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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.

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I agree that in the case of perfect collinearity the interaction is just the square and it is possible to main effects that are not significant but a significant interaction.

If you had perfect collinearity then one approach is to add some small random error to one of the variables, or you could combine them, if this makes sense in your context.

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.

It is, that's exactly what it does.

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  • $\begingroup$ Does this answer your question ? If so, please consider marking it as the accepted answer, and if not please let us know why. $\endgroup$ – Robert Long Aug 21 '20 at 12:07

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