Download Strategic Misspeci cation in Discrete Choice Models Curtis S. Signorino Kuzey Yilmaz

Strategic Misspecication in Discrete Choice Models
Curtis S. Signorino
Department of Political Science
University of Rochester
Kuzey Yilmaz
Department of Economics
University of Rochester
Work in Progress
Comments Welcome
July 17, 2000
Abstract
The most common specication of binary choice models | where the latent variable is a
linear function of the parameters and regressors | is structurally inconsistent with strategic
interaction. We characterize the misspecication induced by these models when used to analyze
data generated by \the simplest strategic model possible" | one that is only partially strategic
and where all action probabilities are monotonically related to each of the regressors. Even
under these ideal conditions, the use of logit and probit is problematic: when actions are the
dependent variable, distributional misspecication and biased and inconsistent estimates result;
when outcomes are the dependent variable, the misspecication is equivalent to omitted variable
bias. In both cases, the misspecication arises due to nonlinear terms that are implicit in the
strategic model, but are not included in typical binary choice specications. Researchers are
recommended to avoid standard logit and probit models if the data generation process is believed
to involve strategic interaction.
Paper previously presented at the 1999 Annual Meeting of the American Political Science Association and at
the 2000 Annual Meeting of the Midwest Political Science Association. The authors would like to thank Paul
Huth, Bradford Jones, Renee Smith, Michael Ward, and David Weimer for helpful comments. Support from the National Science Foundation (Grant # SES-9817947) is gratefully acknowledged. Email: [email protected],
[email protected]
Contents
1
2
3
4
Introduction
The Basic Problem
A Very Simple Strategic Model
Misspecication in the Analysis of Strategic Actions
3
4
5
7
5 Misspecication in the Analysis of Strategic Outcomes
15
6 Concluding Remarks
A Proofs
20
21
4.1 Monotonic Action Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Misspecication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Monte Carlo Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1 Not Necessarily Monotonic Outcome Probabilities . . . . . . . . . . . . . . . . . . . 15
5.2 Strategic Misspecication as Omitted Variable Bias . . . . . . . . . . . . . . . . . . . 16
2
1 Introduction
Multinomial and binomial logit and probit models have become commonplace in the social sciences.
However, recent work by Signorino (1999, 2000) and Smith (1999) suggests that failure to account
for strategic interaction in one's statistical model can result in invalid inferences. Signorino (1999)
demonstrates this with a monte carlo example in which the inferences from logit regressions are far
from (at times completely opposite to) the data generation process.
Although it seems evident that this form of structural misspecication aects the validity of
one's conclusions, Signorino (1999) is not a complete analysis of the misspecication, but more a
warning that the misspecication exists. As of yet, the form of the misspecication has not been
characterized in a way that most practitioners readily understand, or in a manner that allows us
to state when the eects of strategic misspecication should be mild versus severe. In this paper,
we do just that. We examine the misspecication when typical binary choice models | e.g., ones
where the latent variable is a linear function of the regressors | is used to analyze data consisting
of either actions in a game or the resulting outcomes of the game.
This paper is also partly an assessment of the conjecture that nonstrategic statistical models
can be used to conduct partial tests of strategic models. Testing an entire model all at once may be
extremely dicult in terms of the mathematics involved or in terms of the computational burden.
It may also simply be impossible due to lack of data for some components of the model. One of
the more sophisticated methods of (partially) testing models is to derive the comparative statics
of a formal model and then test those, rather than the entire model. The rationale for using logit
(or probit) to test formal and nonformal models is that the relationship between the dependent
variable and the regressors is derived as (or at least assumed to be) monotonic. On the surface,
the use of logit in this situation sounds eminently reasonable. If our theory tells us that, holding
all other variables constant, an increase in one explanatory variable always increases (or always
decreases) the probability of some event, then logit would seem to be an appropriate statistical
tool. We show that this is not the case.
In the interest of keeping the analysis as simple as possible and of creating ideal conditions for
the use of typical binary choice models, we have constructed the \simplest strategic model possible"
and have ensured that the relationship between each of the action probabilities and the regressors is
monotonic. Even under these conditions, the use of logit is problematic. For the actions data, logit
and probit induce a distributional misspecication.1 Even when that is negligible, the estimates of
the eects of regressors | especially for the conditioning variables | are likely to be biased and
By \actions" we refer to the choices of players at their decision nodes. The \outcomes" are the terminal nodes
of the game.
1
3
inconsistent.
Finally, although action probabilities have been constructed to be monotonic in the regressors,
outcome probabilities are not guaranteed to be monotonic. For the outcomes data, the strategic
misspecication induced by logit is equivalent to omitted variable bias. In both cases (actions
and outcomes), the misspecication arises due to nonlinear terms that are implicit in the strategic
model, but are not included in typical binomial regressions linear in the X s. Researchers are
recommended to avoid standard logit and probit models if the data generation process is believed
to involve strategic interaction.
