Synthetic DiD in a 2x2 setting

[Rajdeep Chaudhuri]

Hi,
I am trying to estimate the impact of a policy on certain outcomes. I have one pre-treatment and one post-treatment time period, and one treated unit and 20 control units. Since I do not have more than one pre-treatment units, I cannot check for parallel trends, so I am thinking of trying to use a synthetic DiD. But on reading the reference for the sdid package here , I find that it says

For estimation to proceed, we require at least two pre-treatment periods off of which to determine control units

So does that mean I cannot proceed with the canonical 2x2 setting in the synthetic DiD?

If I can, then the sdid command requires a time variable and a treatment variable. Does it allow both of them to be the same, since in a 2X2 setting they will be the same.

Thanks a lot

[Jeff Wooldridge]

Rajdeep: SDiD was never intended to apply to such cases. It relies on have a sufficient number of pre-treatment periods, both for (effectively) finding a suitable synthetic control. Moreover, inference is based on the assumption that you have many pre- and post-time periods. Having said that, I believe that if you apply sdid in the 2 x 2 setting it will produce the usual did estimator -- which is all you can do in your case. With one treated unit, you're relying on exact inference by regressing the change on the outcome on the treatment dummy. As I discuss in my recent work with Soo Jeong Lee -- almost ready to put in working paper form -- this is identical to obtaining the studentized residual in testing for outliers, and so you have to assume normality and homoskedasticity for this strategy to work.

sort id year
reg c.y treat

If you have a small number of controls, you can add those in the hope of helping with any violations of parallel trends.

[Jeff Wooldridge]

Sorry, that should be d.y, not c.y.

[Rajdeep Chaudhuri]

Thanks Jeff for the answer.
Just asking. in such a setting where I cannot test for parallel trends, how exactly do I go about justifying the DiD coefficient as a causal effect of the policy? Is it possible to do so in such a setting?
Also related to this, if y is a binary variable, your code, I think, will fail to work since the d.y is going to difference between 0 and 1. Is there a way to do what you suggest for binary variables?