Synthetic Control with Nelder-Mead optimization technique

[Gabriele Bertuzzi]

I am trying to replicate the results of this paper (https://www.aeaweb.org/articles?id=10.1257/pol.20170144) with Stata 17. I am working with a panel dataset, I have yearly observations of CO2 emissions for 15 countries starting from 1960 to 2005, as well as other variables (GDP per capita, gas consumption per capita, etc.). Here's an example:

clear
input byte Countryno str13 country int year double(CO2_transport_capita GDP_per_capita gas_cons_capita vehicles_capita urban_pop pop_density)
1 "Australia" 1960  2.092156679764865 12290.4008293509 399.45596141557076 265.71014828338605 81.529   1.33768220454812
1 "Australia" 1961  2.047123914909854 12370.5105471806 407.42153963560054  275.6872237282142 81.941 1.3645652994545905
1 "Australia" 1962  2.094582014522435 12828.9886236666  426.8292682926829 284.27369858994746 82.337 1.3982791611887064
1 "Australia" 1963 2.2264840182648404 13668.1783745589  448.7671232876712 322.27395014835514 82.727 1.4253543860562592
1 "Australia" 1964 2.2906778902122324 13973.7438509531  476.4036894421062  334.7090969445533  83.11 1.4536011350767348
1 "Australia" 1965  2.355988760098349 13947.1397971169    495.69722514928 347.31345144717363 83.485 1.4823685614985096
1 "Australia" 1966  2.357737533258948 14588.7975484633  503.3902669298772   357.810391894425 83.855 1.5166031006339247
1 "Australia" 1967 2.4637681159420293 14907.2601756665   524.281718789728 351.07629388558945 84.217 1.5358681644819911
1 "Australia" 1968   2.58306270297277  15775.699805738  545.1744524939628 363.57207412520023 84.573 1.5632037280501934
1 "Australia" 1969  2.716301068254098 16599.5638645924  571.1489847508766   377.308371394745 84.922 1.5962667430326856
1 "Australia" 1970  2.569760933877029 16750.8665883324  593.3477252738467  389.2886213755416 85.265   1.62802806451193
1 "Australia" 1971 2.5531421504212726 16967.5687971994  603.3856380922934 389.48226699169226   85.6 1.6840008851515822
1 "Australia" 1972 2.5794945738787285 17490.1572460913  626.8498140699703 400.62973275701887 85.681 1.7152415292295276
1 "Australia" 1973 2.8161434977578472 18647.7556983189  664.7234678624812   418.422986029599 85.761 1.7416659073454563
1 "Australia" 1974  2.824455294031917 17863.2021265445  662.9016978794724  430.7955964473754 85.841 1.7863139945068534
1 "Australia" 1975 2.9007413805513567  17681.605486542  680.5585546678184 447.37222988152195 85.921 1.8084427840620647
1 "Australia" 1976  2.963728354592746 18179.9884724148  699.9216133399843 456.16539963239694     86  1.826666493107533
1 "Australia" 1977  3.120772266065389 18103.1988722163  725.4791431792559 469.48559864273113  85.94 1.8473634198091717
1 "Australia" 1978  3.254631564284719 18836.0704930921  743.4183033848726 482.38159509593464 85.881  1.868971531963084
1 "Australia" 1979  3.296127876533003 19297.5627301093  757.5444398511781 489.87601761381137 85.821 1.8892779506137485
1 "Australia" 1980  3.339912877756602 19705.8669871579  742.9213177239313  491.5441921756136  85.76  1.912448094971558
1 "Australia" 1981  3.344275473973337 20401.4029356559  734.5079386346888  504.4437616396864   85.7 1.9430378922978795
1 "Australia" 1982  3.363420740545526 19630.9278058492  742.1926472526025  522.1212016245879  85.64 1.9757103992293974
1 "Australia" 1983  3.286485783069816 20482.0192541605  722.1029344784956  529.0968370977973  85.58  2.000572745141429
1 "Australia" 1984   3.38973237261966 21001.9391952442  730.4426145136388  539.0230886738075  85.52  2.023352381448264
1 "Australia" 1985  3.440157380378221 21281.9813866455  732.3264373651479  549.7593094794391  85.46 2.0512086224177657
1 "Australia" 1986 3.4747540328622084 21371.3875446146  733.8435798831345  553.4644036606795   85.4 2.0851047212423364
1 "Australia" 1987 3.4801000989922466 22680.1385351427  721.9670558722078  554.9016562966794   85.4  2.117061296747068
1 "Australia" 1988 3.5554856582910928 23976.7872215249  735.3528266050496  558.8606060606061   85.4 2.1519857334392043
1 "Australia" 1989  3.604053668284328 24616.5379807033  745.4324864401941  564.8511904761905   85.4  2.188719524100855
