structural static models december 2008 steven stern
DESCRIPTION
Relevant Literature Empirical IO Literature (Berry, BLP, Bresnahan & Riess,Tamer, Aguiregabiria & Mira) Stern Long-Term Care Papers Location Choice (Feyrerra, Bayer)TRANSCRIPT
IntroductionIntroduction
Static Models of Individual BehaviorStatic Models of Individual Behavior Static Models of Equilibrium BehaviorStatic Models of Equilibrium Behavior
Modelling with Estimation in MindModelling with Estimation in Mind EstimationEstimation
ExamplesExamples
Relevant LiteratureRelevant Literature
Empirical IO Literature (Berry, BLP, Empirical IO Literature (Berry, BLP, Bresnahan & Riess,Tamer, Aguiregabiria Bresnahan & Riess,Tamer, Aguiregabiria & Mira)& Mira)
Stern Long-Term Care PapersStern Long-Term Care Papers
Location Choice (Feyrerra, Bayer)Location Choice (Feyrerra, Bayer)
Static Models w/ Single AgentsStatic Models w/ Single Agents
ModellingModelling
EstimationEstimation
ExamplesExamples
ModellingModelling
Utility function and budget constraint Utility function and budget constraint (possibly implied) with errors built into (possibly implied) with errors built into modelmodel
Compute Pr[observed choice] as Compute Pr[observed choice] as statement that error is in range consistent statement that error is in range consistent with observed choicewith observed choice
EstimationEstimation
MLE or MOM with estimation objects MLE or MOM with estimation objects implied by structure of the probability implied by structure of the probability statements associated with modelstatements associated with model
May need simulation methods to integrate May need simulation methods to integrate over relevant subset of error domainover relevant subset of error domain
Example 1: Kinked Budget Set Example 1: Kinked Budget Set AnalysisAnalysis
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
budget constraint
kink point
kink point
indifference curve
indifference curve
Model SpecificationModel Specification
Hausman: hHausman: hikik==ββyyikik++ααwwikik+Z+Ziiγγ+u+uii
Wales & Woodland: specify utility w/ errors Wales & Woodland: specify utility w/ errors built into utility function → indifference built into utility function → indifference curvescurves
Simple example: U= Simple example: U= ββlogL+(1- logL+(1- ββ)logC, )logC, loglogββ~indN(X~indN(Xαα,,σσ22))
Example 2: Heckman Selection Example 2: Heckman Selection ModelModel
Model:Model:
0 iff observed
~,*2
*1
21
222*2
111*1
ii
ii
iii
iii
yy
iidFuuuXy
uXy
Semiparametric SpecificationSemiparametric Specification
Estimate using IchimuraEstimate using Ichimura
iii
iiii
vXhyvXgXy
22222
1212111
22210 : H
InterpretationInterpretation
iidFuu
uXrr
uXww
ri
wi
ririi
wiwii
~,
wiriri
wi
riri
wiwi
iii
XwXruu
uXruXw
rwy
1
1
1
InterpretationInterpretation
iidFuu
uXrr
uXww
ri
wi
ririi
wiwii
~,
wiriri
wi
riri
wiwi
iii
XwXruu
uXruXw
rwy
1
1
1
riwiii XXHXy ,|1Pr
Static Models w/ Multiple AgentsStatic Models w/ Multiple Agents
General Model StructureGeneral Model Structure
EstimationEstimation
ExamplesExamples
General Structure: What is an General Structure: What is an economy?economy?
family in my work;family in my work;
metro area in Feyrerra and Bayer;metro area in Feyrerra and Bayer;
Army unit in Arradillas-Lopez Army unit in Arradillas-Lopez
Notation and StructureNotation and Structure
Define dijk=1 iff ij chooses k, let dij={ dij1, dij2,.., dijK},
and define d/ij to be the set of choices made by other members of the economy other than i.
Objective function of each member i of economy j: Uij(dij,d/ij;β,xij,zj,εij), i=1,2,..,Ij → Pr[dij|d/ij,β,xij,zj]
Pr[dij|d/ij,β,xij,zj]
Define Aij(dij|d/ij,β,xij,zj) = { ε: Uij(dij,d/ij;β,xij,zj,ε)> Uij(d,d/ij;β,xij,zj,ε) d≠ dij}
→ Pr[dij|d/ij,β,xij,zj] = Pr[ε Aij(dij| d/ij,β,xij,zj)]
Note importance of adding randomness to model
Role of InformationRole of Information
Full information: ΩFull information: Ωijij={ ε={ εijij i=1,2,..,Ii=1,2,..,Ijj} → } → issues in existence of an equilibrium or issues in existence of an equilibrium or multiple equilibriamultiple equilibria
Partial information: ΩPartial information: Ω ijij= ε= εijij → each → each member maximizes EUmember maximizes EUijij(d(dijij,d,d/ij/ij;β,x;β,xijij,z,zjj,ε,εijij) ) over the joint density of the other errors over the joint density of the other errors where dwhere d/ij/ij becomes a random vector becomes a random vector
One must be able to solve for One must be able to solve for an equilibrium and, when there an equilibrium and, when there are multiple equilibria, choose are multiple equilibria, choose
among them.among them.
