Supplementary Appendix: Audience Costs and the
Dynamics of War and Peace
Casey Crisman-Cox
∗
Michael Gibilisco
†
Abstract
This document contains the Appendices for “Audience Costs and the Dynamics of War
and Peace.” In Appendix A, we derive the equilibrium constraint, and we characterize
an example equilibrium in Appendix B. In Appendix C, we discuss our implementation
of the CMLE. Appendix D presents Monte Carlo Experiments, and Appendix E con-
tains a discussion of time-invariant covariates. In Appendix F, we provide a table form
of the audience cost parameters and their associated standard errors, and Appendix
G illustrates the distribution of audience costs in autocratic regimes. We discuss the
procedure for substantive effects in Section H. Appendices I and J contain additional
robustness checks and substantive effects, respectively. Finally, Appendix K contains
model fit exercises.
A Characterizing Equilibria
As in Aguirregabiria and Mira (2007), we characterize equilibria with dynamic expected
utilities. Let
v
i
(
a
i
,s
) denote
i
’s net-of-shock expected utility from choosing action
a
i
in
state
s
and continuing to the play the game for an infinite number of periods, and write
v
i
= (
v
i
(
a
i
,s
))
(
a
i
,s
)
∈
A
2
for every country
i
. In other words, given a vector of expected values
v
i
and a vector of random shocks
ε
i
, country
i
chooses action
a
i
in state
s
if and essentially
only if
a
i
= argmax
a
i
∈{
1
,
2
,
3
}
{
v
i
(
a
i
,s
) +
ε
i
(
a
i
)
}
.
Thus,
v
i
is identical to a cut-off strategy for country
i
. Because
ε
i
is distributed type 1 extreme
value,
i
chooses
a
i
in state
s
with probability
P
(
a
i
,s
;
v
i
), where
P
(
a
i
,s
;
v
i
) =
exp(
v
i
(
a
i
,s
))
∑
a
′
i
exp(
v
i
(
a
′
i
,s
))
.
(5)
∗
Texas A&M University Email:
c.crisman-cox@tamu.edu
†
California Institute of Technology. Email:
michael.gibilisco@caltech.edu
1
If
g
is the distribution of
ε
i
, described above, we write country
i
’s average expected utility in
state
s
as
G
(
s,v
i
), which takes the form
G
(
s,v
i
) =
∫
max
a
i
{
v
i
(
a
i
,s
) +
ε
i
(
a
i
)
}
g
(
ε
i
)
dε
i
,
and simplifies to
G
(
s,v
i
) = log
(
∑
a
i
exp(
v
i
(
a
i
,s
))
)
+
C
where
C
is Euler’s constant (McFadden 1978, Corollary p. 82). Consider a profile
v
= (
v
i
,v
j
)
of action-state values. Then country
i
’s iterative value of action
a
i
in state
s
, denoted
Φ
ij
(
a
i
,s,v
;
θ
), is written as
Φ
ij
(
a
i
,s,v
;
θ
) =
∑
a
j
P
(
a
j
,s
;
v
j
)
︸
︷︷
︸
expectation
over
j
’s actions
[
u
ij
(
a
i
,a
j
,s
;
θ
)
︸
︷︷
︸
today’s
payoff
+
δG
(max
{
a
i
,a
j
}
,v
i
)
︸
︷︷
︸
expectation over
tomorrow’s payoff
]
.
(6)
In words, Equation 6 takes a profile of values
v
, supposes countries play according to the
associated choice probabilities, and then returns new expected values of each action in each
state. The iterative value, Φ
ij
(
a
i
,s,v
;
θ
), is comprised of three components. First, country
i
weights its opponent’s actions by the corresponding choice probabilities,
P
(
a
j
,s
;
v
j
). Second,
country
i
receives an immediate payoff,
u
ij
(
a
i
,a
j
,s
;
θ
). Finally, country
i
receives a discounted
expected future payoff,
δG
(max
{
a
i
,a
j
}
,v
i
). The profile
v
is an equilibrium if and only if
Φ
ij
(
a
i
,s,v
;
θ
) =
v
i
(
a
i
,s
) for every country
i
, every action
a
i
, and every state
s
. Hence,
v
is an
equilibrium profile if and only if it is a fixed point of these iterative value functions. Formally,
write Φ
ij
(
v
;
θ
) as Φ
ij
(
v
;
θ
) =
×
a
i
×
s
Φ
ij
(
a
i
,s,v
;
θ
) and Φ(
v
;
θ
) = Φ
ij
(
v
;
θ
)
×
Φ
ji
(
v
;
θ
). Then an
equilibrium is a profile
v
such that Φ(
v
;
θ
) =
v
.
