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Social Interactions

Steven N. Durlauf1 and Yannis M. Ioannides2

1Department of Economics, University of Wisconsin,

Madison, Wisconsin 53706; email: [email protected]

2Department of Economics, Tufts University, Medford,

Massachusetts 02155; email: [email protected]

Annu. Rev. Econ. 2010. 2:451–78

First published online as a Review in Advance on

April 6, 2010

The Annual Review of Economics is online at

econ.annualreviews.org

This article’s doi:

10.1146/annurev.economics.050708.143312

Copyright © 2010 by Annual Reviews.

All rights reserved

1941-1383/10/0904-0451$20.00

Key Words

neighborhood effects, statistical mechanics, social interactions

empirics, endogenous social effects, estimation and identification,

social networks

Abstract

The study of social interactions has enriched both the domain of

inquiry of economists and the way economists conceptualize indi-

vidual decision making. The review aims to introduce the classes of

models that accommodate estimation of social interactions and to

examine the key areas where significant advances have been made

in the identification of social effects. It surveys linear and nonlinear

models and their applications, including results regarding partial

identification. The review also examines conceptual and methodo-

logical links with the spatial econometrics and the social networks

literatures. It considers empirical applications in the context of our

methodological overview.

451

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FurtherANNUALREVIEWS

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“Most people are other people. Their thoughts are some one else’s opinions,

their lives a mimicry, their passions a quotation.”

Oscar Wilde, de Profundis

1. INTRODUCTION

Within economics, the study of social interactions has expanded the domain of inquiry to

incorporate many ideas that are traditionally associated with sociology. Social interactions

analysis also extends the methodological individualism of economics in new directions

through its focus on the feedbacks between individual behaviors and aggregate outcomes.

By social interactions, we refer to interdependences among individuals in which the prefer-

ences, beliefs, and constraints faced by one person are directly influenced by the character-

istics and choices of others. We emphasize the word directly as these interactions do not

occur because of interdependences due to prices, as occurs in an Arrow-Debreu world.

In fact, social interactions often represent externalities. Canonical social interactions

examples include conformity effects, which occur when the utility from a given behavior

increases when others make the same choice, and social networks effects in which

information diffuses via direct contacts. Social interactions have been used to help explain

phenomena ranging from cigarette smoking to the persistence of ghettos and inner-city

poverty.1

Although social interactions models take sociological ideas seriously, they fully pre-

serve the purposeful, choice-based formulation of individual decision making that is the

hallmark of modern economics. These models simply expand the domain of factors

that determine individual decisions. Furthermore, it is also no surprise that a precursor

to the modern literature is Becker (1974). Moreover, social interactions models have

close parallels in game theory, especially quantal response equilibria (McKelvey &

Palfrey 1995).

Although still relatively new, the social interactions literature is sufficiently large and

wide-ranging that a complete survey is beyond the scope of an article of this length (for

related surveys in economics, see Manski 2000, Durlauf 2004, Ioannides & Loury 2004).

We therefore focus on providing a template for understanding the broader literature.

We outline a general theoretical framework for social interactions and then describe the

econometric implementation of social interactions ideas, focusing on identification issues.

We also describe some links between the social networks and spatial econometrics litera-

tures and social interactions.

Section 2 outlines a general discrete choice model of social interactions that illustrates

many of the interesting theoretical properties that social effects have been shown to gener-

ate. Section 3 describes the baseline econometric models that have been used in the study of

social interactions and focuses on what we call classical identification problems involving

the disentanglement of endogenous and exogenous social effects. Some prominent empiri-

cal studies are discussed. Section 4 describes contemporary identification problems, which

include, in particular, environments with self-selection and unobserved group variables

being present. Empirical studies that address these problems, including those that use

Social interactions:

direct

interdependences inpreferences,

constraints, and beliefs

of individuals, whichimpose a social

structure on individual

decisions

1Many of the ideas concerning persistent poverty in Wilson (2009) have been formalized and studied in the social

interactions literature.

452 Durlauf � Ioannides

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experimental data, are reviewed. Section 5 discusses spatial and network approaches to

social interactions. Section 6 concludes.

2. A DISCRETE CHOICE MODEL OF SOCIAL INTERACTIONS

A basic model of social interactions, one that captures many of the interesting implications

of integrating social factors into individual behavior, can be developed in the context of

decisions over a discrete set of choices. We generalize the multinomial choice model of

Brock & Durlauf (2002, 2007) following Durlauf (1997) and Ioannides (2006). We model

a population of I individuals, each of whom chooses oi among unordered alternatives from

the choice set S¼ {1; . . . ; L}. Each agent i is associated with a group g(i), which is the subset

of the population whose behaviors and characteristics enter as direct arguments in i’s

decision problem. Our goal is to describe the population’s behavior in light of these

group-level effects.

For i, each choice ‘,‘ 2 S, produces utility Vi,‘, which is the sum of three distinct

components. The first is hi,‘, private deterministic utility. It is private in that it does not

exhibit direct dependence on the choices of others and is deterministic as it is known to the

modeler. In econometric work, this is operationalized by assuming hi,‘ is known up to some

set of estimable parameters. The second component is deterministic social utility and

captures dependence of the overall utility of a given choice on others’ choices. If individual

i chooses ‘ and j chooses s, then i’s utility is Ji,j,‘,s . We assume that agents do not observe

the choices of others, but rather form beliefs on the choices: pej;sji denotes the probability

i assigns to choice s on the part of j. The social utility components Ji; j;‘;spej;sji are additive

across j for a given ‘. A third component is a random individual utility term, ei,‘, assumed

to be independent across i and ‘. Therefore,

Vi;‘ ¼ hi;‘ þXLs¼1

XI

j6¼i

Ji; j;‘;s pej;sji þ ei;‘: ð1Þ

This utility function allows for an explicit characterization of the equilibrium choice

probabilities once the probability distributions for the random utility terms are specified.

We assume that the ei,‘’s are distributed according to the multinomial logit model with

dispersion parameter B and that expectations are rational (i.e., self-consistency of beliefs).

Thus

Probðoi ¼ ‘Þ ¼ pi;‘ ¼exp

�B�hi;‘ þ

PLs¼1

PIj6¼i

Ji; j;‘;spj;s

��

PL‘0¼1

exp

�B�hi;‘0 þ

PLs¼1

PIj 6¼i

Ji; j;‘0;s pj;s

�� ; ‘ 2 S; i ¼ 1; . . . ; I: ð2Þ

Higher B implies lower variance. At one extreme, B ¼ 0 implies all outcomes are equally

likely because the private random utility density is so diffused that the maximum of

the random utility shocks controls the choice. At the other extreme, B ¼ 1 means that

choices are deterministic as the private random utility terms are all equal to zero with

probability 1. Under the Brouwer fixed-point theorem, at least one such fixed point exists

for each of individual i’s choice probabilities, given by Equation 2.

It is common in the social interactions literature to simplify the interaction structure by

restricting social utility so that an individual only cares about the fraction of the population

www.annualreviews.org � Social Interactions 453

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making the same choice he does. This renders the agent indifferent as to who makes the

choices. Under this simplification, the object of interest is not the matrix with elements pi,‘,

the individual choice probabilities, but rather the aggregate choice probabilities,

p‘ ¼ I�1P

ipi;‘. This leads to

pe‘ ¼ p‘ ¼ I�1XI

i¼1

exp�B�hi;‘ þ Ji;‘p‘

�PLs¼1

exp�Bðhi;s þ Ji;sps

� ; ‘ 2 S; ð3Þ

where Ji,‘ is the social utility weight i assigns to the share among the population making

the choice ‘.

A leading case in the theoretical literature focuses on binary choices with the decision

space S ¼ {1,�1}. These models typically treat social utility as a function of the expected

choice of others; that is, i’s expectation of j’s decision is mei; j ¼ pej;1ji � pej;�1ji .

2 Under self-

consistency of beliefs, mei; j ¼ mj, where me

i; j ¼ Eðoj jFiÞ, the mathematical expectation of

oj, conditional on i’s information set. Let m denote the I vector with the mj’s as elements.

Recalling the hyperbolic tangent function, tanhðxÞ � expðxÞ � expð�xÞexpðxÞ þ expð�xÞ,

mi ¼ tanh ½Bhi þ BJim�; i ¼ 1; . . . ; I; ð4Þwhere hi ¼ hi,1 � hi,�1, and Ji ¼ (Ji1; . . . ; Jij; . . . ; JiI) is an I vector of interaction coeffi-

cients Jij.

To understand the properties of the binary choice model, we consider the case in which

all heterogeneity across agents results from random utility; i.e., we assume that hi,‘ and Ji,‘are constant across agents. Under our preference assumptions, we can focus on symmetric

equilibria, i.e., equilibria in which all the expected values are the same. This means that the

expected value of the average choice of others will be the overall population mean m:

m ¼ tanhðBhþ BJmÞ: ð5ÞThe properties of this special case are straightforward to describe. If BJ > 1, and h ¼ 0,

then the function tanhðBhþ BJmÞ is centered at m ¼ 0, and Equation 5 has three roots:

a positive one (upper), m�þ; zero (middle); and a negative one (lower), m�

�, where

m�þ ¼ jm�

� j. If h 6¼ 0 and J > 0, then there exists a threshold H�, which depends on Band J, such that if Bh < H�, Equation 5 has a unique root, which agrees with h in sign.

