Social Interactions Local Spillovers & Unemployment (Topa 2001)

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The Review of Economic Studies Ltd.

Social Interactions, Local Spillovers and UnemploymentAuthor(s): Giorgio TopaSource: The Review of Economic Studies, Vol. 68, No. 2 (Apr., 2001), pp. 261-295Published by: The Review of Economic Studies Ltd.Stable URL: http://www.jstor.org/stable/2695929

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Review of Economic Studies (2001) 68, 261-295 0034-6527/01/00120261$02.00? 2001 The Review of Economic Studies Limited

S o c i a l Interactions, o c a lSpillovers a n d Unemployment

GIORGIO TOPANew York UniversityFirst version receivedMay 1997;final version accepted July 2000 (Eds.)

I analyse a model that explicitly incorporates local interactions and allows agents toexchange information about job openings within their social networks. Agents are more likely tobe employed if their social contacts are also employed. The model generates a stationary distri-bution of unemployment that exhibits positive spatial correlations. I estimate the model via anindirect inference procedure, using Census Tract data for Chicago. I find a significantly positiveamount of social interactions across neighbouring tracts. The local spillovers are strongerfor areaswith less educated workers and higher fractions of minorities. Furthermore, they are shaped byethnic dividing lines and neighbourhood boundaries.

1. INTRODUCTIONOver the past decade economists have increasingly recognized the importance of non-market interactions in a variety of contexts, such as joblessness, crime and other socialpathologies, peer influences in education, social learning and the diffusion of innovations,localization choices by households and firms, growth and income inequality. One commonfeature in these studies is the assumption that agents' choices and payoffs are affected byother agents' actions not just indirectly through markets, but also directly through imi-tation, learning, social pressure, information sharing, or other non-market externalities.It is also assumed that agents interact locally, with a set of neighbours defined by aneconomic or social distance metric.Benabou (1993, 1996) and Durlauf (1996a, b) incorporate local interactions in thehuman capital accumulation process into endogenous growth models that exhibit neigh-bourhood stratification and persistent and widening income inequality. In the field ofeconomic geography, Audretsch and Feldman (1996) and Rauch (1993) argue that localknowledge spillovers produce agglomeration economies and hence affect the locationdecisions by firms. There exists also a rich theoretical literature that stresses the roleof interactions in models of herds and information cascades (see Banerjee (1992) andBikhchandani et al. (1992)), or in models of social learning (Bala and Goyal (1998), Galeand Rosenthal (1999), Morris (1997)).On the empirical side, a vast and growing literature has attempted to provide a stat-istical estimate of the magnitude of local interactions and neighbourhood effects.1 Glaeseret al. (1996) explain the very high variance of crime rates across U.S. cities through amodel in which agents' propensity to engage in crime is influenced by neighbours' choices:in so doing, they provide estimates for the range of social interactions. Case and Katz(1991) explore the impact of neighbourhood effects on several behavioural outcomes, such

1. Jencks and Mayer (1990), loannides and Datcher (1999) and Brock and Durlauf (1999) give excellentsurveys of the empirical literature.261


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262 REVIEW OF ECONOMIC STUDIESas criminal activity, drug and alcohol use, childbearing out of wedlock, schooling, churchattendance. Ludwig et al. (1999) use the Moving To Opportunity programme as a naturalexperiment to evaluate the magnitude of neighbourhood effects. Bertrandet al. (1999) findthat local social networks have a significant impact on individual welfare participation. Instudies concerning education, there is a long tradition starting with the Coleman report(Coleman et al. (1966)) of studying possible peer influences on educational outcomes:Hanushek et al. (2000) use very detailed data on Texan schools to estimate peer effects instudent achievement, whereas Zax and Rees (1999) use a Wisconsin Longitudinal Studyto estimate the impact of peer influences during school years on subsequent earnings.Sociologists have also argued, long before economists, that "one's neighbours mat-ter" in defining one's opportunities and constraints (see Burt (1992)). Individuals are notconsidered as isolated entities but rather as being part of networks of friends, relatives,neighbours, colleagues, that jointly provide cultural norms, economic opportunities, infor-mation flows, social sanctions and so on. Wilson (1987) argues that adults in a communityinfluence young people by providing role models in terms of the value of education, steadyemployment and stable families. Coleman (1988) considers social networks as a source of"social capital" since they provide valuable information, lower transaction costs, allowmonitoring and enforcement of socially optimal outcomes.The main objective of this paper is to formulate and estimate a model of local interac-tions in the labour market. In particular, I assume that agents exchange information withtheir social contacts about job opportunities and that hiring may occur through informalchannels. The model generates a stationary spatial distribution of unemployment that canbe compared to the empirical unemployment distribution over a set of contiguouslocations in order to estimate the model parameters. I can then test for the existence oflocal information spillovers and provide an empirical measure of their magnitude. Themain innovation with respect to the existing empirical literature is to provide a morestructural approach to the estimation of local interaction effects.The importance of informal channels in finding jobs has been documented, amongothers, by Corcoran, Datcher and Duncan (1980) and Granovetter (1995). Both studiesreport that more than 50%of all new jobs are found through friends, relatives, neighboursor occupational contacts rather than through formal means. This is especially true for low-skill jobs, for less educated workers and for black workers. From a theoretical standpoint,informal hiring channels may coexist in equilibrium with a formal labour market becauseof information asymmetries: Montgomery (1991) analyses a model in which employerscannot perfectly observe the quality of prospective employees and solve the adverse selec-tion problem by relying on referrals from their high-ability workers. The basic assumptionis that there exists assortative matching in agents' social networks, so that high-abilityworkers are more likely to refer individuals like themselves.In my model, agents reside in locations that are linked through an explicit networkstructure. Each agent, while employed, can transmit information about job openings toher unemployed contacts; the same agent, while unemployed, can receive useful tips aboutjobs from her employed contacts. Risk-averse individuals find it in their best interest toengage in such information exchanges in order to (partially) insure themselves againstpossible future unemployment shocks. These mutual insurance arrangements can besustained even in a limited commitment environment.Unemployment in the model evolves according to a Markov process over the set oflocations. At any point in time, employed individuals may become unemployed with someexogenous probability, whereas unemployed individuals may find a job with probability


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TOPA SOCIAL INTERACTIONS 263increasingin the number of neighbourscurrentlyemployed. This process generatespositivespatialcovariancesof unemployment etweennearby ocations.I use data on the spatialdistributionof unemploymentn Chicagoto estimatethestructuralparameters f the model.One of the most striking eaturesof unemploymentin Chicagoin recentyearsis its geographicconcentrationn a few areas,mainlyin theSouth and the West Side: both in 1980 and in 1990,Censustractswith high levels ofunemploymentended to be clustered ogether n geographicallyontiguousareas,ratherthan being spreadaround in a random fashion (see Figures 1 and 2). The change inunemploymentatesbetween1980 and 1990was also spatiallycorrelatedFigure3). Thisgeographic"lumping"s consistentwith thepresenceof local interactions nd informationspillovers.

Unemploymentate,19800-5:::1_n ]:.:: :'.''1.e:1|9: 2 0 2 4 Miles

5-1-7-9 fK-1

7.9-12.1-12.1-17-7-17-7-66.7FIGURE 1

Map of unemployment rate, 1980


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264 REVIEWOF ECONOMICSTUDIES

Unemploymentrate 19900=5iio-56 2 0 2 4 Miles-56-9

9 -913-9-23-423-4-100

FIGURE 2Mapof unemploymentate,1990

The unit of observation n the structuralmodel is a Census tract, and the basicassumptionof the model is that residentsof one tractexchange nformationocallywithresidentsof the adjacent racts.Thislocal interactionmplies hat theemployment ateinone tract is affectedpositivelyby the employmentrate in the neighbouringracts.Thetransitions nto and out of unemployment lsodependon a set of observable ractcharac-teristics.Allowing for tract heterogeneitys imnportantn order to addressthe issue ofpositive sortingof individualsacross locations. In fact, agents may sort into differentneighbourhoodson the basis of their neighbours'characteristics r becausethey havesimilarpreferencesover differentconsumptionbundles(see, for example,BeckerandMurphy 1994)).Suchsorting may inducepositivespatialcorrelationof unemployment ven in theabsence of any local informationspillovers.I try to distinguishthe social interaction


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TOPA SOCIALINTERACTIONS 265

....,...i.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...

