How to generate the Strong Parallel Trend Assumption presented in Callaway, Goodman-Bacon & Sant'Anna 2021

[Ben Hillier]

I'm running a DID model with multiple timer periods and continuous treatment. I currently run the traditional Parallel Trends Assumption presented in Callaway, Goodman-Bacon & Sant'Anna (2021):

E[Yt(0) − Yt−1(0)|D = d] = E[Yt(0) − Yt−1(0)|D = 0]

However, the paper explains how under continuous treatment, stronger parallel trend assumptions are needed. The "Strong Parallel Trends Assumption" is presented as:

E[Yt(d) − Yt−1(0)] = E[Yt(d) − Yt−1(0)|D = d]

Does anyone know if Stata currently has a command that can run this stronger test? The paper writes how "pre-trend tests commonly used to detect violations of parallel trends cannot distinguish between “standard” and “strong” parallel trends". Has this changed since the paper has been published?

Thanks in advance!

[Joseph Nwadiuko]

The last time I checked, currently there is no mathematical way to prove strong parallel trends. It has to do more or less with the assumptions that you are making regarding the relationship among the doses and their counterfactuals. Here's an example of one way strong parallel trends has been asserted:

Continuous treatment DiD - by scott cunningham (substack.com)

[Maxence Morlet]

In addition to Joseph's great response, you should give this paper a read: https://jonathandroth.github.io/asse...view_Paper.pdf