I wanted to share with you a couple graphs and seek your wisdom on the insights that I may be omitting.

According to theory, there are merits to believe that relationship b/w X and Y is moderated by M. Notably, the relationship between X and M is likely to be quadratic. Following Aiken and West (1991), to test for the quadratic interaction I estimate the following model:

y = X + M + X*M + M^2 + X*M^2, in which case X*M^2 would give a test of quadratic relationship. Also, y is of a count type, therefore I use a Poisson estimator.

Now, when I plot the interaction (i.e., impact of X on Y at representative values of M [obtained using Stata's -margins- command]) in terms of average marginal effect I get the two graphs as attached. One is with

*default standard errors*and another one with

*robust SEs*(estimated using so-called "delta method"), as mentioned in the literature to account for the fact that we cannot fully assume Poisson distribution.

The pattern of the confidence intervals on the "robust SE" graph seems somewhat interesting. What can it possibly be an indicator of? E.g., I could have suspected there is another "interacting variable" that might boost or suppress the impact. Or it could be that the magnitude of impact is better predicted closer to value of zero, while beyond some threshold it becomes less efficient.

Any ideas are welcome.