Unemnploymentate,90-SO[I]-66*7--19 2 0 2 4 Miles-1 9-0.60 6-3.23.2-7.87.8-72.9

FIGURE 3Mapof unemploymentate,1990-80

channelfrom otherpossible nter-tractnfluencesby usingadditional nformationon thespecificdimensionsalongwhichagents'social networksareconstructed,and on the geo-graphicboundariesof localcommunitiesdentifiedby residents.For example,sincethereexists a strong degreeof ethnichomogeneityn socialnetworks,one wouldexpectsocialinteractions o be weakerbetween ractsthat haveverydifferentethniccompositions.I estimate hemodelvia the indirect nferencemethodologyof Gourieroux,MonfortandRenault(1993),sinceit is not possible o characterizenalyticallyhe invariantdistri-bution of the contactprocessdescribedabove. The structuralparameters re estimatedindirectly,by minimnizinghe distancebetweenthe actual data and simulationsof thestructuralmodelfor differentparameter alues.In particular, ne usesthe parameters fan auxiliary tatisticalmodel(more readilyestimnablehanthe structural ne) to definea


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266 REVIEW OF ECONOMIC STUDIESchi-square criterion for the indirect estimation.2 In the auxiliary estimation, I also controlfor unobserved tract-specific fixed effects that do not vary over time.The empirical results support the model with built-in local interactions and reject thehypothesis that the observed spatial patterns in unemployment can be explained byobserved tract characteristics alone. The estimated local spillovers are significantly posi-tive: on average, an increase in the employment rate of neighbouring tracts by one stan-dard deviation brings about an increase in expected employment of 0-6 and 1 3 percentagepoints in 1980 and 1990, respectively.3Interestingly, spillovers are stronger in tracts withlower education levels and with higher fractions of non-whites. This is consistent withother direct evidence on informal hiring.4Furthermore, the local interaction channel is found to be weaker across adjacenttracts that have very different ethnic compositions, as well as across local communityboundaries that have been identified by residents. This result lends support to the hypoth-esis that the observed spatial patterns in unemployment are indeed generated by socialinteractions as opposed to other possible inter-tract linkages. Finally, an important caveatis represented by the finding that the introduction of spatially correlated shocks, to mimicthe possible presence of correlated unobservables, greatly reduces the size of the infor-mation spillover. However, the model with correlated shocks does not fit the data as wellas the baseline model.The theoretical and empirical framework developed in this paper may be appliedmore generally, to any of the settings described above in which local interaction effectsmay be present. The wider goal of this paper then is to provide some tools to test for theexistence of interaction effects, to estimate their magnitude, and to simulate the effects ofproposed policy experiments in their presence.

The remainder of the paper is organized as follows. Section 2 presents the structuralmodel and its general properties. Section 3 describes the indirect inference methodologyand the auxiliary model, and discusses important identification issues. The data used inthe estimation and the simulations are described in Section 4. Section 5 reports the empiri-cal results of the indirect inference estimation of the structuralmodel, and provides somenumerical estimates of the magnitude of the spillover effects. Section 6 concludes.2. THE STRUCTURAL MODEL

The starting point of the analysis is the idea that agents are embedded in social networks:in particular, they can exchange information about job opportunities with their socialcontacts, who are more likely to transmit useful information if they themselves areemployed. Thus the probability of finding a job is greater, the higher the employment ratein one's social network.The information exchange activity among agents can be seen as the result of anefficient insurance arrangement with limited commitment. Each agent has an incentive totransmit job information to her contacts when she is employed, in the expectation thatthey will reciprocate when bad times hit. With risk averse agents, such implicit contractsare sustainable even in the absence of formal enforcement mechanisms by means of directpenalties (e.g. social stigma) or the threat of exclusion from the insurance arrangement ifa violation occurs (such contracts have been analysed by Thomas and Worrall (1988) in2. A very good introduction o indirectnferencemethodscan be found in Tauchen 1996).3. See Table4. A onestandard eviation hange n the employment atecorrespondso 8 and12percent-age points n 1980and 1990,respectively.4. See Granovetter1995),JencksandMayer 1990),and Corcoran,DatcherandDuncan(1980).


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TOPA SOCIAL INTERACTIONS 267the context of long-term wage contracts, and by Ligon, Thomas and Worrall (1999) inthe context of informal credit in developing economies).The specific way in which I model the information interaction is through a discrete-time Markov process on a set of locations, that is similar to a contact process (this wasfirst studied by Harris (1974) in the context of interacting particle systems). In the firstpart of this section I present the model and describe its basic properties. The individualunits in this setup are Census tracts, since the empirical analysis will be conducted at thislevel of aggregation. In the second part, I discuss the use of physical distance to determinewho is "close" to whom, and I present some evidence that justifies carrying out the analy-sis at the tract level. In addition, I discuss possible micro-foundations of the model at thelevel of individual agents, and show that this disaggregated framework is observationallysimilar to the former one.

2.1. A model of inter-tract local interactionsThe building blocs of the model are a set of locations, a state variable and a set ofneighbours for each location, and conditional transition probabilities for the state of eachlocation that depend on the state of the neighbours in the previous period.Let S be a finite set of locations. A location in this framework is taken to representa Census tract. Each location i is indexed by a vector of characteristicsXi that are constantover time. These characteristics may affect the probability of gaining or losing jobs ineach location. Time flows discretely from 0 to oo in the model. The state of each tractevery period, Yit, is the employment rate within each area. For computational reasons,the state variable can only take a finite number of values in the interval [0, 1]. In particular,yitE-E {el, . ., eKl, where e1= 0, eK= 1, and ek-ekJ, = 0 1, k = 1, .. , K. Therefore, thestate of the system at any point in time is a configuration of employment rates YtE 1.The evolution of the system is governed by the following conditional transition prob-abilities, that spell out how the state of any one site may change in the next period, giventhe present configuration. If tract i is at full employment (i.e. yit= 1), then it may drop tothe next lower employment rate with a probability that depends on its own characteristics

Pd Pr (yi,t+1= ? 91yit= 1; Xi) = y(Xi), ()5. In the case of sitesarranged n a two-dimensionalntegeratticeand indexedby a pairof integers i,j),this is equivalento defininga distancemetricd s.t. I il,Il) - (i2,j2)1 |il -1 i2 + |IIl -j2-6. Onaverage, Chicago racthas 4 2 adjacent eighbours.Typically,ractsareroughly ectangularreaswith fourneighbours, ne for eachedge.


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268 REVIEW OF ECONOMIC STUDIESand will stay at the present state with probability (1 -Pd). On the other hand, the prob-ability of going from zero employment to the next higher employment rate depends onthe tract's characteristics, but also on the information Iit that residents of tract i mayreceive from their employed social contacts in the neighbouring areas

Pu. Pr (yi,t, I = 0 1Yit= 0; yt, Xi) = cx(Xi) X(Xi) Iit- (2)The flow of information received by tract i is in general an increasing function of theemployment rate in the neighbouring tracts

Iit= f(YNt), P- >) (3where yN 1 N Yit7 Again, tract i may stay at a zero employment rate next periodwith probability (1 -Pu). In the estimation that follows f() is simply the identity function,so the information variable Iit is equal to the average employment rate of the neighboursyN. As for the functions A(-), a(-), and y( ), they are assumed to be linear in theirarguments: e.g. 2(X) .X0+ = l iXl.Finally, if the state of tract i at time t is in the interior of the unit interval, then itswitches to either an upward mode or a downward mode with probability 0.5, and in eachcase the same transition probability as in equations (1) or (2) applies. Therefore, we canwrite:

Pr (yi,t+ 1 = ek-llYit = ek; Xi) = 2 (4)Pr Yi,t+1=ek+lJYit=ek;Yt,Xi)= (5)

Again, the state of tract i may remain unchanged during the next period with probability[12(Pd +PU)]-The transition probabilities (2) and (5) capture the information exchange process.The employment rate in any one tract may rise through two separate channels: one is afunction of tract characteristics that may reflect labour supply or demand conditions inthe area, and is independent of any interaction. The second factor is the informationabout job opportunities received by tract residents from social contacts in the neighbour-ing tracts. The term X(.) captures the "contagion" effect as in the standard contact pro-cess. It is worth noting that the strength of the information exchange channel may also beaffected by the tract residents' characteristics.This allows me to estimate local spillovers inunemployment for different "types" of tracts, in terms for example of the education levelsof its residents or its race composition. Finally, the information exchange process isassumed to affect the probability of an increase in employment, but not the probabilityof a drop in employment in the tract (equations (1) and (4)). In other words, I assumethat information interactions may affect employment opportunities, but do not play arole in the transitions out of employment.Allowing for tract heterogeneity with respect to certain observable characteristics isimportant in order to be able to discuss the implications of positive sorting acrosslocations. Agents may sort into different areas on the basis of some characteristics,whichmay also be related to employment outcomes. Therefore, one could observe positive spa-tial correlations of unemployment that are not related to any information spillovers butare simply due to the spatial correlation patterns of the covariates along which people

7. With a slight abuse of notation, I use Ni to indicate both the set of neighbours of tract i, and thenumber of elements in this set.


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TOPA SOCIAL INTERACTIONS 269sort.8 The absence of a time subscript on the X covariates amounts to assuming that theprocess through which agents lose and find jobs takes place at a higher frequency thanthe process that rules individuals' locational choices and hence the spatial distribution ofthese characteristics across tracts.

