NBER WORKING PAPER SERIES THE SOCIAL MULTIPLIER. Edward L. Glaeser Bruce I. Sacerdote Jose A. Scheinkman

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1 NBER WORKING PAPER SERIES THE SOCIAL MULTIPLIER Edward L. Glaeser Bruce I. Sacerdote Jose A. Schenman Worng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA 0138 September 00 The vews expressed heren are those of the authors and not necessarly those of the Natonal Bureau of Economc Research. 00 by Edward L. Glaeser, Bruce I. Sacerdote, and Jose A. Schenman. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 The Socal Multpler Edward L. Glaeser, Bruce I. Sacerdote, and Jose A. Schenman NBER Worng Paper No September 00 ABSTRACT In many cases, aggregate data s used to mae nferences about ndvdual level behavor. If there are socal nteractons n whch one person s actons nfluence hs neghbor s ncentves or nformaton, then these nferences are napproprate. The presence of postve socal nteractons, or strategc complementartes, mples the exstence of a socal multpler where aggregate relatonshps wll overstate ndvdual elastctes. We present a bref model and then estmate the sze of the socal multpler n three areas: the mpact of educaton on wages, the mpact of demographcs on crme and group membershp among Dartmouth roommates. In all three areas there appears to be a sgnfcant socal multpler. Edward L. Glaeser Bruce I. Sacerdote Department of Economcs 6106 Rocefeller Hall 315 A Lttauer Center Department of Economcs Harvard Unversty Dartmouth College Cambrdge, MA 0138 Hanover, NH and NBER and NBER eglaeser@harvard.edu bruce..sacerdote@dartmouth.edu Jose A. Schenman Prnceton Unversty

3 I. Introducton Emprcal wor n the socal scences frequently attempts to nfer ndvdual behavor from statstcal wor on aggregates. Indvdual labor supply s nferred from changes n the tax schedule. Crme deterrence elastctes are nferred from changes n polcng or punshment. Changes n polces are often seen as our best means of nferrng underlyng economc behavor because varaton n these polces s, n some cases, orthogonal to ndvdual-specfc error terms. However, usng aggregate varaton to nfer ndvdual-level parameters s problematc when there are postve (or negatve) socal nteractons. If one person s proclvty towards crme nfluences hs neghbor s crmnal behavor, then a change n polcng wll have both a drect effect on crme and an ndrect effect through socal nfluence. The presence of postve spllovers or strategc complementartes creates a socal multpler where aggregate coeffcents wll be greater than ndvdual coeffcents (as descrbed by Becer and Murphy, 000). A large body of recent wor (ncludng Katz, Klng and Lebman, 001, and Ludwg, Hrschfeld and Duncan, 001) seems to confrm the exstence of these spllovers n a number of areas. As such, an estmated aggregate elastcty ncorporates both the true ndvdual level response and effects stemmng from socal nteractons. 1 For many purposes, partcularly polcy-related ones, researchers actually want the aggregate coeffcent that ncludes both the ndvdual level response and the socal multpler. In that case, aggregate emprcal wor s approprate. Stll, t s crucal that the emprcal wor s done at the same level of aggregaton as the ultmate polcy. For example, f we want to now the effect of a natonal change n crme polcy, but we wor wth cty-level data, then we wll mss the mpact of all cross-cty nteractons. To adequately nfer state-level effects from cty-level coeffcents, we need to now both the 1 There s a long lterature that dscusses the so-called general equlbrum effects whch may be mssng from some econometrc estmates. In a sense, postve externaltes are just one type of general equlbrum effect.

4 power of socal nteractons and the degree to whch those nteractons decay across jursdctons. We refer to the estmated rato of aggregate coeffcents to ndvdual coeffcents as the socal multpler. So, f wages are regressed on years of schoolng at the ndvdual and at the state level, the rato of these two coeffcents s the socal multpler. It s also true that the same socal multpler can be estmated by regressng aggregate outcomes on aggregate predcted outcomes, where the predctons are based on ndvdual level regressons. In ths paper, we present a theoretcal framewor whch maps ths estmated socal multpler wth underlyng socal nfluence varables. Our theoretcal framewor tells us that f an ndvdual s outcome rses x percent as hs neghbor s average outcome, then the socal multpler roughly equals 1/(1-x) for large enough groups. As such, bg socal multplers do not tend to occur unless the value of x s.33 or hgher. If the spllover wors through the neghbors exogenous characterstcs, not through ther outcomes (.e. your propensty for crme s nfluenced by your neghbors parents characterstcs, not by ther crme level), then typcally the socal multpler s smaller. The presence of sortng wll also mpact the measured socal multpler. If there s sortng on observables and postve socal nteractons, then the ndvdual level coeffcent wll overstate the true ndvdual level relatonshp. The ntuton of ths clam s that wth sortng, one person s educaton wll be correlated wth hs neghbor s educaton and the effect of my educaton (n an ndvdual-level regresson) wll overstate the true mpact of educaton because t ncludes spllovers. The presence of ths bas wll mean that the measured socal multpler wll tend to underestmate the true level of socal nteractons. On the other hand, correlaton between aggregate observables and aggregate unobservables wll cause the measured socal multpler to overstate the true level of socal nteractons. 3