Notice the that function Φ
ij
(
a
i
,s,v
;
θ
) is a weighted sum of current stage utilities and
discounted expected payoffs. When the latter are sufficiently bounded, the continuous function
Φ maps a convex and compact set into itself, so an equilibrium exists. Formally, define
B
≥
0
as
B
= max
i,j,a,s
{|
u
ij
(
a,s
;
θ
)
|}
, and
B
−
and
B
+
where
B
−
=
−
(1 +
δ
)
B
+
δ
log(3) and
B
+
=
(1 +
δ
)
B
+
δ
log(3). Thus,
B
−
and
B
+
represent the smallest and largest possible expected
action-state values, respectively, in any equilibrium. In addition, for all
v
∈
[
B
−
,B
+
]
18
,
Φ(
v
;
θ
)
∈
[
B
−
,B
+
]
18
because the iterative expected utility of each action-state can be no
larger or smaller than
B
+
or
B
−
, respectively, when the
v
∈
[
B
−
,B
+
]
18
. Thus, the continuous
function Φ maps a convex and compact set into itself, so Φ admits a fixed point, an equilibrium.
2
Table 4:
Structural Parameters and Values in the Example Equilibrium
Parameter
x
ij
·
β
(2)
x
ij
·
β
(3)
z
i
·
κ
(2)
z
i
·
κ
(3)
γ
(1)
γ
(2)
γ
(3)
α
i
Value
15
15
−
16
−
16
−
5
5
7
−
15
B Example Equilibrium
We consider a version of the game parameterized by the values in Table 4. We choose the
values for two reasons. First, they are similar in direction and magnitude to those we estimate
in the data. Second, under these values, a symmetric equilibrium exists, and we characterize a
symmetric equilibrium to simplify the exposition. As discussed in the paper, we still maintain
the normalization that
x
ij
·
β
(1) = 0 and
z
i
·
κ
(1) = 0.
Given the values in Table 4, the second column in Table 5 reports a solution to the equation
Φ(
v
)
−
v
= 0, where Φ is defined in Equation 6. When
v
takes on these (non-rounded) values,
the equilibrium constraints are satisfied below a tolerance of 1
e
−
10
, that is,
max
(
i,a
i
,s
)
{|
Φ
ij
(
a
i
,s
;
v
)
−
v
i
(
a
i
,s
)
|}
<
1
e
−
10
.
Although we characterize an equilibrium for specific parameter values, the equilibrium dis-
cussed here will change in a continuous manner for sufficiently small perturbations of the
underlying parameters that enter the constraint equation, Φ(
v
)
−
v
= 0, in a continuously
differentiable manner. To show this, we verify that the Jacobian of the constraint equation,
Φ(
v
)
−
v
, has full rank at the equilibrium of interest.
The Table’s third column reports the corresponding choice probabilities. Using the equi-
librium choice probabilities and the per-period transition function
s
t
+1
= max
{
a
t
i
,a
t
j
}
, we
construct the equilibrium transition matrix,
Q
v
where the entry
Q
v
[
s,s
′
] denotes the proba-
bility that game transitions from
s
to state
s
′
given equilibrium
v
. Specifically, for any state
of hostilities
s
, we have
Q
v
[
s,
1] =
P
(1
,s
;
v
i
)
·
P
(1
,s
;
v
j
)
,
Q
v
[
s,
2] =
P
(3
,s
;
v
i
) +
P
(3
,s
;
v
j
)
,
and
Q
v
[
s,
3] = 1
−
Q
v
[1
,s
]
−
Q
v
[3
,s
]
.
3
Table 5:
Equilibrium Expected Utilities and Choice Probabilities
(
a
i
,s
)
v
i
(
a
i
,s
)
P
(
a
i
,s
;
v
i
)
(1
,
1)
20
.
30
0
.
