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Strategic Misspecification in Regression Models

Curtis S. Signorino University of RochesterKuzey Yilmaz Koc University

Common regression models are often structurally inconsistent with strategic interaction. We demonstrate that this “strategicmisspecification” is really an issue of structural (or functional form) misspecification. The misspecification can be equivalentlywritten as a form of omitted variable bias, where the omitted variables are nonlinear terms arising from the players’ expectedutility calculations and often from data aggregation. We characterize the extent of the specification error in terms of modelparameters and the data and show that typical regressions models can at times give exactly the opposite inferences versusthe true strategic data-generating process. Researchers are recommended to pay closer attention to their theoretical models,the implications of those models concerning their statistical models, and vice versa.

Much of political science research assumes indi-viduals and groups of individuals (e.g., in theform of states) behave strategically. Recent re-

search (Signorino 1999, 2000; Smith 1999) suggests that,when analyzing strategic behavior on the part of individ-uals or states, failure to reflect that strategic interactionin one’s statistical model can result in invalid inferences.Signorino (1999) demonstrates this with a Monte Carloexample in which the inferences from logit regressionsare far from (at times completely opposite to) the strate-gic data-generating process. Signorino (1999), however, isnot a complete analysis of the misspecification, but morea warning and demonstration that the misspecificationexists. As of yet, the form of the misspecification has notbeen characterized in a way that most practitioners readilyunderstand or in a manner that allows us to state when theeffects of strategic misspecification should be mild versussevere. The primary goal of this article is to do exactlythat.

As we demonstrate in this article, strategic misspecifi-cation is really an issue of structural (or functional form)misspecification. Because this type of misspecification isnot well known among political scientists, we begin inthe next section by illustrating the problem of functional

Curtis Signorino is Assistant Professor of Political Science, University of Rochester, 303 Harkness Hall, Rochester, NY 14627([email protected]). Kuzey Yilmaz is Assistant Professor of Economics, Koc University, Rumelifeneri Yolu, 34450 Saryyer, Is-tanbul, Turkey ([email protected])

Article previously presented at the 1999 Annual Meeting of the American Political Science Association, at the 2000 Annual Meetings ofthe Midwest Political Science Association and the Political Methodology Summer Conference. The authors would like to thank Paul Huth,Bradford Jones, Phil Schrodt, Renee Smith, Michael Ward, and David Weimer for helpful comments. Support from the National ScienceFoundation (Grants SES-9817947 and SES-0213771) and from the Peter D. Watson Center for Conflict and Cooperation is gratefullyacknowledged. All proofs, derivations, and graphs used in the analysis but not included in this article are available from the authors athttp://www.rochester.edu/College/PSC/Signorino.

form misspecification as it applies to the classical linearregression model. It is easy to show in this case that func-tional form misspecification is actually a type of omittedvariable bias, where the omitted variables are nonlinearterms in a Taylor series approximation of the true func-tional form.

We then move to the strategic setting and constructwhat we believe is the simplest model possible: a two-player deterrence game. We assume the data in this caserepresents whether a particular outcome (war) has oc-curred or not, and we analyze the misspecification ofusing logit or probit with the ubiquitous linear X B spec-ification of the latent variable equation. As in the OLScase, it is easy to rewrite the strategic misspecification asa form of omitted variable bias, where the omitted vari-ables are nonlinear terms in a Taylor series expansion ofthe true functional form and where the nonlinearity isdue to the players’ expected utility calculations. Becauseof the misspecification, parameter estimates are not onlybiased, but inconsistent. Therefore, throwing more dataat the problem does not make it go away.

In the third section, we analyze the misspecificationwhen the dependent variable is the action taken by thefirst player, the attacker. By construction, this choice is

American Journal of Political Science, Vol. 47, No. 3, July 2003, Pp. 551–566

C©2003 by the Midwest Political Science Association ISSN 0092-5853

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552 CURTIS S. SIGNORINO AND KUZEY YILMAZ

monotonically related to each independent variable. Thiswould seem to be an ideal situation for using logit orprobit. In this case, we not only demonstrate that mis-specification exists, but also mathematically character-ize the specification error in terms of the model pa-rameters and data, showing when it will be better orworse. Interestingly, there are a wide range of very rea-sonable conditions in which logit or probit will incor-rectly indicate that variables in the model have no effector even the opposite effect of their true values. A MonteCarlo example is provided to illustrate the analyticalresults.

Although we frame our analysis in terms of strategicinteraction and discrete-choice models, our argument isnot limited to either, but applies to all parametric esti-mation in the social sciences. In general, when the func-tional form of the statistical model is not consistent withthe data-generating process, misspecification results. Ouranalysis therefore emphasizes that more attention shouldbe paid to the functional relationship of the dependentand independent variables in one’s theory—and that thelack of a strongly-specified theory does not absolve one ofthis critical step. The theory, the hypotheses to be tested,and the statistical model all must reflect each other. If theyare not identical functional specifications, they should atleast be consistent with each other in terms of monotonic-ity conditions, which we discuss at length in the fourthsection.

The Basic Problem Illustratedwith the Classical Linear

Regression Model

The general problem of functional form misspecificationhas been well known for quite awhile within the econo-metrics community, at least in terms of the classical re-gression model and systems of linear equations. However,it has received relatively little attention within politicalscience. Because most political science scholars are nowfamiliar with the classical linear regression model, we il-lustrate the basic functional form problem in this sec-tion using that model and move to the strategic settingthereafter.1

To begin, let us assume that we are analyzing the re-lationship of a single regressor X to a dependent variableY , and that this relationship satisfies all the assumptionsof the classical linear model concerning Y, X , and error

1The illustration in this section is largely based on Kmenta (1986,449–50).

term �, with the exception that Y is a nonlinear functionof X and a parameter �:2

Y = f (X, �) + �. (1)

In analyzing this data, the typical political science scholarmight test for heteroskedasticity, autocorrelation, or anynumber of other deviations from the classical linearmodel. However, invariably she would assume that

Y = B0L + B1L X + �, (2)

i.e., that regression model is a linear function of param-eters B L , where we use the L subscript to denote the lin-ear model. More often than not, she would also assumethat the regression model is a first-order function of someset of substantive explanatory variables X.3 If f (X, �)is nonlinear, but the analyst instead employs X B L (lin-ear in parameters B L and first-order in X), then she hasclearly misspecified the functional form of f (X, �). Thequestion is: Does that matter?

To analyze the misspecification, let us first take theTaylor series expansion of f (X, �) about X . The result-ing Taylor series version of the model will approximate theoriginal nonlinear model, but can be written as linear inthe parameters, which then fits the classical linear frame-work. To simplify notation, we will denote the pth orderTaylor series expansion of f (X, �) about X as fT (X, �).The Taylor series expansion is

fT (X, �) = f (X, �) + (X − X) f ′(X, �) + (X − X)2

2!

