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

June 1st, 20091

Part III: Estimation of Dynamic Games2

Introduction
IFirms often compete each other overtime

Ie.g. chain stores trying expand their networkI

One’s current decision a¤ect its own current and future
pro…tby its decision itself (direct e¤ect)by a¤ecting its own future decision (indirect)
I by a¤ecting its rivals’current and future decisions(indirect)

III

Needs to construct a dynamic game3

Estimation of Dynamic GameI

Dynamic game contains
IIstrategic interaction between agents
dynamic optimization

I

Straightforward estimation (nested …xed point maximumlikelihood) is virtually impossible (don’t even think aboutit!)

I

Need to rely on two-step methods:
Nested pseudo likelihood (NPL) by Aguirregabiria andMira (2007)
I Another two-step method or BBL by Bajari, Benkardand Levin (2008)
4I

Recent ResearchI

Ryan (2009), MIT WP: Studies the impacts ofenvironmental regulation on entry/investment decisions ofU.S. Portland cement industry

I

Sweeting (2009), Duke WP: Studies whether radio
stations reposition their products to deter future entries

ISnider (2009), Minnesota WP: Studies whether airline
companies are engaged in predatory behavior

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Example: Competition between Hotel Chains
ISuzuki (2009) considers a dynamic entry-exit gamebetween hotel chains to recover the impacts of land use
regulation on the cost structure of hotel chains

I

Assume each local market is isolated

I

Hotel chains decide the number of hotels they operate
every period

I

Hotel chains incurII

one-shot sunk entry cost to open a new hotelsunk entry cost is random and private
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Empirical Structure

IThe data
the number of hotels each chain operates for a certain
periodI each hotel’s revenue
I market characteristics
II

By using these data, want to recover the cost parametersthat rationalize both entry/exit behavior and revenue data

7

Idea of Identi…cation

I

Suppose you observe a market with one …rm overtime

I

This observation suggests this market is large enough tosupport one hotel but not moreIRevenue data helps to identify cost level8

The Model: BasicI

Common state space: st = (ht , xt )IChoice variable:
ait 2 A (hit ) = f 2,

I1, 0, 1, 2g \ f0

hit + ait

Notations:i 2 f1, . . . , N g : hotel chaint 2 f1, 2, . . . , ∞g : period
hit : # of hotels chain i operates at t
xt : market characteristics
ait : # of hotels chain i opens/closes at t
97g

The Model: Period Pro…tI

Period pro…t function:
π i (ait , st , vit ) = R (st )(ei + ρi vit ) 1 (ait > 0) ait

δi hit0= Ψ (ait , st , vit ) θ i
where

I

23Ri (st )6 hit77Ψ (ait , st , vit ) = 6
4 1 (ait > 0) ait
51 (ait > 0) ait vitθ i = [1, δi , ei , ρi ]Notations:δi : operating costρi : the stddev of entry
cost
10ei : the mean of entry cost vit N (0, 1)

The Model: Value Function

Vi (st ; σi , σ i ) = E"∞

∑βτ t

τ =t0

Ψ (σi (sτ , viτ ) , sτ , viτ ) θ i σ i= Wi (st ; σ)0 θ iwhereσi (st , vi ):policy function (σi : S"∞

Wi (st ; σ) = ER ! A)

∑ βτ t Ψ (σi (sτ , vi τ ) , sτ , vi τ ) σ i

τ =t

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##

The Model: Markov Perfect Equilibrium
IIn a Markov perfect equilibrium, an equilibrium strategy
σi (s, v ) must be the best response to its rivals’
equilibrium strategy σ i :Vi (s; σi , σ i ) Vi s; σi0 , σ ifor all i, s 2 S and σi0I

Exploiting the linearity of the period pro…t function,Wi (s; σi , σ i ) Wi s; σi0 , σ ifor all i, s 2 S and σi0

