Mini Course on Structural Estimation ofStatic and Dynamic Games
Junichi Suzuki
University of Toronto

June 1st, 2009

About the Instructor

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Name: Junichi SuzukiIA¢ liation: Dept. of Economics, University of Toronto

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Major: Empirical Industrial Organization, Urban
Economics

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Studied in Keio (BA, MA) and Univ. of Minnesota (PhD)Today’s Goal
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Present the very basics of estimating structuralparameters ofstatic games (lecture 1)
single-agent dynamic optimization problem (lecture 2)
I dynamic games (lecture 3)
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Designed for practitioners

IFocus on their implementation rather than proving theirstatistical properties etc

Outline of Each Section
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Construct a basic model

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Clarify the type of data

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Consider three approaches:
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Conventional MLE

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Nested-pseudo likelihood by Aguirregabiria and Mira(2007)

I"BBL" by Bajari, Benkard and Levin (2007)

What Not Covered Today (But Typically Covered
Graduate Empirical IO!)

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Demand: Berry, Levison and Pakes (1992), Nevo (2001),Petrin (2002)

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Productivity: Olley and Pakes (1996), Levinsohn andPetrin (2003)

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Auctions: Haile and Tamer (2003)

What is Structural Estimation?
IConsider a parametric model that characterizes agents’behavior and equilibrium

IThe model should be consistent with economic theory

IEach parameter of the model represents agents’primarycharacteristics
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IPreference
Technology

Structural estimation aims to identify these parametersfrom the dataWhy Structural Estimation?
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Pros
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Can present channels through which policy a¤ects theresulting equilibriumI

Can simulate policy impacts on welfare

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Closely related to economic theory

IAssumptions made are explicit

ConsI

High entry cost (theory, econometrics, numericalmethods, data mining etc..)

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Often require signi…cant amount of computations

Computation
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Need to be familiar with some programming language

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For most cases, STATA is not enoughI

One way is to use matrix-based languages (e.g., Matlab,Gauss)
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Easy to write a program
Speed is slow

Another option is to use primitive languages (e.g.,Fortran, C)
IITime consuming to write a program
Speed is faster

Part I: Estimation of Static Games

Motivation
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Many economic activities involve interaction betweenagentsStore opening of convenience stores
Adoption of technologies: VHS vs Beta, Blue-ray vs HD
DVDI Product type choice: high-end service, low-end service
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Estimation should take into account potential interactions
between agentsI

Need game theoretic models

Model: Simple Simultaneous Static GameIN players: i 2 f1, . . . N g

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Each player’s choice ai 2 A = f0, 1, . . . , K g

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Each player’s payo¤:ui (ai , a i , s, ei ) = π i (ai , a i , s ) + ei (ai )IIs: state variables
ei : choice-speci…c private shock: variables unobservableto econometricians, ei (0) = 0

Examples
IBresnahan and Reiss (1991): A = fEntry, Notg

IMazzeo (2002):
A = fNot, Entry to low end, Entry to high endg

ISeim (2006) :A = fNot, Enter to Mkt 1, . . . , Enter to Mkt Mg

ISuzuki (2009):
A = fNot, Open 1 hotel, . . . ,Open 7 hotelsg

Case 1: Game of Complete Information
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Each player observes not only its own ei but also its
rivals’e iI

ei can be …rm-speci…c (ei 6= ej ) as well asmarket-speci…c (ei = ej )I

Players do not face uncertainty (but econometricians do!)IA pure strategy Nash equilibrium of this game is a set ofstrategies fai (s, e)gNi =1 such that
π i (ai (s, e) , a i (s, e) , s ) + ei (ai (s, e))π i (ai (s, e) , a i (s, e) , s ) + ei (ai (s, e))for all i 2 f1, . . . N g and ai 2 A

Case 2: Game of Incomplete Information

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Each player can observe only its own ei but not e iIOnly the distribution of e i is known