This paper proceeds as follows. In the next section we set up the basic misspecication problem.
We then construct a simple strategic model that is used throughout the paper. Following that, we
address the issue of analyzing strategic actions data and compare the strategic actions model to
typical binomial models. Following that, we turn to strategic outcomes data, relate the strategic
model to the logit model, and identify a number of other problems inherent to using logit to analyze
the strategic outcomes data.
2 The Basic Problem
The basic problem we address here is really that of structural misspecication in binary choice
models.2 In these models, we assume there exists an unobservable (or latent) variable y dened
by
y = g(X; ) + (1)
where g(X; ) is a function of regressors X and parameters , and is a random disturbance from
some density f () with mean zero. Our data, what we actually observe, is whether y is above a
threshold:
0
y = 10 ifif yy >
(2)
0
Based on the specication of g(X; ) and , the exact probability model could be derived for the
regression. By far the most common practice is to assume that g(X; ) = X (i.e., that it is
linear) and that is distributed logistic (for logit) or Normal (for probit). Hereafter when we refer
to \typical" or \traditional" binary choice models, we will refer to models where gL (X; ) = X ,
where the L subscript will be used to denote that it is a linear latent variable specication.
Equation 1 is a structural equation because, once specied, g(X; ) is function that imposes
a relationship on the regressors, parameters, and the latent dependent variable y . It is not an
The results in this paper actually apply to any binomial or multinomial regression model, whether choice based
or not. We simply motivate the analysis with choice-based models.
2
4
empirical relationship that we nd through data analysis, but a theoretical relationship in the
context of which we interpret our estimation results3
This is particularly important because recent work by Signorino (1999, 2000) implies that when
the data generation process is the result of strategic interaction, g(X; ) is nonlinear. Moreover,
the structure of the \game" implies a particular \strategic" function gS (X; ). Given that many
(most?) of our theories assume strategic behavior and given the prevalence of logit and probit
in our data analysis, we would like to know whether assuming a linear latent variable function is
problematic. Specically, we ask the question: What are the eects (on our inferences) of assuming
gL (X; ) when the true model is gS (X; )?
3 A Very Simple Strategic Model
To analyze statistical misspecication, we usually start with the simplest possible model and examine how the misspecication aects the results in that situation. More extensive or general results
can be derived later if time, space, and mathematical tractability allow.
Consider the simple strategic situation depicted in Figure 1(a), which is typical of deterrence
situations, whether in international politics, congress, or market entry. Here, fa1 ; a2 ; a3 ; a4 g are
actions in the game and fY1 ; Y3 ; Y4 g are outcomes (denoted by the numbers in the terminal nodes).
Player 1 must choose between actions a1 and a2 . If she chooses a1 , then the game ends with Y1
as the outcome. If a2 is chosen, then player 2 must choose between actions a3 and a4 , leading
to outcomes Y3 and Y4 , respectively. For each outcome, the observable component of the player's
utility is denoted by Uik , where i indexes the player and k indexes the outcome.4 The game in
Figure 1 is only partially strategic | only player 1 must condition her behavior on player 2's
expected behavior. This is reected in the fact that there is no need for a U21 .
Throughout this paper, we will assume the source of uncertainty | i.e., what makes the strategic
model probabilistic | is based on agent error (see McKelvey & Palfrey 1998, Signorino 2000).
This assumption is implemented simply for mathematical convenience. The reader who does not
nd the behavioral assumptions underlying the agent error model particularly palatable can refer
to Signorino (2000) for two other types of statistical (but strictly Nash) strategic models: one
based on incomplete information concerning outcome payos and another based on variation in the
regressors that is unobserved only by the analyst. The issues addressed here are not limited to the
agent error variant. Indeed, because the primary question addresses the extent to which strategic
misspecication aects inferences in binary choice models, the general conclusions of our analysis
Dubin and Rivers (1989) make the same statement concerning linear regression.
Throughout this paper, all utilities, probabilities, and variables have an implicit observation index. For notational
convenience, we will tend not to include it.