× f ′′(X, �) + · · · + (X − X)p

p!f (p)(X, �) + �p+1,

where f (p)(X, �) is the p-th derivative of f (X, �) withrespect to X , evaluated at X . The remainder, �p+1, canbe thought of as the difference between f (X, �) andfT (X, �), or equivalently as the Taylor series terms fromp + 1 to infinity not incorporated in the p-th orderexpansion. Assuming that limp→∞ �p = 0, higher-order

2Throughout this article, single variables will be denoted by itali-cized, and usually uppercase, letters. Bolded letters will representmultiple variables or parameters. Additionally, all utilities, proba-bilities, and variables have an implicit observation index, which fornotational convenience, we will tend to omit.

3First-order terms are those including only a single variable X j

raised to the first power. For example, a typical first-order linearspecification of X B L would be B0 + B1 X1 + B2 X2 + · · · + Bk Xk .Higher-order terms include quadratic, cubic, and interaction terms.The classical linear model can accommodate higher-order terms asregressors, so long as the model remains linear in the parametersB L . Our general argument does not depend on assuming that X B L

contains only first-order X terms. However, it is by far the mostcommonly used, and it is mathematically convenient.

STRATEGIC MISSPECIFICATION IN REGRESSION MODELS 553

Taylor series expansions will better approximate the orig-inal function.

Assume that p is large enough that �p+1 is negligi-ble. If we replace the true functional form f (X, �) inthe regression model with the p-th order Taylor seriesapproximation of it fT (X, �), and rearrange terms, wecan write the regression model as a linear function of itsparameters:

Y = B0 + B1 X + B2 X2 + B3 X3 + · · · + B p X p + �,

(3)

where

Bk =∞∑

m=k

(mk

)(−X)m−k

m!f (m)(X, �). (4)

In this case, the Taylor series model (Equation 3) is thetrue model (Equation 1)—it is just written in a differentway.

So what is the problem if the researcher employs thetypical first-order linear X B L regression in Equation 2?The true model, Equation 3, shows that the higher-orderterms (X2, X3, etc.) are relevant to the relationship be-tween Y and X—they capture the nonlinearity in f (X, �).However, the specification in Equation 2 omits the higher-order terms. Therefore in this case, functional form mis-specification is equivalent to omitting relevant variables,where the omitted variables represent the nonlinearities inthe relationship between the dependent and independentvariables. Based on what we know about omitted vari-able bias, we can conclude that the estimate of B1L willbe biased to the extent that the higher-order polynomialsof X are correlated with X . In the unlikely event that theomitted and included variables are uncorrelated, modelfit will still suffer to the extent that the omitted variableshave a larger effect on Y . Clearly, the more nonlinear thefunctional form is, the more the higher-order terms willmatter, and thus, the greater the specification error is likelyto be.

We have demonstrated the functional form problemwhen Y is a function of only a single variable. The resultsare similar (actually worse) when Y is a nonlinear functionof multiple independent variables. In that case, the higher-order terms in the Taylor series expansion will consist ofhigher-order polynomials of the independent variables,as well as higher-order polynomials of their interactions.

Strategic Misspecification as OmittedVariable Bias

The problem of strategic misspecification in paramet-ric models is precisely a problem of functional form

misspecification, and its proof is analogous to that justdemonstrated with the classical linear model. In this ar-ticle, we address strategic misspecification in the contextof binary choice models, since binary data is so widelyused throughout political science. However, the conceptsare exactly the same for any binomial, multinomial, orcontinuous-variable regression model. In this section,we first describe the general latent variable frameworkfor binary data and then introduce a simple strategicmodel. Using that model, we demonstrate that strate-gic misspecification is equivalent to omitting relevantvariables.

The Latent Variable Specification

Let us assume there exists a latent (i.e., unobservable)variable y∗ defined by

y∗ = f (X,�) + �, (5)

where f (X,�) is a function of regressors X and param-eters �, and � is a random disturbance from some densityh�(�) with mean zero. Examples of such latent variablesmight be a state’s utility for war versus peace, a senator’sutility for a particular bill in Congress, or the extent towhich a shopper prefers apples instead of oranges. Al-though y∗ is unobservable directly, we often can observewhether y∗ is above some threshold. For example, we canobserve whether a state chooses to go to war, whether asenator votes for a bill, or whether a shopper purchasesan apple or an orange. We denote the observable choiceas y, which will be our data, and for simplicity we set thethreshold to zero:

y ={

1 if y∗ > 00 if y∗ ≤ 0.

(6)

A particular probability model results from the specifica-tion of f (X,�) and �.

Unfortunately, the researcher is not omniscient andmust therefore make an assumption concerning all ofthese elements. Of most importance to us is the as-sumption concerning f (X,�). As with the classical linearmodel, the most common practice is to assume that thefunctional form is both linear in the parameters and afirst-order function of the substantive variables, whichwe will denote as X B L . Hereafter when we refer to the“typical,” “common,” or “traditional” binary specifica-tion, it is precisely to this first-order linear latent variablespecification to which we will refer, and we will use an Lsubscript to denote parameters and equations associated


with it.4 � is generally assumed to be distributed logistic(for logit) or Normal (for probit).

Some mistakenly believe that for logit and probit afirst-order X B L relationship is somehow a “general” ortheory-free specification that “allows the data to speak foritself.” On the contrary, the first-order X B L specificationis a very specific structural relationship—a straightjacket,if you will—that constrains our analysis and affects ourinferences. We are not implying that the first-order X B L

specification is alone in this. Indeed, in the context ofparametric models, any assumption concerning f (X, �)is a structural assumption. Once specified, the assumedform of f (X, �) imposes a relationship on the regressors,parameters, and the latent dependent variable y∗. It isnot an empirical relationship that we find through dataanalysis, but a theoretical or modeling relationship in thecontext of which we conduct our analysis.5 Nevertheless,for too long the first-order linear X B L relationship hasbeen taken for granted.

Historically there may not have been valid theoret-ical reasons for assuming otherwise. However, recentwork (Signorino 1999, 2000) suggests that when thedata-generating process is the result of strategic behav-ior, f (X, �) will most likely be nonlinear. Moreover, thestructure of each “game” implies a particular “strategic”functional form. Given that many of our theories in polit-ical science assume strategic behavior and given the preva-lence of logit and probit with the X B L specification, wewould like to know how bad the misspecification is.

A Very Simple Strategic Model

To analyze statistical misspecification, we usually startwith the simplest possible model and examine how themisspecification affects the results in that situation. Moreextensive or general results can be derived later if time,space, and mathematical tractability allow.

Consider the simple strategic situation depicted inFigure 1(a), which is typical of deterrence situations,whether in international politics, congress, or marketentry. Here, {A, A, R, R} are actions in the game and{SQ, Cap, War} are outcomes. Player 1, the attacker, mustchoose between attacking A and not attacking A. If she

4Again, quadratic or cubic terms sometimes appear in publishedlogit or probit regressions. However, the first-order X B specifica-tion is far more common in political science research, and, becauseof its simplicity, it makes the analysis of the specification error mucheasier. As in the OLS case of the previous section, our general ar-gument does not require this exact specification. Indeed, it doesnot require any particular specification of f (X, �)—only that it bemisspecified.