One final note concerns the nature of the random shocks that are used to simulatethe model. The system is started off at some initial configuration yoe X< Every period, avector (o,of shocks is drawn from a uniform distribution on [0, 1]. Then the state of eachlocation in the model is updated according to the realization of the shock wi, using thetransition probabilities detailed above.9 These exogenous shocks to employment can bei.i.d. in space over the map of the city, or they can be generated according to someautocorrelation structure. The latter option is exploited in the estimation to allow for thepossibility of correlated unobservables driving the spatial correlation patterns observed inthe data.2.1.1. Model properties. In Section 3 the structural model is estimated by comparingsimulated realizations of the cross-sectional distribution of unemployment, drawn by thestationary distribution of the model, with the empirical cross-sectional distribution ofunemployment at a given point in time. Therefore, I focus on the cross-sectional propertiesof the stationary distribution rather than on the dynamic properties of the model.The model described above generates a first-order Markov process on the map oflocations. The state space 3"contains all possible configurations of employment rates overthe set of locations. Since 3"is finite, I can index each state by w = 1, . . ., W where W isthe total number of possible states for the whole system. Let X be the set of all probability

measures on 3". Then a probability measure A eY on v is simply a finite vector ofprobabilities Aw,w = 1,..., W. In particular, the evolution of the system is governed bythe following ruleAt+ = Q It,

where Q is the (Wx W) transition matrix, whose entries qrs denote the transition prob-abilities from state r to state s. These transition probabilities can in principle be calculatedfrom the conditional transition probabilities (1)-(5). A stationary distributionof the pro-cess is a vector v such that v = Q v. It is straightforwardto show that a unique stationarydistribution v(X) exists, for any given choice of tract characteristics X.'0It is very difficult to prove analytical results on the properties of the stationary distri-bution. However, if one considers the case where locations are homogeneous in terms oftheir characteristics,the structuralmodel is very similar to a discrete time contact processdefined over a finite integer lattice. The behaviour of the contact process has been exten-sively studied in the literature on interacting particle systems." Two results are particu-larly interesting from my perspective. According to the first, the states of any two sites on8. One goodexample s education.Highlyeducatedpeoplemaylocate in an areabecause hey enjoythecompanyof other educatedpeople,or because hey attach great importanceo school qualityso theymoveto tractswith good schools.On the other hand, agents'education evelspositivelyaffecttheiremployment

opportunities.9. For example, upposeYit= 1.The shock Oi is drawn rom a U[O, ]. If oit


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270 REVIEW OF ECONOMIC STUDIESthe map exhibit non-negative spatial correlations. The second result states that this corre-lation is bounded above by a quantity that decays exponentially with the distance betweenlocations.Simulations of the structural model show that indeed as one lets the system evolvefor several periods, high or low employment clusters appear on the map of locations. Iftracts are homogeneous, these clusters are equally likely to take place anywhere on themap. Tract heterogeneity implies that low-employment clusters are more likely to occurin certain areas of the map, since tract characteristics affect the transition probabilitiesbetween different employment levels.In my simulation and estimation exercise I take as given the spatial distribution of theX covariates (determined by the sorting process), and I run the local interaction processconditional on that, until it converges to the stationary distribution. The structural param-eters of the y ( ), a(Q), and X( ) functions are estimated off the stationary distribution.The case in which all the spatial correlation is driven by sorting rather than by localinteractions corresponds to the case where )L(*) is identically zero for all values of the Xcharacteristics, so the model delivers a very straightforward way to distinguish the twoeffects.

2.2. Census tracts, physical distance and aggregationIn the structuralmodel, the set of neighbours for each site is defined by physical distance,in that the neighbours of tract i are the tracts immediately adjacent to it. Ideally, onewould like to have individual data on social networks for a large but well delimited set ofagents, with information on the sequence of social ties that connect each agent to anyother agent in the set. Then one could cast the model in terms of individual agents anddefine neighbours as the set of alters to whom each agent is directly connected. Such amodel would predict the emergence of unemployment clusters in the abstract space gener-ated by the map of social ties between agents, which could then be matched to the data.In the absence of such direct data on networks, one has to rely on economic andsociological considerations to make assumptions about the likely dimensions along whichsocial networks are constructed. In general, one can think of several distance metrics aspotentially good candidates to trace out the spatial structure of social networks.12 I wantto argue that physical distance is an important determinant of the way networks areconstructed.The underlying idea is that establishing and maintaining social ties is costly, andthese interaction costs increase with physical distance. For example, transportation costshave both a monetary and a time component that make it harder to maintain activecontacts with persons living far away or simply reduce the frequency of exchanges withsuch persons. In addition, local institutions (such as neighbourhood clubs or associations,churches, PTAs in local schools, local businesses) play an important role in fostering localsocial ties and facilitating information exchanges at the local neighbourhood level.There has been considerable debate in the sociological literatureon whether the notionof a local community has lost its meaning in an era of increasing mobility and an expandingarray of communication devices. However, there is evidence that an important fraction ofsocial contacts takes place at the local level. In a study of Toronto residents in the 1980's,Wellman (1996) finds that about 38%of yearly active contacts in all networks takes place

12. Conley and Topa (2000) define four different metrics, physical distance, travel time, ethnic, and occu-pational distance, and examine the spatial patterns of unemployment with respect to each metric.


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TOPA SOCIAL INTERACTIONS 271between pairs of agents who live less than 1 mile apart; this percentage rises to 64% forcontacts between agents less than 5 miles apart. In a Detroit study that used a 1975 surveyof about 1,200 residents, Connerly (1985) states that 41% of respondents had at least onethird of their Detroit friends residing within one mile. More relevant to this paper, Hunter(1974) reports that out of roughly 800 Chicago residents interviewed during 1967-68,about 49% said that the majority of their friends resided in the same local community.Therefore, identifying social contacts with agents who live physically nearby seems like areasonable approximation for one of the dimensions of social networks.A second question concerns the appropriate scale of analysis. Perhaps the mostrelevant social interactions occur within Census tracts, thus making the assumption thatresidents of one tract exchange information with residents of neighbouring tracts not verymeaningful. However, in the city of Chicago, Census tracts represent fairly small units ofanalysis. Pairwise distances between adjacent tract centroids are less than one mile for allbut a handful of tracts. Typically, tracts contain five or six street blocks. The averagepopulation (16 years and older) in a Chicago Census tract was 2700 in 1980 and 2500 in1990.More importantly, Census tracts are a sub-division of larger units called CommunityAreas. There are 75 such areas in Chicago, and each area contains on average 12 tracts.These areas, such as Hyde Park, Woodlawn, Lincoln Park, Englewood, have names thatChicago residents readily associate with neighbourhoods. In fact, Community Areaboundaries were drawn in the 1920's by a group of Chicago sociologists, such as ErnestBurgess and Robert Park, to represent communities with a distinctive identity. The maincriteria used to establish these Areas were: "(1) the settlement, growth, and history of thearea; (2) local identification with the area; (3) the local trade area; (4) distribution ofmembership of local institutions; and (5) natural and artificial barriers" such as rivers,railroad lines, large roads, parks.'3Even though neighbourhood boundaries change over time, these Community Areasstill represent, in many cases, meaningful and cohesive neighbourhoods. Hunter (1974),in his survey of Chicago residents, asked respondents to name the boundaries of theirneighbourhoods. He identified roughly 200 neighbourhoods, which in all cases representunits larger than Census tracts. In many cases, the boundaries of these neighbourhoodscoincide with the original Community Area boundaries.'4 Therefore, it makes sense toassume that there exist meaningful inter-tract social interactions, and to take Census tractsas the unit of analysis.Finally, I would like to focus on the issue of aggregation. The structuralmodel couldbe set up at the individual level, where sites represent individual agents, and the set ofneighbours is defined as the set of contacts within a neighbourhood of radius r. Thetransition probabilities would then be defined at the single agent's level, determining thetransitions into and out of work. Then the model could be aggregated to the tract levelfor estimation purposes.It is hard to come up with an individual level model that, when aggregated, generatestransition rules as in (1)-(5). However, one can compare the behaviour of such a model,in terms of its spatial properties, with the structural model presented here. Conley and

13. From the Local CommunityFact Book, Erbeet al. (1984).A certaingeographic reawas considereda CommunityArea if it had "a historyof its own as a community,a name,an awareness n the part of itsinhabitantsof common interests,and a set of local businessesand organizationsoriented to the localcommunity".14. In the estimationhat follows,I exploitthese boundarieso try to distinguish pilloversdue to socialinteractionsrom otherpossible ourcesof spatialcorrelation.


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272 REVIEW OF ECONOMIC STUDIESTopa (2001) analyse a model in which artificial agents are placed at the centroid of eachtract, and interact both with agents in their same tract, and with agents in tracts whosecentroids are less than one mile away, on the physical map.15 The transition probabilitiesare defined at the agent's level, and are very similar to the ones described here. Thecontagion term At() is a declining function of physical distance, so the interaction isstronger between residents of the same tract than between residents of two separate tracts.The spatial properties of these two alternative models can be compared by lookingat the Auto-Correlation Functions (ACFs) and the spectral densities generated by thespatial distribution of employment in model simulations, following a procedure described

0-3 , - lTract evel

- - - Individual level0-25

0-2 -

0-15

0-1

0-05-

0

-005 0 2 4 6 8 10 12Physicaldistance(adjacent racts)FIGURE 4

ACF comparison: tract level vs. individual level modelin Section 3.2. Figures 4 and 5 show that one can find parameter values such that theACFs and the spectral densities are quite similar in the two models, thus implying thatthe models can be made to deliver similar spatial correlation patterns.16 Therefore, I con-sider the structural model presented in this section as a viable approximation to a moredisaggregated model in which local interactions are defined at the level of individualagents.

3. ESTIMATION STRATEGYThe objective of the estimation strategy is to estimate the structural parameters of themodel presented in Section 2.1, in order to test for the existence of spillovers generated

15. The numberof agents n each tract s proportional o the actualpopulation n Chicago n 1980. Oneartificial gent correspondso about20 people n real life.16. However, he parameterstimatesand the spillovermagnitudesmay be different n the two setups.ConleyandTopa (2001)examine his issueinmore detail.