3
4
5
1
1
0
0
a1
a2
p1
a1
p2
a2
p1
p2
2
2
2
1
U11
a3
p3
2
1
p4
3
U13
U23
(a) Strategic model
0
a4
4
U14
U24
a3
p3
p4
a4
3
4
X13 β13
X14 β14
0
X24 β24
(b) Utilities specied with regressors
Figure 1: A Very Simple Strategic Model. The simplest strategic model consists of one actor
conditioning her decision on that of another player. Here, player 1 must choose between actions
a
U1 (a2 ) = U1 (a2 ) + 12
= p3 U1 (a3 ) + p4 U1 (a4 ) + 12
= p3 U13 + p4 U14 + 12
(6)
Player 1's observable expected utilities depend on the probability that player 2 chooses actions a3
vs a4 . Let pj be probability that action aj is chosen and pY be the probability that outcome Yk is
realized. Assuming the ij are independent and identically distributed according to some density
f () with nite expectation, then the general form of the equilibrium probabilities is5
k
p1
p4
pY1
pY3
pY4
=
=
=
=
=
1 ; p2 = Pr[U1 (a1 ) > U1 (a2 )]
1 ; p3 = Pr[U2 (a4 ) > U2 (a3 )]
p1
p2 p3
p2 p4
With an appropriately specied f (), the above probabilities can be derived and used in data
analysis (e.g., MLE). For example, if the ij are i.i.d. N(0,1), then the action probabilities will be
Normally distributed. If they are i.i.d. Type 1 Extreme Value, then the resulting action probabilities
will be logistic.
Finally, researchers are often constrained to work with particular forms of dependent variables
at their disposal. The best case here would be to have the actual outcome data available for
analysis (i.e., whether Y1 , Y3 , or Y4 occurred in a given observation). However, data might only be
available for, say, whether player 1 chose a1 vs a2 . Using the appropriate equilibrium probabilities
in maximum likelihood estimation would allow for the analysis of either of these forms of data.
Since the analysis of each of these forms of data is commonplace in the social sciences, we should
understand the implications of strategic misspecication in both. In the following two sections we
examine exactly that. We turn rst to misspecication in the analysis of actions data.
4 Misspecication in the Analysis of Strategic Actions
Suppose our dependent variable was whether player 1 chose a1 versus a2 . For example, in studying
international deterrence, we might have data only on whether a potential attacker (here player
1) was deterred from attacking or not. We might want to examine the relationship of deterrence
success to substantive explanatory variables that we believe aect the incentives of states to engage
5
We do not address here how to derive the probabilities, but only refer the reader to the examples given in
Signorino 1999, 2000.
7
in conict, such as relative capabilities, alliance relationships, and levels of foreign trade. The
typical way to analyze this relationship would be to use logit or probit to regress the explanatory
variables on deterrence success vs failure. As we show in this section, the traditional methods for
conducting this analysis will induce misspecication error.
4.1 Monotonic Action Probabilities
Before we can proceed, we must rst specify the utilities of Figure 1(a) in terms of regressors.
Consider Figure 1(b). Here, the utilities | and, hence, all equilibrium probabilities | are a
function of only three regressors: X13 , X14 , and X24 . As given, the utilities are constructed (1)
to make them as simple as possible, and (2) to try to ensure monotonicity in the probabilities as
a function of the regressors.6 We leave the proof for the appendix and simply state the following
proposition concerning Figure 1(b):
Proposition 1. All action probabilities (p1 , p2, p3, p4) are monotonic in X13 , in X14 , and in X24 .
As such, this is the simplest strategic model possible. Moreover, given that p1 and p2 are monotonically related to each of the regressors, this is a best-case scenario for the argument that monotonic
comparative statics are sucient justication for the use of traditional logit and probit regressions.
4.2 Misspecication
To analyze the misspecication, we recast the model in an equivalent latent variable form. Suppose
our (observable) dependent variable is coded as y = 1 if a2 is chosen and y = 0 if a1 is chosen.
Recall that player 1 will choose a2 if U1 (a2 ) > U1 (a1 ). Let yS = U1 (a2 ) ; U1 (a1 ). Then we observe
y as
0
y = 10 ifif yyS >
(7)
0
S
where the S subscript denotes the strategic latent variable equation. Substituting Equations 5 and
6 into yS and then the utilities from Figure 1(b), the strategic latent variable regression becomes
yS = p3 13 X13 + p4 14 X14 + (8)
where = 12 ; 11 .
In contrast to Equation 8, the typical binary choice regression would be
yL = B0L + B13L X13 + B14L X14 + B24L X24 + (9)
6
The monotonicity claims made here are for \conditional" monotonicity | i.e., that f (x y) is monotonic in x
holding y constant, but that the direction of the monotonicity can change depending on the value of y. Unconditional
monotonicity | where the monotonicity of f (x y) has the same direction for all values of y | is not claimed here.
j
j
8
where the L subscript denotes the linear gL (X; ).
Obviously, the functional form of g(X; ) diers between the two regression models. However, it is
not clear how yL and yS relate to each other | e.g., how the estimators of ij and BijL relate to
each other or how the regressions dier in their predicted probabilities. The traditional yL is linear
in the explanatory variables and coecients. In contrast, X24 24 enters yS through the action
probabilities p3 and p4 . It appears that we have a functional form misspecication. However, it is
not clear how bad that is.