5 We also ntroduce a model where socal nfluence exponentally decays wth socal nfluence. Ths ntroduces a two-parameter model whch can capture both the level of socal nteractons and the degree to whch socal nteractons become less mportant wth socal dstance. The downsde of ths exponental socal nfluence model s that t s not good for dealng wth very large socal groupngs, such as countes and states: exponental decay generally wll mean that people do not sgnfcantly nteract wth people who are outsde of ther county. We then apply our framewor to three dfferent contexts. Frst, we follow Sacerdote (001) and examne the magntude of these nteractons among Dartmouth college roommates. The Dartmouth roommates data have the advantage of lttle unobserved heterogenety and randomzed socal groups. In ths case, we fnd wea socal nteractons n academc achevement, but strong nteractons n socal group membershp. We estmate a socal multpler of 1.4 n groups of eght (floors) and. n groups of 8 (dorms). These estmates are compatble wth a wea degree of socal nfluence that then decays very slowly. Second, we follow Levtt (1999) and loo at the nfluence of demographcs on the crme rate. Levtt (1999) argues that the coeffcents on age from ndvdual level regressons are far too small to suggest large swngs n crme that are related to aggregate changes n the demographc structure. For example, these coeffcents tell us that the baby boom can at best explan one-ffth of the rse n crme between 1960 and Whle ths argument s correct, socal nteractons may help us to understand why demographcs appear to be related to crme n tme seres regressons. Usng crme data we fnd sgnfcant evdence of a socal multpler of 1.7 at the county level,.8 at the state level and 8. at the naton level. These are extremely hgh estmates and we don t necessarly beleve n the level of socal nteractons that they mply. Stll, they certanly suggest that the aggregaton level s crucal. In fact, there s a slght dfference between the Becer and Murphy defnton of socal multpler and our own, although our defnton represents a monotonc transformaton of the socal multpler as they defne 4

6 Fnally, we follow Rauch (1993) and Acemoglu and Angrst (1999) and turn to the ssue of human captal spllovers. In ths case, we use ndvdual coeffcents to create a predcted wage for an aggregate (such as a state or Publc Use Mcrosample Area, or PUMA) and then regress actual wage on predcted wage. Usng ths approach, we estmate a socal multpler of 1.67 at the PUMA level and.17 at the state level. We agree wth Mans s (1993) generally pessmstc vew of the ablty to dentfy socal nteracton parameters, at least n the absence of true randomzaton. However, our results suggest that socal nteractons may be large, and that coeffcents at dfferent levels of aggregaton dffer sgnfcantly, ether because of socal nteractons or because of non-random sortng across dfferent areas. Whle we reman cautous n nterpretng our parameter estmates, we do beleve that our evdence casts doubt on the use of aggregate changes to mae nferences about ndvdual level parameters. II. A Framewor We follow Glaeser and Schenman (00), and present a smple framewor whch wll connect coeffcents from regressons run at dfferent levels of aggregaton wth underlyng socal nteracton parameters. The smplest algebrac representaton of the socal multpler can be shown wth a global nteracton model, where one person s acton depends on the average acton n a group. One partcularly smple model of ths type s that A = θ + A, where A s N 1 j j G ( ), j the acton of person, G() refers to person s group whch s of sze N, s the socal nteracton parameter and θ reflects the exogenous forces ncreases the level of the t. Goldn and Katz (00) use the term n a way that could encompass ether defnton. 5

7 acton. 3 We assume that groups represent a non-overlappng partton of the entre economy, so that f G( j) then j G() θ = +, where β X ε. In general, we wll assume that X s the value of attrbute for person and mpact of attrbute and ε s a person-specfc random effect. β s the drect Ths model mples that A j = θ j N 1 N j G ( ) j G ( ) and A = j (1 )( N 1+ ) N 1+ θ θ. The term captures j G ( ), j (1 )( N 1+ ) the fact that f person A has an ntrnscally hgher propensty to do an actvty ths wll both have a drect mpact on hs actvty level, but wll also ndrectly mpact hs actvty through ts nfluence on the other ndvduals n the group. When N s large, ths term wll be neglgble (as long as s bounded away from one). When N=, ths term becomes, whch s greater than one whenever > The smplest case occurs when the values of θ are ndependent wthn a group, and where the ncluded X regressors are ndependent of the error term. In that case, an ndvdual-level regresson where A s regressed on an exogenous varable X yelds a coeffcent estmate of β 1+. An aggregate regresson where group (1 )( N 1+ ) level average outcomes are regressed on group level average β coeffcent estmate of. 1 X characterstcs yelds a We defne the socal multpler as the rato of the group level coeffcent to the ndvdual level coeffcent, or the amount that the coeffcent rses as we move from ndvdual to 3 Ths equaton can be justfed by assumng that people maxmze A A A + θ j. N 1 j G ( ), j 6

