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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?

Any suggestions?

Thanks!

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The purpose of the parallel trends assumption is that it allows you to use changes in the control group to estimate what changes in the treatment group would have been in the counterfactual case where treatment had not occurred (and it is counterfactual - you can't observe it or generally test for it directly since it's an assumption about a counterfactual; I suspect here you mean you're testing for prior trends as a check on the plausibility of parallel trends).

In this case, the trend you're describing shows that the treated group could not have taken the counterfactual path implied by the parallel trends assumption, so you're right that it cannot hold.

Or at least it cannot hold in the level of the proportion. Parallel trends is partially a functional form assumption - if PT holds for the outcome, it won't hold for the log of the outcome, and vice versa, for example. The assumption also means different things depending on the linearity of the model - logit, LPM, etc. I'm framing this all to explain why it's even possible to rescue this design when PT fails so clearly in your original tests.

So, in this case, an intuitive first step is to use a form that would allow your treatment group proportion to drop without becoming negative (and to evaluate the plausibility of PT in the alternative PT form implied by that structure). This working paper suggests that fractional logit does work for DID. Be careful to read through what they say about what parallel trends actually means in this context (as it will change how you'd check for it in prior trends) and how to interpret the regression results to get the DID estimate.

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