difference-in-difference with single treated unit

[Liz Gooch]

Hi - I am investigating a quantitative problem with one treated unit and a single time for the treatment onset and around 21 untreated units using the synthetic control method. I was wondering if someone could shed some light on whether there are any bias or inference issues with applying a difference-in-difference methodology to this situation? I am particularly interested in knowing if methodological problems/challenges may arise from the singularity of the treated unit.

Appreciatively,
Liz

[George Ford]

synthetic is the way to go. lots of issues with a single treated unit in more traditional regression based approaches.

[Jeff Wooldridge]

Unfortunately, the validity of SC requires many post-treatment periods -- otherwise the inference is not valid. I've proposed a different approach based on the representation of DiD as a cross-sectional regression. You average the pre-treatment periods and subtract, for each unit (treated or otherwise), from the average of the post-treatment period. In this case, the single time period. Now you have a single cross section, and you can regress the differenced variable on a constant and the treatment dummy. If you believe the classical linear model assumptions hold for this cross-sectional equation, you can use the usual t statistic. With a single treated unit, this is identical to "outlier" analysis, which makes sense: you are testing whether the treated unit is an outlier compare with the controls. The coefficient on the treatment dummy is the Studentized residual from outlier analysis. I discuss this in Section 9.5 of my introductory econometrics book. In case it is useful, I'm attaching slides from a workshop I've taught. In Section 5, I show how to collapse the data and do the simple OLS regression. With 21 control units, you could even include some covariates. I have the Stata code, too, but it's pretty simple.

[George Ford]

I was thinking "single time for the treatment onset" meant a non-staggered treatment.

Looking forward to reviewing the slides.

.

[Jeff Wooldridge]

Definitely not staggered. Things are even easier in the common timing case, which is Liz’s setting.

[George Ford]

Liz, to clear things up, does "single time for the treatment onset" mean 1 period during treated period or just one treatment?

[Jeff Wooldridge]

Ah, I see how one could interpret what Liz described as common timing with many treated periods and not just a single treated period. The method I described still works, but SDID also would work unless the number of treated periods is small.