I have a very basic question regarding a DiD-Design. If the subjects in this design select themselves into treatment/non-treatment group because they anticipate the treatment (e.g., they want to avoid/be part of a policy change), the DiD won't work (i.e., "random" assignment with respect to the treatment is required) - otherwise, the parallel trends assumption would be violated, or? I read a couple of textbooks, but somehow I was not able to get a clear answer.
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The non-randomness of assignment is not a problem per se (it's part of the design) - it depends on whether you think that the self-assignment choice is likely to be predictive of different time-courses of change independent of treatment. In other words, if you withheld treatment from the the people who elected to be in the treatment group, would you expect to see their results (in terms of change) match those of the self-selected non-treatment group. If not - then you've violated the expectation of parallel trends.