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TOPA SOCIAL INTERACTIONS 2730-35 , , I , ,

Tract evel- - - Individual evel

0-3-

t 025 -

d02 -

0-15 // X N015 ' ' *


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274 REVIEW OF ECONOMICSTUDIESvector of q parameters of the auxiliary model, with q' p. First, one estimates the auxiliarymodel using the actual data. These parameter estimates p depend on the true value 00.Then one simulates the structural model for different values of 0 in the parameter space0. For each choice of 0, one can estimate the auxiliary model using the outcome of thesimulations (in this case, a simulated unemployment realization over the map of the city).This estimation yields parameter estimates P(0) that depend on the specific value of 0used for that simulation. The parameters of interest 0 are estimated by minimizing thedistance between i(00) and p(O), according to some metric to be specified. In the remain-der of this section, I first of all present the indirect inference procedure and describe theauxiliary model; secondly, I discuss identification issues.3.1. Indirect inference methodologyThe data (y,, xn) used in the estimation are cross-sectional data on unemployment andtract characteristicsover the city of Chicago. The outcome variable y, is the (n x 1) vectorof unemployment rates for all tracts i = 1,.. , n. The covariates x, are a (n x M) matrixof exogenous variables.The auxiliary model is a statistical approximation to the structural model. LetJn(yn xn p) be the GMM criterion used in the auxiliary estimation; when performed onthe actual data, this yields the following parameter estimates

pn = argmin Jny(Yn,n, p). (6)pE-The auxiliary model is possibly misspecified, since I assume that the structural model is

a,sthe true model. One can show that p, -4 p as n * oo, where p r(0O). The functionr: e - is defined as follows:r(0) = argmin Joo(G,0, p). (7)

pEM

where Joo(G, , p) = lim,OO J(Yn 9Xn Ip), and G is the distribution of the random shocksthat determine the stochastic process of y. The limit criterion Joo(G,0o, p), evaluated atthe true value 00, is assumed to have a unique minimum at p.Turning now to the simulations, for each value of Oe 0E draw H simulated realiza-tions of y out of the stationary distribution of the structural model, PYhX, 0), h1,.. , H. I then perform the auxiliary estimation on the simulated outcomes, for eachvalue of 0 and for given x,: this yields

pn (0)= argminJn(Yn 0),xn,p)- (8)peSAgain, for the simulated estimator of p, one has pn(0) 4 r(0), VOe0; in particular forthe true value 0 n, ^(0O) -4 p.The indirect inference method simply evaluates mn(0) -n- P(0) over the parameterspace e, and picks the value 0* that minimizes this distance mnh(0). The indirect inferenceestimator nHis then the solution to the following minimum chi-square problem

on = argmmi[n _ 1h =1 p(o)] Qn[Pn Hh= lPI(0) 9Oee L H - HIAs is the case for a standard GMM procedure, the optimal weighting matrix Qn n thequadratic criterion (9) is Q = V-1, where V, is the covariance matrix of the auxiliary


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TOPA SOCIAL INTERACTIONS 275parameters p. The indirect inference estimator is consistent and asymptotically normal. Astandard chi-square specification test for the structuralmodel is also available: the statisticicKis equal to the minimized value of (9), scaled by a function of H, and is distributed asa 2(q -p), where q = dim p and p = dim 0.

Finally, Gourieroux, Monfort and Renault (1993) provide a test of hypotheses onthe structural parameters 0. Let 0 be partitioned into 0= [ ] , and consider the nullhypothesis Ho: 01 = 0. Define the constrained indirect estimator OH,o as the estimate thatresults from minimizing (9), subject to the constraint 01 = 0. The test statistic Kcis definedas the difference between the constrained and the unconstrained optimal value of (9), andis distributed as a x2(dim 01).3.2. The auxiliary modelSince I am particularly interested in the spatial properties of the stationary distributiongenerated by the structural model, I need to consider an auxiliary statistical model thatmimics these spatial characteristicswell.'8 An obvious choice is a Spatially Auto-Regress-ive model (SAR), since theoretical results and simulations on the contact process suggestthat the stationary distribution exhibits positive spatial correlations, decreasing in thedistance between locations. A SAR of order D can be defined as follows

Yi>d= idYi +?i (10)where the superscript Nd refers to the average level of unemployment in the tracts atdistance d from tract i.19

I use a criterion, based on spectral decompositions, to compare the spatial propertiesof a contact process and of a SAR model, and to choose the order of the SAR that bestfits the contact process. Let the contact process be the "true" model, and let the SAR bean approximating model to the truth. For a given order D of the Spatial Auto-Regression,I estimate via maximum likelihood the parameters of the SAR that best fit the truemodel.20I can then compare the fit across different possible approximating models (i.e.SAR models of different orders). The result is that the goodness of fit improves as oneadds higher order terms to the SAR: Figure 6 reports the spectral density of a typicalcross-sectional distribution generated by the contact process and compares it to that ofseveral SARs. In most cases, a SAR of order 6 or higher fits the spectra generated by acontact process remarkably well: therefore, I use a SAR(6) as the basis of my auxiliarymodel.The statistical model in (10) needs to incorporate a set of M covariates xi in order tocontrol for tract characteristics that may both affect the probability of finding and losingjobs, and be dimensions along which agents sort into locations. The structural model ofSection 2.1 also lets the local interaction term 24() be itself a function of local character-istics. To capture this feature in the auxiliary model, I add interaction terms between y_118. Consistency f the indirect nference stimatordoes not dependon the specific hoice of the auxiliarymodel. However,the closenessof fit betweenthe structuraland the auxiliarymodels affects efficiency: ee

Tauchen 1996).19. This is definedas yNd= Wi(d) y, whereWi(d) is the i-th row of a weightingmatrixW(d) constructedas follows. Wi(d)givespositiveequalweights o all tracts hat are at distanced fromi. If a tractsharesan edgewith i on the physicalmap,d= 1; if a tract is adjacent o one of these immediateneighbours f i (but is notadjacent o i), d= 2, and so on. The weightsadd up to unity.20. See Hansenand Sargent 1993).The details of the construction re available romthe authoruponrequest.


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276 REVIEW OF ECONOMIC STUDIESCPvs. SAR(1) CP vs. SAR(5)0.6 0.6

0-4 0.4

O. D= |~I.2 0.2

G ~~~~~~~~~~~0-4 -2 0 2 4 -4 -2 0 2 4CP vs. SAR(6) CPvs. SAR(8)0.6 0.6

0.4 - 0.4

0.2 0.2

- 4 -2 0 2 4 -2 0 2 4FIGURE 6

Spectrum comparison: Contact Process (solid) vs. SARs

and a subset of the covariates. Let xi be a Jx 1 vector of regressors, with J< M. Let-= [y,..., y6 ] Equation (10)can be rewritten asyi = (.i) + (xi) ,B+ -i, i= 1... ,n. (1

The following assumptions are made concerning the covariance structure of the errorterms: E(8i) = 0, E(,c2) = ui, E(8cie) = 0, Vi?j.. In other words, I assume no cross-corre-lation between different units, but I allow for heteroskedasticity. Another observationconcerning the error terms is in order. The data used in the estimation are taken fromtwo Census years, 1980 and 1990, thus creating a short panel. Therefore, all the variablesin (11) should be indexed by a time subscript. However, the short panel feature allows meto incorporate an unobserved, time-invariant component into the error term:Eit= pi + uit. The term rpiattempts to capture features of a specific location (unobservableto the econometrician), that may attract or turn away agents with certain characteristicsthat may be in turn correlated with the ability to find jobs. Insofar as these unobservedcharacteristics are really time-invariant over the decade under consideration, I can elimin-ate them by first-differencing the data. I follow this approach in the estimation, so theauxiliarymodel (11) is to be taken in its first-differencespecification from now on, unlessotherwise noted.21Hence, all time subscripts are suppressed.In equation (11), the yN covariates are correlated with the error term E?, since theyare endogenous. Therefore, I use instruments for these variables: one obvious choice is touse the exogenous variables in the neighbouring tracts to i. In particular, I use obser-vations in neighbours up to a distance 3 from tract i: so the instruments are

21. Therefore, y' really denotes Yit Yit 1; xi denotes xit - xit; and the error term actually denotes thedifference in the time-varying part of the original error term, uit uit-1 .


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TOPA SOCIAL INTERACTIONS 277Xi[N ... .N3] The complete set of instruments can then be defined asZi[ [ N] . These variables are assumed to be uncorrelated with the error terms

E(zjEj)= 0. (12)Equation (12) provides moment conditions that I can use to estimate the auxiliary modelof equation (11) via a GMM procedure.The use of IV is not strictly necessary here. The auxiliary model is only a statisticalmodel that summarizes certain properties of interest of the structural model, and can bemisspecified. Thus, an equally valid procedure would be to estimate (11) via ordinary leastsquares. This procedure would yield biased estimates for 4 and ,B, but it would not affectthe consistency of Vn'. However, using IV can be interpreted as adding several spatialcross-correlations of y and x to the auxiliary model (as it is, the model only includesautocovariances of y). This is roughly equivalent, in a time-series context, to adding cross-moments involving leads and lags of y and x.Finally, I augment the auxiliary model bv a certain number of raw spatial momentsof the unemployment variable. The rationale for doing so is to add flexibility to theauxiliary model, in order to be better able to detect signs of misspecification of the struc-tural model. As Tauchen (1996) shows, the X2test of the over-identifying restrictions ismore likely to fail to detect misspecification if the auxiliary model is not flexible enough.In addition to the mean and variance, I consider the spatial covariances up to distanceD, since the structural model delivers implications in terms of the spatial distribution ofunemployment: these are the pairwise autocovariances for pairs of tracts at given distanced, whereas the 4 parameters in the SAR measure the autocovariance between unemploy-ment in tract i and the average unemployment in the neighbouring tracts. These additionalauxiliary parameters can be estimated together with those of the SAR using the sameGMM framework. Let v, be the vector of moments to be estimated

yr= [A, a 2, C15 .. CD] 5where Cd = Cov (yi yY), d= 1, ... , D. Here yd indicates the unemployment rate in tractsthat are at distance d from tract i. So the complete vector of auxiliary parameters is p[? yr . I also need to define

