Joint model with interaction terms vs. separate regressions for a group comparison After gathering valuable feedback from previous questions and discussions, I have came up with the following question: Suppose that the goal is to detect effect differences across two groups, male vs. female for example. There are two ways to do it:


*running two separate regressions for the two groups, and employ Wald test to reject (or not) the null hypothesis $H_0$: $b_1-b_2=0$, where $b_1$ is the coefficient of one IV in male regression, and $b_2$ is the coefficient of the same IV in female regression.


*pool the two groups together, and run a joint model by including a gender dummy and an interaction term (IV*genderdummy). Then, the detection of the group effect will be based on the sign of interaction and the t-test for significance.
What if Ho is rejected in case (1), i.e. group difference is significant, but the coefficient of interaction term in case (2) is statistically insignificant, i.e. group difference is insignifant. Or vice versa, Ho is not rejected in case (1), and interaction term is significant in case (2). I have ended up with this outcome several times, and I was wondering what outcome would be more reliable, and what is the reason behind this contradiction.
 A: The first model will fully interact gender with all other covariates in the model. Essentially, the effect of each covariate (b2, b3... bn). In the second model, the effect of gender is only interacted with your IV. So, assuming you have more covariates than just the IV and gender, this may drive somewhat different results. 
If you just have the two covariates, there are documented occasions where the difference in maximization between the Wald test and the Likelihood ratio test lead to different answers (see more on the wikipedia). 
In my own experience, I try to be guided by theory. If there is a dominant theory that suggests gender would interact with only the IV, but not the other covariates, I would go with the partial interaction.
A: Anytime two different procedures are used to test a particular hypothesis there will different p-values.  To say one is significant and the other is not can be just making a black and white decision at the 0.05 level.  If one test gives a p-value of 0.03 and the other say 0.07 I would not call the results contradictory.  If you are going to be that strict in thinking about significance it is easy to have either situation (i) or (ii) arise when boardline significance is the case.
As I mentioned in response to the previous question my preference for looking for an interaction is to do one combined regression.
A: In the second case, standard software would suggest you a t-stat with t-student pvalues whereas for the first case the Wald tests may have two options. Under errors normality assumption Wald statistic follows an exact Fisher statistic (which is equivalent to the t-stat as it assumes error's normality). Whereas under asymptotic normality, Wald statistic follows a Chi2 distribution (which is analague to the a t-stat following a normal distribution asimptotically) What distribution are you assuming ? Depending on this your p-values risk to give you different results. 
In Textbooks you will find that for bilateral single tests (one parameter) both, t-student and Fisher statistics are equivalent.
If your sample is not large then comparing  comparing chi2 and t-stat pvalues would yield different results for certain. In that case assuming an asymptotic dsitribution would not be reasonable. IF your sample is rather small then assuming normality seems more reasonable, this implies t-stat and Fisher pvalues for case 2 and 1 respectively.