8 group level regressons. In ths case, the socal multpler wll equal ( N 1+ ) or N 1+. As N grows large, ths value approaches (1 )( N 1) + 1. When s small, 1 ths coeffcent wll be close to one and as approaches one, the socal multpler approaches nfnty. The socal multpler can also be estmated by regressng group level outcomes on the group level outcome that would be predcted usng ndvdual coeffcents. If ths procedure s followed, the coeffcent estmated by regressng average communty (1 )( N 1+ ) outcome on β 1+ j G ( ) coeffcents estmated usng ndvdual level regressons, agan equals. X j / N, the outcome predcted by ( N 1+ ) One generalzaton of our assumpton s to assume that there s sortng across groups, at least on the bass of observables. To nclude sortng wthn the framewor, we X X + µ =, where X represents a group level average and µ represents an ndvdual specfc component whch s ndependent across people. We mantan the assumpton that the values of X are ndependent across characterstcs (although not Var( X ) across people) and ndependent of the error term. We use the notaton σ =, Var( X ) where σ represents the share of the varaton n observable characterstcs whch s due to the group level component. Ths new assumpton does not change the group level regresson coeffcent, whch β remans. However, allowng a group-specfc correlaton of characterstcs does 1 7

9 nfluence the ndvdual level coeffcents because now the ndvdual-level coeffcent ncludes the mpact of a correlaton between the ndvdual s X value and the values of hs neghbors. If we see a correlaton between the school outcomes of chldren and the schoolng of ther parents, ths may all occur because parental schoolng nfluences chldren s outcomes drectly. Alternatvely, t may n part reflect the fact that well-schooled parents lve together and as a result, the chldren of the more educated have more successful peers. j X The estmated coeffcent from an ndvdual level regresson now equals + σ (1 )( N 1) β 1+. The bas due to the cross-person correlaton roughly (1 )( N 1+ ) equals σ --the product of the degree of sortng and the amount of socal nfluence. If ether sortng or socal nfluence s unmportant, then ths term s small and can be gnored, but we thn n many cases, both of these terms wll be bg. The socal multpler now equals N 1+ (1 )( N 1)(1 + σ ) +, whch wll approach 1 as N gets large. If we now the value of σ, then we can nfer the sze of (1 )(1 + σ ) the socal nfluence parameter. Ths formula mples that for hgh levels of N, the presence of sortng on observables wll always cause the socal multpler to declne. As such, the measured socal multpler wll tend to understate the true level of socal nteractons, prmarly because the ndvdual level coeffcent s based upwards. 4 We now consder the case where there s sortng across neghborhoods on the bass of unobservable characterstcs. To formalze ths, we assume that ε = ε + ν, where ε 4 If the socal multpler s estmated by regressng an aggregate outcome on a predcted aggregate outcome, where the predcted value s formed usng ndvdual level coeffcents then the estmated multpler wll N 1+ Var( X ) agan equal, but n ths case t s necessary that σ = for each of (1 )( N 1)(1 + σ ) + Var( X ) the observable varables. 8

10 represents a communty specfc average level of the unobserved shoc. Furthermore, we assume that ε and the X n queston covary. In partcular, we assume that Cov( X, ε ) λ =. The person-specfc shocs are stll assumed to be ndependent of each Var( X ) other. In ths case, the person specfc coeffcent wll equal + σ (1 )( N 1) N 1 β 1+ + λσ. The bas created by sortng on (1 )( N 1+ ) N 1+ observables wll approxmately equal λσ as N grows large. The aggregate coeffcent now becomes 1 σn β + λ. The socal multpler, n ths case, equals: 1 σn + 1 σ λ σn ( N 1) 1+ β σ σ N + 1, or λ ( N 1) σ β 1+ λ β (1 )(1 + σ ) + σλ β as N gets large. The socal multpler wll rse wth λ f and only f 1 > (1 + σ ). The reason that the socal multpler can ether rse or fall wth ths form of sortng s that sortng mpacts both the mcro and macro coeffcents. If 1 > (1 + σ ) then the macro-coeffcent wll ncrease wth sortng more than the mcro-coeffcent and an ncrease n sortng causes the socal multpler to rse. If ths condton does not hold, whch really only occurs when the form of socal nteractons are very ntense (and the degree of sortng on observables s qute hgh), then ncreases n sortng mean the mcro-coeffcent ncreases sgnfcantly through the sortng related bas and ths causes the coeffcent to ncrease. In general, we wll rarely now the value of λ. In some cases, such as the Dartmouth College Roommates data set descrbed below, we now that roommates are randomly assgned and n that case λ equal zero. However, n other cases, t mght be qute hgh 9

