identi–cation of structural parameters in dynamic discrete...

Identification of Structural Parametersin Dynamic Discrete Choice Games

with Fixed Effects Unobserved Heterogeneity

Victor Aguirregabiria, Jiaying Gu, & Pedro Mira

IAAE CONFERENCE 2019 - NICOSIA

Jue 26, 2019

Aguirregabiria, Gu, & Mira (IAAE CONFERENCE 2019 - NICOSIA)FE Dynamic Games Jue 26, 2019 1 / 24

Introduction / Motivation

Empirical Dynamic Discrete Games

Dynamic Discrete Choice games are useful tools for empirical analysisof strategic and social interactions that involve dynamic(investment) decisions.

Important applications:- Firms’entry/exit and investment.- Dynamic social interactions and peer effects.- Electoral competition (Sieg and Yoon, 2017)- Labor supply within the family (Eckstein and Lifshitz, 2015)



Structural Parameters and Unobserved Heterogeneity

Two types of structural parameters are particularly important:- Switching or Adjustment cost parameters (dynamics)- Strategic or social interaction parameters (game)

Identification of these parameters depends crucially on thespecification of Unobserved Heterogeneity (UH).

Misspecifying Persistent UH can imply substantial biases in theestimation of parameters capturing dynamics. Spurious Dynamics.

Misspecifying UH common to players can imply substantial biasesin the estimation of parameters capturing strategic or socialinteractions between players. Spurious positive spillover.



Fixed Effects - Suffi cient Statistics Approach

We study the identification of these structural parameters in dynamicgames where N players are observed playing the game at M marketsand T periods of time, where T is small and M is large.

We consider a Fixed Effects model for the time-invariantPlayer-Market UH.

- The joint distribution of the UH and the initial values ofendogenous variables is unrestricted.

We consider a Suffi cient Statistics - Conditional Likelihoodapproach (Chamberlain, 1985, Honore & Kyriadzidou, 2000).

Is there a vector of statistics such that conditional on this vector theprobability of the observed history of players’choices does not dependon UH but still depends on structural parameters?



Two main challenges for Dynamic Games

[1] Continuation values. If players are forward-looking, theirdecisions depend on continuation values and these are nonlinear functionsof UH and endogenous state variables.

For single-agent dynamic discrete choice models, Aguirregabiria, Gu,and Luo (2018) show that we can get identification.

[2] Multiple equilibria. The model implies only bounds onprobabilities of choice histories.

The standard Suffi cient Statistic - Conditional ML approach needs tobe extended to deal with this case.



Outline

1. Model

2. Identification2.1. Model with unique predictions2.2. Model with multiple equilibria (bounds)

3. Conclusions and Extensions


Model

– – – – – – – – – – – – – – – – – – – – – – – – – – – –

1. Model– – – – – – – – – – – – – – – – – – – – – – – – – – – –


Model

Model: Basic Features

For today’s talk, I will focus on a dynamic game with:- Two players: i ∈ {1, 2}- Binary choice: yimt ∈ {0, 1}- Logit i.i.d. transitory shocks εimt- The only endogenous state variables are (y1m,t−1, y2m,t−1)- Complete information

At every period t, players choose simultaneously their actions (y1mtand y2mt) to maximize: Et [∑∞

s=0 δsim Uim,t+s ].

Let ∆Uimt ≡ Uimt (1, yjmt )− Uimt (0, yjmt ).

∆Uimt = αim + γi yjmt + βii yimt−1 + βij yjmt−1 − εimt


Model

Model: Best response equations

The equilibrium of the dynamic game can be described by:

y1mt = 1{

ε1mt ≤ α1m + γ1 y2mt + β11 y1mt−1 + β12 y2mt−1+v1(y2mt , αm)

}

y2mt = 1{


}

v1(y2mt , αm) and v2(y1mt , αm) are the "differential continuationvalues": difference between future values of choosing 1 versus 0.

p(α1m , α2m , y1m0, y2m0) is unrestricted, i.e., FE model.


Model

Multiple Equilibria

There is a region in the space of (ε1mt , ε2mt ) with non-uniquepredictions about the outcome (y1mt , y2mt ):


Model

Stackelberg Dynamic Game

If β12 = γ1 = 0, then player 1 is the leader and its current or futuredecisions are not affected by player’s 2.

y1mt = 1{

ε1mt ≤ α1m + β11 y1mt−1 + v1(αm)}

Player 2 is the follower such that:

y2mt = 1{


}This model has unique predictions for the choice probabilities P(ymt| ymt−1; αm) of any outcome ymt .


Identification

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2. Identification– – – – – – – – – – – – – – – – – – – – – – – – – – – –


Identification

Data

The researcher observes panel data for the players playing the gameat M markets and T time periods:

Data = { y1mt , y2mt : m = 1, 2, ...,M ; t = 0, 1, ...,T}

M is large and T is small.

Given these data and the restrictions from the model, the researcheris interested in the estimation of the structural parameters β11, β22,β12, β21, γ1, γ2.

We denote these structural parameters using the vector β.


