ oncompleted education is significantly stronger among men who lived near acollege than among those who did not. These results suggest that changes in theinstitutional structure of the education system can affect the mapping betweenindividual ability and education outcom es, leading to a violation of assump tionssuch as ind epend ence or homoskedasticity n eeded for a conv entional IV estima-tor to yield a consistent estimate of the average marginal return to education.

A closely-related alternative to IV estimation of a random coefficients modelis a control function approach, first proposed in the schooling context by Garen1984).19 Th e basic idea of this appro ach is to mak e so me assumptions about thenature of the covariances between the unobserved ability components a and iand the observable variables S and Z i, and include additional terms in theearnings model that capture these relationships. To illustrate, assume that allunobserved ability and taste c om pone nts are m ean-independent of Z . Assumefurther that the conditional expectations of the unobserved ability componentsa and b, are linear in the schooling residual:

Eq uatio ns 5), 9b), and 9c) imply that

19 Willis and Rosen 1979) present a mod el with only two levels of edu cation th at includes both a


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1140 DAVID C RDThis equation can be estimated by a two-step proced ure in which th e estimatedresidual from the reduced form schooling equatio n is substituted for t i , and& Si is substituted for tiSi.To illustrate the connection between this approach and conventional IV,suppose that k 0. In this case it is well known that the inclusion of thereduced form residual associated with the endogen ous regressor Si leads toan OLS estimate of the coefficient on schooling that is identical to the oneobtained by IV using Z i as an instrument for Si. U nd er th e assumption thatE [a i S i ,Z i l ti the addition of to the estimated earnings function purgesthe observed relationship between log earnings and schooling of any effect ofai.'' How ever, stan dar d IV will not eliminate the influence of the heterogene itycomponent (bi b ) s i unless E [(b i b ) ~ ,Z i l is orthogonal to Z i (as is the caseif (9b) holds and ti is homoskedastic). Assuming that E[ (b i Si ,Z i ] is linearin the schooling residual t i , however, th e add ition of lisias a second controlvariable is sufficient to eliminate endogeneity biases, even if the reduced formschooling residual is heteroskedastic.In the case where changes in the instrumental variable affect the entiremapping between unobserved abilities an d schooling outcomes, the assumptionthat E[ (b i b ) S ,, Z i ] ti is problematic, since cov[bi, t Z i ] potentiallyvaries with Z i, as does v ar [t i lZ i]. Nevertheless, a simple extension of th econtrol function ap pro ach may be app rop riate if Z i is an indicator variable (asin the college proximity example). Specifically, replace (9b) and (9c) with

According to (9b') and (9c1), the conditional expectations of the unobservedability components are still linear functions of the schooling residuals within aparticular institutional setting, but a change in the education system (fromZ i 0 to Z i 1) is allowed to shift the relationship between schooling andability. Th ese assump tions imply that

This model adds four control functions to the earnings model: the schoolingresidual, its interaction with Z i, its interaction with Si , and a three-way in terac-tion with S,Zi.A more radical alternative to IV is maximum likelihood estimation of astructural mod el of earnings an d schooling, based o n a co mplete specification of2 0 estimation of the second step equation provides consistent coefficient estimates but the~

standard errors have to be adjusted f i r the f i rs t s tage est imation error , as in Murphy and Tope1(1985).


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R E T U R N TO SCHOOLING 1141the unobservable com pone nts in th e earnings function and the utility function.An advantage of this approach is that the earnings function can be made quitegeneral-for example, by allowing th e returns to differ ent years of schooling tovary in a flexible manner with individual ability. Similarly, the choice functioncan be precisely specified, rather than approximated as in equation 3). Perhapseven more importantly, by fitting such a model to panel data it may be possibleto recover information on the dynamic process by which individuals learn theirabilities and modify their schooling choices over time, as in recent studies byKeane and Wolpin (1999) and Cameron and Taber (2000). This is a promisingline of inquiry bu t is beyond th e sco pe of this pap er.

D . What Does W Estim ate?If the returns to education vary across individuals and conditions such asthose described earlier a re no t satisfied, wha t do es a conventional instrum entalvariables estimator estimate? This question was addressed by Imbens andAngrist (1994) in the context of a dichotom ous instrument that is indep end ent ofindividual characteristics, and analyzed in more detail by Angrist and Imbens(1 99 3, Angrist, Imbens, and Rub in (1996), and Angrist, Graddy, an d Imbens(1995). To illustrate the key results, consider a schooling reform indexed by Zithat affects one of two otherwise identical populations. Suppose that a givenindividual would have schooling level Sf and earnings yf if he o r she attend edth e regula r school system (i.e. if Z i 0), whereas the same individual wouldhave a schooling outcom e Sf ASi if he or she atten ded a reform ed school 6.e .if Z i 1 lZ 2Let Pi denote individual i 's marginal return to schooling. Ignoringsecond and higher-order terms, the effect of the schooling reform on earningsfor individual i is

A log yi .A Si .The probability limit of an IV estimator of the return to schooling formed bypooling rando m samples from the Z 0 and Z 1 populations and using Z i asan instrument for schooling is(10) plim bi, cov[log yi Z i /cov[ Si Zi

where expectations are taken over the joint distribution of the observable andunob servable characteristics in the two populations. By assum ption, these distri-butions are the same, implying that E[Si Zi 11E [S i Zi 01 E[A Si], and