5See Dubin and Rivers (1989) for a similar statement concerninglinear regression.

does not attack A, then the game ends with the status quo(SQ) as the outcome. If she attacks, then player 2, thedefender, must choose between resisting R and not resist-ing R, leading to outcomes War and capitulation (Cap),respectively. For each outcome, the observable compo-nent of the player’s utility is denoted by Ui (k), where iindexes the player and k refers to the outcome. The gamein Figure 1 is only partially strategic—only player 1 mustcondition her behavior on player 2’s expected behavior.This is reflected in the fact that there is no need to specifyU2(S Q).

Throughout this article, we will assume the sourceof uncertainty—i.e., what makes the strategic modelprobabilistic—is based on agent error (see McKelvey andPalfrey 1998; Signorino 1999, 2000). This assumption isimplemented simply for mathematical convenience. Thereader who does not find the behavioral assumptions un-derlying the agent-error model particularly palatable canrefer to Signorino (2000) for two other types of statisti-cal (but strictly Nash) strategic models: one based on in-complete information concerning outcome payoffs; andanother based on variation in the regressors, which is un-observed only by the analyst. The issues addressed here arenot limited to the agent error variant. Indeed, because theprimary question addresses the extent to which strategicmisspecification affects inferences in binary choice mod-els, the general conclusions of our analysis in this articleapply to the payoff perturbation and unobserved regres-sor variation models as well.

In the current context, player i’s true utility for ac-tion j is denoted by U ∗

i ( j ) and is divided into two compo-nents: an expected utility component Ui ( j ) that is observ-able by everyone (including the analyst) and a randomcomponent � j that is observed only by player i. Referringstill to Figure 1(a) above, if player 1 attacks, then player2’s utilities for not resisting R and resisting R are

U ∗2 (R) = U2(R) + �r

= U2(Cap) + �r , (7)

U ∗2 (R) = U2(R) + �r

= U2(War ) + �r . (8)

As we noted before, player 2’s calculations are nonstrategicbecause she does not condition on the attacker’s behavior.In contrast, player 1’s utilities for actions A and A are

U ∗1 (A) = U1(A) + �a

= U1(S Q) + �a , (9)

U ∗1 (A) = U1(A) + �a

= pr U1(R) + pr U1(R) + �a

= pr U1(Cap) + pr U1(War ) + �a . (10)


FIGURE 1 A Very Simple Strategic Model.The simplest strategic model consists of one actorconditioning her decision on that of another player. Here,player 1 must decide whether to attack (A) or not attack (A)player 2. If player 1 does not attack, then the status quo (S Q)results. If player 1 attacks, then player 2 must choose betweenresisting (R) or not resisting (R), leading to (War ) andcapitulation (Cap), respectively. Figure 1(a) depicts the gamein its more general form. Figure 1(b) shows an example wherethe individual choice probabilities are all monotonicallyrelated to the variables that comprise the utilities.

U1(SQ)

U1(Cap)

U2(Cap)

U1(War)

U2(War)

1

2

A A

R R

(a) Strategic deterrence model

pa

pr

1

2

A A

R R

0

X13 β13

0

X14 β14

X24 β24

(b) Utilities specified with regressors

Player 1’s calculations are strategic. Player 1’s observableexpected utility for attacking depends on the probabilitythat player 2 resist or not. Let p j be probability that ac-tion j is chosen and pk be the probability that outcomek is realized. Assuming the � j are independent and iden-tically distributed according to some density f (�) withfinite expectation, then the general form of the equilib-rium probabilities is6

pa = 1 − pa = pr

[U ∗

1 (A) > U ∗1 (A)

],

pr = 1 − pr = pr

[U ∗

2 (R) > U ∗2 (R)

],

ps q = pa ,

pcap = pa pr ,

pwar = pa pr .

With an appropriately specified f (�), the above prob-abilities can be derived and used in data analysis (e.g.,MLE). For example, if the � j are i.i.d. N(0, �2), then theaction probabilities will be Normally distributed. If they

6For examples of how to derive such probabilities, see Signorino(1999, 2000). Derivations of results in this article are also availableupon request from the authors.

are i.i.d. Type 1 Extreme Value, then the resulting actionprobabilities will be logistic.

For regression analysis, we must also specify theutilities of Figure 1(a) in terms of regressors. ConsiderFigure 1(b). Here, the utilities—and, hence, all equilib-rium probabilities—are a function of only three regres-sors: X13, X14, and X24. We have constructed the utili-ties (1) to make them as simple as possible and (2) totry to ensure wherever possible the monotonicity of theprobabilities as a function of the regressors. Althoughwe will discuss this in more detail in a later section, itis important to note here that all of the action prob-abilities ( pa , pa , pr , pr ) are monotonic in each of theregressors.

Finally, researchers are often constrained to workwith particular forms of dependent variables at their dis-posal. The best case here would be to have the actual out-come data available for analysis (i.e., whether S Q, Cap,or War occurred in a given observation). However, datamight only be available for, say, whether player 1 choseA vs. A. Using the appropriate equilibrium probabili-ties in maximum-likelihood estimation would allow forthe analysis of either of these forms of data. Because ofthe prevalence of both forms of data in political science


research, we analyze the specification error for the modeloutcomes in this section and for the attacker’s actions inthe third section.

War Data: Aggregating the StrategicModel’s Outcomes

Assume now that the true data-generating process is thedeterrence model shown in Figure 1(b). Numerous anal-yses in international relations have employed logit orprobit to analyze data where the dependent variable de-notes simply “War” vs. “Not War.” Indeed, this type ofdata, where at least one of the categories aggregates mul-tiple outcomes in some “original” model or process, isactually quite common in political science more broadly.In the context of the deterrence model, the “Not War” cat-egory would include both the status quo and capitulationoutcomes (SQ and Cap, respectively).

Notice that we have three outcomes in the strategicmodel (SQ, Cap, War), but only two outcomes (War,NotWar) in the available data. Before we can assess the mis-specification of a typical logit or probit model in thiscontext, we need to transform the strategic model inFigure 1 into an equivalent binary model—one where theprobabilities over the aggregated outcomes are consistentwith the strategic model.

We illustrate this transformation using Figure 2.Figure 2 shows a progression of models in which theprobabilities are consistent with the underlying strate-gic model in Figure 1, here assuming � is distributedType 1 Extreme Value. Figure 2(a) displays the multi-nomial logit model with the same number of outcomesand the same outcome probabilities as the strategicmodel. In some ways it looks strange — and it should!We have taken a two-player strategic model and aggre-gated it into a single decisionmaker model, where thedyad appears to be the “decisionmaker.” Nevertheless,the single decisionmaker model in Figure 2(a), with theaccompanying specification of the utilities, is equiva-lent to the strategic model in terms of the outcomeprobabilities.

Now consider a dependent variable yS that is codedyS = 1 if War occurred and yS = 0 if either SQ orCap occurred. If we aggregate the SQ and Cap out-comes in Figure 2(a), the resulting binary model canbe expressed as in Figure 2(b). Again, even thoughFigure 2(b) is a binary model, the probabilities are en-tirely consistent with (i.e., derived from) the strategicdata-generating process. We will use the binary model inFigure 2(b) as our referent model for the remainder of thissection.

