程序代写代做代考 ECON6300/7320/8300 Advanced Microeconometrics Cross-Sectional Dependence

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,thesocialeffectparameter􏰛γ+βη􏰜isnot 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
􏰉 Itcanbe􏰛γ+η􏰜̸=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+β􏰁ω(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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