Potentially non random assignment to treatment in DiD settings

[Mansour Mohseni]

Hi
I have two questions regarding the design of the DiD setting. I appreciate your guidance.

1. In a setting where treated subjects chose to be treated (such as being in a policy or making a decision), if we could construct a suitable control group or use a flexible DID model that helps us to create a pre-parallel trend, is the estimation of ATT credible?

2. In a setting where an agency makes some subjects treated (like a regulatory policy), but the treated subjects may know they are going to be treated or the treatment is not pure exogenous, while the assumption of pre-parallel trends holds (using a suitable control), is the estimation of ATT credible?

[Jeff Wooldridge]

The second situation is typically called a violation of "no anticipation" because the units under consideration may change behavior in expectation of being put in the treated group, say. This is difficult to deal with. Sometimes a gap is left between the first control period and the first treated period to reduce the possibility of anticipatory behavior. It's fairly easy to test with enough pre-treatment periods: in a flexible regression equation estimated by TWFE, put in next periods time-varying treatment indicator and do a t test.

The DiD setting allows selection into treatment, but it's limited to differences in levels -- that is the import of the parallel trends assumption. That, saw, low earning workers are more likely to be choice for a job training program is allowed with DiD. Or high unemployment census tracts are selection to become enterprise or opportunity zones. But what's not allowed if if selection is based on the trajectory of Y_t(0) prior to the intervention. So selection as an EZ can't be correlated with the trend in unemployment even though it can be correlated with the level. Of course, the PT assumption can be tested, to a degree, by having enough pre-treatment periods.

DiD certainly doesn't require "purely exogenous" treatment assignment. It wouldn't be very useful if it did.

[Mansour Mohseni]

The second situation is typically called a violation of "no anticipation" because the units under consideration may change behavior in expectation of being put in the treated group, say. This is difficult to deal with. Sometimes a gap is left between the first control period and the first treated period to reduce the possibility of anticipatory behavior. It's fairly easy to test with enough pre-treatment periods: in a flexible regression equation estimated by TWFE, put in next periods time-varying treatment indicator and do a t test.

The DiD setting allows selection into treatment, but it's limited to differences in levels -- that is the import of the parallel trends assumption. That, saw, low earning workers are more likely to be choice for a job training program is allowed with DiD. Or high unemployment census tracts are selection to become enterprise or opportunity zones. But what's not allowed if if selection is based on the trajectory of Y_t(0) prior to the intervention. So selection as an EZ can't be correlated with the trend in unemployment even though it can be correlated with the level. Of course, the PT assumption can be tested, to a degree, by having enough pre-treatment periods.

DiD certainly doesn't require "purely exogenous" treatment assignment. It wouldn't be very useful if it did.

Dear Professor Wooldridge
Thank you so much for your time and response.

Regarding the first setting, in a context where subjects such as firms are not sure about the potential consequences of making a decision, like the financial outcome, can we consider the decision as some kind of treatment and create a suitable control group (assuming PT) for firms that made the decision, and estimate the impact of making that desion using DiD?

[Mansour Mohseni]

The second situation is typically called a violation of "no anticipation" because the units under consideration may change behavior in expectation of being put in the treated group, say. This is difficult to deal with. Sometimes a gap is left between the first control period and the first treated period to reduce the possibility of anticipatory behavior. It's fairly easy to test with enough pre-treatment periods: in a flexible regression equation estimated by TWFE, put in next periods time-varying treatment indicator and do a t test.

The DiD setting allows selection into treatment, but it's limited to differences in levels -- that is the import of the parallel trends assumption. That, saw, low earning workers are more likely to be choice for a job training program is allowed with DiD. Or high unemployment census tracts are selection to become enterprise or opportunity zones. But what's not allowed if if selection is based on the trajectory of Y_t(0) prior to the intervention. So selection as an EZ can't be correlated with the trend in unemployment even though it can be correlated with the level. Of course, the PT assumption can be tested, to a degree, by having enough pre-treatment periods.

DiD certainly doesn't require "purely exogenous" treatment assignment. It wouldn't be very useful if it did.

Dear Professor Wooldridge

I was wondering whether using methods such as synthetic DiD could allow us to study settings as you mentioned; more specifically, if the selection is based on the trend, does creating a synthetic control, ensuring the PT, lead to a correct estimation of ATT?