Inference Procedures

[azev77]

Placebo methods for SE (use MSPE of Placebo units/time to estimate SE)
(3 types from Douchenko, Imbens):

• Placebo unit (Random Assignment of the Unit)
• Placebo time (Random Selection of the Treatment Period)
• Placebo unit & time

In-place placebo (permutation) tests for p-values: use the placebo treatment effects to form an empirical placebo distribution and then use that empirical distribution to compute p-values.
(Note: this method gives p-values. P-values can be inverted following Chernozhukov et al to get CIs.)
This is the approach from ADH, and Galiani & Quistorff (Synth_Runner).

• Re-estimate SCM for all donors to get ATET_{t,j=1, ..., J}, this is the empirical placebo distribution for the treated unit ATET_{TR, t}.
  The two-sided p-value is:
  The one-sided p-value (for positive effects):
  Can assess significant of joint post-treatment effects similarly (comparing treated w/ placebo).
  Post-treatment differences may be large just because of poor match quality

Other inference methods:

• Jackknife+, Predictive Inference with the Jackknife+
• Jackknife (same paper as Jackknife+)
• Jackknife Naive (in-sample pre-treatment errors, same paper as Jackknife+)
• Jackknife-minmax (same paper as Jackknife+)
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• Conformal inference Moving Block @VC2015 @kwuthrich: paper, code, Github: scinference
• Conformal inference IID @VC2015 @kwuthrich: paper, code, Github: scinference
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• Practical and robust t-test @VC2015: paper, Github: scinference
  Need long Post-treatment Pd, only for Avg post effect (not ATT_t),
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• CFT prediction intervals using non-asymptotic bounds, @mdcattaneo, Feng, @titiunik (2019): Paper, Code
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• Synthetic Learner @dviviano: paper, replication code
• Kathleen Li: paper JASA 2020, no code
• GSynth has: Parametric bootstrap, Nonparametric bootstrap, Jackknife
• AugSynth has: Conformal/Jackknife/Jackknife+
• SCUL: nice Placebo inference figures
• Non-asymptotic Randomization Tests with Staggered Adoption, Shaikh and Toulis (2020): Paper