1. science
2. physics
3. thermodynamics

Identi…cation of Games of Incomplete Information
with Multiple Equilibria and Unobserved Heterogeneity
Victor Aguirregabiria
University of Toronto and CEPR
Pedro Mira
CEMFI, Madrid
This version: April 20, 2015
Abstract
This paper deals with identi…cation of discrete games of incomplete information when we
allow for three types of unobservables for the researcher: (a) payo¤-relevant variables that are
players’private information; (b) payo¤-relevant variables that are common knowledge to all the
players; and (c) non-payo¤-relevant variables which are common knowledge to the players and
determine the selection between multiple equilibria. The speci…cation of the payo¤ function is
nonparametric, and the probability distributions of the unobservables is also nonparametric but
with …nite support (i.e., …nite mixture model). We show that, in a model with N players, J + 1
choice alternatives, L points of support in the distribution of common knowledge unobservables,
and with N
3 and L
(J + 1)int[(N 1)=2] , all the structural functions of the model are
identi…ed under the same type of exclusion restrictions that we need for identi…cation without
unobserved heterogeneity. In particular, we can separately identify the relative contributions
of payo¤-relevant and "sunspot" unobserved heterogeneity to explain players’ behavior. We
discuss the identi…cation of counterfactuals. Finally, we illustrate these results using Monte
Carlo experiments.
Keywords: Discrete games of incomplete information; Multiple equilibria in the data; Unobserved heterogeneity; Sunspots; Finite mixture models.
JEL codes: C13, C35, C57.
A previous version of this paper was titled "Structural Estimation in Games when the Data Come fromMultiple
Equilibria". We would like to thank comments from Nicolai Kumino¤, Sokbae Lee, John Rust, Steven Stern, Artyom
Shneyerov, Ken Wolpin, two anonymous referees, and from participants in seminars at Concordia, Copenhagen,
Duke University, University College of London, the Econometric Society World Congress in Shanghai, and the UPenn
conference in honor of Kenneth I. Wolpin.
1
Introduction
Multiplicity of equilibria is a prevalent feature in games. An implication of multiplicity of equilibria
in the structural estimation of games is that the model implies more than one probability distribution of the endogenous variables conditional on structural parameters and exogenous variables. The
standard criteria used for estimation, such as likelihood or GMM criteria, are no longer functions of
the structural parameters but correspondences, and this makes the application of these estimation
methods impractical in many relevant cases. A substantial part of the recent literature on the
econometrics of games of incomplete information proposes simple two-step estimators that deal
with these issues.1 These two-step methods assume that the only unobservables for the researcher
are variables that are private information of the players, so there are no unobservables that are
common knowledge to players. A second key assumption maintained in almost all applications of
two-step methods is that the same equilibrium (or equilibrium "type") has been played in all the
observations in the data. The model may have multiple equilibria for the true value of the structural parameters, but only one of them is present in the data.2 A weaker version of this assumption
establishes that we can partition the data into a number of subsamples according to the value of an
exogenous variable such that the same equilibrium is played within each subsample. That is, the
potential multiplicity of equilibria in the data should be explained by some observable exogenous
variable(s). Under this assumption, structural parameters in these models are identi…ed given the
same type of exclusion restrictions as in games with equilibrium uniqueness (see Aguirregabiria and
Mira, 2002a, Pesendorfer and Schmidt-Dengler, 2003, and Bajari et al., 2010).
The assumption that all the data have been generated from a single equilibrium is very strong.3
In the context of most empirical games of incomplete information, uniqueness of the equilibrium
in the data, together with the assumption that there are no common knowledge unobservables,
imply that the actions of players are independent of one another conditional on observables. This
implication is likely to fail in most datasets. One possible interpretation of failure of this conditional independence is that common knowledge unobservables are present. Aguirregabiria and Mira
(2007), Arcidiacono and Miller (2011), and Grieco (2014) extend sequential estimation methods to
allow for common knowledge unobservables in games of incomplete information. A recent paper by
de Paula and Tang (2012) relaxes the assumption of a unique equilibrium in the data. De Paula
1
See Aguirregabiria and Mira (2007), Bajari et al. (2007), and Pesendorfer and Schmidt-Dengler (2008) as seminal
contributions in this literature. Other recent contributions to this topic in the context of games of incomplete
information are Sweeting (2009), Aradillas-Lopez (2010), and Bajari, Hong, Krainer, and Nekipelov (2010).
2
See Bajari, Hong, and Nekipelov (2013) for a recent survey of this literature.
3
Otsu, Pesendorfer and Takahashi (2014) propose statistical tests for this assumption.
1
and Tang interpret failure of independence in terms of multiple equilibria and show that it is actually helpful to identify the sign of the parameters that capture the strategic interactions between
players. However, de Paula and Tang assume that the model does not contain common knowledge
unobservables. A relevant question is whether it is possible to separate empirically the contribution
of unobservables that a¤ect the selection of an equilibrium in the data (i.e., non-payo¤ relevant
unobservables or "sunspots") from the contribution of unobservables that are payo¤-relevant. Distinguishing between these two alternative explanations can be important in empirical applications.
For instance, they can generate di¤erent predictions when we use the estimated model to make
predictions or counterfactual experiments. Furthermore, authors in di¤erent areas of Economics
have suggested that multiplicity of equilibria may be necessary to explain important aspects of
economic data. This type of arguments have been used to explain macroeconomic ‡uctuations
(Farmer and Guo, 1995), regional variation in the density of economic activity (Krugman, 1991,
and Bayer and Timmins, 2005, 2007), local market variation in …rms’strategic behavior (Sweeting,
2009, and Ellickson and Misra, 2008), or black-white wage inequality (Moro, 2003).
In this paper, we study the identi…cation of games when we allow for three types of unobserved
heterogeneity for the researcher: payo¤-relevant variables that are private information of each
player (PI unobservables); payo¤-relevant variables that are common knowledge to all the players
(PR unobservables); and variables that are common knowledge to all the players and are not
payo¤-relevant but a¤ect the equilibrium selection ("sunspots" or SS unobservables). As far as
we know, this is the …rst paper to study identi…cation of games with these three di¤erent sources
of unobservables. The speci…cation of the payo¤ function is nonparametric, and the probability
distribution of common knowledge unobservables is also nonparametric but with …nite support (i.e.,
…nite mixture model). The model is semiparametric because we assume that the researcher knows
the distribution of the private information unobservables up to a scale parameter.
We show that, in a model with N players, J + 1 choice alternatives, L points of support in
the distribution of common knowledge unobservables, and with N
3 and L
(J + 1)(N
1)=2 ,
all
the structural functions of the model are identi…ed under the same type of exclusion restrictions
that we need for identi…cation without unobserved heterogeneity. In particular, we can separately
identify the relative contributions of payo¤-relevant and "sunspot" unobserved heterogeneity to
explain players’ behavior. We also study the identi…cation of counterfactual experiments using
the estimated model. The identi…cation of the probability distribution of the "sunspot" unobserved heterogeneity is particularly important for the identi…cation and implementation of these
2
counterfactuals.
Most of our identi…cation results in this paper are based on a sequential approach. In a …rst
step, we consider the nonparametric identi…cation of players’ strategies (de…ned as Conditional
Choice Probabilities) and the distribution of common knowledge unobservables in the context of a
nonparametric …nite mixture model. In a second step, we study the identi…cation of payo¤s and
the separate identi…cation of payo¤-relevant (PR) and non-payo¤-relevant (SS) common knowledge
unobservables. The strongest identi…cation assumptions are in the …rst step, while the main identi…cation condition in the second step is a exclusion restriction on players’payo¤ functions. We show
with an example that the conditions for the identi…cation of the …nite mixture model in the …rst
step are su¢ cient but not necessary. In particular, when using a non-sequential identi…cation approach, the exclusion restrictions in the payo¤ function can help us to relax some of the restrictions
that we use to identify the …nite mixture model in the …rst step of the sequential approach.
We also …nd an issue in the implementation of the sequential identi…cation approach. In the
…rst step, the distribution of the unobservables (conditional on a given value of the observable
exogenous variables) is identi…ed up to label swapping of the types. We can identify the distribution
of the unobservables for each value of the exogenous variables but, without further assumptions,
we cannot “match” unobservable types across di¤erent values of these exogenous variables. The
implementation of the instrumental-variables-like identi…cation in the second step requires that
the researcher be able to match mixture components across games with di¤erent values of the
instruments. We refer to this identi…cation issue as the problem of matching unobservable types.
We provide su¢ cient conditions for the identi…cation of this matching problem which are weaker
than assuming independence between unobservable and instruments.
The rest of the paper is organized as follows. Section 2 introduces the class of models. Section
3 presents our identi…cation results. In section 4, we illustrate our identi…cation results and explore
the power of the proposed tests using several Monte Carlo experiments. We summarize and conclude
in section 5.
3
2
Model
Consider a game that is played by N players which are indexed by i 2 I = f1; 2; :::; N g. Each
player has to choose an action from a discrete set of alternatives A = f0; 1; :::; Jg. The decision
of player i is represented by the variable ai 2 A. Each player chooses his action ai to maximize
his expected payo¤. The utility or payo¤ function of player i is
a real-valued function; a
x 2 X, ! 2
i
2 AN
1
i (ai ; a i ; x; !; "i ),
where:
i (:)
is
is a vector with choice variables of players other than i; and
, and "i are vectors of exogenous characteristics of players and of the environment
(market). The variables in x and ! a¤ect players’utilities and they are common knowledge for all
players. The vector "i represents characteristics that are private information of player i. Variables
! and "i are unobservable to the researcher and x is observable.
In addition to these payo¤ relevant state variables, there are also common knowledge, non-payo¤
relevant state variables that do not have a direct e¤ ect on the payo¤ of any player, but they a¤ect
players’ beliefs about behavior of other players, or more speci…cally, they a¤ect players’ beliefs
about which equilibrium, from the multiple ones the model has, is the one that they are playing.
We denote these non-payo¤ relevant variables as sunspots. We represent these sunspot variables
using the vector , that contains variables unobserved to the researcher.
EXAMPLE 1: Coordination game within the classroom (Todd and Wolpin, 2012). In an elementary
school class the students and the teacher choose their respective levels of e¤ort, ai 2 A. Each
student has preferences on her own end-of-the-year knowledge,
i.
The teacher cares about the
aggregate knowledge of all the students. There is a production function that determines end of
the year knowledge of a student. According to this production function, a student’s knowledge
depends on her own e¤ort, the e¤ort of her peers, teacher’s e¤ort, and exogenous characteristics
of the student, the classroom, and the school. This type of game is an example of Coordination
Game (Cooper, 1999) and its main feature is the strategic complementarity between the levels of
e¤ort of the di¤erent players. Coordination games typically have multiple equilibria, and these
equilibria can be ranked in terms of the levels of e¤ort of the players. In this example, we can
distinguish three di¤erent types of unobservables from the point of view of the outside researcher.
The …rst type consists of payo¤-relevant common knowledge unobservables (PR unobservables, !),
e.g., classroom, school, teacher, and students characteristics that enter in the production function
of students’knowledge and are known to all the players but not to the researcher. The second type
consists of private information unobservables (PI unobservables, "i ), e.g., part of the students’and
teacher’s skills, and their respective costs of e¤ort, are private information of these players, and
4
they are also unknown to the researcher. Finally, in the presence of multiple equilibria, we may
have that two classes with exactly the same (payo¤ relevant) inputs have selected di¤erent types of
equilibria. Apparently innocuous variations in the initial conditions in the class may a¤ect students’
and teachers’beliefs about the e¤ort of others, and therefore a¤ect the selected equilibrium. Part
of these non-payo¤ variables a¤ecting beliefs are unobservable to the researcher (SS unobservables,
).
Assumption 1 contains basic conditions on the structural model that are standard in the empirical literature of discrete games of incomplete information.4
ASSUMPTION 1. (A) Payo¤ functions f
mation component, i.e.,
i
i
: i 2 Ig are additively separable in the private infor-
= ei (ai ; a i ; x; !) + "ei (ai ), where "ei
f"ei (ai ) : ai 2 Ag is a vector of
J + 1 real valued random variables. (B) "ei is independently distributed across players and indepen-
dent of common knowledge variables x, !, and , with a distribution that is absolutely continuous
with respect to the Lebesgue measure in RJ+1 .
It is well known that a player’s optimal choice or best response is invariant to any a¢ ne transformation of his payo¤ function. This necessarily implies that the set of equilibria of a game is
also invariant to any a¢ ne transformation of the players’payo¤ functions. Therefore, we need to
acknowledge that we can identify the payo¤ function only up to an a¢ ne transformation.5 Given a
baseline choice alternative, say alternative 0, for any ai 6= 0 de…ne the "normalized" payo¤ function,
i (ai ; a i ; x; !)
variables "i (ai )
[ei (ai ; a i ; x; !)
["ei (ai )
"ei (0)]=
i
ei (0; a i ; x; !)]= i , and the "normalized" private information
where
2
i
V ar("ei (1)
"ei (0)). For the rest of the paper,
we describe the model in terms of the normalized payo¤ functions
variables "i .
i
and the private information
ASSUMPTION 2. The model is semiparametric in the sense that the researcher knows the distribution function, G, of the (normalized) vector of private information variables "i
A
f"i (ai ) : ai 2
f0gg.
