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I understand the unconfoundedness (selection on observables) assumption to be that the expected value of an outcome is independent of treatment after controlling for observed regressors (since these may collectively explain selection into treatment): $\mathbb{E}[y |D,x]=\mathbb{E}[y|x]$.

It seems like the assumption of parallel trends is saying a similar thing: that all individuals have the same 'latent' characteristics over time regardless of whether they were treated or not.

Not sure if I'm misunderstanding here, but what is the difference between the two? When would you invoke one assumption over the other in a treatment effects study?

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You're pretty much correct - the two assumptions are very similar.

The only difference is that the Parallel Trends assumption allows for differences between the treatment group and the control group (conditional on controls) so long as these differences are stable over time.

Assuming a binary treatment, only two periods, and abstracting from control variables, we can write this using potential outcomes notation as: $$E[Y_{it}(0)|D_i,T_t]=\alpha+\phi D_i +\delta T_t$$

That is, there cannot be an interaction between the treatment group assignment and time, absent treatment.

This allows for a level difference between the treatment and control groups, which is less restrictive than the Selection-on-Observables assumption. Consequently, settings in which there are unobservable factors that influence the outcome variable, but these unobservable characteristics can (conditional on your controls) be expected to be constant, are candidates for Difference-in-Difference designs, but not selection-on-observables.

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    $\begingroup$ Good answer +1. This answer could be improved by explicitly stating the assumption with the controls included although I think the more simple case here is instructive as well. Also, it would be nice to know what $T_t$ is? If it is simply a time-variable measuring time in some unit it would seem the trend here is assumed linear (obviously this is of no consequence if it is assumed that there are only two time periods but this is an unnecessary restrictive assumption - the parallel trend assumption is often assumed in research designs with several pre- and post-treatment periods). $\endgroup$ Commented May 21 at 20:55
  • $\begingroup$ Both fair comments. Would it be enough to write: $$E[Y_{it}(0)|D_i] = \alpha + \delta_t+ \phi D_i$$ Where I more explicitly define $D_i$ to be the treatment group. $\endgroup$ Commented May 22 at 6:45

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