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Each player makes its decision based on its belief about
the distribution of its rivals’decisions

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Need to employ a Bayesian Nash equilibrium as anequilibrium conceptPure Strategy Bayesian Nash Equilibrium
1. a set of strategies fai (s, ei , σ i ( ))gNi =1 and2. equilibrium beliefs fσi (ai , s )gNi =1

such that∑ σ i (a i ,s ) [πi (ai (s, ei , σ i ( )) , a i , s )a i+ei (ai (s, ei , σ i ( )))]
∑ σ i (a i ,s ) [πi (ai , a i , s ) + ei (ai )]for all i 2 f1, . . . N g and for all ai 2 A andσi (a, s ) =Z1(a = arg max ∑ σ i (a i , s ) [π i (ai (s, ei , σ i ( ))a i+ei (ai (s, ei , σ i ( )))]dF (ei )Estimation
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Want to recover the structural parameters of
π (ai , a i , s ) from the data

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Data should consist of …rms’decisions fai gN
i =1state variables s, coming from several markets

IMaximum likelihood is the most straightfoward way

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Stick with a simple entry model

IStart with a mere regression and examine why it isproblematicandExample: Entry Model
IConsider the following entry model:"π (ai , a i , s ) = ai α1 + α2 ln Pop
aiα3∑ aj 6=ij!

2 f0, 1g

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One’s pro…t depends on the number of rival …rms andlocal market size

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Each …rm has two options: "enter" (ai = 1) or "notenter" (ai = 0)

+ ei#Estimation: Reduced-Form Regression
IConsider the following reduced form regression:y = β1 + β2 ln Pop + η

where y =1 if y > 00 otherwise

IParameter estimates will be consistentI

β2 does not re‡ect the direct impacts of population
increase on pro…ts (β2 6= α2 )I

Rather, it also includes the impacts of its rivals’entry
triggered by population increase

Estimation: Ignoring Interaction
INext consider the following regression:y= α1 + α2 ln Pop

α3∑ aj 6=ij

where y =!

+ ei

1 if y > 00 otherwise

IIn a game of complete information, resulting estimatesare inconsistent since αj and ei are correlated

IWhen ei is high, its rivals are less likely to enter, causing
underestimation of α3I

In a game of incomplete information, resulting estimates
are incorrect since player i does not know the value of aj
when it makes its own decision

Estimation: Taking Interaction Into Account
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Want to estimate the model by explicitly taking into
account interaction between players

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Possible multiple equilibria are one of the main obstacles

IGames of complete information:
IIfor a given error term fei gNi =1 , more than one pair ofentry decisions satisfy the conditions for N.E.

Game of incomplete informationI

more than one belief and entry policy satisfy theconditions for B.N.E.

Dealing with Multiple Equilibria
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When a model has multiple equilibria, likelihood is not
well-de…ned

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Several ways to deal withI

Look at a variable that is unique to all equilibria (e.g.,
the total number of entrants)

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Impose some arbitrary selection rule (e.g., pick the onethat maximizes total pro…t)

IBound estimators

Computational Issue: A Game of Complete
Information
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Assume that the model has the unique equilibrium

IA game of COMPLETE information often requires thecalculation of highly complicated integrals

ITo calculate the chance of certain events, need to …nd all
combinations of fei gN
i =1 that leads this event andcalculate the integrals

IOften requires simulation to calculate the integralComputational Issue: A Game of Incomplete
Information
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Calculation of the likelihood in a game of INCOMPLETE
information requires the calculation of equilibrium belief

ITo evaluate the likelihood function for certain parametervalues,calculate the equilibrium belief as a …xed point of thebest response function
I calculate the probability that each player picks the choice
I take log and summing them upIIThis algorithm is called a nested …xed-point algorithm

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Note that …nding the …xed point for every set ofparameter can be computationally super costly!