If Bh > H�, then Equation 5 has three roots: one with the same sign as h and two with

the opposite sign. If J < 0, then there is a unique equilibrium that agrees with h in sign.

The relationship between the number of equilibria and the parameters J, h, and B can

be given some intuition. Holding B constant, it is not surprising that multiple equilibria

emerge when the strength of conformity effects, measured by J, is large relative to the

strength of private incentives, measured by h. The role of the measure of random utility

dispersion B is less obvious. Higher Bmeans lower heterogeneity. The degree of heterogeneity

in turn determines how private and social incentives interact to produce equilibria. With

large heterogeneity, then relatively large fractions of the population will experience draws

such that either ei;‘ � ei;‘

0 or ei;‘

0 � ei;‘ is large. This means in turn that a relatively high

fraction will have their decisions controlled by their idiosyncratic payoffs. But this means

that a relatively small percentage of the population remains that can engage in self-consistent

2This may be seen a special case of Equation 1, where Ji; j;‘;1 ¼ Jij; Ji; j;‘;�1 ¼ �Jij, so thatPj6¼i

Ji; j;‘;spj;sji ¼PjJijmj:


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bunching because of social utility effects. In other words, if enough agents make choices

driven by symmetrically distributed payoff differences, this implicitly delimits the magnitude

of the social utility terms because it delimits the possible support ofm.

Brock & Durlauf (2002, 2007) show that these properties extend to the multinomial

choice model, when hi;‘ ¼ h; 8i; ‘. If BJL

> 1, then at least three equilibria exist, whereas ifBJL

< 1, the equilibrium aggregate choice probabilities are unique. The dependence of the

threshold for multiplicity on the number of choices L occurs because, when the ei,‘’s areindependent across i, the probability that the idiosyncratic draw for one of the possible

choices dominates the agent’s decision is increasing in the number of choices.

Similar properties hold for alternative interaction structures. Blume (1993) conceptual-

izes agents as located on a two-dimensional integer lattice. He assumes that agents interact

locally, which amounts to replacing the social utility component in Equation 1 with

JP

jj�ij¼1Efojg. It is straightforward to see that the mean choice level for the population is

characterized by Equation 5. This example illustrates a general principle about discrete

choice models of social interactions, namely that their qualitative properties are often

independent of the specifics of the interaction structure. This property, called universality

in the physics literature, helps justify the analysis of particular interaction structures

because their properties are not tied to their detailed specification.

Universality provides a segue between social interactions models and statistical mechan-

ics models in physics, in which the property was originally conceptualized. Statistical

mechanics models study the macroscopic properties of a large number of interacting

objects (e.g., atoms). The Brock-Durlauf model is mathematically equivalent to a variant

of the Curie-Weiss model of magnetism, whereas the Blume model is a variant of the Ising

model of magnetism. Another example of the application of statistical mechanics models

to economics is Topa (2001), who employs contact processes to study local interactions in

unemployment. Despite these examples, we believe the mathematics of statistical mechan-

ics has yet to be fully exploited by economists.

One must be careful in translating statistical mechanics models into social science

contexts. One reason is that statistical mechanics models take as primitives the conditional

probabilities that relate objects in a system, whereas socioeconomic contexts require that

these conditional probabilities derive from individual decision problems. The importance

of this distinction is demonstrated by Horst & Scheinkman (2006). In a social interactions

model in which agents are arranged on integer lattices, in which the individual payoff

structure is defined in terms of values for Ji,j,‘,s, the authors show that the equilibrium

conditional probabilities that relate the choices may have a different pattern of zero and

nonzero values, when agents observe one another’s choices. Therefore, the conditional

probability structure for a given statistical mechanics model does not always reveal the

payoff interdependences needed to generate it. Another reason is that exact versus approx-

imate statistical mechanics models can have substantively different interpretations in social

science contexts. Brock & Durlauf (2001a) show that the exact solution to the Curie-Weiss

model without approximation corresponds to a social planner’s problem, whereas the

mean field approximation represents an exact (to social scientists) noncooperative model

of binary choice.

Discrete choice models do provide a tight connection between theory and econometrics,

although this link has not usually been exploited in empirical work. Our canonical social

interactions model treats Ji,j,‘,s as a primitive. One can identify models that produce


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different types of social interactions as equilibrium outcomes (e.g., Bernheim 1994,

Streufert 2000, Akerlof & Kranton 2002). This type of work, however, has not been

directly integrated into the abstract theoretical structures we outline above.

3. ECONOMETRIC MODELS OF SOCIAL INTERACTIONS: CLASSICALIDENTIFICATION ISSUES

In this section we consider the problem of identifying social interactions in various

empirical contexts. This leads to variations of the classical identification problem of

simultaneous equations systems that reflect the specific structure of social interactions

models. The statistical models allow us to study both the identifiability of a role for

some type of social influence on behavior and the identifiability of types of social

influences (for estimation issues, see Krauth 2006, Aradillas-Lopez 2007, Conley &

Topa 2007, de Paula 2009).

The distinction between types of social interactions did not arise in our baseline theo-

retical model of social interactions because effects other than those associated with the Jij’s

are subsumed in the private deterministic utility term, as we were considering individuals

within a common group. Manski (1993) first emphasized the difficulties of distinguishing

types of social interactions and developed an important dichotomy. Endogenous effects

capture social interactions that directly occur between one agent’s decisions and those of

others. An essential feature of this type of interaction is that the choices are simultaneously

determined. A canonical example is peer effects in classroom effort; one student works

hard based on whether others do so (or are expected to do so). Of course, one can equally

well imagine that individuals are affected by the personal characteristics of others. The

educational level of the parents of classmates matters if these levels help form the aspira-

tions of the student (and affects students’ net utility from effort): These are contextual

effects.

In discussing the econometrics of social interactions, we assume that the unobserved

heterogeneity is independent across agents and unpredictable given individual and group

characteristics. We relax these assumptions in the next section.

Our analysis of identification focuses on the two main classes of econometric models:

linear-in-means models and discrete choice models. It will be evident that the connections

between the linear-in-means models and economic theory are tenuous; Brock & Durlauf

(2001c) show that one can reverse engineer a linear-in-means model to represent an agent’s

optimal behavior. Discrete choice models, of course, provide a tighter link between theory

and econometrics, although the empirical literature has typically eschewed fully exploring

these links.

Our discussion of identification considers individual observations across groups, g(i).

Outcomes for the linear-in-means model are denoted as yi, whereas oi denotes outcomes

for discrete choice models; xi denotes individual-level characteristics, and zg(i) denote

group-level characteristics.

3.1. Linear Models of Social Interactions

The linear-in-means model is a standard regression model:

yi ¼ a0 þ axi þ yzgðiÞ þ bmgðiÞ þ ei; ð6Þ


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where mg(i) is the expected average choice in g(i), xi is an r vector of individual effects, zg(i)is an rg vector of contextual effects, and ei is unobserved individual heterogeneity. Applying

an expectations operator to both sides of Equation 6 provides an expression for mg(i) in

terms of observables that produces a reduced form for yi:

yi ¼ a01� b

þ axi þ b1� b

axgðiÞ þ y1� b

zgðiÞ þ ei: ð7Þ

Variants of Equation 6 and especially Equation 7 have been used in many contexts to

study social interactions. Datcher (1982) deserves credit for pioneering the empirical

approach along the lines of Equation 7. Durlauf (2004) provides an overview of the

many studies that have estimated regressions of this type and surveys the kinds of

variables that have been purported to represent evidence of social interactions. Promi-

nent examples include the following. Brooks-Gunn et al. (1993) relate IQ and behav-

ioral problems at 36 months of age, high school dropout rates, and nonmarital fertility.

Aizer & Currie (2004) and Bertrand et al. (2000) argue that individual use of social

services/public assistance is affected by the usage rates of others with whom one inter-

acts, and Weinberg et al. (2004) and Bayer et al. (2008) argue that individual market

outcomes are influenced contemporaneously by the labor or market status of neighbors

(see also Ioannides & Loury 2004 for a review of the literature). Corcoran et al. (1992)

claim that individual labor market outcomes are influenced by growing up in a poor

neighborhood, and Burke et al. (2003) contend that physicians’ following each others’

practices produces geographical variations in medical care practices. Gaviria & Raphael

(2001) explore endogenous peer effects in drinking and smoking among teens. An espe-

cially interesting example of this type of regression is Mas & Moretti (2009), who

employ a data set that measures supermarket employee productivity in 10-min intervals;

their data set is also impressive because the set of peers for a given worker regularly

changes due to differences in shift composition and because the spatial orientation of

workers in a store is known. This allows for analyses of such questions as whether

frequent interactions induce stronger social effects and whether physical proximity to

others matters.