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1142 D A V I D C A R D

Note that if E[P;AS,] E [P ,] .E [A S ,] , then the IV estimator provides aconsistent estimate of the average marginal return to education E [ Pi]. Thiswill be true if the schooling reform induces an equal change in schooling for allindividuals, o r mor e generally if E[ AS , p i] is inde pend ent of pi . Otherwise,und er the assumption that AS, 0 for all i, the ratio on the right-hand side ofequation (11) can be interpreted as a weighted average of the marginal returnsto education in the population, where the weight for any particular personis the relative size of the increment in his or her schooling induced by thereform (AS,/E[AS ,l). Imbens and Angrist (1994) referre d to this weightedaverage as the local average treatment effect (LATE). The bias in LATErelative to dep ends on the covariance of Pi and AS,. An IV procedure basedon a school reform that leads to bigger changes in the education choices ofpeople with relatively high marginal returns to education will tend to produce anover-estimate of the average marginal return to education.In this light it is interesting to reconsider the effects of a supply-side changethat causes a proportional reduction in the marginal cost of schooling, asdescribed by equation (4 ). In the unreformed school system, individual iwith ability parameter b, and cost parameter r, 7, obtains schooling S,(b, r,)/k , whereas the sam e individual in the reformed school system wouldobtain schooling S, (b, 0r,)/k. T he induced change in schooling for individ-ual i isASi r , ( l 0 ) /k F ( l 0 ) /k 7 , (1 0 ) /k ,

which is positive (assuming r, 0). Th us th e mo noton icity assum ption nee dedfor a LATE interpretation is satisfied. Individual i's marginal return to school-ing in the absence of the intervention isPi bi k,Si

Substituting these expressions into equa tion ( 1 0 , the probability limit of the IVestimator isplim biL (u;k,/k ubT(l k , /k ) ) /? .

If individuals with higher returns to schooling have lower discount rates, thena 0 and the IV estimator may be positively or negatively biased relative top.23A positive bias arises because the marginal return to schooling is decreasing

2 3 ~ fl is the sa me for all (in which case evelyone gets the sa me incre men t to schooling), thenrr vhl i 0, and the IV estimator is consistent for p Alternatively, if earnings are linear in


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1144 DAVID CARDship in th e affect ed cohort (relative to the comp arison cohorts) is AS,, = .rrlti ti .Assume that the earnings function is

log yic= a. @,Sic a, , eic,where el, is an erro r compone nt capturing determinants of earnings other thanability or education. Suppose that individuals in different co horts have the sam edistributions of ability and tastes, so that

Finally, assume that el, = e, eic whe re e, is a ran dom effect represe nting th einfluence of unobserved factors that are com mon across individuals in the s am ecohort." In this case, an IV estimator based o n mem bership in cohort T has

@ , . A S i ] e, Average(e, I c T )plim bi, = S ,where Average(e, c T ) represents a weighted average of the cohort effectsfor the comparison cohort^. ^ Although o ne can think of e, as a random variablethat averages to 0 across many cohorts, the IV estimator is based on the gap inearnings between a particular cohort T and a fixed set of comparison cohorts.T o the ex tent that e, is a "bad draw," or the average of the cohort effects in thecomparison sample is far from 0, the IV estimator may give a misleadingestimate of the return to education. Moreover, the conventional sampling errorof the IV estimator makes no allowance for any inherent uncertainty associatedwith the variance of e,.These considerations suggest that an IV procedure that implicitly comparesmany subgro ups of individuals-say, youn ger versus older cohorts in severaldifferent regions-may be m ore reliable than o ne that relies on a single affectedsubgroup. They also illustrate the importance of identifying interventions orchanges on the supply side of the education market that generate large changesin schooling, since the bias associated with a particular realization of the cohorteffects is {e, Average(e, c f T)}/AS, where AS is the difference in meanschooling between the affected cohort and the comparison cohorts. If anestimate of a (the standard deviation of e,) is available, it may be useful tomake an assessment of the potential magnitude of any biases associated with a"ba d draw" on e, by com paring th e magnitud es of b,, to u,/AS. If a,/lASl islarge relative to Ibi,l, a cross-cohort comparison is not a particularly attractivebasis for inferring the causal effect of schooling.

AII example of such a factor is the state of the business cycle at the beginning of the cohort slabor market career, which may exert a permanent effect on the cohort s earnings (see Beaudry andDiNardo (1991)).


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1145E T U R N T O S C H O O L I N G3 I N S TR U M E N T A L V AR IA B L ES E S TI M AT E S O F T H E R E T U R N T O

SCHOOLINGI now turn to a selective review of recent studies that have used institutional

features of the schooling system to identify the re turn to s ~ h o o li n g . ' ~ able I1summarizes eleven recent studies that estimate the return to schooling usinginstrumental variables based o n this idea. For each study I report bo th O LS andIV estimates derived from th e same samp le with the same control variables.T h e first entry in the table is Angrist a nd K rueger's (1991) land m ark study ofcompulsory schooling and ed ucation, which uses an individual's qua rter of birth(interacted with year of birth or state of birth in some specifications) as aninstrument for schooling. Angrist and Krueger observed that U.S. men bornfrom 1930 to 1959 with birth da tes ear lier in th e year have slightly less schoolingthen men bor n later in the year-an effect they attrib ute to compulsoryschooling laws. Specifically, they note that children born in the same calendaryear generally start school at the same time (e.g. in September of the year theyturn 6). As a result of this institutional feature, individuals born earlier in theyear reach the minimum school-leaving age at a lower grade than people bornlater in th e year, allowing those w ho want t o d ro p out as soon as legally possibleto leave school with less education. Assuming that quarter of birth is indepen-den t of taste a nd ability factors, this phen om eno n g ene rates exogenous variationin education that can be used in an IV estimation scheme.

