Difference in Difference with flexible parallel assumptions

[Philip Gigliotti]

cross posted from stackexchange and econjobrumors

https://stats.stackexchange.com/ques...wth-assumption

https://www.econjobrumors.com/topic/...-anything-goes

I've been very interested in the argument in this highly cited paper by mora and reggio 2012.

They argue that by estimating a difference-in-differences model with unit-specific linear time trends, you remove the necessity of parallel trends assumption. Instead the identifying assumption is parallel growths, which means the second derivative must be similar, instead of the first.

Here's the article:

https://e-archivo.uc3m.es/bitstream/...pdf?sequence=1

Here are slides from a lecture on the topic:

http://www.stata.com/meeting/spain12...ora_Reggio.pdf

In the lecture they say you can now identify treatment effects even if there are differing pre or post treatment trends.

What does this mean in practice? Does it mean anything goes and you can now estimate a diff in diff with no consideration of trends? What's the trade off?

Specifically I'm struggling to understand what parallel growths would mean and why it is a weaker assumption. If the first derivative (velocity) is different why and when would the second derivative (acceleration) be the same? Is it more plausible that the second would be true than the first?

[Philip Gigliotti]

So I figured it out and I'd like to thank one of the authors Ricardo mora for his assistance in explaining.

The difference in difference model requires that the dependent variable have the same trend in in both control and treatment groups in the absence of treatment. This is usually tested by checking for the same trends in the pre treatment period.

In order for the parallel trends assumption to hold, the slope of the trend lines for the two groups, or the first derivative must be similar.

You can relax this assumption by controlling for group specific or panel specific linear time trends. This controls for the first derivative. Now the group trends can have different slopes, but they must be linear trends so that the second derivative is zero. This is a weaker assumption, the trends must not have the same slope they must just be linear or similar in their non linearities. This is likely to be true unless there was a significant shock that impacted one group and not the other.

The authors extend this to a family of parallel assumptions. If you control for group or panel specific quadratic trends, the first and second derivatives can be different, and now only the third derivative must be the same. In the case of a quadratic or linear trend in both groups, the third derivative would be zero and the assumption would hold. Adding cubic trends extends the logic one degree further and so on and so forth.

This paper is a huge contribution for causal inference techniques in formalizing something that people have been doing without solid theory for many years. It was recently accepted for publication in econometrics reviews, a top 5 econometrics journal.

[Jesse Wursten]

I think one caveat is that the trends must be consistently estimated and that you potentially add bias by mis-estimating these trends. Imagine you only have a short pre-treatment period. If you use this short timespan to estimate linear, let alone higher order trends, then this might lead to very misleading results. In the end diff-in-diff is just a comparison of four expected means (pre/after, control/treated). Incorrectly factoring in trends can have a large impact of the values of these means.

I'm personally more a fan of using synthetic control group methods, which use a data driven approach to generate an artificial control group out of multiple potential "donor" control groups. It is also very fragile with short pretreatment periods though.