One topic that has received particular attention concerns the effects of school and

classroom peers. For example, Hoxby (2000) examines peer effects in the classroom, and

Hanushek et al. (2003) find that growth in academic achievement is associated with the

average achievement of school peers. Controlling for peers’ achievement, Hoxby &

Weingarth (2006) conclude that peers’ race, ethnicity, income, and parental education have

no or very weak effects, and Henry & Rickman (2007) study peer academic achievement

and learning for preschoolers. Lavy et al. (2008) find that the proportion of low-achieving

peers has a negative effect on the performance of regular students.

Numerous studies have appeared that have failed to find evidence of social interactions.

Among recent studies, Oreopoulos (2003) fails to find evidence that differences in neigh-

borhood quality affect longer-run labor market outcomes, which contrasts with findings

such as Corcoran et al. Employing methods similar to Mas &Moretti, Guryan et al. (2007)

find little evidence of workplace effects in the context of golf tournaments. We do not

attempt to adjudicate the disparate results in the extant literature, which may result from

factors ranging from different socioeconomic contexts to different model specifications,

but merely observe that, taken at face value, the empirical evidence for social interactions is

not decisive.


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The regression literature on social interactions suffers from serious measurement prob-

lems. One difficulty with empirical studies based on Equation 7 (and, for that matter, its

nonlinear analogs we discuss below) is that economic theory does not dictate the appropri-

ate empirical measures of contextual variables that a researcher ought to use. As a result,

one finds Bertrand et al. (2000) using the product of welfare usage and own-ethnic group

intensity to explain individual welfare usage, whereas Aizer & Currie (2004) use the

utilization rate of an individual’s language group to measure social effects on public

prenatal-care utilization. Similarly, the empirical literature does not typically consider

how social variables should interact with individual decisions. If the reason why utilization

of social services depends on the usage of others is because of information transmission, as

argued by Bertrand et al., then it is unclear why the percentage of users is the appropriate

variable, as opposed to some nonlinear transformation, as presumably one only needs one

neighbor to provide the information. While considering this type of problem in studying

social effects in marriage markets, Drewianka (2003) argues that a higher marriage rate in

a community may reduce the propensity of unmarried people to marry as a higher rate

hampers search.

Cooley (2009) analyzes this measurement problem from a different vantage point,

namely the discrepancy between observable measures and causal ones. She argues that,

from the vantage point of theory, causal sources of peer effects between students in a

classroom involve two unobservables, endogenous effort and contextual ability, whereas

work on classroom peer effects typically uses observable classroom achievement as the

measure of outcomes and various ad hoc measures of student characteristics for contex-

tual efforts. Cooley demonstrates that absence of attention to the link between

unobserved causal factors and observable proxies can render reduced-form regressions

uninterpretable.

A distinct measurement problem arises because theory does not provide guidance as to

the appropriate measure of groups. Hence one finds the use of zip codes in Corcoran et al.

(1992) and census tracts and blocks in Weinberg et al. (2004) to determine neighborhoods.

Akerlof (1997) argues that social interactions are best understood as occurring in a social

space that may have many dimensions; this follows naturally when one considers the

overlapping effects of factors such as physical proximity, ethnicity, gender, and education

on the ways in which individuals interact. Conley & Topa (2002) are unusual in seeking

to identify the appropriate axes to situate the actors in social space, finding that ethnicity is

of particular importance in defining social interactions for labor market outcomes in

Chicago.

3.2. The Reflection Problem

A fundamental identification issue for the linear-in-means model arises from the fact

that the model is a variant of a simultaneous equations system in which the expected

choices of one agent appear as arguments in the equations describing the choices of

others. The possibility that this interdependence could cause identification to fail was

first recognized in a fundamental paper by Manski (1993), who dubbed the failure of

identification the reflection problem. To understand the problem, assume that the

within-group averages of the individual characteristics map one to one to the contextual

effects, xg(i) ¼ zg(i), so that both are r vectors. In this case, the reduced form for out-

comes (Equation 7) becomes


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yi ¼ a01� b

þ axi þ baþ y1� b

xgðiÞ þ ei: ð8Þ

In this case, there are 2r þ 1 coefficients corresponding to 2r þ 2 structural parameters.

Identification thus fails. In contrast, a necessary condition for identification is that there

exists at least one individual characteristic whose group-level average is not a contextual

effect, as shown originally in Brock & Durlauf (2001c); this corresponds to the classical

row condition for identification in simultaneous equations systems. Durlauf & Tanaka

(2008) give a sufficient condition for identification, which corresponds to the rank condi-

tion in simultaneous equations. These conditions, of course, require prior knowledge on

the part of the researcher to produce exclusion restrictions and therefore fall prey to the

problem that social interactions models are typically open-ended,3 which means that

the presence of particular group-level determinants of behavior is logically independent of

the presence or absence of various individual-level determinants.

Once one moves away from a linear cross-section structure, the reflection problem may

not arise, even if there is a one-to-one correspondence between individual and contextual

effects. For example, non-linear-in-means variants generically cannot exhibit it. Brock &

Durlauf (2001c) show that the reflection problem does not occur for

yi ¼ a0 þ axi þ yzgðiÞ þ bm�mgðiÞ

�þ ei; ð9Þ

for @2m�mgðiÞ

�=@m2

gðiÞ 6¼ 0, relative to the space of twice differentiable functions, outside of

nongeneric cases. The intuition is straightforward; the reflection problem requires linear

dependence between group outcomes and certain group-level aggregates, which is ruled

out by the nonlinearity in Equation 9. Furthermore, the reflection problem does not arise in

linear models with dynamic forms of interactions, e.g.,

yit ¼ a0 þ axi;t þ yzgðiÞ;t þ bmgðiÞ;t�1 þ ei;t: ð10ÞThese examples in which the reflection problem fails to arise should be understood with

caution. The reflection problem occurs when there is linear dependence between regres-

sors; multicollinearity can still be a problem in models such as Equations 9 and 10, even

if linear independence holds, so that parameter estimates are highly imprecise. Further-

more, the examples given above in which the problem does not arise rely on parametric

modeling assumptions. For cross-section data, Manski (1993) provides a nonidenti-

fication result for nonparametric contexts that parallels the reflection problem in linear

models.

3.3. Variance-Based Approaches

The fact that endogenous social interactions help amplify differences in the average group

behavior has been used as another route for identification. Glaeser et al. (2003) introduce

the concept of a social multiplier to measure this effect. The social multiplier approach can

also be useful in delivering a range of estimates for the endogenous social effect when

individual data are hard to obtain, as in the case of crime data. Glaeser et al. (1996)

motivate their study of crime and social interactions by the extraordinary variation in the

3Brock & Durlauf (2001b) introduce the idea of model open-endedness in the context of economic growth models,

but the idea is implicit in earlier critiques of econometric practice such as Sims (1980).


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incidence of crime across U.S. metropolitan areas over and above differences in fundamen-

tals. They use the idea that the convergence of sample means is slowed down by social

interactions as a basis for assessing whether cross-city differences in crime rates are large,

once one has controlled for some city-wide characteristics. This approach only requires

city-level data. Formally, Glaeser et al. (1996) work with a statistic, the covariance of

per-capita outcomes across agents, a measure of social interactions, that is derived as a

function of the proportion of agents in a city who are either law-abiders or criminals. This

covariance generates overdispersion in cross-city crime rates relative to the absence of

social interactions. They show that the estimated index is highest for petty crimes but

declines for more serious crimes to become negligible for the most serious ones. Across

cities, the implied level of interactions is roughly constant. Because the statistic is related to

the expected size of the interacting group of criminals, their results suggest that more

serious crimes are associated with smaller group interactions.

Glaeser et al. (2003) extend the idea of variance comparisons to formalize the idea of a

social multiplier to measure how endogenous social interactions amplify private incentives.

They compute the social multiplier by comparing the coefficient vector produced by a

within-group regression across individuals,

yig ¼ kþ pxi þ �ig; ð11Þwith the coefficient produced by a regression across group-level averages,

�yg ¼ k0 þ p0�xg þ ��g: ð12Þ

Social multipliers are defined as the ratios of the coefficients in p0 to their counterparts

in p.4 This approach, of course, requires the existence of at least one individual effect, xi.

The social multiplier approach has also proven useful in uncovering b, even when no

individual or contextual effects are present. We return to this approach in the next section.5

3.4. Discrete Choice Models of Social Interactions

Discrete choice models of social interactions have been applied to a wide range of contexts,

although the model is used more frequently than its linear-in-mean analog. Early examples

include Crane (1991) and Evans et al. (1992) on teenage behaviors; more recent examples

range from cigarette smoking (Krauth 2005, Nakajima 2007) to medical practices (Burke

et al. 2003). As in the case of linear models, one finds conflicting results in the literature.