The standard equilibrium concept in static games of incomplete information is Bayesian Nash
equilibrium (BNE). We assume that the outcome of this game is a BNE. Under this assumption, a
4
In a recent working paper, Liu, Vuong, and Xu (2013) study identi…cation of binary choice games of incomplete
information relaxing the assumptions of additive separability and independence between players’private information.
Also, Wan and Xu (2014) study identi…cation of a semiparametric binary game with correlated private information.
These two papers assume that there is common knowledge unobserved heterogeneity or multiple equilibria in the
data.
5
In this paper, we consider that the researcher has data only on players’choices and state variables. Some of our
normalization assumptions can be relaxed when the researcher has data on a component of the payo¤ function such
as …rms’revenue.
5
player’s strategy is a function only of payo¤-relevant variables, i.e., a function of (x; !; "i ). If the
game has multiple equilibria, then the sunspot variables in
a¤ect the selection of the equilibrium
and therefore the outcome of the game. We …rst describe a BNE and then we incorporate the
equilibrium selection mechanism when the model has multiple equilibria. Let
i 2 Ig be a set of strategy functions where
i
RJ into A. Associated with a
is a function from X
set of strategy functions we can de…ne a vector of choice probabilities P(x; !; )
(ai ; i) 2 A
f0g
Ig such that:
Pi (ai j x; !;
Z
i)
= f i (x; !; "i ) :
fPi (ai jx; !;
1 f i (x; !; "i ) = ai g dG("i )
i)
:
(1)
where 1f:g is the indicator function. These probabilities represent the expected behavior of player
i from the point of view of the other players, who do not know "i . By the independence of private
information across players in Assumption 1(B), players’actions are independent once we condition
on common knowledge variables (x; !) and players’s strategies, such that Pr(a1 ; a2 ; :::; aN jx; !; ) =
QN
i=1 Pi (ai jx; !; i ).
Given beliefs
Let
i
about the behavior of other players, each player maximizes his expected utility.
(ai ; x; !) + "i (ai ) be player i’s (normalized) expected utility if he chooses alternative ai (not
necessarily an optimal choice) and the other players behave according to their respective strategies
in
. By the independence of private information in Assumption 1(B), we have that:
i
(ai ; x; !)
a
P
N
i 2A
1
Q
j6=i Pj (aj jx; !;
j)
i (ai ; a i ; x; !)
(2)
DEFINITION: A Bayesian Nash equilibrium (BNE) in this game is a set of strategy functions
such that for any player i and for any (x; !; "i ),
i (x; !; "i ) = arg max
ai 2A
n
i
(ai ; x; !) + "i (ai )
o
(3)
We can represent this BNE in the space of players’choice probabilities. This representation is
convenient for the econometric analysis of this class of models. Let
and let P(x; !;
be a set of BNE strategies,
) be the vector of choice probabilities associated with these strategies. Solving
the equilibrium condition (3) into the de…nition of choice probabilities in (1), we get that for any
6
(ai ; i) 2 A
f0g
Pi (ai j x; !;
i)
I:
=
=
Z
Z
1 ai = arg max
1 "i (k)
= Gai
"i (ai )
i
(ai ; x; !)
i
(k; x; !) + "i (k)
i
k2A
(ai ; x; !)
i
dG("i )
(k; x; !) for any k 6= ai
dG("i )
(k; x; !) for any k 6= ai
i
where Gai (:) is the CDF of the vector of random variables f "i (k)
is known given the CDF G. As shown in equation (2), the function
(4)
"i (ai ) : for any k 6= ai g, that
i
depends on other players’
strategies only through their choice probabilities associated with . Therefore, the right hand side in
equation (4) de…nes a function
i (ai jx; !; P i ),
where P
i
fPj (aj ) : (aj ; j) 2 A f0g I i g. This
function can be evaluated at any set of choice probabilities P i , not just equilibrium probabilities.
For arbitrary P i , we have that the function i (ai jx; !; P i ) is de…ned as:
!
!)
Z (
P Q
1 ai = arg max
Pj (aj )
dG("i )
i (ai jx; !; P i )
i (k; a i ; x; !) + "i (k)
k2A
= Gai
We call the functions
i best
P
a
i
Q
j6=i
a
i
j6=i
!
Pj (aj ) [ i (ai ; a i ; x; !)
i (k; a i ; x; !)]
for any k 6= ai
!
(5)
response probability functions because they provide the probability that
an action is optimal for player i given that he believes that his opponents behave according to their
probabilities in P i . Therefore, the vector of equilibrium probabilities P (x; !) is such that every
player’s choice probabilities are best responses to the other players’probabilities. That is, P (x; !)
is a …xed point of the best response mapping
(x; !; P)
P (x; !) =
f
i (ai jx; !; P i )
: (ai ; i) 2 A
(x; !; P (x; !))
Ig:
(6)
EXAMPLE 2: (i) Binary Probit game. A = f0; 1g and "i (1) has a standard normal distribution.
In this case, G1 (:) is equal to the CDF of the standard normal, (:). Then,
!
!
P Q
Pj (aj )
i (1; a i ; x; !)
i (1jx; !; P i ) =
a
i
(7)
j6=i
(ii) Multinomial Logit game. A = f0; 1; :::; Jg and the "ei ’s are iid extreme value type 1 distributed
7
such that,
(
exp
i (ai jx; !; P i )
P Q
a
=
PJ
k=0 exp
=
1+
k6=ai
i (ai ; a i ; x; !)
i
(
P
j6=i Pj (aj )
)
P Q
a
j6=i Pj (aj )
i
(8)
i (k; a i ; x; !)
i
exp
)=
1
(ai ; x; !)
i
= Gai ()
(k; x; !)
Assumption 1 implies that the best response probability mapping
entiable. Therefore, by Brower’s …xed point theorem, the mapping
(9)
is continuously di¤er(x; !; :) has at least one
equilibrium. That equilibrium may not be unique. For some values of (x; !) the model may have
multiple equilibria. The set of BNE associated with (x; !) is de…ned as
(x; !)
fP : P =
(x; !; P)g :
Under our regularity conditions, the set of equilibria
(10)
(x; !) is discrete and …nite for almost all
games (x; !; G). Furthermore, each equilibria belongs to a particular "type" such that a marginal
perturbation in the payo¤ function implies also a small variation in the equilibrium probabilities
within the same type. The following de…nitions and lemma establish these results formally.
DEFINITION [Regular BNE]. Let f (x; !; P) be the function P
(x; !; P) such that an equi-
librium of the game for given (x; !) can be represented as a solution to the system of equations
f (x; !; P) = 0. We say that a BNE P is regular if the Jacobian matrix @f (x; !; P )=@P0 is
non-singular.
DEFINITION [Equilibrium types]. Let
f (x; !; P) as f (
(x;!) ; P).
be the vector of players’ payo¤ s f i (ai ; a i ; x; !)g
N
associated to (x; !). The vector
the equilibrium mapping
(x;!)
(x;!)
is a point in the Euclidean space RN (J+1) . Given that
depends of (x; !) only through
Let
0
1
and
(x;!) ,
we can represent the function
be two vectors of payo¤ s in this Euclidean space, e.g.,
vectors of payo¤ s associated to two di¤ erent values of (x; !). And let P
associated with
0
and
1,
respectively. We say that P
0
and P
1
0
and P
1
be BNEs
belong to the same type of
equilibrium if and only if there is a continuous path fP(t) : t 2 [0; 1]g that satis…es the condition
f
(1
for every t 2 [0; 1], such that P(0) = P
way the equilibria P
0
t)
0
0
+t
1
; P(t) = 0
and P(1) = P 1 , and this path connects in a continuous
and P 1 .
8
LEMMA 1 [Doraszelski and Escobar (2010)].6 Under the conditions of Assumption 1, for almost
all games/payo¤ s
(x;!) :
(A) all equilibria are regular; (B) the number of equilibria is …nite; and
(C) each equilibria belongs to a particular type.
Based on Lemma 1, we index equilibrium types by
2 f1; 2; :::g, and we use
(x; !) to represent
the set of indexes for the equilibrium types in the set of equilibria (x; !). That is, we can represent
the set of equilibria associated to a game with payo¤s
(x;!)
(x; !), or in the space of indexes of equilibrium types using
either in the space of CCPs using
(x; !).
EXAMPLE 3: Consider a simple version of the coordination game within the classroom that we
have introduced in Example 1 above. A student’s choice set A is binary: ai = 0 represents low
e¤ort and ai = 1 indicates high e¤ort. There are N students in class. The teacher’s combination of
skills and e¤ort is considered exogenous and represented by the scalar variables x and !, where x
is observable to the researcher and ! is unobservable. A student’s utility function is a combination
of the production function of the student’s end of the year knowledge and her cost of e¤ort. It has
the form
where
0,
i
= ei (ai ; a i ; x; !) + "ei (ai ), and more speci…cally,
8
1 X
>
>
aj
+
x
+
'
!
+
x
>
0
0
0
0
>
j6=i
<
N 1
i =
>
>
1 X
>
>
aj
: 1 + 1 x + '1 ! + 1 x
j6=i
N 1
0,
'0 ,
0,
1,
1,
'1 , and
1
+ "ei (0) if ai = 0
(11)
+ "ei (1) if ai = 1
are parameters. This speci…cation establishes that a
student’s payo¤ depends on his own e¤ort, the teacher’s e¤ort-skills, the average e¤ort of the other
students, and his own private information cost of e¤ort (or skills). Suppose that "ei (0) and "ei (1)
are normal random variables, independently distributed across students with zero mean and with
V ar("ei (1) "ei (0)) = 2 . The normalized payo¤ is i (1; a i ; x; !)+"i (1) with i (1; a i ; x; !) = +
P
x+' !+ x N 1 1 j6=i aj , with
( 1 0 )= ,
( 1 0 )= , ' ('1 '0 )= ,
( 1 0 )= ,
and "i (1)
("ei (1)
"ei (0))= . In this simple model all the students are assumed identical except for
their private information variables, and therefore they all have the same best response probability
function
(1jx; !; P i ). Furthermore, every student perceives the other students as identical and
believes that all other students have the same probability of high e¤ort P (x; !), i.e., we assume
that the equilibrium is symmetric. Then, the best response probability function of any student in
6
Doraszelski and Escobar (2010) study dynamic games of incomplete information. The equilibrium concept that
they use is Markov Perfect. Our static model and our equilibrium concept (BNE) are equivalent to the ones in
Doraszelski and Escobar when the time discount factor is zero. Our Lemma 1(A) comes from their Theorem 1.
Lemma 1(B) corresponds to their Corollary 1, that in turn comes from Haller and Laguno¤ (2000). Lemma 1(C) is
a corollary of Proposition 2 in Doraszelski and Escobar.
9
this model is:
(1j x; !; P ) =
Suppose that x > 0 and
( +
x+' !+
x P (x; !))
(12)
> 0 such that there are positive synergies between the teacher’s ef-
fort/skills and students’ e¤ort. Then, the model is a Coordination Game and the best response
probability function has an S form as shown in Figure 1.
Figures 1 and 2 come from this example when the parameter values are
' = 0, and
= 2:0,
=
7:31,
= 6:75, and the variable x that represents teacher’s e¤ort-skills is an index in the
interval [0; 1]. Figure 1 presents the equilibrium mapping when teacher’s e¤ort is x = 0:52. For
this level of teacher’s e¤ort the model has three equilibria with low, middle, and high probability
of high students’e¤ort. The high and low e¤ort equilibria are (Lyapunov) stable, while the middle
e¤ort equilibrium is unstable.
Figure 2 illustrates how the two stable equilibrium types vary when we change teacher’s e¤ort. In
this example with + < 0, teacher’s e¤ort is a substitute of student’s own e¤ort in the production
function of knowledge, i.e., the equilibrium probability of high e¤ort declines with teacher’s e¤ort
x. This …gure also shows that each equilibrium type exists only for values of x within an interval.
The low-e¤ort equilibrium type exists only for values of x in the interval [0:48; 1], while the highe¤ort equilibrium type exists only for values of x in the interval [0; 0:60]. The fact that some type
of equilibria appear or disappear for di¤erent values of x introduces a discontinuous relationship
between teacher’s e¤ort x and students’e¤ort as represented by the equilibrium P .
10
Figure 1: Coordination Game. Three Types of Equilibria
Best response function: (P ) = (2:0 7:31 x + 6:75 x P )
Teacher’s e¤ort: x = 0:52; Set of Equilibria: f0:054, 0:521, 0:937g
Figure 2: Coordination Game. Equilibrium Types
Best response function: (P ) = (2:0 7:31 x + 6:75 x P )
Multiple equilibria for x 2 [0:48; 0:60]
11
Assumption 1 has some other implications that are relevant for the identi…cation and estimation
of the model. For instance, the best response probabilities
i (ai jx; !; P i )
are bounded away from
zero and one. This is needed for the uniform convergence of the log-likelihood function and for
consistency and asymptotic normality of the maximum likelihood estimator. The following Lemma
2 is also an implication of Assumption 1 and it plays an important role in the identi…cation of the
structural model.