Zi- Ei(0, 0)Yi -2 _ 2 2g(4s,p)-YiY'12C (13)

IyD2-C

where iY jiN i2 ... yf xN . The GMM estimator pn6MM is obtainedviaminim-ization of a quadratic criterion Jn(p) based on g(41, p), with the standard properties, andthe optimal choice of a weighting matrix is SA?',where S is the covariance matrix of thelimitingdistribution f 1/ S In?=g(4i5 p).22

22. Conley (1999) providesa robust estimator or the covariancematrix S that does not rely on anyspecificassumptionson the structureof the error terms,but only uses informationon economic distancesbetween ocations.Use of thisrobustestimator oes not lead to significantlyifferent esults hanmorestandardestimators. Therefore, in what follows, I use Sn = 1/n n= g(4i, pn) *g(4, pn)


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278 REVIEW OF ECONOMIC STUDIES3.3. IdentificationIdentification issues are crucial in this empirical exercise. This paper argues that localspillovers, generated by local social interactions, may explain part of the spatial patternsof urban unemployment observed in the data. The estimation methodology basicallyworks by matching empirical spatial moments with simulated moments out of the localinteraction model, to estimate the structural parameters of the model and thus give ameasure of the magnitude, if any, of the local spillovers. However, there may be otherfactors that give rise to the same patterns of positive spatial correlation in unemployment,but may be indistinguishable from the social interaction channel that I focus on.Formally, this discussion can be cast in terms of Manski (1993)'s analysis of identifi-cation issues in the context of endogenous social effects, such as peer influences, neigh-bourhood effects, social interactions, contagion, and the like. Restricting oneself forsimplicity to linear models, one can posit the following population relationship

y=a?4+E(yIA)+E(xIA)y?+x1 +, (14)where y is a scalar outcome variable, x and ? are attributes that directly affect the unit'soutcome (observed and unobserved,respectively), and A are attributes that define the indi-vidual unit's "reference group". In the present context, y is unemployment, x are tractcharacteristics, and A is the distance metric that specifies which tracts are neighbours ofeach tract i. Each term in (14) represents a separate effect: in addition to the direct effect,Bof observable characteristics on outcome, 4 expresses an "endogenous social effect" thatthe mean outcome in the reference group has on the individual unit's outcome, whereas yrepresents an "exogenous or contextual effect" of the mean observable attributes in theneighbouring tracts on tract i's unemployment rate. Furthermore, the error term E maydepend in part on unobservable characteristics that are themselves spatially correlated.In the context of the linear model (14), Manski (1993) shows that the endogenoussocial effects 4 can be identified if one assumes that there are neither exogenous effects(y = 0) nor correlated effects (Eei?j = 0). I make these assumptions both in the structuraland in the auxiliary model. Specifically, in the structural model the conditional transitionprobabilities (1)-(5) only depend on tract i's own characteristics and on the neighbours'employment rate in the previous period, while the neighbours'characteristics do not playa role. In addition, the evolution of the system is ruled by random draws a) from a vectorof uniform distributions that are i.i.d. in time and in space by construction.In the special case where exogenous and correlated effects are absent, 4 and ,B areidentified if the conditional expectation E(x IA) varies non-linearly with A, andVar (xIA)> 0 (Proposition 2 in Manski (1993)). In my model, x are tract characteristics,whereas A is the physical distance metric. Therefore, Manski's identification conditionsare satisfied. In practice, I estimate E(x IA1) non-parametrically as a local average

I /Ni xj 1, where Ni is the set of neighbours at distance 1 from i.Furthermore, it is worth noting that even if neighbours' characteristics did play arole in the evolution of the system, identification would still be ensured by the non-linearit-ies implied by the stationary distribution of the structural model, as well as by the asym-metric way in which tract characteristics and information enter those conditionaltransition probabilities (this point has been made by Brock and Durlauf (1999, p. 31-33)).Identification conditions for the structuralparametersin ( ), ac( ), and y( ) can alsobe expressed in terms of the indirect inference methodology. Intuitively, one wants to ruleout the possibility that the chi-square criterion (9) may be minimized by more than oneset of parameter values. Formally, this requirement amounts to assuming that the limit


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TOPA SOCIAL INTERACTIONS 279criterion JCO(G, 9o,p) has a unique minimum at p, that the binding function r( ) in (7) isone-to-one and that ar/aO is of full column rank. The rank condition can be locallytested and is in fact satisfied at the point estimates reported in Section 5.23Finally, with regard to the identification of the auxiliary model, I assume that theerror terms are uncorrelated in the cross-section, E(Ei?j) = 0, Vi ?j, and are uncorrelatedwith the neighbouring tracts' characteristics: E(x7E?) = 0. The latter condition concernsthe validity of the instruments used in the auxiliary estimation. Both conditions can beeasily tested.The assumptions made here, regardingthe absence of exogenous or correlated effects,may be considered too strong. For example, suppose criminal activity in one locationspills over to neighbouring locations, because of physical contiguity. An increase in crimemay induce certain types of residents to move out, who are more likely to be employedthan people who stay. Alternatively, the rise in crime may have an adverse effect on localbusinesses and employers, forcing them to leave. Thus the crime level in the neighbouringtracts would impact unemployment within those tracts, and have a contextual effect onthe unemployment rate in tract i (y? 0), generating positive spatial correlation in unem-ployment that is not due to social interactions. Another example is the location of schools.Suppose high school quality in tract j induces agents with high ability or motivation tolocate in tract j as well as in neighbouring tract i. This may affect attributes such asgraduation rates or school drop-out rates in both tracts, while at the same time reducingunemployment in both tracts. Again one would observe a non-zero exogenous effect (y? 0)in that attributes of neighbourj would be associated with outcomes of tract i . Finally, asan example of correlated effects, one can think of positive sorting inducing positive corre-lation of certain unobserved attributes across neighbouring areas, that may directly affectemployment outcomes.These are potentially serious violations of the identification assumptions. Therefore,I follow three alternative strategies in the estimation of the structural model, to try todistinguish spillovers that are due to social interactions across neighbouring tracts fromother possible inter-tract influences. The first two approaches exploit indirect informationthat may be available on the dimensions along which social networks are constructed,while the third addresses the issue of generic correlated effects across tracts.There is considerable evidence in sociology on the extent of assortative matching inagents' social networks. In particular Marsden (1987,1988), using data from the 1985General Social Survey, shows that network homogeneity with respect to race and ethnicityis very high.24Quite simply put, individuals are more likely to interact with people of thesame race or ethnicity than with members of different groups. The idea then is to dividethe set Ni of tracts physically adjacent to i into two subsets: those that are ethnically"close" to tract i, and those that have a very different ethnic composition. Let me call thelatter subset EDNi.25 If social networks follow racial and ethnic lines, then one can expect

23. This is done by running a very long simulation of the structural model at the estimated value 0 H, andnumerically evaluating the matrix of partial derivatives ap/IO at the optimum OH. I can then test whether thismatrix has full rank.24. For example, the likelihood of observing a social contact between two black persons is 4 2 times higherthan that generated by pure random matching, given the relative proportions of the different ethnic and racialcategories in the population.25. Operationally, I use the same procedure as in Conley and Topa (2000) to calculate pairwise ethnicdistances between tracts. I consider nine different racial and ethnic groups to construct this metric. I then definea set EDNi of "Ethnically Distant Neighbours" of i as those tracts in Ni whose ethnic distance from i is abovea certain threshold: 85 for 1980, 91 for 1990 (the maximum possible ethnic distance is 100v i). These thresholdsroughly correspond to the 55-th percentile of the distribution of pairwise ethnic distances.


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280 REVIEW OF ECONOMIC STUDIESvery little social contact and information xchangebetween ract i and tracts n EDN,26In practice, modify the conditional ransitionprobability 2) in the followingway

PU = a2(X,)+ X(X,3 N Yjt + XEDN(Xi) ED SYk-l kt- (15)N1 EDN, -The other upward ransitionprobability 5) is modifiedaccordingly. f local spilloversaremostly generatedby social exchanges,one would expect the local interaction hanneltobe weakerwith tract i's ethnicallydistant neighbours han with its other neighbours nNi. On the otherhand,other nter-tractnfluences uch ascrimeshouldnot be so affectedby racialand ethnic distance: n the data, the spatialcorrelationof crimeacrossadjacenttracts s 0-75 for areas that are ethnicallyclose, and 0-79for ethnicallydistanttracts.Intermsof equation 15),thishypothesis ranslates nto XEDN(')


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TOPA SOCIAL INTERACTIONS 281TABLE 1