FIGURE 2 Multinomial and BinomialEquivalents of theStrategic Model.Figure (2a) is a multinomiallogit model that producesoutcome probabilities that areequivalent to the strategicmodel’s. Figure (2b) is theequivalent binomial model andwill be used as the referentmodel for comparison with thetypical logit regression. The Vk

terms are based on the strategicmodel’s utilities. Notice thatthe resulting binary modellooks nothing like a typicallogit regression.

( )ba VV ee 11ln + V3 V4

psq pcap pwar

SQ Cap War

(a) Equivalent MNL Model

0 ( )311ln4VVV eeeV ba ++−

SQ or Cap War

pwar psqcap

(b) Equivalent Logit Model

V1a = U1(SQ ) + U2(Cap)

V1b = U1(SQ ) + U2(W a r)

V3 = pr U1(Cap) + pr U1(W a r) + U2(Cap)

V4 = pr U1(Cap) + pr U1(W a r) + U2(W a r)

Before proceeding, we should point out that manyrecent studies use multinomial logit or probit to modelthe strategic decisions of dyads in crises. These studies as-sume the dyad makes a “decision” among a number of op-tions, including war. However, no consideration is given


in these studies to how one would aggregate an underlyingmultiplayer strategic model into a dyad-as-player model.The above exercise of aggregating the strategic model intoequivalent multinomial and binomial logit models illus-trates two important points. First, the specific functionalform of the aggregated models depends on the specifi-cation of the original strategic model (e.g., the numberof players, their sequence of choices, and their utilities)and on which actions or outcomes are aggregated. Second,the resulting specifications of the aggregated multinomialand binomial logit models look nothing like the specifica-tions in typical multinomial and binomial logit analyses.In fact, unless the researcher can prove otherwise, oneshould generally expect such aggregation of strategic be-havior to be nonlinear in both the regressors and param-eters. The fact that past studies using multinomial logitand probit all employ traditional X B L specifications forthe “utilities” should be a red flag concerning the effectsof strategic misspecification on the conclusions reachedin those studies.

Specification Error

We can now express the model in Figure 2(b) with itsequivalent latent variable representation, and then usethat in our analysis of specification error. Figure 3 dis-plays three latent variable specifications for y∗. The first,y∗

S , displays the latent variable equation implied by the ref-erent strategic model, having substituted the utility spec-ification of Figure 1(b). At the expense of repeating theobvious, y∗

S represents the true strategic data generatingprocess. If one believed the strategic model generated thedata and only had this aggregated binary data on hand,then y∗

S is the binary model one should employ.The regression equation for y∗

S is clearly not a typicallogit regression, which is denoted by y∗

L in Figure 3. Theaction probabilities pr and pr are nonlinear functions ofX24 and �24. Similarly, the RS term is a nonlinear functionof all of the regressors and coefficients.

To demonstrate the misspecification more clearly, wereplace pr , pr , and RS with Taylor series approximations,here second-order and taken about zero. The resulting re-gression equation is denoted by y∗

T in Figure 3.7 The cross-product and higher-order terms, along with the second-order Taylor series remainder �3, are collected into the RT

term to help in comparing the models. Note that, as spec-

7We could have also taken the Taylor expansion of the entire equa-tion y∗

S . However, it was mathematically more convenient to takethe Taylor expansion of only pr and pr . The general result is thesame either way.

ified, the Taylor model y∗T is equivalent to the strategic

model y∗S —it is simply written in a different way.

Comparing the Taylor and logit regressions, y∗T and

y∗L , respectively, we see that both contain a constant

and first-order terms (i.e., X13, X14, and X24). The mostimportant difference, however, is that the Taylor regres-sion contains the RT term, which includes the second-order and higher effects, whereas the typical logit regres-sion does not. If the strategic model generated the data,then the second-order and higher terms in RT are rele-vant variables—and the logit regression omits these vari-ables. Hence, the typical logit regression will induce omittedvariable bias in the estimated parameters of the includedvariables.

The logit equation y∗L is essentially a first-order Tay-

lor expansion, minus a Taylor expansion remainder �2.It is an attempt to capture the linear effects of thevariables. However, if the data is generated by a nonlin-ear (e.g., strategic) process, then the Taylor remainder �2

becomes a relevant variable, which is omitted from theequation. The resulting omitted variable bias means thatnot even the linear effects are correctly estimated. In gen-eral, the greater the nonlinearity implied by the strate-gic model, the larger the Taylor series remainder �2 andthe greater the omitted variable bias in the typical logitregression.

One might be tempted to look for salvation in a lackof correlation between the included and excluded regres-sors. However, the excluded higher-order terms are allfunctions of the included first-order terms, and there-fore some correlation will likely be present. In fact, asYatchew and Griliches (1985) demonstrate for probit, al-though correlation of the included and excluded vari-ables will exacerbate the omitted variable bias, it is notnecessary for bias to result. It is, therefore, unlikely thatany of the effects will be correctly estimated by logiteven for this simplest of models. Given this, any in-ferences based on the estimated parameters or on thepredicted probability of War will almost certainly beinvalid.

Characterizing the SpecificationError in the (Almost) Ideal Case

We have just demonstrated that strategic misspecifica-tion in discrete choice models is equivalent to omittedvariable bias. We did so based on a binary aggregation ofthe outcomes. The strategic equation in Figure 3 showsthat the relationship between the dependent variableand the regressors in that case is nonlinear. In fact, itis actually nonmonotonic as well. Some might therefore


FIGURE 3 Strategic, Taylor, and Logit Models of War.The figure summarizes the three versions of y∗, representing thelikelihood that the attacker and defender will go to war with eachother. The first equation is the strategic model, derived from thedata-generating process. The second equation is the Taylor seriesapproximation of the strategic model. The third equation is themodel commonly estimated to test hypotheses concerning X13, X14,and X24. RS is a nonlinear term that results from aggregating theoriginal strategic model into a binary model. RT is the higher-orderremainder for the Taylor series approximation. Notice that thecommonly estimated Logit model omits these relevant terms.

argue that we have stacked the deck against typical logitor probit models and that more sophisticated researcherswould only use logit or probit to analyze data where theybelieved the dependent and independent variables weremonotonically related.8 Although we would argue (andhave argued) that countless scholars have implicitly con-ducted analyses where the misspecification is similar tothat in the previous section, in this section we examinewhether the typical logit or probit model is misspecifiedin the “ideal” case—when the strategic binary dependentvariable is monotonically related to all regressors.

Specification Error

Consider again the simple strategic model in Figure 1(b).Suppose now that we do not have data on the defender’s

8Roughly speaking, Y is a monotonically increasing (decreasing)function of X if, holding all other variables constant, Y alwaysincreases (decreases) as X increases. Y and X have a nonmonotonicrelationship if, holding all other variables constant, as X increases,Y sometimes increases and sometimes decreases.

actions, but only on whether the potential attacker de-cided to attack or not (A vs A). We might want toexamine the relationship of deterrence success to sub-stantive explanatory variables that we believe affect theincentives of both states to engage in conflict, again rep-resented by X13, X14, and X24. As already noted, weconstructed the model so that the attacker’s choice prob-abilities pa and pa are monotonically related to each ofthe regressors.

