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I'm aware of the basics of the synthetic control method, and that permutation tests are commonly used for inference.

Are there papers that develop asymptotic theory? i.e. if the regression approach to create weights is used (as in the synth command), and the number of control groups and pre-treatment time periods goes to infinity, are there papers proving consistency or asymptotic normality?

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There is some work on the asymptotic consistency like the Robust Synthetic Control by Amjad et al (2018), also the study of Firpo and Possebom 2016 discusses asymptotic inference.

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  • $\begingroup$ Yeah. Thanks for this. It looks like there isn't anything of the "Consistent and Asymptotically Normal" result as GMM or MLE do. I guess also asymptotics aren't really relevant because there are often few controls and time periods. Thanks anyway! $\endgroup$
    – Michael Gmeiner
    Commented Aug 16, 2021 at 20:47
  • $\begingroup$ @MichaelGmeiner oh I thought you will be happy with just some asymptotic inference, as you say it is very unlikely it would be possible to have asymptotic normality with this sort of model $\endgroup$
    – 1muflon1 ♦
    Commented Aug 16, 2021 at 20:56
  • $\begingroup$ Oh, I'm happy you sent these! It's good to be aware that there is some work on the subject. $\endgroup$
    – Michael Gmeiner
    Commented Aug 16, 2021 at 22:17
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    $\begingroup$ Also @1muflon1 I don't know if you care, but I found another. tandfonline.com/doi/full/10.1080/… $\endgroup$
    – Michael Gmeiner
    Commented Aug 17, 2021 at 13:32

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