# Difference-in-differences using two time series

I hope the group will be able to help on the following. I have the following policy-evaluation problem: stock exchange A reduced their trading costs after period X (say, after 2008), thus managing to spur their trading activity. Estimation of the actual effect is however impossible because of the lack of a counterfactual. Neighbouring exchange B, with similar characteristics and a similar trend in trading activity, does not apply any such or other confounding policies after the treatment period, thus representing a natural candidate as a control group.

While this setup seems perfect for a difference-in-differences estimation, I am left with an excruciating doubt: in both the treatment and the control groups I have a sample of 1. Can one use the argument that if the series are stationary and ergodic, one can estimate a Conditional Average Treatment Effect on the Treated (that is, a treatment effect on the sample of the treated, in this case exchange A?).

I haven't found similar applications, and perhaps for a reason...

Waiting for your expert comments!

• You might wish to read Abadie's work on synthetic controls. It's a 1 TS against 1 TS constructed from 'many' time series, but may interest you. He's been doing it for about two decades, but just published a retrospective of the method: Abadie, A. (2021). Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects. Journal of Economic Literature, 59(2), 391–425. Commented Oct 20, 2021 at 1:41
• Have you considered doing something like changepoint detection on the difference in trading activity between exchanges to test if there is a significant change around the introduction of the policy. Commented Jun 22, 2022 at 23:29

## 1 Answer

If you have a single treatment group $$(i=1)$$ and a single control group $$(i=2)$$ for many time periods before and after treatment it is perfectly acceptable to use a diff-in-diff methodology. (So on net, you'd need: $$T >= 2, I >=2, n >=4$$ ) Additional control groups will help with the robustness and persuasiveness but are not required.

However, if you have only a single sample (n=1), then you cannot reasonably do a diff-in-diff examination, and I think that should be very clear.

Plug: You may want to consider the Economics StackExchange for Diff-in-Diff approaches, it is pretty discipline-specific.

• SE standing for? Commented Dec 10, 2018 at 16:58
• It is clear, to me. What I was asking was slightly different, that is: if one has a long enough time series, can one exploit time series property (such as ergodicity, that is the fact that the mean over time would be equal to the cross-sectional mean of all possible unrealised alternatives) to run the evaluation? Commented Dec 10, 2018 at 17:09
• As I haven't found such applications, i believe the answer would be "no", but tried to throw a pebble in the pond. Commented Dec 10, 2018 at 17:09