370EstimatingtheGPSandthedose–responsefunction

6.2Outputfromdoseresponse

Beforerunningdoseresponse,wehavetodecideaboutthetreatmentlevels,whichestimatetheaveragepotentialoutcome.FollowingHiranoandImbens(2004),wefocusonthevalues10,20,...,100,whichwestoretoa10-dimensionalvectornamedtp(seebelow).Theoutputfromrunningdoseresponseisasfollows:

.uselotterydataset.dta,clear

.quigeneratecut=23ifprize<=23

.quireplacecut=80ifprize>23&prize<=80.quireplacecut=485ifprize>80

.matrixdefinetp=(10\\20\\30\\40\\50\\60\\70\\80\\90\\100)

.doseresponseagewownhsmaletixbotowncollworkthenyearwyearm1yearm2>yearm3yearm4yearm5yearm6,outcome(year6)t(prize)gpscore(pscore)>predict(hat_treat)sigma(sd)cutpoints(cut)index(p50)nq_gps(5)>t_transf(ln)dose_response(dose_response)tpoints(tp)delta(1)

>reg_type_t(quadratic)reg_type_gps(quadratic)interaction(1)bootstrap(yes)>boot_reps(100)filename(\analysis(yes)graph(\detail********************************************ESTIMATEOFTHEGENERALIZEDPROPENSITYSCORE********************************************(outputomitted)Theoutcomevariable``year6′′isacontinuousvariable

Theregressionmodelis:Y=T+T^2+GPS+GPS^2+T*GPS

SourceSSdfMSNumberofobs

F(5,196)

Model2945.927385589.185477Prob>FResidual38378.9633196195.811037R-squared

AdjR-squared

Total41324.8907201205.596471RootMSE

year6

prizeprize_sqpscorepscore_sqprize_pscore

_cons

Coef.-.2254371.0003537-103.3373131.949.549993331.26845

Std.Err..0748156.000166948.3707679.40569.21976616.955419

t-3.012.12-2.141.662.504.50

P>|t|0.0030.0350.0340.0980.0130.000

======2023.010.01220.07130.047613.993

[95%Conf.Interval]-.3729839.0000245-198.7312-24.65021.116583517.55138

-.0778902.0006828-7.943281288.5482.983403144.98552

Bootstrappingofthestandarderrors

...............................................................................>.....................

TheprogramisdrawinggraphsoftheoutputThisoperationmaytakeawhile(filegraph_output.gphsaved)EndoftheAlgorithm

M.BiaandA.Mattei371Theestimatedcoe?cientsoftheregressionoftheoutcome,earningssixyearsafterwinningthelottery,theprize,andthescoreareshownbecausewehaverequiredadetailedoutput.Otherwise,doseresponseprovidesonlyagraphicoutput,suchasthatshownin?gure1.Figure1showsboththeestimateddose–responsefunctionandtheestimatedtreatment-e?ectfunction,whichcanbeinterpretedasaderivate,becausewehavespeci?edatreatmentgapequalto1(delta(1)).OnlyinformationconcerningtheGPSestimationisprovidedwhendetailisnotspeci?edandtheanalysis()optionissettono.Dose Response Function200E[year6(t+1)]?E[year6(t)]020406080Treatment level100?2000?1000100E[year6(t)]500010000150002000025000Treatment Effect Function204060Treatment level80100Dose ResponseUpper boundLow boundTreatment EffectUpper boundLow boundConfidence Bounds at .95 % levelDose response function = Linear predictionConfidence Bounds at .95 % levelDose response function = Linear predictionFigure1.Estimateddose–responsefunction,estimatedderivative,and95%con?dencebands(Continuedonnextpage)372EstimatingtheGPSandthedose–responsefunction

TheresultsgeneratedbydoseresponsearestoredinanewStata?le,whichwehavenamedoutput.This?lehas10observationsand6variables:treatmentlevel,containingthetreatmentlevels,atwhichweestimatetheaveragepotentialoutcome;treatmentlevelplus,containingthe#-shiftedtreatmentlevels,where#isequalto1;doseresponse,theestimateddose–responsefunction;sedoseresponsebs,thestandarderrorsoftheestimateddose–responsefunction;diffdoseresponse,thees-timatedtreatment-e?ectfunction;andsediffdoseresponsebs,thestandarderrorsoftheestimatedtreatment-e?ectfunction.Thegraphicoutputisalsostoredtoanew?le,whichwehavenamedgraphoutput.

7Acknowledgments

WethankFabriziaMealli,GuidoImbens,andKeisukeHiranofortheirinsightfulsug-gestionsanddiscussions,andGuidoImbensandKeisukeHiranoforprovidingthedata.

8References

Becker,S.O.,andA.Ichino.2002.Estimationofaveragetreatmente?ectsbasedonpropensityscores.StataJournal2:358–377.Bia,M.,andA.Mattei.2007.Applicationofthegeneralizedpropensityscore.Eval-uationofpubliccontributionstoPiedmontenterprises.POLISWorkingPaper80,UniversityofEasternPiedmont.Hirano,K.,andG.W.Imbens.2004.Thepropensityscorewithcontinuoustreat-ments.InAppliedBayesianModelingandCausalInferencefromIncomplete-DataPerspectives,ed.A.GelmanandX.-L.Meng,73–84.WestSussex,England:WileyInterScience.Holland,P.W.1986.Statisticsandcausalinference.JournaloftheAmericanStatisticalAssociation8:945–960.Imbens,G.W.,D.B.Rubin,andB.I.Sacerdote.2001.Estimatingthee?ectofunearnedincomeonlaborearnings,savings,andconsumption:Evidencefromasurveyoflotteryplayers.AmericanEconomicReview91:778–794.Je?reys,H.1961.TheoryofProbability.3rded.Oxford:OxfordUniversityPress.Leuven,E.,andB.Sianesi.2003.psmatch2:StatamoduletoperformfullMahalanobisandpropensityscorematching,commonsupportgraphing,andcovariateimbalancetesting.BostonCollegeDepartmentofEconomics,StatisticalSoftwareComponents.Downloadablefromhttp://ideas.repec.org/c/boc/bocode/s432001.html.Rosenbaum,P.R.,andD.B.Rubin.1983.Thecentralroleofthepropensityscoreinobservationalstudiesforcausale?ects.Biometrika70:41–55.Rubin,D.B.1976.Inferenceandmissingdata.Biometrika63:581–592.

M.BiaandA.Mattei

Abouttheauthors

373

MichelaBiaisaresearchassistantatLaboratorioRevelli,CentreforEmploymentStudies,CollegioCarloAlberto,Turin,Italy.

AlessandraMatteiisastatisticsresearchassistantintheDepartmentofStatistics,“GiuseppeParenti”,UniversityofFlorence,Italy.OnFebruary25,2008,shewonacompetitiveexamtoworkasanassistantprofessorofstatisticsinthefacultyofpsychologyattheUniversityofFlorence.