11 and we can only guess about the extent that ths nfluences the measured socal multpler. We now turn to three varatons on ths model. Frst, we consder the case of a contnuous outcome where the externalty depends on the nnate characterstcs of the ndvduals and not ther actons (or outcomes). Ths s partcularly useful n the case of human captal spllovers where we thn that wages (and productvty) may well be a functon of human captal n the area. In ths case the model becomes A = θ + θ. The expected value of the ndvdual level coeffcent on X N 1 j j G ( ), j wll equal β f there s no correlaton across people n the value of correlaton, the ndvdual level coeffcent equals both across people n the value of mcro-level coeffcent equals X. When there s ( 1+ σ )β. When there s correlaton X and sortng on the bass of unobservables, the ( 1 σ ) β + σλ. + When there s no sortng on unobservables (.e. whether there s correlaton n observables or not), the expected value of the group-level coeffcent equals ( 1+ )β. Thus, wthout sortng the socal multpler equals 1 + and wth sortng the socal 1+ multpler equals. Notce that pure nput externaltes sgnfcantly decrease the 1+ σ possblty of very large multplers. Ths occurs because the feedbac effects, whch are the ey to large multplers n output-based externalty models, are absent n ths case. When there s sortng on unobservables, the expectaton of the aggregate coeffcent equals N(1 σ ) (1 + ) β + λ, and thus the socal multpler equals N(1 σ ) + σ 1+ λ Nσ 1+ λ 1+ whch approaches 1+ σ β Nσ + 1 σ 1 + as N gets large. Just as 1+ σ β before, the measured socal multpler has the potental to loo very large f there s sgnfcant sortng on the bass of unobservable characterstcs. 10

12 In the case of the Dartmouth roommates, the ey dependent varables are dscrete and ths requres further assumptons. Followng Glaeser and Schenman (00) (but not Broc and Durlauf, 001), we wll assume that observed outcomes are dscrete, but that ndvdual actons are stll contnuous. As such, each ndvdual stll chooses a level of A, but then ths A s translated nto a zero-one outcome. As such, each ndvdual stll chooses a level of A, and ths contnuous A s what nfluences neghbors, but the observable outcome taes on a value of one, f and only f A> for some fxed cutoff. The correct approach to ths problem would be to postulate a normal or logstc dstrbuton for ε and then to estmate the parameters usng maxmum lelhood. However, the purpose of ths short paper s to gve an easly usable method for calculatng the relatve sze of socal multplers wth smple calculaton. As such, we proceed wth the approxmaton that the probablty of tang the acton equals p + ( A A), where A s the natonwde average level of A. Essentally, we are assumng that the dstrbuton of the error term s approxmately unform, wth densty one, n the relevant regon of estmaton. 5 In that case, the algebra descrbng the estmated coeffcents s the same as n the contnuous case, and we can use the prevous dscusson wthout alteraton. The Deprecaton of Socal Influence over Socal Dstance Ths global nteractons approach helps to mae the pont that a macro-coeffcent does not necessarly mply much about a mcro-relatonshp. Unfortunately, ths smple approach does not help us to understand the degree to whch nteractons deprecate over space. As such, t gves us no gudance about what the mpact of a polcy evaluated on cty-level data wll have on the country as a whole. In order to have a framewor whch helps us understand the relatonshp between effects at dfferent levels of aggregaton, we 5 Assumng unformty s, of course, a very strong assumpton, but assumng that the densty equals one condtonal upon normalty s nnocuous, snce A can always be rescaled so that ths s true. 11