For example, Crane (focusing on residential neighborhoods) finds strong social effects,

whereas Evans et al. (focusing on schools) do not; similarly, Nakajima concludes that peer

effects play a larger role in smoking than does Krauth.

The econometrics of discrete choice models with social interactions is studied by

Brock & Durlauf (2001a,c, 2006, 2007). These papers show that the reflection problem

does not, under relatively weak assumptions, apply in multinomial choice models when the

analyst knows the distribution of the unobserved error terms; the standard assumption in

4For example, suppose that the true data-generating process is yig ¼ a0 þ axi þP

j6¼iyij þ �ig, where a is scalar. Then,one can show that p0

p ¼ 1ð1�bÞð1þsbÞ, where s ¼ Varð�xgÞ

VarðxiÞ :5Although the literature on the social multiplier has so far rested on linear models in static settings and emphasizes

measuring the strength of social interactions, the concept may be extended to dynamic and nonlinear settings

(Brock & Durlauf 2001a), to complete versus incomplete information (Bisin et al. 2006), and to more complex

interaction topologies (Ioannides 2006).


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theoretical models that the errors are logit is inessential. Intuitively, identification holds so

long as the individual and contextual effects exhibit sufficient variation so as to imply that

choice probabilities are nonlinearly related to them. Further, Brock & Durlauf show that

for the binary choice case identification holds even if the error distribution is unknown,

following Manski (1988).

A number of variations of our baseline discrete choice model have been studied.

Soetevent & Kooreman (2007) use Brock & Durlauf (2001a) as a building block with an

inessential conceptual difference. Soetevent & Kooreman’s model involves interactions in

relatively small groups of given sizes in which choices of other individuals are assumed to

be fully observed, and therefore an individual’s payoff depends on the actual choice of

others in her group, as opposed to expected choices as in Brock & Durlauf. Like Brock &

Durlauf, Soetevent & Kooreman focus on pure Nash equilibria with binary outcomes,

oi ¼ 1,�1, and estimate the model in effect as a system of simultaneous equations by

means of simulation methods.6

Their empirical application examines the individual behavior of high school teenagers

in almost 500 school classes from 70 different schools using data from the National School

Youth Survey of the Netherlands. In their baseline model, endogenous social interactions

effects are strong for behavior closely related to school (truancy), somewhat weaker for

behavior partly related to school (smoking, cell-phone ownership, and moped ownership),

and absent for behavior far away from school (asking parents’ permission for purchases).

Intragender interactions are generally much stronger than cross-gender interactions. When

school-specific fixed effects are allowed, social interactions effects are insignificant, with

the exception of intragender interactions for truancy. This work is also significant because

its simulation-based estimation method allows the authors to account for the potential

multiplicity of noncooperative Nash equilibria and the identification problems it poses.

Another interesting variation of the discrete choice model is developed in Aradillas-

Lopez (2008, 2009), who considers the question of identification of social effects when

agents’ expectations are not necessarily correct. Aradillas-Lopez (2008) focuses on the

empirical implications of social interactions when beliefs are only required to be consistent

with the iterated elimination of dominant strategies. Probability bounds are established for

social interactions parameters that depend on the number of iterations. Aradillas-Lopez

(2009) extends this reasoning to consider environments in which agents’ behavior is

required to obey a criterion called weakly consistent equilibrium. This imposes that the

choices of other agents always lie in the support over which an agent forms beliefs about

these behaviors. These approaches provide a link between the social interactions literature

and developments in behavioral economics.

In terms of empirical work, the discrete choice model has been extended to consider

dynamic contexts and in turn has been applied to duration and optimal stopping problems.

Sirakaya (2006) uses a national U.S. sample to identify the risk factors for recidivism

among female, male, black, white, and Hispanic felony probationers. She assumes that

the individual hazard function for time to recidivism depends on various individual and

neighborhood characteristics, as well as social interactions among probationers. Because a

probationer is required to live in the jurisdiction that passed the probation sentence, the

neighborhood of probationer i, g(i), is assumed to be the jurisdiction. Sirakaya’s results

6With many agents, the two formulations are equivalent, but the finite-agent case raises issues of identification that

are common in models of systems of simultaneous outcomes (see Tamer 2003).


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point to social interactions as one of the most significant factors affecting recidivism within

all gender, ethnicity, and race groups, with unobserved neighborhood-level heterogeneity

being negligible.

A number of other studies are closely related to Sirakaya’s methodological frame-

work. Irwin & Bockstael (2002) argue that the development of adjacent parcels of land

in exurbia may confer social benefits, if proximity is desired, but also costs, if conges-

tion and environmental degradation arise; their empirical work finds the second effect is

dominant. They estimate net social interactions effects in exurban land-use development

that are negative. A second example is Costa & Kahn (2007), who use a longitudinal

data set of Union Army soldiers and a cross-sectional data set of the population of

Andersonville to examine the role of social networks in ensuring the survival of Union

Army soldiers in captivity. They estimate hazard functions for individual survival prob-

abilities as functions of the number of friends, individual characteristics, and camp

conditions and use the cross-sectional data to estimate a probit model of the probability

of survival as a function of the number of friends and of demographic characteristics.

They find that friends had a statistically significant positive effect on survival probabil-

ities and that the closer the ties were between friends as measured by such identifiers as

ethnicity, kinship, and the same hometown, the bigger the impact of friends on survival

probabilities was. De Paula (2009) uses the same data and finds evidence of bunching in

desertions that is consistent with social interactions. His analysis is based on a sophisti-

cated optimal stopping problem with endogenous social interactions whose econometric

properties are carefully examined.

4. IDENTIFICATION: CONTEMPORARY PERSPECTIVES

In our discussion of social interactions, a best-case assumption on the error structure is

assumed, namely that individual unobservables are exchangeable across individuals and

groups. Exchangeability in turn is operationalized as independence. Much of modern

econometric practice, of course, has developed in a systematic effort to avoid such assump-

tions (see Heckman 2001). In this section, we explore the implications of alternative error

structures, due to self-selection and group-level unobservables.

4.1. Identification of Social Interactions with Self-Selection toGroups and Sorting

For many contexts, economic agents choose their group memberships, and these choices

are influenced by social interactions effects. Endogenous group formation naturally affects

the distribution of the unobserved heterogeneity within groups, which depends on both

endogenous and contextual effects, as well as individual specific characteristics, i.e., in

Equation 6, Efei jxi; zgðiÞ;mgðiÞ; i 2 gðiÞg 6¼ 0, the standard self-selection problem. Because

self-selection into a group means that each member of the group has been influenced by a

common set of social factors, this leads to dependence in the conditional expectations

across individuals, which Manski (1993) terms correlated effects.

The social interactions literature has followed the broader econometric literature’s

strategies to address self-selection. A number of studies have employed instrumental vari-

ables methods (see Evans et al. 1992). Instrumental variable approaches are problematic

due to theory open-endedness, which is discussed above. Social interactions models do not


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naturally generate valid instruments; rather instruments are produced via intuition. This

contrasts with contexts such as Euler equation estimation, in which the theory, which

implies combinations of forecast errors, itself generates instruments for the analyst.

A second strategy follows the control function approach pioneered by Heckman (1979).

From this vantage point, self-selection is addressed by accounting for the behaviors that

generate it. Following Heckman, suppose that evaluation of the attractiveness of a group g

may be expressed in terms of an unobservable latent quality variable Q�i;gðiÞ. Individual i

evaluates group g by means of observable attributes Wi,g(i), which enter with weights z andunobservable component #i,g(i),

Q�i;gðiÞ ¼ zWi;gðiÞ þ #i;gðiÞ; ð13Þ

and chooses the group with the highest Q�i;gðiÞ. Random shocks ei, from Equation 6 and

#i,g(i) above, are assumed to have zero means, conditional on (orthogonal to) regressors

[xi, zg(i),Wi,g(i)], across the population. Under various parametric and semiparametric

assumptions on the joint distribution of�ei; #i;gðiÞ

�; d�z;Wi;gðiÞ;Wi;�gðiÞ

�, an object that con-

verges to a value proportional to Efei j xi; zgðiÞ; i 2 gðiÞg can be constructed and used as a

regressor in Equation 6, now corrected for selection bias:

yi ¼ a0 þ axi þ yzgðiÞ þ bmgðiÞ þ kd�z;Wi;gðiÞ;Wi;�gðiÞ

�þ xi: ð14Þxi is now uncorrelated with the other regressors. Note that the selection correction term

depends on the attributes of all groups in the opportunity set, not only the one that is

chosen (see Brock & Durlauf 2001c, 2006 for further discussion).

