with Python
 Synthetic Difference in Differences
 twitter @asas_mimi 1

Table of Contents 1. Difference in Differences 2. Synthetic Control Method 3. Synthetic Difference in Differences 4. Python pkg : pysynthdid 2

Original Paper: Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. https://www.nber.org/system/files/working_papers/w25532/w25532.pdf 3

Difference in Differences (DID)
 4 pre post outcome If there had been no intervention, the outcome of the treatment group would probably have been here counterfactual treatment group control group pros Excellent interpretability cons Parallel trend assumption ● DID requires the selection of a control group that has parallel trends in the treatment group and the pre-intervention period. ● This selection may be arbitrary in some cases. e.t.c.

setup
 binary treatment observable outcomes T pre
 T post
 N tr
 N co
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DID as a two-way fixed effect regression
 Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. unit fixed effect time fixed effect 6

Synthetic Control Method (SC)
 ωSC vertical regression Estimating the ω that regresses the treatment group from the control group in the pre-term ωSC Synthetic Control counterfactual observable outcome ADH restrictions (Abadie, Diamond and Hainmueller (2010)) SUM(ω) = 1 , non-negativity, no intercept Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. 7

Synthetic Control Method (SC)
 SC regression formula : regression with time fixed effects and sc weights omit New!! Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. 8

Synthetic Difference in Differences (SDID)
 ωSDID vertical regression + horizontal regression - Estimating the ω that regresses the treatment group from the control group in the pre-term - The basic idea is the same as ADH SC. However, an intercept term and an L2 penalty term are introduced. λSDID non-negativity sum(λ)=1 with intercept L2 penalty Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. 9

Synthetic Difference in Differences (SDID)
 The method of estimating ω is slightly different from the classical one(ADH). non-negativity sum(λ)=1 with intercept L2 penalty Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. intercept “unexposed pre-trends perfectly match the exposed ones; rather, it is sufficient that the weights make the trends paralle” L2 penalty “to increase the dispersion, and ensure the uniqueness, of the weight” Even if we don't get a perfect match, we can correct it later with unit fixed effect (DID). 10

Synthetic Difference in Differences (SDID)
 ωSDID vertical regression + horizontal regression - Estimating the λ that regresses the post-term from the pre-term λSDID Objective variable: average outcome of post term non-negativity sum(λ)=1 with intercept Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. 11

Synthetic Difference in Differences (SDID)
 Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. 12

Synthetic Difference in Differences (SDID)
 pre post outcome treatment group synthetic control Synthetic Control Method ATTSC pre post outcome treatment group synthetic control Synthetic Difference in Differences 　 λ intercept ATTSDID Adjusting the reference point by λ 13

Reproduction experiment in Python
 My notebooks are here:
 https://github.com/MasaAsami/pysynthdid
 
 These implementations were based on the following R package:
 https://github.com/synth-inference/synthdid 
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Data
 The following section examines the Tobacco Tax and Health Protection Act of 1989 (California), famous dataset for Synthetic Control Method. see: 1988 California Proposition 99 ( https://en.wikipedia.org/wiki/1988_California_Proposition_99 ) 15

setup
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Setup & fitting
 1989 2000 1970 1988 California = treated Control (donor pool) 17

plot : sdid.plot(model=“hogehoge”)
 Difference in Differences Synthetic Control Method 18

plot : sdid.plot(model=“sdid”)
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Comparison of results
 pysynthdid result Arkhangelsky, Dmitry, et al. (2019) table 1: Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. 20

Comparison of ω
 Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. ω is able to confirm that the results were mostly consistent with the original paper. Also, it matches the results of the R package perfectly. typo?? 21

Comparison of λ
 Arkhangelsky, Dmitry, et al. Synthetic difference in differences. No. w25532. National Bureau of Economic Research, 2019. λ is almost identical to the original paper 22

Robust check for scale differences in donor pools
 In following note, we checked how the estimation results change with changes in the scale of the donor pool features. see: https://github.com/MasaAsami/pysynthdid/blob/main/notebook/ScaleTesting_of _DonorPools.ipynb Due to the L2 penalty term, mixing donor pools with extremely different orders may influence the results. In such a case, it may be better to consider logarithmic transformation, etc., as in a normal regression model.
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Thank you
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