ECON6300/7320/8300 Advanced Microeconometrics Cross-Sectional Dependence
Christiern Rose 1 1University of Queensland
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Introduction
The paper studies identification of endogenous social effects.
The propensity of an individual to behave in some way varies with the prevalence of that behaviour in some reference group containing the individual.
Such phenomena have been often called “social norms”, “peer influences”, “neighbourhood effects”, “conformity”, “imitation”, “contagion”, “epidemics”, etc.
Endogenous social effects in Economics:
The output chosen by each firm is a function of aggregate
industry output
When decision making is costly, people may want to imitate
the behaviour of other persons who are better informed.
Manski (1993) studies identification of endogenous social effects
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Three Hypotheses
Three hypotheses on the common observation that individuals belonging to the same group tend to behave similarly.
Endogenous effects: the propensity of an individual to behave in some way varies with the behaviour of the group.
Exogenous effects: the propensity of an individual to behave in some way varies with the exogenous characteristics of the group.
Correlated effects: individuals in the same group tend to behave similarly because they have similar individual characteristics or face similar institutional environments.
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Three Hypotheses: High school achievements
Endogenous effects: individual achievement varies with the average of average achievements in the youth’s school, ethnic group, or other reference group ⇒ social multiplier, reflection
Exogenous effects: individual achievement depends on the average of individual characteristics in the reference group
Correlated effects: similar family background, same teachers, etc.
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Setup
Each member i in the reference group g is characterised by a value for (yig,xg,zig,uig).
yig ∈ R: scalar individual outcome (exam score, etc)
xg ∈ RJ : characteristics of the reference group (school,
race, etc.)
(zig,uig)∈RK ×R:individualcharacteristics
Econometricianobserves(yig,xg,zig),butnotuig.
Assumption on DGP:
y =α+βE[y |x ]+E[z |x ]′γ+z′ η+u ig iggiggigig
with E[uig|xg,zig] = xg′ δ.
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Setup
The regression model is given as
E[y |x ,z ]=α+βE[y |x ]+E[z |x ]′γ+x′δ+z′ η
iggig igg igg g ig β measures the endogenous effects: E[yig|xg] is the
average exam score in the reference group
γ measures the exogenous effects: E[zig|xg] is the average socio-economic characteristics in the reference group, i.e., average family income, average parents’ education, etc.
δ measures the correlated effects
η measures direct effects
We are interested in the identification of (α, β, γ, δ, η).
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Identification
Rewrite the structural model:
E[y |x ,z ]=α+βE[y |x ]+E[z |x ]′γ+x′δ+z′ η
iggig igg igg g ig
We assume that the conditional expectation functions above E[yig|xg,zig], E[yig|xg], and E[zig|xg] are all identified, and focus on identification of (α, β, γ, δ, η).
Integrate zig out:
E[yig|xg] = α + βE[yig|xg] + E[zig|xg]′(γ + η) + xg′ δ
If β ̸= 1, we can isolate E[yig|xg] on the LHS:
α ′γ+η ′δ
E[yig|xg] = 1−β +E[zig|xg] 1−β +xg 1−β Plug the last equation into the one in the first bullet point.
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Identification
Then, we have the reduced form regression
α ′γ+βη ′ δ
E[yig|xg,zig]= 1−β +E[zig|xg] 1−β +xg 1−β +z′ η
ig
Even if the four parameters are all identified, we cannot identify all the structural parameters (α, β, γ, δ, η) without further assumptions, as we have 5 structural parameters but only 4 reduced form parameters.
But, we can still learn about presence of social effects (i.e. β = γ = 0 or not).
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Identification of composite social effects
Result 1: Provided that β ̸= 1, the four composite parameters above, α ,γ+βη, δ ,η are identified
1−β 1−β 1−β
if the regressors (1, E [zig |xg ], xg , zig ), are linearly
independent.
Result 1 implies that
we identify the presence of social effects because if γ+βη ̸= 0, it must be that either γ ̸= 0 or β ̸= 0.
1−β
ifE[zig|xg]islinearin(1,xg,zig),wecannotevenidentify
γ+βη. 1−β
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Identification of composite social effects
Specifically,thesocialeffectparameterγ+βηisnot 1−β
identified if either
zig is a function of xg
E[zig|xg] does not depend on xg, or E[zig|xg] is linear in xg
Therefore, the social effect parameter is identified only when E[zig|xg] is nonlinear in xg.
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Identification of composite social effects
Let’s investigate each case.
First, if zig = h(xg), the reduced form regression is written
as
E[yig|xg,zig]= 1−β +E[h(xg)|xg] 1−β +xg 1−β
+h(xg)′η
α ′γ+η ′δ
= 1−β +h(xg) 1−β +xg 1−β
Now we have only 3 reduced form parameters, and 5
structural parameters
Itcanbeγ+η̸=0,whenη̸=0butγ=β=0(nosocial
α ′γ+βη ′ δ
1−β effects).
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Identification of composite social effects
Second, if E[zig|xg] = E[zig], we have
α ′γ+βη ′ δ
E[yig|xg,zig]= 1−β +E[zig] 1−β +xg 1−β +z′ η
ig
α+E[zig]′[γ+βη] ′ δ ′ = 1−β +xg 1−β +zigη
The intercept can be nonzero if α ̸= 0 and γ = β = 0. Finally, if E[zig|xg] = xg′ κ, we have
α ′ κ[γ+βη]+δ ′ E[yig|xg,zig]= 1−β +xg 1−β +zigη
where κ[γ+βη]+δ can be nonzero if δ ̸= 0 and γ = β = 0. 1−β
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Identification of pure endogenous effects
Empirical studies of endogenous effects typically assume that γ = δ = 0; no exogenous and no correlated effects.
