I am not sure your understanding of positivity is right: It is simply a matter of whether there exists any stratum where nobody is treated (or nobody is untreated).
The distinction between “random” and “structural” violations was inherited from a course that did not insist as strongly on distinguishing between statistics and causal inference. I think the necessary assumption for identification is structural positivity, and that “random violation of positivity” is simply an apparent violation of the assumption due to sampling. I will update the text to make this clear.
I’ll keep thinking about your question 1. Possibly there are other people who would be better suited to answer it than me..
Thank you!
I am not sure your understanding of positivity is right: It is simply a matter of whether there exists any stratum where nobody is treated (or nobody is untreated).
The distinction between “random” and “structural” violations was inherited from a course that did not insist as strongly on distinguishing between statistics and causal inference. I think the necessary assumption for identification is structural positivity, and that “random violation of positivity” is simply an apparent violation of the assumption due to sampling. I will update the text to make this clear.
I’ll keep thinking about your question 1. Possibly there are other people who would be better suited to answer it than me..