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.