88
(2
,
1)
18
.
14
0
.
10
(3
,
1)
16
.
12
0
.
01
(1
,
2)
34
.
43
0
.
02
(2
,
2)
38
.
37
0
.
84
(3
,
2)
36
.
61
0
.
15
(1
,
3)
36
.
03
0
.
80
(2
,
3)
34
.
44
0
.
16
(3
,
3)
33
.
31
0
.
04
In this example, the equilibrium transition matrix takes the following form:
Q
v
=
0
.
78 0
.
19 0
.
03
0
.
00 0
.
73 0
.
27
0
.
63 0
.
28 0
.
09
.
Notice these values are rounded to the second digit. Without rounding,
Q
v
[2
,
1]
>
0.
An invariant or stationary distribution
π
v
describes the distribution of states in the long
run, where
π
v
(
s
) is the probability of observing state
s
from the path of play given equilibrium
v
. Given the transition matrix
Q
v
,
π
solves the equation
π
v
Q
v
−
π
v
= 0
.
(7)
Because
P
(
a
i
,s
;
v
i
)
>
0 for all actions
a
i
and all states
s
, every state
s
′
can be reached from
any state
s
. Thus, the Markov chain described by transition matrix
Q
v
is aperiodic, so a
unique invariant distribution exists. In this example,
π
v
takes on the following values:
π
v
= (0
.
41
,
0
.
44
,
0
.
14)
.
With the invariant distribution in hand, we can compute the probability that
i
initiates a
dispute along the path of play given equilibrium
v
:
f
i
(
v
) =
π
v
(1)
·
(1
−
P
(1
,
1;
v
i
))
.
(8)
In other words,
i
initiates a dispute when the current state is
s
and
i
plays a non-peaceful
action
a
i
>
1. In our numerical example,
f
i
(
v
) = 0
.
05.
4
Effects on Conflict Initiation.
We examine how
i
’s probability of initiating a conflict
changes as we make the country’s audience cost parameter more negative. To ease exposition,
we focus on country 1, while noting that the results are symmetric. Specifically, we compute
−
∂f
1
∂α
1
=
−
∂f
1
∂v
·
∂v
∂α
1
.
The gradient
∂v
∂α
1
can be computed using the Implicit Function Theorem and the equilibrium
constraint Φ(
v
)
−
v
= 0, where
∂v
∂α
1
=
−
∂
Φ
∂α
1
·
(
∂
Φ
∂v
−
I
18
)
−
1
and
I
k
is the
k
×
k
identity matrix. Finally, the gradient
∂f
1
∂v
can be computed using the
product rule, where
∂P
(1
,
1;
v
i
)
∂v
has a closed form solution and
∂π
v
(1)
∂v
can also be computed using
the Implicit Function Theorem and the equation
πQ
v
−
π
= 0 although we use numerical
derivatives here in our analysis. In the example under consideration,
−
∂f
1
∂α
1
= 0
.
022. Using
linear interpolation, increasing the magnitude of country 1’s audience costs by 25% changes
1’s probability of initiating from 0
.
05 to 0
.
14. The direction of this effect generally matches
the equilibria estimated from the data and reported in Table 2.
Likewise, we can consider how country 2’s audience costs affect country 1’s initiation
probability, that is,
−
∂f
1
∂α
2
=
−
∂f
1
∂v
·
∂v
∂α
2
.
where we compute the component gradients as described above. In our equilibrium of interest,
−
∂f
1
∂α
2
=
−
0
.
023. Again, using linear interpolation, decreasing
α
2
to
−
16 from
−
15 suggests
that
f
1
(
v
) drops to 0
.
03 from 0
.
05, and this effect matches the equilibrium effects found in
the data (see Table 2).
Competing Effects on Peace.
On the one hand, larger audience costs (more negative) for
country
i
encourages the country to initiate conflict in the peace state, i.e.,
−
∂f
1
∂α
1
>
0. On the
other hand, if country
i
has enhanced audience costs, it’s rival is less like to initiate conflict.
What are the total effects of larger audience costs on peace? We compute
∂π
v
∂α
i
, and find
that
−
∂π
v
∂α
i
<
0, so increasing country
i
’s audience costs discourage peace in this equilibrium.
