Structural Econometrics in Industrial
Organization
Dynamics

Marc Rysman
Boston University

February 16, 2009

Many important economic decisions are dynamic

I Consumers:

Buy today or wait (learning)Experience goods (experimentation)
I stockpiling from sales
II

I Firms:

Entry and exit
Investment
I Product introductions
II

I Note that many of these involve discrete decisions.

Traditional continuous choice with a single agent
I Consumers chooses consumption and savings:

max E
ct

I"∞X

#β τ −t U(ct ) Ωt s.t.ct ≤ rt qt +yt ;qt+1 = rt qt +yt −ctτ =tWhere qt is savings, rt is savings payoff, yt is income, Ωt isinformation.

I The Bellman equation is:V (Ωt ) = max U(ct ) + βE [V (Ωt+1 )|Ωt ]ct

I Use derivative based arguments to get rid of V and derive

the Euler equation:
U 0 (ct ) = βE[U 0 (ct+1 )|Ωt ]Dynamic Discrete Choice
Rust (1987, Econometrica)

I Consider Harold Zurcher, chief mechanic for the public bus

system in Madison, WI
I Zurcher decides when to overhaul engines.
I After overhaul, engines are like new.
I This is an “optimal stopping problem”.

Zurcher’s problem

I State variable: Number of miles on the bus, s (followsMarkov process)
I Infinite, discrete time. Discount rate: β.
I Choice variable: Whether or not to overhaul, i ∈ {0, 1}.
I Cost function C(s, i, θ) + i :IIOverhaul: C(s, 1, θ) = θ0
No overhaul: C(s, 0, θ) = theta1 + theta2 s

I Stationary problem. The Bellman equation is:V (s, ) = max C(s, i, θ) + i + βE V (s0 , 0 )|s, i, i∈{0,1}Simplification

I We want to solve for the value function but  makes theproblem very hard.

Assumption of Conditional Independencep(s0 , 0 |s, , i) = p1 (0 |s0 )p2 (s0 |s, i)
I  has no dynamic content. s and i are sufficient to predict

future states.
I In practice, we assume  is iid.

 ∼ EV .

I Study:

ZEV (s) =EV (s, )Empirical approach
I Bellman equation:!
XEV (s) = ln
exp C(s, i, γ) + βE EV (s0 )|s, ii
I Probabilities of i follows logit form:

exp (C(s, i, θ) + βE [EV (s0 )|s, i])
0k ∈{0,1} exp (C(s, k , θ) + βE [EV (s )|s, k ])P(i|s) = P

I If we knew EV , we could estimate θ. If we knew θ, wecould solve for EV .

Algorithm
The Nested Fixed Point Algorithm

1. Discretize EV . Estimate discrete transition matrix p(s0 |s, i).
2. Pick γ.
3. Pick values for EV .
4. Solve!
0EV = lnX
exp C(s, i, γ) + βE EV (s0 )|s, ii

5. If d(EV 0 , EV ) > cutoff, go to 3.
6. Construct likelihood:
JXexp C(sj , ij , γ) + βE EV (s0 )|sj , ij
PL(γ) =0 )|s , kexp
C(s,k,γ)+βEEV(sjjk∈{0,1}
j=1
7. Pick γ to raise L(γ). Go to 2.

Issues
I Exogenous persistent heterogeneity.
IUse EM algorithm,
I

Heckman and Singer (Econometrica, 1984), Cameron andTrivedi (Textbook, Sec. 10.3.7)

I Endogenous heterogeneity (learning)
ICan be solved with simulation
IAckerberg (IER, 2003), Crawford and Shum (Econometrica,2005)

I Large state spaces
IRandomization
I

I

Rust (Econometrica, 1997), Hotz and Miller (ReStud 1993),
Imai, Jain and Ching (2008 SSRN).

Conditional choice probabilities
I

Hotz and Miller (ReStud 1993), Aguirregabiria and Mira(Econometrica, 2002)

I Applications to games
I

Bajari, Benkard and Levin (2007, Econometrica),
Aguirregabiria and Mira (2007, Econometrica), Pakes,
Ostrovsky and Berry (2007, RAND)

Bajari, Benkard and Levin, 2007

Estimating dynamic games introduces two important problems:
1. Enormous state spaces
I

Benkard (ReStud, 2004) takes a month to solve once!

2. Multiple equilibria
IMultiple solutions to fixed point algorithms means wecannot construct a likelihood function.

BBL addresses both of these problems.

Basic idea

I Estimate in reduced form choices and state transitions as afunction of state variables.
I Use simulation to calculate value function at any given

state resulting from possible choices.
I Estimate structural parameters in payoff function taking

value function as given.

