Understanding Regression versus Variance Tests for Social Interactions
Steven N. Durlauf1
Hisatoshi Tanaka2
November 26, 2006
Abstract
This paper considers the relationship between the use of regressions and the use of
variance contrast methods to uncover social interactions. We illustrate how these
methods employ different identifying assumptions and are therefore complementary
approaches. We also provide formal identification results that extend existing ones for
the two methods.
JEL Classification Codes: C12, C21, C23, Z13
Keywords: Social Interactions; Peer Group Effects; Variance Contrasts; Identification
1
Department of Economics, University of Wisconsin-Madison, 1180 Observatory Drive,
Madison, WI 53706-1393. E-mail: sdurlauf@ssc.wisc.edu
2
Department of Economics, University of Wisconsin-Madison, 1180 Observatory Drive,
Madison, WI 53706-1393. E-mail: htanaka2@wisc.edu
1. Introduction
This paper is designed to understand the relationship between two approaches
to the empirical analysis of social interactions.
The first approach is the use of linear
regressions to uncover social interactions. This approach estimates social interaction
effects through changes in the conditional (linear) expectation of individual outcomes
when group level variables are included in the conditioning set. In econometrics this
method was first analyzed in Manski (1993) who dubbed it the linear in means model
since the group level variables are typically averages of individual variables within the
group. This model is also studied in Brock and Durlauf (2001a,b), Cohen-Cole (2006)
and Graham and Hahn (2005), among others. Durlauf (2004) surveys empirical studies
of social interactions based on such regressions.
The second approach, originally proposed by Glaeser, Sacerdote, and
Scheinkman (1996) and developed into a full inferential procedure by Graham (2005),
who refers to this approach as variance contrasts, uses intragroup variation in behavior
to infer social interactions. The idea in this approach is that social interactions will
induce intragroup dependences in behavior that affect sample variances for intragroup
outcomes. There has been relatively little work evaluating social interactions using
variance contrasts. One reason is that with the exception of Graham's work, there is
1
little formal statistical theory for the approach.
The purpose of this paper is to clarify the relationship between regression and
variance contrast tests. We clarify how these tests employ different informational
assumptions to achieve identification of social interaction. Specifically, we show that
regression and variance contrast approaches correspond to identification via exclusion
restrictions in the one case and covariance restrictions on errors in the other. As such,
the two approaches replicate the classes of identification strategies that appear in the
classical simultaneous equations literature, see Fisher (1966) for a classic treatment.
The reminder of the paper is organized as follows. In Section 2 we define a
general linear in means model and describe identification from the regression
perspective.
Section 3 provides identification analysis from the perspective of
variance contrasts.
Section 4 concludes.
As Graham’s analysis focuses on peer
effects in classrooms, we will use that example to motivate discussion.
2. The linear in means regression model
We consider the analysis of a cross-section data set which consists of observations
on individuals in a set of groups. The number of students in classroom g is m g with
support t1    t k . Individual decisions  g ,i obey
2
 g ,i  k  X g ,i c  J g  Ygd  z g ,i
(1)
or
g ,i  k  X g ,i c  JEg g  Ygd  J g  Eg g   z g ,i .
Where g  mg
1

mg
i 1
(2)
g ,i denotes average behavior within a group, and the expectation
E g  g is conditioned on information shared by all agents in the group.
The
conversion of the individual behavioral equation into one where the behavior of i
depends on the expected average behavior places the model in the form studied in
Manski (1993), Brock and Durlauf (2001a,b), etc. Note that for this equation to have a
self consistent solution E g  g it is necessary that | J | 1 .
The regression approach to social interactions assumes that z g ,i is orthogonal
to 1, X g ,i , Yg .
However, no other assumptions are placed on the unobservables.
Unobserved group characteristics are allowed so long as this orthogonality condition
holds. Identification of social interactions in this model requires that one is able to
construct an empirical proxy for E g  g that is not collinear with X g , i or Yg . Brock
and Durlauf (2001a,b) show that this requires that X g , i is not linearly dependent on
Yg , which is equivalent to the classical rank condition for simultaneous equations
models. The sufficient condition for identification, which has not stated previously, is
given in Proposition 1.
3
Proposition 1
Suppose the following conditions are satisfied: (i) X g , i consists of subvectors X 1g ,i
and X g2,i . In particular, dim( X 1g ,i )  0 . (ii) Yg consists of subvectors Yg1 and Yg2 ,
 1
where the second vector, the contextual effects, is defined by Y  X  :
 m
g

2
g
while the first vector, the exogenous effects,
2
g
mg
X
i 1
2
g ,i

,


is linearly independent of X 1g . (iii) z g ,i
is orthogonal to (1, X g ,i , Yg ) . Then the parameters (k , c, J , d ) are identified.
3. The variance contrast approach
Following Graham (2005), the variance contrast approach uses a simplified
version of (1).
g ,i  k  J g  z g ,i  k  JEg  J g  Egg   z g ,i .
(3)
Relative to the regression approach, all observable individual and contextual controls
are ruled out.
It is therefore evident why the regression approach cannot work for this
model: there are no instruments to construct E g  g in (3).
Can identification still be achieved? Graham (2005) shows that the answer is
yes when additional structure is placed on z g ,i . Specifically Graham assumes that
z g ,i   g   g ,i
(4)
where  g is a group level unobservable and  g ,i is an individual level unobservable
4
with mean zero; it is assumed that these are identically distributed and uncorrelated
across i and g. This assumption is motivated by the substantive assumption that students
are randomly assigned and hence exchangeable.
We work with a slightly more general
form than Graham, i.e. we employ a covariance structure for z g ,i of the form
1

