Complete Entry Pooling Large Support Partial

Econometric Analysis of Games 1

Abi Adams

HT 2017

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

RecapAim: provide an introduction to incomplete models and partialidentification in the context of discrete games

1. Coherence & Completeness

2. Basic Framework

3. Pooling Outcomes

4. Large Support

5. Partial Identification

6. Adding Structure

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Recap

I Last lecture discussed the rank and order conditions thatcharacterise when linear simultaneous systems are pointidentified.

I This week, we will review identification results in theeconometric analysis of discrete games, which are alsosimultaneous systems

I The existence of multiple equilibria results in additionalidentification issues

I After discussing games of complete information, we willreturn to multiplicity in the context of social interactions

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Completeness & CoherenceI Structural model: a set of structural equations and a

distribution between observed and unobservedexplanatory variables specified by an economic theory

h(y ,X , ε) = 0 (1)

with ε ∈ Ω

I The model is coherent if for each ε ∈ Ω there exists atleast one value of y that satisfies the structural equations

I The model is complete if for each ε ∈ Ω there exists atmost one value of y that satisfies the structural equations

I With a complete and coherent model, a unique reducedform is guaranteed

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Completeness & Coherence

I Incoherent and incomplete models can arise insimultaneous systems

I We start with a simple abstract example, and then willconsider these issues in the context of an entry game

I Consider the following simultaneous system

y1 = I(y2 + ε1 ≥ 0)

y2 = θy1 + ε2(2)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Completeness & Coherence

I We have that

y1 = I(θy1 + ε1 + ε2 ≥ 0) (3)

I With unrestricted errors, the model is incomplete: let−θ ≤ ε1 + ε2 < 0:

y1 = I(θy1 + ε1 + ε2 ≥ 0)

ε1 + ε2 0→ y1 = 0θ + ε1 + ε2 ≥ 0→ y1 = 1

(4)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Completeness & Coherence

I With unrestricted errors, the model is incoherent: let0 ≤ ε1 + ε2 < −θ:

y1 = I(θy1 + ε1 + ε2 ≥ 0)

ε1 + ε2 ≥ 0→ y1 6= 0θ + ε1 + ε2 < 0→ y1 6= 1

(5)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Outline

1. Coherence & Completeness

2. Basic Framework

3. Pooling Outcomes

4. Large Support

5. Partial Identification

6. Adding Structure

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Entry Games

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Entry Games

I Large literature on the estimation on binary games in theempirical IO set-up

I How to measure the competitiveness of a market and thetoughness of competition?

I Bresnahan & Reiss (1991): don’t need toprice/quantity/cost data but can infer from information onequilibrium market structure

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Entry Games: Data

I A firm’s decision to enter depends on its profit, whichdepends on whether the other firm entered the market

I Let profits be given by:

πi = xiβ +4iyj + εi (6)

for i = 1,2

I The distribution of ε ≡ (ε1, ε2) is given by (known) F , withmean normalised to zero and variances to 1

I Start by assuming complete information: realisations of xand ε observed by all players

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Entry Games: Data

I Similar set-up to Ciliberto & Tamer’s (2009) analysis of theairline industry

I Each market is a particular route, the firms being differentairlines

I xi gives the various market and firm specific variables thataffect demand (e.g. population size, income &c) and costsfor firms

I Linearity relaxed in some specifications, but additiveseparability of observables and unobservables typicallyassumed

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Payoff Matrix

y?i = xiβ +4iyj + εi

yi = I(y?i ≥ 0)(7)

Player 20 1

Player 1 0 0, 0 0, x2β2 + ε21 x1β1 + ε1, 0 x1β1 +41 + ε1, x2β2 +42 + ε2

I Nash equilibrium concept

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Payoff Matrix: 4i < 0

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Incomplete

I For certain values of the error term, the model predicts twopossible solutions

I Absent a rule for equilibrium selection, the model isincomplete — cannot write down uniquely determinedlikelihood function because we cannot assign separateprobabilities to (0,1) and (1,0)

I Structural parameters often under-identified from entrydata

I An equilibrium selection mechanism, λ, specifies whichequilibria is selected with which probability

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Incomplete

I How can we deal with this?

