Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Dynamic Oligopoly
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Paul Schrimpf
UBC
Economics 565
October 8, 2014
Dynamic
Oligopoly
Paul Schrimpf
References
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Reviews:
• Aguirregabiria (2012) chapters
• Ackerberg, Caves, and Frazer (2006) section 3
• Aguirregabiria and Mira (2010)
• Doraszelski and Pakes (2007)
• My notes from 628
• Key papers:
• Ericson and Pakes (1995), Aguirregabiria and Mira
(2007), Bajari, Benkard, and Levin (2007)
Dynamic
Oligopoly
Paul Schrimpf
Introduction
1 Introduction
Model
Identiﬁcation
2 Model
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
3 Identiﬁcation
4 Estimation
5 Example: Dunne et al. (2013)
6 Generalizations and extensions
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Section 1
Introduction
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Introduction
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Section 2
Model
Dynamic
Oligopoly
Model primitives I
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• N players indexed by i
• Discrete time index by t
• Player i chooses action ait ∈ A; actions of all players
at = (a1t , ..., aNt )
• State xt = (x1t , ..., xNt ) ∈ X observed by econometrician
and all players at time t
• Private shock εit ∈ E
• Payoﬀ of player i is Ui (at , xt , εit )
• Discount factor β
Dynamic
Oligopoly
Assumptions I
Paul Schrimpf
Introduction
1
A is a ﬁnite set
Model
2
Payoﬀs additively separable in εit ,
Identiﬁcation
Ui (at , xt , εit ) = u(at , xt ) + εit (ait )
Estimation
Example:
Dunne et al.
(2013)
3
Generalizations
and
extensions
xt follows a controlled Markov process
F(xt+1 |
References
It
) = F(xt+1 |at , xt )
|{z}
all information
at time t
4
The observed data is generated by a single Markov
Perfect equilibrium
5
β is known
6
εit i.i.d. with CDF G, which is known up to a ﬁnite
dimensional parameter
Dynamic
Oligopoly
Paul Schrimpf
Assumptions II
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Each of these assumptions could be (and in some papers has
been) relaxed; relaxing 6 is probably most important
empirically
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Value function and equilibrium I
• Strategies α : (X × E )N →AN
• αi is the strategy of player i
• α−i is the strategy of other players
• Value function given strategies: Vαi (xt , εit )
• Integrated value function
∫
¯ α (x) =
V
Vαi (xt , εit )dG(εit )
)
∫ (
α
=
max vi (xt , ait ) + εit (ait ) dG(εit )
ait ∈A
where vαi is the choice speciﬁc integrated value
function,
[
]
u(ait , α−i (xt , ε−it ), xt )+
α
vi (ait , xt ) =Eε−i
¯ α (xt+1 )|ait , α−i (xt , ε−it ), xt ]
+βEx [V
i
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Value function and equilibrium
II
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
• Markov perfect equilibrium: given α−i , αi maximizes vi
[
αi (xt , εit ) ∈ arg max Eε−i
ai
u (ai ,[α−i (xt , ε−it ), xt ) + εit (ai )+ ]
¯ α (xt+1 )|ait , α−i (x, tε−it ), xt
+βEx V
i
• Conditional choice probabilities
References
(
)
Pαi (ai |x) =P ai = arg max vαi (j, x) + εit (j)|x
∫
=
{
j∈A
}
1 ai = arg max vαi (j, x) + εit (j) dG(εit ).
j∈A
]
Dynamic
Oligopoly
Value function and equilibrium
III
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
• Equilibrium condition in terms of conditional choice
probabilities:
vPi (ait , xt ) =
)
u(ait , a−i , xt )+
¯ α (xt+1 )|ait , a−i , xt ]
+βEx [V
P−i (a−i |xt )
i
a−i ∈AN−1
Generalizations
and
extensions
References
(
∑
where
N
∏
P−i (a−i |x) =
P(aj |x).
j̸=i
Let
∫
Λ(a|vPi (·, xt )) =
{
}
1 ai = arg max vPi (j, x) + εit (j) dG(εit ).