The effect is relatively small, however, where
−
∂π
v
∂α
i
=
−
0
.
009. Notice that audience costs
discouraging peace is an effect we do not generally see in the equilibria estimated in the data,
5
i.e., those effects reported in Table 2.
Standing Firm in Crises.
We examine as to whether larger audience costs increase or
decrease a country’s probability of standing firm in a crisis. Conditional on the path of play
beginning in state
s
= 2,
i
’s probability of maintaining or escalating the dispute is
g
i
(
v
i
) = 1
−
P
(1
,
2;
v
i
)
.
In our example equilibrium,
i
’s probability of maintaining or escalating the crisis is
g
i
(
v
i
) =
0
.
98. As above, we can compute the effects of larger (more negative) audience costs on this
probability for country 1. In our example,
−
∂g
i
(
v
i
)
∂α
1
= 0
.
012, demonstrating that larger audience
costs for country 1 encourage it to stand firm in crises. And similar results hold when looking
at the war state
s
= 3. When increasing country 2’s audience costs, country 1’s conditional
probability of maintaining or escalating the crisis also increases, where
−
∂g
i
(
v
i
)
∂α
1
= 0
.
004, but
the effect is substantially smaller.
Probability of Receiving Audience Costs.
In an equilibrium
v
, country
i
’s long-term
probability of backing down and receiving an audience cost can be computed as
h
i
(
v
) =
π
v
(2)
·
P
(1
,
2;
v
i
)
·
(1
−
P
(1
,
2;
v
j
)) +
π
v
(3)
·
(1
−
P
(3
,
3;
v
i
))
·
P
(3
,
3;
v
j
)
.
(9)
In Equation 9,
π
v
(2) denotes the probability that the equilibrium path of play is in the crisis
state, and
P
(1
,
2;
v
i
)
·
(1
−
P
(1
,
2;
v
j
)) is the probability that country
i
receives an audience
cost in the same state. Likewise,
π
v
(3) denotes the probability that the path of play is in the
war state, and (1
−
P
(3
,
3;
v
i
))
·
P
(3
,
3;
v
j
) is the probability that country
i
receives an audience
cost war. In our example equilibrium,
h
i
(
v
) = 0
.
01. In addition, larger audience costs (more
negative) for country
i
decreases this probability even further, that is,
−
∂h
i
(
v
)
∂α
i
<
0, which
matches the effects from the estimated equilibria.
C Implementation
In this Appendix, we detail our implementation of the CMLE. Our data contains 125 countries
in 179 games, and solving the constrained optimization in Equation 4 requires estimating more
than 3
,
347 parameters, where 3
,
222 are expected utility constraints. The high-dimensionality
raises some questions about feasibility. To perform the optimization we use the program
IPOPT (Interior Point OPTimizer), which is an open-source, industrial optimizer used to
6
solve problems with potentially hundreds of thousands of variables (W ̈achter and Biegler
2006). IPOPT is particularly well suited to the large problem here. In time trials, IPOPT
had better convergence and performance properties than other optimizers such as KNITRO
and a version of the Augmented Lagrangian Method. Throughout, we set our convergence
criterion to IPOPT’s default at 1
e
−
6
.
A general drawback of interior-point methods is that they require accurate representations
of the Hessian of the Lagrangian for the problem in Equation 4. In our experiments, numerical
approximations using finite differences substantially inflate the estimator’s variance. To work
around this, we compute the Hessian and all other derivatives using the program ADOL-C
which implements an algorithmic differentiation (AD) routine (Griewank, Juedes and Utke
1996). In our set-up, we supply only the log-likelihood and constraint function, and the
AD program produce the derivatives by repeatedly applying the chain rule to the supplied
functions. We implement the estimator using Python 2.7 on Xubuntu 14.04 using the pyipopt
software developed by Xu (2014) to call IPOPT within Python and the pyadolc package
developed by Walter (2014) to use the AD routines discussed above. Asymptotic standard
errors are estimated using Silvey (1959, Lemma 6, p. 401).
For comparison, we also simulate standard errors using a parametric bootstrap from Davi-
son and Hinkley (1997). Overall, the bootstrapped standard errors closely match the analytical
ones; however, countries involved in only one dyad with few (one or two) non-peaceful states
have, on average, larger bootstrapped standard errors, associated with their audience cost
parameter, than analytical ones. These countries, e.g., Ghana, comprise only 10% of those in
our sample.
