Hello,

I am running a model with county fixed effects and time fixed effects using repeated cross-sectional data. My treatment variable is a three-level variable and thus, the set-up of my model is not exactly a difference-in-difference model, I think. But let me know if I am wrong.

I wanted to ask what would be the identifying assumption in this model. In DID, we need to meet the common trends assumption but in this case since I don't have a binary treatment indicator, I was not sure how the common trends assumption would look like. My understanding is that conditional on county and time fixed effects and other controls I include, the error term (epsilon_ict when i = individual, c= county, t=time) is uncorrelated with the treatment variable. Is this correct? If so, is this assumption different from the common trends assumption when I have a three-level treatment variable rather than a binary variable?

To give you a better understanding of my model, here is the code I'm using:

Code:
svy: reg outcome i.lag_treatment i.county i.time i.covariates
The covariates include individual characteristics as well as county-specific linear time trends. The survey set up takes into account the clustering of standard errors.

I would really appreciate it if you can provide me with some help on this. Thank you in advance!