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I am sorry if this is very trivial and a repetition. I could not find a direct question on the website that addresses my question

I am studying the relationship between X1 (independent variable) and Y (outcome) variable. X2 is my moderating variable

What is the difference between these two following regression equations :

Y = b1 + b2X1 + b3X2 +b4 X1X2 ... (1)

Y = b1 + b2X1 +b3 X1X2 ....... (2)

X1 and X2 are not correlated and Since X2 is my moderating variable , I am not interested in the main effects of the moderating variable..in that case would equation 2 be okay where I take only the interaction effects? Is equation 2 okay to use? If not, what are the problems. Any inputs will help. Mainly I am trying to understand if dropping main effects estimation of the moderating variable will be of any issue?

I did read this - The regression analysis that tests for an X1 by X2 interaction must also include the X1 and X2 variables as predictors. In fact, the X1 × X2 product term represents an interaction only in the context of a model that also includes X1 and X2 as predictors (Cohen, 1978).

However I have seen models that do not include the moderating variable as a predictor. When does one do this. How do I interpret this?

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In the first model, the coefficient on $X_2$ (i.e., $b_3$) corresponds to the expected slope of $X_2$ on $Y$ when $X_1=0$. Omitting this term as in the second model is essentially forcing the coefficient on $X_2$ to be equal to zero. If that coefficient is indeed zero in the population, then there is no harm in setting it to zero, but typically researchers don't know the value of slopes in the population (otherwise they wouldn't need data in the first place). If the coefficient is set to zero (or any value) and the population value is far from that value, all the other coefficient estimates will be badly biased. There is no practical reason to do this.

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In general Cohen is right. As long as your are not completely confident then sticking to that rule is the safest thing you can do.

There are exceptions (there are always exceptions), in the sense you do not add the main effect directly. However, they are usually adding the main effects in an indirect way to the model. So these "exceptions" aren't exceptions at all. See for example: M.L. Buis (2012) "Stata tip 106: With or without reference", The Stata Journal, 12(1), pp. 162-164.

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