P (a ; x; !)
i
i
LEMMA 2 (Based on Hotz and Miller, 1993). Let
(ai ; x; !) be the (normalized)
P
Q
P (a ; x; !)
i
i
a i [ j6=i Pj (aj )]
i
expected payo¤ function as de…ned in equation (2) above, i.e.,
i (ai ; a i ; x; !).
Under Assumption 1, for any value of (x; !) there is a one-to-one mapping G()
between the J (normalized) expected payo¤ s of player i, f
best response CCPs of player i, f
ai 2 A
f0gg = G(f
P (a ; x; !)
i
i
i (ai jx; !; P i )
: ai 2 A
P (a ; x; !)
i
i
: ai 2 A
: ai 2 A
f0gg. That is, f
f0gg) where G()
f0gg and the J
i (ai jx; !; P i )
:
[G1 (); G2 (); : : : ; GJ ()], each
component Ga () is the CDF de…ned in equation (4), and G is invertible.
Under Assumption 2, the researcher knows mapping G. Therefore, this assumption together
with Lemma 2 imply that the researcher can uniquely determine expected payo¤s from players’
best response CCPs. This result is the starting point for our analysis of identi…cation in the next
section.
3
3.1
Identi…cation
Data and data generating process
Suppose that the researcher observes M di¤erent realizations of the game; e.g., M di¤erent classroomyears in our example of game within the classroom, or M di¤erent local markets in a game of market
competition. We use the index m to represent a realization of the game. For the sake of concreteness in our discussion, we consider that these multiple realizations of the game represent the same
players playing the game at M di¤erent markets. For every market m, the researcher observes the
vector xm and players’ actions fa1m , a2m , :::; aN m g. For the asymptotics of the estimators, we
consider the case where the number of players N is small and the number of realizations of the
game is large (e.g., the number of markets M goes to in…nity). As stated in Assumption 2, we
assume that the distribution of the normalized private information unobservables, G, is known to
the researcher. We study the nonparametric identi…cation of the distribution of common knowledge
unobservables and of the normalized payo¤ functions
12
i
using these data.
Let
be the vector of players’payo¤ functions in the population under study. Assumption 3
summarizes all the conditions that we impose on the Data Generating Process (DGP).7
ASSUMPTION 3: The DGP is described by the following conditions. (A) The realizations of the
unobservable variables in ! m and the observable exogenous variables in xm are independent random
draws from a joint probability distribution Fx;! . (B) Let F! (!jx) be the conditional probability
f! (1) ;
function associated to the joint probability Fx;! . The distribution F! has …nite support
! (2) , :::, ! (L) g, i.e., …nite mixture model. (C) The variable
m,
that represents the equilibrium
type selected in market m, is independent of f"im g and independently distributed over m with
probability distribution
( jxm ; ! m ) on
(xm ; ! m ), where
( jxm ; ! m )
(D) The observed vector of players actions in market m, am
draw from a multinomial distribution Pr(am j xm ; ! m ;
Pr(am jxm ; ! m ;
(E) The vector P(
m)
(xm ; ! m )
(
fPi
m)
m)
=
YN
i=1
m)
(
Pi
m)
Pr(
m
=
jxm ; ! m ).
fa1m , a2m , ..., aN m g, is a random
such that
(aim jxm ; ! m )
(aim jxm ; ! m ) : (ai ; i) 2 A
f0g
Ig containing the
population CCPs of every player in market m, is a regular equilibrium of the game, i.e., there is an
equilibrium type
m
2
(xm ; ! m ) such that P(
m)
(xm ; ! m ) =
(xm ; ! m ; P(
m)
(xm ; ! m )).
Let Q(ajx) be the probability distribution of observed players’actions conditional on observed
exogenous variables: Q(ajx)
Pr(am = a j xm = x). This probability distribution Q is identi…ed
from the data under very mild regularity conditions. Furthermore, this probability distribution
contains all the information from the data that is relevant to identify the structural parameters of
the model, f , F! ,
g. According to the model and our assumptions on the DGP, we have the
following relationship between Q and the structural parameters f , F! , g:
Q(ajx)
=
X
!2
subject
X
2 (x;!)
to P( ) (x; !) =
F! (!jx) ( jx; !)
N
Q
i=1
( )
Pi (ai jx; !)
(13)
(x; !; P( ) (x; !))
The system of equations in (13), including the equilibrium conditions that de…ne implicitly the
equilibrium CCPs, summarizes all the restrictions imposed by the model on the data for identi…cation of the structural parameters. Therefore, given Q, the primitive functions are identi…ed if this
system of equations has a unique solution for f , F! , g.
7
Note that in the description of the DGP we do not need to specify the distribution of the vector of unobservable
sunspots m but only of the selected equilibrium type m .
13
DEFINITION (Identi…cation): Suppose that the distribution Q is known to the researcher. The
model is fully (point) identi…ed i¤ there is a unique value f , F! ,
g that solves the system of
equations (13).
Since the two types of common knowledge unobservables, ! and , have …nite supports, we
can de…ne a scalar random variable
g(!; ) with discrete and …nite support that contains the
same information as (!; ). Variable
represents the combination of all the common knowledge
P
unobserved heterogeneity. Let H( jx) be the PDF of , i.e., H( jx) = !; 1f = g(!; )g F! (!jx)
( jx; !). We also present identi…cation conditions for f , Hg without the separate identi…cation
of F! and .
We are interested in two main questions: under which conditions is the payo¤ function identi…ed? and under which conditions is it possible to separately identify the relative contribution of
payo¤-relevant common knowledge unobservables (PR) and ’sunspots’(SS) as competing explanations for non-independence of players’actions in the data?
We follow a sequential approach to derive conditions for identi…cation. We assume the probability function Q(ajx) to be known. In the …rst step, given Q(:j:), we obtain conditions for the
identi…cation of the CCPs Pi (ai j x; ) and the probability distribution H( jx) from the system of
equations:
Q(ajx) =
X
H( jx)
N
Q
i=1
Pi (ai j x; )
(14)
By Lemma 2, the identi…cation of the CCPs Pi (:j:) is equivalent to the identi…cation of the equilibrium expected payo¤ function
P (a ; x;
i
i
payo¤ function
given that the expected payo¤
i (ai ; a i ; x; !)
). In the second step we consider the identi…cation of the
P (a ; x;
i
i
) is known. Finally, in
step 3, we derive conditions for the identi…cation of the distributions F! (!jx) and ( jx; !) given
the payo¤ function
i
and the distribution H( jx).
We are interested in conditions for point identi…cation as well as in testing three di¤erent null
hypotheses: (a) no common knowledge unobserved heterogeneity, i.e.,
unobservables, i.e.,
3.2
is constant; (b) no SS
is constant; and (c) no PR unobservables, i.e., ! is constant.
Model without PR or SS unobserved heterogeneity
Before we present our identi…cation results for the model with the two sources of unobserved
heterogeneity, it is helpful to discuss the identi…cation of the model without any of these two
sources of heterogeneity. This case is a useful benchmark of comparison, and it illustrates the
importance of exclusion restrictions for the identi…cation of payo¤s.
14
Consider the model without any form of common knowledge unobserved heterogeneity, either
payo¤ relevant or sunspots. In this restricted version of the model, ! m is a constant across markets,
and
m
is a deterministic function of the observable xm , i.e.,
m
= h(xm ). Therefore, the probability
distribution that describes the equilibrium selection is degenerate: for any ( ; x), we have that
( jx) = 1f = h(x)g, where 1f:g is the indicator function. Conditional on the value of x only one
equilibrium is selected. This condition is a soft version of the assumption "only one equilibrium
is played in the data" that has been one of the cornerstones of the recent literature exploiting the
two-step approach.8
3.2.1
Step 1: Identi…cation of equilibrium CCP’s
Without common knowledge unobservables, players’actions are independent conditional on observQ
ables x such that Q(ajx) = N
i=1 Qi (ai jx) where Qi is the marginal distribution of ai conditional
on x. According to the model, this marginal distribution is the equilibrium CCP for player i:
Pi (ai jx;
= h(x)) = Qi (ai jx). If x has a discrete and …nite support, the probabilities Qi can
b i (ai jx) = PM 1faim = ai ;
be consistently estimated using simple frequency estimators, e.g., Q
m=1
PM
xm = xg= m=1 1fxm = xg. The case of continuous variables in x is slightly more complicated
because multiplicity of equilibria may generate discontinuity points in the CCP function. The researcher does not know, ex-ante, the number and the location of these discontinuity points, and
this complicates the application of smooth nonparametric estimators, such as kernel or sieve estimators.9 However, the discontinuity of the probability function Q does not imply that the model is
not identi…ed. Müller (1992) and Delgado and Hidalgo (2000) study nonparametric estimation of a
regression function with ’change-points’or discontinuities when the location of these points is unknown to the researcher. They propose variations of standard kernel methods and show consistency
and asymptotic normality.
3.2.2
Step 2: Identi…cation of payo¤ s
Given that Pi (ai jx;
= h(x)) = Qi (ai jx), Lemma 2 implies that we can uniquely identify equilib-
rium expected payo¤s f
P (a ; x)
i
i
: ai 2 A
f0gg from fQi (ai jx) : ai 2 A
8
f0gg if we invert the
Most papers using the two-step approach assume, more or less explicitly, that only one equilibrium is present in
the DGP. Aguirregabiria and Mira (2007) refer to the possibility that di¤erent equilibria might be selected across
subsamples de…ned by the value of common knowledge variables, as long as the sample partition is known by the
researcher.
9
If the model has multiple equilibria this function may be discontinuous if only because some equilibria can appear
and disappear when we move continuously along the space of x. This point is illustrated in Figure 2. For any value
of x in the interval [0:48; 0:60], the model has multiple equilibria. However, the model has a unique equilibrium for
values x < 0:48 or x > 0:60.
15
P (a ; x)
i
i
one-to-one mapping G(). That is, in equilibrium
= Gai1 (fPi (ai jxg) and we can treat the
expected payo¤s hereafter as known. The problem of identi…cation is that of recovering the payo¤
function
from the system of equations:
P
i (ai ; x)
Q
where Q i (a i jx)
values
i (ai ; a i ; x)
j6=i Qj (aj jx).
=
a
Q i (a i jx)
i
i (ai ; a i ; x)
(15)
Because of strategic interactions, there are multiple payo¤
P (a ; x)
i
i
for every
P
that is identi…ed, so a discrete game is severely under-
identi…ed relative to a standard discrete choice - random utility model. Some restrictions on payo¤s
are needed to restore identi…cation. In this literature, exclusion restrictions have been the most
common type of identifying restrictions (see Bajari et al., 2010). Suppose that x = fxc ; zi : i 2 Ig
where zi 2 Z and the set Z is discrete with at least J + 1 points. Furthermore, suppose that
depends on (xc ; zi ) but not on z
i (ai ; a i ; x)
di¤erent values of z
i
the primitive payo¤s
However, the probabilities Q
i
fzj : j 6= ig. Then, for …xed (xc ; zi ) and
i
i (ai ; a i ; x)
on the right-hand-side of (15) are constant.
and the expected payo¤s do vary with z
P (a ; xc ; z )
i
i
i
the payo¤s and equilibrium behavior of other players. Let
collecting
P (a ; xc ; z ; z )
i
i
i
i
c
i (ai ; a i ; x ; zi )
payo¤s
c
i (ai ; x ; zi )
for all z i , and let
= Q i (xc ; zi )
where Q i (xc ; zi ) is a matrix with dimension jZjN
6
6
6
6
4
(1)
i
(1)
(a
i
i
Q i (a
Q
(1)
i
Q i (a
(1)
i)
(2)
c
x ; zi ; z i )
j xc ; zi ; z
j
(jZjN
i
1
(2)
i
(2)
(a
i
i
Q i (a
Q
..
.
j xc ; z i ; z
because z
be the jZjN
be the (J +1)N
1
1
changes
i
1 vector
1 vector collecting
for all a i : Then, equations (15) can be written in vector form as
P
c
i (ai ; x ; zi )
2
i
)
(2)
i
) Q i (a
We can recover the vector of payo¤s
(J + 1)N
1
(1)
i)
(2)
c
x ; zi ; z i )
j xc ; zi ; z
j
..
.
j xc ; zi ; z
(jZjN
i
c
i (ai ; x ; zi )
c
i (ai ; x ; zi )
1
)
1
(16)
that is de…ned as
((J+1)N
i
((J+1)N
(a
i
i
:::
Q i (a
:::
Q
((J+1)N
i
) ::: Q i (a
1)
1)
1)
..
.
3
(1)
i)
(2)
c
x ; zi ; z i )
j xc ; zi ; z
j
j xc ; zi ; z
(jZjN
i
1
)
)
7
7
7;
7
5
from (16) as long as matrix Q i (xc ; zi ) has full
column rank.
3.2.3
Step 3: Identi…cation of the equilibrium selection function
Given the identi…cation of the payo¤ function, we know the form of the equilibrium mapping
(x; P) and we can compute all the equilibria that the model has for each value of x in the sample.