Angrist a nd Kreuger's emp irical analysis confirms that the quarterly patte rn inschool attainment is paralleled by a similar pattern in earnings. As shown inTable 11, their IV estimates of the return to education are typically higher thanthe corresponding OLS estimates, although for so me cohorts and specificationsthe two estimators are very close, and in no case is the difference between theIV and OLS estimators statistically significant.Angrist and Krueger's findings have attracted much interest and some criti-cism. Bound, Jaeger, and Baker (1995) pointed out that several of Angrist andKrueger's IV m odels (specifically, tho se tha t use interactions betw een qu ar ter ofbirth and state of birth as predictors for education) include large numbers ofweak instruments, and are therefore asymptotically biased toward the corre-sponding OL S estimates. This weak instruments bias is not as serious for thespecifications reported in Table 11, which rely on a more parsimonious set ofinstruments. Moreover, to the extent that Angrist and Krueger's IV estimatesare above the corresponding OLS estimates, one might infer that asymptoticallyunbiased estimates of the causal effect of education are even higher. This isconfirmed by the findings of Staiger and Stock (19971, who re-analyzed the 1980Census samples used by Angrist and Krueger and computed a variety ofasymptotically valid confidence intervals for standard IV and limited informa-

2 9 ~ h edea of using insti tut ional features of the school system to overcome problems ofendogeneity and unobserved ability is also proving useful in studies of the effect of school quality.


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46 D VID CARD


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TABLE 11-Coiztiizued

Author10. Maluccio 1997)

11. Duflo 1999)

Sample and Instlumcnt

Bicol Multipurpose Survey rural Philippines):male and female wage earners age 20-44 in1994, whose families were interviewed in 1978.Instruments are distance to nearest highschool and indicator for local private highschool. Controls include quadratic in age andindicators for gender and residence in arural community.

1995 Intercensal Survey of Indonesia: men born1950-72. Instruments are interactions of birthyear and targeted level of school buildingactivity in region of birth. Other controls aredummies for year and region of birth andinteractions of year of birth and childpopulation in region of birth. Second IVadds controls for year of birth interactedwith regional enrollment rate and presenceof water and sanitation programs in region.

Schoohng CoeflicientsOLS IV

Models that do not control forselection of employmentstatus or location

Models with selectioncorrection for locationand employment status

Model for hourly wage

Model for monthly wage withimputation for self-employed.

oter Sec tcxt for sources and more ~n form stio n n indlvldual studies.


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1149ETURN TO SCHOOLINGtion maximum likelihood (LIML) estimates. Staiger and Stock's preferred LIMLestimates, utilizing quarter of birth interacted with state of birth and year ofbirth as instruments, are reported in row 2 of Table 11. These are uniformlyabove the correspo nding two-stage least squares estimates, and 50-70 percenthigher than the OLS estimates.A second criticism of Angrist and Krueger's findings, raised by Bound andJaeger (1996), is that quarter of birth may be correlated with unobserved abilitydifferences. Bound and Jaeger examine the schooling outcomes of earliercohorts of men who were not subject to compulsory schooling institutions andfind some evidence of seasonal patterns. They also discuss evidence from thesociobiology and psychobiology literature that suggests that season of birth isrelated to family background. To the extent that children born earlier in theyear have poorer family backgrounds, one might expect them to have lowercompleted education and lower earnings. Moreover, if the quarterly patterns ofeducation and earnings are solely attributable to differences in family back-ground, then Angrist and Krueger's IV estimators have the same bias as IVestimators based directly o n family backgroun d.30 Althou gh th e C ensus dataused by Angrist an d Krue ger co ntain n o inform ation o n family backgroun d, it ispossible to use other data sources to examine differences in family backgroundby qu arter of birth fo r similar cohorts. Fo r example, I used 1940 Censu s data t ocompare the mean levels of parental education by quarter of birth for childrenunder one year of age.31 T h e me an levels of m other's education for thesechildren are 9.04, 8.95, 8.97, and 8.95 for quarters I-IV, respectively (withstandard errors of about 0.05). The corresponding means of father's educationar e 8.61, 8.50, 8.52, and 8.58. Th ese com parisons give no indication that childrenborn in th e first qua rter com e from relatively disadvantaged family backgrounds,and suggest that th e seaso nal patter ns identified by Angrist and K rueger a re notsimply attributable to differences in family backgro und.The third study summarized in Table 11, by Kane and Rouse (1993), isprimarily concerned with the relative labor market valuation of credits fromregular (4-year) and junior (2-year) colleges. Their findings suggest that creditsawarded by the two types of colleges are interchangeable: in light of thisconclusion they measure schooling in terms of total college credit equivalents.In analyzing the earnings effects of college credits, Kane and Rouse compareOL S specifications against I V mo dels that use th e distance to th e nearest 2-yearand 4-year colleges and state-specific tuition rates as instruments. Their IVestimates based on these instruments are 15-50 above the corresponding OL Sspecifications.

