三重差分法運(yùn)行和示例

三重差分法及運(yùn)行雙重差分法的關(guān)鍵假設(shè)是實(shí)驗(yàn)組與對(duì)照組的時(shí)間效應(yīng)一樣。這個(gè)假設(shè)只有通過(guò)足夠長(zhǎng)的時(shí)間序列數(shù)據(jù)才能檢驗(yàn)。需要指出的一點(diǎn)是，即使干預(yù)發(fā)生之前兩組時(shí)間序列一致，也不能保證干預(yù)發(fā)生后兩組時(shí)間序列是一致的。有可能在干預(yù)發(fā)生的同時(shí)在實(shí)驗(yàn)組或者對(duì)照組中又發(fā)生了其他影響產(chǎn)出的事件，則干預(yù)發(fā)生后兩組的時(shí)間趨勢(shì)是不一致的。簡(jiǎn)單的雙重差分估計(jì)是有偏的。如圖3所示，如果對(duì)照組是虛線所示，則雙重差分估計(jì)是無(wú)偏的。但如果對(duì)照組是上方的實(shí)線，則雙重差分估計(jì)法是有偏的，偏差部分是在.時(shí)刻，該實(shí)線與虛線之間的距離。解決這個(gè)問(wèn)題有兩個(gè)思路:第一個(gè)是尋找更多的對(duì)照組，把多個(gè)對(duì)照組加權(quán)構(gòu)造成一個(gè)虛擬的對(duì)照組，使得雖然每個(gè)對(duì)照組都與實(shí)驗(yàn)組的時(shí)間趨勢(shì)不一樣，但加權(quán)后的虛擬對(duì)照組的時(shí)間趨勢(shì)與實(shí)驗(yàn)組的一樣。這個(gè)方法被稱作綜合控制法(SyntheticControlMethod)。Abadie&Gardeazabal(2003)(14)用這個(gè)方法研究了恐怖沖突對(duì)經(jīng)濟(jì)發(fā)展的影響。解決這個(gè)問(wèn)題的第二個(gè)思路是估算出這個(gè)因?yàn)闀r(shí)間趨勢(shì)不同而帶來(lái)的偏差，然后從雙重差分結(jié)果中減去這個(gè)偏差即可。這被稱作三重差分法(Difference-in-differences-in-differences,DDD)。三重差分法的思路是，既然兩個(gè)地區(qū)(分別指實(shí)驗(yàn)組和對(duì)照組)的時(shí)間趨勢(shì)不一樣，那么我們可以分別在兩個(gè)地區(qū)尋找一個(gè)沒有受到干預(yù)影響的人群/行業(yè)，通過(guò)對(duì)這兩組的雙重差分估算出時(shí)間趨勢(shì)的差異，然后再?gòu)脑瓉?lái)實(shí)驗(yàn)組和對(duì)照組的雙重差分估算值中減去這個(gè)時(shí)間趨勢(shì)差異。Gruber(15)就使用了這種方法。

Yutime圖3時(shí)間趨勢(shì)差異造成的估計(jì)偏差三重差分法例證：YutimeTwoLowns,XandY.Ineachtown:Poolcross-sectionsofnnaleandfemalehourlywages,onefromtheyearbeforeawagediscriminationpolicytpokandonefrarri街程yearafter,Considerthefollowingmodeh切妨當(dāng)=曲-hRafterifemale+既X」十+■睥件/七f苗wd%+Xtfemalci+禺a(chǎn)>ftgfemol電Xj4-也where

Inthismadel,theaffectofthe-policyiscapturedby/57:thedifference-Fn-difference-in-differences'estimator.Fortobeinterpretedasthecausaleffectofthepolicyonwag-esintownX,E^lfemnfei^fter^X,)=0hastebeassuiried.Doesthismeanthatoncegender^d-ateofpoEicyfandtownhavebeencontrolled：fortthereexistnounobservedfac-torgwhichchangeoverbothtown日"dtimethataffectwages?Whatyouproposeheretsactuallydifferencetndiffersr?t>eindifferences(DDD)insteadoftheusualdifferenceindifferences(seethesel&ciurenotesby偵i■!m「：anthefirsttwopage&)ThismethodcanpotentiallyaccounUortheunobservedtrendsinwagesofwomenacrossyaurtwotownsandthewagecjluiigesatbotrmaleandfemaleworkersintnetreatmenttown.IntHh$seriseDDDisamorerobustapprcacntotheDDapproachyouhadinmindpreviously.L&lybeaverageincomeandindexperiods1,2,femalesandmenF\M.sndtownsX、YhthenyourDDDcoefficientgivesyouinyforwarnenintownX