Equation 14 integrates information about the group selection process into the behav-

ioral equation for outcomes and in a way that provides additional information on social

interactions. The additional regressor d�z;Wi;gðiÞ;Wi;�gðiÞ

�affects the possibility of the

reflection problem’s arising. To see why, consider two possibilities. First, suppose that

groups are chosen according to the individuals’ characteristics xi and the group’s charac-

teristics zg(i). The additional regressor will then take the form d�z; xi; zgðiÞ; z�gðiÞ

�, which is

an individual specific random variable (its group-level average does not appear in the

equation). Second, suppose that groups are chosen according to expected average behavior

in the group. In this case d will functionally depend on mg(i). Because d is almost always a

nonlinear function, its introduction transforms the linear-in-means model to a nonlinear

one. Both these changes are sufficient to produce identification, even if the model were not

identified under random assignment. The ability of self-selection to facilitate identification

is not surprising. Group choice naturally contains information about the determinants of

the choice. What is perhaps surprising is that randomized group assignment would work to

the detriment of identification.

Ioannides & Zabel (2008) is the first empirical study to use self-selection to facilitate

identification of social interactions. Studying housing demand, Ioannides & Zabel merge

the neighborhood cluster subsample with confidential information from the National

American Housing Survey. Their estimation of the demand equation for housing structure

allows for contextual effects at the level of the immediate neighborhood and at the census

tract of residence, and for endogenous social effects, defined as the mean housing demand

among immediate neighbors. Their estimation results confirm that the endogenous neigh-

borhood effect is significant and stronger after controlling for the endogeneity of neighbor-

hood choice.


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Self-selection also suggests that evidence of social interactions may derive from the

prices for group membership. For example, social effects of residential neighborhoods will

be reflected in housing prices, via standard hedonic price arguments. Not surprisingly, in

view of Brock & Durlauf’s (2001c) results on the ability of self-selection to facilitate

identification, Nesheim (2002) shows that preferences and the technology of social inter-

actions, measured as children’s schooling as a function of the mean parental education in

the neighborhood and of parental characteristics, may be identified. Ioannides (2008)

shows how certain functional forms produce a closed-form solution for the neighborhood

hedonic price in terms of the mean parental education, which should facilitate empirical

implementation.7

A related strategy for integrating group choice with social interactions involves estima-

tion of equilibrium models of neighborhood formation in which social interactions play a

role in locational decisions. Bayer et al. (2007) employ a discrete choice model to estimate

preferences over neighborhood composition with respect to housing choices. Calabrese

et al. (2006) estimate a sophisticated structural model that explicitly addresses the political

economy of local educational expenditure in which neighborhood composition is shown to

affect neighborhood choice.

One general feature of the various approaches to sorting and social interactions is that

they assume that once groups are formed, agents in a common group experience similar

interactions. Weinberg (2007) makes an original argument that interactions may occur at a

lower level of aggregation than the level at which groups are formed. For example, within a

classroom, friendships may occur among students with similar characteristics. Weinberg

provides empirical evidence that this type of phenomena occurs. He further makes the

argument that the capacity of intragroup sorting into subgroups will depend on group size,

as larger groups will have more opportunities for individuals to find similar types with

which to form bonds. Social interactions in social networks, which we discuss further

below, carry this notion further.

4.2. Group-Level Unobservables

The second basic problem facing econometric analyses of social interactions is the pres-

ence of unobserved group effects. In a classroom, for example, differences in teacher

quality will simultaneously affect all students in a classroom. From the perspective of our

basic social interactions models, these may be understood as replacing the initial error

assumptions in Equation 6 with ag(i) þ ei, where ag(i) is a common group-specific shock.

Of course, one can imagine more elaborate changes in the error structure; these have not

been much studied.

In our view, unobserved group effects represent the most difficult hurdle to the con-

struction of persuasive evidence of social interactions because, unlike self-selection, there is

typically no economic reasoning to facilitate modeling the influences. There is thus no

natural solution to group effects analogous to the joint modeling of group formation and

behaviors within groups. The methods that exist for overcoming their presence are thus

essentially statistical.

7Bayer et al. (2007) report some nonstructural hedonic regressions of housing prices on neighborhood characteristics

[see also Bayer & Ross (2009) who use neighborhood prices to construct a control function to proxy for unobserved

neighborhood characteristics].


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One approach uses panels to difference out the group effects; Hoxby (2000) is a good

example. Intuitively, variation in zg(i) over time induces variation in mg(i) and therefore

produces identification in the differenced model. For Hoxby’s model, zg(i) is the percentage

of a student’s own ethnic group in a classroom. Brock & Durlauf (2007) show that this

argument generalizes to panels with discrete data, using ideas in Chamberlain (1984). This

justifies the approach taken by Arcidiacono & Nicholson (2005) in evaluating the effects

of peers on specialty choice in medical school. Differencing, of course, assumes that ag(i) isitself not time varying. Furthermore, the approach treats the individual fixed effects

as nuisance parameters. Arcidiacono et al. (2007) consider a linear model in which the

spillover effects operate through the fixed effects of a student’s peers, making fixed effects

for one agent influence other agents. They provide conditions under which these effects can

be exploited in a panel to facilitate identification of social interactions.

Differencing becomes problematic for models with nonadditive error structures. Cooley

(2008) shows how social interactions may be identified for a class of models with nonad-

ditive group effects, exploiting results in Imbens & Newey (2009). Cooley applies the

strategy to school data in North Carolina, finding strong evidence of intraethnic group

social interactions in achievement. Cooley’s approach does require the use of instru-

mental variables, which, of course, constitutes a second general strategy for dealing with

unobserved group effects, albeit one that we argue above can be hard to justify because of

theory open-endedness. Given that unobserved group effects are undertheorized, open-

endedness is especially hard to overcome for this case. However, as the careful discussion

in Cooley shows, it is not impossible given a specific context.

4.2.1. Variance-based approaches to uncovering social interactions with unobserved

group effects. An alternative strategy for uncovering social interactions in the presence

of unobserved group effects is based on the variance-covariance structure of outcomes.

Graham (2008) extends earlier variance comparisons such as Glaeser et al. (1996), who do

not allow for unobserved group effects. Following Graham, we rule out individual contex-

tual effects and assume that behaviors depend on the realized rather than expected choice

of others, producing the model

yi;gðiÞ ¼ kþ b�ygðiÞ þ ei;gðiÞ ¼ kþ bmgðiÞ þ b½�ygðiÞ �mgðiÞ� þ ei;gðiÞ: ð15Þ

Obviously, relative to Equation 15, no instruments exist formg(i). However, Graham shows

that identification can still be achieved. He does this by employing an approach that echoes

the classical literature on identification in simultaneous equations models, in which the

structure of the variance-covariance matrix of the reduced-form system augments identifi-

cation via exclusion restrictions. The key to Graham’s method is the assumption that

unobserved group effects are random rather than fixed, which imposes structure on the

variance-covariance matrix of outcomes. The random-effects assumption can be unappeal-

ing in contexts where groups are endogenously formed.

To see how a random-effects assumption on group unobservables facilitates identifica-

tion, decompose the residuals in Equation 15 as

ei;gðiÞ ¼ agðiÞ þ �i;gðiÞ; ð16Þwhere ag(i) is a group-level unobservable, and �i,g(i) is an individual-level unobservable;

each has mean zero and constant variance across i and g, s2a and s2� , respectively; we assume


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the two variables are uncorrelated with each other. Let the variance-covariance matrix of

the ei,g(i)’s be Vn. These assumptions imply that the variance of average outcomes in groups

of size n equals

Var½�yg� ¼ In � bn1n

� ��1

Vn In � bn1n

� ��1

: ð17Þ

This expression indicates how the parameters b; s2a ; and s2� can be recovered from size-

specific group variances; e.g., calculating Equation 17 for different n’s reveals these param-

eters. Graham (2008) and Durlauf & Tanaka (2008) discuss ways to modify Graham’s

baseline assumptions and preserve identification. An especially valuable study is Davezies

et al. (2006), which provides a systematic discussion of the use of the variance-covariance

matrix of residuals for the linear-in-means model to facilitate identification; it also pro-

vides results for binary choice models. A nice feature of their work is the clarification of

when homoskedasticity assumptions are needed.

Graham applies his methods to data from Project STAR, a class-size reduction experi-

ment in Tennessee. In that experiment, kindergarten students and teachers were randomly

assigned to large and small classrooms. Random assignment of teachers ensures that the

distribution of their characteristics is similar across the two types of classrooms, and

Equation 17 may be exploited. Graham (2008) reports differences in peer-group quality

that constitute evidence of social interactions.

4.2.2. Variance components and social interactions. The use of variance information has

been exploited in a different way by Solon et al. (2000). Their analysis focuses on the case

in which one cannot observe either family-specific or group-specific effects. Solon et al.

propose a variance decomposition of individual outcomes to bound the contribution of

group effects. One can formulate their analysis in terms of a variance components model

(Searle et al. 2006):

yi;f;n ¼ mf þ nn þ of;n þ ei;f;n; ð18Þwhere mf denotes a family effect, nn denotes a group effect, of,n denotes an interactive

effect between family and group, and ei,f,n denotes an idiosyncratic effect. Assuming

that these components are orthogonal (and such a decomposition always exists), one

can calculate the variance contribution of nn to the overall variance of yi,f,n. Solon et al.

do not use this decomposition directly; instead they use sibling versus neighbor covari-

ances to bound the variance of nn, finding little evidence that neighborhoods affect

education.