To assess the misspecication, we transform yS into a form that is directly comparable with
yL . Since the main problem in comparing the two models is the nonlinear p3 and p4 terms in yS ,
we use a rst-order Taylor series expansion of p3 and p4 about the mean of X24 . Greatly abusing
notation, we denote p3 and p4 evaluated at E (X24 ) by p3 and p4 , respectively.
The rst order Taylor approximations to p3 and p4 , taken about E (X24 ), are
p~3 = p3 ; p3 p4 [X24 ; E (X24 )] 24
p~4 = p4 + p3 p4 [X24 ; E (X24 )] 24
(10)
(11)
Substituting these into Equation 8 gives
yT = fp3 ; p3 p4 [X24 ; E (X24 )] 24 g 13 X13 +
fp4 + p3 p4 [X24 ; E (X24 )] 24 g 14 X14 + (12)
where the T subscript indicates that it is the strategic regression but with the Taylor series approximation of p3 and p4 .
We now take the Taylor regression and write the explanatory variables X13 , X14 , and X24 as
consisting of a mean and a deviation. Let Xij = mij + uij , where mij is the mean of explanatory
variable Xij and uij is a random disturbance with E [uij ] = 0.7 The Taylor regression can then be
rewritten as
yT = B0 + B13 X13 + B14 X14 + B24 X24 + (13)
where
B0
B13
B14
B24
=
=
=
=
=
;p3p4 (14 m14 ; 13 m13 )24 m24
p3 13
p4 14
p3 p4 (14 m14 ; 13 m13 ) 24
p3 p4 (14 u14 ; 13 u13) 24 u24 + (14)
(15)
(16)
(17)
(18)
7
We are not implying that the Xij are measured with error here. Rather, we are simply writing any observation
of a random variable as consisting of the mean of the random variable plus some deviation from that mean.
9
Notice that is the \new" error term in the above regression. Denote the variances of , uij , and
as 2 , ij2 , and 2 , respectively. Assuming the uij are independent of and of each other, then
E () = 0 and
;
2 + 2 2 2 2 + 2
2 = p24 p23 132 13
14 14 24 24
Cov(X13 ; ) = Cov(X14 ; ) = Cov(X24 ; ) = 0
We can no
(19)
when is Normally distributed and to
B0L = qB30
2
2 B13L = qB313
2
2 B14L = qB314
2
2 B24L = qB324
2
2 (21)
when is logistic.
The eect of strategic misspecication can be seen in the hypothesis tests and in the predicted
probabilities that result from the above estimation. Notice, for example, that whenever X13 and
X14 both have mean zero, the estimate of B24L will tend to be zero. In other words, simply because
our data are centered around zero, which is not uncommon, traditional logit and probit will lead
us to believe that X24 has no eect on whether player 1 chooses a2 . By construction, we know that
inference is false here.
Although it is dicult to assess the eect of strategic misspecication on other types of hypothesis tests simply with reference to the parameter estimates, we should expect the unconditional
monotonicity of typical binary choice models to result in invalid hypothesis tests concerning the
direction of the eect of regressors. For example, in typical logit and probit analyses it is common
to interpret the sign on parameter estimates (in combination with their signicance) as an hypothesis test concerning the direction of the eect of a substantive variable. However, the conditional
nature of strategic choice generally results in only a conditionally monotonic relationship between
regressors and the dependent variable | and often not even that. We will see shortly an example
of just this misspecication.
Another potential eect of the misspecication concerns the variance of . Equation 19 shows
that 2 is composed of two components: variance due to and variance due to the regressors that
results from the structural misspecication. The larger the latter, the greater the extent to which
's distribution diverges from 's and the greater the rescaling downwards of the estimated BijLs. In
examining omitted variable bias in probit, Griliches and Yatchew (1985) suggest that this rescaling
downward can aect hypothesis tests.
Finally, because the parameter estimates in any of these choice models (strategic or nonstrategic)
are dicult to interpret by themselves, analysts typically interpret the predicted probabilities for a
better understanding of the relationship between the dependent variable and the regressors | not
only for the direction of the relationship but the relative magnitude. We would, of course, like to
know the extent to which strategic misspecication aects the predicted probabilities. Although
the misspecication can be expressed in a fairly general form, its mathematical complexity does
not allow for a simple (and useful) analytical expression. However, we provide the reader with a
sense of the misspecication in the following section.
11
4.3 Monte Carlo Analysis
To illustrate the results of the previous section, we present the results of two monte carlo analyses. In the rst monte carlo analysis, each replication of the analysis involved generating N=2000
observations based on the behavioral assumptions of the strategic model in Figure 1. The explanatory variables were taken from a uniform distribution on [;2; 2] and the coecients were set to
13 = 14 = 24 = 1. was drawn from a logistic distribution with mean zero. The strategic and
logit regressions were run and their estimates saved. These steps were replicated 2000 times to
form densities of the parameter estimates.
