# Parallel trend assumption in diff-in-diff and fixed effects models

What happens if the parallel trend assumption between control and treatment groups (diff-in-diff) or between individuals (individual-level fixed effects model) is not fulfilled?

Andrew Gelman, a prominent Professor of Statistics and Political Science has a recent post that you may find useful.

But, first, I recommend you examine his earlier post from a few years back about how Difference-in-difference estimators are a special case of lagged regression.

He makes the argument that:

[Difference in Difference] just a special case of lagged regression where the lag is restricted to have a coefficient of 1. In educational research, this is sometimes called the analysis of “gain scores.”

NOTE: As an aside, he suggests that lagged-regressions may "just be better than difference-in-difference," since DiD may "limit your statistical efficiency and range of applicability,":

That said, as Gelman acknowledges, difference-in-difference models can be useful when you have "a model with error terms for individual units" since "differencing makes the error terms drop out" and thus can give a "cleaner estimator."_

This aside should be read in the context of the question you asked - namely, how to respond when the assumptions underlying Diff-in-Diff are not satisfied.

For this, please read the aside above n the context of the follow-up post update this May by Gelman which quotes a paper addressing your concern:

Angrist and Pischke (2009) show that difference-in-differences and the lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming the parallel trend will overestimate the effect; in contrast, if the parallel trend is correct, then mistakenly assuming ignorability will underestimate the effect.

So, as they say: "mistakenly assuming the parallel trend will overestimate the effect" - and, in such cases, the lagged-dependent variable version of the DiD regression model may be preferable. It is recommended you test your data for these assumptions and select between these two options accordingly.

• I've always been a bit confused about the parallel trends assumption in DiD analysis. Can't we use interaction terms to account for varying trends in the control and treatment groups and still obtain an unbiased estimate of the treatment effect? Feb 28 at 19:44