One limitation of this approach is that it reduces the vector of social interactions to a

scalar so that one cannot tell which social factors matter. In fact, it is possible for different

social factors to cancel each other out. Another limitation is that the presence of of,n can

make interpretations problematic, although Solon et al. argue that one can assume the term

is positive due to sorting across neighborhoods. For these reasons, Oreopoulos (2003) is

interesting. This paper uses data from Toronto that allow one to trace the adult labor

market outcomes of children who grew up in different public housing projects; these pro-

jects differ in terms of neighborhood characteristics such as crime. Random assignment

across public housing projects eliminates correlations between individual and neigh-

borhood characteristics that cloud the variance decomposition (Equation 18). That said,


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Oreopoulos also finds that social interactions fail to play a large role in explaining differ-

ences in adult economic outcomes.

4.2.3. Partial identification with nonlinear models. A final approach to unobserved group

effects is due to Brock & Durlauf (2007), who develop partial identification results for

binary choice models under what they argue are weak assumptions about the distribution

of the unobserved group effects. This work exploits a fundamental difference between

endogenous effects and unobserved group effects: Only endogenous effects can produce

multiple equilibria. Hence, if evidence of multiple equilibria can be generated, it represents

evidence of endogenous social interactions.

Brock & Durlauf operationalize this idea in several ways. First, they consider the use of

pattern reversals in group-level outcomes. A pattern reversal occurs when the rank order of

average outcomes between two groups is the reverse of what one would predict given the

observed individual and contextual effects for the groups. Brock & Durlauf demonstrate

that, under various shape restrictions on the probability density of the unobservables,

pattern reversals can only occur because of multiple equilibria and hence endogenous

social interactions. For example, if the distribution of unobservables shifts monotonically

in observables, then pattern reversals cannot occur without social interactions. A second

approach involves the bimodality of linear combinations of contextual effects. Here

Brock & Durlauf show that, conditioning on a given average outcome, the cross-section

distribution of certain linear combinations of contextual variables must be unimodal.

This last result corrects a misconception that multiple equilibria induce multimodality of

average outcomes conditional on contextual effects, whereas multiple equilibria do induce

a mixture density of outcomes; not all mixtures are multimodal.

Brock & Durlauf (2010) extend this perspective to the study of social effects in technol-

ogy adoption. In this analysis, individuals are motivated to adopt a new technology by

both individual observable and unobservable characteristics (which may be correlated

across individuals) as well as by the percentage of the population that has adopted, which

defines the social interactions effect. They provide conditions under which a higher adop-

tion rate by individuals whose observable fundamentals predict a lower adoption rate than

another group can only occur because of social effects. This pattern reversal does not occur

because of multiple equilibria but rather because of the potential for social effects to lead

agents with different private incentives to bunch and adopt simultaneously. As such, the

analysis complements the more structural one of de Paula (2009), as well as a subtle

investigation by Young (2009) on the observational differences between learning and

conformity models in dynamic environments.

Although partial identification arguments of the Brock-Durlauf type have not been used

empirically, one does find analyses that may be interpreted in this way. For example, Card

et al. (2008) evaluate Schelling’s classic tipping model of segregation by a search for

discontinuities in the relationship between changes in neighborhood racial composition

and initial racial composition. We conjecture that this achieves partial identification of the

effect of neighborhood composition on individual utility so long as one imposes a continu-

ity assumption on the density of unobservables, which is what Brock & Durlauf (2010) call

a shape assumption.

For many forms of social interactions, endogenous group formation can produce assor-

tative matching. One intuition follows from the Becker (1973) marriage model and can be

generalized in a number of ways (Durlauf & Seshadri 2003): Social interactions can


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represent complementarities and mean that stratification is efficient. One example for this

is human capital complementarity across workers. In other contexts, such as neighbor-

hoods, social interactions can produce incentives to stratify regardless of whether the

stratification is efficient; Becker & Murphy (2000) give a nice exposition of this property

for residential neighborhoods. These theoretical results suggest that another strategy for

uncovering social interactions could involve the evaluation of degrees of stratification

across groups. Although there does exist a literature on the identification of complemen-

tarities via stratification patterns (see Siow 2009 for a recent example), this method has yet

to be used to uncover social interactions per se. The closest example of this strategy is Card

et al. (2008), discussed above, as it uses changes in the degree of segregation as the basis for

inferences. It should be noted that methods for estimating neighborhood choices as exem-

plified in Epple & Sieg (1999) are explicitly designed to explain the intragroup distribu-

tions of characteristics, so our suggestion should be understood as complementary to this

structural style of work.

Although we focus on partial identification in social interactions contexts, there is a

broader literature on partial identification in games, much of which is driven by multiple

equilibrium issues. Tamer (2003) is a seminal contribution.

4.3. Experiments

As is true for much of economics, concerns over the assumptions needed to achieve

identification with observational data have led to substantial interest in experiments and

quasi-experiments as ways to uncover social interactions. The experimental approach has

seen little direct work on econometric issues, relying on the existing treatment effect

methods to study social interactions. One exception is Hirano & Hahn (2009), who study

design issues for social interactions experiments. Another is Graham et al. (2008), who

formulate measures of changes in group composition on group outcomes that generalize

treatment effect calculations to explicitly consider social interactions.

At one end of the scale, interest in the effects of residential neighborhood effects led the

Department of Housing and Urban Development in 1994 to implement the Moving to

Opportunity (MTO) demonstration in Baltimore, Boston, Chicago, Los Angeles, and New

York. The program provides housing vouchers to a randomly selected group of families

from among residents of high-poverty public housing projects. Within this subsidized

group, families in turn were randomly allocated between two subgroups: one that received

unrestricted vouchers and another that received vouchers that could only be used in census

tracts with poverty rates below 10% (these users are termed the experimental group).

Members of the experimental group also received relocation counseling.

The data from the MTO program have been used by a number of researchers to

evaluate social interactions. Kling et al. (2007) is the state-of-the-art study both in terms

of methodology and coverage across all study sites; earlier analyses include Ludwig et al.

(2001) and Kling et al. (2005). Following Kling et al. (2007), social interactions are

inferred when evidence is found that movement to a low-poverty neighborhood affects

socioeconomic outcomes. In doing this, the authors calculate both intent to treat and

treatment on the treated effects, corresponding to the effects of receipt of the voucher and

use of the voucher, respectively.

In what is the best overview of the MTO program, Kling et al. (2007) find mixed

evidence on the effects of movement to lower-poverty communities, even though the


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housing voucher lottery caused otherwise similar groups of families to reside in very

different neighborhoods. For adults, their analysis revealed little evidence that movement

to low-poverty neighborhoods affected economic self-sufficiency although some improve-

ments occurred in mental health. In terms of the children, they find beneficial effects for

teenage girls with respect to education, risky behavior, and physical health. Alternatively,

for teenage boys, relocation to lower-poverty neighborhoods was associated with deterio-

ration of a number of socioeconomic outcomes, in particular with respect to criminal

behavior. The evidence that effects of housing vouchers appear to accrue from changes in

neighborhood characteristics rather than from moves per se suggests that interventions

that substantially improve distressed neighborhoods could have effects at least as large as

those observed from moving to lower-poverty neighborhoods.

Although the MTO program is a unique evidence source, there are a number of limita-

tions to the use of existing findings of program effects to either draw policy implications or

even to infer the empirical salience of social interactions. One general problem is that only

one-quarter of eligible families signed up for MTO, which also suggests that decisions are

influenced by unobservables. These statistics also suggest that the respective populations

are not representative. In terms of policy, one criticism is due to Sobel (2006), who argues

that different respondents’ outcomes are correlated, which is known as interference among

respondents. Using the MTO demonstration as a concrete context, Sobel provides a causal

framework for policy evaluation when interference is present. He characterizes the proper-

ties of the usual estimators of treatment effects, which are unbiased and/or consistent in

randomized studies without interference. When interference is present, the difference

between a treatment group mean and a control group mean does not estimate an average

treatment effect, but rather the difference between two effects defined on two distinct

subpopulations. This result means that a researcher who fails to recognize it could infer

that a treatment is beneficial when in fact it is not. Kling et al. (2007) reject the argument of

interference by pointing to the fact that 55% of household heads who signed up for MTO

had no friends and 65% had no family in their baseline neighborhoods. However, to the

extent that interference did not occur, then it is unclear how to extrapolate the program to

larger policy interventions. In particular, Durlauf (2004) argues that moving large numbers

of poor families to more affluent communities will induce general equilibrium effects in

terms of the location decisions of other families and the ability of schools in these neighbor-

hoods to provide needed services, for example. One can additionally imagine that the

commitment of affluent families to public schools would be strained by a massive influx

of poor families into their communities.