ISee Seim (2006) for its implementation

Example:I

Let’s go back to the simple example:"π (ai , a i , s ) = ai α2 ln Popaiα3∑ aj 6=ij!

+ ei#2 f0, 1gI

Assume …rms are symmetric and play the sameequilibrium strategy and hence the same equilibrium belief
σIConsider applying MLE

Nested Fixed Point Algorithms
ITo evaluate the likelihood for a given (α1 , α2 , α3 ), need
to …nd equilibrium belief …rst
σ (α) = Φn 1

α2 ln Pop

α3 ∑k =0nk

σ k (1σ )n k k

INote that you might …nd more than one σ (α) thatsatis…ed this equation

INext evaluate the resulting likelihood by calculating
Li = σ (α)1(ai =1) (1 σ (α))1(ai =0)ln L = ∑ [1 (ai = 1) ln σ (α) + 1 (ai = 0) ln (1!

σ (α))]

Di¢ culty in Nested Fixed Point Algorithms
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Calculating equilibrium belief for a given parameterrequires solving all solutions for a system of nonlinear
equations

INo algorithm guarantees to …nd all solutions

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Need to rely on generic methods such as homotopy
methodI

When the model has multiple equilibria, likelihoods are
not well-de…ned

Two-Step Methods

INested …xed point algorithm is not practical when gamesinvolve many players and large choice sets

ITwo step methods avoid this computation problem at theexpense of e¢ ciency (but not consistency!)

IYou can apply similar idea to the estimation of
single-agent dynamic optimization problem as well as
dynamic games

Step 1: Estimate Reduced-Form Policy Functions
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Estimate each agent’s choice probabilities conditional onstate variables in a ‡exible way

IIn practice, people use logit/probit by adding statevariables and their interaction terms

ICan use more ‡exible semiparametric method as well. SeeBajari et al.

IThis policy function should represent their equilibriumstrategy

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Implicitly assume that players always pick the sameequilibrium even under multiple equilibria

Step 2: Estimate Structural Parameters
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Assume its rivals follow the policy function estimated inthe …rst step

IFor each possible choice, we can calculate choice-speci…cexpected payo¤

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That transforms the model into the one of single-agent
discrete choice model

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Estimation only involves multinomial probit/logit

INo need to …nd the …xed point anymore

Step 1: Estimating Policy Functions
IConsider the following a ‡exible logit/probit:= β1 + β2 ln Pop + β3 (ln Pop )2 + η
1 if y > 0where y =0 otherwiseyI

Assuming symmetry, can calculate the probability of entry
conditional on populationp̂ (Pop ) = Pr β1 + β2 ln Pop + β3 (ln Pop )2 + η > 0= 1I

Φβ1β2 ln Pop

β3 (ln Pop )2Can calculate the distribution of its rivals’entry decisions!
b ∑ aj 6=i = k = n p̂ (Pop )k (1 p̂ (Pop ))n k
PrkjStep 2: Estimating Structural Parameters
INow we can estimate structural parameters

IEstimate the following binomial discrete choice model"! #N 1b ∑ aj 6=i = k k + ei
y = α1 + α2 ln Pop α3 ∑ Prk =0jI

Note that we transformed a model with interactions
between players into single-agent discrete choice modelI

We are going to use the same trick again and again

Nested Pseudo Likelihood Approach

IAguiregabiria and Mira (2007) suggests iterating thistwo-step method

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Iteration does not help to increase asymptotic e¢ ciencyIIn …nite sample, iteration might help to improve e¢ ciency

Implementing NPL
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Using this updated-policy function, maximize the(pseudo) likelihood and obtain new updated parameterestimates

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Using parameter estimates and policy function as given,calculate each player’s best response

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Check if updated policy functions are close enough to theprevious policy function

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Iterate this process until you get convergenceSummaryI

Study very basics of estimation of static games

IAs games become complicated, brute-force estimation
becomes impracticalI

Two step method works at the expense of e¢ ciency