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!