Then, the reduced form regression reduces to α ′βη ′
E[yig|xg,zig]= 1−β +E[zig|xg] 1−β +zigη Result2:Providedβ̸=1andγ=δ=0,thecomposite
parameters α , βη , and η are identified if 1−β 1−β
(1, E [zig |xg ], zig ) are linearly independent. Moreover, the
endogenous effect β is identified if η ̸= 0.
As before, β is not identified, if η = 0 or E[zig|xg] is linear in (1, zig ), i.e.,
if zig is a function of xg,
if E[zig|xg] does not depend on xg, or
if E[zig|xg] is linear in xg and xg is linear in zig.
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Tautological models
We have seen that even when the parameters are not identified, we could have some testable restrictions. Recall the reduced model
α ′γ+βη ′ δ E[yig|xg,zig]= 1−β +E[zig|xg] 1−β +xg 1−β
+z′ η ig
where γ+βη ̸= 0 implies some social effects, either 1−β
γ ̸= 0 or β ̸= 0.
But, some specifications of zig and xg may lead to a tautological model that is consistent with any observed behaviour.
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Tautological models
For example, if zig = h(xg), then the structural model
E[y |x ,z ]=α+βE[y |x ]+E[z |x ]′γ+x′δ+z′ η
iggig igg igg g ig
=⇒E[y |x ]=α+βE[y |x ]+E[z |x ]′γ+x′δ+z′ η igg iggigggig
which always holds with β = 1 & α = γ = δ = η = 0. So, E[yig|xg] = E[yig|xg]…. (We regress the outcome on itself!)
Similarly, if xg = h(zig), then
E[y |x ,z ]=α+βE[y |x ]+E[z |x ]′γ+x′δ+z′ η
iggig igg igg g ig
=⇒E[y |z ]=α+βE[y |x ]+E[z |x ]′γ+x′δ+z′ η ig ig ig g ig g g ig
which always holds with α = β = γ = δ = 0. So, E[yig|zig] = z′ η and therefore only testable restriction is
the linearity assumption.
ig
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Prior knowledge on reference group
The econometrician must know, a priori, how individuals form reference groups i ∈ g with xg for studying social effects.
To see this, suppose the econometrician tries to infer the reference group from the observed behaviour, i.e., the econometrician forms xg using the observed characteristics zig.
Then, xg is determined by zig. So, the model becomes tautological.
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Estimation Strategies
Typically, researchers assume that there is no correlated effect (δ = 0) and there is either only exogenous effects (β = 0,γ ̸= 0) or only endogenous effects (β ̸= 0,γ = 0).
Studies of exogenous effects use two stage method to estimate (γ, η) restricting (β = δ = 0).
Under the parameter restrictions, the reduced form model becomes
E[y |x ,z ]=α+E[z |x ]′γ+z′ η ig g ig ig g ig
Stage 1: estimate E[zig|xg] nonparametrically.
Stage 2: regress yig on 1, E[zig|xg], and zig.
Note here that often xg is discrete, and E[zig|xg] is simply the cell average of zig.
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Estimation Strategies
Studies of endogenous effects also use two stage method to estimate (β, η), restricting (γ = δ = 0).
The structural model reduces to
E[y |x ,z ]=α+βE[y |x ]+z′ η
iggig igg ig
Stage 1: estimate E[y|x] nonparametrically
Stage 2: regress yig on 1, E[y|xg], and zig.
Many nonparametric estimates E[y|xg] are in the form of weight average (LOWESS), i.e., E[y|x] := ig ωig(x)yig
Then, the representation above has the form of the spatial correlation model
y =α+βω(x)y+z′η+u.
ig igig ig ig ig
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Nonparametric endogenous effects model
The regressions E[yig|xg,zig] does not have to be linear.
Forsomeunknownfunctionf :R×RK →R,wehave
E[yig|xg,zig] = f(E[yig|xg],zig)
The endogenous effects can be measured by the
difference
f(E[yig|xg],zig)−f(E[yig|x ̃g],zig)
at two different points xg and x ̃g, holding zig at a certain
point.
Manski (1993) does not provide identification conditions, but discusses conditions under which the endogenous effects cannot be nonparametrically identified; see the reference.
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Demand analysis
The endogenous social effects model can be used for demand analysis:
individual demand yig for a product varies with price p(xg), which is partly determined by aggregate demand in the relevant market xg.
So, the individual demand model can be written as E[yig|xg,zig] = D[p(xg),zig]
where zig is individual characteristics and D is the mean demand.
Equilibrium price p(x) is determined by the aggregate demand and supply condition of market g,
p(xg) = π{E[yig|xg]m(xg),s(xg)}
where m(xg) is the size of market g and s(xg) is the supply condition.
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Demand analysis
Then, the individual demand model is
E[yig|xg,zig] = D[π{E[yig|xg]m(xg),s(xg)},zig]
which is different from the endogenous effects model we have studied,
E[yig|xg,zig] = f(E[yig|xg],zig)
But, if we assume that m(xg) and s(xg) do not depend on xg, all markets have the same size and homogenous supply conditions, then the demand model can be written as
E[yig|xg,zig] = D[E[yig|xg],zig]
and analysed in the framework of endogenous social effects model.
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