Figure 2 displays the densities of the estimates from the regressions. The top row displays the
estimates from the strategic regressions. All three distributions are centered around one, indicating
that the strategic model was able to recover the correct estimates on average. Of note is that the
distribution for ^24 is wider than the distributions for ^13 and for ^14 . This is because, with the
actions data, we have direct information on what player 1 chose, but not on what player 2 chose.
All we can do is infer what player 2 chose, based on the structure of the model and player 1's
choice. That uncertainty is translated into the density of ^24 .
The second row in Figure 2 displays the densities for the logit estimates. What are our expectations concerning the logit regressions using this data? It turns out that fairly well approximates a
logistic distribution in this case, also with approximately the same variance as . Therefore, distributional misspecication should not be an issue. Because the explanatory variables are uniformly
distributed between ;2 and 2, m13 = m14 = m24 = 0. The estimators in this case should therefore
converge to B0L = 0, B13L = :48, B14L = :48, and B24L = 0. Indeed, we see from Figure 2 that
the logit densities are centered approximately over these values. Moreover, as we derived in the
previous section, on average logit suggests that X24 has no eect on player 1's decision to choose
a2 .
To understand what these estimates imply for the predicted probabilities, Figure 3 displays (a)
the logit predicted Pr[a2 ] as a function of X24 and X13 , (b) the strategic predicted Pr[a2 ] as a
function of X24 and X13 , and (c) the strategic predicted Pr[a2 ] as a function of X24 and X14 . In
all cases, the third variable not graphed is held constant at its mean. Figure 3(a) reects logit's
estimate that X24 has no eect on Pr[a2 ] | e.g., for any value of X13 , increasing X24 has no eect
on Pr[a2 ]. Figures 3(b) and (c) present a slightly dierent story. In the model that generated the
data, X24 clearly aects player 1's decision concerning a2 . Moreover, using logit to make inferences
concerning the direction of X24 's eect would be problematic here (even if the estimate were not
zero). As we noted previously, the unconditional monotonicity of logit means the direction of a
regressor's eect is determined solely by its estimate's sign. In contrast, Figure 3(b) shows that for
12
Figure 2: Densities of Parameter Estimates from Strategic and Logit Regressions on Actions Data.
Data was generated using the strategic model's behavioral assumptions and setting 13 = 14 =
24 = 1. As the plots indicate, the strategic regression (top row) recovered the parameters fairly
well. The bottom row shows the estimates from the logit regressions. The logit estimators in this
case should converge to B0L = 0, B13L = :48, B14L = :48, and B24L = 0, which appears to be the
center of the displayed densities. Note that, on average, we would infer from logit analysis that X24
has no eect on the dependent variable, which we know is false.
13
small X13 , increasing X24 increases the probability of a2 , while Figure 3(c) shows that for small X14 ,
increasing X24 has the opposite eect. Finally, we see in Figures 3(a) and (b) that the magnitude
of logit's error in predicting Pr(a2 ) is not negligible, but also not huge | at most 0.15.
The second set of monte carlo regressions employed exactly the same assumptions and parameterizations as the rst, with the exception that the range of the Xij s was increased: now uniformly
distributed between ;10 and 10. Figure 4 displays the same three plots as before, but with the
data from the new monte carlos. Figure 4(a) is very similar to its counterpart in Figure 3, although
over a larger range. As before, we would infer from this plot that X24 has no eect on player
1's decisions. The contrasting inferences displayed in Figure 3 are magnied here. Not only do
Figures 4(b) and (c) emphasize that X24 aects player 1's decision to select a2 (and in dierent
ways depending on the values of X13 and X14 ), but the magnitude of logit's error is even higher
here: o by more than .4 at times.
5 Misspecication in the Analysis of Strategic Outcomes
The previous section analyzed misspecication under the conditions generally considered ideal for
using logit | e.g., when the dependent variable is monotonically related to all regressors. In this
section, we still assume the \true" model is the simple strategic model in Figure 1. However, the
analyst is now assumed to have data representing which of the outcomes occurred: Y1 , Y3 , or Y4 . As
before, we are interested in characterizing the misspecication when a typical binary choice model
with g(X; ) = X is used to estimate the eects of the explanatory variables. In particular, we
focus on the use of logit in this section.
5.1 Not Necessarily Monotonic Outcome Probabilities
In Section 3, we specied the utilities of the simple strategic model so that all action probabilities
are monotonically related to the regressors. We saw there that using the traditional binary choice
model specication to test hypotheses was still problematic, especially concerning inferences about
X24 in particular and predicted probabilities in general. Using logit to analyze strategic outcomes
is even more problematic. Even when the action probabilities are all monotonic in the regressors,
the outcome probabilities may not be. The following proposition applies to the strategic model in
Figure 1(b) and its proof is given in the appendix:
Proposition 2. Outcome probability pY1 is monotonic in all variables. Outcome probabilities pY3
and pY4 are monotonic in X13 and in X14 , but not in X24 .
