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Multiplicative Interactions

Interaction models have been popular in the social sciences. However, recent reviews [1, 2, 3] have revealed serious problems with regard to their construction and interpretation. In particular, the exclusion of constituent variables has led to the specification of incorrect models, in the sense of a mismatch between model form and the objective of the researcher formulating the model. This Demonstration illustrates the consequences of not including all constituent variables for the estimation of marginal effects.

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Generally, multiplicative interaction terms should be used whenever the marginal effect of a variable is hypothesized to be conditional on the level of some other variable (or even ). In its simplest form, such a general model with two interacting variables may look as follows:
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In this model, and are called the main effects of the constituent variables , respectively, while is called the interaction effect. In order to illustrate the meaning of the interaction term , consider how the expected value of changes when , change by some constant . Then
,
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From this result, two cases arise. First, if , the interaction term drops out so that the expected change in or only depends on , but not on the value of , , respectively. Second, if , the expected change or depends on and the value of , respectively.
Thus, whenever inference suggests that the interaction effect is significantly different from zero, the estimated marginal effect of each constituent variable is not constant anymore, but becomes conditional on the level of that constituent variable with which it interacts. If and are both binary variables, is the effect of when is absent, is the effect of when is absent, and is the effect of the joint presence of and . This Demonstration shows the example of a model where both and are continuous, and is a binary variable that takes on the value of 1 when a condition is present and the value of 0 when it is absent. The estimated regression function for the complete model given by is shown by the two dark blue lines in the figure, one for each state of , and the coefficients are provided in the table underneath. The true data-generating parameters can be changed in the controls under the "parameters" heading. For the initial values set there, the hypothesis may read as follows: "The relationship between and is negative if is present, and positive if is absent".
Case 1: omitted
Consider the case where the constituent variable is omitted from the model, which is equivalent to imposing the constraint . The resulting model looks as follows:
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Its estimated regression function can be obtained by checking the appropriate box in the controls under the "specification" heading. Before explaining the effects of this omission, let us see why researchers would want to leave out . Some have argued that can be omitted either if (a) it has no independent effect on or (b) if it has no effect on when is zero. The justification given in (a) is never valid because the estimated marginal effect of is conditional on the value of as was shown above. This conditionality implies that the effect of on can be zero, but need not be zero for this to be the case. For example, at the initial settings of this Demonstration where , the marginal effect of is zero when (here an out-of-sample value because ).
The justification given in (b) presupposes a strong theoretical argument, but the conditions that must hold in such a case before can be omitted are virtually never fulfilled. There are two such conditions, namely that the population parameter must be zero, when is ratio-scaled and zero. Why does need to be ratio-scaled? Only ratio-scaled variables, such as age and GDP, have a natural zero point so that the quotient of two values does not depend on the chosen unit measure. Any variable measured on a lower scale such as interval, as most political science indexes are, can be subjected to linear transformations. In order to illustrate the effect, let be the linear transformation of such that . The full model can now be written as
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Thus, if , but a constant is added (as [1] shows for the case of the Polity index), the intercept and the coefficient on will change. If is transformed with and , all coefficients will change. To see this, you can vary both and under the "transformation" heading. If is not ratio-scaled, it is impossible for the researcher to know the effect of the peculiar scaling of on the size of ex ante. In consequence, there cannot be a theoretical argument for omitting either. However, even if is ratio-scaled, and the researcher has a strong theoretical argument, it should still be tested whether is in fact zero, which requires the specification of . In this case, the researcher may as well include all constituent variables in the model, also with a view to the fact that even if the null hypothesis cannot be rejected at an appropriate significance level, it may in fact not be exactly zero. But if , the estimates of the marginal effects of for each state of under model , and will be very close to , as you can convince yourself of by adjusting the respective control in this Demonstration. This leads us to the effects of omitting by constraining to be zero on the remaining coefficients.
If is not exactly zero, and or , which is virtually always true in the social sciences, all OLS estimators will be biased and inconsistent. This is a clear case of omitted variable bias. Before, the intercept was when and when . With omitted, only one intercept can exist for both states of , namely . This also forces the marginal effects to change from when and when to when and when . More precisely,
,
,
and ,
where the 's are the coefficients from and the 's are the coefficients from the regression of on and , including an intercept:
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Case 2: omitted
When omitting the constituent variable , the logic is very similar, but there are some particularities. The resulting model looks as follows:
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Before, the intercept was when and when . With omitted, the intercept changes to when and when . The marginal effect changes from when and when to 0 when and when . More precisely,
,
,
and ,
where the 's are the coefficients from and the 's are the coefficients from the regression of on and , including an intercept:
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However, when , it follows that because , so we get the interesting result that and , which are just the intercept and the marginal effect for under the complete model . Thus, in the case of being omitted, we will still be able to estimate the appropriate functional relationship when , but not when . Put differently, we can only test the first half of the hypothesis formulated above, but not the second part.
Case 3: and omitted
Finally, consider the case where both constituent variables and are omitted, so that the following model results:
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Such a specification has been presented, for example, in [4] on p. 127, which tests the interactive effect between public opinion and attention to foreign policy on defense spending. For illustrative purposes, let be defense spending, public opinion, and a binary indicator of attentiveness. Again, there is only a single intercept, , for each state of . As the marginal effect of is now given by , it will be zero whenever . In the case of , it reduces to . More precisely,
,
and ,
where the 's are again the coefficients from and the 's are the coefficients from the regression of on , and on , including the intercepts:
,
.
Thus, the only case in which the estimated marginal effect of under is very close to the estimated marginal effect of under when arises when has no effect on in the absence of , that is, if . As for the case of , it arises only in the improbable situation where . In all other cases, any configuration is possible.
Conclusion
If a multiplicative interaction model is the appropriate specification, never forget to include all constitutive terms.
References
[1] T. Brambor, W. R. Clark, and M. Golder, "Understanding Interaction Models: Improving Empirical Analyses," Political Analysis, 14(1), 2006 pp. 63–82.
[2] B. F. Braumoeller, "Hypothesis Testing and Multiplicative Interaction Terms," International Organization, 58(4), 2004 pp. 807–820.
[3] C. Ai and E. C. Norton, "Interaction Terms in Logit and Probit Models," Economic Letters, 80(1), 2003 pp. 123–129.
[4] B. D. Jones, Reconceiving Decision-Making in Democratic Politics: Attention, Choice, and Public Policy, Chicago: University of Chicago Press, 1994.
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