1 "Australia" 1990 3.5974005426279363 24053.3955943188  758.4485294548523  571.0572429906541   85.4  2.221352980227276
1 "Australia" 1991  3.469682943763018 23827.3894926357  716.7900948854432  558.4201388888888   85.4 2.2498470510133686
1 "Australia" 1992 3.5124321234638467 24496.2443671283   711.689054015433  569.1538021726701 85.566  2.277312783931895
1 "Australia" 1993 3.5399332088073807 25263.6270165308  710.4771608082866  571.2911077149487 85.748  2.299701912187756
1 "Australia" 1994 3.5939512741528983  26380.661060725  715.5418650238029  580.3921568627451 85.928 2.3241737500488133
1 "Australia" 1995  3.699092518813634 27463.3054981186  716.9654714475432  589.4244604316547 86.106  2.352420499069289
1 "Australia" 1996 3.7851564633280543 28405.3919163355  712.9048113156027  606.3989071038251 86.283 2.3835309737969097
1 "Australia" 1997 3.8159529081384673 29473.0494263664  705.7838742776908  613.5783783783784 86.504  2.410345859963813
1 "Australia" 1998 3.7763882208326653 30442.3119275183  700.2298113409224  625.8930481283423 86.727 2.4355987139268187
1 "Australia" 1999 3.8058755151643244 31649.6552334523  703.0539997886506  624.7296587926509 86.947 2.4635851242466447
1 "Australia" 2000  3.876677282932178 31808.5517218924  711.0113298177831  627.3394255874674 87.165 2.4931335667703682
1 "Australia" 2001 3.7943646010405394 32638.4236134744  670.8391284191006  625.0515463917526 87.378 2.5269775978548092
1 "Australia" 2002  3.834841283572672 33289.3632545641  695.2176435266698  632.3193337057539 87.541  2.558009970972235
1 "Australia" 2003 3.8013812238004765 34283.1060245603  689.8579571157152  643.8241526429509 87.695 2.5897712924514793
1 "Australia" 2004 3.8936971491598515 34903.0118501845  723.2429424565518  654.8718055803696 87.849 2.6199705817268266
1 "Australia" 2005 3.8500009806421245 35506.2815141701  691.9410830211623  665.8915468459458     88 2.6547778660036707
2 "Belgium"   1960  .7658281994985736 9337.64958633288 117.98779678437369 101.61746939279202  92.46  278.8997257769653
2 "Belgium"   1961  .7905097023632974 9698.11677816021 125.43625029235793 110.89324250087887 92.554 279.28092626447295
2 "Belgium"   1962  .8047217864216322 10159.8373809861   129.818325922735 120.92189288397697 92.679   280.865326020719
2 "Belgium"   1963    .80733968655844 10431.6745352363 134.12603326024217  132.7233453700913 92.835 282.84582571602704
2 "Belgium"   1964  .8338564485200807 11092.3387976455 148.64397560575352 146.25797908098343 92.988   285.405240706886
2 "Belgium"   1965  .8939452328574116 11461.5755596638 164.31262849802303 166.33193998746714 93.137     287.8732480195
2 "Belgium"   1966    .83544933267435 11744.4608089791    160.47764191697 191.28019239205966 93.284     289.7014015844
2 "Belgium"   1967  .9028293628498347  12132.742006814  175.9734457531585 184.34234472433045 93.428   291.194393662401
2 "Belgium"   1968 1.0178031337940165 12590.2722584439 193.37219906607467 215.12476401019103 93.569  292.1998781230961
2 "Belgium"   1969 1.0522461463947042 13450.3643813944  209.7235422814272  226.0206896497199 93.707 292.90067032297395
2 "Belgium"   1970 1.1703114965290944 14252.7596107932 228.15895812863673 240.91097689027924 93.843 293.66240097501503
2 "Belgium"   1971  1.210565893551664 14668.6369542211 238.59829908772332 250.82734842591438 93.976   294.728823887873
2 "Belgium"   1972  1.289244334970804 15573.9057384289 259.29051401409623  262.5025994158916 94.106   295.825716026813
2 "Belgium"   1973 1.3159893735397854  16700.203463619  263.7111310938931  274.1272628220217 94.233   296.709323583181
2 "Belgium"   1974 1.2719470992801272 17212.4412778826   256.640653660061  285.4861688142071 94.358   297.623400365631
2 "Belgium"   1975 1.3560255900088767   16833.34979536  282.1227055210342 296.81304680894175  94.48   298.446069469835
2 "Belgium"   1976 1.4391600438653538 17881.7145359953   291.702361332652  309.0315571467624   94.6  298.9335770871419
2 "Belgium"   1977 1.5075748004294451 18214.3594171213 302.53221703624627 322.64766786805035 94.806 299.26873857403996
2 "Belgium"   1978 1.5468212214115018 18889.6595527448  315.3604632089284 332.88905294628063 95.005   299.512492382693