The fact that the typical logit functional form is unconditionally monotonic in the regressors should
15
Figure 5: Monotonic Action Probabilities May Yield Nonmonotonic Outcome Probabilities. The
plot shows the probability of Y4 as a function of X24 for the strategic model (solid line) and for a
logit model that is linear in the X (dashed line). In this case, the monotonic action probabilities of
the simple strategic model have produced nonmonotonic outcome probabilities. Logit that is linear
in the X cannot model this nonmonotonicity.
be a red ag that misspecication will exist if we use it to analyze data where the true relationship
between the outcomes and regressors is not guaranteed to be even conditionally monotonic.
Figure 5 gives an example of this. Here, 13 = 4, 14 = ;2, and 24 = 2. Data was generated
using these parameters, allowing the Xij to vary uniformly between ;5 and 5, and, of course, using
the behavioral assumptions of the strategic model. The logit and strategic regressions were run.
Then, using the estimated coecients, the X13 and X14 were held constant at one, while X24 was
varied from ;5 to 5. The solid line in Figure 5 displays the strategic probability pY4 and the dashed
line shows logit's estimated probability. Again, logit's S-curve varies monotonically from zero to
1. However, it cannot model the functional form of the strategic model, which is nonmonotonic in
this case.
5.2 Strategic Misspecication as Omitted Variable Bias
To assess the misspecication when logit is employed in this context, we again need the strategic
and logit regressions to be in comparable forms. In the previous section, the strategic latent variable
model was already framed as a binary choice model, which allowed us to compare it to the typical
latent variable model with a linear g(X; ). In contrast, we now have three outcomes to consider
in the strategic model.
16
It is not uncommon to see logit or probit analyses performed on data where the dependent
variable can be thought of as Y vs Not Y . Usually Y represents some particular phenomenon of
interest and Not Y is a category that includes everything else that could have occurred. In that
sense, the NotY category is an aggregation of all the other categories. As an example, suppose the
model in Figure 1 represented a deterrence situation, where player 1 was a nation deciding whether
to attack another nation (player 2). If attacked, the second nation would have the opportunity
to defend itself. The outcomes of this situation would be a Status Quo if the rst nation did
not attack, Capitulation by nation 2 if nation 1 attacked and nation 2 backed down, and War if
nation 1 attacked and nation 2 defended itself. Many (most?) analyses in international relations
have employed logit or probit to analyze data where the dependent variable denotes simply "War"
vs "Not War." In the context of the model just outlined, the "Not War" category includes both
the Status Quo outcome and the Capitulation outcome. Because of the prevalence of this type
of analysis, we analyze misspecication in situations where outcomes are aggregated and analyzed
using a binary model.
Referring to Figure 1, consider a dependent variable that is coded y = 1 if Y4 occurred and
y = 0 if Y1 or Y3 occurred. As we will see, this implies a dierent gS (X; ) than in the previous
section. The question is: What is the eect of using logit with gL (X; ) = X when the true
structural model is gS (X; )? A prerequisite for answering that question is that we nd a way of
transforming the strategic model in Figure 1 into an equivalent binary choice model based on the
above dependent variable.
Figure 6 shows the progression of models that does exactly this. Figure 6(a) can be considered
an \intermediate" multinomial model in that aggregations of its outcomes produce exactly the same
outcome probabilities as in the strategic model. For example, aggregating Y1a and Y1b forms the
multinomial model in Figure 6(b), which has the same number of outcomes and the same outcome
probabilities as the strategic model in Figure 1(b). Aggregating outcomes Y1 and Y3 in Figure 6(b)
produces the binomial model in Figure 6(c). Finally, Figure 6(d) is the same probability model as
Figure 6(c), but standardized in a way that will allow for easier comparison to the traditional latent
variable specication for logit. Notice that, although we have constructed a binomial logit model
in Figure 6(d), the implied gS (X; ) is consistent with the original strategic model and nonlinear.
We will use Figure 6(d) as the referent version of the strategic model for subsequent analysis.
