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Let's say I'm running a linear regression that has the form $y = \beta_0 + \beta_1A+\beta_2B+\beta_3AB +\epsilon$.

If $\beta_3$ is positive, does this imply a positive correlation between $A$ and $B$? (Conversely, a negative correlation if $\beta_3$ is negative?)

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    $\begingroup$ This is a nice example of a question where simulations can help. Generating some random $A,B,y$ data, you should be able to find some counter-examples pretty quickly. Also, note that bilinear interpolation is typically done on grids where $x$ and $y$ are orthogonal, and the "interaction term" can have arbitrary sign. $\endgroup$ – GeoMatt22 Sep 28 '16 at 4:28
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    $\begingroup$ Actually, just the very idea of creating a dataset according to this model ought to answer the question definitively. Why not generate uncorrelated variates $(A_i,B_i)$, independent errors $\epsilon_i$, and choose any values of the betas you wish. Is there any obstacle to computing a value of $y_i$ for each $i$? If not, then the value of $\beta_3$ implies nothing about the correlation of $A$ and $B$. (cc @GeoMatt22) $\endgroup$ – whuber Sep 28 '16 at 14:16
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No, a non-zero $\beta_3$ does not imply $A$ and $B$ are correlated. It implies $y$ is correlated with $AB$.

Simple example:

Imagine we have data on visits by people to a gas station.

  • Let $A$ be the volume of someone's gas tank in gallons.
  • Let $B$ be the price of gas at the time of the visit.
  • Let $y$ be the spending on gas this visit.

$A \cdot B$ is how much it would cost to fill the person's gas tank. $AB$ is almost certainly correlated with $y$, the spending on gas this visit.

A positive $\beta_3$ in this trivial example does not imply that the size of someone's gas tank is correlated with the price of gas. A positive $\beta_3$ would mean that spending $y$ is positive related to the carrying capacity of someone's gas tank measured in dollars (i.e. $AB$).

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Here is a potential applied counterexample: suppose $A$ is gender, $B$ are years of schooling and $y$ are labor-market earnings. So, after, say, 12 years of primary and secondary school and a three-year Bachelor degree, you would have completed 15 years of schooling.

Then, it is not totally off to assume that $A$ and $B$ are uncorrelated - in the past, men used to have higher degrees, nowadays, if anything, women. So there probably was a moment in the (not so distant) past when gender and years of schooling were uncorrelated, and the correlation certainly is not strong today.

And yet, it is not difficult to make a case that $\beta_3\neq0$, as an additional year of schooling may have a differential effect on earnings for men than for women.

This would, for example, be the case when there is wage "discrimination" (in quotation marks as it is a hotly debated issue) mostly in jobs for more highly educated employees. Anecdotical evidence suggests that this may be the case, as male executives tend to be better paid than female ones. On the other hand, salaries in jobs that require less education may be more frequently determined by broad agreements between unions and employers' associations (at least in, for example, continental Europe), leaving less room for wage discrimination.

(The quotation marks could for example be justified by the fact that this simple story does not account for sectors, experience, etc.)

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  • $\begingroup$ What is "schooling" in your example? I looked in a dictionary and the word seems to have multiple meanings. $\endgroup$ – ttnphns Sep 28 '16 at 8:11
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    $\begingroup$ Thanks, I made an edit. I'm too entrenched in economics jargon... $\endgroup$ – Christoph Hanck Sep 28 '16 at 8:13
  • $\begingroup$ Apologies for my downvote, this answer just adds so much information that is not needed for an explanation that it confused me. In addition to this it makes us assume that A and B could be uncorrelated, which is not really the case, and therefore it struck me as counterintuitive. $\endgroup$ – Dennis Jaheruddin Sep 28 '16 at 16:38
  • $\begingroup$ Well...if you care to explain what parts are not needed for the example? Also, it is in the nature of assumptions that they may not hold, and it is made fairly clear that this particular one may or may not hold. But other than that there is no need to apologize, if you feel a downvote is warranted. $\endgroup$ – Christoph Hanck Sep 28 '16 at 16:43

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