In terms of the question of the empirical salience of social interactions, MTO studies

suffer from the lack of attention to mapping between treatment parameters and the under-

lying social interactions. A range of neighborhood characteristics generate social interac-

tions effects; the role model effects of affluent neighbors are quite distinct from the support

mechanisms that are generated by proximity to friends and relatives. MTO families who

moved to low-poverty neighborhoods did not simply move from worse to better neighbor-

hoods; rather they traded off one vector of neighborhood attributes for another. This also

renders the interpretation of positive evidence of voucher effects problematic. One exam-

ple is discussed by Kling et al. (2007): the reduction in asthma among children who move

to low-poverty neighborhoods. This could result from stress reduction; to the extent this is

generated by neighborhood characteristics, this is a social interactions effect. Another

reason for lower asthma rates, however, could be reduced exposure to vermin infestations,


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induced by changes in housing quality, in turn induced by the requirement that the

vouchers were used in low-poverty neighborhoods, which is not a social interactions effect.

A second class of experimental studies has been conducted in laboratories, such as

Falk & Ichino (2006). Their experiment involves workers who are assigned in pairs to

stuff envelopes, with a control group being provided by subjects working alone in a room.

These authors find that standard deviations of output are significantly smaller within pairs

than between pairs, and average output per person is greater when subjects work in pairs.

They also predict theoretically that peer effects are asymmetric; low-productivity workers

are more sensitive to the behavior of high-productivity workers as peers, and they find

indirect evidence that this holds. Although the findings of the effects of working in pairs

versus working singly are quite clean, they do not reveal why the effects are occurring.

A third class of studies involves natural experiments, in which the assignment of indi-

viduals to groups has occurred in a way that facilitates identification of social effects.

Sacerdote (2001) exploits the effects of randomly assigned freshman-year room- and

dormmates at Dartmouth College. Sacerdote estimates Equation 6 with an individual’s

grade point average as a dependent variable, as a function of an individual’s own academic

ability and social habits, and the academic ability and grade point average of roommates,

finding that peers have an impact on each others’ grade point average and on decisions to

join social groups such as fraternities. He does not, however, find residential peer effects in

decisions such as choice of college major. Interestingly, peer effects are smaller the more

directly a decision is related to labor market activities.

A final class of studies follows the spirit of natural experiments but does not rely on

identifying contexts where random assignment holds, but rather on other data features.

Young & Burke (2001) use differences in the terms of tenant/landowner agriculture con-

tracts across Illinois to evaluate the importance of social norms on contract terms. They

find that contract terms tend to focus on simple crop-sharing rules that cannot be

explained by standard contract theory.

Although we emphasize some limitations of experimental studies, this does not imply

either that the studies are uninformative or that their utility is lower than observational

studies. Rather, we seek to combat the presumption that experimental studies trump

observational ones and that randomization should be regarded as the gold standard for

empirical work.

5. SOCIAL INTERACTIONS IN SOCIAL NETWORKS

Our discussion of empirical work above focuses on social interactions models in which

group influences are generated by relatively crude aggregates, normally the average char-

acteristics or behaviors of others. In contrast, social networks models provide a primary

focus on the microstructure of interactions emphasizing heterogeneity in these interactions

across individual pairs. Jackson (2008) reviews the new theories. Here are examples of

social interactions analyses that have been enriched by network thinking.

5.1. Models

One reason social networks matter in social interactions analysis is because they facilitate

identification by breaking the reflection problem. This was originally recognized in Cohen-

Cole (2006) and is systematically explored in Bramoulle et al. (2009). Using Y to represent


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the I-column vector of individual actions, we may generalize the linear-in-means model

(Equation 6) to allow for an arbitrary set of pairwise interactions. In a linear setting

(Equation 6), this amounts to replacing bmg(i) with bJY. Following Bramoulle et al., we

employ the former and further assume that Jik ¼ 1, if i is influenced by k, and Jik ¼ 0,

otherwise. In addition, the endogenous effect is defined as bJY. In other words, each

pairwise interaction has an endogenous interaction weight b or a weight zero, and every

contextual effect has a weight y or zero, depending on whether the agents are directly

linked. Recalling that i is a vector of ones, the vector counterpart of Equation 6 is thus

Y ¼ a0iþXaþ JXyþ bJYþ e; ð19Þwhere a and y are r-column vectors of parameters, and X is an I � r matrix of individual

effects. This system of equations retains a single parameter b for the endogenous social

effect but allows for heterogeneous interactions among individuals.

If jbj < 1, then I � bJ is invertible, and provided that each individual is socially con-

nected with at least one other individual, Equation 19 yields the reduced form:

Y ¼ 1

1� ba0iþXaþ

X1k¼0

bkJkþ1Xðabþ yÞ þX1k¼0

bkJke: ð20Þ

Bramoulle et al. (2009, proposition 1) use Equation 20 to obtain the unique solution for

EfJY j Xg and establish that identification of social effects in Equation 19 requires that

both ay þ b 6¼ 0 and matrices I, J, and J2 be linearly independent. Moreover, when no

individual is isolated, the matrices I, J, and J2 are linearly dependent if and only if

EfJY j Xg is perfectly collinear with regressors (i,X, JX). Otherwise, the network structure

is rich enough to make the variables (i, X, JX, J2X, J3X; . . . ;) serve as appropriate instru-

ments that allow endogenous and exogenous effects to be identified. Bramoulle et al. also

obtain identification results for the model with correlated effects, as in Equation 21 below.

Why does this condition allow for identification? Intuitively, accounting for network

structure means that each individual i’s behavior is associated with a set EfYgðiÞg ¼ JEfYgof variables that need to be instrumented. Consider two other agents, j and k, such that

Jij ¼ 0 (i and j are not socially connected) is imposed a priori. When Jik ¼ 0 has not been

imposed a priori, one can use the individual characteristics for j to instrument for EfYgðkÞgso long as j does directly affect k. In other words, by setting certain coefficients equal to

zero, network structure produces exclusion restrictions that are analogous to the rank and

order condition in simultaneous equations analysis.

There exists a close relationship between social interactions and spatial econometrics

models that was only recently recognized by researchers.8 Lee (2007), Lee et al. (2010),

and Lin (2008) estimate a version of the social interactions problem with a network

structure, which is specified in terms of groups g ¼ 1; . . . ; G:

Yg ¼ bJgYg þXaþ JgZyþ a0gig þ eg; ð21Þwhere Z is an ng � r matrix of contextual effects, Yg is the column ng vector of outcomes

for each member of group g, ng is the group’s size, Jg is group g’s social weights matrix, ig isa column ng vector of ones, eg is a column ng vector of shocks, and a0g is a group-specific

8The canonical version of the linear social interactions model, in vector form (Equation 19), with y ¼ 0, coincides

with the classic Cliff-Ord model of spatial autocorrelation with one spatial lag, JY, but no spatial error structure (see

Cliff & Ord 1981).


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fixed effect. The network structure in Equation 21 generalizes the standard spatial auto-

correlation (SAR) model. The shocks eg’s may have a general variance-covariance matrix.

However, the spatial econometrics literature has typically assumed that

eg ¼ rJg;eeg þ Rg; ð22Þwhere Jg,e is a matrix of coefficients that corresponds to network structure in the errors

for group g, which may or may not be the same as Jg (see Lee et al. 2010), and Rg denotesan ng column of independently and identically distributed shocks, which allows for infor-

mation from the covariance matrix of errors to assist identification.

A comparison of Equation 19 and Equations 21 and 22 indicates how network and

spatial models are interrelated. Each assumes prior knowledge of the members of each

agent’s group. Each assumes that interactions within a group are homogeneous; i.e., the same

weight is assigned to each member of the group. The key differences lie in the introduction of

an error structure corresponding to the network structure in the spatial econometrics case.

5.2. Empirics of Social Interactions in Social Networks

As emphasized above, network models require prior knowledge of the network structure.

Relatively few data sets have this feature. For this reason, economists have been especially

interested in the Add Health data set. Add Health, the National Longitudinal Study of

Adolescent Health (http://www.cpc.unc.edu/addhealth), is a U.S.-based study designed to

facilitate exploration of the causes of health-related behaviors of adolescents in grades 7

through 12, the structure of adolescent friendship networks, and individual outcomes in

young adulthood. It seeks to examine how social contexts (families, friends, peers, schools,

neighborhoods, and communities) influence adolescents’ health and risk-taking behaviors.

It has become a major resource for social networks analysis. In the remainder of this

section we describe some examples of interesting studies using this data set.

Fryer & Torelli (2006) use academic achievement as a definition of “acting white” by

nonwhite students to examine its relationship to popularity. They measure popularity in terms

of a network-specific spectral popularity measure, which identifies popularity of the members

of a group with a measure of the intensity of the social connections among the members of

that group. Specifically, this study can be seen as a simplified version of Equation 19, in

which, instead of the full system of simultaneous equations, achievement as an outcome is

related to a popularity statistic based on J in Equation 19. They demonstrate that there are

large racial differences in the relationship between popularity and academic achievement.

