Social Interaction. The Manski re ection problempnikolov/papers/Manski.pdf · Social Interaction....

Social Interaction. The Manski reflection problem


Giulia Camilotti

February 13, 2012

Plamen Nikolov

Text Box


Table of contents

Introduction

Manski reflection problem

The linear in means modelSolutions to the reflection problem

Hierarchical models

Spatial models of social interaction

Age at marriage

Conclusion


Introduction

Social interaction

how people affect each others’ behavior


Introduction

When social interaction matters?

theory: network analysisempirics:

I education (Bramoulle et al 2009, Sacerdote 2001, Comola2008)

I adoption of new technolgies (Conley and Udry 2004, 2010;Bandiera and Rasul 2006);

I social norms (fertility and birth control: Krishnan 2001;Kohler 2000, 2001)

I ...

Cross section data



social interaction

Manski (1993):

I endogenous effect

I exogenous (contextual) effect

I correlated effect → no social interaction



Example

dropping out school

I endogenous effect: individual behavior varies with groupbehaviorex: imitation of poorly performing students living in myneighborhoodYoung 1998, cited in Wydick et al 2011:

1. pure conformity2. instrumental conformity3. informational conformity



Example

I exogenous effect: individual behavior varies with theexogenous characteristics of the groupex: parents of my classmates are poor: I observe low rewardsfrom education



Example

I correlated effect: individuals behave similarly because theyhave similar characteristics or face similar institutionalenvironment. Ex:

1. bad teacher2. poor neighborhood → poor parents transmit lower cognitive

skills



The reflection problem

I endogenous effect:group behavior affects individual behavior orgroup behavior is simply the aggregation of individualbehaviors?

I how to distinguish between different effects?

I policy implications



Models of social interactions (Blume et al. 2011)

1. the linear-in means model

2. hierarchical models

3. spatial models of social interaction


The linear in means model

the linear in means model

ωi = k + cxi + dyg + Jmeig + εi (1)

whereme

ig = E (ω|g) ≡ mg : group behavior → endogenous effectyg = x = E (x |g) : group characteristics → exogenous effectfor each g and i ∈ gE (εi |xi , xg , yg , i ∈ g) = 0for each i; j, g, h such that i 6=j and g 6= hcov(εiεj | xi , yg , i ∈ g , xj , yh, j ∈ h) = 0no correlated effect




Assuming that everybody knows everything in the model (exceptthe idiosincratic error of the others),expected average behavior in the group is

meig = mg = Ex([E (ω|x , g)]) = k + cE (x |g) + dyg + Jme

igx (2)

mg =k + (c + d)yg

1− J(3)




Plug (3) into (1) get the reduced form:

ωi =k

1− J+ cxi +

Jc + d

1− Jyg + εi = πi + π2xi + π3yg + εi (4)

⇒ identification fails = reflection problem:if we observe that individual behavior is correlated to groupbehavior, it is simply because mg is linear dependent on yg



Solutions to the reflection problem

How to break the reflection problem?

⇒ Under which condition the model is identified?

1. relax assumption yg = xg

2. relax assumption of the linearity of E (ω|x , g)

3. change model specification ! ⇒ hierarchical and spatial modelof social interaction




Relax assumption yg = xg

⇒ find an individual characteristic whose group average level /∈ yg

mg =k + cxg + dyg

1− J(5)

Problem:

I difficult in practice




Relax assumption of the linearity of E (ω|x , g)

⇒ bionomial choice model (Brock and Durlauf 2001)No reflection problem: nonlinear relationship between groupcharacteristics and group behavior(transform linear combination of some regressors into probabilities)⇒ median; Moffitt’s model (2004), etc...


Hierarchical models

Hierarchical models

⇒ alternative conceptualization of the impact of social interactionon individual behavior

ωig = kg + cgxi + εi (6)

kg = k + dyg + Jmg (7)

cg = c + y ′gΨ + mgψ (8)

ωig = k + cxi + dyg + Jmg + y′gΨxi + mgψxi + εi (9)


Hierarchical models

Hierarchical models

impose self-consistency and yg = xg

mg =k + (c + d)yg + y ′gΨyg

1− J − ψyg(10)

⇒ no reflection problem (no linear dependence between m and theregressors)BUT if m is highly correlated with individual determinants ⇒estimates imprecise


Hierarchical models

correlated effect

I unobserved group heterogeneity

1. instrumental variables: instrument which is not an individual orexogenous determinant of ω (Cooley 2008)

2. application: Wydick 2011

I self selection (?)



Spatial models of social interaction (Bramoulle et al 2009,2010)

information on the network structure: adjacency matrix A isknown,whereAij = 1

Piif j affects i

Aij = 0 otherwiseModel (in matrix notation):

ω = kι+ cx + dAx + JAω + ε (11)

with E [ε | x] = 0



Spatial models of social interactions

The model is identified if I,A,A2 are linearly independentNB: assumption of intransitivity in the network.However, identification with transitive networks possible if :

I individual excluded from his own network

I at least two groups of different size



Correlated effect

Model extended to take into account correlated effects.The model is identified if I,A,A2,A3 are linearly independentNB: the network structure is taken as exogenous


Age at marriage

age at marriage

I early marriage: is it a coordination problem?

I intervillage network for marriage ⇒ endogenous effect?

I early marriage associated with low level of income andeducation;

I age at marriage varies across ethnic groups;


Conclusion

Conclusion

I social interactions give rise to issues of identification;

I severity reflection problem varies;

I different model specifications and assumptions possible: →data?

I empirical investigation of networks ? → endogenous networkformation

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