Then, we can identify the equilibrium selection function h(x) at every value of x observed in our
sample. More precisely, for every value of x, the index h(x) should be the value that uniquely
16
solves the following optimization problem.
h(x) = arg min
2 (x)
where Q(x) and P( ) (x) are JN
Q(x)
P( ) (x) ,
(17)
1 vectors of choice probabilities for every player and choice
alternative conditional on x such that Q(x) contains the empirical probabilities estimated from the
data, and P( ) (x) contains the equilibrium probabilities implied by the model for the equilibrium
type .
3.2.4
Testing the hypothesis of "no common knowledge unobserved heterogeneity"
This version of the model imposes the following testable restrictions on the probability distribution
Q
Q: Q(ajx) = N
i=1 Qi (ai jx). These restrictions are direct implications of the assumption of "no
common knowledge unobserved heterogeneity". Therefore, this assumption can be easily tested
using a test of the null hypothesis of independence of players’actions conditional on x. For instance,
for a binary choice game with two players the testable restriction is:
Q(1; 0jx)
Q1 (1jx)
Q(1; 1jx)
=
=
Q(0; 0jx)
Q1 (0jx)
Q(0; 1jx)
3.3
3.3.1
(18)
Model with both PR and SS unobserved heterogeneity
Step 1: Identi…cation of equilibrium CCP’s and mixing distributions
By Lemma 2, the vector of expected payo¤s f
if the vector of CCPs fPi (ai jx; ) : ai 2 A
P (a ; x;
i
i
) : ai 2 A
f0gg is identi…ed if and only
f0gg is identi…ed. The identi…cation of CCPs is based
on the set of restrictions:
L (x)
Q(ajx) =
X
=1
H( jx)
N
Q
i=1
Pi (ai j x; )
(19)
where L (x) represents the number of points in the support of the distribution H( jx). This
system of equations describes a nonparametric …nite mixture model. The identi…cation of this class
of models has been studied by Hall and Zhou (2003), Hall, Neeman, Pakyari and Elmore (2005),
Allman, Matias, and Rhodes (2009), and Kasahara and Shimotsu (2014), among others. In all
these papers, identi…cation is based on the independence between the N variables fa1 ; a2 ; :::, aN g
once we condition on (x; ) and it does not exploit any variation in the exogenous variables in
x, e.g., independence assumptions between x and . Therefore, the analysis that follows applies
separately for every value of x and for notational simplicity we drop x as an argument.
17
Clearly, in equation (19) the necessary order condition for identi…cation is (J + 1)N
JN L + (L
1 >
1), i.e., the number of restrictions or known probabilities Q should be greater or
equal than the number of unknown parameters in the choice probabilities and in the distribution
of the unobservables . The basic intuition from this order condition is that the assumption of
independent marginals can deliver identi…cation if the number of variables and/or their support
are su¢ ciently large. Hall and Zhou (2003) studied nonparametric identi…cation for a mixture
with two branches, L = 2 in our notation. They showed that the model cannot be identi…ed for
N = 2, even if J is made large enough to satisfy the order condition. However, for any N > 3
they showed that the primitives of the model, fH(1); Pi (:j = 1), Pi (:j = 2) : i = 1; 2; :::; N g, are
identi…ed under very mild regularity conditions (Theorem 4.3 in Hall and Zhou, 2003). Allman et
al (2009) study the more general case with L
2 branches. They establish that a mixture with
L components is "generically" identi…ed if N > 3 and L
(J + 1)int[(N
1)=2] ,
where int[:] is the
integer or ‡oor function10 . Note that the upper bound to the number of identi…able branches not
only increases with the number of variables (players) N but also with the size of support of these
variables. Generic identi…cation here means that the set of primitives for which identi…ability does
not hold has measure zero.
The following Proposition 1 is an application to our model of Theorem 4 and Corollary 5 in
pages 13-14 of Allman et al (2009).
PROPOSITION 1. Consider the model in equation (19). Let fY1 ; Y2 ; Y3 g be three random variables
that represent a partition of the vector of players’ actions (a1 ; a2 ; :::; aN ) such that Y1 is equal to
the action of one player (if N is odd) or two players (if N is even), and variables Y2 and Y3
evenly divide the actions of the rest of the players. For j = 1; 2; 3, let PYj ( )
fPr(Yj = yj j ) :
for any value yj g be the vector with the probability distribution of Yj conditional on the unobserved
component
. Suppose that: (a) N > 3; (b) L 6 (J + 1)int[(N
1)=2] ;
(c) H( ) > 0 for any
= 1; 2; :::; L ; and (d) for j = 1; 2; 3, the L vectors PYj ( = 1), PYj ( = 2), ..., PYj ( = L )
are linearly independent. Then, the distribution H and players’CCPs Pi ’s are uniquely identi…ed,
up to label swapping.
To illustrate the conditions for identi…cation of the mixture components and weights in Proposition 1, consider the following examples. In an binary choice game with three players, the model
is step 1-identi…ed if the DGP has two mixture components, but no more. A binary choice game
with …ve players is identi…ed in step 1 with up to 4 mixture components, e.g., there might be a
10
The ‡oor function int[x] is the the greatest integer less than or equal to x.
18
binary payo¤-relevant unobservable with two di¤erent equilibria being played at each of the two
values of the payo¤-relevant unobservable.
In general, the true number of mixture components, L , is not known by the researcher. This
is particularly relevant in our model because the support of
depends on the number of equilibria
of the model that are selected in the DGP, which is an endogenous object. Therefore, it seems
reasonable not to impose restrictions on the number of mixture components for
but to identify
it from the data. Kasahara and Shimotsu (2014) provide conditions for identi…cation (and estimation) of a lower bound on the number of mixture components. The following Proposition 2 is an
application to our model of identi…cation results in Kasahara and Shimotsu (2014).
PROPOSITION 2. Let ai be a variable that is deterministic function of variable ai and that may
imply some information reduction with respect to ai , e.g., ai 2 f0; 1; 2g and ai = 1fai
1g. Given
a de…nition of (a1 ; a2 ; :::; aN ), let S1 and S2 be any pair of random variables such that they are
a partition of the N variables (a1 ; a2 ; :::; aN ), e.g., S1 = fa1 g and S2 = fa2 ; :::; aN g. Let Je1 and
Je2 be number of points in the supports of S1 and S2 , respectively, and let C(S1 ;S2 ) be the Je1
Je2
matrix describing the joint distribution of (S1 ; S2 ). Then the rank of C(S1 ;S2 ) is a lower bound of
the true number of mixture components L . Furthermore, if the rank of C(S1 ;S2 ) is strictly lower
than min[Je1 ; Je2 ], then the bound is tight and the number of components is exactly identi…ed as
L = rank(C(S1 ;S2 ) ).
From Proposition 2, lower bounds on the number of mixture components are easily identi…able.11
Clearly, di¤erent de…nitions of variables S1 and S2 are possible and di¤erent lower bounds may be
obtained depending on the researcher’s choice. Intuitively, S variables with larger supports may
give more accurate lower bounds but their distributions will be estimated with less precision in any
given sample than those of S variables which use data reduction.
EXAMPLE 4: (i) Two-player game. As shown in Hall and Zhou (2003), the parameters of this
model are not uniquely identi…ed if L
2. However, using Proposition 2 we can identify the
number of components L , or at least a lower bound. With only two players we can set S1 = a1
and S2 = a2 without any data reduction, and matrix C(S1 ;S2 ) has dimension (J + 1)
If C(S1 ;S2 ) is full rank, then we can say that L
(J + 1).
J + 1. Otherwise, we have that L is exactly
identi…ed as the rank of C(S1 ;S2 ) . For instance, in a two-player binary choice game we have that
11
See Kasahara and Shimotsu (2014), section 2.2, for the derivation of the bound. In section 3 of that paper, they
describe a fairly simple sequential algorithm for estimation of the bound based on the rank tests of Kleibergen and
Paap (2006). The estimator allows the researcher to aggregate information from di¤erent choices of S1 and S2 .
19
jC(S1 ;S2 ) j = Q(0; 0) Q(1; 1)
Q(1; 0) Q(0; 1). If this determinant is zero, then the rank of C(S1 ;S2 )
and the value of L are equal to 1. In this particular example the identi…cation of the bound
on L is equivalent to the test of "no common knowledge unobserved heterogeneity" that we have
described above at the end of section 3.2.1.
(ii) Three-player binary choice game. By Proposition 1, this model is step 1 identi…ed if the
DGP has two mixture components, but no more. De…ne S1 = fa1 ; a2 g and S2 = fa3 g such that
Je1 = 4 and Je2 = 2. If the rank of C(S1 ;S2 ) is 2, then we can tell that the number of components is
at least 2. If the rank of C(S1 ;S2 ) in the data is 1 then the number of components is exactly 1 such
that the model does not have unobserved heterogeneity.
(iii) Five-player binary choice game. This game is identi…ed in step 1 with up to L = 4 mixture
components, e.g., there might be a binary payo¤-relevant unobservable and two di¤erent equilibria
being played at each of the two values of the payo¤-relevant unobservable. In this case we might
set S1 = (a1 ; a2 ; a3 ), S2 = (a4 ; a5 ) and C(S1 ;S2 ) would be 8
4. With 4 mixture components in the
DGP, the rank of this matrix would be 4 and the researcher would obtain this as a lower bound on
the unknown true number of components.
(iv) Five player game with J + 1 = 3 choice alternatives. The maximum number of components
that can be identi…ed is 9. If we set ai = 1(ai
1) for i = 1; 2; 3, S1 = (a1 ; a2 ; a2 ) and S2 = (a3 ; a4 ),
then C(S1 ;S2 ) is 8 9: If the DGP had 6 components the rank of C(S1 ;S2 ) would be 6 which is smaller
than min[8; 9] so the bound is tight and the researcher would know this to be the exact number of
components. Note that when the bound is not tight, it still is at least as large as the maximum
number of components that can actually be identi…ed in step 1.12
3.3.2
Step 2: Identi…cation of payo¤ function
Given the identi…cation of the CCPs Pi (ai j x; ), the application of Lemma 2 implies identi…cation
of expected payo¤s
P (a ; x;
i
i
). Then, the identi…cation of the payo¤ function
is based on the
system of equations:
P
i (ai ; x;
)=
P
a
i
Q
j6=i
!
Pj (aj jx; )
i (ai ; a i ; x; !)
(20)
Suppose that the conditions of Propositions 1 and 2 hold such that the distribution H and the
CCPs fPi (ai jx; )g are identi…ed, and the number of mixture components for the unobserved het12
To see this, suppose the number of players is even. Then we partition the set of players into two subsets with
N=2 players each and we get min[Je1 ; Je2 ] = (J + 1)N=2 which is larger than the maximum number of identi…able
components (J + 1)(N 1)=2 : If the number of players is odd then min[Je1 ; Je2 ] = (J + 1)(N 1)=2 :
20
erogeneity, L (x), is known to the researcher.13 However, the researcher has not identi…ed yet
which part of this heterogeneity is PR and which part is SS. It should be clear that the worst-case
scenario for the identi…cation of the payo¤ function
i
is when all the unobserved heterogeneity
is payo¤ relevant, i.e., L (x) = L! (x). Our identi…cation strategy is agnostic but allows for this
worst-case scenario. Therefore, as a working hypothesis, we allow the payo¤ function to depend
freely on the whole unobserved component , i.e.,
i (ai ; a i ; x;
). Note that this working assump-
tion does not introduce any bias in the estimation of the payo¤ function. Furthermore, once the
payo¤ function has been recovered we will be able to identify whether for two di¤erent values of
the payo¤ function is the same, and therefore these two values of
represent variation in non-
payo¤-relevant unobserved heterogeneity. That procedure will be part of the identi…cation of the
probability distributions of ! and
in step 3.
The identi…cation of players’payo¤s is based on a similar identi…cation argument as in section
3.2.1 for the model without unobserved heterogeneity. We assume that the vector of observable
state variables is x =fxc ; zi : i 2 Ig where, for every player i, variable zi enters in the payo¤ function
of this player but not in the payo¤s of other players. However, a di¢ culty arises in the model with
unobserved heterogeneity that we did not have in section 3.2.1. As mentioned in Proposition 1, the
identi…cation of the distribution H and of CCPs Pi ’s is up to label swapping, and "pointwise" or
separately for each subpopulation de…ned by a value of the observable x. In order to implement the
identi…cation argument in Step 2, the researcher needs to be able to "match" mixture components
which correspond to the same value of ! across di¤erent subpopulations of observables. If the
researcher makes an assignment which (incorrectly) matches mixture components corresponding
to di¤erent values of !, then the system of equations (16) which exploits exclusion restrictions is
not satis…ed at the true payo¤s, and the estimation of payo¤s in step 2 will be inconsistent. The
following example illustrates this matching problem.
EXAMPLE 5: Consider a three-player binary choice game. Suppose that in step 1 the researcher
has identi…ed L = 2 mixtures or points in the support of the unobservable , that we represent
as
A
and
B.
The observable exogenous variables zi are binary: zi 2 Z = f0; 1g for i = 1; 2; 3.
Here we concentrate in the identi…cation of player 1’s payo¤. For any value of (z1 ; ), we have a
system of four equations to identify the four unknowns
1 (1; a 1 ; z1 ;
) for a
1
2 f(0; 0), (0; 1),
(1; 0), (1; 1)g. For notational simplicity, we omit the arguments (a1 ; z1 ) in the payo¤ functions.
