Dif in Dif on countries, is it necessary to cluster by country? I have a question regarding Dif-in-Dif (DD) studies. If I want to compare a policy that is implemented differently in three different countries - thus three different treatment groups - using a DD approach, is it necessary to cluster standard errors by country (instead of for example by industry)?

From my understanding this would make a DD approach almost impossible as the amount of clusters can not be high enough (aka not higher than 42 or whatever number one likes to go by). With other words all DD studies comparing policy effects between 2 or 3 different countries (or states) would be invalid.
You should cluster on level where you believe the autocorrelation or heteroskedasticity occurs.

If you don’t have enough clusters at a given level, for example, if you believe the above mentioned problems occur on country level but you have only 3 countries, you are correct you should not be using clustered standard errors which by rule of thumb require at least about 40 clusters to be properly estimated. However, clustered errors are not the only type of error adjustment there is. For example, you can instead of clustered errors use Newey-West errors or bootstrapped errors (of course these have their own advantages and disadvantages as well). The above mentioned are just examples other ways of adjusting errors exist as well.