Selection of indicators for the interaction term in structural equation models with interaction

Kenny and Judd (1984, Psychological Bulletin 96: 201-210) suggested using structural equation models to model interaction effects since they allow correction for measurement error. They proposed using all possible products of the indicators of the two interacting variables as indicators for the interaction term. Joreskog and Yang (1996, Advanced Structural Equation Modeling. Mahwah, NJ: Lawrence Erlbaum, pp. 57-88.) defended that this is not necessary; one product variable is sufficient to estimate the interaction effect. However, they did not specify which indicators should be chosen if there is more than one possibility. We prove that the optimal choice is to select the indicators with the highest reliabilities. But this is only true if certain assumptions hold. We go on to show that one can get very different results depending on the indicators chosen for the interaction term if the indicators are not congeneric which is often the case. These methodological arguments will be illustrated by a study of the purchasing or boycotting of certain products for environmental reasons.