I Find a coarser unique prediction — e.g. focus on number ofentrants — useful in a symmetric world but doesn’t alwayswork (Bresnahan and Reiss, 1991; Berry, 1992)

I Exclusion & large support assumptions (Tamer, 2003)

I Complete the model by assuming an equilibrium selectionrule or impose more structure on the game to obtain aunique equilibrium

I Sacrifice point estimates and only find bounds onparameters

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Outline

1. Coherence & Completeness

2. Basic Framework

3. Pooling Outcomes

4. Large Support

5. Partial Identification

6. Adding Structure

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Pooling Multiple Equilibria Outcomes

I A common strategy taken is to aggregate the monopolyoutcomes

I Transform model from one that explains each entrantsstrategy to one that predicts the number of entrants:N = 0,1,2

I Essentially turns the problem into an ordered discretechoice model

I Multiplicity not necessarily an impediment to analysis —here some quantities of interest are invariant acrossequilibria

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Pooling Multiple Equilibria Outcomes

I Likelihood function would then include the probabilitystatements

Pr(N = 0|x) = Pr(x1β1 + ε1 < 0, x2β2 + ε2)

= Fε1,ε2(−x1β1,−x2β2)

Pr(N = 2|x) = Pr(x1β1 +41 + ε1 < 0, x2β2 +42 + ε2)

= Fε1,ε2(−x1β1,−x2β2)

Pr(N = 1|x) = 1− Pr(N = 0|x)− Pr(N = 2|x)(8)

I Joint distribution of 4i determines the specific functionalform for these probability statements

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Pooling Multiple Equilibria Outcomes

I However, lose information about firm heterogeneitiesunless firms are symmetric

I We then have

Pr(N = 0|x) = Pr(ε1 < −xβ)

Pr(N = 1|x) = Pr(−xβ < ε < −xβ −4)

Pr(N = 2|x) = Pr(−xβ −4 < ε)

(9)

I With ε ∼ N(0,1) we get a very simple ordered probit model

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Pooling Multiple Equilibria Outcomes

I While convenient, this strategy imposes strong restrictionson firm level heterogeneity

I Typically want to allow for firms to asymmetric impacts onone another

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Outline

1. Coherence & Completeness

2. Basic Framework

3. Pooling Outcomes

4. Large Support

5. Partial Identification

6. Adding Structure

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Large Support

I Uses similar ideas to those we have discussed in thecontext of special regressors and identification ofsimultaneous systems

I Intuition: find a regressor for which the actions of all butone player are dictated by dominant strategies — turn theproblem into a discrete choice problem by the single agentwho does not play dominant strategies

I For i = 1 or i = 2, there exists a regressor with βik 6= 0 thatis excluded from the other firm’s covariates and haseverywhere positive density

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Large Support

I Point identification of (β1, β2,41,42) then guaranteed if x1and x2 have full column rank

I Can find extreme values fo xik such that only uniqueequilibria in pure strategies are realised, e.g. asx1k → −∞:

Pr(ε1 < −x1β)→ 1Pr(ε1 < −x1 −41β)→ 1

(10)

I Let x?2 be such that:

x?2β2 6= x?2 b2 (11)

which is guaranteed by the full rank condition on x2

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Large Support

I We then have that, as x1k →∞

Pr ((0,0)|x) = Pr(ε1 < −x1β1, ε2 < −x?2β2)

≈ Pr(ε2 < −x?2β2)

6≈ Pr(ε2 < −x?2 b2)

(12)

which implies that β2 is identified

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Large SupportI Let x?1 be such that:

x?1β1 6= x?1 b1 (13)

which is guaranteed by the full rank condition on x1

I We then have that

Pr ((0,0)|x) = Pr(ε1 < −x?1β1, ε2 < −x2β2)

6= Pr(ε1 < −x?1 b1, ε2 < −x2β2)(14)

which implies that β1 is identified

I Could also introduce a separate large support regressorfor firm 2

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Identification at Infinity

I This strategy is often called identification at infinity —using independent variation in one regressor while drivinganother to take extreme values on its support identifies theparameters

I Used in many different applications — Heckman (1990) &c

I However, note that there are important implications forinference: this will lead to asymptotic convergence ratesthat are slower than parametric rates as the sample size(i.e. number of games) increases (see Kahn & Tamer(2010); Bajari et al (2011))

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Outline

1. Coherence & Completeness

2. Basic Framework

3. Pooling Outcomes

4. Large Support

5. Partial Identification

6. Adding Structure

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Partial Identification

I Typically covariates’ support is not rich enough to admit astrategy based on Tamer (2003)

I Can instead rely on partial information and use bounds forestimation — while a model may not make exactpredictions, it may still meaningfully restrict the range ofpossible outcomes

I Advocated by, e.g., Manski (1995)

I These results can be useful for testing particularhypotheses or illustrating the range of possible outcomes

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Partial Identification

I Given that the model is incomplete, the probability ofoutcome (0,1) and (1,0) cannot be written as a function ofthe structural parameters

I The model instead provides upper and lower bounds onthese probabilities

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(15)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(16)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(17)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(18)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≥ PL,01(β,4)

= Pr(ε1 < −x1β1,−x2β2 < ε2)

+ Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 −42 < ε2)(19)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≤ PU,01(β,4)

= Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 < ε2 < −x2β2 −42

)+ PL,01

(20)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Bounds

Pr((0,1)|x) ≤ PU,01(β,4)