13 wll need a model where the level of socal nfluence changes wth the degree of socal proxmty. In order to present a smple model whch captures the deprecaton of socal nfluence, we assume that d A = θ + δ d = 1 1 A d, where A d s the acton taen by the person who s exactly d unts of socal dstance from the actor. We wll thn of people as beng organzed on a lne, and a group of sze N as ncludng N people who are closest to one another on the lne. Indvduals could be located on a multdmensonal lattce and could nteract wth any number of neghbors. 6 We consder the smplest case of a lne wth undrectonal socal nfluences. 7 In other words, we assume that people are only nfluenced by people who are behnd them n ths lne,.e. person follows person 1 and person 0, but person 1 only follows person zero. Ths one-sded feedbac ensures that the ndvdual level regresson yelds an unbased estmate of that N = 1 A N = β (as long as there s no sortng). Aggregatng yelds the formula: (( + δ ) A δθ ) N ( + δ ) 1 δ N 1 + N N = 1 1 ( + δ ) 1+ 1 δ N θ. We assume 1 > + δ. 8 The coeffcent from an aggregate regresson equals N ( + δ ) β 1 and thus the socal multpler s 1 δ N(1 δ ) 1 N 1 1 ( + δ ) 1 δ N(1 δ ) + equals 1+ / when N equals., whch converges to 1 δ 1 δ as N gets large and III. Example # 1: Dartmouth Roommates 6 We have dscussed socal structure of ths nd n our prevous wor, e.g. Glaeser and Schenman (00). 7 If the structure s a b-drectonal crcle, then even determnng A X j s not straghtforward hgher values of any ndvdual X varable wll have both a drect effect and an ndrect effect through the nfluence of ths X on the peers who then n turn nfluence the ndvdual n queston. 8 Ths assumpton s necessary to guarantee that actons have fnte varance. 1

14 In ths secton, we follow Sacerdote (001) and loo for the presence of socal nteractons among Dartmouth College roommates. The advantage of Dartmouth roommates s that they are essentally randomly assgned. 9 As such, t provdes one example of a stuaton where socal connecton s random and not the result of sortng. Thus, the socal multpler methodology seems most lely to be cleanly applcable n ths case. There are three natural unts of aggregaton wthn Dartmouth College: the room, the floor and the dormtory. The average room contans.3 students. The average floor contans eght students and the average dormtory contans twenty-seven students. Our exogenous varables are gender, verbal Scholastc Apttude Test (SAT) score, math SAT score, hgh school grade pont average (GPA), famly ncome and a dummy varable that taes on a value of one f the ndvdual dran beer n hgh school. The data on GPA, SAT scores and famly ncome comes from the Dartmouth admssons department. The data on beer consumpton comes from the Survey of Incomng Freshmen (sponsored by UCLA) whch s flled out by thousands of enterng college students. We frst examned the determnants of college GPA. We found no evdence of a socal multpler n ths case (results not shown). The coeffcents on ndvdual level regressons were the same as the coeffcents on aggregate regressons. For example, a 100 pont ncrease n math SAT score rased freshman year GPA by.13 ponts n an ndvdual level regresson,.1 ponts n a room level regresson and.10 ponts n a floor level regresson. These results are not a surprse Sacerdote (001) also found that no nfluence of roommate bacground characterstcs on freshman year grades. 10 In ths case, there s lttle evdence for socal nteractons or a socal multpler. 9 In fact, the assgnment s only condtonal wthn blocng group, where blocng groups are defned by answers to a pre-college survey. Sacerdote (001) controls for ths non-random element of assgnment, but fnds no effect of ths control. For smplcty, therefore, we wll gnore ths mnor element of selecton. 10 Sacerdote (001) does, however, fnd a correlaton between the grade of two roommates whch s some evdence for spllovers. As such, the actual magntude of ntra-room spllovers remans somethng of a puzzle. 13

15 We then turn to the area of fraternty or sororty membershp. Ffty-one percent of Dartmouth undergraduates jon a fraternty or sororty. Table 1 shows the results from estmatng lnear probablty models n the case of fraternty membershp. At the ndvdual level, ndvduals who dran beer n hgh school are 10.4 percent more lely to jon a fraternty or sororty. There are also more surprsng ndvdual level coeffcents. Hgher math scores ncrease fraternty membershp a 100 pont ncrease n math SAT score leads to a 5 percent greater lelhood of jonng a fraternty. Hgher hgh school GPAs also ncreases the lelhood of jonng a fraternty. People from rcher famles are also more lely to jon a fraternty. 11 Men are more lely to jon fraterntes than women are to jon sorortes. When we aggregate to the room level, the mpact of drnng beer, gender and famly ncome both ncrease slghtly. The mpact of GPA and math SAT score declne. None of these changes are statstcally sgnfcant. Aggregatng to the floor and then dormtory level causes beer drnng to become even more mportant (to a statstcally and economcally sgnfcant degree), but the other varables become nsgnfcant. These regressons llustrate both the potental and the problems wth socal multpler analyss. The coeffcent on beer rses wth the level of aggregaton, just as the model predcts. The other coeffcents just bounce around. As such, we wll focus our analyss on the changes n the coeffcent on past beer drnng. If we use the global nteracton model, we estmate dfferent values of for each of the dfferent levels of aggregaton. The formula for the socal multpler s N 1+. At the room level, the estmated socal multpler s less than one (1 )( N 1) + (although we can t reject small postve multplers). At the floor level, the socal multpler 1.4 and the group sze s eght. Together these mply that equals.38, whch stres us as a reasonable number. At the dormtory level, the estmated socal multpler s.3 and the group sze s 57. These mply that the socal multpler equals.56. In fact, 11 The magntude of these effects are almost the same f we use a probt rather than a lnear probablty model. 14