D Monte Carlo Experiments
In this Appendix, we describe a Monte Carlo experiment in which we use simulated data to
evaluate the performance of the method and our implementation as a function of the number
of dyads and time periods. The results of this experiment are important for two reasons. They
demonstrate that firstly, the parameters of interest are identified, and secondly, the estimation
procedure accurately recovers the model’s parameters for numbers of dyads and time periods
that are similar to our dataset used in the study.
In this experiment,
x
s
ij
= (1
,x
1
ij
) for all
s
and
z
i
= (
z
1
i
), where
x
1
ij
and
z
1
i
are random
variables. In addition, we vary the number of countries
N
to be values in
{
10
,
20
,
30
}
and
T
to be values in
{
20
,
80
,
150
,
250
}
. We consider every possible combination of countries, that
7
Table 6:
Coefficients used in the first Monte Carlo experiment analyzing the performance of
the CMLE as a function of
N
and
T
.
Coefficient
β
(2)
β
(3)
κ
(2)
κ
(3)
γ
(1)
γ
(2)
γ
(3)
α
i
Value
(
−
1
,
1) (
−
2
,
2)
−
0
.
5
−
1
−
0
.
5
0
0
.
5
−
2 +
2(
i
−
1)
N
−
1
is,
D
=
{{
i,j
} |
i,j
∈ {
1
,...,N
}
,i
6
=
j
}
and
D
=
(
N
2
)
. These values capture those in the
real-world application below in which
T
= 180 and
D
≈
(
20
2
)
.
The experiment is conducted as follows. We fix the coefficients used throughout to those
in Table 6. For each fixed value of
N
and
T
, we first generate control variables
x
1
ij
∼
N
(0
,
1)
and
z
1
i
∼
U
(0
,
1). Then for each unordered dyad
k
, we compute an equilibrium
v
k
by solving
the system of equations generated from Eq. 6. Next, we generate
T
periods of data using
the computed equilibrium, the associated conditional choice probabilities in Eq. 5, and the
transition
s
kt
+1
= max
{
a
kt
i
k
,a
kt
j
k
}
. The initial state
s
k
1
is drawn from
{
1
,
2
,
3
}
with equal
probability. After a suitable burn-in period, we combine the generated data and estimate the
parameters of the model by solving the constrained optimization problem in Eq. 4 using the
tools described in the previous section. The procedure is repeated 50 times for each value of
N
and
T
.
8
Table 7:
Summary of Monte Carlo Experiment
N T
ˆ
β
(2)
1
ˆ
β
(2)
2
ˆ
β
(3)
1
ˆ
β
(3)
2
ˆ
κ
(2)
1
ˆ
κ
(3)
1
ˆ
γ
(1)
ˆ
γ
(2)
ˆ
γ
(3)
10
20
-1.10
(0.24)
0.98
(0.15)
-2.02
(0.28)
2.02
(0.16)
-0.43
(0.20)
-1.02
(0.28)
-0.62
(0.46)
0.07
(0.32)
0.45
(0.19)
10
80
-1.05
(0.11)
1.00
(0.08)
-2.05
(0.11)
2.01
(0.08)
-0.45
(0.09)
-0.95
(0.12)
-0.45
(0.22)
0.02
(0.16)
0.48
(0.08)
10 150
-1.02
(0.10)
1.00
(0.06)
-2.00
(0.08)
2.02
(0.06)
-0.50
(0.06)
-1.01
(0.10)
-0.50
(0.19)
0.03
(0.11)
0.50
(0.07)
10 250
-1.01
(0.06)
0.99
(0.04)
-1.99
(0.08)
2.00
(0.04)
-0.50
(0.04)
-1.01
(0.07)
-0.57
(0.16)
0.03
(0.11)
0.49
(0.05)
20
20
-1.03
(0.13)
1.02
(0.09)
-1.98
(0.13)
2.01
(0.07)
-0.48
(0.11)
-1.02
(0.14)
-0.49
(0.29)
-0.01
(0.19)
0.49
(0.10)
20
80
-1.01
(0.06)
0.99
(0.03)
-2.00
(0.06)
1.99
(0.04)
-0.48
(0.04)
-0.99
(0.08)
-0.50
(0.09)
-0.02
(0.06)
0.49
(0.05)
20 150
-1.00
(0.04)
1.01
(0.02)