thetimechangeinyforwomenintownY

inmmiX1^7=—)thetimechangei一(歹inyforwarnenintownX

thetimechangeinyforwomenintownY

inmmiXThekeyidentifyingassumptioninDDmodelsisthatth?treatmentstateshavesimilartrendstothecontrolstatesmtheabsenceoftreatment.尊Withonlyonetreatmentwndcontrolgroup,graphyourresu[tsrandlookattrendsinperiodswithbeforethetreatment.。Withmanytreatmentandcontrolgroups:?andabinarytreatment,estimatetreatmenti-mpacts,atdifferentdatesq匕知=%+A+12伊比[十J+，知j=_EwhereisnowanindicatorforwhetherthetreatmentgotswitchedonHyeart\Thisestimatesqkadsandmlagsofthetreatment.Theieadsshouldallhezero,?includestatespecifictrends.三重差分法回歸及運(yùn)行命令TheDfDmsttiodcanbaimplementedaccordingtothetablebelow,wfierethelowerrightcellIsthaDJDestimator.Hilt§28=1Difforencet=2J12E/L2一&22*=1!Ail如-如1Change如一際如-i/12（機(jī)1一戒1）一（§12一如）RunningsregressionanalysisgivesthesameresuIt.ConsidertheQLSmodwly=為+#i『+禺S++E政上面的二■差分法實(shí)際上運(yùn)用的是OLS做的回歸，我們之前說(shuō)過(guò)，倍差法是相當(dāng)于two-wayfixedeffectmodel，里面包括個(gè)體效應(yīng)時(shí)間效應(yīng)，而對(duì)于一個(gè)這樣的paneldata,我們可以運(yùn)用LSDVB過(guò)添加個(gè)體和時(shí)間虛擬變■來(lái)回歸，或者運(yùn)用demeanedvariables回歸來(lái)消除個(gè)體和時(shí)間效應(yīng)，再加上那些交互效應(yīng)后就可以像其他fixedeffect回歸一樣.二重差分法一般是在同一個(gè)?。ǖ貐^(qū)）區(qū)分treatment和control組的，而三重差分法則包括另一個(gè)未受到政策沖擊的?。ǖ貐^(qū)），來(lái)區(qū)分treatment和control組的，當(dāng)然三重差分法要穩(wěn)健得多。EmpiricalMethodsinAppliedEconomicsLectureJorn-SteffenPischke接下來(lái)，我們來(lái)區(qū)分一下二■差分和三■差分在Stata的運(yùn)行過(guò)程1992年，美國(guó)新澤西州通過(guò)法律將最低工資從每小時(shí)4.25美元提高到5.05美元，但在相鄰的賓夕法尼亞州最低工資卻保持不變。CardandKrueger收集了兩個(gè)州的快餐店在實(shí)施新法前后雇傭人數(shù)的數(shù)據(jù)，并使用雙重差分法進(jìn)行估計(jì)。注：fte:fulltimeemployment人數(shù)；treated=1,表示快餐店在新澤西州，否則在賓夕法尼亞州；t=1，表示時(shí)間為1992年11月，否則為1992年2月；bk=1，表示BurgerKing快餐品牌；kfc=1，表示肯德基快餐品牌；roys=1，表示RoyRogers快餐品牌；wendys=1，表示W(wǎng)endy's快餐品牌。以下黑色字體的code可以直接在Stata上執(zhí)行use"repec/bocode/c/CardKrueger1994.dta2.單的二■差分：2.單的二■差分：idtresLt&dftebkWE日叫f10NJ311□0QQi11SJ401000120NJ131fl00fl21NJ1Z.510Q0130國(guó)J12aS01□00317-50100Q40BJ16001a□41NJ200Q1Q05QNJ2=0-fluQ1g051NJ.25001006DNJ30010061tiJ60Q100B0■000i□B1NJ27-S0□0iadifffte,t(treated)p(t)DIFFERENCE-IttDIFFEMNCES^STIMATiarRESULTSNumberofobservationsintheDIFF-IN-DIFF:801BeforeControl:70Treated:326404After113203^7155646Outcomavar+fteS?Err-It|P>lt|BeforeControlTHewt兮dDiff(T-C)19.94917.065-2.3B41.135-2.540,011**AfterControlTr&ated.Diff(T-C)17.54217.5730.0301.1430.030.979Diff-in-Eiff2.9141.6111+810.071*R-square:0-01、MeansandStandardErrorsjareeatimatedbylinearregressionInfer&nce:*攵*p<0.01;士*p<0.05/太p<0-1*這里DD10%水平下顯著3.簡(jiǎn)單的三重差分（快餐品牌6義作為第二個(gè)處理組）：difffte,t(treated)p(t)ddd(bk)

TRIPLEDIFFERENCE(DDD)EST工MMTONRESULTSNotationofDDD:Control(A)treated=0a.ndb)E=1Control(E)treated.=0andbJc=0Treated(A)treated=1andblc=1Treated(B)treated=1a.ndb)匚=0Numberofobservationsint;heDDD:801BeforeAfterControl(A):343589Control(B):444286Treated(A):L33132265Treated(A):L93168381404397Outcoinev^r.fteS.Err.|t|P>|t|BeforeControl(A)25.654Control(B)15,540Treated(A)1B.547Treated(B)16.044Dxff(T-C)-7.￡122.2063.450.001***AfterControl(A)22.193Control(E)13.667Treated(A)19.913Treated(B)15.930Diff(T-C)-4,5432.2142.050.040**DDD3.0693.1250.980,326*這里DDD表不顯著*afieri=1tfdateaftergend&r-wagediscriminationpolicy;Difdatebeforegender-wagediSCrimiMationpolicy.Thegernderwagediscriminationpolicy