To analyze whether the typical logit or probit modelis misspecified in this situation, we again recast the modelin an equivalent latent variable form. Suppose our ob-servable dependent variable is coded as y = 1 if player1 attacks (A) and y = 0 if player 1 does not attack(A).Recall that player 1 will attack if U ∗

1 (A) > U ∗1 (A). Let

y∗S = U ∗

1 (A) − U ∗1 (A). Then we observe y as

y ={

1 if y∗S > 0

0 if y∗S ≤ 0

, (11)

where the S subscript denotes the strategic latent variableequation. Substituting Equations 9 and 10 into y∗

S and


then the utilities from Figure 1(b), the strategic equationbecomes that shown at the top of Figure 4, where � =�a − �a .

Now consider the typical binary choice regression,y∗

L , displayed in the third equation in Figure 4. Obviously,the functional form differs between the two regressionmodels. However, it is not clear how y∗

L and y∗S relate to

each other—e.g., how the estimators of �i j and Bi j L re-late to each other or how the regressions differ in theirpredicted probabilities. The traditional y∗

L is linear in theexplanatory variables and coefficients. In contrast, X24�24

enters y∗S through pr and pr as part of the expected util-

ity calculation. Although it appears that we have a func-tional form misspecification, it is not obvious how badit is. Rather than simply showing that the structural mis-specification is equivalent to omitted variable bias, we willinstead characterize it in a slightly different way, so we canassess just how bad the specification error will be underdifferent conditions.

Since the main problem in comparing the two modelsis the nonlinear pr and pr terms in y∗

S , we use a first-orderTaylor series expansion of pr and pr about the mean ofX24. Let mi j be the mean of variable Xi j . Then we canwrite Xi j = mi j + ui j , where ui j is the deviation of Xi j

from its mean, with E [ui j ] = 0. Greatly abusing nota-tion, we denote pr and pr evaluated at m24 by pr and pr ,respectively.9 With the Taylor expansions of pr and pr ,we can rewrite the strategic equation as its Taylor seriesequivalent, y∗

T . This is displayed as the second equation inFigure 4, with its coefficients Bi j and error term � rewrit-ten at the bottom of Figure 4 in terms of the data andparameters.

To restate the obvious yet again, y∗T represents the

strategic data-generating process. It also happens to be ina form that is comparable to the first-order linear model.Therefore, we can now make a number of statementsconcerning the effects of estimating the first-order lin-ear model y∗

L , when the data has been generated by oursimple strategic model y∗

T .

When will strategic misspecification not be a problem?Perhaps the first question we should ask is: when willstrategic misspecification not be a problem in our ex-ample? Consider the Taylor coefficients Bi j and the er-ror term � at the bottom of Figure 4. Notice that when�24 = 0, the Taylor model reduces to y∗

T = 12 �13 X13 +

12 �14 X14 + �. This is simply a linear latent variable modelof the same form as the linear logit model, so there is nomisspecification. In terms of the underlying choice model,�24 = 0 implies that X24 has no effect on the attacker’s cal-

9In other words, pr = pr |X24=m24 and pr = pr |X24=m24 .

culations. Since X24 enters the attacker’s decision throughits assessment of the probability that the defender will re-sist, this implies that the attacker does not condition herchoice on the defender’s expected behavior. To put it sim-ply, there is no strategic misspecification if there is nostrategic interaction.

If, on the other hand, the attacker takes into accounthow the defender will respond if attacked, but we estimatea typical binary choice model, then the statistical modelwill be structurally inconsistent with the data-generatingprocess and some form of misspecification will exist. Toassess this misspecification, we will assume �24 �= 0 forthe remainder of the article.

Distributional misspecification and inconsistent esti-mates. Consider now the “new” error term, �, in theTaylor regression. Perhaps one of the more glaring prob-lems is that � and � cannot have the exact same distri-bution. As Figure 4 shows, � is composed of two compo-nents: the original error term �, and some function of theui j , determined by the underlying strategic model.

The first potential problem arises if the explanatoryvariables are correlated. Recall that correlated explanatoryvariables implies the ui j are correlated (by definition ofthe ui j ). It is straightforward to show that if the ui j arecorrelated with each other, then the Xi j will be correlatedwith �. Estimated parameters will therefore be biased andinconsistent.

Let us now assume the ui j are independent of � and ofeach other, then E (�) = 0 and the explanatory variableswill be uncorrelated with �. Denote the variances of �, ui j ,and � as �2

� , �2i j , and �2

� , respectively. Then

�2� = p2

r p2r

(�2

13�213 + �2

14�214

)�2

24�224 + �2

� . (12)

The second potential problem is that � is an additive andmultiplicative function of � and the ui j . Because of this,there is no reason to expect that � and � will even sharethe same density but with different variances. Estimatedparameters will therefore also likely be inconsistent dueto this.

Finally, because � is composed of both � and the ui j

terms, �’s variance will always be larger than �’s. As Equa-tion 12 displays, the difference between �’s and �’s vari-ances will depend on the variance in the regressors andon the magnitude of the �i j parameters. The larger thevariances of the regressors and the greater the magnitudeof the parameters, the greater will be the difference inthe variance of � versus �. In examining omitted variablebias in probit, Yatchew and Griliches (1985) note that thisform of misspecification can affect hypothesis tests.

Having said this, the distributional misspecification isa matter of degree. If the ui j are symmetrically distributed


FIGURE 4 Strategic, Taylor, and Logit Models of DeterrenceSuccess vs. Failure.The figure summarizes the three versions of y∗, representingthe attacker’s propensity for attacking. The first equation isthe strategic model, derived from the data-generatingprocess. The second equation is the Taylor seriesapproximation of the strategic model. The third equation isthe model commonly estimated to test hypothesesconcerning X13, X14, and X24. The remaining equations givethe values of the Taylor coefficients (B0, B13, B14, B24) anddisturbance � in terms of the original parameters and data.Here, mi j is the mean of explanatory variable Xi j , p j is thedefender’s probability p j evaluated at m24, and ui j is thedeviation of Xi j from its mean mi j . The Logitmisspecification can be assessed based on assumptionsconcerning the parameters and data.

and do not have a large variance relative to �, then � willtend to be distributed similarly to � and distributionalmisspecification will have less effect on inferences.

Consistent estimates, but wrong inferences. Althoughthe preceding would seem to offer enough indictmentof strategic misspecification, of more interest to us ishow the structural misspecification affects our inferenceswhen the parameters are estimated consistently—or atleast close to it. For the sake of examining other aspectsof strategic misspecification, we will now give the typicalfirst-order linear specification the “benefit of the doubt”and assume that the distributional misspecification is neg-ligible. In this case, the Taylor regression y∗

T takes the samefunctional and distributional form as the logit regressiony∗

L . As usual, the regressor and variance parameters canonly be estimated to scale. When � is logistic, consis-tent estimation (e.g., MLE) of the logit parameters will

converge to

B0L = B0√3

�2 �2�

B13L = B13√3

�2 �2�

B14L = B14√3

�2 �2�

B24L = B24√3

�2 �2�

. (13)

If we now substitute the equations for the Bi j at the bot-tom of Figure 4 into the above equations, we can de-termine the effects of using the logit model with thedeterrence data.

