Synthetic Control: Math Explained

This is a follow up video to my explanation of synthetic control. I focus on the math behind the method.
I use the following paper for the explanation:
Andersson, Julius J. "Carbon taxes and CO 2 emissions: Sweden as a case study." American Economic Journal: Economic Policy 11.4 (2019): 1-30.
If you are more curious about the method, check out:
Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. "Synth: An R package for synthetic control methods in comparative case studies." Journal of Statistical Software 42.13 (2011).
Disclaimer: If you use R, always use the improveSynth function of the following package: cran.r-project...
The algorithm of Abadie's synth package seldom converges!!!

[@redhoJAMBI]

nice and clear explanation, hope you make some tutorial on how to use it in R

[@Alex-ko1ci]

Have you already found a relevant source that explains how to use it in R?

[@disintegrators6940]

​@@Alex-ko1cihave you?

[@Mo-cr7ry]

@@disintegrators6940 have you?

[@anatsintsadze21]

@@disintegrators6940have you found it by any chance?😊

[@user-wr4yl7tx3w]

If you have a simple example illustrating the exact steps in calculation, that would be really helpful. May be an idea for a video.

[@zmandrik7]

Great video. Overarching concept and math was very easy to follow! Thanks

[@viscaernesto]

Very clear! Thank you very much!

[@AnlMeydan]

Hello, thank you for the great video. I'm curious about this; if the intervention group consists of a large number of people and it's difficult to regulate the control group, can we still apply the synthetic control method? Or is there a possibility of obtaining unhealthy results?

[@FinAndEcon]

Yes you can, you would just make a separate synthetic control for every person in the intervention group,

[@xuyang2776]

Thank you very much. It's really clear. But I have a question. The initial value of matrix V is randomly set, right? Then we can get a vector W*(V). Taking it back to (y1- Wy0)^2, we can get the value of the difference. At second round, we set another V, and repeat the above procedure and get a new value of difference....After several iteration, we can choose the best V which corresponding to the smallest value of differenc. My question is that is there any algorithm in searching V ? I guess the value of V will not be randomly selected at very round.Thanks again.

[@kristianwichmann9996]

The idea is to iterate on V as well. So Both V and W are tweaked in each iteration, eventually reaching equilibrium (which should exist and be unique since this afaics is a convex optimization problem)

[@xuyang2776]

Thanks, but I want to know in the process of reaching equilibrium, how to searching value of matrix V at very round is the best? Or which way is the fast in the process of reaching equilibrium.
@@kristianwichmann9996

[@axe863]

@@kristianwichmann9996 This is a great idea but it lacks even moderate rigor. I can probably easily incorporate nonsense that shares the same trend dynamics and it would be included.... Maybe add false positive controls like knockoff generation; bilevel controls; residual analysis (residual non-stationarity; clustering in the extremes; residual features etc) etc etc. This is basically a simplistic sparse tracking with a different application.

[@user-wr4yl7tx3w]

But isn’t the counterfactual only as good as your construction of your model of synthetic control?

[@FinAndEcon]

Absolutely.