= Pr(−x1β1 ≤ ε1 < −x1β1 −41,−x2β2 < ε2 < −x2β2 −42

)+ PL,01

(21)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Partial Identification

I Can derive similar expressions for the probabilities of otheroutcomes

I The information about the parameters can then berepresented as:

PL,00(θ)PL,01(θ)PL,10(θ)PL,11(θ)

≤

Pr((0,0)|x)Pr((0,1)|x)Pr((1,0)|x)Pr((1,1)|x)

≤

PU,00(θ)PU,01(θ)PU,10(θ)PU,11(θ)

(22)

I The identified set, ΘI , is the set of all parameters thatsatisfy these inequalities

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Example: Ciliberto & Tamer (2009)

I Interested in measuring competitive pressures in the airlineindustry, i.e. the impact on profit of a firm entering aparticular market

I Want to allow for heterogeneity across firms — i.e. theentry decision of American Airlines has a different effect onthe profit of its competitors than the entry of a low costairline

I Some important policy questions regards rules on airportpresence on number of routes served &c

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Example: Ciliberto & Tamer (2009)I Each observation is a route served between two airports

I Cross section study and thus assume firms are in long runequilibrium

I Firm profit function given by:

y?im = Smαi + Zimβi + Wimγi +∑j 6=i

θij yjm +

∑j 6=i

Zjmθij yjm + εim

(23)

I S is a vector of common market characteristicsI Z is a matrix of firm characteristics that enter into the profit

functions of all firmsI W are firm specific characteristics such as cost variables

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Example: Ciliberto & Tamer (2009)

I Have the (more complicated!) set of moment equalities PL,1(θ)...

PL,2N (θ)

≤ Pr(y1|x)

...Pr(y2N |x)

≤

PU,1(θ)...

PU,2N (θ)

(24)

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Example: Ciliberto & Tamer (2009)

I The estimator uses the following objective function

Q(θ) =

∫ [|| (Pr(y |X )− PL(X , θ))− ||

+|| (Pr(y |X )− PH(X , θ))+ ||]

dFx

(25)

where

(A)− =

a11(a1 ≤ 0)...

a2N 1(a2N ≤ 0)

(26)

and (A+) defined similarly

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Example: Ciliberto & Tamer (2009)

I Note that Q(θ) ≥ 0 and Q(θ) = 0 iff θ ∈ ΘI

I To estimate ΘI , take the sample analog of Q(·)

QN(θ) =1N

∑m

[|| (Pn(Xm)− PL(Xm, θ))− ||

+|| (Pn(Xm)− PH(Xm, θ))+ ||] (27)

where Pn(Xm) can be estimated non-parametrically and PLand PH are computed via simulation

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Example: Ciliberto & Tamer (2009)

I Unless the number of firms is very small, the upper andlower bound probabilities do not have a convenient closedform

I Common problem with more complicated models:likelihood or moment inequalities do not have a closedform that can adapt standard Maximum Likelihood or GMMmethods for

I Proceed by simulation: calculate an upper and lowerbound for each equilibrium probability for every X for aparticular guess of the parameter values

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Simulation ProcedureI 1. Draw R simulations of the firm unobservables, εr —

these draws and stored and held fixed throughout theoptimisation procedure

I NB can allow for correlation in these draws if each firm’sdraw is random normal by transforming the error termsusing the Cholesky Decomposition of the covariance matrix

I 2. For a given X , a particular draw of the error term, εr andan initial guess of the parameter vector, θ, calculate thevector of firm’s profits for a particular set of entry decisions,y j

I 3. If each firm is earning non-negative profits, then theoutcome y j is an equilibrium

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Simulation Procedure

I 4. If this equilibrium is unique (for no other y j is it true thatall firms profits are positive), then add 1

R to the lower andupper bound for this outcome

I 5. If this equilibrium is not unique, then add 1R to the upper

bound only

I 6. Repeat steps 1-5 for each Xm and εr , r = 1, ...,R

I 7. Calculate the objective simulation estimates of PL andPH

I 8. Repeat steps 1-7 as search for the value of θ thatminimises QN

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Data

I 2001 Airline Origin and Destination Survey

I Markets defined as trips between airports — data setincludes 2742 marets

I Focus on American, Delta, United, Southwest, ‘Medium’Airlines and Low Cost Carriers

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Results Summary

I Heterogeneity in profit functions

I Competitive effects of large and low-cost airlines aredifferent

I Competitive effects of an airline increasing in its airportpresence

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Results

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Results

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Results

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Results

Abi Adams

TBEA

Complete Entry Pooling Large Support Partial

Conclusion

I To finish, note that many other economic models withsimilar structures

I (Especially!) with strategic interactions, need to consider ifyour model is complete and consider the implications foridentification

I Identification can be achieved through a number ofstrategies; which one is preferable will depend on thecontext and data to hand

Abi Adams

TBEA