I am trying to test for parallel trends when using diff-in-diffs, where my outcome variable is nonnegative (basically a percentage -- the share of subjects with some property). I run into a simple-looking problem which I never saw addressed in the standard literature.
Suppose that for all $t < T$, the value of the outcome variable for the untreated group is $x_t$ and for the treated group it is $x_t-\delta$. Then, for $t=T$, it is $x_T$ for the untreated group, but it cannot be $x_T-\delta$ for the treated group, because this value is negative. So in practice the difference between the values at $t=T$ is less than $\delta$.
Strictly speaking, this is a rejection of parallel trends, but are there ways around it, given that the assumption is only in one data point, where there is a strict limit on the value?