-2.00
(0.04)
2.01
(0.03)
-0.50
(0.03)
-1.00
(0.03)
-0.50
(0.08)
0.00
(0.04)
0.50
(0.03)
20 250
-1.00
(0.03)
1.00
(0.02)
-2.00
(0.03)
2.00
(0.03)
-0.50
(0.03)
-1.00
(0.04)
-0.50
(0.04)
0.00
(0.04)
0.50
(0.03)
30
20
-1.01
(0.08)
0.99
(0.05)
-2.01
(0.12)
1.99
(0.05)
-0.49
(0.08)
-0.97
(0.13)
-0.57
(0.12)
0.02
(0.10)
0.49
(0.07)
30
80
-1.00
(0.04)
1.00
(0.02)
-2.00
(0.04)
2.00
(0.02)
-0.50
(0.03)
-1.00
(0.04)
-0.50
(0.07)
0.01
(0.05)
0.50
(0.03)
30 150
-1.00
(0.02)
1.00
(0.02)
-1.99
(0.02)
2.00
(0.02)
-0.50
(0.02)
-1.01
(0.03)
-0.51
(0.05)
0.00
(0.05)
0.50
(0.02)
30 250
-1.00
(0.02)
1.00
(0.02)
-2.00
(0.03)
2.00
(0.01)
-0.50
(0.02)
-1.00
(0.03)
-0.51
(0.03)
0.00
(0.02)
0.50
(0.01)
Table 7 reports the means and standard errors in parentheses for the parameters
β
(
s
),
κ
(
a
i
), and
γ
(
s
). Due to space concerns, we do not report the audience cost parameters. In
addition, Figures 5 and 6 summarize the results. In these figures, we graph the CMLE’s bias
and variance, respectively, averaged over the four different sets of coefficients. More specif-
ically, to produce the upper-left graph of Figure 5, we first compute the expected bias of
ˆ
β
(
s
) for each
s
= 1
,
2, and then we averaged these values for each specification of
N
and
T
.
The upper-left graph of Figure 6 averages the variance of
ˆ
β
(
s
). The remaining graphs are
produced in a similar manner. Most importantly, the bias and the variance of the constrained
ML estimator decreases as we increase
N
and
T
. This monotonic relationship is especially
9
0.00
0.01
0.02
50
100
150
200
250
Time Periods
Absolute Average Bias
D
45
190
435
Average over:
β
0.00
0.01
0.02
0.03
0.04
0.05
50
100
150
200
250
Time Periods
Absolute Average Bias
D
45
190
435
Average over:
κ
0.00
0.02
0.04
0.06
50
100
150
200
250
Time Periods
Absolute Average Bias
D
45
190
435
Average over:
γ
0.00
0.03
0.06
0.09
0.12
50
100
150
200
250
Time Periods
Absolute Average Bias
D
45
190
435
Average over:
α
Figure 5:
The average bias of the constrained ML estimator by four different types of
coefficients as functions of the number of countries
N
and time periods
T
. In the analysis,
D
=
(
N
2
)
. Note that the average bias over
β
is
1
4
∑
3
s
=2
|
β
(
s
)
−
E[
ˆ
β
(
s
)]
|
1
.
10
0.00
0.01
0.02
0.03
0.04
50
100
150
200
250
Time Periods
Variance
D
45
190
435
Averaged over:
β
0.00
0.02
0.04
0.06
50
100
150
200
250
Time Periods
Variance
D
45
190
435
Averaged over:
κ
0.00
0.05
0.10
50
100
150
200
250
Time Periods
Variance
D
45
190
435
Averaged over:
γ
0.00
0.05
0.10
0.15
0.20
0.25
50
100
150
200
250
Time Periods
Variance
D
45
190
435
Averaged over:
α
Figure 6:
The average variance of the constrained ML estimator by four different types of
coefficients. In the analysis,
D
=
(
N
2
)
.
pronounced with the estimator’s variance. Even though increasing
N
means estimating an
additional audience cost parameter and more equilibrium constraints, the additional informa-
tion still attenuates the estimator’s bias and variance. With a very small number of countries,
i.e.