16 our estmates are suffcently mprecse that we cannot reject the null hypothess that these two values are the same. Stll, we fnd the general pattern of coeffcents ncreasng wth the level of aggregaton. Of course, logcally, we expect the socal multpler to ncrease wth the sze of the group. Presumably, the bgger the group, the greater the share of socal nfluences beng ncluded. Stll, the global nteractons model gves us lttle ablty to actually nterpret the extent to whch the socal multpler changes wth the level of aggregaton. The local nteractons model s meant to remedy ths lac. Usng the local nteractons formula for the socal multpler, and gven an average floor 8 sze of eght, ths mples that 1 1 ( + δ ) = (1 ) δ δ and the dormtory level 8 regressons tell us that 1 1 ( + δ ) = (1 ) δ δ. Together these equatons / 8 ( δ + ) + (8/ 8)( δ + ) mply that: = 1. 88, whch mples that δ + =. 95, 8 7 8( δ + ) + ( δ + ) and usng and hence =. 14 and δ =. 81. These numbers mply that each ndvdual has only a small nfluence on hs neghbor, but ths nfluence deprecates qute slowly over tme. The overall socal multpler s qute hgh, and as N gets large, t approaches.8. Whle the standard error bands surroundng our estmates are suffcently large to mae us qute cautous about acceptng these numbers, they stll suggest that the methodology does provde estmates that are at least plausble. IV. Example # : Crme In our prevous wor (Glaeser, Sacerdote and Schenman, 1996), we have focused on socal nteractons n crmnal behavor. There s a large amount of anecdotal behavor supportng the exstence of these nteractons, and t seems reasonable to expect to fnd a 15

17 socal multpler n the level of crme. As we have no data set featurng randomzed nteractons n ths context, we wll have to use exstng data on the level of crme to produce prelmnary estmates of the socal multpler n crmnal behavor. Indvdual level crme rates do not exactly exst. There are data on people who are arrested and people who go to prson. And there s self-reported data on crmnal behavor. Self-reported data s problematc for two reasons. Frst, people do not always report ther llegal actvtes honestly. Second, standard self-reported nformaton on crmnal behavoral (e.g. the Natonal Longtudnal Survey of Youth) does not contan crme data that s closely comparable to nformaton about crme rates. Because of these problems, we used natonwde arrest rates by age to form our basc ndvdual level estmates. These data correspond to arrests, not crmes. In order to mae these two sets of numbers comparable, we multply the age-specfc arrest rate by the natonal rato of reported crmes to arrests. In other words, we ensure that at the natonal level, our predcted crme measure s the same as the actual crme level. Nonetheless, our use of arrest rates wll be problematc f the rato of crmes to arrests dffers across age categores. Stll, because ths wor s meant to be exploratory, we wll go ahead wth ths nformaton. A further ssue s that snce our only ndependent varable s age, we may mss many possble sources of strategc complementartes n the level of crme. These ndvdual crme rates provde us wth a predcted level of crme n each neghborhood. As descrbed above, we use the ndvdual level coeffcents to predct an aggregate crme measure,.e. a π ( a ) p( a), where π (a) represents the arrest rate n each age category a and p(a) represents the share of the populaton n that age category. We gnore any ssues that mght come from aggregatng a dscrete varable. In Table II regresson (1), we report the results from regressng actual crme rates on predcted crme rates at the county level. The coeffcent s 1.7. In regresson (), we show that the state level socal multpler s.8. In prncple, both of these numbers mght be based because of sortng on observables or unobservables. In fact, the sortng 16

18 by age across countes and states s qute low. Sortng on unobservables (or observable varables that are not ncluded n the regresson) s lely to be hgher, and as a result t maes sense to tae these results warly, as they may well overstate the true socal multpler. Fnally, n regressons (3) and (4), we loo at the socal multpler that s estmated usng tme seres data at the naton level. In these regressons, we follow Levtt (1999) closely and n a sense merely duplcate hs evdence showng that, f mcro-level coeffcents are used, then aggregate changes n demographcs can explan lttle of the changes n aggregate crme. In our framewor, ths observaton shows tself n an estmated socal multpler of 8.16 for crme as a whole and 4.47 for homcdes. These hgh socal mulplers tell us that crme rates are movng around very qucly, gven the farly modest changes n aggregate demographc compostons. We have our doubts about the nterpretaton of these estmates. It s at least as lely that these hgh estmates are due to a correlaton between demographcs and unobservable elements. Stll, ther hgh values contnue to provde some evdence that socal nteractons are mportant n the level of crme and more generally that socal multplers are worth worryng about. We have estmated three dfferent socal multplers at dfferent levels of aggregaton. The estmated socal multpler rses substantally wth the level of aggregaton, so one mght thn that the exponental model could be useful n nterpretng ths data. However, the exponental model s actually pretty hard to use when addressng such large aggregatons. To mae the pont, the dfference between state and county level 86,700 5,07, ( δ + ) ( δ + ) coeffcents mples that =. 6, whch mples 86,700 86,699(1 δ ) + ( δ + ) that δ + > However, f δ + >. 999, then the estmated socal multpler at the county level requres that < The basc problem s that wth an exponental model, there should be lttle socal nteractons beyond close neghbors and as such, dfferences between county, state and naton-wde socal multplers are dffcult to wor wth. Future wor wll hopefully come up wth a better model for addressng the deprecaton of socal nfluence n large groups. 17