EstimationEstimation
Use Pr[error in appropriate area consistent Use Pr[error in appropriate area consistent w/ choice]w/ choice]
Much emphasis on Tamer (Heckman Much emphasis on Tamer (Heckman logical inconsistency property)logical inconsistency property)
Use moments or likelihood Use moments or likelihood
Moments EstimationMoments Estimation
DefineDefine DDjkjk=Σ=Σii1[ε1[εijij A Aijij(d(dijkijk| d| d/ij/ij,β,x,β,xijij,z,zjj)] )] with with conditional expected value conditional expected value ΣΣiiPr[εPr[εijij AAijij(d(dijkijk| d| d/ij/ij,β,x,β,xijij,z,zjj)])]
Minimize quadratic form in deviations Minimize quadratic form in deviations
between Dbetween Djkjk and its conditional moment and its conditional moment
Moments EstimationMoments Estimation
Issue: What does the deviation between Issue: What does the deviation between the sample and theoretical moments the sample and theoretical moments represent? (What if added an error urepresent? (What if added an error u jj?)?)
Example 1: My Long-Term Care Example 1: My Long-Term Care ModelsModels
Economy is family with n children and n+2 Economy is family with n children and n+2 choiceschoices
Value to family member i of choice k is Value to family member i of choice k is VVjikjik=Z=Zj0j0ββkk+X+Xjkjkδ+Qδ+Qjikjikλ+uλ+ujikjik
Equilibrium mechanisms Equilibrium mechanisms →→ probabilities of probabilities of observed choicesobserved choices
In most recent paper, we model utility In most recent paper, we model utility function of each family member as function of each family member as UUjiji= β= β11logQlogQjj+ (β+ (β22εε22)logX)logXjiji+ (β+ (β33εε33)logL)logLjiji+ (β+ (β44+ε+ε44)t)tjiji+ u+ ujiji
Choices: XChoices: Xjiji, L, Ljiji, H, Hjiji, t, tjiji subject to a budget constraint. subject to a budget constraint.
Construct subsets of the domain of the Construct subsets of the domain of the errors consistent with each observed errors consistent with each observed choice and the maximize the probability of choice and the maximize the probability of errors being in those subsets.errors being in those subsets.
Divorce Model w/ Private Divorce Model w/ Private InformationInformation
UUhh=θ=θhh +ε +εhh-p; U-p; Uww= θ= θww +ε +εww+p+p θθjj=Xβ=Xβjj+e+ejj, j=h,w, j=h,w VVjj[U[Uhh, U, Uww]] Bargaining mechanismBargaining mechanism Data: {X,H,D}Data: {X,H,D}
Indifference Curves
-6
-4
-2
0
2
4
6
8
10
-4 -2 0 2 4 6
u(w)
u(h)
V(h) = -1
V(h) = 0
V(h) = 1
V(h) = 2
V(h) = 3
Divorce Probabilities for Different Decision Makers
0%
20%
40%
60%
80%
100%
-1 0 1 2 3 4 5
Husband's information about happiness = theta(h)+theta(w)+epsilon(h)
Prob
abili
ty o
f div
orce
No planner: Asymmetricinfo w/ caring
No planner: Asymmetricinfo w/ no caringOmniscient planner
Limited planner: Caring
Limited planner: Nocaring
Efficient and Inefficient Divorce Probabilities
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-1 0 1 2 3 4 5
theta(h)+theta(w)+eps(h)
Prob
abili
ty
efficient
inefficient
FeyrerraFeyrerra Economy is a set of school districts in Economy is a set of school districts in
metro areametro area 3 school types: public, private Catholic, 3 school types: public, private Catholic,
private non-Catholicprivate non-Catholic Households differ in income, religious Households differ in income, religious
preferences, and idiosyncratic tastes for preferences, and idiosyncratic tastes for Catholic schools and neighborhoods Catholic schools and neighborhoods
Public school choice depends on Public school choice depends on residence; private does notresidence; private does not
FeyrerraFeyrerra
s=school qualitys=school quality κ=neighborhood qualityκ=neighborhood quality c=consumptionc=consumption ε=idiosyncratic preference for particular ε=idiosyncratic preference for particular
neighborhod/school choiceneighborhod/school choice
Utility:Utility: U(κ,s,c,ε) = sU(κ,s,c,ε) = sααccββκκ1-α-β1-α-βeeεε
FeyrerraFeyrerra Budget constraint: Budget constraint:
c+(1+tc+(1+tdd)p)pdhdh+T=(1-t+T=(1-tyy)y)ynn+p+pnn Production of school quality: Production of school quality:
s = qs = qρρxx1-ρ 1-ρ
q = y(S) q = y(S) where S is set of households who where S is set of households who attend particular school, and y(S) is the attend particular school, and y(S) is the average income of those attending. average income of those attending. sskjkj=R=Rkjkjssjj
FeyrerraFeyrerra
Funding for schools: Funding for schools: for private, x=T; for private, x=T; for public, for public, x=((tx=((tdd(P(Pdd+Q+Qdd))/(n))/(ndd))+AID))+AIDdd
FeyrerraFeyrerra
Household decision problem Household decision problem
Majority rule voting Majority rule voting
EquilibriumEquilibrium
EstimationEstimation
Adding DynamicsAdding Dynamics
Issues w/ modeling dynamic equilibriumIssues w/ modeling dynamic equilibrium
Data needs much greaterData needs much greater
Significant computation problemsSignificant computation problems
Pitfalls of Ignoring StructurePitfalls of Ignoring Structure
Macurdy Criticism of HausmanMacurdy Criticism of Hausman
Feyrerra Errors Interpretation Feyrerra Errors Interpretation ProblemProblem
Linear probability model Linear probability model