[Taiba Chau]

Sorry to clarify what would be the main economics behind it because I seem to be confused. The control group would be the non-ethnic individuals in the countries while the treatment group would be the ethnic individuals in the countries.
For this regression:
regress Mental_Health i.Country##i.Wave##i.non_ethnic Gender Age Marital People_IN_House Educ Employ Inc

I am confused about how I would then write this in an econometric regression form. Would it be:

[Clyde Schechter]

In the DDD model, there are two groups, mathematically. These are the ethnic and non-ethnic subsets of the population. From a mathematical point of view, neither one is the "control" group. They are just two groups that get contrasted with each other. It is in our interpretations of these results that one group gets perceived as the "control" group and the other as the "treatment" or "intervention" group. If, in your mind, the non-ethnic individuals are the control group, then the equation becomes simplest if you use a formulation in which the variable is ethnic (1 for the ethnic group, 0 for the non-ethnic group). With that specification, you can write the regression equation directly by plugging the regression coefficients in for the corresponding deltas without having to change any signs.

There are, however, some errors in the equation as you have written it. There should be no tau_c term in the equation, because that is accounted for by the Country variable. There can be a gamma_t term--but that does not correspond to the regression you have run. You only represent time there with the PostPolicy variable--you do not have a separate variable for calendar time. And it is best not to have both as they would be colinear and cause a mess in interpreting which is contributing to what. So ditch the gamma_t term. You also don't have a mu_i term in the model you regressed. I cannot tell from what you have written in the thread so far whether such a term is even possible in your data--it would be possible only if you have the same people responding to the survey in each wave. If you do have the same people responding in each wave, then you really should be including a mu_i term--and the way you would do that is by going to a multi (or at least 2) level model, not -regress-. If you have different people responding in each survey wave, then no mu_i term is even possible, so you should just forget it. As for omega_ict, I don't know what that is supposed to even be. If you modified your model to include such a term, it would just be colinear with all of the other variables anyway and would make a mess out of interpretation. Finally, because you do have multiple observations in each combination of country, ethnic,, and post-policy, your residual error term, epsilon, should not bear any subscripts: it is not constant within each combination but varies at the single-observation level. So no subscript there.

Finally, a trivial observation: there needs to be another + before delta₆.

Added: The equation you posted (except for the mistaken omission of a + before delta₆ is the equation of a fairly complicated multi-level model with crossed random effects. Your model is a simple regression without any multi-level structure. If you were intending to do a multi-level model, you have to start over and the code will be different. To be honest, I do not recommend you do that. First, I don't think having random effects at the ethnicity level and the time level makes statistical sense because each of those has only two levels. Second, the code required is substantially more complicated to write, and even more substantially complicated to interpret. Given how you are struggling with interpreting the simple regression, I suspect this thread would go well beyond 100 posts before it got ironed out. Frankly, I'm not up for that! I think that the regression model you have done, possibly augmented by person-level random effects if you actually have the same people responding at successive survey waves, is a better model for the data and you should leave it at that.