j∈A
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Value function and equilibrium
IV
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Then the equilibrium condition is that
Pi (a|x) = Λ(a|vPi (·, x))
or in vector form P = Λ(vP )
• Equilibrium existence:
• If Λ : [0, 1]N|X| →[0, 1]N|X| is continuous, then by Brouwer’s
ﬁxed point theorem, there exists at least one
equilibrium
• Λ need not be continuous, see Gowrisankaran (1999)
and Doraszelski and Satterthwaite (2010)
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Section 3
Identiﬁcation
Dynamic
Oligopoly
Identiﬁcation I
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
• As in single-agent dynamic decision problems given G,
β, and Eε [u(0, α−i (x, ε−i ), xt )] = 0, we can identify the
expectation over other player’s actions of the payoﬀ
function,
∑
Eε [u(ai , α−i (x, ε−i ), x)] =
P(a−i |x)u(ai , a−i , x)
a−i
References
• See Bajari et al. (2009), which builds on Hotz and Miller
(1993) and Magnac and Thesmar (2002)
• Hotz and Miller (1993) inversion shows
∗
∗
vαi (a, x) − vαi (0, x) = q(a, P(·|x); G)
for some known function q
Dynamic
Oligopoly
Identiﬁcation II
Paul Schrimpf
Introduction
Model
Identiﬁcation
• Use normalization and Bellman equation to recover vαi
∗
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
∗
vαi (0, x) = E[u(0, α−i
(x, ε−i ), x)] +
{z
}
|
=0
∗
+ βE[max
vαi (a′ , x′ ) + ε(a′ )|a, x]
′
a ∈A
∗
= βE[max
vαi (a′ , x′ )
′
a ∈A
|
∗
− vαi (0, x′ ) + ε(a′ )|0, x] +
{z
}
≡M(x,P(·|x),G)
+
∗
βE[vαi (0, x′ )|0, x]
∗
M is known; can solve this equation for vαi (0, x), then
∗
∗
vαi (a, x) = vαi (0, x) + q(a, P(·|x); G)
∗
Dynamic
Oligopoly
Identiﬁcation III
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
∗
∗
• Recover E[u(ai , α−i
(x, ε−i ), x)] from vαi using Bellman
equation
∗
∗
E [u(ai , α−i
(x, ε−i ), x)] =vαi (ai , x)−
[
]
α∗ ′ ′
′
− βE max
v
(a
,
x
)
+
ε(a
)|a,
x
i
′
a ∈A
• Need exclusion to identify u(a, x)
• Without exclusion order condition fails
∑
Eε [u(ai , α−i (x, ε−i ), x)] =
P(a−i |x)u(ai , a−i , x)
a−i
If X ﬁnite, then left side takes on|A||X| identiﬁed values,
but u(a, x) has |A|N |X| possible values
Dynamic
Oligopoly
Identiﬁcation IV
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
• Assume u(a, x) = u(a, xi ) where xi is some sub-vector of
x. u identiﬁed if
Eε [u(ai , α−i (x, ε−i ), x)] =
∑
a−i
References
has a unique solution for u
P(a−i |x)u(ai , a−i , xi )
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Section 4
Estimation
Dynamic
Oligopoly
Estimation I
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Can use similar methods as in single agent dynamic
models
• Maximum likelihood
max
θ∈Θ,P∈[0,1]N
Tm ∑
M ∑
N
∑
(
)
log Λ aimt |vPi (·, xmt ; θ)
m=1 t=1 i=1
P
s.t.P = Λ(v (θ))
• Nested ﬁxed point: substitute constraint into objective
and maximixe only over θ
• For each θ must solve for equilibrium – computationally
challenging
• Λ not a contraction
• What to do when equilibrium not unique?
Dynamic
Oligopoly
Estimation II
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• MPEC (Su and Judd, 2012): solve constrained
optimization problem
ˆ
• 2-step estimators: estimate P(a|x)
from observed
actions and then
max
θ∈Θ
Tm ∑
M ∑
N
∑
ˆ
log Λ(aimt |vPi (·, xmt ; θ))
m=1 t=1 i=1
• Can replace pseudo-likelihood with GMM (Bajari,
Benkard, and Levin, 2007) or least squares (Pesendorfer
and Schmidt-Dengler, 2008) objective
• Unlike single agent case, eﬃcient 2-step estimators do
not have same asymptotic distribution as MLE1
Dynamic
Oligopoly
Estimation III
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
• Nested pseudo likelihood (Aguirregabiria and Mira,
2007): after 2-step estimator update
(k−1)
ˆ (k−1) )), re-maximize pseudo likelihood to
ˆ(k) = Λ(vPˆ (θ
P
(k)
ˆ
get θ
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
• Asymptotic distribution depends on number of
References
iterations; if iterate to convergence, then equal to MLE
1
In single agent models eﬃcient 2-step and ML estimators have the
same asymptotic distribution but diﬀerent ﬁnite sample properties.
Dynamic
Oligopoly
Paul Schrimpf
Incorporating static parameters
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Often some portion of payoﬀs can be estimated
without estimating the full dynamic model
• E.g. Holmes (2011) estimates demand and revenue from
sales data, costs from local wages, and only uses
dynamic model to estimate ﬁxed costs and sales
• Bajari, Benkard, and Levin (2007) and Pakes, Ostrovsky,
and Berry (2007) are methodological papers that
incorporate a similar idea
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Section 5
Example: Dunne et al. (2013)
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Dunne et al. (2013) “Entry, Exit
and the Determinants of Market
Structure” I
• Market structure = number and relative size of ﬁrms
• Classic question in IO: how does market structure
aﬀect competition?