19 V. Example # 3: Schoolng and Earnngs In the case of schoolng and earnngs, we turn to the varant of the model where spllovers occur across θ s (.e. nputs) not As (.e. outputs). Ths framewor corresponds more closely to the dea that ndvdual earnngs are a functon both of ther own schoolng and of the schoolng of ther neghbors. In ths case, the socal multpler 1+ should equal σ λ. We wll use the approach of regressng aggregate β outcomes on predcted aggregate outcomes. We wll use the Indvdual Publc Use Mcro-Sample from 1990, and nclude all adults between 18 and 60 years of age. Our ndvdual coeffcents are found by regressng ndvdual wages (n levels) on gender and race dummes, martal status, a thrd order polynomal n educaton, and a fourth order polynomal n age. All of our coeffcents n ths frst stage regresson looed qute standard and we don t report them to save space. In Table III, we report our results from aggregatng wages and predcted wages up to the Publc Use Mcrosample Area (PUMA) and State level. These are the two levels of geography that are avalable n the 1990 census. PUMAs on average have 8,800 members. The average state has.8 mllon members. In regresson (1), we fnd a PUMA level socal multpler of In regresson (), we fnd a State level socal multpler of.17. As we would expect, the socal multpler rses wth the level of aggregaton. However, just as n the case of crme, the exponental model s hard to use wth aggregatons of ths sze. These socal multplers may be based upwards because of sortng on unobservables; however, sortng on observables at least s stronger at the PUMA than at the state level. Stll, these dfferent coeffcents should stand as a warnng aganst usng coeffcents from one level of aggregaton to nform us of effects at a dfferent level of aggregaton. 18

20 Furthermore, these results contnue to suggest that there are human captal spllovers, as suggested by a wde body of other research (e.g. Rauch, 1993, Lucas, 1988, Acemoglu and Angrst, 1999). However, the results from these regressons mply a much larger human captal spllover than the prevous research. To us, ths dscrepancy serves to emphasze the fact that unobserved heterogenety may be drvng our results. VI. Concluson Often emprcal wor treats the level of aggregaton as rrelevant. Routnely, state or natonal polcy nterventons are used to nfer underlyng ndvdual-level parameters n contexts as dverse as labor supply or the returns to schoolng. If postve spllovers or strategc complementartes exst, then these forms of nference are mproper. State-level regressons yeld approprate answers to questons about state-level polces, but not necessarly anythng else. The exstence of a socal multpler means that n many contexts, aggregate level coeffcents wll tend to radcally overstate the true ndvdual level response. Ths paper has presented a bref analyss of the socal multpler. We presented a seres of smple models, all of whch tell us how to nfer socal nteractons varables from the level of the estmated socal multpler. In prncple, these models can be computed effcently by usng maxmum lelhood, but n many contexts, an unbased measure of the socal multpler can be estmated by comparng ordnary least squares coeffcents found at dfferent levels of aggregaton. In the emprcal sectons of the paper, we found evdence for a socal multpler at three dfferent levels of aggregaton. Usng Dartmouth roommates data, where roommates are randomzed, we found that the mpact of at least one predetermned varable had a bgger mpact on jonng a fraternty or sororty at hgher levels of aggregaton. In ths case, our results were compatble wth our model of exponentally declnng socal nfluence. Usng crme data, we found evdence for a very large socal multpler n the level of crme. We do not necessarly tae the estmates as beng precse, but they are large 19

21 enough to support the dea that a socal multpler exsts. Fnally, usng data on wages and human captal varables, we found further evdence for large socal multplers n the case of wages and human captal. The pattern supports the dea that researchers need to be careful about how socal nteractons can potentally mae nference very dffcult, especally when state level varaton s used as the source of dentfcaton. 0

