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If I have two regressors that should theoretically have opposite signed coefficients, should I switch the sign (i.e. multiply by -1) of one of the variables before creating an interaction term?

In my work, I found the 'correct' signs for the main regressors, one positive and one negative in the non-interacted regression. When I created an interaction term between the two, the main effects kept their original signs but the interaction term was insignificant.

I switched the signs of one of the variables (multiplied each observation by -1) and re-ran the regressions. Obviously, without the interaction term, the absolute value of the coefficient on that variable remained the same but the sign had switched. However, when I added the new interaction term, the coefficient on the interaction term was now significant.

Note that these are all continuous variables and interaction terms were based on mean-centered variables as in Balli & Sorensen (2013).

I'm not sure why this would happen. Is this acceptable? Is it normal? Should I be concerned about my data?

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    $\begingroup$ What you describe cannot happen, because the significance of the regression results will be wholly unaffected by any constant rescaling of any of the variables. Because your procedure rescaled one of the regressors by $-1$ and (consequently) the interaction also by $-1$, it should not have changed any p-value whatsoever. $\endgroup$ – whuber Jul 27 '16 at 19:46
  • $\begingroup$ I agree with whuber, algebraically it is not possible. Something else probably happened (like some observations ended up as missing, perhaps, or some other undiagnosed change). Without a reproducible example or a very careful accounting of what you did I doubt anyone will be able to confidently suggest what happened. $\endgroup$ – Glen_b -Reinstate Monica Jul 28 '16 at 5:11
  • $\begingroup$ Thank you for your comments. You're correct that the procedure should have simply switched the sign of the interaction term as well. The problem is that I had mean-centered the interaction term, meaning that some observations which had previously been positive were now negative (or vice versa). E.g. if the mean was 5, an observation of 3 would now become -2, creating an interaction term that is no longer simply the negative of the original interaction term. What I find odd is that it is this (incorrect) interaction term that is significant in the model. Spurious I suppose. $\endgroup$ – Simon Jul 28 '16 at 21:25
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Whenever you add a new variable in a regression analysis, all coefficients may change. They will always change unless the added variable is uncorrelated to all the previous variable. The extreme case where two or more variables are highly correlated is multicollinearity.

And about your question of switching signs of one of the variables, it will just change the signs of some of your coefficients. In fact, applying any linear transformation to any variable won't change anything; coefficients will just get transformed in the same way to keep yielding the same predictions for the same situation.

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