Summary statistics of all variables1980 1990 1990-80

Variables Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.Unemployment rate 11 71 8 16 14-93 12 06 3 22 8 95Segregation index 87 20 13 03 91-82 11 52 4 62 14-32Non-white persons (%/o) 50 55 39-63 57 91 37 14 7.36 14-59Hispanic persons (%l) 7 19 1292 19 53 26 36 12-34 17 24High-school grad. or more (%) 53 65 1694 62 53 17 20 8 88 1023College grad. or more (%l) 10-86 12 91 15-62 17 37 4 77 8 62Crime index 22 59 15-51 19 39 13 50 -3 20 8 04Females (%) 52 91 6 63 51 81 5 29 -1 10 5 68Persons 18-24 y.o. (%) 1935 670 1594 706 -341 6 19Persons 0-24 y.o. (%) 41 74 12-03 38-01 11 97 -3 73 7 74Persons per household (average) 2 77 0 62 2 82 0 67 005 040Vacant housing units (%) 736 5 99 1048 8 18 3 12 8 14Median gross rent 227 42 55 22 252 07 73 22 24 65 51 77Average housing value 10 05 13 52 26 18 22-30 16 13 19 60Employed in prfs/mngr jobs (%) 9.59 891 1277 1155 318 712Out of labour force-males (%) 30 29 13 22 3048 1403 0.19 1208Out of labour force-females(%) 51 01 12-56 45 82 13 73 -5 19 11-87Same county workers (%) 83 89 12 02 94 60 5 46 10-71 12-41Median travel time to work 27-26 6 38 27 99 6 52 0-73 5 96See the Appendix for a definition of all the variables and units.

covariates. Summary statistics for all variables are presented in Table 1 and precise defi-nitions are given in the Appendix.First of all, I consider a set of sorting variables, i.e., variables that may affect thedecisions by different types of individuals to locate in a given area. These include averagehousing values in the Census tract, median gross rents, the fraction of vacant housingunits in the area, a crime index, the fraction of persons with managerial or professionaljobs, the percentage of non-white persons, the percentage of Hispanic persons, a segre-gation index, and the number of persons per household.Secondly, I consider variables that may be linked more directly to the probability ofbeing employed. These include the percentage of persons with at least a high school dip-loma, the fraction of persons with at least a college degree, the age composition in thetract (to proxy for potential experience), the fraction of females 16 years and older, andthe percentage of males and females out of the labour force in the tract.Finally, there is a relatively large literature in urban and labour economics that dis-cusses the spatial mismatchhypothesis. This literature aims at explaining the high unem-ployment levels in mostly black, inner city neighbourhoods by local labour marketconditions. The basic idea is that during the 1970's and 1980's many jobs (especially low-skill ones in the service industry) have moved from central city areas to the suburbs. Inaddition, the contention is that there is low residential mobility and a certain degree ofhousing segregation for inner-city blacks. For example, it may be very costly for a blackhousehold to relocate to the suburbs in a mostly white neighbourhood, where the socialcapital provided by a black community would be missing. Several authors, such as Holzer(1991) and Ihlanfeldt and Sjoquist (1990,1991) have analysed this issue empirically. Bynow, there is a certain consensus that physical proximity to jobs explains a portion ofblack/white unemployment differences, for instance. Therefore, I include the median com-muting time to work for workers who live in each tract, and the fraction of residents whogo to work in the same county.


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282 REVIEW OF ECONOMIC STUDIESNumerical simulations of the structural model were performed in order to deliverrealizationsyh(xn, 0) to be used in the estimation of 0. The simulations use the conditionaltransition probabilities (1)-(5), where 2(.), ac(), y( ) are linear functions of tract covari-ates X. The outcome variable y takes values in the discretized [0, 1] interval E, and stands

for the employmentrate. Therefore, in order to make the simulated yn(xn, 0) comparablewith Ynfrom the data, I operate the transformation yn(xn, 9) = 100 (1 -Jy) on the outcomey of each simulation.For a given value of the parameters 0, the system is simulated starting at full employ-ment: i.e. the initial configuration of employment rates over the map of locations is y_1-28The X covariates that enter the conditional transition probabilities are taken from thedata and stay fixed at either their 1980 or 1990 level throughout the simulation. I let thesimulated process run for 1300 iterations. This seems a large enough number to let theprocess converge to its stationary distribution. I then draw H "snapshots" out of thisdistribution: these deliver the sample of H simulated realizations yn(xn, 9) used in theindirect inference procedure. It is essential, of course, to use the same sequence of shocks{t = 1 for all the different simulations involving different values of 029Computational limitations imply that the dimensionality of the parameter space 0has to be rather small. Therefore, instead of letting the transition probabilities of thestructuralmodel depend on the full set of M covariates described above, I use a subset ofL characteristics, with L < M. In particular, I choose the three covariates that contributethe most to reducing the spatial correlation in raw unemployment rates. In practice, I runthe auxiliary regression (11) without interaction terms, both in levels and in first differ-ences, and compute fitted residuals Cj=_ ,B.I estimate non-parametrically the Auto-Correlation Function (ACF) of these residuals, as a function of distance, following amethodology developed in Conley and Topa (2000). Then, I repeat this estimation exclud-ing from the set of covariates one variable at a time, to see how the ACF changes follow-ing the exclusion of each variable. It seems that the three variables that bring about thelargest reductions in spatial correlation of unemployment are the percentage of non-whitepersons in the tract (NW), the fraction of high school (for 1980) and college (for 1990)graduates (HI and CL, respectively), and the crime variable (CR).Therefore, in the simulations, I specify the X(-) functions for 1980 and 1990 as follows

X(Xi)80O + Xnw NW80 +ed HII80 Xcr ' CR8,X(Xi)90-Xo+ Xnw NW70+ Xed CL9i0+ Xcr CR70; (16)

the a(-) and y( ) functions are defined exactly in the same way. Finally, to further reducethe number of structural parameters to be estimated, I impose symmetry and pro-portionality restrictions on a(.) and y(*)Yo= &Xo,

ynw= -8anwYed= i50ed,Ycr = _3ac X

28. The initial condition does not really matter, since there exists a unique stationary distribution.29. The auxiliary estimation actually uses first-differenceddata. To make the simulations y consistent withthis, I use the following strategy. For each value of 0, I first run the simulation fixing the X characteristics attheir 1980 levels. This yields a simulated unemployment realization for 1980, Y(0)80. I then repeat the simulationfor the same 0 (and the same sequence of shocks) using the 1990 values of X. This delivers Y(0)go. I thencompute Y(0)go-80 Y(0)go -Y(0)80. This is the simulated counterpart to the dependent variable of the auxiliaryregression on the data.


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TOPA SOCIAL INTERACTIONS 283where 3 > 0. All these manipulations imply that the actual vector of structural parametersto be estimateds 0 =_ onw Xed Xcr a0 anw aed acr 3] . This vector is augmentedby

EFDN or XNCA in the model specifications that allow the intercept of the local interactionterm to differ for ethnically distant neighbours or for neighbours in different CommunityAreas and different "Hunter neighbourhoods".I use a simulated annealing algorithm to minimize the indirect inference criterion (9)with respect to the parameters 0. This algorithm is explicitly designed to find a globalminimum of functions that may present multiple optima and discontinuities. Therefore,it is particularly well suited for the problem at hand. Very briefly, simulated annealingperforms a random search over the parameter space, and accepts not only downhill movesbut also uphill moves. The probability of accepting an upward move depends positivelyon a "temperature" parameter that decreases as the search progresses. At the beginningof the search, the algorithm is allowed to make large upward moves, and thus searchesover the whole parameter space. As the temperature drops, the algorithm concentrates onmore promising regions, but the random nature of the search still allows it to escape localminima. Simulated annealing has proven to be very effective in this particular problem.30

5. EMPIRICAL RESULTSAs a preliminary step, I report in Table 2 the results of the estimation of the auxiliarymodel, performed on the actual unemployment data in first differences. It is worth notingthat, even after controlling for tract characteristics as well as time-invariant tract-specificfixed effects, the unemployment rate of neighbouring tracts has a positive and significantparameter estimate, at least for tracts up to a distance 2 from tract i. Even though theauxiliary parameters are not the actual parameters of interest, they still give an indicationthat unemployment is characterizedby positive spatial correlations: this in turn is consist-ent with the model of local interactions.Figure 7 also reports the non-parametric ACF estimates for the dependent variableand the fitted residuals of the auxiliary regression. The solid line refers to the actualunemployment rate in first differences. The dashed line refers to the fitted residuals,ii- (Yi) 'GMM (x1) GMM. Finally,theintermediateine shows theACF of a differ-ent set of fitted residuals, calculated excluding from the covariates the direct effect ofunemployment in the neighbouring tracts: ij=yi - (xi) GMM. The circles, diamonds andasterisks highlight the portions of the ACFs where one can reject the null hypothesis ofspatial independence, at the 5%significance level.31It is interesting to note that both the raw unemployment variable and the fittedresiduals 'i display a positive and statistically significant amount of spatial correlation,that declines monotonically with distance. On the other hand, the fitted residuals Ei donot exhibit any positive spatial correlation and are close to being white noise (in space),except for a small amount of negative autocorrelation at distances 1 and 2. Therefore, itseems that the assumption of no cross-correlation of the residuals across individual unitsis a reasonable one.Tables 3 and 4 present the results of the indirect inference estimation of the structuralparameters. The model is estimated under four different specifications. The first is thebaseline model, with conditional transition probabilities as in equations (1)-(5). The

30. For a more detaileddescription f the algorithm, ee Goffe (1996)and Goffeet al. (1994).I amverygrateful o Bill Goffe for sharinghis MATLABsimulated nnealing outineswithme.31. See Conleyand Topa (2000)for the detailsof the ACF estimationand the bootstrap est of spatialindependence.