13
Note that we allow for the number of mixtures L (x) to vary with the vector of exogenous observables x.
21
Table 1
Matching unobservable types across di¤erent values of the instruments
Panel I: Consistent Matching
Unobserved type
Q
P2 (1jz 1 ;
Probs
z
1
1 (a 1 j z 1 ; A )
P3 (1jz 1 ; A )
A)
A
1 (a 1 ; A )
1 P
1 ( A)
A )]
Estimated
P (z ;
1
1
A)
a
1
[Q
1(
True
1 (a 1 ;
(0,0)
0.70
0.60
1.04
(0,0)
6.0
6.0
(0,1)
0.50
0.75
1.00
(0,1)
2.0
2.0
(1,0)
0.90
0.45
0.96
(1,0)
2.0
2.0
(1,1)
0.80
0.70
0.42
(1,1)
-1.0
-1.0
Unobserved type
Q
P2 (1jz 1 ;
Probs
z
1
1 (a 1
B)
j z 1; B )
P3 (1jz 1 ; B )
A)
B
1 (a 1 ; B )
1 P
1 ( B)
B )]
Estimated
P (z ;
1
1
B)
a
1
[Q
1(
True
1 (a 1 ;
(0,0)
0.20
0.15
1.03
(0,0)
2.0
(0,1)
0.10
0.50
-0.15
(0,1)
0.0
0.0
(1,0)
0.60
0.05
0.63
(1,0)
-2.0
-2.0
(1,1)
0.50
0.40
-0.40
(1,1)
-3.0
-3.0
B)
2.0
Panel II: Inconsistent Matching
Q
P2 (1jz 1 ;
Probs
z
1
(0,0)
0.70
Unobserved type A [ represents incorrect matching]
(a
j
z
Estimated 1 (a 1 ; A )
1
1; A)
1
P
P3 (1jz 1 ; A )
a 1
[Q 1 ( A )] 1 P1 ( A )
A)
1 (z 1 ; A )
0.60
True
1 (a 1 ;
1.04
(0,0)
-15.1
6.0
2.0
(0,1)
0.50
0.75
1.00
(0,1)
11.1
(1,0)
0.60
0.05
0.63
(1,0)
8.1
2.0
(1,1)
0.50
0.40
-0.40
(1,1)
-4.1
-1.0
Q
P2 (1jz 1 ;
Probs
z
1
(0,0)
0.20
Unobserved type B [ represents incorrect matching]
(a
j
z
Estimated 1 (a 1 ; B )
1
1; B )
1
P
a 1
[Q 1 ( B )] 1 P1 ( B )
B ) P3 (1jz 1 ; B )
1 (z 1 ; B )
0.15
1.03
(0,0)
1.4
True
1 (a 1 ;
2.0
(0,1)
0.10
0.50
-0.15
(0,1)
1.6
0.0
(1,0)
0.90
0.45
0.96
(1,0)
-2.0
-2.0
(1,1)
0.80
0.70
0.42
(1,1)
0.4
-3.0
22
A)
B)
2
6
6
4
2
This system of equations is,
3
P (z = (0; 0); )
1
1
P (z = (0; 1); ) 7
1
1
7
P (z = (1; 0); ) 5 =
1
1
P (z = (1; 1); )
1
1
Q
6 Q
6
4 Q
Q
1 (0; 0j0; 0;
)
(0;
0j0;
1;
)
1
1 (0; 0j1; 0; )
1 (0; 0j1; 1; )
Q
Q
Q
Q
1 (0; 1j0; 0;
)
(0;
1j0;
1;
)
1
1 (0; 1j1; 0; )
1 (0; 1j1; 1; )
Q
Q
Q
Q
where, with some abuse of notation, Q
1 (1; 0j0; 0;
)
(1;
0j0;
1;
)
1
1 (1; 0j1; 0; )
1 (1; 0j1; 1; )
1 (a2 ; a3 jz2 ; z3 ;
Q
Q
Q
Q
3 2
)
7 6
1 (1; 1j0; 1; ) 7 6
5 4
1 (1; 1j1; 0; )
1 (1; 1j1; 1; )
1 (1; 1j0; 0;
3
(0; 0); )
7
1 (a 1 = (0; 1); ) 7
5
1 (a 1 = (1; 0); )
1 (a 1 = (1; 1); )
(21)
1 (a 1 =
) represents the probability P2 (a2 jz2 ; z3 ; )
P3 (a3 jz2 ; z3 ; ). In step 1, the researcher has identi…ed CCPs for all the players, and using the
inversion property the expected payo¤ of player 1,
P,
1
is also identi…ed. Therefore, given an
assignment of unobserved types across the di¤erent values of z
P(
1
vector
[Q
1(
)]
) and the matrix Q
1
P(
1
1(
1,
the researcher constructs the
) and solves in equation (21) for the vector of payo¤s as
1(
)=
). A key condition for the consistency of this estimator is that the matching of
unobserved types is correct. Table 1 presents a numerical example. Panel I illustrates the case
when the researcher makes a correct matching of unobserved types such that the estimator of
payo¤s is consistent. Panel II presents the case when the researcher makes an incorrect assignment.
After step 1, the researcher does not know the correct assignment of the unobservables types
and
B
A
across the four di¤erent values of (z2 ; z3 ). Suppose that he makes a correct assignment
for (z2 ; z3 ) = (0; 0) and (z2 ; z3 ) = (0; 1), but for values (z2 ; z3 ) = (1; 0) and (z2 ; z3 ) = (1; 1) the
researcher swaps the correct types. Therefore, in the estimation of payo¤s the researcher solves
the incorrect system of equations. That is, in the system of equations for
come incorrectly from
for
B.
P(
B)
1
and Q
1 ( B ),
A
the two bottom rows
and the opposite occurs in the system of equations
We see that the estimated payo¤s are very seriously biased for the two unobserved types.
The bias is not just in the level or/and the scale of the payo¤s but the whole pattern of strategic
interactions is inconsistently estimated.
Step 2 identi…cation requires identi…cation of a correct assignment of unobservable types along
the di¤erent values of the observable instruments z0s. Independence between unobservables and
instruments is a su¢ cient condition to identify a correct assignment. The following Proposition
provides su¢ cient conditions for identi…cation of the payo¤ function based on this independence
assumption.
PROPOSITION 3. Suppose that the conditions of Proposition 1 hold such that the distribution H
23
and the CCPs fPi (ai jx; )g are identi…ed. And suppose that: (i) (exclusion restriction) x = fxc ; zi :
i 2 Ig where zi 2 Z and the set Z is discrete with at least J +1 points, and
i (ai ; a i ; x;
) depends
on (xc ; zi ; ) but not on fzj : j 6= ig; (ii) (rank condition) for any value of (xc ; zi ; ), the matrix
Q
Q i (xc ; zi ; ) with dimension jZjN 1 (J + 1)N 1 and elements the probabilities j6=i Pj (aj jx; )
(with …xed (xc ; zi ; ) and every possible value of fzj : j 6= ig) has full column rank; and (iii) H( jx)
depends on xc but not on fzi : i 2 Ig, and H( jxc ) 6= H( 0 jxc ) for any two values
support of H(:jxc ). Under these conditions, the payo¤ functions
i
and
0
in the
are identi…ed.
Proof: In the Appendix.
Independence of the unobservables and the instruments may be a strong assumption, in particular for equilibrium type. As illustrated in …gure 2 above, the set of possible equilibria may
vary over the support of the instruments, and this may generate a change in the distribution of the
sunspot unobservable. However, it is important to emphasize that this independence assumption
is testable after step 1. Furthermore, as we show next, independence is su¢ cient but not always
necessary to identify the labels of the unobserved types.14 Lemma 3 establishes weaker su¢ cient
conditions for identi…cation of a correct assignment. To understand Lemma 3, it is important to
take into account that there may be ways of assigning unobservable types that are not the "correct
assignment" but that also provide a consistent estimation of payo¤s in step 2. In this sense it is
useful to distinguish between consistent and inconsistent assignments. The true label assignment is
(of course) consistent. An assignment which is not the true one is consistent if it matches correctly
the mixture components corresponding to the same value of the payo¤-relevant unobservable !,
but not the mixture components corresponding to di¤erent equilibrium types. Note that if multiple
equilibria or "sunspots" were the only source of unobserved heterogeneity, then all label assignments are consistent.15 In that case, step 2 identi…cation is as simple as in the model with no
heterogeneity.16 In the general case, a consistent assignment and true payo¤s are not identi…ed in
the second step without further assumptions. However, if the exclusion restrictions provide suf…cient over-identifying power the additional assumptions required for identi…cation will be weak.
This follows from the following Lemma.
LEMMA 3. Suppose the true mixture components have been identi…ed in step 1 up to label swapping.
14
For instance, in one of our Monte Carlo experiments in section 4 we do not assume independence but use the
monotonicity of CCPs with respect to the unobservables to obtain identi…cation of equilibrium types.
15
We can test for this hypothesis after step 1, and without additional conditions for the identi…cation of payo¤s in
step 2. We describe this test in section 3.3.4 below.
16
The "matching problem" described here may arises when sequential identi…cation/estimation approaches are used
in other structural models, e.g., the single-agent mixture dynamic models considered in Kasahara–Simotsu (2009),
Aguirregabiria and Mira (2010), Arcidiacono and Miller (2011).
24
Given an arbitrary label assignment, consider the system of equations
P
c
i (ai ; x ; zi ;
) = Q i (xc ; zi ; )
(ai ; xc ; zi ; )
where: Q i (xc ; zi ; ) is the matrix with dimension jZjN 1 (J +1)N 1 and elements the probabilities
Q
c
P (a ; xc ; z )
i
i
i
j6=i Pj (aj jx; ) with …xed (x ; zi ; ) for every possible value of fzj : j 6= ig; and
is the vector of expected payo¤ s. Under exclusion restrictions, a consistent label assignment is
identi…ed in step 2 if and only if: (a) for any consistent label assignment, matrix Q i (xc ; zi ; )
has full column rank for every player i, i.e., the system of equations has a unique solution; and
(b) for every inconsistent label assignment, there is at least one player i for which the rank of the
augmented matrix [
P (a ; xc ; z ;
i
i
i
) j Q i (xc ; zi ; )] is larger than the rank of matrix Q i (xc ; zi ; ),
i.e., the system of equations does not have a solution.
Proof: In Appendix. By contradiction and the Rouché-Capelli theorem. If (b) does not hold there
is another assignment such that there is at least one alternative vector of payo¤s which rationalizes
mixture components.
Under exclusion restrictions, conditions (a) and (b) in Lemma 3 are necessary and su¢ cient
for identi…cation in the second step. The number of label assignments is …nite. Under condition
(b), if the researcher considers an inconsistent label assignment, then the system of equations
P (a ; xc ; z ;
i
i
i
) = Q i (xc ; zi ; )
(ai ; xc ; zi ; ) will not have a solution. Under condition (a), if a
label assignment leads to a system which has a unique solution, then that assignment is consistent
and the solution is the true vector of payo¤s. This provides the set of conditions for identi…cation
in Proposition 4 below.
PROPOSITION 4. Suppose that the conditions of Proposition 1 hold such that the distribution
H and the CCPs fPi (ai jx; )g are identi…ed. And suppose that: (i) (exclusion restriction) x =
fxc ; zi : i 2 Ig where zi 2 Z and the set Z is discrete with at least J +1 points, and
i (ai ; a i ; x;
)
depends on (xc ; zi ; ) but not on fzj : j 6= ig. Then, the payo¤ functions are identi…ed if and only
if conditions (a) and (b) of Lemma 3 are satis…ed.
Proof: In the Appendix.
Under exclusion restrictions alone, conditions (a) and (b) of Lemma 3 become necessary and
su¢ cient for identi…cation of the model. If exclusion restrictions are just-identifying given the true
label assignment, condition (b) is very restrictive and the mixture structural model is unlikely to
be identi…ed. If exclusion restrictions provide over-identifying power then the model is identi…ed
25
under extended rank conditions (a)-(b). Rank conditions (b) allow the researcher to distinguish
between consistent and inconsistent assignments whereas rank conditions (a) are more standard and
identify payo¤s from a consistent assignment. Conditions (a)-(b) are much weaker than alternative
su¢ cient conditions. However, although "minimal", conditions (a)-(b) may not be very useful in
practice when the number of label assignments that need to be checked becomes very large. To
be practical, the sequential approach may require stronger assumptions such as the independence
assumption in Proposition 3.
