¹ It should be noted that Equation (2) imposes no restriction on the relationship between \(A\) and \(P\) and therefore permits treatment assignment to depend on the pretest, including through \(P \to A\). From an identification perspective, DiD does not require the absence of \(P \to A\) as a separate condition; rather, it requires the parallel trends assumption to hold under the actual treatment assignment mechanism. Thus, the parallel trends assumption can, in principle, hold even when \(P \to A\) (see Footnote 2 for the corresponding mathematical conditions). In practice, however, such dependence may make the assumption less plausible and more difficult to assess (Kim & Steiner, 2021).