Figure 7(a) displays the regression equation for the latent dependent variable yS implied by the
strategic referent model and with the utility specication of Figure 1(b). The regression equation
for yS is clearly not a typical logit regression as in Figure 7(c) | i.e., gS (X; ) is not the same
functional form as gL (X; B ). The action probabilities p3 and p4 are nonlinear functions of X24 and
24 . Similarly, the RS term is a nonlinear function of all of the regressors and coecients. If one
17
pY1a
pY4
pY1b
pY1
pY3
1a
1b
3
4
V1a
V1b
V3
V4
(
1
ln eV1a + eV1b
)
(a) “Intermediate” MNL Model
pY13
13
(
ln eV1a + eV1b + eV3
)
pY3
pY4
3
4
V3
V4
(b) Equivalent MNL Model
pY4
pY13
4
13
V4
0
(c) Equivalent Logit Model
pY4
4
(
V4 − ln eV1a + eV1b + eV3
)
(d) Referent Logit Model
V1a
V1b
V3
V4
=
=
=
=
U11 + U23
U11 + U24
p3 U13 + p4 U14 + U23
p3 U13 + p4 U14 + U24
Figure 6: Multinomial and Binomial Equivalents of the Strategic Model. Figure (a) is an intermediate multinomial model that is used in forming Figure (b). Figure (b) is a multinomial model that
produces outcome probabilities that are equivalent to the strategic model's. Figures (c) and (d) are
equivalent binomial models. (d) will be used as the referent model for comparison with the Taylor
and logit regressions. The Vk terms are based on the strategic model's utilities.
18
(a) Strategic
yS =
p3 13 X13 + p4 14 X14 + 24 X24 + RS + = ; ln(3) + 13 13 X13 + 13 14 X14 + 32 24 X24 + RT + = B0L + B13L X13 + B14L X14 + B24L X24
+ (b) Taylor
yT
(c) Logit
yL
h
RS = ; ln 1 + eX24 24 + ep3 X13 13 +p4 X14 14
i
1 X 2 2 ; 1 X 2 2 ; 1 X 2 2
RT = ; 36
13 13 36 14 14 9 24 24
; 181 X13 X14 13 14 ; 19 X13 X24 13 24 + 92 X14 X24 14 24
Figure 7: Latent Variable Models for the Dependent Variable: Y4 versus (Y1 or Y3 ). The gure
displays three specications of the latent dependent variable, y . Equation (a) is the strategic
model, which is consistent with the data generation process. RS is the nonlinear term that appears
in Figure 6(d). Equation (b) corresponds to the strategic model, but with the p3 , p4 , and RS
replaced by their second-order Taylor series approximations. RT collects all of the second-order
terms. Finally, Equation (c) is the logit model analysts would normally run to test their hypotheses
concerning X13 , X14 , and X24 .
believed the strategic model generated the data, then yS is the binary choice model one should
employ.8
As in the previous section, we deal with p3 , p4 , and RS by replacing them with Taylor series
approximations, here second-order and taken about zero. The resulting regression equation is
denoted by yT and shown in Figure 7(b). The cross-product terms and a constant are collected
into the term RT to help in comparing the models.
Comparing the logit and Taylor regressions, we see that both contain the same rst-order
eects, albeit multiplied by a constant for each variable. The biggest dierence is that the Taylor
regression contains the RT term, which includes the second-order eects. Moreover, in contrast to
the regression equations for actions data, here there are second-order eects for all combinations
of the regressors.
It is important to note that if the strategic model generated the data, then each of the secondorder terms in RT is a relevant variable. Hence, not only will standard logit regression fail to model
the nonlinear terms, but it will also induce omitted variable bias on the parameters of the included
variables. It is likely that the omitted second-order terms will be correlated with the included terms.
8
Alternatively, one could run a multinomial version of the strategic model.
19
However, as Yatchew and Griliches (1985) demonstrate for probit, correlation of the included and
excluded variables is not necessary for bias to result. It is, therefore, unlikely that any of the eects
will be correctly estimated by logit in the case of outcomes data | even for this simplest of models.
Given this misspecication and given that any nonmonotonicity in the strategic model cannot be
captured by logit's unconditional monotonicity, inferences based on the predicted probability of Y4
will almost certainly be invalid.
6 Concluding Remarks
To recap, we have attempted to characterize the strategic misspecication induced by using the
binary choice specication most commonly used in practice | one where the latent dependent
variable is a linear function of the X s. In the interest of keeping the analysis as simple as possible
and of creating ideal conditions for those interested in applying logit to comparative statics, we
have constructed the \simplest strategic model possible" and have ensured that the relationship
between all action probabilities and the regressors is monotonic. Even under these conditions,
the use of logit is problematic. For the actions data, logit and probit induce a distributional
misspecication. Assuming that is negligible, the estimates of the eects of regressors | especially
for the conditioning variables | are likely to be biased and inconsistent.
Even when the action probabilities are guaranteed to be monotonic in the regressors, outcome
probabilities are not guaranteed to be monotonic as well. For the outcomes data, the strategic
misspecication induced by logit is equivalent to omitted variable bias. In both cases (actions
and outcomes), the misspecication arises due to nonlinear terms that are implicit in the strategic
model, but are not included in typical binomial regressions linear in the X s. Researchers are
recommended to avoid standard logit and probit models if the data generation process is believed
to involve strategic interaction.