Interpreting the sign and magnitude of regressioncoefficients are two common practices in political scienceresearch. As B13 and B14 show (Figure 4), the effects ofX13 and X14 will tend to be estimated correctly by thelogit model, at least in terms of the direction of theireffect on the attacker’s decision. In contrast, inferencesconcerning X24 will depend on idiosyncracies of the data.


In particular, the estimate of B24L is a factor of �14m14 −�13m13. So, for example, if we center our data aroundzero, which is not uncommon, traditional logit or probitwould lead us to believe that X24 has no effect on whetherthe attacker chooses to attack. By construction, we knowthat inference is false. Similarly, if �14m14 < �13m13, thenlogit and probit would lead us to believe that X24 has theopposite effect it actually does.

Finally, because the parameter estimates in any ofthese choice models (strategic or nonstrategic) are diffi-cult to interpret by themselves, analysts typically interpretthe estimated probabilities for a better understanding ofthe relationship between the dependent variable and theregressors—not only for the direction of the relationshipbut the relative magnitude. We would, of course, like toknow the extent to which strategic misspecification affectsthe estimated probabilities. Although the misspecificationcan be expressed in a fairly general form, its mathematicalcomplexity does not allow for a simple (and useful) ana-lytical expression. We therefore provide the reader with asense of the misspecification in the following example.

Monte Carlo Example

To demonstrate that the simplifying assumptions of theTaylor series approximation are not unreasonable andto show how the predicted probabilities can differ be-tween the strategic and logit models, we present the re-sults of a simple Monte Carlo analysis. Each replica-tion of the analysis involved generating N = 2000 ob-servations based on the behavioral assumptions of thestrategic deterrence model in Figure 1. The explanatoryvariables were randomly drawn from a uniform dis-tribution on the interval [−2, 2], and the coefficientswere set to �13 = �14 = �24 = 1. The � disturbances inEquations 7–10 were drawn from a Type 1 Extreme Valuedistribution with variance �2/6, resulting in a logistic �

with variance �2/3. The strategic and logit regressionswere run and their estimates saved. These steps werereplicated 2000 times to form densities of the parame-ter estimates.

Parameter estimates. The densities (not shown here) ofthe strategic estimates were all approximately normal andcentered around one, indicating that the strategic modelwas able to recover the correct estimates on average.10

What are our expectations concerning the logit estimatesusing this data? It turns out that �fairly well approximatesa logistic distribution in this case, with approximately the

10Plots of the densities for all parameter estimates in this sectionare available upon request from the authors.

same variance as �. Therefore, distributional misspecifi-cation is not a big concern. Next, note that because theexplanatory variables are uniformly distributed between−2 and 2, m13 = m14 = m24 = 0. Given the true parame-ter values and the characteristics of the data, the logit esti-mators should converge to B0L = 0, B13L = .48, B14L =.48, and B24L = 0. In fact, our Monte Carlo analysis pro-duced mean estimates of �̂0L = .000, �̂13L = .49, �̂14L =.49, and �̂24L = .000. Again, the direction is correct con-cerning the variables in the attacker’s utilities. However,researchers would incorrectly infer that X24 has no effecton the attacker’s decision to attack. Moreover, althoughthe signs of �̂13L and �̂14L are correct, we will next showthat the picture is more nuanced than the logit modelallows.

Estimated probability of attacking. To understand whatthe parameter estimates imply for the estimated probabil-ities, Figure 5 displays the attacker’s estimated probabilityof attacking based on (a) the logit model and (b) thestrategic model, both as a function of X24 and X13.11 Inboth cases, the third variable, X14, is held constant at itsmean (zero).

Turning to Figure 5(a), we see that the researcher em-ploying the logit model would infer from the estimatedprobabilities that X24 has no effect on the attacker’s choice.No matter what X13’s value is, changing X24 has no effecton the probability of attacking—the probability remainsconstant and the first-difference is zero. The researcherwould also conclude that increasing X13 always increasesthe probability of attacking, and by the same amount re-gardless of the value of X24.12

Now consider the estimated probabilities in Figure5(b). The strategic model paints quite a different pic-ture. Moreover, how X13 and X24 affect the probabilityof attacking is actually quite intuitive. Recall the strategicmodel as specified with the regressors in Figure 1(b), andrecall that the graph was produced holding X14 at its mean,zero. In the context of the strategic model, the observableutilities for the attacker are U1(SQ) = 0, U1(Cap) = X13,andU1(War) = 0. For the defender, they areU2(Cap) = 0and U2(War) = X14.

To interpret the effects of X13 and X14, it may be help-ful to consider the three regions (A, B, C) in Figure 5(b)

11The graphs are based on exactly the same Monte Carlo as pre-viously detailed, but with the Xij drawn from U[−10, 10] insteadof U[−2, 2]. All inferences remain the same as before, includingthose pertaining to the estimated probabilities. The larger intervalfor the Xij simply emphasizes the point made here. Plots for bothsets of Monte Carlos are available upon request from the authors.

12Although not shown here, logit produces similar inferences whenthe plot is a function of X14 and X24, holding X13 at zero.


FIGURE 5 Estimated Probabilities of Attacking.Figures (a) and (b) display the estimated probability that the attacker will attack, based on the logitand strategic models, respectively. The probabilities are plotted as a function of X13 and X24,holding X14 at its mean (zero). As (5a) displays, logit would lead us to believe that X24 has no effecton the attacker’s decision, which contrasts with the (true) strategic model (5b). Figure (5b) alsoshows that the direction of the effect of X24 depends on X13. The magnitude of the logit model’serror is also fairly high for certain values of (X13, X24): at times off by more than .4.

(a) Logit (b) Strategic

that are “almost flat.” Region A reflects a situation wherethe defender’s (observable) utility for war (X24) is muchlower than his utility for capitulation. In this case, he willlikely capitulate. The attacker knows this, and her utilityfor capitulation (X13) is much higher than her utility forthe status quo. Therefore, in region A, the attacker willalmost certainly attack. Region B again reflects a situationwhere the defender will back down if attacked. However,here the attacker’s utility for the defender’s capitulationis actually much lower than her utility for the status quo.Therefore, there is almost no chance the attacker will at-tack. Finally, consider region C. In this case, the defender’sutility for war is very high, and he will likely defend himselfif attacked. Knowing this, the attacker then has a choicebetween the status quo and war, both of which have anobservable utility of zero. Because the observable utilitiesare so close (relative to the size of the variance), the analystis almost completely uncertain as to whether the attackerwill attack or not.