22 References Acemoglu, D. and J. Angrst (1999) How Large are the Socal Returns to Educaton? Evdence from Compulsory Schoolng Laws, NBER Worng Paper # Becer, G. and K. Murphy (000) Socal Economcs. Cambrdge: Harvard Unversty Press. Broc, W. and S. Durlauf (001) Dscrete Choce wth Socal Interactons, Revew of Economc Studes 68(): Glaeser, E. and J. Schenman (001) Non-Maret Interactons, Advances n Economc Theory, forthcomng. Glaeser, E., Sacerdote, B. and J. Schenman (1996) Crme and Socal Interactons, Quarterly Journal of Economcs CXI (): Goldn, C. and L. Katz (00) The Power of the Pll: Oral Contraceptves and Women s Career and Marrage Decsons, Journal of Poltcal Economy, forthcomng. Katz, L., Klng, J. and J. Lebman (001) Movng to Opportunty n Boston: Early Results of a Randomzed Moblty Experment, Quarterly Journal of Economcs CXVI (): Levtt, S. (1999) The Lmted Role of Changng Age Structure n Explanng Aggregate Crme Rates, Crmnology 37(3): Lucas, R. (1988) On the Mechancs of Economc Development, Journal of Monetary Economcs (1): 3-4. Ludwg, J., Hrschfeld, P. and G. Duncan (001) Urban Poverty and Juvenle Crme: Evdence from a Randomzed Housng-Moblty Experment, Quarterly Journal of Economcs, CXVI (): Mans, C. (1993) Identfcaton of Endogenous Socal Effects: The Reflecton Problem, Revew of Economc Studes 60(3): Rauch, J. (1993) Productvty Gans from Geographc Concentraton of Human Captal: Evdence from the Ctes, Journal of Urban Economcs 34(3): Sacerdote, B. (001) "Peer Effects Wth Random Assgnment: Results for Dartmouth Roommates" Quarterly Journal of Economcs CXVI ():

23 Table I Socal Multplers n Fraternty Partcpaton Dartmouth Roommate Data: Effect of Bacground Characterstcs on Partcpaton n Fraterntes at the Indvdual Level and Three Levels of Aggregaton Column (1) shows the OLS regresson of ndvdual fraternty partcpaton on own use of beer n hgh school, own SAT scores, own hgh school GPA and own famly ncome (self reported). Column () regresses the average partcpaton at the dorm room level on dorm room averages of hgh school beer use, SAT scores, HS GPAs, and famly ncome. Columns (3) and (4) ncrease the level of aggregaton to the dorm floor and dorm buldng respectvely. (1) () (3) (4) Member of fraternty or sororty Room average level membershp Floor average membershp Dorm average membershp Dran beer n hgh school (0.058) (0.0399) (0.081) (0.1930) Male (0.056) (0.086) (0.0540) (0.038) SAT verbal score (0.000) (0.0003) (0.0006) (0.0011) SAT math score (0.000) (0.0003) (0.0006) (0.0014) Hgh school GPA (0.0001) (0.000) (0.0003) (0.0005) Famly Income ' (0.000) (0.0003) (0.0006) (0.0013) Constant (0.1455) (0.66) (0.4594) (1.141) R-squared Observatons Average group sze Notes: Data are for Dartmouth Freshmen. Roommates and dormmates are randomly assgned as descrbed n Sacerdote [001]. SAT scores are from Dartmouth Admssons data. Famly ncome, use of beer, and hgh school GPA are self reported on the UCLA Hgher Educaton Research Insttute's Survey of Incomng Freshmen. Standard errors n parentheses.

24 Table II Regresson of Crmes Rates on Predcted Crme Rates Predcted crme rates for countes (or states or US) are formed by multplyng percentage of persons n each of eght age categores by the crme rate for persons n that age category. Data are from Census Bureau and Unform Crme Reports. Expected crme rate condtonal on age s based on age dstrbuton of arrestees for the U.S. Columns (1)-(4) are cross sectonal and srme data are for 1994 and demographc (age) data are for Columns (5) and (6) are the tme seres data for the US as a whole. County Crme Rate (1) () (3) (4) State US Crme Crme Rate Rate US Homcde Rate Predcted crme rate (or homcdes) (0.088) (1.070) (0.998) (0.637) Constant (0.004) (0.045) (0.043) (0.000) R-squared Observatons Average Group Sze 86,700 5,07,000 6,75,000 6,75,000 3

25 Table III Regresson of Wages on Predcted Wages Predcted wages are formed by regressng ndvdual level wages on gender, race dummes, martal status, educaton, educaton squared and educaton cubed, age, age squared, age cubed and age to the fourth power. We then aggregate to the PUMA (state) level and regress mean wages on the mean of predcted wages. (1) () PUMA State Mean Wages Mean Wages Predcted wages (.070) (.46) Constant ( ) (675.3) R-squared Observatons Average group sze 8,800,80,000 4