² Under the DGP specified in Equation (2), the parallel trends assumption is equivalent to \((\beta - 1)\{\mathbb {E}[P \mid A = 1] - \mathbb {E}[P \mid A = 0]\} = 0\). This equality holds if either \(\beta = 1\) or \(\mathbb {E}[P \mid A = 1] = \mathbb {E}[P \mid A = 0]\). Here, under the stipulated DGP, \(\beta = 1\) means that a one-point increase in \(P\) shifts the conditional mean of the untreated potential posttest outcome \(Y(0)\) by one point, so that the between-group mean difference in \(Y(0)\) equals the between-group mean difference observed at pretest. Alternatively, the parallel trends assumption holds when \(\mathbb {E}[P \mid A = 1] = E[P \mid A = 0]\). This condition is satisfied, for example, when treatment assignment is independent of \(P\), as in RCTs. We emphasize that these conditions are not additional identifying assumptions but rather the model-specific algebraic form of the parallel trends assumption under the stipulated DGP; they should not be interpreted as implying either that DiD is generally appropriate whenever \(P \to A\) or that \(\beta = 1\) is generally plausible in nonrandomized settings.