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Social Interaction. The Manski reflection problem
Social Interaction. The Manski reflection problem
Giulia Camilotti
February 13, 2012
Social Interaction. The Manski reflection problem
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
Social Interaction. The Manski reflection problem
Introduction
Social interaction
how people affect each others’ behavior
Social Interaction. The Manski reflection problem
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. The Manski reflection problem
Manski reflection problem
social interaction
Manski (1993):
I endogenous effect
I exogenous (contextual) effect
I correlated effect → no social interaction
Social Interaction. The Manski reflection problem
Manski reflection problem
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
Social Interaction. The Manski reflection problem
Manski reflection problem
Example
I exogenous effect: individual behavior varies with theexogenous characteristics of the groupex: parents of my classmates are poor: I observe low rewardsfrom education
Social Interaction. The Manski reflection problem
Manski reflection problem
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
Social Interaction. The Manski reflection problem
Manski reflection problem
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
Social Interaction. The Manski reflection problem
Manski reflection problem
Models of social interactions (Blume et al. 2011)
1. the linear-in means model
2. hierarchical models
3. spatial models of social interaction
Social Interaction. The Manski reflection problem
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
Social Interaction. The Manski reflection problem
The linear in means model
The linear in means model
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)
Social Interaction. The Manski reflection problem
The linear in means model
The linear in means model
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
Social Interaction. The Manski reflection problem
The linear in means model
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
Social Interaction. The Manski reflection problem
The linear in means model
Solutions to the reflection problem
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
Social Interaction. The Manski reflection problem
The linear in means model
Solutions to the reflection problem
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...
Social Interaction. The Manski reflection problem
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)
Social Interaction. The Manski reflection problem
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
Social Interaction. The Manski reflection problem
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 (?)
Social Interaction. The Manski reflection problem
Spatial models of social interaction
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
Social Interaction. The Manski reflection problem
Spatial models of social interaction
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
Social Interaction. The Manski reflection problem
Spatial models of social interaction
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
Social Interaction. The Manski reflection problem
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;
Social Interaction. The Manski reflection problem
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