Structural Static Models
December 2008
Steven Stern
Introduction
 Static
Models of Individual Behavior
 Static Models of Equilibrium Behavior
 Modelling
with Estimation in Mind
 Estimation
 Examples
Relevant Literature
 Empirical
IO Literature (Berry, BLP,
Bresnahan & Riess,Tamer, Aguiregabiria &
Mira)
 Stern
Long-Term Care Papers
 Location
Choice (Feyrerra, Bayer)
Static Models w/ Single Agents
 Modelling
 Estimation
 Examples
Modelling
 Utility
function and budget constraint
(possibly implied) with errors built into
model
 Compute Pr[observed choice] as
statement that error is in range consistent
with observed choice
Estimation
 MLE
or MOM with estimation objects
implied by structure of the probability
statements associated with model
 May
need simulation methods to integrate
over relevant subset of error domain
Example 1: Kinked Budget Set
Analysis
3.5
3
2.5
budget constraint
2
kink point
kink point
1.5
indifference curve
indifference curve
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Model Specification
 Hausman:
 Wales
hik=βyik+αwik+Ziγ+ui
& Woodland: specify utility w/ errors
built into utility function → indifference
curves
 Simple example: U= βlogL+(1- β)logC,
logβ~indN(Xα,σ2)
Example 2: Heckman Selection
Model
 Model:
y1*i  X 1i 1  u1i
y2*i  X 2i  2  u2i
u1i , u2i  ~ iidF
y1*i observed iff y2*i  0
Semiparametric Specification
y1i  X1i 1  g  X 2i  21   v1i
y2i  h X 2i  22   v2i
Semiparametric Specification
y1i  X1i 1  g  X 2i  21   v1i
y2i  h X 2i  22   v2i
 Estimate
using Ichimura
H 0 : 21  22
Interpretation
wi  w X i  w   uiw
ri  r  X i  r   uir
u
w
i

, uir ~ iidF
Interpretation
wi  w X i  w   uiw
ri  r  X i  r   uir
u
w
i

, uir ~ iidF
yi  1wi  ri 

 1u


 1 w X i  w   uiw  r  X i  r   uir
w
i
 uir  r  X i  r   w X i  w
Interpretation
wi  w X i  w   uiw
ri  r  X i  r   uir
u
w
i

, uir ~ iidF
yi  1wi  ri 

 1u


 1 w X i  w   uiw  r  X i  r   uir
w
i
 uir  r  X i  r   w X i  w
Pr yi  1 | X i   H  X i  w , X i  r 
Static Models w/ Multiple Agents
 General
Model Structure
 Estimation
 Examples
General Structure: What is an
economy?
 family
in my work;
 metro
area in Feyrerra and Bayer;
 Army
unit in Arradillas-Lopez
Notation 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 Information
 Full
information: Ωij={ εij i=1,2,..,Ij} →
issues in existence of an equilibrium or
multiple equilibria

Partial information: Ωij= εij → each
member maximizes EUij(dij,d/ij;β,xij,zj,εij)
over the joint density of the other errors
where d/ij becomes a random vector
One must be able to solve for
an equilibrium and, when there
are multiple equilibria, choose
among them.
Estimation
 Use
Pr[error in appropriate area consistent
w/ choice]
 Much
emphasis on Tamer (Heckman
logical inconsistency property)
 Use
moments or likelihood
Tamer Problem
y1*i  1 y2i  X i 1  u1i
*
y21
  2 y1i  X i  2  u2i
Tamer Problem
y1*i  1 y2i  X i 1  u1i
*
y21
  2 y1i  X i  2  u2i
*
y21
1, 2  0
1, 2  0
X i 1
X i 2
y1i*
Moments Estimation
 Define
Djk=Σi1[εij Aij(dijk| d/ij,β,xij,zj)]
with conditional expected value
ΣiPr[εij Aij(dijk| d/ij,β,xij,zj)]

 Minimize
quadratic form in deviations
between Djk and its conditional moment
Moments Estimation
 Issue:
What does the deviation between
the sample and theoretical moments
represent? (What if added an error uj?)
Example 1: My Long-Term Care
Models
 Economy
is family with n children and n+2
choices
 Value to family member i of choice k is
Vjik=Zj0βk+Xjkδ+Qjikλ+ujik
 Equilibrium mechanisms → probabilities of
observed choices
 In
most recent paper, we model utility
function of each family member as
Uji= β1logQj+ (β2ε2)logXji+ (β3ε3)logLji+ (β4+ε4)tji+ uji
 Choices:
Xji, Lji, Hji, tji
subject to a budget constraint.
 Construct subsets of the domain of the
errors consistent with each observed
choice and the maximize the probability of
errors being in those subsets.
Divorce Model w/ Private
Information
 Uh=θh
+εh-p; Uw= θw +εw+p
 θj=Xβj+ej, j=h,w
 Vj[Uh, Uw]
 Bargaining mechanism
 Data: {X,H,D}
Indifference Curves
10
8
6
V(h) = -1
u(h)
4
V(h) = 0
2
V(h) = 1
V(h) = 2
0
-4
-2
0
2
-2
-4
-6
u(w)
4
6
V(h) = 3
Divorce Probabilities for Different Decision Makers
Probability of divorce
100%
No planner: Asymmetric
info w/ caring
80%
No planner: Asymmetric
info w/ no caring
60%
Omniscient planner
40%
Limited planner: Caring
20%
Limited planner: No
caring
0%
-1
0
1
2
3
4
Husband's information about happiness =
theta(h)+theta(w)+epsilon(h)
5
Efficient and Inefficient Divorce Probabilities
0.9
0.8
0.7
Probability
0.6
0.5
efficient
0.4
inefficient
0.3
0.2
0.1
0
-1
0
1
2
3
theta(h)+theta(w)+eps(h)
4
5
Feyrerra
 Economy
is a set of school districts in
metro area
 3 school types: public, private Catholic,
private non-Catholic
 Households differ in income, religious
preferences, and idiosyncratic tastes for
Catholic schools and neighborhoods
 Public school choice depends on
residence; private does not
Feyrerra
 s=school
quality
 κ=neighborhood quality
 c=consumption
 ε=idiosyncratic preference for particular
neighborhod/school choice
 Utility:
U(κ,s,c,ε) = sαcβκ1-α-βeε
Feyrerra
 Budget
constraint:
c+(1+td)pdh+T=(1-ty)yn+pn
 Production of school quality:
s = qρx1-ρ
q = y(S)
where S is set of households who attend
particular school, and y(S) is the average
income of those attending.
skj=Rkjsj
Feyrerra
 Funding
for schools:
for private, x=T;
for public,
x=((td(Pd+Qd))/(nd))+AIDd
Feyrerra
 Household
 Majority
decision problem
rule voting
 Equilibrium
 Estimation
Adding Dynamics
 Issues
 Data
w/ modeling dynamic equilibrium
needs much greater
 Significant
computation problems
Pitfalls of Ignoring Structure

Macurdy Criticism of Hausman

Feyrerra Errors Interpretation Problem

Linear probability model
Value of Thinking thru Structure
 Policy Analysis
 Discipline
 Clarity
 Fun