# Regression analysis with interaction and decomposed main effect

I'm trying to figure out if I need to include the main effect in a regression analysis with an interaction if I am already including a decomposed version of the main effect.

For example, let:

-INT is an 0/1 indicator variable to use in the interaction
-AVG_Overall is an average value of variable X for all individuals in groups A and B
-AVG_A is the average value of variable X for individuals in group A ONLY
-AVG_B is the average value of variable X for individuals in group B ONLY
-Y is the dependent variable


Is it appropriate to estimate the following regression?:

$$Y_i = \beta_0 + \beta_1INT_i*AVG\_Overall_i + \beta_2AVG\_A_i + \beta_3AVG\_B_i + \beta_4INT_i + u_i \tag{1}$$

This specification omits the main effect of AVG_Overall. However, because AVG_A and AVG_B are the average variables of the two 2 subsets that comprise the overall sample used to calculate AVG_Overall, it seems reasonable that they are capturing the main effect. My thinking is that if the individuals in each subset have an identical effect on Y, then $$\beta_2=\beta_3$$ and they would also be the same as the coefficient on the main effect AVG_Overall would have been. If, however, individuals in subsets A and B have a different effect on Y, then $$\beta_2$$ and $$\beta_3$$ should be different.

An alternative specification is to include the main effect of AVG_Overall along with AVG_A and AVG_B, i.e.:

$$Y_i = \beta_0 + \beta_1INT_i*AVG\_Overall_i + \beta_2AVG\_A_i + \beta_3AVG\_B_i + \beta_4INT_i + \beta_5AVG\_Overall_i + u_i \tag{2}$$

However, AVG_Overall is by definition collinear with both AVG_A and AVG_B, making it difficult to distinguish the different effects of AVG_A and AVG_B ($$\beta_2$$ and $$\beta_3$$). Empirically I'm finding that including the main effect of AVG_Overall changes the magnitude of $$\beta_2$$ and $$\beta_3$$ and makes them difficult to interpret, although their significance remains basically unchanged.

I have theoretical reasons why I think $$\beta_2$$ and $$\beta_3$$ should differ, and also why it doesn't make sense to interact the INT variable with both AVG_A and AVG_B, but I'm concerned that I might be running into some econometric problems.

Please help! Does specification (1) seem appropriate? If so (or if not), can you point me to some resources that explain the statistics behind this?

Thanks!

• Possibly I'm missing something, but I don't understand why you want to use averages at all. If you have a dataset with individual values of $X$ and $Y$, why not use the full data in your regression? – Adam Bailey Jul 9 at 9:23
• There are individual values of X but not of Y, so if I ran it with individual observations for each X then there would be many repeated values of Y. I went down that path using clustering, but in the end clustering is meant to address correlated values of Y--not identical values. – elci Jul 9 at 17:47
• what do you mean by main-effect ? And what is the objective of your study/quarry ? – Subhash C. Davar Aug 3 at 5:08