N
= 10, increasing the number of time periods in the observation may increase the esti-
mator’s bias, especially concerning the action-specific cost parameters,
κ
(
a
i
). However, with
a larger number of countries or dyads, this non-monotonicity disappears.
Finally, the experiment provides some quality control on our specific implementation.
When
T
= 20, the convergence rate of the procedure is approximately 50%. This is the same
across values of
N
and
D
. In contrast, when
T >
20, the convergence rate is 100%, and this is
consistent across values of
N
and
D
. In addition, Figure 7 graphs the time until convergence.
There is exponential growth in computational time as we increase
N
or the number of ordered
dyads. (Recall that adding an unordered dyad means estimating an additional 18 auxiliary
11
2.0
2.5
3.0
3.5
4.0
50
100
150
200
250
Time Periods
log
10
CPU Time (seconds)
D
45
190
435
Average Time to Estimate
Figure 7:
CPU time of the CMLE.
parameters.) Nonetheless, even with 435 dyads the average estimation time in Monte Carlos
is approximately three hours.
E Time-Invariant Covariates
Our model and subsequent estimation procedure do not allow for time-varying covariates.
More precisely, we have constructed utility functions that are dependent on the state of conflict
and actions taken, and we do not incorporate observed variables into the state space. Readers
may be concerned that the independent variables included in the model exhibit considerable
or even moderate fluctuations over time or that even smaller changes are correlated with
observed actions and states. Both of these concerns are unwarranted given our data, however.
Namely, we observe very few and very minimal changes in our independent variables, and the
changes that do exist are not correlated with the actions chosen and states of conflict.
First, we first examine whether our independent variables change over time. Our indepen-
dent variables come in two types, and the first type varies by country. These variables include
polity2, military personnel per capita, GDP per capita, CINC score and population. For each
variable, we compute its means in each country between 1993 and 2007, and then compute
country-year deviations from these mean values. The second type varies by dyad, and these
variables include trade dependence and whether the dyad has an alliance. We repeat the
same process for these variables except we use direct-dyads as observations. Figure 8 displays
12
Figure 8:
Country-Year Deviations from Mean Values of Independent Variables
0.00
0.25
0.50
0.75
1.00
−10
−5
0
5
10
Deviation from Mean Polity2
Density
0
100
200
300
400
500
−0.02
0.00
0.02
0.04
Deviation from Mean CINC score
Density
0
100
200
300
400
−0.02
−0.01
0.00
0.01
0.02
Deviation from Mean Mil. Per. (pc)
Density
0.0
0.1
0.2
0.3
−60
−30
0
30
Deviation from Mean GDP (pc)
Density
0
1
2
3
4
5
−0.4
0.0
0.4
Deviation from Mean log Pop.
Density
0
25
50
75
100
−0.1
0.0
0.1
0.2
Deviation from Mean Trade Dep.
Density
0
2
4
6
8
−1.0
−0.5
0.0
0.5
1.0
Deviation from Mean Alliance
Density
histograms of these deviations for each variable and illustrates that observed deviations from
the mean are relatively small across our dataset.
Second, we then attempt to explain our independent variables using observed states and
actions in a panel data analysis. Specifically, we regress the country-specific variables on the
number of conflict states (
s
t
>
1) in which a country is involved in a given year, the number
of hostile actions a country takes in a given year (
a
t
i
>
1), and the number of hostile actions
other countries take against it (
a
j
>
1). We also include a lagged dependent variable, lags of
these observed actions and states, and country and year fixed effects. We repeated the same
process with trade-dependence and alliance presence for the directed dyads in our data except
we include dyad and year fixed effects.
Models 1-4 in Table 8 display regression results with country-year observations, and Models
13
5-6 displays a similar regressions with directed dyad-year observations.
1
The main takeaway
should be and that observed actions and states have very little if any influence on our key
independent variables. Furthermore, the coefficients on the lagged values are close to 1, which
is to be expected if these variables do not change. While one out of the 30 coefficients of
interest are significant at the
p < .
1 level, this is not robust to various model and standard-
error specifications.
1
The number of observations vary across models due to missing data in the dependent variable. This is
not a problem in the main analysis because we average across years 1993-2007.
