Synthetic Differences in Differences : When can we use it?

[Rajdeep Chaudhuri]

I have an individual level dataset, which has data across 3 years: 2 years prior to the treatment and 1 year after the treatment. However, the treatment occurred at the State level. I have around 30 states: 1 treated and 29 control. Since synthetic DiD requires panel data, I cannot proceed with State as the panel, since I have an individual level dataset. My question is, is there a workaround to this problem? I think that something like averaging on to the State does not work, because I have quite a few categorical and binary variables as my outcomes. So am I doomed, or is there any way I can account for pre-treatment divergence like we could using synthetic controls?

[Jeff Wooldridge]

Rajdeep: SDiD is not well suited for this setup because you have so few pre-treatment periods. Do you have repeated cross sections, so that you obtain a new sample of individuals in each period?

[George Ford]

Since you have individuals, you might think about something like this: The flow through of cost changes in competitive telecommunications: Theory and evidence, Empirical Economics, 30, 555–573 (2005), https://link.springer.com/article/10...181-004-0233-5. (When this paper was written, the procedure took days to finish. I wonder what it would take nowadays--a few minutes probably).

The paper uses a double bootstrap. Don't think I've seen the double bootstrap used for DID, but it seems like a decent approach to problems like yours. Some thought might be put into the fact you have one treated state, but my guess is this approach would be as suitable (if not better) than other approaches for one treated unit (all with their own problems) when you have longitudinal data and the treatment is at the state level.

Jeff has good ideas on these situations, so I'm staying tuned.

[Jeff Wooldridge]

There are two options that I see. Use the disaggregated data in a standard diff-in-diffs and compute heteroskedasticity-robust standard errors. The problem is that the standard errors reflect only sampling error, not the fact that the treatment is the same for all individuals in a state. Unfortunately, clustering at the state level. Instead, you can aggregate the data into an N = 30, T = 3 panel data set. Then, for each state, subtract the average outcome over the first two periods from the outcome in t = 3. Call this variable Ydot₃. Then use a cross-sectional regression of Ydot₃ on the treatment dummy, D. If normality and homoskedasticity hold, you can use this as having an exact t distribution with 28 degrees of freedom. This is actually the studentized residual from outlier analysis.

A student of mine, Soo Jeong Lee, and I just posted a working paper proposing this method. It draws on ideas from Donald and Lang (2007, REStat). We show how to transform the data so that serial correlation across time is automatically handled. When applied to the California smoking data popularized in the synthetic control literature, it works well.

After you collapse the data to state-level panel data, the code is simple. This assumes the years are adjacent. Otherwise, create a time variable that runs 1, 2, 3:

xtset state year
gen ydot_3 = y - (L.y + L2.y)/2 if year == 3
reg ydot_3 d

The above produces the usual DiD estimator and, assuming normality and homoskedasticity are reasonable, a valid t statistic and confidence interval. Note that any pattern of serial correlation is allowed, which is one of our innovations over Donald and Lang.

Here's the paper link: Lee and Wooldridge (2025)

[Rajdeep Chaudhuri]

Thanks Jeff and George for your feedback.

Do you have repeated cross sections, so that you obtain a new sample of individuals in each period?

Yes. This is a repeated cross section.
Also, if my outcomes are binary, does that affect the method posted in #4?

[Jeff Wooldridge]

In theory, yes, but you have little choice. Working in your favor is that the binary variable becomes a proportion when aggregated, and by the central limit theorem, should have an approximately normal distribution because there are many units per state. As I discuss in my 2003 AER Papers and Proceedings article, the heteroskedasticity due to different state sizes is safely ignored if you have many observations per state. Not also that the transformation in the Lee and Wooldridge paper means ydot_3 will take on negative and positive values, typically making normality a better approximation.

[Rajdeep Chaudhuri]

Thanks a lot Jeff for your comments. It has been quite a help.

[George Ford]

I've used this approach. Very simple and clean.

I'd appreciate Jeff's thoughts on this idea, based on the paper mentioned above. You're collapsing the data to a single observation (the mean) for a state. But, this is one value for the treated state has no variance (the variance of the cross section is not the one you want); in the paper, the statistic of interest was average revenue per minute for a firm. Some solutions to the one-treated unit focus on this issue. That one value will be different when the sample is different. By bootstrapping the individuals, you can get a distribution of the treated's mean, and the test will be non-parametric.