3 As noted in Card (19991, IV estim ators of the re turn to schooling using parental education as aninstrument tend to be substantially above the corresponding OLS estimators.3 Th e 1940 Census, which was conducted in April, reports m onth of birth for children u nder oneyear of ag e. T he re ar e 19,089 childre n und er 1 year of age in the public use file, of whom 98.4


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1150 DAVID CARDTwo subsequent studies by Card (1995) and Conneely and Uusitalo (1997)examine the schooling and earnings differentials associated with growing upnear a college or university. In my 1995 study I found that when collegeproximity is used as an instrument for schooling in the National LongitudinalSurvey (NLS) Young Men sample, the resulting IV estimator is substantiallyabove the corresponding OLS estimator, although rather imprecise. Consistentwith the idea that accessibility matters more for individuals on the margin ofcontinuing their education, college proximity is found to have a bigger effect forchildren of less-educated parents.32 This suggests an alternative specificationthat uses interactions of college proximity with family background variables asinstruments for schooling, and includes college proximity as a direct controlvariable. The IV estimate from this interacted specification is somewhat lower

than th e estimate using college proximity alone, but still ab out 30 percent abovethe OLS estimate.The Conneely-Uusitalo (1997) study utilizes a very rich Finnish data set thatcombines family background information, military test scores, and administra-tive earnings data for men who served in the army in 1982. Like Kane andRouse (1993) and Card (1995) they find that IV estimates of the return toschooling based on college proximity exceed the corresponding OLS estimatesby 20-30 percen t, depending on what other controls are added to the m odel. Itis worth noting that all three of these studies report models that control for afairly detailed set of family background characteristics. Such controls are desir-able if families that live near colleges have different family backgrounds, and iffamily background has some independent causal effect on earnings. Conneelyand U usitalo s IV estim ate controlling for par enta l ed ucation and e arnings isbelow t he I V estimate that excludes these controls, but is still above the simplestOLS estimate without family background controls. Despite the rather large sizeof their sample (about 22,000 observations) and the very high quality of theirunderlying data, however, Conneely an d U usitalo s IV estimates are somewhatimprecise, and are not significantly different from their OLS estimates.33

T he next gro up of four stud ies in Table I1 uses c oh ort differences as a sourceof identification of the retu rn to schooling. H arm on and W alker (1995) study th ereturns to education among British male household heads using changes in thelegal minimum school-leaving age as instruments for completed education.Their instruments distinguish between three cohorts of m en: those born b efore1932, who faced a minim um school-leaving age of 14; those b orn from 1933 to1957, who faced a minimum age of 15; and tho se bo rn after 1957, wh o faced aminimum age of 16 As shown in Table I1 the IV estimate based on these cohortdummies is considerably above the corresponding OLS estimate (2.5 times3 Kling (1999) re-analyses the same data and reports similar findings, including significantdifferences in the distribution of schooling attainment between those near and far from college in

the lowest quartiles of family background.


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RETURN TO SCHOOLING 1151higher) and is relatively precise. Several features of their estimation strategysuggest the need for caution in the interpretation of these findings, however.Most importan tly, th e 1947 lami change-which is the ma jor source of identifi-cation in their results-came just after World W ar I1 and may cap ture othercohort differences between those who attended school before, during, and afterW W I I . ~ ~oreover, Harmon and Walker do not allow for systematic inter-cohortgrowth in educational attainment, other than that attributable to the lawchanges in 1974 and 1973. Both these factors may affect their IV estimators,which pool many cohorts of men, rather than relying on comparisons betweencohorts who a tten ded school just before and just after the law changes.Th e seventh study in Table 11, by Ichino and W inter-Ebm er (1998), focuses o nthe disruptive effects of World War I1 on the schooling of children in Austriaand Germany born between 1930 and 1935. They argue that W WII had aparticularly strong effect on the education al attainmen t of children who reach edtheir early teens during the war and lived in countries directly subject tohostilities. Using data for 14 countries they find relatively big differences incompleted education for children in the 1930-35 cohort in countries that weremost heavily affected by the war (e.g. Germany, Austria, and the U.K.) butrelatively small differences for this cohort in other places (e.g. the U.S. andIreland). When they use an indicator for tlie 1930-35 cohort as an instrumentfor schooling (measured by a single dummy variable indicating more than aminimal level of schooling) they find that the earnings advantage roughlydoubles from its OLS value in both Austria and Germany, although the IVestimates are imprecise. W hile one might be co ncerned tha t the 1930-35 coho rtsuffered other disadvantages besides their disrupted education careers, theseresults are comparable to Harmon and Walker's (1995) in terms of the magni-tude of the IV /OL S gap. For their G erman sample Icl~ ino nd Winter-Ebmeralso consider a second IV estimator that uses cohort and father's veteran statusas instruments. The resulting estimate is slightly larger and substantially moreprecise than the o ne based o n cohort alone.Study number 8 in Tahle 11, by Lemieux and Card (1998), also uses acohort-specific difference in educational attainment attributable to WWII. Inthis case, the differential is associated with educational benefits offered toCan adian veterans.36 Lemieux and Ca rd note that the fraction of veterans wasmuch higher among English-speaking Canadians than French-speaking Canadi-ans. Moreover, after the war, French-speaking colleges in Quebec made few

4As noted below, Ichino and Winter-Ebmer (1998) find that the educational attainments ofchildren horn between 1930 and 1935 were substantially below those of children born just earlier o rlater in many European countries.5 Their specifications control for age and sulvey year. One can infer the presence of importantcohort effec ts from the fact th at the ir survey year effects show a 0.5 year rise in educationalattain me nt betwee n surveys in 1979 and 1986, colltrolling for age and the school-leaving ageindicators.A similar pack age of educ ation benefits-known as the G 1 Bill-was availab le to U.S.