2 "Belgium"   1979 1.6358016981875803 19542.5887184122  317.2094664890131 345.55101368279344 95.196  299.7257769652649
2 "Belgium"   1980 1.6076286594851814 20262.2052655668  298.5016495182895 368.63715141363156 95.381 300.03046922608195
2 "Belgium"   1981  1.565070308476068 18990.1341313303 275.38340165343647  351.4398926963628 95.535    300.18281535649
2 "Belgium"   1982  1.583758129189007 18401.4500015241  270.4868143765467   353.998679891537 95.637 300.30469226081703
2 "Belgium"   1983  1.618382388752699 17862.9249534197  261.2748997516113 357.70419332063165 95.737 300.30469226081703
2 "Belgium"   1984 1.6214507174361354 17901.2810668614 262.69936842566676  361.7995398817316 95.835 300.21328458257193
2 "Belgium"   1985 1.6605283584160693 17741.8355187227 253.69465023815445  366.7867940539882  95.93 300.36563071297996
2 "Belgium"   1986 1.8100101776314586 18683.4485994807 275.81107468669836  371.0647494869762 96.023 300.48750761730696
2 "Belgium"   1987 1.8631777118961923 19444.1440230618  287.4298623517943 384.33704087199527 96.115    300.73126142596
2 "Belgium"   1988  2.007743344956969 20955.3244914384 296.31383169535945  395.3409759276103 96.204 301.70627666057294
2 "Belgium"   1989  2.042726800786943 22174.1502075456 289.50369487014945 408.70781462449804 96.292   302.803168799513
2 "Belgium"   1990  2.006545552245982 23044.8914109019  273.5924860487396  422.0320664586743 96.377 303.68677635588097
2 "Belgium"   1991  2.049080782361033 23491.9390406788 273.67722839534184  434.1785151380354 96.461 304.84460694698396
2 "Belgium"   1992  2.179159352197347 24233.7621192089  288.9949565750982 438.74492854084696 96.542 306.06337599024994
2 "Belgium"   1993 2.2172696149278965 24355.8971645373  281.5218442209436  446.5189709140522 96.622 307.28214503351603
2 "Belgium"   1994  2.248012303369359 25267.6176101506  280.6555377865264 459.45553062990786   96.7 308.22669104204795
2 "Belgium"   1995 2.2294980147109382 26120.0587547085  279.4764546759331 464.96773876905803 96.777 308.86654478976203
2 "Belgium"   1996  2.324588345532089 26172.5966638384   269.675877950546 472.83883668998845 96.851 309.47592931139496
2 "Belgium"   1997  2.325845218340193 27685.6357886652 248.98723093295564  481.1320373336638 96.924   310.207190737355
2 "Belgium"   1998  2.360088319052577 28587.6796746813  246.2019043795712 489.42050882637574 96.996 310.87751371115195
2 "Belgium"   1999  2.372286916857211 29682.3192116252 234.09954158928946 500.08665303013044 97.065 311.57830591102993
2 "Belgium"   2000 2.3714181197414947 31589.3844335182 218.99768320936474 510.48086165754745 97.128   312.370505789153
2 "Belgium"   2001   2.43910263576683 31735.7962491225   212.510098118226  516.9614093498608 97.184  313.4369287020109
2 "Belgium"   2002 2.4194832274164226 32457.0473545754 201.97845982472296  519.9616755821838 97.239   314.838513101767
2 "Belgium"   2003 2.4999679553066643 31523.1214697546 202.86941194759166  533.4391873479485 97.292  314.3935405240707
2 "Belgium"   2004 2.5448278820247734 31763.9643330839  185.3924384642482  546.7320233563054 97.345 315.04442413162707
2 "Belgium"   2005 2.4592940079783427 31678.7111483935 168.15196127504228  559.4894743535823 97.397           315.5444
3 "Canada"    1960 2.7997082362290397 11758.0418752576  657.0436141943978  292.4517783973028 69.061   1.94220369440525
3 "Canada"    1961 2.7639428602703737 11934.4537737086  659.6245416233376  300.8422619336053 69.668 2.0092351578213474
3 "Canada"    1962 2.8403352315461476 12445.8107005251  682.6582142473408  309.1797386760763 70.494 2.0469543663557856
3 "Canada"    1963  2.938726007171483 12820.8184647129  711.6642058637418 318.96505103082416 71.307  2.085443354656233
3 "Canada"    1964 3.0566623544631306 13459.0420509267  735.9379042690815 328.62051925079186 72.107  2.125141996874694
3 "Canada"    1965  3.175119422705559 14110.3285865656  765.7790425856286  337.4232635414848 72.892 2.1639608907891454
3 "Canada"    1966  3.316041500399043 14758.4593706042  796.1392657621708 346.40827724868734 73.642  2.204649249849618
3 "Canada"    1967  3.425436018028611 14920.3096577685  822.1634332745443 352.50873923684514 74.155 2.2446777976820833
end