3.3.3
Step 3: Identi…cation of distributions for the two types of heterogeneity
Suppose that the conditions in Propositions 1 and 4 hold such that the researcher has identi…ed
the distribution H( jx) and the payo¤ functions
distributions F! (!jx) and
i
depends on ! but not on ; and (2) by de…nition,
H( jx) = 1f = g(!; )g F! (!jx) ( jx; !).
i (x)
Now, we want to identify the probability
( jx; !). There are two sets of restrictions that we can exploit to
identify these distributions: (1) the payo¤
Let
i.
be the matrix with dimension J(J + 1)N
1
L (x) that contains all the payo¤s
f i (ai ; a i ; x; )g for a given value of x. More speci…cally, each column corresponds to a value of
and it contains the payo¤s
of
i (ai ; a i ; x;
) for every value of (ai ; a i ) with ai > 0. If two values
represent the same value of !, then the corresponding columns in the matrix
equal. Therefore, the number of distinct columns in the payo¤ matrix
i (x)
i (x) should
be
should be equal to
L! (x). That is, we can identify the number of mixtures L! (x) as:
L! (x) = Number of distinct columns in
The information in matrix
i (x)
i (x)
(22)
not only identi…es the number of points in the support of the
PR unobservables !, but it also identi…es the inverse of the one-to-one mapping
= g(!; ) such
that we know the value of (!; ) that corresponds to each value of . In particular, we identify the
columns of
i (x)
that are di¤erent and the ones that are the same, and we can label each column
(i.e., each value of ) with two values, one for ! and other for . Let g
represent the inverse of the one-to-one mapping
1(
)
[g! 1 ( ); g
= g(!; ), where g! 1 ( ) and g
1(
1(
)]
) are the
components of the inverse mapping that correspond to ! and to , respectively. Without loss of
generality we can make g! 1 (1) = 1 and then for
2 we generate g! 1 ( ) as:
8
1 0
if 9 0 < such that column = column
>
< g! ( )
g! 1 ( ) =
>
: max g! 1 ( 0 ) + 1 otherwise
0
<
26
0
(23)
And without loss of generality we can make g
1(
) = . Note that at this stage, without solving
the equilibrium (and applying an homotopy method) we cannot establish whether two vectors of
CCPs for two di¤erent values of ! correspond to the same equilibrium type or not. Therefore, for
the moment we consider that they are di¤erent equilibrium types.
EXAMPLE 6. Suppose that L (x) = 7 such that
i (x)
= 1; 2; :::; 7. Suppose the number of distinct columns of
has seven columns that we label as
i (x)
is 4: columns 1, 2, and 4 are equal
to each other, and columns 6 and 7 are also equal to each other. Then, it is clear that L! (x) = 4.
We start with the …rst column of matrix
i (x),
= 1, and make g! 1 (1) = 1. We continue with
i.e.,
the second column and …nd that it is equal to column 1, and then g! 1 (2) = 1. The third column
is di¤erent to columns 1 and 2, and then g! 1 (3) = max
way applying equation (23) to
0 <3
g! 1 ( 0 ) + 1 = 2. We proceed in this
= 4; 5; 6; 7 to obtain: g! 1 (1) = g! 1 (2) = g! 1 (4) = 1;g! 1 (3) = 2;
g! 1 (5) = 3; and g! 1 (6) = g! 1 (7) = 4. As described above, we make
Given the identi…cation of the inverse mapping g
1(
)
=g
[g! 1 ( ); g
1(
1(
)= .
)], the probability dis-
tribution of the payo¤ relevant heterogeneity, F! (!jx), is identi…ed as:
F! (!jx) =
LP
(x)
=1
Taking into account that
1 ! = g! 1 ( )
H( jx)
(24)
= , the probability distribution of multiple equilibria heterogeneity,
( jx; !), is identi…ed as:
( jx; !) =
LP
(x)
H( jx)
1
1
1 g! ( ) = g! ( )
=1
(25)
H( jx)
EXAMPLE 7. Consider the model in Example 6 above. Given the form of g! 1 ( ) for this example
we have that: F! (1jx) = H(1jx) + H(2jx) + H(4jx), with ( jx; ! = 1) =
H( jx)
H(1jx)+H(2jx)+H(4jx)
for
2 f1; 2; 4g; F! (2jx) = H(3jx), with (3jx; ! = 2) = 1; F! (3jx) = H(5jx), with (5jx; ! = 3) =
1; and F! (4jx) = H(6jx) + H(7jx), with ( jx; ! = 4) =
H( jx)
H(6jx)+H(7jx)
for
2 f6; 7g.
PROPOSITION 5. Under the conditions of Propositions 1 and 3, the one-to-one mapping
g(!; ) and the probability distributions of the unobservables, F! (!jx) and
=
( jx; !), are nonpara-
metrically identi…ed.
3.3.4
Testing hypotheses on unobserved heterogeneity
As a Corollary of Proposition 5, we have simple tests for the null hypotheses of no PR or no SS unobserved heterogeneity. If there is not SS unobserved heterogeneity, then the number of points in the
27
support of !, L! (x), should be equal to the points of support of
Therefore, taking into account that L (x) = cols(
i (x))
for any value of x in the sample.
and that L! (x) = distinct_cols(
i (x)),
testing for the null hypothesis of "no SS unobserved heterogeneity" is equivalent to testing for:
H0 : For every value of x, cols(
i (x))
= distinct_cols(
i (x)).
(26)
If there is not PR unobserved heterogeneity, then for any value of x in the sample the number of
points in the support of ! should be equal to the 1. This implies that testing for the null hypothesis
of "no PR unobserved heterogeneity" is equivalent to testing for:
H0 : For every value of x, distinct_cols(
i (x))
= 1.
(27)
As we have mentioned above, in footnote 15, we can test for the null hypothesis of no PR
unobserved heterogeneity just after step 1, without additional conditions for the identi…cation of
payo¤s. Consider matrix Q i (xc ; zi ) that is obtained by stacking vertically, for all
Q i (xc ; zi ; ). Similarly, consider the augmented matrix [
have also stacked vertically the vector of expected payo¤s
P (a ; xc ; z )
i
i
i
P (a ; xc ; z ;
i
i
i
; matrices
j Q i (xc ; zi )] where we
) for all . Under the null
that multiple equilibria is the only source of common-knowledge unobserved heterogeneity, matrices
Q i (xc ; zi ) and [
P (a ; xc ; z )
i
i
i
j Q i (xc ; zi )] have the same rank. This is because, under the null
hypothesis, the vector of expected payo¤s
P (a ; xc ; z )
i
i
i
can be written as a linear combination
of the columns in matrix Q i (xc ; zi ; ), where the coe¢ cients of this linear combination are the
payo¤s f i (ai ; a i ; xc ; zi ) : for any a i g that do not depend on any observable.
Therefore, the tests for these null hypotheses can be described in terms of tests of the rank
of a matrix of statistics. This type of tests have been proposed and developed by Cragg and
Donald (1993, 1996, 1997) and Robin and Smith (2000), among others, and have been applied
to di¤erent econometric problems such as detecting the number of factors in factor models, the
number cointegration relationships, or the identi…ability of IV estimators.
3.4
Identi…cation without a sequential approach
All the previous identi…cation results are based on the sequential approach. The exclusion restrictions we exploit in step 2 are quite natural in the estimation of games, and they are necessary for
nonparametric identi…cation even in games without unobserved heterogeneity. However, the conditions for the identi…cation of the nonparametric …nite mixture in step 1 are more stringent and
may rule out some interesting applications. An important question is whether these restrictions
are really necessary for identi…cation. More precisely, if we do not follow a sequential approach to
28
identify/estimate the model but we estimate jointly all the structural functions, is it possible to
obtain identi…cation even when the conditions in Proposition 1 are not satis…ed? In this section
we study this issue. We conclude that when the exclusion restrictions that are needed to identify
the payo¤ function in step 2 provide over-identifying restrictions, these may help identify the mixture components even when Step 1 identi…cation conditions are not satis…ed. In particular, some
two-player games will be identi…ed.
As we have shown above, once the mixing distributions H and the payo¤ vectors
have been
identi…ed, disentangling "payo¤ relevant" and "sunspot" heterogeneity in step 3 (i.e., identifying
repeated columns in the matrices
i (x))
does not require any additional assumptions. Therefore,
our discussion of sequential versus joint identi…cation concentrates on steps 1 and 2. The following
de…nitions and Lemma formalize the relationship between sequential and joint identi…cation.
DEFINITIONS: Let f ; Hg be the structural parameters of the model, and let P be the vector of
players’choice probabilities. (i) f ; Hg are jointly identi…ed i¤ , given the population Q(ajx), there
is a unique pair f ; Hg which satis…es the sets of conditions:
L (x)
(a)
Q(ajx) =
X
H( jx)
P
Q
=1
and
(b)
P (a ; x;
i
i
) =
a
i
j6=i
N
Q
i=1
Pi (ai jx; )
!
Pj (aj jx; )
for any (a; x)
(28)
i (ai ; a i ; x;
) for any (x; )
(ii) f ; Hg are sequentially identi…ed i¤ : (Step 1 identi…cation) given the population Q(ajx) there
is a unique pair fH; Pg (up to label swapping) that satis…es conditions (28)(a); and (Step 2 identi…cation) given the true CCP’s P and the expected payo¤ s
P
obtained from them by Lemma 2,
consider every possible label swap and the associated system of equations (28)(b); then the system
either has no solution, or else has the vector of payo¤ s
as its unique solution.
LEMMA 4: (A) Sequential identi…cation implies joint identi…cation. (B) The converse of (A) is
not true in general. In particular, joint identi…cation implies step 2 identi…cation but do not imply
step 1 identi…cation. (C) Sequential identi…cation and joint identi…cation are equivalent i¤ the Step
2 systems are just identi…ed, i.e., if jZj = J + 1.
Part A of this Lemma is almost trivial. Part (B) follows from the fact that joint and sequential
identi…cation embody exactly the same set of restrictions, (28)(a) and (28)(b). The identi…cation
argument in step 2 essentially starts from the equilibrium conditions, inverts them using Lemma 2
29
and rearranges the equations to exploit exclusion restrictions.17 This leads to system (20) which is
linear in the unknown payo¤ vectors. Step 2 identi…cation says that, given the players’strategies
recovered from the data, there is a unique vector of payo¤s that is consistent with best response
equilibrium behavior.
If the exclusion restrictions give just identi…ed systems in Step 2, then models which are not
step 1 identi…ed will not be jointly identi…ed. This is because, for each one of the many mixtures
of CCP’s that can explain the data in step 1, we will be able to obtain at least one payo¤ vector
satisfying the structural restrictions of step 2. Furthermore, in this case joint identi…cation will
also fail as seen in Lemma 3. However, if the Step 2 systems are over-identi…ed it can be the case
that, of all the mixtures that are observationally equivalent in step 1, only the true DGP can be
rationalized by a payo¤ vector. When exclusion restrictions provide a su¢ cient number of overidentifying restrictions, the more restrictive assumptions in Propositions 1 and 3B are not necessary
for identi…cation. In particular, as illustrated in the following example, two-player games can be
identi…ed and the assumption of independence between unobservables and z’s is not necessary for
identi…cation.
EXAMPLE 8: Consider a two-players binary choice game. For simplicity, the vector of observed
exogenous variables x includes only the player-speci…c variables z1 and z2 associated to the exclusion
restrictions: x = (z1 ; z2 ) with zi 2 Z. The number of mixture components of the random variable
is L
2, that we assume is the same for di¤erent values of (z1 ; z2 ). This model does not satisfy
the identi…cation condition in Proposition 1 because N < 3. Suppose that the model satis…es
the exclusion restriction but not the independence between zi and
in Proposition 3. Under
these conditions, the structural parameters in this model are: the 4 jZj L payo¤s
1 (1; 1; z1 ;
),
2 (1; 0; z2 ;
), and
2 (1; 1; z2 ;
); and the (L
1 (1; 0; z1 ;
),
1) jZj2 free probabilities in the
distribution H ( jz1 ; z2 ). To identify these parameters, the model imposes the following restrictions:
Q(a1 ; a2 jz1 ; z2 ) =
for (a1 ; a2 ) 2 f0; 1g
L
X
=1
H ( jz1 ; z2 ) P1 (a1 j z1 ; z2 ; ) P2 (a2 j z1 ; z2 ; )
(29)
f0; 1g, and with P 0 s satisfying the equilibrium restrictions:
Pi (1 j z1 ; z2 ; ) =
( i (1; 0; zi ; ) + Pj (1jz1 ; z2 ; ) [ i (1; 1; zi ; )
i (1; 0; zi ;
)])
(30)
The number of restrictions implied by the model is equal to the number of free probabilities in Q,
17
Lemma 2 combines the equilibrium conditions of this game and invertibility properties of a broad class of econometric discrete choice models.
30
that is 3 jZj2 . Therefore, the order condition of identi…cation is satis…ed if:
3 jZj2
| {z }
# Restrictions
4 jZj L + (L
1) jZj2
|
{z
}
# Parameters
And this condition is equivalent to 4 L + (L
4L = (4
4) jZj
(31)
0, and equivalent to fL
L )g. This condition is satis…ed for L = 2 and jZj
3 and jZj
4, and for L = 3 and jZj
12. In general, there is a continuum set of values of the primitives for which the rank condition of
identi…cation is also satis…ed - see the numerical examples in Section 4 for an illustration.
