1. Structural Static Models December 2008 Steven Stern

2. Introduction • Static Models of Individual Behavior • Static Models of Equilibrium Behavior • Modelling with Estimation in Mind • Estimation • Examples

3. Relevant Literature • Empirical IO Literature (Berry, BLP, Bresnahan & Riess,Tamer, Aguiregabiria & Mira) • Stern Long-Term Care Papers • Location Choice (Feyrerra, Bayer)

4. Static Models w/ Single Agents • Modelling • Estimation • Examples

5. 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

6. 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

7. Example 1: Kinked Budget Set Analysis

8. Model Specification • Hausman: hik=βyik+αwik+Ziγ+ui • Wales & Woodland: specify utility w/ errors built into utility function → indifference curves • Simple example: U= βlogL+(1- β)logC, logβ~indN(Xα,σ2)

9. Model: Example 2: Heckman Selection Model

10. Semiparametric Specification

11. Semiparametric Specification • Estimate using Ichimura

12. Interpretation

13. Interpretation

14. Interpretation

15. Static Models w/ Multiple Agents • General Model Structure • Estimation • Examples

16. General Structure: What is an economy? • family in my work; • metro area in Feyrerra and Bayer; • Army unit in Arradillas-Lopez

17. 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]

18. 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

19. 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

20. One must be able to solve for an equilibrium and, when there are multiple equilibria, choose among them.

21. Estimation • Use Pr[error in appropriate area consistent w/ choice] • Much emphasis on Tamer (Heckman logical inconsistency property) • Use moments or likelihood

22. Tamer Problem

23. Tamer Problem

24. 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

25. Moments Estimation • Issue: What does the deviation between the sample and theoretical moments represent? (What if added an error uj?)

26. 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

27. 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.

28. 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}

32. 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

33. 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

34. 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

35. Feyrerra • Funding for schools: for private, x=T; for public, x=((td(Pd+Qd))/(nd))+AIDd

36. Feyrerra • Household decision problem • Majority rule voting • Equilibrium • Estimation

37. Adding Dynamics • Issues w/ modeling dynamic equilibrium • Data needs much greater • Significant computation problems

38. Pitfalls of Ignoring Structure • Macurdy Criticism of Hausman • Feyrerra Errors Interpretation Problem • Linear probability model

39. Value of Thinking thru Structure • Policy Analysis • Discipline • Clarity • Fun