Interpretation of interaction terms with categorical moderator in regression

[Tony Shayan]

I am estimating a model with interaction terms and have difficulty interpreting the results. My model estimates the effect of "historic irrigation" on current "social capital" using district-level data.

Theory suggests that this effect should be weaker in districts experiencing higher "climate instability." To demonstrate this, I categorize districts into three bins based on climate instability percentiles: low instability (<25), medium instability (25-75), and high instability (>75). Then, I estimate the following model:

social capital = a0 + a1historic irrigation + a2controls + b2bin2 + b3bin3 + c2*(bin2 × historic irrigation) + c3*(bin3 × historic irrigation) + error term

I exclude the first bin (<25) and its interaction with historic irrigation from the regression for two reasons:

1. I intend to compare medium and high instability regions against the most stable regions.
2. Practically, I need a baseline category for the estimation.

1- First question is about interpreting the coefficient "a1". Can I interpret a1 directly as the effect of historic irrigation in the most stable districts?

I also created a graph (see below) to visualize the results. The corresponding coefficients plotted are:

• a1 for the low instability bin (<25)
• a1 + c2 for medium instability (25-75)
• a1 + c3 for high instability (>75)

Note that coefficients b2 and b3 are not statistically significant.

2- Given this, do you think the point estimate for the low instability bin (<25) is directly comparable to the other two bins?

3- If my current approach has issues, what revisions or alternative methods would you recommend?

Thank you in advance for your advice.

[Carlo Lazzaro]

Tony:
why not creating a three-level categorical variables for the climate instability percentile and then interacting it with irrigation?

[Tony Shayan]

Thanks a lot for your response. The reason I use dummy variables is primarily because I use "acreg" to correct for the spatial standard errors. When I use a simple "reg" command and interaction features (# operator) in Stata, I get the same result. To my knowledge, "acreg" does not have this feature. Apart from that, as far as I understood, creating dummies is a valid method that allows each interval to impact the outcome variable differently. Does that make sense?

[Carlo Lazzaro]

Tony:
if the community-contributed module (as FAQ kindly recommend you to mention it) -acreg- does not support -fvvarlist- notation, you can create interactions the old way (that is, by hand). Unfortunately, -margins- and -marginsplot- do not work with interactions created by hand.

[Tony Shayan]

Thanks again for your feedback. Sorry I missed mentioning the community-contributed module.
As you said, I created interactions the old way and implemented my analysis. Apart from the implementation, I am not 100% sure about my interpretation—that's why I was hoping someone in the community would give a hint or an insight.