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1152 DAVID CARDchanges to accommodate returning veterans, whereas universities in the rest ofCanada set up transitional programs that allowed many veterans-includingthose who had not finished high school-to en ter college. This comb ination offactors mea nt th at veteran education benefits had little or n o effect on French-speaking Canadians, but a potentially large effect among English-speakers. Inview of the difference, Lemieux and Card use the interaction of English-speak-ing ethnicity with a co hort effect for men who were age 19-22 in 1946 as aninstrument for education. The advantage of this strategy is that it allows forarbitrary cohort effects in observed earnings and educational attainment, andrelies on differences between English and French speakers in one cohortrelative to others to identify the return to education. Instrumental variablesestimates based o n this strategy are 20-100 percent above th e correspondingOLS estimates of the return to education, with the more precise estimatesnearer the bottom of this range.The fourth cohort-based study, by Meghir and Palme (1999), examines edu ca-tion a nd earnings outcom es of Swedish men b orn in th e late 1940 s and early1950 s who were affected by t he introduction of a new education system thatraised the minimum years of schooling by 2 and instituted other changes. Thenew system was introduced on a municipality-by-municipality basis over the1950-62 period. By 1961 abo ut one-half of municipalities were operating und erth e new system and in 1962 th e system was implem ented nationally. Meghir andPalme use a simple dummy variable indicating whether an individual attended areformed school system as an instrument for schooling. Their reduced-formmodels suggest that average years of schooling are about 0.8 years higher forme n who atten ded the reformed schools than for those who did not, controllingfor year of birth, father s education, a nd cou nty of residence .To evaluate the earnings impacts of this extra education Meghir and Palmeuse two samples: a small sample of me n bo rn fro m 1945 to 1955 and interviewedin the 1991 Swedish Level of Living Survey (SLLS); and a larger sam ple of me nborn in 1948 and 1953 and included in the Individual Statistics (IS) data set.Both surveys ar e linked to administrative e arnings records an d the refo re providerelatively precise earnings data. The IS data set also includes a battery of testinforma tion obtained o n individuals at age 12-13. Using the SLLS da ta set, theyobtain an OLS estimate of 0.028 for the return to schooling, and a correspond-ing IV estim ate of 0.036, although t he lat ter is relatively imprecise.37 M eghirand Palme s results for th e IS sample are derived from a mo del that includesdummy variables for each of 6 education levels. Only the first 4 of thesedumm ies are treated as endo geno us in their IV specifications, via the inclusionof generalized residuals from an ordered probit m ode l of schooling.38 I sum ma -rize their results by reporting the OLS and IV estimates of the earningspremium for the highest education level that is treated as endogenous. As in th e

7 They also report a control function estimate from a model that includes the reduced form


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R TURN TO S HOOLING 1153simpler linear models from the SLLS sample, the IV estimate from thisspecification is slightly above the corresponding OLS estimate, although theOLS/IV difference is not significant.The final two studies in Table I1 are both based on data for developingecono mies. Study num ber 10, by Maluccio (1997), applies th e school proximityidea to data from the rural Philippines. Maluccio combines education andearnings information for a sample of young adults with data for their parents'households, including the distance to the nearest high school and an indicatorfor the presence of a local private high school. These variables have a relativelystrong effect on completed education in this sample. Maluccio estimates OLSand conventional IV m odels using school proximity as an instr um ent, as well asIV models that include a selectivity correction for employment status andlocation. Both IV estimates are substantially above the corresponding OLSestimates. Maluccio's analysis suggests that the reliability of his schoolingvariable is somewhat lower than in conventional U.S. or European data sets,accounting for some of the gap between the IV and OLS estimates. Malucciodoes not present OL S or IV models that con trol for family background. Rat her ,he presents IV models that use parental education and wealth as additionalinstrumen ts for edu cation, leading to slightly smaller but some wha t m or e preciseIV estimates.The final study in Table 11, by DuFlo (1998), examines the education andearnings trends associated with a school building program in Indonesia in the1970's. The program set a target number of primary schools to be built in eachof Indonesia's 281 districts, based on the enrollment rate of primary-school agechildren in the district in 1972. DuFlo shows that average educational attain-ments rose more quickly in districts that had a greater program intensity,measured by the target number of new schools per primary-school age studentin the district in 1971. She also argues that the program had a bigger effect (onaverage) for children who entered school later in the 1970's, and no effect forchildren wh o finished primary school before 1974. Based on th ese considerationsshe uses interactions of year of birth with program intensity in the district ofbirth as instrume nts fo r schooling. H e r samp les include individuals age 2-24 in1974: those who were age 13-24 are presum ed to have been unaffected by theschool building program (and are ther efore assigned a 0 value for th e programintensity variable). T h e presence of thes e unaffected individuals in eachdistrict allows her to include unrestricted district-specific fixed effects in theearnings m odels.DuFlo7sbasic OLS and IV models are fit to a sample of wage-earners (as of1991). The OLS estimate of the return to schooling using hourly wages of thissam ple is 0.078, while the IV estim ate is slightly smaller-0.064 perc ent. T h emagnitude of the OLS estimate is unaffected by the addition of controls forregion-specific enrollment rates prior to the school-building program and mea-sures of spending on region-specific water and sa nitation progra ms. Th e addition


