Dear All:

I suddenly found myself confused in interpreting multiple interactions of categorical variables in a single specification. I have two questions and I sincerely appreciate your help. For illustration, let's consider the below specification:

Code:
regress temperature i.timeint i.region i.region#timeint b3.regionclass b3.regionclass#timeint lowjobs c.perc_black##c.perc_black##c.perc_black c.perc_unemploy##c.perc_unemploy, vce(cluster tractfips) allbaselevels
In the above specification:
(1) Say region is a categorical variable consisting of 6 categories indicating which region in the U.S. the panel belongs to.
(2) regionclass is another categorical variable consisting of 4 categories.
(3) timeint is a dummy variable indicating which year the observation within a panel belongs to, e.g., timeint = 1 if 2020 and 0 if 2018.

I have two questions:

A) How would I interpret the coefficients on the interaction term i.region#timeint? In this case, assume 1.region and timeint = 0 (2018) is the reference category. My understanding has been that the coefficient on 2.region#timeint will indicate the change in temperature for a station in region == 2 in 2020 compared to 2013. Likewise, per my understanding, I would interpret the coefficient on 1.regionclass#timeint as the change in temperature for a station in regionclass == 1 in 2020 compared to 2013. Is my interpretation correct? Based on contrasting information on other websites, I am now confused if my interpretation is correct.

When I calculate the marginal effects as:
Code:
margins timeint, dydx(region)
Stata provides the linear predictions for each of the six regions in 2018 and 2020. When I calculate the differences, they are exactly similar to the raw coefficients the software provides for each combination of region and timeint (excluding reference categories of course). This again makes me think that the raw coefficients above can indeed be interpreted as the increase/decrease in temperature (response outcome) for a specific combination of region and timeint compared to the base.


(B) What would be the reference category for the interaction term i.region#timeint? As I stated above, I am interpreting 2.regionclass#timeint as the change in temperature for a station in region == 2 in 2020 compared to 2013. But does the fact that the coefficient for 1.region cannot be estimated (and thus Stata uses it as a reference category) change my above interpretation. My guess would be yes but am unclear especially how the presentation of the Stata output changes when timeint#i.region is used in the command instead of i.region#timeint (used above).

Thanks a lot for your insights and help!