Structural Static ModelsStructural Static Models

December 2008December 2008Steven SternSteven Stern

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

iii

iiii

vXhyvXgXy

22222

1212111

Semiparametric SpecificationSemiparametric Specification

Estimate using IchimuraEstimate using Ichimura

iii

iiii

vXhyvXgXy

22222

1212111

22210 : H

InterpretationInterpretation

iidFuu

uXrr

uXww

ri

wi

ririi

wiwii

~,

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

Tamer ProblemTamer Problem

iii

iiii

uXyy

uXyy

2212*21

1121*1

Tamer ProblemTamer Problem

*1iy

*21y

iii

iiii

uXyy

uXyy

2212*21

1121*1

1iX

2iX0, 21

0, 21

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

Value of Thinking thru StructureValue of Thinking thru Structure

Policy AnalysisPolicy Analysis

DisciplineDiscipline

Clarity Clarity

Fun Fun