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1154 DAVID C RDcolumn of Table 11. DuFlo also reports results based on monthly earnings, withimputed values for self-employed workers. In these specifications the estimatedreturns to schooling are somewhat lower: the OLS estimate is 0.057, the IVestimate without added controls is 0.064, and the IV estimate with controls fordistrict-level enrollment rates before the school-building program and district-level water and sanitation programs is 0.049. As in many of the other studiessummarized in Table 11, none of the IV estimates is significantly different fromthe corresponding OLS estimate.In addit ion to the 11 studies included in Tab le 11 a few other relatively recentstudies have used IV techniques to estimate the return to schooling. Oneinnovative example is Hausm an and Taylor (1 98 0, which used th e m eans ofthre e time-varying co variates (age an d indicators for th e incidence of bad healthand unemployment) as instruments for education in a panel data model ofearnings outcomes for prime-age men . Hausm an and Taylor find that the re turnto schooling rises from about 7 percent in OLS specifications to 12-13 percentin their IV specifications. Although subsequent researchers have not directlyfollowed Hausman and Taylor's methodology, their use of mean age as aninstrument for schooling is equivalent to using a linear cohort variable, and isthus similar in spirit to several of the studies in Table 11.Another pair of studies not reported in Table 11, by Angrist and Krueger(1992, 1 9 9 3 , examines the effect of draft avoidance behavior on the educationand earnings of men who were at high risk of being drafted under the lotterysystem used during the Viet Nam war. During one phase of the draft, enrolledstudents could obtain draft exemptions, and many observers have argued thatdraft avoidance led to higher college enrollment rates among men with thehighest probabilities of being drafted. If this was true, one could use draftlottery numbers-which we re random ly assigned by day of birth-as instru -ments for education. While Angrist and Krueger (1992) reported initial IVestimates based on this idea, subsequent research (Angrist and Krueger (1995))showed that the link between lottery numbe rs and com pleted education is quiteweak. In fact, the differences in education across groups of men with differentlottery numbers are not statistically significant, suggesting that draft avoidanceby those with low lottery numbers had a negligible effect on their schoolingbehavior.39A third pair of recent papers not included in Table I1 are studies of thereturns to education in Ireland and Italy by Callan and Harmon (1999) andBrun ello an d Miniaci (1999). Both of these pap ers describe institutional changesin the education systems of the two countries that potentially affected theschooling attainments of more recent cohorts. In each case, however, theschool reform instrumen t is com bined with ot he r instrumen ts based onparen tal education a nd /o r socioeconomic class. Callan and Har mo n report that

3 y ~ h eonventional IV estimates are typically equal to or just above the OLS estimates. Angrist


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1155ETURN TO SCHOOLINGthe Irish schooling reform variables (which capture changes in the school-leav-ing age and the introduction of free secondary education) have no significanteffect on schooling attainment. Thus, the schooling reform variables by them-selves are not useful instruments for education, and Callan and Harmon's IVestimates are driven by their family background instruments. Brunello andMiniaci do not report a comparable test of the effects of the Italian reforms.Nevertheless, an examination of inter-cohort trends in their data set suggeststhat th e edu cation refo rm they discuss (a 1969 law tha t op ene d universities to allstudents regardless of high school curriculum) had little discontinuo us effect oneducationa l attainment. Even if the reform had som e effect, it is likely to beoverpowered in th e reduced form mo dels by th e effects of par enta l education.In light of the potential correlations between family background and abilitydifferences, IV estimators based on family background do not shed much lighton th e m agnitude of any ability or endogeneity biases in the return to education(see Card (1999) for an extended discussion).

nterpretationn interesting finding that emerges from the studies in Table I1 is thatinstrumental variables estimates of the return to schooling typically exceed thecorrespon ding OL S estimates-often by 20 percen t or more. If o ne assumes on

a priori grounds that OL S methods lead to upward-biased estimates of th e tru ecausal effect of schooling, th e even larger I V estimates o btain ed in many recentstudies present something of a puzzle. A number of explanations have beenoffered for this pattern . T he first explanation-proposed by Griliches (1977) andech oed by Angrist and Krue ger (1991)-is that ability biases in the O LSestimates of the return to schooling are relatively small, and that the gapsbetween the I V and OLS estimates in Table I1 reflect th e downward bias in theOLS estimates attributable to measurement errors. The imprecision of most ofthe IV estimates in Table I1 makes it difficult to rule out this explanation on astudy-by-study basis. Since measurement error bias by itself can only explain a10 percen t g ap between OL S and IV , however, it seem s unlikely that so manystudies would find large positive gaps between their IV and OLS estimatessimply because of m easurem ent error.41A second explanation is that the IV estimates are ev en further upward biasedthan the corresponding OLS estimates by unobserved differences between thecharacteristics of the treatm ent and comparison group s implicit in the IVscheme. A two-stage least squares estimator based on a quasi-experimentalcomparison can be interpreted as an estimate of the return to schooling usinggrouped data, where there are only two groups. As in other situations where a

