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:

.

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

,

.

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:

.

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

.

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:

.

**Case 2: **** omitted**When omitting the constituent variable

, the logic is very similar, but there are some particularities. The resulting model

looks as follows:

.

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:

.

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:

.

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.

If a multiplicative interaction model is the appropriate specification, never forget to include all constitutive terms.

[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.