# Differences in Differences with Small Violations of Parallel Trends

I am interested (for no particular reason) in estimating a hypothetical Differences-in-Differences model with one period of treatment. However, we observe small non-linear violations of the parallel trends assumption.

For instance, consider wages for IN vs. OH before and after IN implements a new tax policy. We see that prior wages in IN are equal to wages in OH for T = $$-\infty, ..., -3,-2,-1$$. However, we see that prior wages are $$\epsilon > 0$$ higher in OH over the period $$T = -5-k, -5$$, with $$k$$ some small-ish natural number. Here, parallel trends does not hold, but we might still expect to be able to observe the impacts of a change in tax policy.

Is there a way to still properly estimate a Differences in Difference estimator under these situations? I imagine that we should see the same point estimate for treatment at time $$T = 0$$, with a penalty term being added to the standard errors of the point estimate which is a function of $$\epsilon$$. That said, I have been unable to find any papers which cover this issue.

Related papers include: https://jonathandroth.github.io/assets/files/HonestParallelTrends_Main.pdf

However, as far as I can tell, that paper deals with differences in trends after treatment, while I am interested in differences in trends prior to treatment.

• Very helpful, thank you -- I think the first option is closer to what I am aiming at. Is there a way to get an upper bound on the bias if we observe a stochastic difference in trends? Something like, for $t < 0$ (i.e. pre-treatment) and $\sigma^2$ small: $$Y^1_t - Y^0_t = u_t \sim N(0,\sigma^2)$$ Where $Y^1_t$ is the treated group and $Y^0_t$ untreated? I'm somewhat interested in cases of papers where the previous trends look fairly parallel, but clearly differ in small ways. Thanks again! Nov 3, 2022 at 1:45