I use the synth package developed by Hainmueller, Abadie and Diamond to perform synthetic control. If I specify the option "nested", a nested optimization will be performed using the constrained quadratic programming routine and Stata's ml optimizer.

tsset Countryno year 
synth CO2_transport_capita GDP_per_capita gas_cons_capita vehicles_capita urban_pop CO2_transport_capita(1989) CO2_transport_capita(1980) CO2_transport_capita(1970), trunit(13) trperiod(1990) xperiod(1980(1)1989) unitnames(country) nested allopt technique(bfgs) resultsperiod(1960(1)2005) fig

I want to perform synthetic control using the Nelder-Mead maximization technique, but the option "technique()" allows only to perform one among the following techniques: modified Newton-Raphson, Davidon-Fletcher-Powell, Broyden-Fletcher-Goldfarb-Shanno and Berndt-Hall-Hall-Hausman. Is there a way to perform synthetic control using Nelder-Mead in Stata?

PS. As this is my first question on this forum, any feedback on how to post a question here is highly appreciated!

[Jared Greathouse]

Do you know how to program the Nelder-Mead method?

And as someone who's research is in part dedicated to synthetic controls, why do you want this technique? What makes it better then the nested version?


EDIT: Based off a quick read, the authors don't even mention NM in their paper

[Gabriele Bertuzzi]

Do you know how to program the Nelder-Mead method?

And as someone who's research is in part dedicated to synthetic controls, why do you want this technique? What makes it better then the nested version?


EDIT: Based off a quick read, the authors don't even mention NM in their paper

Thank you for your reply.

The author performs the synthetic control using the synth package on R. When he runs the synth command, he specifies method = "all", which means that synth runs the results over all algorithms (NM is included among them). From the R output, I can see that the Nelder-Mead method is the one selected.

I want to replicate exactly the results of the paper, whereas if I run the code I attached in my first post, there's still a small difference. I was hoping that there was a quick way to implement NM in synth in Stata, without programming the whole method by scratch.

[Jared Greathouse]

Nope there's no pre-canned way to do this. You'd need to know Mata to do this. But on all honesty, if the results are similar enough, it really doesn't matter, so long as the underlying method is sound. In my experience, R will have a different set of functionalities than Stata, and we sort of have to accept those differences sometimes unless we're gonna go nuts and start writing new optimizers or algorithms.

[Gabriele Bertuzzi]

Thank you for your answer!
The difference is very small, and the cost to write the optimizer from the scratch outweighs the benefit, at least for me (I'm only a beginner in Mata).