# Interpreting interaction term on highly correlated variables

Somebody at a meeting today made the following comment about a Marketing Mix Model (Linear Regression) we run every year.

• We should account for the high collinearity of the two Marketing Variables (two Marketing variables we include in our model)
• We should always include the interaction of these two Marketing variables in the model, to better interpret the synergistic effect

I feel there's something wrong about these two statements being put together. Let me simplify the problem:

Let's denote $$X_1$$ and $$X_2$$ as two Marketing variables, and we want to estimate their contribution to total sales $$Y$$. We then run a linear model $$Y = \beta_0 + \beta_1X_1 + \beta_2X_2$$ We find that $$X_1$$ and $$X_2$$ are highly correlated between one-another. This should, in theory, impact the coefficients and the individual t-statistics due to collinearity.

Now if we do the model such as $$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_1 X_2$$ In the extreme case of perfect collinearity I would assume that the interaction term $$X_1 X_2$$ it's the same as $$X_1^2$$ or quadratic relationship to dependent variable $$Y$$. So there could be situations where the coefficients for $$X_1$$ and $$X_2$$ individually are not significant but $$X_1 X_2$$ is significant (where significant means a low t-statistic). Please correct me if this is not true.

Now my question is. Does this last model really make sense at all, from an interpretation standpoint?

Even without perfect collinearity issues I don't know if we can infer that the interaction is really measuring the linear combination of both variables. Meaning when I execute both marketing tactics together I'm not getting the simply the synergistic effect. Instead, I'm actually capturing the quadratic effect of just one of the variables. In this sense I can't really interpret the interaction term. Also I'm not quite sure how $$X_1 X_2$$ is providing new information to the model given that their variance explained should be near identical.