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Is it ok to interpret the sign of an interaction term coeficiente in a logit or probit model with random effects?

For example,

Y=b0+b1*X1+b2*X2+b3*X1*X2

being Y binary (e.g. 1 if company i has reduced the employees' monetary incentives in year t, 0 otherwise), X1 is continuous (e.g. return of company i in year t), and X2 is binary (e.g. 1 if company i is a big one).

Suppose b0<0; b1<0; b2>0; b3<0

I believe b3 can be interpreted as the incremental relation between the likelihood of reducing employees' monetary incentives and returns for big companies.

Nonetheless, I have seen some warnings regarding the interpretation of interaction terms in non-linear models. However, I am just interested in the sign and its association with a positive/negative effect in the likelihood of Y and not in marginal effects.

I would also like to ask if there is a meaningful interpretation of the intercept sign in this kind of models. For instance, if b0<0, is it likely that companies have a propensy not to reduce employees' monetary incentives (when not taking into account the rest of the regressors)? I am not sure if this makes any sense.

Thanks in advance.

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Yes the sign does matter.

Think of the simple case with two predictors that are binary (0 or 1)

If b3 is positive, when both x1 and x2 are 1 then they are associated with higher values of y.

If b3 is negative then when x1 and x2 are 1 then they are associated with lower values of y.

So the sign shows whether the interaction term is associated with increases or decreases in the value of the dependent variable.

This means in the logit case the sign says whether the interaction term is contributing to make the y equal to 1 or 0.

For a meaningful interpretation you can calculate odds ratios for all parameters. And so if say the odds ratio for b3 = +2. Then when x1 and y1 are BOTH 1, y is 2 times more likely to be 1. If it were -2 then it would be 2 times LESS likely.

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    $\begingroup$ See Data Colada #57 on the interaction point, this post also cites the relevant literature. The sign can be deceiving about the direction of the effect. Even if you do that correctly, the other tricky point is what is happening to the random effect and how that can enters into the marginal effect. $\endgroup$ Commented Nov 3, 2017 at 16:38
  • $\begingroup$ Yes this is a good point. It's important to keep the non-interaction terms in your equations to probe the equations $\endgroup$ Commented Nov 3, 2017 at 16:42
  • $\begingroup$ Thank you for your post @DimitriyV.Masterov. I have read it but I am still in doubt. Since I am just interested in the interaction term coefficient to determine the sign of the incremental effect on the impact of X1 caused by the event that the dummy variable X2 assumes the value 1, while keeping the other variable (X3) constant, I guess in this case the interpretation of the total effect is clear and equivalent to the sign of b3. As I am not concerned in the total/marginal effect as exemplified in Data Colada#57 where both X1 and X2 are changed, isn't it possible to interpret it like I did? $\endgroup$ Commented Nov 3, 2017 at 19:34
  • $\begingroup$ @DDDD To quote the abstract of the Ai and Norton paper cited by DC, "The magnitude of the interaction effect in nonlinear models does not equal the marginal effect of the interaction term, can be of opposite sign." $\endgroup$ Commented Nov 3, 2017 at 20:53

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