The effects of X13 and X24 can now be thought ofas how they change the equilibrium probabilities as the

utilities move from one extreme to the other. What is theeffect of X24? It depends. When X13 > 0, then increasingX24 decreases the probability of attack. On the other hand,when X13 < 0, increasing X24 increases the probability ofattack. What is the effect of X13? Again, it depends. Ingeneral, increasing X13 always increases the probability ofattack. However, when X24 is small, X13 has a large effecton the probability, whereas when X24 is large, X13 hasvery little affect on the probability. Finally, it should beremembered that, because this is an equilibrium model,the effect of each variable depends on the values of theother variables. We have demonstrated how X13 and X24

affect the probability of attack when X14 is held constantat zero. The relationship may be very different when X14

is held constant at a different value.To summarize, in contrast to the researcher using a

logit model, a researcher employing the strategic modelwould conclude that (1) X24 affects the probability ofattacking, (2) the value of X13 affects the direction ofX24’s effect, (3) X24 affects the magnitude of X13’s ef-fect. It should also be noted that the difference between


the logit and strategic estimated probabilities is off attimes by more than .4. Moreover, the researcher em-ploying the strategic model would be able to interprether results in the context of a causal, equilibrium-basedmodel.

The Monte Carlo examples presented here were in-tended to provide a concrete demonstration of the analy-tical results previously derived. They also suggest that thesimplifying assumptions made in characterizing the spec-ification error may not be especially egregious. In fact, aswe proceeded through our analysis of the specificationerror, we repeatedly gave logit the benefit of the doubt.When those assumptions are violated, the specificationerror should only be worse. Finally, the graphs in Figure 5raise another important issue, one concerning functionalform and hypothesis specification and testing. Althoughthe topic deserves a separate monograph of its own, itis relevant to the analysis here, so we provide at least acursory discussion of it in the next section.

Theories, Functional Form,and Hypothesis Testing

Thus far, we have framed the problem under considera-tion as one of structural or functional form misspecifi-cation. To most practitioners, this may have previouslyseemed to be a rather arcane technicality, since so little at-tention is generally paid to it in political science methodstraining. However, functional form specification is alsointimately related to hypothesis specification and test-ing. We have argued that greater attention should be paidto substantive theory and its implications for functionalform. Yet, the converse is true as well: greater attentionshould be paid to the implications of our hypotheses con-cerning the functional form of our statistical model andits relationship to the theories we think we are testing.

The Functional Form Impliedby Hypotheses

In an ideal world, a researcher would have a well-specifiedtheory. Hypotheses would naturally follow from the the-ory. And the theory would suggest the appropriate func-tional form for the statistical model used to test the hy-potheses. All three—theory, hypotheses, and statisticalmodel—would be consistent with each other. When theyare not consistent with each other, it raises a red flag thata serious problem exists in the research design. After all,hypotheses are supposed to reflect aspects of a theory to betested, and the statistical model must reflect the hypothe-

ses in order to actually test them. Thinking about it thisway—where a well-specified theory provides a functionalform that drives the hypotheses and statistical model—the role of functional form in hypothesis specification andin the statistical model seems obvious.

What may not be so obvious are the implications ofthe above when a strong theory does not exist to specifythe functional form and hypotheses. Most political sci-ence research still falls into this category. Often scholarsinvest a great amount of time thinking through the logicof what they are studying—in a very real sense, buildingmodels in their minds. However, it remains that, becausethe model is not written formally (e.g., using mathemat-ical equations or formal logic notation), the paths to theconclusions reached and the relationships between thevarious parts of the model often remain unclear.

Almost invariably in this case, hypotheses arespecified as unconditionally monotonic relationships—relationships where as one explanatory variable increases,the dependent variable always increases (or always de-creases), regardless of the values of the other explanatoryvariables. Indeed, often a laundry list of unconditionallymonotonic hypotheses is presented. A simple example re-lated to international relations might be:

H1: The stronger a nation is, the more likely it is toenter into war.

H2: The more democratic a nation is, the less likelyit is to enter into war.

Taken together, and in the context of the typical logitor probit model, hypotheses H1 and H2 imply that thefunctional relationship between the regressors and thedependent variable is unconditionally monotonic. Theimplied model is that the stronger a nation is, the morelikely it is to go to war, regardless of how democratic it is;and, similarly, that the more democratic a nation is, theless likely it is to go to war, regardless of how strong it is.

If scholars truly believe the theory being tested isconsistent with the jointly unconditionally monotonicrelationships, then the common first-order X B L func-tional form is exactly what they should use. However,many if not most theories analyzed in political sciencesuggest more complicated relationships. The problem isthat many scholars automatically employ a laundry listof monotonic hypotheses and the first-order X B L func-tional form, without realizing the implied limitationsconcerning the theory being tested and the effects of mis-specification on their inferences.

Although it is beyond the scope of this article to assesswhy this is the case, two rationales are frequently offered.


The first is the belief that the first-order X B L specifi-cation is somehow a “general” specification. Apparentlybecause of its simplicity (and perhaps widespread use?),many seem to view the first-order X B L functional form asa more general functional form, where researchers can atleast correctly estimate the first-order effects of variables.This is incorrect. As we have demonstrated, in the contextof parametric estimation (whether MLE or Bayesian), it iscertainly not a more general specification—it is a specificstructural specification, which is quite restrictive. Indeed,when the data-generating process is not first-order lin-ear, then it is a misspecification, and our inferences areaffected.

To further illustrate why unconditional monotonic-ity is restrictive, consider the following two plausible sit-uations that violate it. In the first, suppose (1) that waralways decreases as democracy increases, regardless of thestrength of the nation, but (2) that greater strength leadsto less war for democratic regimes and to more war for au-thoritarian regimes. In this case, war and democracy arestill unconditionally monotonically related, but war andstrength are now conditionally monotonically related—i.e.,war and strength are monotonically related for any givenlevel of democracy, but the direction of that relationshipchanges depending on the level of democracy. Now con-sider a second example, where the likelihood of war ismoderate for authoritarian regimes, very high for regimesthat have some moderate level of democratic institutions,but very low for those nations with the highest level ofdemocracy. In this case, war is a nonmonotonic function ofdemocracy. All of these relationships seem plausible, butare inconsistent with the typical first-order linear model.

The second rationale for the laundry-list approach isthat the researcher lacks a strongly specified theory to pro-vide a functional form. In many ways, this rationale seemsreasonable. If one lacks a well-specified theory prior toconducting the empirical analysis, what rationale is therefor specifying a complicated relationship between the de-pendent variable and regressors? Historically, researcherswithout a firm theoretical justification for including in-teraction and higher-order polynomial terms have riskedaccusations of data mining. The lack of a well-specifiedtheory is then held up as an excuse for the list of mono-tonic hypotheses.

Ironically, within this category there exists a largegroup of positivist scholars who recognize the impor-tance of competition, incentives, and institutions, andwho use statistical analysis to uncover general explana-tions (or “processes“) of political behavior. However, thelack of a well-specified theory is not a “get out of jailfree” card. Because the hypotheses and statistical modelare formally specified, but the theory is left loosely speci-

fied, one can only conclude one of two things: (1) eitherthe statistical model and hypotheses are inconsistent withwhatever (more complicated) theory the researcher hasin mind, or (2) they are consistent with a theory that onlyallows for unconditionally monotonic relationships. Inother words, the researcher in this position must eitheraccept that the statistical model does not reflect the the-ory or that her theory is far more simplistic and restrictivethan she might want to admit.