θi120

Nested Fixed Point Algorithm

I

Brute-force method is hopeless in most dynamic games

I

Evaluating the likelihood function even once could takemore than a day

I

Plus, multiple equilibria are prevalent

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Using BBL

ISuzuki (2009) used BBL for estimating structural costparametersI

It does not requireIII

discretization of continuous state variables
calculating the inverse matrix

Three continuous variables with 20 intervals would be a203 203 matrix
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Estimation Step 1: Revenue Function Estimationln rikt = γi + η 1 + xt0 η 2

η 3 ln (Σj hjt )η 4 ln hit + eikt

IExploit the panel structure

IMarket speci…c …xed e¤ects (at least partially) take careof the endogeneity of hit

I

Chain-speci…c e¤ects are also taken into accountI

The transition function (population, establishments andtrend) are estimated by AR1 regression
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Estimation Step 2: Policy Function Estimation

yit = αi + αm + α2 ln xt
α5 (ln (Σj hjt ))α3 ln hit

α4 (ln hit )2α6 (ln (Σj hjt ))2 + ω itI

Estimate the policy function by using an ordered logit

IUse market-dummy to take into account the impact ofunobservable market-speci…c characteristics on theirentry-exit decision

I

Take into account the change in choice set
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Estimation Step 3: Recovering Cost Parameters
I

Recover chain i’s structural cost parameters θ̂ i thatrationalizes the estimated revenue function and theestimated policy function

I

Each chain’s policy function must be the best response toits rivals’policy functions

IThe observed policy must beat other possible policies
when its rivals’play observed policies

IBBL shows that we can exploit this property to identify
structural parameters

I17
No data are used once policy functions are estimatedImplementing BBL Step by Step
I

Focus on chain i’s structural parameters in market mI

NI
Generate chain i’s fake policies fσmi gm =1 by slightly
perturbing the observed policy function
IHow we should make perturbations is an open questionIDi¤erent perturbation will a¤ect the e¢ ciency ofestimators

IMight make a sense to consider a perturbation with clear
intuition (e.g., never enter)18

Implementing BBL Step by Step
I

Want to approximate the expected sum of future pro…twhen they follow certain strategies

IAssuming other chains follow the observed policy σ̂j 6=i ,simulate the model and calculate Wi (st ; σ ) for theobserved policy σ̂i as well as σmi

I

Simulate economy for T periods

IT should be large enough so that βT is su¢ ciently small

I

The number of simulation should be very large since itneeds to capture entry/exit behavior
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Exploiting Linearity

I

Note that linearity assumption allows us to separatesimulation from estimation

ISimulation calculates Wi (st ; σ)INo structural parameters are directly involved

IOtherwise, need to conduct simulations for each θ20Computational Tips in BBL
I

Under linearity, you can separate simulation fromoptimizationI

BBL requires running a large number of simulations

IGood thing is each simulation is independentI

Can save time by using multiple processors/computers at
the same time

IMatlab has a toolbox for parallel computation
21Implementing BBL Step by StepI

Try to …nd the set of parameters that minimizes thefollowing loss function:NIθi= arg min ∑ (min fgm (θ ) , 0g)2gm (θ ) =θm =1

Wi (s; σi , σ i )Wi s; σi0 , σ iθiIWhen the observed policy beats a fake policy, it adds zeroto the loss function

I

When a fake policy beats the observed policy, it adds
squared pro…t-di¤erence to the loss function
22

Conducting Counterfactual Experiments

IA motivation of doing structural estimation is to conduct
counterfactual experiments

I

Using structural parameters, the model can calculate anequilibrium under alternative policies

ICan compare two competing policies by comparing thewelfare level they bring23

Conducting Counterfactual Experiments
INote that in counterfactual experiments, you need tosolve the model explicitly

I

In complicated models, solving an equilibrium might bevirtually impossible

I

You might need to make a model simple in simulations

I

Also multiple equilibria are prevalent in dynamic games

IThe impacts of policies might not be pinned down24

SummaryI

Go over the very basics of estimating a dynamics
entry-exit modelI

Nested …xed point algorithm turns out to be virtually
impossible

I

BBL with linearity assumption signi…cantly reducescomputational burden without hurting consistency

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