1 am grateful to Marc o Ma nacord a for his analysis of the Bank of Italy survey data.4 1 One caveat to this conclusion is the possibility that measurement errors are larger, or more


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1156 DAVID CARDmicro-level regression mo del is comp ared to a grou ped regression, grouping mayaccentuate any inherent biases in the micro-level model by reducing the vari-ance in the independent variable (schooling) by more than it reduces thecovariance of th e in dep end ent variable with th e bias terms.42 This possibility cannever be ruled out on a priori grounds, and may be especially worrisome forquasi-experimental comparisons based on treatment and comparison groupsthat differ in terms of family background or measured ability. The only remedy(short of conducting randomized experiments) is to compare the results from awide variety of diffe rent quasi-experiments, and to en su re tha t observabledifferences between th e treatm ent and com parison group are taken into consid-erati on whenever possible.A third possibility, suggested in a recent overview of the retu rns to educationliterature by Ashe nfelter, Harm on , and Oo sterb eek (1999), is specification

searching. They hypothesize that in com parin g altern ative IV specifications,researchers tend to favor those that yield a higher t statistic for the estimatedreturn to schooling. If minor changes in specification have little effect on theprecision of the IV estimator, but generate a range of point estimates, thisbehavior will lead to a positive bias in the distribution of reported IV estimates,and to a positive correlation across reported specifications between the IV-OLSgap and the sampling error of the IV estimate. Ashenfelter, Harmon, andOosterbeek find such a positive correlation, and conclude that there may besome spec if ication sea rch b ias in the l i t e r a t ~ r e . ~ ~A final explanation is that there is underlying heterogeneity in the returns toeducation, and that many of the IV estimates based on supply-side innovationstend to recover returns to education for a subset of individuals with relativelyhigh retu rns to ed ucation. Institutional fe atur es like compulsory schooling or th eaccessibility of schools are most likely to affect the schooling choices ofindividuals who would otherwise have relatively low schooling. If the mainreason that these individuals have low schooling is because of hig her-than-ave r-age osts of schooling, rather than because of lower-than-average returns toschooling, then local average treat me nt effect reasoning suggests that IVestimators based on compulsory schooling or school proximity will yield esti-mated returns to schooling above the average marginal return to schooling inthe population, and potentially above the corresponding OL S estimates. U nd erthis scenario, both the OLS and IV estimates are likely to be upward-biased42 Consider a model for individuals who belong to groups (indexed by j : y,, = a bx,, el j. Th easymptotic bias in the m icro-level OLS regression coefficient is cov[x,,, e,,]/var[x,,]. If th e mo del isfit by weighted O LS to grouped d ata, the corresponding bias is cov[x,, e,]/var[x,], where xi d enote sthe mean of x,, in group j and el is the mean residual in group j The ecological fallacy problem

arises because although var[e,] may be small, the bias in the grouped data regression may be quitelarge.43 ~c ro sshe 22 pairs of OLS and IV estimates in Table I1 there is a positive but insignificantcorrelation between the ratio of the IV to the OLS estimates, and the ratio of the standard error of


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1157E T U R N TO S HOOLINGestimates of the average marginal return to education. For policy evaluationpurposes, however, the average marginal return to schooling in the populationmay be less relevant tha n th e average return for the group who will be impactedby a proposed reform. In such cases, the best available evidence may be IVestimates of the return to schooling based on similar earlier reforms. Thepattern of results in Table I1 suggests that the OLS estimate, even if upwardbiased as an estimate of the average causal effect of education, may be arelatively conservative estimate of the causal effect for groups typically affectedby supply side reforms.

4. ON LUSIONSThis paper reviews the recent literature that has attempted to measure the

causal effect of education on labor market earnings by using institutionalfeatures on the supply side of the education system as exogenous determinantsof schooling outcomes. The idea of using supply-side shocks to identifydemand-side parameters is a cornerstone of structural econometric methodol-ogy. Thus, I believe it is helpful to place the returns to education literature in astan dard supply and dem and framew ork, as first suggested by Beck er (1967).Such a framework immediately focuses attention on the rather special condi-tions that are required in order for the labor market to be characterized by aunique return to education. More generally, different individuals finish theirschooling at a point where the marginal return to the last unit of education maybe either above or below the average marginal return in the population as awhole.supply and demand framework leads to a somewhat richer econometricmodel for schooling and earnings than is usually adopted in the appliedliterature. In particular, the implied data generation process for earnings hasboth a random intercept (reflecting differences across individuals in the amountthey could earn at every level of schooling) and a random education slope(reflecting differences across individuals in the marginal return to education).Although one can still estimate a standard human capital earnings function bystandard OLS or IV methods, the parameter estimates must be interpretedcarefully. Even IV estimation based on ideal instruments (observable factorsthat are by assumption in epen ent of individual abilities) will typically recover aweighted average of returns to education for people whose education choiceswere affected by the instrument, rather than the average marginal return toeducation in th e population.The recent literature that uses supply-side features to instrument schoolingchoices tends to find IV estimates of the return to schooling that are at least asbig and sometimes substantially bigger than the corresponding OLS estimates.In many cases the IV estimates are relatively imprecise, and none of theempirical strategies is based o n tru e rand om ization. Thus, n o individual study is


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1158 D VID C RDfrom the recent IV literature are remarkably consistent with Griliches' (1977)assessment of a much earlier set of studies, and point to a causal effect ofeducation that is as big or bigger than the OLS estimated return, at least forpeople whose schooling choices ar e affected by th e supply-side innova tions thathave been studied so far.