2 1
Var ( Z g | mg  m)   


1
1  1
1


1  1
2 


  


1  1

  
1   


  1


,
(5)
where Z g  (Z g ,1 ,, Z g ,mg )' , the matrices are m m ,  2  Var( g ) ,  2  Var ( g ,i )
and   Corr ( g ,i ,  g ,i ) for i  i .
Following Graham, these moments are assumed
to be independent of group size; this assumption is key in producing identification.
Recall that Condition (iii) of Proposition 1 assumes that, in the regression
approach, any group effects are random rather than fixed. Therefore, the variance
approach uses the same orthogonality condition as the regression approach. In the
variance contrast approach, however, the covariance structure (5) is also imposed to
achieve identification; this is not needed for identification in the regression approach.
While Graham develops identification for a particular estimator 3 , it is
straightforward to prove identification for this simultaneous equations system based on
3
Graham’s estimate of J is defined via
2
Var[ g | mg  t j ]  Var[ g | mg  tl ]
 1 
.

 
1
(t j  1) Var[( g ,i   g ) | mg  t j ]  (tl  1) 1Var[( g ,i   g ) | mg  tl ]
1 J 
5
covariance restriction assumptions. This derivation, because it is not estimator-specific,
can be generalized to alternative specifications, as we will demonstrate with an example.
Proposition 2
Suppose that (i) the covariance structure of the errors in (3) is such that (5) holds, and
(ii) |  | 1 . If m g has at least two support points, then J is identified.
Pf. Define m m matrices H m and I m
1
1
Hm  


1
1  1
1  1
  

1  1
O
1


1

 ,
Im 





1
O
,
(6)
so that (5) may be rewritten as
Vm  aH m  bI m
where
(7)
Vm  Var ( Z | m) , a   2   2 and b  (1   ) 2 .
~ ~
Observational equivalence of ( J , a~, b ) and ( J , a, b) with respect to the
covariance structure may be written as the requirement that
1
J
J




Var (W | m)   I m  H m  Vm  I m  H m 
m
m




1
1
1
~
~

 ~

J
J
  I m  H m  Vm  I m  H m  (8)
m
m




~
~
where W  (1 ,, m )' and Vm  a~H m  b I m holds for all values of the support of m.
This requires
6
~
1
1

J
J
J
~ 





Vm   I m  H m   I m  H m  Vm  I m  H m 
m
m
m





~


J
 Im  Hm 


m


1
1  
(Vm H m  H mVm )  
 H mVm H m ,
m
 m 
2
 Vm 
where  
(9)
~
1 J
. Note that the particular variance structure of Vm is not needed for
1 J
(9). Since H m H m  mH m , (9) implies
1
(1   ) 2
~
Vm H m  Vm H m 
(mVm H m  H mVm H m ) 
H mVm H m
m
m
 (1   )
 Vm H m 
H mVm H m
m
(10)
~
H mVm H m  H mVm H m   (1   ) H mVm H m   2 H mVm H m ,
(11)
and
both of which are also true under general Vm .
~
~
Substitute Vm  aH m  bI m and Vm  a~H m  b I m into (11), this indicates that
~
a~m  b
 
must hold for any m in the support if observational equivalence holds.
am  b
2
~
~
This implies that a~   2 a and b   2b , which implies Vm   2Vm . Substitute
~
Vm   2Vm into (9) to obtain
1
1  
O  (1   )Vm 
(Vm H m  H mVm )  
 H mVm H m .
m
 m 
2
2
(12)
Since Vm  aH m  bI m , (12) simplifies to
(1   2 )bI m  (1   2 )[ a(m  1)  b]H m .
(13)
Since I m and H m are linearly independent, (13) is satisfied only if   1 , or
~
equivalently J  J , which completes the proof.
While this proof is dependent on the assumption that the moments of the
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variance covariance matrix of individual outcomes does not depend on the size of the
group, its structure also indicates how prior information on this matrix can substitute for
the assumption. In Durlauf and Tanaka (2006) we are studying how prior information on
social network structure can substitute for the size invariance assumption to produce
identification.
4. Conclusion
The upshot of this discussion is that regression and variance contrasts methods
should be regarded as complementary methods in uncovering social interactions. The
appropriateness of one approach versus another will depend on context.
For the
regression approach, the fundamental information requirement is that there exist
observable individual outcome determinants whose group level analogs do not directly
affect behavior. The variance contrast approach requires stronger a priori assumptions
on the variance covariance structure of the unobservables. As pointed out by Graham,
a key assumption is size invariance, i.e. the moments of the variance covariance matrix
do not depend on the size of the group. For the classroom context, this requires that
the variances of unobserved teacher quality and unobserved student characteristics are
8
not affected by the number of students. It further requires that the covariation between
students is unaffected by classroom size. This seems to us a very strong assumption;
we note that the Project Star data used by Graham is drawn from an experiment to study
how classroom size affects student outcomes, so that the experiment presupposed that
the assumption is false.
One limitation of both methods is that neither uses the information on social
interactions that presumably is embedded in the choice of groups.
The utility of
information on self-selection in identifying social interactions has been explored in
Brock and Durlauf (2001b) and Ioannides and Zabel (2003); Ekeland, Heckman, and
Nesheim (2004) and Nesheim (2002) illustrate how hedonic price approaches can be
used. Our conjecture is that the joint analysis of group formation and subsequent
behavior is the next step in empirical social interactions research.
9
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