The potential for exclusion restrictions to help with joint identi…cation can also be seen in the
general order condition for joint identi…cation (without the restriction of independence between
and z):
(J + 1)N
1
jZjN
L
N jZj (J + 1)N
1
J + (L
1) jZjN
where the left-hand-side counts the number of equations or probabilities in the Q(ajz1 ; : : : ; zn )
and the RHS is the sum of the number of payo¤s and the number of mixing weights. We can
ignore the existence of non-excluded regressors xc without loss of generality because introducing
them would just multiply each of the 3 terms of the order condition by the same factor, i.e., the
cardinality of xc : The exclusion restriction makes the number of payo¤ parameters linear in the
number of players N whereas the number of restrictions on the data remains exponential in N:
Excluded variables with su¢ ciently large support should provide identi…cation as long as L <
(J + 1)N . Note that this requirement that the number of outcomes of the game be su¢ ciently large
relative to the number of mixture components is weaker than the condition for Step 1 identi…cation,
L < (J + 1)int[(N
1)=2] .
However, an implication of this is that a two-player binary choice game
with both PR-unobserved heterogeneity and multiple equilibria is not identi…ed unless we impose
restrictions on the equilibrium selection. To see this consider the simplest case where ! is binary
and two equilibria are present in the DGP for each (z; !). In this case L! = 2 and L = 4 which
does not satisfy the order condition. If the researcher knows that multiplicity of equilibria occurs
for only one of the two values of !; then L = 3 and the model may be identi…ed.
To wrap up this discussion, we review here the role of di¤erent sources of identi…cation, namely:
(a) independence of private information across players; (b) equilibrium conditions; (c) regularity of
the equilibria present in the DGP; (d) exclusion restrictions on payo¤s; (e) independence between
excluded variables and common-knowledge unobservables. The sequential approach is not feasible
in some models which are identi…ed because Step 1 identi…cation only exploits independence. Joint
31
identi…cation exploits all sources of identi…cation. The regularity condition is necessary for identi…cation. It rules out singularities in rank conditions but it provides no identifying power beyond
that.18 Equilibrium conditions are the key element of the structure that is used to recover payo¤s.
However, equilibrium conditions per se are insu¢ cient to deliver identi…cation. Exclusion restrictions on payo¤s are needed. Models which are not amenable to the sequential approach can still be
identi…ed and estimated when exclusion restrictions are su¢ ciently over-identifying. Independence
between excluded variables and common-knowledge unobservables is not necessary for sequential or
joint identi…cation, but it is very helpful to implement the sequential approach. Note that parametric assumptions on payo¤s can play the same role as exclusion restrictions providing restrictions
both across markets and across players. Finally, the restrictions on the number of players and
actions which are necessary for Step 1 identi…cation may be relaxed if the researcher is willing to
impose exclusion restrictions in the mixing distributions, e.g., restrictions on equilibrium selection,
or independence between ! and x. However, plausible restrictions on equilibrium selection may
not be easy to justify. In some cases the researcher might know that a unique BNE exists. Another
practical di¢ culty is that, even if the researcher is willing to impose restrictions on () or F! , this
does not translate always in restrictions on H() which is the relevant mixing distribution in Step 1.
3.5
Predictions and counterfactual experiments
Let (
0;
F!0 ,
0
) be the true structural functions in the population under study. Suppose that
the researcher has consistently estimated these primitive functions and is interested in using these
estimates to make predictions about the e¤ects on players’ behavior (choice probabilities) of a
counterfactual change in the economic environment or in the primitives of the model. We consider
three di¤erent types of prediction exercises in a given market m:
0;
F!0 ,
0
0;
F!0 ,
0
) constant.
, keeping xm = x0 and
0
constant.
(a) Change in xm from observed x0 to counterfactual x 2 X , keeping (
(b) Change in xm from observed x0 to counterfactual x 2
= X , keeping (
(c) Change in
from estimated
0
to counterfactual
Of course, the interesting case is when the set
) constant.
(x ; !) of equilibrium types in the counterfactual
scenario contains multiple equilibria. Otherwise, we know that the single equilibrium is selected
with probability one and we have a very standard exercise of prediction or counterfactual experiment
using a structural model.
Prediction exercise (a) is relatively straightforward. Given that x belongs to the support set
18
It can be shown that if the regularity condition is not satis…ed, the jacobian matrix of equilibrium CCP’s with
respect to payo¤s is non-singular and the rank condition for joint identi…cation does not hold.
32
X , the researcher has identi…ed the payo¤ functions and the distribution of the unobservables
(including the distribution of the ’sunspot’) for this value x . That is, there is a market m0 6= m
in our (large) sample such that xm0 = x . Therefore, the counterfactual CCPs and distributions
of unobserved heterogeneity in market m when the vector of observables is x are just the factual
CCPs and distribution of unobservables in market m0 .
Prediction exercises (b) and (c) are more challenging. The main problem comes from the
extrapolation of the equilibrium selection distributions from (:jx0 ; !) to (:jx ; !). To identify this
type of predictions or counterfactual experiments we exploit the assumption that the set of possible
equilibria
(x; !) and the probability distribution
vector of payo¤ values
(:j
(x;!) ).
(x;!) ,
i.e., with some abuse of notation,
(x; !) =
(
vectors of payo¤s are the same, i.e.,
0
(x0 ;! 0 )
have the same probability distribution
1 (:)
(x;!) )
0 (:)
Suppose two realizations of the game, one with payo¤ function
(x0 ; ! 0 ) and the other with payo¤ function
and (:jx; !) =
and state variables
and state variables (x1 ; ! 1 ), and suppose that the
1
(x1 ;! 1 ) .
=
Then, these two realizations of the game
over equilibrium types. Therefore, if the counterfactual
and value x are such that there is a market m0 in the sample such that
payo¤ function
(x ;!) ,
(:jx; !) depend on (x; !) only through the
0
(xm0 ;!)
=
then the implementation of predictions (b) and (c) is very similar to the prediction exercise
in (a): the counterfactual CCPs and distributions of unobserved heterogeneity in market m are
just the factual CCPs and distribution of unobservables in market m0 with
When there is not any market m0 in the sample such that
0
(xm0 ;!)
=
0
(xm0 ;!)
(x ;!) ,
=
(x ;!) .
the identi…cation
and estimation of the counterfactual is substantially more complicated. However, we still can
exploit the smoothness properties of function (:j ) within a neighborhood of the counterfactual
payo¤s
. Identi…cation is based on the idea that when the sample size increases, we can get
sample values of
0
(xm0 ;!)
that are arbitrarily close to
(x ;!) .
Let
(
be the set of equilibrium CCPs for the vector of counterfactual payo¤s
of equilibrium CCPs for an arbitrary vector of payo¤s
6=
)
fP : P =
. And let
(
; P)g
( ) be the set
. Using the de…nition of equilibrium
type, and with the help of an homotopy method, we can determine which equilibria in
( ) belong to the same type. We say that P 2
(
) and P 2
(
) and
( ) belong to the same type of
equilibrium if and only if there is a continuous path fP(t) : t 2 [0; 1]g that satis…es the condition
P(t)
[(1
Suppose that we make this exercise using
t)
+t
and
; P(t)] = 0
0
(x;!)
ulation. And suppose that there is at least one value of
types for
0
(x;!)
for multiple values of (x; !) in the pop0
(x;!)
is identical to the set of equilibrium types for
33
(32)
for which the set of equilibrium
, i.e.,
(
0
(x;!) )
=
(
). More
0
generally, let
0
(x;!)
be the set of payo¤ vectors
associated to values of (x; !) for which the set
of equilibrium types is identical to the set of equilibrium types for
n
0
0
(x;!) :
(x; !) 2 X
and
(
:
0
(x;!) )
=
(
)
o
A …rst condition that we need to make this inter- or extrapolation is that the set
But even if
( j
0
(x;!) )
0
is not empty, it does not mean that for
and ( j
(:j
0
[f
2
0
0
is not empty.
the probability distributions
) are identical. In general, these distributions will be di¤erent. However, we
can assume that the distribution function (:j
of) set
0
(x;!)
(33)
(x;!) )
is a smooth function within the (convex hull
g. Given these conditions, we can approximate the counterfactual distribution
) using the following kernel approximation:
b( j
) =
P P
x2X !2
1
n
0
(x;!)
P P
1
x2X !2
2
n
0
0
(x;!)
o
2
( j
0
o
0
(x;!) )
K
0
(x;!)
K
0
(x;!)
b
b
!
!
(34)
where b is a bandwidth parameter and K(:) a kernel function. For the consistency of this estimator,
we need that for any " > 0, Pr inf
0
(xm ;!)
over m
b
goes to in…nity. Under this condition, we can use b( j
>"
goes to zero as the sample size
) to estimate consistently counterfactual
experiments (b) and (c).
4
Monte Carlo experiments
In this section we illustrate the previous identi…cation results using two experiments.
4.1
Main features of the DGP
Our experiments are based on a static game of market entry where N …rms (potential entrants)
decide whether to be active or not in a market. We index …rms by i and markets by m. Let
aim 2 f0; 1g be the binary variable that represents the entry decision of …rm i in market m.
The pro…t of an inactive …rm is normalized to zero. If active in the market, a …rm’s pro…t is a
function
i (a im ; zim ; ! m ) + "im
where: a
im
is the vector of other …rms’entry decisions; zim is an
observable variable that a¤ects the …xed cost of …rm i in market m (e.g., the geographic distance
of market m to the headquarters of …rm i); ! m is a market characteristic that is unobservable to
the researcher but common knowledge to all the …rms; and "im is a component of the …rm i’s …xed
cost that is private information of this …rm and unobservable to the researcher. Let zm be the
34
vector (z1m ; z2m ; :::; zN m ). Each variable zim takes jZj values uniformly spaced between 0 and 1,
i.e., zim 2 Z = f jZj0 1 ; jZj1 1 ; :::
jZj 1
jZj 1 g.
We assume that "im is i.i.d. across …rms and markets,
and independent of (zm ; ! m ) with a standard normal distribution. We consider the following
speci…cation of the pro…t function:
i (a im ; zim ; ! m )
where f i ;
i; i
=
i
+
i
zim + ! m +
i
P
j6=i ajm
(35)
: i = 1; 2; :::; N g are parameters. Therefore, equilibrium CCPs satisfy the following
equilibrium conditions: for every …rm i:
Pi (z; !) =
i
+
i
zi + ! +
i
P
j6=i Pj
(z; !)
(36)
The PR unobservable ! m has a probability distribution with two points of support f! A ; ! B g =
P
f 0:75; +0:75g. The probability of ! m = ! A conditional on zm = z is F!A (z) = f0 +f1 N 1 N
i=1 zi ,
where f0 and f1 are parameters. In our experiments we do not have multiple equilibria unobserved
heterogeneity.
The researcher has a dataset of M markets, and for each market m he observes faim ; zim :
i = 1; 2; :::; N g. He is interested in the estimation of the pro…t functions and the distribution
of the unobservables. We consider that the researcher does not know the functional form of the
pro…t function or of the probability function F!A (z), and therefore, she estimates a model with a
nonparametric speci…cation of
i (a im ; zim ; ! m )
and of F!A (z). Similarly, the researcher does not
know the number of points in the support of the unobserved heterogeneity, L , and the nature of
this heterogeneity, PR versus non-PR. Instead, the researcher estimates the value of L , or a lower
bound, and tests for the existence of multiple equilibria (non-PR) unobserved heterogeneity.
We have implemented two experiments that we label as SEQ and NONSEQ. In Experiment
SEQ, the model is sequentially identi…ed. The number of players is N = 4, and the unobserved
heterogeneity ! m is independent of the observable state variables zm . The main purpose of this
experiment is to illustrate the identi…cation power of the sequential approach and the proposed
tests. In Experiment NONSEQ, we consider a model that is not identi…ed if we follow a sequential
approach. The number of players is N = 2, such that there is not identi…cation in the …rst step of the
sequential approach. Furthermore, the unobserved heterogeneity ! m is correlated with observable
state variables zm . The purpose of this experiment is to present an example where all the primitives
of the model are jointly identi…ed despite there is not sequential identi…cation. Table 2 provides a
summary of the DGPs in our Monte Carlo experiments.
35
Table 2
Summary of DGPs in Monte Carlo Experiments
Common features in the two experiments
Payo¤ function:
Distribution zim :
Distribution
!m:
P
zhim +! m + i
j6=i ajm
i
jZj 1
0
1
i.i.d. Uniform jZj 1 ; jZj 1 ; ::: jZj 1
i=
i+ i
Support
F!A (z)
# equilibria in data:
# markets (M ):
# MC replications:
Experiment SEQ
1=
4.2
1:00;
2=
f 0:75; +0:75g
= f0 + f1 N
1
1
PN
i=1 zi
50 and 200 for each possible value of
z
1,000
Experiment NONSEQ
N =4
jZj = 3
0:80; 3 = 0:60; 4 = 0:40
1 = 2 = 3 = 4 = 3:0
=
1
2 = 3 = 4 = 0:5
f0 = 0:20 and f1 = 0:25
N =2
jZj = 5
1 = 1:00; 2 = 0:40
1 = 2 = 3:0
=
=
=
1
2
3
4 = 0:5
f0 = 0:20 and f1 = 0:25
Experiment SEQ
Table 3 summarize the results form our …rst experiment. In table 2, we have organized the results in
four panels that correspond to four steps in the sequential estimation and testing method. Panel A
presents our results for the rank test in Proposition 2 and the estimation of number of branches L
of unobserved heterogeneity. We have used variables S1 = fa1 ; a2 g and S2 = fa3 ; a4 g to construct
b (S ;S ) with the empirical joint distribution of (S1 ; S2 ) conditional on a value of z.
the 4 4 matrix C
1 2
b (S ;S ) , and the determinants of its 3
We calculate the determinant of C
1 2
3 and 2
2 submatrices.
