Let's say I have three variables:

Variable A, B and C, where C is the product of A and B. Both A and B are continuous variables.

If I regress Y onto A and B, A is significant and B is not.

Then, if I regress A, B and A*B (which is variable C), then A and C are significant. Which means that C has some explanatory power on top of A alone, right?

How do I interpret the coefficient of the product of the two variables?

Does it make sense to have such an interaction?



Paraphrasing the discussion in Wooldrigde's introductory econometrics textbook, consider the following example:

It may make sense for the partial effect of the dependent variable with respect to an explanatory variable to depend on the magnitude of yet another explanatory variable.

Example (not exactly continuous, but the logic is very similar for "truly" continuous variables): $$ price = \beta_0 + \beta_1 squarefeet +\beta_2 bedrooms + \beta_3 squarefeet\cdot bedrooms + \beta_5bathrooms+ u, $$ Now, the partial effect of $bedrooms$ on $price$ (holding other variables fixed) is $$\beta_2 + \beta_3 squarefeet$$ If $\beta_3> 0$, then this implies that an additional bedroom yields a higher increase in housing price for larger houses. To assess the partial effect, evaluate the equation at interesting values of $squarefeet$, such as the mean value in the sample.

Note: The parameters on the original variables can become tricky to interpret. For example, the above equation shows that $\beta_2$ is the effect of $bedrooms$ on $price$ for a house with $squarefeet=0$...

  • $\begingroup$ Thanks very much for the reply. How do you deal with the problem with the beta 2 you mention in the end? $\endgroup$
    – adrCoder
    Jun 18 '15 at 14:04
  • $\begingroup$ there is no single solution, what people typically do is, as mentioned, evaluate partial effects at the mean, $\endgroup$ Jun 18 '15 at 14:05
  • $\begingroup$ Hello, I will accept your answer + upvote, and I will go read the Wooldridge book and search more, thanks ! $\endgroup$
    – adrCoder
    Jun 18 '15 at 14:19
  • 2
    $\begingroup$ You can also rescale the continuous variables before estimation, so that the zero of $\tilde x_i = x_i-\bar x$ corresponds to average value of $x$. That way $\beta_2$ is interpretable. $\endgroup$
    – dimitriy
    Jun 18 '15 at 17:16
  • $\begingroup$ @Dimitriy V. Masterov Can you please provide some more information on what you are saying ? $\endgroup$
    – adrCoder
    Jun 19 '15 at 9:22

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