14
Table 8:
Predictors of Independent Variables
Polity2
CINC
Mil. Per. (pc) GDP (pc)
Log Pop.
Trade Dep.
Ally
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Model 7
Dep. Var., lag
0
.
71
∗∗∗
0
.
97
∗∗∗
0
.
74
∗∗∗
0
.
91
∗∗∗
0
.
90
∗∗∗
0
.
92
∗∗∗
0
.
80
∗∗∗
(0
.
03)
(0
.
05)
(0
.
04)
(0
.
02)
(0
.
02)
(0
.
08)
(0
.
02)
Confl. states
0
.
09
0
.
00
0
.
00
−
0
.
04
0
.
00
0
.
00
−
0
.
01
(0
.
08)
(0
.
00)
(0
.
00)
(0
.
04)
(0
.
00)
(0
.
00)
(0
.
01)
Confl. states, lag
−
0
.
01
0
.
00
0
.
00
−
0
.
04
0
.
00
0
.
00
0
.
00
(0
.
07)
(0
.
00)
(0
.
00)
(0
.
03)
(0
.
00)
(0
.
00)
(0
.
00)
Conf. Acts against
0
.
00
0
.
00
0
.
00
0
.
03
†
0
.
00
†
0
.
00
0
.
01
(0
.
02)
(0
.
00)
(0
.
00)
(0
.
01)
(0
.
00)
(0
.
00)
(0
.
00)
Conf. Acts against, lag
−
0
.
01
0
.
00
0
.
00
0
.
02
0
.
00
0
.
00
0
.
00
(0
.
03)
(0
.
00)
(0
.
00)
(0
.
01)
(0
.
00)
(0
.
00)
(0
.
00)
Conf. Acts taken
−
0
.
04
0
.
00
0
.
00
0
.
00
0
.
00
0
.
00
0
.
01
(0
.
03)
(0
.
00)
(0
.
00)
(0
.
01)
(0
.
00)
(0
.
00)
(0
.
00)
Conf. Acts taken, lag
−
0
.
02
0
.
00
0
.
00
0
.
00
0
.
00
0
.
00
0
.
00
(0
.
02)
(0
.
00)
(0
.
00)
(0
.
01)
(0
.
00)
(0
.
00)
(0
.
00)
N
1736
1750
1750
1730
1750
5004
5012
Notes:
∗∗∗
p <
0
.
001;
∗∗
p <
0
.
01;
∗
p <
0
.
05;
†
p <
0
.
1
Clustered Standard Errors in Parenthesis
15
Summary statistics of the variables used in the analysis are found in Table 9.
Table 9:
Summary of main variables
Min Mean Median St. Dev.
Max
Directed Dyadic Variables
X
ij
min(Polity 2) -10.00
-1.03
-1.38
5.30 10.00
log(Cap. Ratio)
-7.49
0.00
0.00
2.39
7.49
sqrt Trade Depend.
0.00
0.07
0.04
0.07
0.46
Alliance
0.00
0.31
0.00
0.46
1.00
log(distance)
1.61
6.65
6.53
1.19
9.23
Country Specific Variables
Z
i
log(GDP pc +1)
0.14
2.00
1.71
1.21
4.74
log(Mil. Per. pc +1)
0.00
0.01
0.00
0.01
0.05
log(Pop.)
6.11
9.45
9.29
1.47 14.05
F Audience Cost Parameters
This section contains the point estimates on the 125 audience cost parameters we estimated.
Table 10:
Audience Cost Estimates
Country
Audience Cost St. Err.
p
Nepal
-0.87
7.61
0.91
Bangladesh
-1.86
1.12
0.10
Nigeria
-2.18
2.15
0.31
Cambodia
-2.32
6.53
0.72
Lebanon
-2.43
0.77
<
0
.
01
Benin
-2.46
8.80
0.78
Azerbaijan
-2.64
0.93
<
0
.
01
Armenia
-2.90
0.93
<
0
.
01
Pakistan
-3.61
0.90
<
0
.
01
India
-3.80
0.89
<
0
.
01
Cameroon
-4.03
2.36
0.09
Djibouti
-4.52
10.85
0.68
Saudi Arabia
-4.75
2.01
0.02
16