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284 REVIEW OF ECONOMIC STUDIESTABLE 2

Auxiliary regression on the dataDependent variable:Unemployment rate, 1990-80

Variable name rho-gmm S.E. Included?Unemployment rate (nbs-1) 0 6959 0.1637 YesUnemployment rate (nbs-2) 04351 0.1444 YesUnemployment rate (nbs-3) -0.3236 0 1760 YesUnemployment rate (nbs-4) -0 0891 0 2634 YesUnemployment rate (nbs-5) 0 3877 0 3880 YesUnemployment rate (nbs-6) -0 2317 0 3160 YesUnemployment rate (nbs-1) xNon-white persons (%o) 0 0013 0 0133 YesUnemployment rate (nbs-1) xCollege grad. or more (%) -0 0462 0 0166 YesUnemployment rate (nbs-1) xCrime index 0 0136 0 0139 YesConstant 0-5943 0 8129 NoSegregation index 0 0151 0 0240 NoNon-white persons (%) 0.0111 0 0538 YesHispanic persons (%) -00524 00258 NoHigh-school grad. or more (%/6) -0 0336 0 0377 NoCollege grad. or more (%/6) -0 0269 0 0671 YesCrime index -0 0420 0 0681 YesFemales (%) -0 1780 0.0798 NoPersons 18-24 y.o. (%) -0 1732 0-0749 NoPersons 0-24 y.o. (%) 0 1073 0 0927 NoPersons per household (average) -0 2168 0-1313 NoVacant housing units (%o) 0.0758 0 0497 NoMedian gross rent 0 0022 00078 NoAverage housing value 0 0100 0 0141 NoEmployed in prfs/mngr jobs (/o) -0-2467 0 0967 NoOut of labour force-males (%) -0-0628 0.0340 NoOut of labour force-females (%o) -0 0610 0.0443 NoSame county workers (%o) 0 0235 0 0307 NoMedian travel time to work 0 1615 0 0738 NoEstimated moments of theDependent variable

rho-gmm S.E. Included?Mean 3 0466 02577 YesVariance 48 0445 8 0315 YesSpatial Cov(l) 170045 3 5405 YesSpatial Cov(2) 11-8436 2 3545 YesSpatial Cov(3) 8.4629 1 8718 YesSpatial Cov(4) 6 3632 1 4791 YesSpatial Cov(5) 5 7835 1 3146 YesSpatial Cov(6) 3 9162 1.2448 YesJ test (chi-square, 45 d.f.): 40 1099 (p-value = 0.6788).Adjusted R2. 02468.The last column indicates the parameters that are used in the indirectinference criterion.

second takes into account the possibility that local interactions may be weaker acrosstracts that are ethnically very distant, as in equation (15): in practice, only the interceptof the 2(4) term is allowed to vary, but the additional parameter )ODN is estimated separ-ately for 1980 and 1990. The third model specification is similar to the previous one, butconsiders tracts that belong to different Community Areas and different neighbourhoods


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TOPA SOCIAL INTERACTIONS 285

-- -- Resid GMM0 5 0 5%signif.

- - - Resid (X only)0 5% signif.0-4 Rawunempl.* 5%signif.

0-3-

Q002

0.1

0~~~~~~~~~010 C or0-1 ~

-0 210 2 4 6 8 10Physicaldistance(adjacent racts)FIGURE 7

ACF for unemployment rate, 1990-80

according to Hunter (1974). Here a single additional intercept parameter XN CA isestimated.These three model specifications are simulated using a sequence of shocks {a}T=lthat are i.i.d. over time and, more importantly, across space. The final specificationimposes a certain spatial correlation structure on these shocks, to try to mimic the pres-ence of correlated unobservables and see how it may affect the results. In particular, theshocks now follow a SAR(l) structure, with autocorrelation parameter set equal to 0 5.The structural estimates of the baseline model are reported in the first column ofTable 3,32 The i, test of the over-identifying restrictions indicates that one cannot rejectthe null hypothesis of correct specification of the structural model, even at high signifi-cance levels (the p-value is 0.36). More importantly, a joint test of the null that all the Aparameters are equal to zero can be rejected at the 10%level (using the test statistic 4cdescribed in Section 3.1). So the estimation of the structuralmodel supports the hypothesisthat an important portion of the spatial autocorrelation of unemployment observed in thedata can be attributed to local interactions that take place across neighbouring tracts.Tract characteristics alone (operating through the a and y parameters) are not sufficientto fit the observed spatial distribution of unemployment.The signs of the a parameter estimates are quite intuitive. The probability of anupward transition in the employment rate depends positively on the education level in thetract, and negatively on the fraction of non-whites and on the crime rate in the tract.32. The auxiliary parametersp based on simulations performed at the optimal value nHare very close tothe auxiliary parameters p from the data. These comparisons are reported in a table available from the authorupon request.


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Structuralparameter estimatesParameter Baseline Ethnicities Comm. areas lambda 0 Corr. shocks lambda 0

lambda_0 0-2495 0.2057 0 1966 0 3001(0-0104) (0.0103) (0.0211) (0.0254)lambda_EDN(80) 0-0024(0.0017)lambda_EDN(90) -0 2094(0.0226)lambda_NCA -0 0159(0 0046)lambda_ed -01049 -0 1558 -0 0922 -0 1691(0 0109) (0 0175) (0 0182) (0 0312)lambda_nw 0 0910 0 1101 0 2718 -0-1657(0 0107) (0 0090) (0 0328) (0 0157)lambda_cr -0 2592 -0 3262 -0 3583 -0 3510(0 0257) (0 0175) (0 0165) (0 0470)

alpha_0 0 3919 0 3724 0 3719 0 6505 0 7528 0 2313(0 0170) (0 0139) (0 0225) (0 0158) (0 0352) (0 0064)alpha_ed 0 0778 0-0814 0-0926 0 0737 0 1662 0 1319(0 0046) (0-0068) (0-0095) (0 0054) (0 0274) (0 0174)alpha_nw -0-1248 -0 1017 -0 1234 -0 1109 0-0600 -0 0614(0-0040) (0-0052) (0 0064) (0-0049) (0-0054) (0 0031)alpha_cr -0 0401 -0 0289 -0 0392 -0-2473 -0 2009 -0-0866(0 0048) (0 0016) (0 0029) (0-0096) (0-0238) (0 0063)delta 0 6483 0 6687 0 6784 0 4169 0 5754 0 3128(0 0171) (0 0113) (0 0118) (0 0066) (0 0274) (0-0045)K-test 12 0666 6 0698 10 3123 20 0642 18 4197 18 7334p-value (0 3586) (0 7329) (0 4135) (0 1695) (0 0723) (0 2261)Kc-test(Ho: lambda =0) 7 9976 13 9944 9 7519 0 3137p-value (0 0917) (0 0297) (0.0826) (0 9889)Kc-test(Ho: lambda_EDN= 0) 5 9967p-value (0 0499)Kc-test(Ho: lambda_NCA = 0) 1 7542p-value (0 1853)Standard errors in parentheses.

Furthermore, I can characterizehow the information spillovers vary with tract character-istics: the interaction term X( ) has a positive intercept, is decreasing in the educationlevel and in the crime rate, and is increasing in the percentage of minorities in the tract.Thus the spillover effects are stronger for areas with lower education levels and witha higher percentage of non-white residents. This is consistent with empirical findingsreported by Corcoran, Datcher and Duncan (1980) and by Granovetter (1995). This litera-ture focuses on the nature of informal hiring channels and is based on detailed data onemployees' work history (in particular, how they were hired). The authors find that infor-mal contacts used to acquire jobs are more important for younger workers, low-skilledjobs, less educated workers and minorities. An alternative explanation for the reportedsigns of X,A, and Xed could be related to the choice of physical distance as a proxy forsocial distance, in the structural model. Since social networks of poorer, less educatedagents tend to be more geographically concentrated (see Fischer (1982)), it is quite intuit-ive that the local interaction effect is stronger for tracts with these characteristics, since I


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TOPA SOCIAL INTERACTIONS 287am focusing on the geographic component of social networks. Finally, the sign of X,, mayindicate that local networks and institutional loci of social interactions such as neighbour-hood organizations tend to break down in the presence of high crime rates in the neigh-bourhood, perhaps because crime raises the cost of maintaining local social ties.

Columns 2 and 3 in Table 3 report the parameter estimates for the model specifica-tions that take into account ethnic distance and neighbourhood boundaries. Again, theoverall ic, specification test does not reject the structural model in either case, with veryhigh p-values. The ethnic distance version fits the data particularly well. As for the jointtest of the null hypothesis that the X parameters are all equal to zero, one can reject thenull even more strongly than in the baseline model (the p-values are now 0 03 and 0.08).The estimates of the X and a parameters are similar to the baseline model.The interesting result is that the signs of the additional parameters )EDN and 2?NCAgo in the direction expected according to the theory: the local interaction channel is indeedweaker across tracts with very different ethnic compositions (at least for 1990), and fortracts that are separated by neighbourhood boundaries. The first result is especially strong,since one can reject he nullhypothesis hat XODN = 0 at the 5% evel.Therefore, there is some evidence that inter-tract influences follow ethnic lines andneighbourhood boundaries that were identified by local residents. Since social networksexhibit strong assortative matching with regardto ethnicity and are thought to be strongerwithin, rather than across neighbourhoods, this evidence lends support to the idea thatthe local spillovers detected here are indeed caused by social interactions rather than byother factors that may generate positive spatial correlations.The last two columns of Table 3 introduce a note of caution. Although the pointestimate of the 2(4) intercept is quite positive and significant, one can no longer reject thenull hypothesis that the )( ) term is identically equal to zero. However, the model withcorrelated shocks fails to pass the overall Kc,specification test, and in general provides aworse fit of the data than the model specifications with i.i.d. shocks. Figures 8 and 9contain the simulated maps of unemployment in the ethnic distance specification withi.i.d. shocks and in the correlated shock specification, respectively: these can be comparedto the actual data, in Figure 3. The signs of the parameter estimates remain unchanged,with the notable exception of )L, which reverses sign: thus the interaction effect wouldbe weaker for tracts with higher fractions of minorities. It may be that the degree ofautocorrelation introduced in the shocks used to simulate the model is too high and simplywashes out everything else. This issue certainly warrants further examination.In Table 4, I use the parameter estimates of the structural model to perform severalthought experiments, aimed at evaluating the magnitude of the spillovers generated bylocal interactions. I consider different types of tracts: the first is an artificial average tract,in the sense that it has the city-wide averages of unemployment, education, presence ofnon-whites, and crime. The other four tracts are real ones, taken from four differentneighbourhoods:33 Woodlawn and Grand Boulevard are characterizedby high unemploy-ment, high poverty rates, relatively low education levels and relatively high crime rates,and are mostly non-white. On the other hand, Brighton Park and Lake View have lowunemployment, low poverty rates, low crime, high education levels, and are mostly white.The contrast between Grand Boulevard and Lake View is especially strong, whereas thedifferences between Woodlawn and Brighton Park are not as marked.34

