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I estimated a model for which I tested the assumption that the effect of my variable of interest is homogeneous for observations belonging to two different groups (for simplicity assume rural/urban).

Results show that, on the one hand, both the main effects are not statistically different from zero*, on the other the interaction term is statistically significant. This denotes a cross-over interaction where the effect changes direction in sign according to the group being considered.

Specification ONE:

In this particular example I constructed the dummy as 1 if the observation is a rural household, 0 otherwise. As anticipated above the only statistical significant coefficient is that of the interaction term, which is negative (-0.62), thus the overall effect for rural households coincides with the interaction's coefficient.

I thus reject the assumption that the effect of my variable of interest is homogeneous for the two groups. However, I am interested to see the extent of this switch in sign of the coefficient of interest.

Specification TWO:

Conversely, if I construct the dummy in the opposite way (i.e 1 if the observation is urban, 0 otherwise) and estimate the same model. I get the following results:

  • interaction: statistically different from zero (0.62);
  • main effect of variable of interest: statistically different from zero (-0.63);
  • main effect of the dummy defining the groups: not statistically different from zero.

The overall effect of the interest variable for urban households equals -0.01.

However, two questions arise:

  1. Given the detection of a cross-over interaction in the first specification, is it worth computing and showing the results re-computing the dummy for rural/urban as in specification two above? As a reader I would like to see at what extent the effect is positive (something I cannot see from the results of specification one).
  2. Can I still talk of cross-over interaction even if in specification two I get a rather very small but still negative and significant effect?

Note: Statistical significant results are those for which p<0.05; for not statistically significant results p>0.1.

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When a cross product is in the model the effects of other variables are really simple effects: they are effects when the other variables are 0. You might find your results more interpretable if you use contrast coding (-1,1) and center your other variable.

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  • $\begingroup$ I see your point. However, the contrast coding will allow me to explore the differences in mean of the two groups. Moreover, its interpretation is straightforward only for farily balanced designs. I do not see how this is able to provide an answer to my question. What I am interested at is the change in sign and the magnitute of the overall effect by group, not difference in groups' means. $\endgroup$ – Caserio Mar 16 '17 at 8:32

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