20
A Proofs
All proofs or derivations in this section pertain to Figure 1, with U11 = U23 = 0, U13 = X13 13 ,
U14 = X14 14 , and U24 = X24 24 .
Proof of Proposition 1.
;
p4 : The probability of action a4 is p4 = eX24 24 = eX24 24 + 1 . dp4 =dX13 = dp4 =dX14 = 0.
dp4 =dX24 = 24 p4 p3 . Since p3 and p4 are always nonnegative, sign (dp4=dX24 ) = sign (24 )
8X24 . Therefore, p4 is monotonic in X24 .
p3 : The probability of action a3 is p3 = 1= eX24 24 + 1 . dp3 =dX13 = dp3 =dX14 = 0.
dp3 =dX24 = ;24 p4 p3 . Since p3 and p4 are always nonnegative, sign (dp3 =dX24 ) = sign (;24 )
8X24 . Therefore, p3 is monotonic in X24 .
p2 : The probability of action a2 is p2 = ep3 X13 13 +p4 X14 14 =
;
; p X +p X e 3 13 13 4 14 14
+1 .
1. dp2 =dX13 = 13 p3 p2 p1 . Since the action probabilities are always nonnegative, sign (dp2 =dX13 ) =
sign (13 ) 8X13 . 13 is constant, so p2 is monotonic in X13 .
2. dp2 =dX14 = 14 p4 p2 p1 . sign (dp2 =dX14 ) = sign (14 ) 8X14 . Therefore, p2 is monotonic
in X14 .
3. dp2 =dX24 = 24 p4 p3 p2 p1 (X14 14 ; X13 13 ). The action probabilities are always nonnegative, so sign (dp2 =dX24 ) = sign [24 (X14 14 ; X13 13 )] 8X24 . We hold X14 and X13
constant, so (X14 14 ; X13 13 ) does not vary over X24 . Therefore, p2 is monotonic in
X24 .
p1 : The probability of action a1 is p1 = 1=
; p X +p X e 3 13 13 4 14 14
+1 .
1. dp1 =dX13 = ;13 p3 p2 p1 . Since the action probabilities are always nonnegative, sign (dp2 =dX13 ) =
sign (;13 ) 8X13 . 13 is constant, so p1 is monotonic in X13 .
2. dp1 =dX14 = ;14 p4 p2 p1 . sign (dp1 =dX14 ) = sign (;14 ) 8X14 . Therefore, p1 is monotonic in X14 .
3. dp1 =dX24 = ;24 p4 p3 p2 p1 (X14 14 ; X13 13 ). The action probabilities are always nonnegative, so sign (dp1 =dX24 ) = sign [;24 (X14 14 ; X13 13 )] 8X24 . (X14 14 ; X13 13 )
does not vary over X24 . Therefore, p1 is monotonic in X24 .
Proof of Proposition 2.
21
pY1 : Since pY1 = p1 , see the proof for Proposition 1.
pY3 : The probability of outcome Y3 is
p3 X13 13 +p4 X14 14
pY3 = p2p3 = ep3eX13 13 +p4X14 14 + 1 eX24 124 + 1
1. dpY3 =dX13 = 13 p23 p2 p1 .
sign (dpY3 =dX13 ) = sign (13 ). Therefore, pY3 is monotonic in X13 .
2. dpY3 =dX14 = 14 p4 p3 p2 p1 .
sign (dpY3 =dX14 ) = sign (14 ). Therefore, pY3 is monotonic in X14 .
3. dpY3 =dX24 = 24 p4 p3 p2 [p3 p1 (X14 14 ; X13 13 ) ; 1].
sign (dpY3 =dX24 ) = sign f24 [p3 p1 (X14 14 ; X13 13 ) ; 1]g. Since
p3p1 (X14 14 ; X13 13 ) ; 1 may change sign over X24 , pY3 is not monotonic in X24 .
pY4 : The probability of outcome Y4 is
p3 X13 13 +p4 X14 14
X24 24
pY4 = p2p4 = ep3eX13 13 +p4X14 14 + 1 eXe24 24 + 1
1. dpY4 =dX13 = 13 p4 p3 p2 p1 .
sign (dpY4 =dX13 ) = sign (13 ). Therefore, pY4 is monotonic in X13 .
2. dpY4 =dX14 = 14 p24 p2 p1 .
sign (dpY4 =dX14 ) = sign (14 ). Therefore, pY4 is monotonic in X14 .
3. dpY4 =dX24 = 24 p4 p3 p2 [p4 p1 (X14 14 ; X13 13 ) + 1].
sign (dpY4 =dX24 ) = sign f24 [p4 p1 (X14 14 ; X13 13 ) + 1]g. Since
p4p1 (X14 14 ; X13 13 ) + 1 may change sign over X24 , pY4 is not monotonic in X24 .
22
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