Identification

Identification of Stackelberg model

Let ym ≡ (y1mt , y2mt : t = 0, 1, ..,T ) be a market history :

P (ym | αm , β) = pαm (y1m0, y2m0)

T

∏t=1

exp { y1mt [α1m + β11 y1mt−1 + v1(αm)] }1+ exp { y1mt [α1m + β11 y1mt−1 + v1(αm)] }

exp { y2mt [α2m + γ2 y1mt + β21 y1mt−1 + β22 y2mt−1 + v2(y1mt , αm)] }1+ exp [α2m + γ2 y1mt + β21 y1mt−1 + β22 y2mt−1 + v2(y1mt , αm)]

where pαm (y1m0, y2m0) = probability of initial condition given αm .


Identification

Identification of Stackelberg model [2]

The log-probability of a market history has the following form:

lnP (ym | αm , β) = s(ym)′ g(αm) + c(ym)′ β

where s(ym) and c(ym) are vectors of statistics.

This structure implies that s(ym) is a suffi cient statistics for αm .

If the elements in the vector [s(ym)′, c(ym)′] are linearly independent,then the maximization of the Conditional log-likelihood functionimplies the identification of the vector of parameters β.


Identification

Identification of Stackelberg Dynamic Game [3]

The switching costs parameters (β11, β22) and the lagged strategiceffect β21 are point identified.

The vectors of suffi cient statistics is:

s(y) =[y10, y20, y1T , y2T , T(1), T(2), T(1,2)

]where T(1) ≡

T

∑t=1y1t ; T(2) ≡

T

∑t=1y2t ; T(1,2) ≡

T

∑t=1y1ty2t

And:c(y)′ β = β11 C(1,1) + β22 C(2,2) + β21 C(2,1)

where C(i ,j) ≡T

∑t=1yit yjt−1


Identification

Examples of identifying pair of histories

Examples of histories and identified parameters with T=3A = {y0, a, b, y3}; B = {y0, b, a, y3}

y0 a b y3 lnP (A)− lnP (B)

Case 1:(00

) (00

) (10

) (10

)β11

Case 2:(00

) (00

) (01

) (01

)β22

Case 3:(00

) (00

) (10

) (01

)β21


Identification

Conditional Likelihood

Consider the Conditional Likelihood (CL) function:

`c (β) =M

∑m=1

lnP (ym | s(ym), β)

Given the structure described above, the CL function has the form:

`c (β) =M

∑m=1

c(ym)′ β− ln[

∑y: s(y)=s(ym )

exp{c(y)′ β

}]

This log-likelihood function is globally concave in the vector β suchthat it is straightforward to compute the CMLE of β using standardgradient methods (Newton, or BHHH).


Identification

Identification of Model with Multiple Equilibria

Using the bounds for players’choice probabilities we can constructbounds for the probability of a market history P (ym | αm , β).

L(ym | αm , β) ≤ lnP (ym | αm , β) ≤ U(ym | αm , β)

We can obtain (non sharp) bounds that have a logit structure andwith the following form:

L(ym | αm , β) = s(ym)′ g(αm) + cL(ym)′ β

U(ym | αm , β) = s(ym)′ g(αm) + cU (ym)′ β

Very importantly, these lower and upper bounds depend on theincidental parameters exactly in the same way, s(ym)′ g(αm).


Identification

Identification of Model with Multiple Equilibria [2]

Let A and B be two market histories such that:s(A) = s(B)

cL(A)− cU (B) 6= 0 OR cU (A)− cL(B) 6= 0Then, we have that:

[cL(A)− cU (B)]′ β ≤ ln(

P (A)P (B)

)≤ [cU (A)− cL(B)]′ β

These inequalities provide set identification of the structuralparameters.

Interestingly, there are pairs of histories such that:

cL(A)− cU (B) = cU (A)− cL(B) 6= 0such that there is point identification of some structural parameters,or linear combination of structural parameters.


Identification

Dynamic Game with Multiple Equilibria

The parameters β11, β22, γ1 and γ2 are set identified.

The vectors of suffi cient statistics is:

s(y) =[y10, y20, y1T , y2T , T(1), T(2), C(1,2), C(2,1)

]with either C(1,2) = C(2,2) or C(2,1) = C(1,1)And:

cL1(y)′ β = β11 C(1,1) + β22 C(2,2) + γ1 T(1) + γ2 T(1,2)cL2(y)′ β = β11 C(1,1) + β22 C(2,2) + γ1 T(1,2) + γ2 T(2)cU (y)′ β = β11 C(1,1) + β22 C(2,2) + γ1 T(1,2) + γ2 T(1,2)


Identification

Example of identifying pair of histories

Examples of histories and identified parameters with T=3Lower Upper

A =(00

)(00

)(11

)(11

)B =

(00

)(10

)(01

)(11

) β11 + γ1 + γ2 β11 + γ1


Identification

Summary & Conclusions

We study the identification of dynamic discrete choice games with aFE specification of UH and using short panels.

Relative to reduced form panel data models, the identification shouldcontrol for the continuation values and deal with multiple equilibria.

We extend the Suffi cient Statistics approach to deal with bounds onthe probabilities of market histories.

We find positive (and some negative) identification results.

Things to do: Inference; Empirical application; Relax the logitassumption.


identi–cation of structural parameters in dynamic discrete...

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