I found only one published article doing so, but econometrics scholars absolutely reject this idea.
Hi Fabio, if you are analysing the interaction of X and Z (XZ), it is necessary to include both X and Z, otherwise the interaction effexc might be confounded with the main effects of the lacking variable (see classical texts on interaction such as...
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2013). Applied multiple regression/correlation analysis for the behavioral sciences. Routledge.
Aiken, L. S. & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Sage.
Jaccard, J., & Turrisi, R. (2003). Interaction effects in multiple regression (Vol. 72). Sage.
Simple answer: you can't.
If you are including X and XZ without including Z as a regressor you are almost automatically facing a problem of omitted variable bias.
This is the case, because you do not know whether the coefficient on the interaction term does capture the "effect" of the interaction and/or the effect of the omitted variable, i.e. Z.
Dear Prof. La Rosa,
Theoretically speaking if you are dealing with the long run this question is very important as it has to do with the transversality condition (see for e.g. Lucas, Prescott and Stokey, Macroeconomic Dynamics, HUP book). In the short run and in cross section I think the answer has to come from the model you are intrepreting. Correlations carry information by themselves as for e.g. the CAPM tells us. However there is a lot of debate in the growth and development economics literature which deals with the specific importance of correlations as well as levels. However, you can refer to my RG page and the stock pricing and banking convergence papers which show that the predictive power increases when you include both, which means that you can also include the second variable in scale also. SKM QC
Adding variable(s) and or interaction between variables to the model should be determined by theory and previous research. You may lastly depend on logical basis
It makes no sense to include an interaction between two factors (qualitative variable|) and to exclude the main effect of either factor. See the following paper for the reasons: Nelder, J. A. (1977). "A Reformulation of Linear Models". Journal of the Royal Statistical Society 140 (1): 48–77. doi:10.2307/2344517. Some software (e.g. GenStat) allows you to fit a two-way effect between two factors without explicitly fitting the main effect(s), but it is then automatically interpreted as including the main effect(s) along with the interaction. Other software (e.g. SAS) actually uses models with no main effect nonsensically in the calculation of Type III and IV sums of squares; see Nelder JA, Lane PW (1995) The computer analysis of factorial experiments: in memoriam Frank Yates. The American Statistician 49, 382-385. However, if you consider the variation of the effect of a variate (quantitative variable) among the levels of a factor to be an "interaction" between the factor and the variate, then it can make sense to fit a model with such an interaction but not the main effect of the factor (but you still need the "main" effect of the variate).