# 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...

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.