Figure 3 presents the empirical distribution (using 1; 000 replications) of these determinants, that
we denote them as det 4, det 3, and det 2, for the case of z = (0:5; 0:5; 0:5; 0:5). We have obtained
the same qualitative results for other values of z. We have used a Chi-square test with a 1%
signi…cance level to test for the null hypothesis that the determinant is zero.19 Based on this test,
19
Using the Monte Carlo distribution of the statistic det k (for k = 2; 3; 4), we calculate its standard error, se(det k),
and the statistic x2k = [det k=se(det k)]2 . Then, we consider that under the null hypothesis that the true value of the
determinant is zero, the statistic x2k is asymptotically distributed as a chi-square with one degree of freedom. For the
decision to reject/no reject the null hypothesis we use a signi…cance level of 1%.
36
c (z). If we reject that det 4 is zero,
we obtain an estimate of the number of mixture components L
c (z) = 4; if we do not reject that det 4 is zero but reject det 3 = 0, then L
c (z) = 3; if we do
then L
c (z) = 2; and …nally if we
not reject that det 4 and det 3 are zero but reject that det 2 = 0, then L
c = 1, i.e., no unobserved heterogeneity.
do not reject that det 4, det 3, and det 2 are zero, then L
Panel A in table 3 presents the results of this test and the empirical distribution of the estimate of
c (z). We can see that the estimate is very precise around the true value. Having a non-negligible
L
distance between ! A and ! B de…nitely helps in the precise estimation of the number of mixtures.
However, in this experiment this distance seems reasonable for values that we can …nd in actual
applications.20
Panel B summarizes the properties of the estimator of the probability H(
A jz)
and of players’
CCPs in the …rst step of the sequential method. We report the Bias and the Root Mean Square
Error (RMSE) from the Monte Carlo distribution of the estimates, as well as these statistics as
percentages of the true value of the parameter. We report results only for two values of the vector
z and for player 1, but again the results for other parameters are qualitatively identical. The
estimates are quite precise, especially taking into account that the estimation in this …rst step is
independent across z, and in this experiment we have relatively small samples of 50 observations
and 200 observations (markets) for each value of z.
Panel C presents results from the estimation of players’ payo¤s in step 2. It is important to
note that in this experiment there is not independence between unobserved heterogeneity and z.
It is clear that we do not need independence between
and z to identify the parameters in step 1
up to label swapping of the unobserved types. As we have explained in section 3.2.2, the problem
appears in the estimation of payo¤s in the step 2 because we need to keep the unobserved type
constant when we vary the z variables. However, in this experiment, the strict monotonicity of the
CCPs with respect to
provides a way to the labelling of unobserved along di¤erent values of z.
For every value of z, we can assign the label
A
to the branch of the …nite mixture associated to
the lower value of players’CCPs. Again, the estimates of payo¤s are quite precise even when only
50 observations per value of z.
In panel D, we present results from a Chi-square test of the null hypothesis of multiple equilibria.
20
In this experiment, the values of ! A = 0:75 and ! B = +0:75, together with the other parameter values, imply
an average distance of 0:4 between CCPs of a player in branch A and B.
37
Table 3. Experiment SEQ
Sample Sizes: 50 and 200 markets per value of z. Monte Carlo Simulations = 1; 000
c (z)
Panel A. Test of rank of C(S1 ;S2 ) and empirical distribution of L
c (z)
% of cases
Estimate of L
Tests
50 obs per z
200 obs per z
50 obs per z
200 obs per z
Reject det 4 = 0
det 4 = 0 & reject det 3 = 0
Accept det 4 = det 3 = 0 & reject det 2 = 0
Accept det 4 = det 3 = det 2 = 0
Accept
1.62%
0.35%
4
4
5.62%
2.72%
3
3
90.20%
95.70%
2
2
2.58%
1.23%
1
1
Panel B. Step 1: Estimation of H( jz) and CCPs
Bias (% true)
RMSE (% true)
Parameter
50 obs per z
200 obs per z
50 obs per z
200 obs per z
H( A jz = [0; 0; 0; 0])
P1A (z = [0; 0; 0; 0])
P1B (z = [0; 0; 0; 0])
H( A jz = [1; 1; 1; 1])
P1A (z = [1; 1; 1; 1])
P1B (z = [1; 1; 1; 1])
0.0042 (2.1%)
0.0000 (0.0%)
0.1282 (64.0%)
0.0607 (30.3%)
0.0003 (2.9%)
-0.0001 (-1.0%)
0.0070 (65.4%)
0.0034 (31.5%)
0.0006 (0.4%)
-0.0011 (-0.8%)
0.0523 (40.6%)
0.0258 (20.1%)
-0.0041 (-0.9%)
0.0015 (0.3%)
0.1126 (25.0%)
0.0591 (13.1%)
0.0069 (1.6%)
0.0004 (0.1%)
0.0837 (19.5%)
0.0405 (9.4%)
0.0005 (0.1%)
-0.0027 (0.0%)
0.0522 (6.4%)
0.0276 (3.3%)
Panel C. Step 2: Estimation of players’payo¤s
Bias (% true)
RMSE (% true)
Parameter
50 obs per z
200 obs per z
50 obs per z
200 obs per z
A (a
1
A (a
1
B (a
1
B (a
1
1
1
1
1
= [0; 0; 0]; z1
= [1; 1; 1]; z1
= [0; 0; 0]; z1
= [1; 1; 1]; z1
= 0:5)
= 0:5)
= 0:5)
= 0:5)
-0.0235 (-3.1%)
-0.0022 (-0.3%)
0.1438 (19.1%)
0.0717 (9.5%)
-0.1824 (-8.1%)
-0.0376 (-1.6%)
1.1030 (49.0%)
0.5074 (22.5%)
0.0153 (2.0%)
0.0047 (0.6%)
0.3518 (46.9%)
0.1684 (22.4%)
-0.0157 (-2.0%)
-0.0049 (-0.6%)
0.2006 (26.7%)
0.1005 (13.4%)
Panel D. Test of multiple equilibria
% Rejections
50 obs per z
200 obs per z
Chi-square test Ho :
A
i
=
B
i
38
4.3
Experiment NONSEQ
The main purpose of this experiment 1 is to present an example of a two-player binary choice
game that does not satisfy the conditions for step 1- identi…cation in Proposition 1, but that is
jointly identi…ed. The DGP is related to Example 8: two players, binary choice, jZj = 5, and
L = 2 components in the …nite mixture. For each value of (z; !), there is only one equilibrium in
the data. We compute each equilibrium in the DGP by iterating in the best response probability
mapping (36), simultaneously for the two players, using as starting value the vector of probabilities
(P1 ; P2 ) = (0:99; 0:01). The model has 65 parameters: 40 parameters, parameters in players’payo¤
functions, and 25 parameters in the distribution F! . We estimate this model by full maximum
likelihood using a nested pseudo likelihood estimation method. Table 4 presents Bias and Root
Mean Square Error (RMSE) of the ML estimates of structural parameters for two di¤erent sample
sizes with 50 and 200 observations per value of z.
Table 4. Experiment NONSEQ
Sample Sizes: 50 and 200 markets per value of z. Monte Carlo Simulations = 1; 000
Parameter
F!A (z = [0; 0])
F!A (z = [0; 1])
F!A (z = [1; 0])
F!A (z = [1; 1])
A (a
2
1
A (a
2
1
B (a
2
1
B (a
2
1
5
= 0; z1
= 1; z1
= 0; z1
= 1; z1
= 0:5)
= 0:5)
= 0:5)
= 0:5)
Bias (% true)
50 obs per z 200 obs per z
RMSE (% true)
50 obs per z 200 obs per z
-0.0014 (0.7%)
0.0008 (0.4%)
0.0720 (36.0%)
0.0267 (13.3%)
0.0015 (0.4%)
0.0006 (0.1%)
0.1010 (31.0%)
0.0554 (17.0%)
-0.0087 (-2.6%)
0.0015 (0.4%)
0.1546 (49.1%)
0.0662 (20.3%)
0.0020 (0.6%)
-0.0005 (-0.1%)
0.0639 (14.2%)
0.0270 (6.0%)
-0.0115 (-2.1%)
0.0038 (0.7%)
0.1705 (31.0%)
0.0808 (14.7%)
0.0514 (4.9%)
-0.0199 (-1.9%)
0.2426 (23.1%)
0.1071 (10.2%)
0.0294 (3.1%)
0.0085 (0.9%)
0.3743 (39.4%)
0.1197 (12.6%)
-0.0099 (-2.2%)
-0.0031 (-0.7%)
0.0936 (20.8%)
0.0414 (9.2%)
Conclusion
In empirical applications of games of incomplete information, we typically …nd that conditional
on observable exogenous variables players’ actions are correlated. One possible interpretation of
this correlation is that common knowledge unobservables are present. Some of these unobservables
may be payo¤ relevant while others may be ’sunspots’that a¤ect players’beliefs and the selected
equilibrium but do not have a direct e¤ect on players’ payo¤s. This paper is motivated by the
following question: is it possible to separate empirically the contribution of unobservables that a¤ect
39
the selection of an equilibrium in the data (i.e., non-payo¤ relevant unobservables or "sunspots")
from the contribution of unobservables that are payo¤-relevant? Is it possible to conclude that we
need ’sunspots’to explain players’observed behavior?
We investigate this question by studying semiparametric identi…cation of games when we allow
for three types of unobserved heterogeneity for the researcher: payo¤-relevant variables that are
private information of each player (PI unobservables); payo¤-relevant variables that are common
knowledge to all the players (PR unobservables); and variables that are common knowledge to all
the players, are not payo¤-relevant but a¤ect the equilibrium selection ("sunspots" or SS unobservables). We show that if the number of action pro…les in the game is su¢ ciently large relative to the
number of mixture components then the model is nonparametrically identi…ed under the same type
of exclusion restrictions that we need for identi…cation without unobserved heterogeneity. However, we also show that implementation of a sequential identi…cation/estimation approach requires
that the researcher be able to match mixture components across games with di¤erent values of
the excluded variables. This matching requirement is quite easily handled if we assume statistical
independence of the excluded variables ("instruments") and common-knowledge unobservables. It
is fairly common for applied researchers to make this kind of identi…cation assumption. In the
case of independence between player-speci…c excluded variables and unobservable market types,
it may be straightforward to interpret and defend the assumption in a particular application and
structural model. On the other hand, to the extent that we want to be agnostic about equilibrium
selection it may be harder to justify independence between excluded variables and equilibrium type.
If the researcher is unwilling to make the independence assumption, we have shown that the …nite
mixture game is still identi…ed if exclusion restrictions have su¢ cient over-identifying power but
sequential estimation will be less straightforward.
Our results show that it is possible to separately identify the relative contributions of payo¤relevant and "sunspot" type of unobserved heterogeneity to observed players’ behavior. As De
Paula and Tang (2012) and others have shown, multiplicity of equilibria can help identify some
elements of the structure such as the sign of strategic interactions. However, without the exclusion
restrictions that are needed to identify payo¤s even if multiple equilibria are not present it does
not seem possible for the researcher to ascertain ex-ante that the correlation between the actions
of players is induced by the occurrence of multiple equilibria in the data.
40
APPENDIX: Proofs of Lemmas and Propositions
Proof of PROPOSITION 3. Under condition (iii) we have that, for given value of xc , we can
assign the labels for the unobserved
such that H(
= 1jxc ) < H(
= 2jxc ) < :::: < H(
=
L (xc )jxc ). Therefore, for a …xed value of xc but di¤erent z 0 s values we are able to "match" the
unobserved types, i.e., type
= 1 is the one with the lowest probability, type
= 2 has the second
lowest probability, and so on. Now, for a given value of (ai ; xc ; zi ; ), the system of equations
P
Q
P (a ; x; )
i
i
a i [ j6=i Pj (aj jx; )] i (ai ; a i ; x; ) can be written in matrix form as:
P
c
i (ai ; x ; zi ;
where:
P (a ; xc ; z ;
i
i
i
) is the jZjN
P i (xc ; zi ; ) is the jZjN
and
1
(J +1)N
(ai ; xc ; zi ; ) is the (J + 1)N
) = P i (xc ; zi ; )
1
1
1
1 vector f
(ai ; xc ; zi ; )
P (a ; z ; z ;
i i
i
i
matrix with rows f
Q
) : for every z
j6=i Pj (aj jx;
) : for every a
1 vector f i (ai ; a i ; xc ; zi ) : for every a
i
2 ZN
i
2 AN
i
2 AN
1 g.
1 g;
1 g;
Under
rank condition (ii), we have that P i (xc ; zi ; )0 P i (xc ; zi ; ) is a non-singular matrix and we can
uniquely identify the vector of payo¤s
(ai ; xc ; zi ; ) as:
(ai ; xc ; zi ; ) = P i (xc ; zi ; )0 P i (xc ; zi ; )
41
1
P i (xc ; zi ; )0
P
c
i (ai ; x ; zi ;
)
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