Finally, it is often conjectured that typical logit or pro-bit techniques should be fine for testing models where onecan show (e.g., via an analysis of comparative statics) thatthe relationship being analyzed is monotonic. However,as we noted previously, the received wisdom requires theimportant qualification that the relationship is not justmonotonic, but unconditionally monotonic. When thedata-generating process implies unconditionally mono-tonic relationships between the regressors and dependentvariable of interest, then the first-order X B L specificationis appropriate. Nevertheless, it is incumbent upon the re-searcher to determine (i.e., prove) that the theory impliessuch a relationship before using that specification.

Is Unconditional Monotonicity Likelythe Rule or the Exception?

If most relationships were unconditionally monotonic,then, given current practices, researchers would have littleconcern about functional form. Unfortunately, in manyareas unconditional monotonicity may be the exception,rather than the rule.

Consider again our simple model in Figure 1(b). It iseasy to show that, using the least restrictive definition ofmonotonicity, all action probabilities ( pa , pa , pr , pr ) aremonotonic in all of the regressors. As we noted before, thiswould seem to be an ideal situation for the argument thatmonotonic comparative statics are sufficient justificationfor the use of traditional logit and probit specifications.Then why did we find misspecification?

pr and pr are functions only of X24 and are indeedunconditionally monotonic in X24.pa and pa are also un-conditionally monotonic in X13 and X14. However, pa

and pa are conditionally monotonic in X24. Assuming ourdata consisted of the attacker’s actions, the typical laun-dry list of unconditionally monotonic hypotheses wouldbe inconsistent with the data-generating process, and thefirst-order X B L regression, as a reflection of the hypothe-ses, would be misspecified. Indeed, that is exactly what wesaw in the third section; and Figure 5(b) clearly displaysthe conditional monotonicity of pa and X24: for X13 < 0,


the probability of attack monotonically increases in X24,whereas for X13 > 0, it monotonically decreases in X24.

Now suppose our data represented whether war oc-curred or not (i.e., War vs. {SQ or Cap}). Again, it is easy toshow that ps q is unconditionally monotonic in X13 andX14, but conditionally monotonic in X24, and that pcap

and pwar are unconditionally monotonic in X13 and X14,but nonmonotonic in X24. Even when each player’s indi-vidual choice probabilities are monotonic in the regres-sors, the outcome probabilities may be nonmonotonic inthe regressors. Again, this result is inconsistent with theunconditional monotonicity assumption one often findsin lists of hypotheses.

The deterrence model in Figure 1 is one of the verysimplest strategic models possible, and, yet, a numberof the relationships described by it are nonmonotonicor conditionally monotonic. It does not seem unreason-able to assume that much of the behavior we analyzein political science is at least as complex, if not more,resulting in similar nonmonotonic and conditionallymonotonic relationships. This would imply that the tra-ditional practice of specifying hypotheses as a list of un-conditionally monotonic relationships may be woefullyinconsistent with much of the political behavior we ana-lyze, let alone the loosely-specified theories we think weare testing.

Concluding Remarks

To recap, we have characterized the strategic misspecifi-cation that arises from using the typical logit or probitspecification—one where the latent dependent variableis a linear function of the parameters and a first-orderfunction of the explanatory variables. In the interest ofkeeping the analysis as simple as possible and of creat-ing ideal conditions for those interested in applying logitto comparative statics, we have constructed the “simpleststrategic model possible” and have ensured that the rela-tionship between all action probabilities and the regres-sors is monotonic.

Even under these conditions, the typical first-orderlinear specification is problematic. Ultimately, strategicmisspecification is a functional form problem. Strategicmodels often imply a nonlinear and possibly nonmono-tonic relationship between the latent dependent variableand the independent variables, in contrast to the first-order X B L specification commonly found in multino-mial and binomial logit and probit models. We haveshown that as a functional form misspecification, strategicmisspecification is equivalent to omitting relevant vari-

ables, where the omitted variables are nonlinear higher-order terms associated with expected utility calculations,and often with data aggregation as well.

So what is a poor researcher to do? Our use of Taylorseries expansions as approximations for the strategic re-gressions suggests one possible solution to the functionalform problem: run polynomial regressions with interac-tion terms. Data-mining critiques aside, there are at leasttwo other problems with this approach. Most problematicis that increasing the order (and, therefore, the extent towhich the polynomial approximates the true functionalform) combinatorially increases the number of param-eters that must be estimated, becoming impractical toimplement with only a relatively small number of inde-pendent variables. For example, a typical first-order spec-ification with six variables requires that we estimate onlyseven parameters—i.e., the coefficients for each indepen-dent variable, plus the constant. On the other hand, thefull second-order specification for those same six vari-ables requires that we estimate 28 parameters, and thethird-order specification requires that we estimate 84 pa-rameters. Secondly, as defined here, a polynomial regres-sion really is an exercise in curve fitting—i.e., of cor-rectly approximating the strategic “curve,” in order thatfunctional form misspecification not bias estimates. Assuch, although it may allow us to estimate the functionalform correctly, it does not allow us to say anything aboutcausality.

One approach would be to develop better theory andthen derive hypotheses and statistical models from thattheory, thereby ensuring consistency of one’s theory andstatistical analysis. Another approach would be to use“theory-less” statistical techniques that do not impose somuch structure on the analysis. The benefit of the struc-tural approach is that a particular causal relationship isanalyzed. A problem with it is that, if the structure ismisspecified, then specification error results. The benefitof the nonstructural approach is that (by definition) itdoes not rely on structural specification. The problem isthat it has no theoretical (or causal) purchase. Still an-other approach would be to use both methods iteratively:using nonlinear techniques to determine the functionalrelationship of dependent and independent variables, andthen developing and estimating structural models that areinformed by the previous stage. In our opinion, the worstsituation would be to continue the current practice ofusing the very simplest of structural statistical models—the first-order X B L specification—to analyze behaviorthat we believe a priori cannot be consistent with thatspecification, and which we now know produces invalidinferences.


References

Dubin, Jeffrey A., and Douglas Rivers. 1989. “Selection Bias inLinear Regression, Logit, and Probit Models.” SociologicalMethods and Research 18(2/3):360–90.

Kmenta, Jan. 1986. Elements of Econometrics. 2nd ed. New York:Macmillan.

McKelvey, Richard D., and Thomas R. Palfrey. 1998. “QuantalResponse Equilibria for Extensive Form Games.” Experimen-tal Economics 1(1):9–41.

Signorino, Curtis S. 1999. “Strategic Interaction and the Statis-

tical Analysis of International Conflict.” American PoliticalScience Review 93(2):279–98.

Signorino, Curtis S. 2000. “Structure and Uncertainty in Dis-crete Choice Models.” Presented at the 1999 Summer Polit-ical Methodology Meeting.

Smith, Alastair. 1999. “Testing Theories of Strategic Choice: TheExample of Crisis Escalation.” American Journal of PoliticalScience 43(4):1254–83.

Yatchew, Adonis, and Zvi Griliches. 1985. “SpecificationError in Probit Models.” Review of Economics and Statistics67(1):134–39.

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