• Here: how is market structure determined? Entry and
exit
• Sunk entry costs
• Fixed operating costs
• Expectations of proﬁts (nature of competition)
• Like Bresnahan and Reiss (1991) summarize with proﬁts
as a function of number of ﬁrms, π(n)
• Estimate dynamic model of entry and exit to determine
relative importance of factors aﬀecting market
structure
• Context: dentists and chiropractors
Dynamic
Oligopoly
Model I
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Similar to Pakes, Ostrovsky, and Berry (2007)
• State variables s = (n, z)
• n = number of ﬁrms, z = exogenous proﬁt shifters
• Follow a ﬁnite state Markov process
• Parameters θ
• Proﬁt π(s; θ) (leave θ implicit henceforth)
• Fixed cost λi ∼ Gλ
• Discount factor δ
• Value function
V(s; λi ) = π(s) + max{δVC(s) − δλi , 0}
where VC is expected next period’s value function
[
[
] ]
VC(s) =Ecs′ π(s′ ) + Eλ′ max{δVC(s′ ) − δλ′ , 0}|s |s
Dynamic
Oligopoly
Model II
Paul Schrimpf
Introduction
• Probability of exit:
Model
Identiﬁcation
px (s) = P(λi > VC(s)) = 1 − Gλ (VC(s)).
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Assume λ exponential, Gλ = 1 − e−(1/σ )λ , then
[
(
) ]
VC(s) = Ecs′ π(s′ ) + δVC(s′ ) − δσ 1 − px (s′ ) |s
• Let Mc be the transition matrix, then
VC =Mc [π + δVC − δσ (1 − px )]
VC =(I − δMc )−1 Mc [π − δσ (1 − px )]
(1)
• Pakes, Ostrovsky, and Berry (2007): use non parametric
estimates of Mc and px in (1) to form VC
Dynamic
Oligopoly
Model III
Paul Schrimpf
Introduction
Model
• Here: use non parametric estimate of Mc and form VC
by solving
Identiﬁcation
[
]
VC = Mc π + δVC − δσ Gλ (VC)
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Potential entrants:
• Expected value after entering
[
(
) ]
VE(s) = Ees′ π(s′ ) + δVC(s′ ) − δσ 1 − px (s′ ) |s
• Cost of entry κi ∼ Gκ
• Entry probability
pe (s) = P (κi < δVE(s)) = Gκ (δVE(s))
• As before can use Bellman equation in matrix form to
solve for VE
Dynamic
Oligopoly
Empirical speciﬁcation I
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
• Data: U.S. Census of Service Industries and Longitudinal
Business Database
• 5 periods – 5 year intervals from 1982-2002
• 639 geographic markets for dentists; 410 for
chiropractors
• Observed average market-level proﬁts πmt
• Number of ﬁrms nmt , entrants, emt , exits xmt , potential
entrants pmt
• Market characteristics zmt = (popmt , wmt , incmt )
References
• Proﬁt function
5
∑
πmt =θ0 +
θk 1{nmt = k} + θ6 nmt + θ7 n2mt +
k=1
+ quadratic polynimal in zmt +
+ fm + εmt
Key assumption: εmt independent over time
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Empirical speciﬁcation II
• Transition matrix Mc
• Deﬁne ˆ
zmt = estimate value polynomial in zmt in proﬁt
function
• Discretize ˆ
zmt into 10 categories and use sample
averages to estimate transition probabilities
• Fixed (Gλ ) and entry costs (Gκ )
c ) and VE(σ
c ) as described above
• VC(σ
• Log-likelihood
( (
))
c mt (σ ); σ +
(nmt − xmt ) log Gλ VC
(
(
))
c mt (σ ); σ +
∑ +xmt log 1 − Gλ VC
( (
))
L(σ , α) =
κ c
+e
log
G
VE
(σ
);
α
+
mt
mt
m,t
(
(
))
c mt (σ ); α
+(pmn − emt ) log 1 − Gκ VE
Dynamic
Oligopoly
Results
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Proﬁt function:
• Decreasing with n increasing in w, inc, pop
• Compare ﬁxed eﬀects and OLS estimates in depth
• Seems unneeded to me; there is no reason you want to
use the OLS estimates anyway
• More relevant concern is assumption of εmt
independent over time — this is empirically testable, but
they do not do anything about it
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
ˆ
π(n)
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Dynamic
Oligopoly
Subsidies to entry and ﬁxed
costs
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
• Health Professional Shortage Areas (HPSA) have entry
Generalizations
and
extensions
• Entry cost subsidy = change distribution of entry costs
References
subsidies
for all markets to the distribution estimated for HPSA
markets
• Fixed cost subsidy = reduce mean of ﬁxed cost by 8%
(chosen to generate similar number of ﬁrms as HPSA
subsidy)
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
2
2
FIXME: The numbers in this table are diﬀerent in the published paper.