Dept. of Economics 3880, Uniuersiy of California Berkeley, Berkeley, C A94720-3880. U.S.A.Marzuscrkt receired April, 1999; final recision receiued Jzlrze, 2000.

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RETURN TO SCHOOLING 1159BOUND, OHN,DAVIDA. JAEGER, ND RE G I N AM. BAKER (1995): Problems with Instrumen talVariables Estimation when the Correlation between the Instruments and the EndogenousExplanatoly Variables is Weak, Journal of the American Statistical Association 90, 443-450.BRUNELLO,IORGIO,AND RAFFAELEMINIACI 1999): The Econo mic Retu rns to Schooling forItalian Men . An Evaluation Based on Instrumental Variables, Labour Economics 6, 509-519.CALLAN,IM, NDCOLMH A RM O N1999): The Econom ic Re turns to Schooling in Ireland, Labo~lrEconomics 6, 543-550.CAMERON, AND CHRISTOPHERTEP HEN , TAB ER 2000): Borrowing Constraints and the Returns toSchooling, Unpublished Working Paper, Northwestern University De partm ent of Econom ics.CARD,DA VID 1995): Using G eographic Variation in College Proximity to Estim ate the R eturn toSchooling, in Aspects of Labour Market Behacio~lr:Essays in Honour of John Vanderkamp ed , byLouis N. Christofides, E . Ke nnet h G rant, and R obert Swidinsky. Toro nto: University of Toro ntoPress, 201-222.(1999): The Causal Effect of Education on Earnings, in Handbook of Labor EconomicsVolum e 3A, ed , by Orley Ashenfelter and David Ca rd. Amsterdam and New York: NorthHolland.CARD,DAVID,AND TH OM ASLE M IE UX2000): Drop out and Enrollm ent Tre nds in the Post-WarPeriod: What W ent Wrong in the 1970s? National Bureau of Economic Research WorkingPaper 7658.CO NNE ELY ,AREN , NDROOPEUUSITALO1997): Estimating He terog eneo us Trea tme nt Effects inthe Becker Schooling Model, Unpublished Discussion Pap er, Princeton University IndustrialRelations Section.DUFLO,ESTH ER 1998): Schooling and Lab or M arket Consequen ces of School Construction inIndonesia: Evidence from an Unusual Policy Experime nt, Unpublished W orking Paper, Mas-sachusetts Institute of Technology Department of Economics.FR EE MA N,IC HA RDB. (1986): Dem and for Education, in Handbook of Labor Economics ed , by

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You have printed the following article:

Estimating the Return to Schooling: Progress on Some Persistent Econometric Problems

David Card

Econometrica, Vol. 69, No. 5. (Sep., 2001), pp. 1127-1160.

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[Footnotes]

15 Instrumental Variables Methods for the Correlated Random Coefficient Model: Estimatingthe Average Rate of Return to Schooling When the Return is Correlated with Schooling

James Heckman; Edward Vytlacil

The Journal of Human Resources, Vol. 33, No. 4. (Autumn, 1998), pp. 974-987.

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19 Education and Self-Selection

Robert J. Willis; Sherwin Rosen

The Journal of Political Economy, Vol. 87, No. 5, Part 2: Education and Income Distribution. (Oct.,1979), pp. S7-S36.

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27 The Effect of Implicit Contracts on the Movement of Wages Over the Business Cycle:Evidence from Micro Data

Paul Beaudry; John DiNardo

The Journal of Political Economy, Vol. 99, No. 4. (Aug., 1991), pp. 665-688.

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Using Maimonides' Rule to Estimate the Effect of Class Size on Scholastic Achievement

Joshua D. Angrist; Victor Lavy

The Quarterly Journal of Economics, Vol. 114, No. 2. (May, 1999), pp. 533-575.

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Income, Schooling, and Ability: Evidence from a New Sample of Identical Twins

Orley Ashenfelter; Cecilia RouseThe Quarterly Journal of Economics, Vol. 113, No. 1. (Feb., 1998), pp. 253-284.

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The Effect of Implicit Contracts on the Movement of Wages Over the Business Cycle:Evidence from Micro Data

Paul Beaudry; John DiNardo

The Journal of Political Economy, Vol. 99, No. 4. (Aug., 1991), pp. 665-688.

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Problems with Instrumental Variables Estimation When the Correlation Between theInstruments and the Endogeneous Explanatory Variable is Weak

John Bound; David A. Jaeger; Regina M. Baker

Journal of the American Statistical Association, Vol. 90, No. 430. (Jun., 1995), pp. 443-450.

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The Returns to Schooling: A Selectivity Bias Approach with a Continuous Choice Variable

John Garen

Econometrica, Vol. 52, No. 5. (Sep., 1984), pp. 1199-1218.

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Estimating the Returns to Schooling: Some Econometric Problems

Zvi Griliches

Econometrica, Vol. 45, No. 1. (Jan., 1977), pp. 1-22.

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