Insignificant Interaction effect in applied econometrics (multinomial logit regression) I'm currently writing my master's thesis where I'm applying a multinomial logit regression with interactions effect. 

The model I'm using is 
$$
P(y=j) = exp(xb_j)/(1+∑exp(xb_h))
$$ 
where 
$$
xb_j =x_1b_{ij} + x_2b_{2j} + x_1x_2b_{3j} + XB + \epsilon
$$
  $x_1$ and $x_2$ are dummy variables ($x_1$ is coded 0/1 for man/woman).

I use marginal (or incremental) effects to report my results, since I'm interested in the effect of a unit change on the probability of af given outcome of my y-variable. 

My question is regarding my results, where I get an insignificant marginal effect of the interaction term, which I guess indicates that there is no difference between how men and women are affected by a unit change in $x_2$. But when I look separately at men and women, I get that the marginal effect of $x_2$ is significant for men ($\frac { \partial P(y=j) } {\partial x_2 }$ when $x_1=0$) and insignificant for women ($\frac { \partial P(y=j) } {\partial x_2 }$ when $x_1=1$). (I use three different significance levels (0.1, 0.05 and 0.01))

How do I interpret an insignificant effect from the interaction, but significant in a subgroup (men)?
It is not 100% clear what procedure you used to do the subgroup tests, so I will cover a few bases. Below, I suggest exploratory tests to better understand the results. The logic in this response is the same as you would follow in a linear model. 

If you are taking the results from the single model and doing post hoc tests of the coefficents, be sure that you are interpreting the model correctly. Because men are the implicit reference category, the constant represents the men's average (ceteris paribus) and the constant plus b1 is the women's average. Similarly, b2 is men's "slope" with respect to x2, while b2 + b3 is women's "slope" with respect to x2. So, to test whether x2 matters at all for women, you would have to test b2 AND b3 together. 

That said, it sounds to me like, after running the full model, you ran the model separately for men and women (minus the sex-related terms) to investigate further. Here's a few things to investigate:


How different are the other parameters for men and women in the separate models from the main model? 


Are there similar point estimates?
Are the same variables significant?
Do the models have comparable model fit?

What kind of variation do you have in your data? You can do descriptive and exploratory analyses to understand this.


Present a descriptive table of the frequencies by sex, x2, and j. If every "cell" created in this way had exactly the same count, you would be rather unlikely to get a disparity between the subgroup models and the full model (unless the data generating processes for men and women are entirely different, and/or there is heteroskedasticity by sex).


That is, my hunch is that you have "micronumerosity" for some of the women in certain outcomes, and thus you cannot get a small enough standard error to distinguish the interaction term.
Another way of looking at this is figuring out the realm of common support for your model.

Run descriptive statistics (mean, SD) on the other variables separately for men and women and look for discrepancies. If there are substantial differences, it could mean something other than sex is driving the difference, and/or that collinearity between that other variable and sex is complicating the estimation. 


If the covariates are normally distributed, you could run a series of t-tests in addition to eyeballing the means. (If you wanted to be more formal about this and take into account multiple hypothesis testing, you could do various kinds of balance checks, essentially using the x variables to predict sex.)

Graphs of the area of common support or distributions of other variables conditional on sex can help show what you find.



To write up the results, I'd suggest presenting a table with the coefficients for the full model without the interaction, the full model with the interaction (showing the interaction is not significant), and the models by gender. 

For the discussion section, you can mention that you looked into the role of the interaction term. You can show your results about micronumerosity and covariate balance by sex (whether or not they seem to explain the problem). All of these are exploratory analyses, and so they can't stand on their own as definitive explanations. But they help you address what should be done in future analyses (e.g. that a larger N overall is needed or that oversampling people in small cells would help, or finding other ways to increase the precision of the model).