Chiro estimates look the same, but dentists changed.
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Section 6
Generalizations and extensions
Dynamic
Oligopoly
Paul Schrimpf
Generalizations and extensions
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
• Unobserved state variables
• Multiple equilibria
• Continuous time
• Non-additively separable preferences over-time (such as
Epstein-Zin preferences)
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Ackerberg, D., K. Caves, and G. Frazer. 2006. “Structural
identiﬁcation of production functions.” URL
http://mpra.ub.uni-muenchen.de/38349/.
Aguirregabiria, Victor. 2012. “Empirical Industrial
Organization: Models, Methods, and Applications.” URL
http://www.aguirregabiria.org/barcelona/book_
dynamic_io.pdf.
Aguirregabiria, Victor and Chun-Yu Ho. 2012. “A dynamic
oligopoly game of the US airline industry: Estimation and
policy experiments.” Journal of Econometrics 168 (1):156 –
173. URL http://www.sciencedirect.com/science/
article/pii/S030440761100176X.
Aguirregabiria, Victor and Pedro Mira. 2007. “Sequential
Estimation of Dynamic Discrete Games.” Econometrica
75 (1):pp. 1–53. URL
http://www.jstor.org/stable/4123107.
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
———. 2010. “Dynamic discrete choice structural models: A
survey.” Journal of Econometrics 156 (1):38 – 67. URL
http://www.sciencedirect.com/science/article/
pii/S0304407609001985.
Bajari, Patrick, C. Lanier Benkard, and Jonathan Levin. 2007.
“Estimating Dynamic Models of Imperfect Competition.”
Econometrica 75 (5):pp. 1331–1370. URL
http://www.jstor.org/stable/4502033.
Bajari, Patrick, Victor Chernozhukov, Han Hong, and Denis
Nekipelov. 2009. “Nonparametric and Semiparametric
Analysis of a Dynamic Discrete Game.” Tech. rep. URL
http://www.econ.yale.edu/seminars/apmicro/am09/
bajari-090423.pdf.
Bresnahan, Timothy F. and Peter C. Reiss. 1991. “Entry and
Competition in Concentrated Markets.” Journal of Political
Economy 99 (5):pp. 977–1009. URL
http://www.jstor.org/stable/2937655.
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Doraszelski, Ulrich and Ariel Pakes. 2007. “Chapter 30 A
Framework for Applied Dynamic Analysis in IO.” In
Handbook of Industrial Organization, Handbook of Industrial
Organization, vol. 3, edited by M. Armstrong and R. Porter.
Elsevier, 1887 – 1966. URL http://www.sciencedirect.
com/science/article/pii/S1573448X06030305.
Doraszelski, Ulrich and Mark Satterthwaite. 2010.
“Computable Markov-perfect industry dynamics.” The
RAND Journal of Economics 41 (2):215–243. URL
http://onlinelibrary.wiley.com/doi/10.1111/j.
1756-2171.2010.00097.x/full.
Dunne, Timothy, Shawn D. Klimek, Mark J. Roberts, and
Daniel Yi Xu. 2013. “Entry, exit, and the determinants of
market structure.” The RAND Journal of Economics
44 (3):462–487. URL
http://dx.doi.org/10.1111/1756-2171.12027.
Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Ericson, Richard and Ariel Pakes. 1995. “Markov-perfect
industry dynamics: A framework for empirical work.” The
Review of Economic Studies 62 (1):53–82. URL http://
restud.oxfordjournals.org/content/62/1/53.short.
Gowrisankaran, Gautam. 1999. “A dynamic model of
endogenous horizontal mergers.” The RAND Journal of
Economics :56–83URL
http://www.jstor.org/stable/10.2307/2556046.
Holmes, T.J. 2011. “The Diﬀusion of Wal-Mart and
Economies of Density.” Econometrica 79 (1):253–302. URL
http://onlinelibrary.wiley.com/doi/10.3982/
ECTA7699/abstract.
Hotz, V. Joseph and Robert A. Miller. 1993. “Conditional
Choice Probabilities and the Estimation of Dynamic
Models.” The Review of Economic Studies 60 (3):pp.
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Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
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Dynamic Discrete Decision Processes.” Econometrica
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Games1.” Review of Economic Studies 75 (3):901–928. URL
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Dynamic
Oligopoly
Paul Schrimpf
Introduction
Model
Identiﬁcation
Estimation
Example:
Dunne et al.
(2013)
Generalizations
and
extensions
References
Su, C.L. and K.L. Judd. 2012. “Constrained optimization
approaches to estimation of structural models.”
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