Modeling Interaction Effects with Product Variables

The dominant method of modeling an interaction effect of two independent variables on a dependent variable is to include a product variable in a linear estimation equation. Textbook writers and researchers almost always interpret the coefficient for the product variable as describing both how the first independent variable affects the effect of the second independent variable, and vice versa. As a result, writers often claim that interaction effect are "symmetrical", "two sides of the same coin", or are equal in magnitude. This interpretation of the meaning of the product variable's coefficient is wrong, because it fails to appreciate that different kinds of interaction effects imply the same estimation equation. In this note, I show that researchers need strong theory or knowledge before they can interpret the meaning of the coefficient for a product variable. Specifically, the coefficient for the product variable conflates two different kinds of interaction effects, and we need additional information to solve the "identification problem" of separating them. I also discuss how theory or knowledge are needed to correctly specify product-variable models of interaction effects, and address the question of why researchers rarely use non-deterministic models of interaction effects.