Calvo-Armengol et al. (2009) estimate individual school performance as a function of

the topology of their friendship networks, defined as embedded in different schools, while

controlling for individual characteristics, such as family background controls, residential

group variables, contextual effects, and network component fixed effects. Specifically, they

estimate Equation 21 imposing b ¼ 0 and assuming a given adjacency matrix for the social

network J, that is, Ji,j ¼ 1, if agents i and J are directly connected, and Ji,j ¼ 0, otherwise.

The error structure consists of network component-specific fixed effects plus a vector of

shocks e that satisfy

e ¼ aNg þ fJgeþ R; ð23Þwhere Jg is the group g–specific adjacency matrix, and Ng is a vector whose i element

is the number of direct links for each person in group g(i). Because the error structure


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represents individual outcomes that are not explained for by individual characteristics,

X, and contextual effects, JX, it proxies for correlated effects and peer effects. Their

estimation of the stochastic structure of Equation 21 with Equation 23 thus subsumes

endogenous social effects into the estimation of (a, f) above. Their results suggest that

a one-standard-deviation increase in their centrality index translates into approximately

7% of a standard deviation in education outcomes. Interestingly, these authors’ esti-

mates of the effect on a school performance index of the number of each individual’s

friends and of the peer effect, (a, f), change little if social interactions are allowed to

be directional.

6. CONCLUSIONS

Scholarly interest in social interactions has rapidly expanded in many areas of economics

and has led to numerous methodological and empirical advances. For theorists, key chal-

lenges include the development of analytically tractable frameworks that allow for the

analysis for social interactions contexts in ways that accommodate increasingly rich forms

of intertemporal as well as cross-section heterogeneity.

In parallel, we see a need for a more constructive role for computer simulations when

analytical tractability is impossible. For theorists as well as econometricians, a primary

challenge is the complete integration of the measurement of social interactions with the

joint processes of group formation and subsequent behaviors within groups. The social

interactions literature has paid inadequate attention to markets both in the sense of model-

ing the mechanisms by which groups are formed and in understanding how social influ-

ences manifest themselves in equilibria. For example, socially determined effort in a firm is

manifested in wages.

Social forces, whether manifested through neighborhoods, schools, communities, or

firms, impact much of what constitutes modern economic life. We therefore conjec-

ture that social interactions become an increasingly important area of research.

SUMMARY POINTS

1. The integration of social interactions into theoretical economic models deserves

attention.

2. The identification of social interactions relies on features of different types of

data.

3. Relationships between social interactions models and spatial and network models

have only recently attracted serious scholarly attention.

FUTURE ISSUES

1. The integration of richer interactions structures into social interactions models

will benefit from endogenizing links.

2. The integration of endogenous network formation and the identification of social

interactions deserve attention in future research.


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3. The use of price data to infer social interactions effects continues to be an

important topic.

4. The use of transition versus steady-state data to infer social interactions effects

should attract attention.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings

that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

Durlauf’s work was supported by the National Science Foundation grant SES-0518274,

the University of Wisconsin Graduate School, and the Laurits R. Christiansen Chair in

Economics. The authors thank Yann Broumoulle, Jane Cooley, Andres Aradillas-Lopez,

and Adriaan Soetevent for insightful comments and Nonarit Bisonyabut and Xiangrong Yu

for excellent research assistance. Durlauf thanks Buz Brock for years of collaboration and

conversation on social interactions that have profoundly affected his thinking.

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Develops statistical

mechanics models

that parallel social

interactions models.

Develops identification

conditions for linear

social networksmodels.

Along with Brock &

Durlauf 2001c and

2007, develops theory

and econometrics of

discrete choice models

of social interactions.

Develops a well-

delineated general

equilibrium model of

neighborhood choice

that is estimable.

Pioneering empirical

paper in the effort to

measure social

interactions effects.


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introduces the

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the literature.


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RELATED RESOURCE

Bisin A, Benhabib J, Jackson M, eds. 2010. Handbook of Social Economics. Amsterdam: North-

Holland. In press. Presents a definitive collection of studies of different aspects of social

interactions.


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Annual Review of

Economics

Contents

Questions in Decision Theory

Itzhak Gilboa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Structural Estimation and Policy Evaluation in Developing Countries

Petra E. Todd and Kenneth I. Wolpin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Currency Unions in Prospect and Retrospect

J.M.C. Santos Silva and Silvana Tenreyro . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Hypothesis Testing in Econometrics

Joseph P. Romano, Azeem M. Shaikh, and Michael Wolf . . . . . . . . . . . . . . 75

Recent Advances in the Empirics of Organizational Economics

Nicholas Bloom, Raffaella Sadun, and John Van Reenen . . . . . . . . . . . . . 105

Regional Trade Agreements

Caroline Freund and Emanuel Ornelas . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Partial Identification in Econometrics

Elie Tamer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Intergenerational Equity

Geir B. Asheim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

The World Trade Organization: Theory and Practice

Kyle Bagwell and Robert W. Staiger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

How (Not) to Do Decision Theory

Eddie Dekel and Barton L. Lipman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Health, Human Capital, and Development

Hoyt Bleakley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Beyond Testing: Empirical Models of Insurance Markets

Liran Einav, Amy Finkelstein, and Jonathan Levin. . . . . . . . . . . . . . . . . . 311

Inside Organizations: Pricing, Politics, and Path Dependence

Robert Gibbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Volume 2, 2010

vi

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Identification of Dynamic Discrete Choice Models

Jaap H. Abbring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Microeconomics of Technological Adoption

Andrew D. Foster and Mark R. Rosenzweig . . . . . . . . . . . . . . . . . . . . . . 395

Heterogeneity, Selection, and Wealth Dynamics

Lawrence Blume and David Easley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Social Interactions

Steven N. Durlauf and Yannis M. Ioannides. . . . . . . . . . . . . . . . . . . . . . . 451

The Consumption Response to Income Changes

Tullio Jappelli and Luigi Pistaferri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Financial Structure and Economic Welfare: Applied

General Equilibrium Development Economics

Robert Townsend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Models of Growth and Firm Heterogeneity

Erzo G.J. Luttmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Labor Market Models of Worker and Firm Heterogeneity

Rasmus Lentz and Dale T. Mortensen . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

The Changing Nature of Financial Intermediation and the

Financial Crisis of 2007–2009

Tobias Adrian and Hyun Song Shin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Competition and Productivity: A Review of Evidence

Thomas J. Holmes and James A. Schmitz, Jr. . . . . . . . . . . . . . . . . . . . . . . 619

Persuasion: Empirical Evidence

Stefano DellaVigna and Matthew Gentzkow . . . . . . . . . . . . . . . . . . . . . . 643

Commitment Devices

Gharad Bryan, Dean Karlan, and Scott Nelson . . . . . . . . . . . . . . . . . . . . 671

Errata

An online log of corrections to Annual Review of Economics

articles may be found at http://econ.annualreviews.org

Contents vii

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AnnuAl Reviewsit’s about time. Your time. it’s time well spent.

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Annual Review of Statistics and Its ApplicationVolume 1 • Online January 2014 • http://statistics.annualreviews.org

Editor: Stephen E. Fienberg, Carnegie Mellon UniversityAssociate Editors: Nancy Reid, University of Toronto

Stephen M. Stigler, University of ChicagoThe Annual Review of Statistics and Its Application aims to inform statisticians and quantitative methodologists, as well as all scientists and users of statistics about major methodological advances and the computational tools that allow for their implementation. It will include developments in the field of statistics, including theoretical statistical underpinnings of new methodology, as well as developments in specific application domains such as biostatistics and bioinformatics, economics, machine learning, psychology, sociology, and aspects of the physical sciences.

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Radu V. Craiu, Jeffrey S. Rosenthal•Build, Compute, Critique, Repeat: Data Analysis with Latent

Variable Models, David M. Blei•Structured Regularizers for High-Dimensional Problems:

Statistical and Computational Issues, Martin J. Wainwright

•High-Dimensional Statistics with a View Toward Applications in Biology, Peter Bühlmann, Markus Kalisch, Lukas Meier

•Next-Generation Statistical Genetics: Modeling, Penalization, and Optimization in High-Dimensional Data, Kenneth Lange, Jeanette C. Papp, Janet S. Sinsheimer, Eric M. Sobel

•Breaking Bad: Two Decades of Life-Course Data Analysis in Criminology, Developmental Psychology, and Beyond, Elena A. Erosheva, Ross L. Matsueda, Donatello Telesca

•Event History Analysis, Niels Keiding•StatisticalEvaluationofForensicDNAProfileEvidence,

Christopher D. Steele, David J. Balding•Using League Table Rankings in Public Policy Formation:

Statistical Issues, Harvey Goldstein•Statistical Ecology, Ruth King•Estimating the Number of Species in Microbial Diversity

Studies, John Bunge, Amy Willis, Fiona Walsh•Dynamic Treatment Regimes, Bibhas Chakraborty,

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