ModelI J firms, discrete, infinite time
I state space st ∈ RLI actions i ∈ I (ctns or discrete)
I private shock nujt ∼ G(·|st )I flow profit πj (ijt , st , νjt ).
I Before realizing νjt :V (st ) = E"∞X#βτ −tπj (iτ , sτ , νjτ st )

τ =twherest+1 ∼ P(st+1 |it , st )

Model II

I Markov Perfect Equilibrium
I Denote strategies as σ(s, νj )I Bellman equation conditioning on strategies:


Z00Vj (s, σ) = E πj (σ(s, νj ), s, νj ) + Vj (s , σ)dP(s |σ(s, νj ), s) sFirst stage of estimation

I Assume data is generated by a single MPE so we canestimate σ(s, νi ) from observed data.
I Estimate σ(s, νj ), P(s 0 |it , st ).
I Now we can calculate the value of being at any given states by simulation up to a set of parameters.
I Rather than solve for exact value function from equilibrium

conditions, we are using observed choices and statetransitions to approximately obtain value function.

Calculate value function

1. Draw νjt for all firms for T periods into the future
2. Calculate choices in period t from σ(s, νjt ).
3. Calculate st+1 from P(s0 |it , st ).
4. Go to 2.
5. Repeat ns times (go to 1).
6. Calculate:
1Vbj (st , σ) =nsns XTX

β τ −t πj (itk , stk , νjtk , θ)

k =1 τ =t

It will be handy if πj (it , st , νjt , θ) = πj (it , st , νjt )θ

Second stage
Method 1

I For each firm in each period, calculate optimal choice frommodel.
IFor continuous choices, find:i ∗ s.t.

Id Vbj (st , σ(i ∗ ))
= 0.
dijt

For discrete choices, find:i ∗ s.t.Vbj (st , σ(i ∗ )) ≥ Vbj (st , σ(i))∀iLet ξ = i ∗ (θ) − i data .
I Form moments m = z 0 ξ, objective function obj = m 0 wm.
I

I May be very computationally costly in practiceSecond stage
Method 2

I Determine an alternative strategy σ 0 (s, νjt ).
IFor instance, add random normal terms to parameters in σ.

I Use revealed preference to derive moment inequalities.

gj (s, σ, σ 0 , θ) = Vj (s, σj , σ−j , θ) − Vj (s, σj0 , σ−j , θ)
I Form objective function:

obj =X
2min g(s, σ, σ 0 , θ), 0s∈dataI Pick θ to minimize this function.

Example: Rust 1987
I Estimate choice probabilities using logit model:

σ(sjt , νjt ) = 1Iif

α0 + α1 sjt + νjt > 1

νj ∼ EV⇒ P(ijt |sjt ) =exp(α0 + α1 sjt )1 + exp(α0 + α1 sjt )I Estimate state transitions:sjt+1 = γ0 + γ1 sjt + ujt
I Now compute value of state sjt+1 up to parameters θ.

Rust Example: Computing V
I Draw ujt ,jt for T periods into the future, for each state

observed in the data ns times.
I Simulate future realizations of sjt , ijt . Compute:nsxjt1 =T

1 X X τ −t k
β 1{ijt = 1}
nsk =1 τ =t
nsxjt2 =T

1 X X τ −t k
β 1{ijt = 0}
nsk =1 τ =t
nsxjt3 =T

1 X X τ −t kβ sjt
nsk =1 τ =t

I Value function is:hV (sjt , σ, θ) = xjt1xjt2ixjt3 θRust Example: Estimate θ
I Add random numbers to α0 , α1 to create σ 0 (sjt , νjt ).
I Compute V (sjt , σ 0 , θ)
I Value of observed strategy minus value of alternativestrategy:hg(sjt , σ, σ 0 , θ) = xjt1 − xjt10

xjt2 − xjt20

ixjt3 − xjt30 θI Objective function:

obj =X

2min g(sjt , σ, σ 0 , θ), 0sjt ∈data

I In practice, this also includes a summation over many

alternative strategies.

Applications

I Typically, these models are followed by simulations ofmarket equilibrium under alternative policy regimes in thespirit of Ericson and Pakes (ReStud 1995) and Pakes andMcGuire (RAND, 1994).
I Ryan (2008): Entry and investment by US cement

producing plants
I Stahl (2009): Consolidation of US local television industry

after deregulation.
I Snider (2009): Predatory pricing by US airlines.

Outstanding issues

I If there is any persistent unobserved heterogeneity, thefirst stage results is not right for any individual agent.
IAnalog to conditional independence assumption

I Many papers do not sufficiently concern themselves withobtaining causal parameters in the first stage.
I

Just regressing choice variables on state variables does notmean we have estimated a causal effect.