Calculating the percentage effect of a treatment using synthetic control method

[Joe Zonda]

Hello,
I am using the Synthetic Control Method (synth) developed by Abadie Aberto et. al. I would like to express the effect of the policy as a percentage.
The following is my data:

* Example generated by -dataex-. To install: ssc install dataex
clear
input byte index str3 isocode int year float prgdp
1 "AUS" 2000  41.45803
1 "AUS" 2001   42.6604
1 "AUS" 2002  43.46615
1 "AUS" 2003  44.71541
1 "AUS" 2004  45.55513
1 "AUS" 2005  46.13455
1 "AUS" 2006   47.0964
1 "AUS" 2007   47.9088
1 "AUS" 2008  47.88457
1 "AUS" 2009  47.93401
1 "AUS" 2010  48.21926
1 "AUS" 2011  49.25634
1 "AUS" 2012  49.72223
1 "AUS" 2013  50.21233
1 "AUS" 2014  50.57065
1 "AUS" 2015  51.24196
1 "AUS" 2016   51.7074
1 "AUS" 2017  52.53539
1 "AUS" 2018   52.9948
1 "AUS" 2019  52.20425
2 "KOR" 2000  22.61658
2 "KOR" 2001 23.551487
2 "KOR" 2002  25.21583
2 "KOR" 2003 25.868595
2 "KOR" 2004  27.08259
2 "KOR" 2005  28.12902
2 "KOR" 2006  29.50116
2 "KOR" 2007  31.11384
2 "KOR" 2008  31.95508
2 "KOR" 2009  32.10069
2 "KOR" 2010  34.14795
2 "KOR" 2011 35.235477
2 "KOR" 2012 35.884182
2 "KOR" 2013 36.810196
2 "KOR" 2014 37.792206
2 "KOR" 2015  38.68932
2 "KOR" 2016  39.70417
2 "KOR" 2017  40.86813
2 "KOR" 2018  41.99446
2 "KOR" 2019  42.80603
3 "CAN" 2000  42.17812
3 "CAN" 2001  42.52756
3 "CAN" 2002  43.39205
3 "CAN" 2003  43.73918
3 "CAN" 2004  44.62534
3 "CAN" 2005  45.55576
3 "CAN" 2006   46.2203
3 "CAN" 2007  46.61373
3 "CAN" 2008  46.50728
3 "CAN" 2009  44.60006
3 "CAN" 2010   45.4376
3 "CAN" 2011  46.33575
3 "CAN" 2012  46.63475
3 "CAN" 2013  47.21461
3 "CAN" 2014  48.06879
3 "CAN" 2015    47.899
3 "CAN" 2016  47.90493
3 "CAN" 2017  48.89198
3 "CAN" 2018  49.61724
3 "CAN" 2019  50.08624
4 "ISR" 2000 30.907154
4 "ISR" 2001  30.37002
4 "ISR" 2002  29.78265
4 "ISR" 2003  29.58852
4 "ISR" 2004  30.46429
4 "ISR" 2005  31.08056
4 "ISR" 2006 32.132683
4 "ISR" 2007 33.252167
4 "ISR" 2008 33.510857
4 "ISR" 2009  33.05777
4 "ISR" 2010 34.158024
4 "ISR" 2011  35.15149
4 "ISR" 2012 35.380398
4 "ISR" 2013  36.31908
4 "ISR" 2014 37.118385
4 "ISR" 2015  37.38509
4 "ISR" 2016  38.24815
4 "ISR" 2017  38.96732
4 "ISR" 2018  39.64619
4 "ISR" 2019  40.37104
5 "CHL" 2000 14.743486
5 "CHL" 2001 15.070696
5 "CHL" 2002 15.234614
5 "CHL" 2003   15.6445
5 "CHL" 2004 16.569315
5 "CHL" 2005  17.41039
5 "CHL" 2006 18.315748
5 "CHL" 2007   19.0101
5 "CHL" 2008 19.471315
5 "CHL" 2009 18.964655
5 "CHL" 2010  19.86565
5 "CHL" 2011 20.870377
5 "CHL" 2012  21.76971
5 "CHL" 2013  22.42965
5 "CHL" 2014  22.58499
5 "CHL" 2015  22.83477
5 "CHL" 2016 22.919737
5 "CHL" 2017  22.86397
5 "CHL" 2018 23.438625
5 "CHL" 2019 23.407227
end

synth prgdp prgdp(2000) prgdp(2002) prgdp(2005) prgdp(2010) prgdp(2012), trunit(5) trperiod(2014) fig keep(trial_prgdp, replace)

I am aware that to estimate the actual (absolute) treatment effect, we do:

use trial_prgdp.dta, clear

gen effect= _Y_treated - _Y_synthetic

However, I am not sure whether divide the "effect" by the _Y_synthetic as below:

gen percentage_change = 100*(effect/_Y_synthetic)

Or perhaps divide the effect by the _Y_treated as below:

gen percentage_change = 100*(effect/_Y_treated)

Please help clarify lest I mess up.

[George Ford]

How do you want to describe the result? The treatment raised the outcome by ___% over the counterfactual. (divide by Ysynth). Or, absent the treatment, the outcome would have been __% lower than its observed value? (divide by Ytreated). Also, divide by the average of the two as an avoidance strategy. Just be clear what you did and pick one, and state both values in the paper so the reader can make his own calculation if they don't like your choice.

Also, the order of the calculation tends to lean you one way or the other since it establishes the sign of the difference.

[Joe Zonda]

George Ford thank you very much for your clear explanation.
This has helped.