33. Table5 presents ome summary tatistics or the four tractsusedin theexperiments.34. Thelatterpairof locationswaschosenbecause heyhavenon-empty ets of ethnicallydistantneigh-bours(EDNi),andneighbours ot in the sameCA/HunterneighbourhoodNCAi).


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288 REVIEW OF ECONOMIC STUDIES

Unemnploymentate, 0-80:SIM EDN) 2 0 2 4 MilesLii-66.7--1.93.2-7.87.8-72.9

FIGURE 8Mapof simulated nempl.rate,1990-80(EDN)

The firstthoughtexperiments to raisethe information lowI,t available o residentsof tract i by one standarddeviation,and calculate ts expected mpacton the unemploy-ment ratein the tract.Forthetwo pairsof actual racts,I alsocalculate heeffect of givingeachtract the levelof information hat it would receive f it had the sameneighboursasthe other tract in the pair.The other threeexperiments onsist of raisingthe educationlevel in the tract,decreasinghe proportionof non-whites,and decreasinghe crimerateby one standarddeviation.For the baselinemodel,raising he informationavailable o an average ractby onestandarddeviation(about 8 percentagepoints in 1980,about 12 points in 1990)bringsabout a decrease n expectedunemployment f 0.63 and 126 percentagepoints,, espect-ively.This is a rathersmalleffect,but it is roughlyof the samemagnitudeas theeffectof


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Unemnploymentate, 0-80:SIM COR) 2 0 2 4 Miles

-3-2-7-87-8-72-9FiGuRE 9

Map of simulated nempl.rate,1990-80(COR)raisingeducation n the tractby one standarddeviation.Thelargestchanges n unemploy-ment ratesare thoseimpliedby a 1 s.d. change n thepercentage f non-whitesandin thecrimerate. Theimpactof crime n particulars verystable acrossall modelspecifications.The informationspillover grows larger as one looks at the two poorer tracts,especially he one in Grand Boulevard.Here,the decrease n expectedunemploymentsroughlyone percentagepoint in 1980 and one and a half percentagepointsin 1990. Thespillover s stronger n GrandBoulevardbecausethis tract is characterized y lower edu-cation levels and a higherfractionof non-whites han the averagetract.3 Te fact thatthe local interaction ermX(~)dependson these characteristics lsoimpliesan asymmetryin the spillovereffectsbetween Grand Boulevardand LakeView: for a givenchangein

35. Although hehigher hanaverage rimerate nthe tract endsto reduce hemagnitude f thespiHlover.


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Magnitude of spilloversBaseline Ethnicities Comm. areas Corr. shocks

Experiment 1980 1990 1980 1990 1980 1990 1980 1990Average tract:Raise Info. by 1 s.d. 0 63 1 26 045 1-14 0 68 1 38 0 11 0 39Raise Educ. by 1 s.d. 0 81 0 80 048 047 1 30 1 27 1 28 1 29Lower nw by 1 s.d. 469 422 361 317 121 100 385 360Lower Crime by 1 s.d. 2 74 2 33 3 65 3 00 3 34 2 85 407 3 54Grand Boulevard:Raise Info. by 1 s.d. 0.91 1.53 0 66 1 27 1.11 1 84 -0 24 -0 05Same Info. as Lake View 2 86 4 77 2.10 3 99 3 68 5 81 -0 79 -0 17Raise Educ. by 1 s.d. 0 86 0 89 049 0 62 1 28 1 28 1.50 1 44Lower nw by 1 s.d. 5 05 444 3 74 3 29 1 14 1 40 4.68 3 58Lower Crime by 1 s.d. 3 08 227 3 86 2 68 3 34 2 55 493 3 55Lake View:Lower Info. by 1 s.d. -0 57 -0 77 -0 36 -0 56 -0 57 -0 69 -0 15 -0 41Same Info. as Grand Blvd. -1 89 -2 68 -1 17 -1 91 -2 17 -2 58 -0 48 -1 39Raise Educ. by 1 s.d. 0 85 0 84 0 50 0 56 1.39 1.39 1 29 1 18Lower nw by 1 s.d. 4 85 4 15 3 81 3 30 1 30 1.39 3 83 295Lower Crime by 1 s.d. 2 78 205 3 86 2-82 3 54 2 85 406 293Woodlawn:Raise Info. by 1 s.d. 078 1 27 0.59 1 37 091 1.42 -0 11 024Same Info. as Brghtn. Pk. 0 61 1.19 0 45 2 55 0 61 1 29 -0 08 0 22Raise Educ. by 1 s.d. 072 067 035 041 111 098 128 110Lower nw by 1 s.d. 442 348 324 276 085 099 430 288Lower Crime by 1 s.d. 2 85 1 88 3 64 2 62 3 01 2 03 4 50 2 85Brighton Park:Lower Info. by 1 s.d. -0 51 -1 21 -0 30 -1 22 -0 42 -1 27 -0 25 -0 64Same Info. as Woodlawn -0 39 -1 13 -0 22 -2 70 -0 21 -1 11 -0 19 -0 60Raise Educ. by 1 s.d. 0 89 0 83 0.59 0.51 1.51 1 37 1 26 1.24Lower nw by 1 s.d. 485 423 389 331 1 52 1.09 3 51 333Lower Crime by 1 s.d. 2 58 221 3 58 3 06 3.76 3 06 3 74 3 28Each cell reports the expected change in employment, in percentage points, following each experiment.

information, the size of the spillover effect for Lake View is roughly half the size of theeffect for Grand Boulevard (in absolute value). Thus a tract in Lake View would notsuffer from a decrease in information levels as much as a tract in Grand Boulevard wouldgain from an increase in information of the same size.Finally, giving the tract in Grand Boulevard the same information as the tract inLake View has a large effect on unemployment: the expected unemployment rate woulddecreaseby almost three percentage points in 1980, and by more than four points in 1990.Again, the reverse experiment on Lake View would raise unemployment by a smalleramount, because of the different tract characteristics that affect the strength of the localinteractions.The magnitudes of these effects remain roughly unchanged across the differentmodel specifications with i.i.d. shocks: the size of the local interaction spillover islargest under the specification with neighbourhood boundaries (except for the Wood-lawn/Brighton Park exchange). With correlated shocks, the spillover effect is greatlyreduced for the average tract, and reverses sign in some cases, especially those regardingGrand Boulevard. This counter-intuitive result may be due to the sign of 2,. under thisspecification.


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TABLE5

Summary

statisticsfor

selectedCensustracts

1980

1990

GrandBlvd.

LakeView

Woodlawn

BrightonPk.

GrandBlvd.

LakeView

Woodlawn

BrightonPk.

Tract3801

Tract612

Tract4202

Tract5801

Tract3801

Tract612

Tract4202

Tract5801

Unemploymentrate

2582

342

2313

5-1

35-46

5-69

3318

931

Median

household

income

441

1727

551

178

287

2204

752

1705

Poor

households(%o)

5409

1523

48-11

127

6888

839

4306

1429

Householdson

welfare(%)

19-43

076

1408

0-72

1588

005

793

0-23

High-schoolgrad.ormore(%)

3044

6558

4668

5253

4146

88-46

5503

51-85

Collegegrad.ormore(%)

283

2706

67

7-36

3-27

5812

1783

549

Non-white

persons(%)

9961

3423

9152

11.8

99

1009

8789

2921

Crimeindex

44-2

183

31

196

405

7-2

129

13-1

Seethe

Appendixfora

definitionofallthe

variablesandunits.


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292 REVIEW OF ECONOMIC STUDIES6. CONCLUSION

This paper analyses a model of local social interactions in the context of urban unemploy-ment, and provides empirical estimates of the size of the spillovers generated by localinteractions. The underlying idea is that agents exchange information within their socialnetworks about job opportunities. There exists a local positive feedback in that agents aremore likely to transmit useful information about jobs if they themselves are employed.This feedback generates positive spatial correlations of unemployment.Insofar as the dimensions along which social networks are constructed are at leastpartially observed by the econometrician, the information exchange process generatesobservable implications that can be used to detect the existence of local interactions. Inthis paper, I argue that physical distance is an important determinant of social ties, anduse the observed spatial patterns in unemployment according to

Social Interactions Local Spillovers & Unemployment (Topa 2001)

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Transcript of Social Interactions Local Spillovers & Unemployment (Topa 2001)