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Hi Simon,
thank you so much for your reply.
I tried your code but it gives me the following error message:

"invalid matrix stripe;
c.treatment_village##c.period
error in expression: _b[c.treatment_village##c.period]"

So I tried to generate the treatment variable:

gen treatment=treatment_village*treatment_period

And run:
ritest treatment_village _b[treatment], cluster(village): xtreg y i.treatment i.period, fe cluster(village)

Resulting however in a p-value = 1 which seems not right at all.