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366EstimatingtheGPSandthedose–responsefunction
programs.Speci?cally,ourprogramsdonotallowustoconsiderafunctionofthetreat-mentvariableorafunctionoftheGPSintheestimationoftheconditionalexpectationoftheoutcome,giventhetreatmentandtheGPS.However,wegetqualitativelysimilarresults.
6.1Outputfromgpscore
We?rstchoosethequantilesofthetreatmentvariabletodividethesampleintogroups.FollowingHiranoandImbens(2004),wedividetherangeofprizesintothreetreatmentintervals,[0–23],(23–80],and(80–485].Thenwerungpscoreusingthespeci?cationappliedbyHiranoandImbens(2004).Theoutputlookslikethefollowing:
.uselotterydataset.dta
.quigeneratecut=23ifprize<=23
.quireplacecut=80ifprize>23&prize<=80.quireplacecut=485ifprize>80
.gpscoreagewmaleownhsowncolltixbotworkthenyearwyearm1yearm2yearm3>yearm4yearm5yearm6,t(prize)gpscore(pscore)predict(hat_treat)sigma(sd)>cutpoints(cut)index(p50)nq_gps(5)t_transf(ln)detailGeneralizedPropensityScore
******************************************************Algorithmtoestimatethegeneralizedpropensityscore******************************************************
Estimationofthepropensityscore
Thelogtransformationofthetreatmentvariableprizeisused
T
1%5%Pu???%
Percentiles1.6094382.2838512.4200122.835211
3.457834.1430084.8754265.1288925.720607
logloglogloglog
Largest5.5987925.7206075.7786436.183716likelihoodlikelihoodlikelihoodlikelihoodlikelihood
=====
Smallest.1301507.13015071.6094381.67818
Obs
SumofWgt.Mean
Std.Dev.VarianceSkewnessKurtosis
-
2372373.558185.9553768.9127448-.01658893.452439
initial:feasible:rescale:rescaleeq:Iteration0:
(couldnotbeevaluated)
(outputomitted)Iteration4:loglikelihood=-307.68186
M.BiaandA.Mattei
NumberofobsWaldchi2(13)Prob>chi2
Std.Err..0048563.1351124.060835.0397666.0182546.1645602.0464566.010379.0162758.0166256.0158217.0153635.0110455.4693959.040709
z3.133.240.320.940.240.77-0.030.60-0.760.721.53-1.41-0.454.9321.77
P>|z|0.0020.0010.7520.3490.8120.4400.9750.5500.4490.4720.1260.1590.6510.0000.000
===
23737.220.0004
367
Loglikelihood=-307.68186
T
eq1
agewmaleownhsowncolltixbotworkthen
yearwyearm1yearm2yearm3yearm4yearm5yearm6_conseq2
_cons
.886297.0151905.4379826.0192025.0372805.0043423.1270879-.0014367.0062064-.0123161.0119446.0242245-.0216437-.00500212.315546
Coef.
[95%Conf.Interval].0056724.1731672-.1000319-.0406607-.031436-.1954442-.09249-.014136-.044216-.0206411-.0067855-.0517555-.02665091.395547.806509
.0247086.702798.1384368.1152217.0401206.44962.0896166.0265488.0195839.0445302.0552344.0084682.01664673.235545.9660851
Testfornormalityofthedisturbances
Kolmogorov-Smirnovequality-of-distributionstestNormalDistributionofthedisturbances
One-sampleKolmogorov-Smirnovtestagainsttheoreticaldistribution
normal((res_etreat-r(mean))/sqrt(r(Var)))
SmallergroupDP-valueCorrectedres_etreat:0.0517Cumulative:-0.0420CombinedK-S:0.0517TheassumptionofNormalityis
0.2810.4340.5500.517
statisticallysatisfiedat.05level
Estimatedgeneralizedpropensityscore
1%5%P%
Percentiles.0131817.0869414.1272663.2255553.3536221
Smallest.0003053.0011738.0131817.0163113
Obs
SumofWgt.Mean
Std.Dev.
237237.3196603.1222106.0149354-.77235012.510499
Largest
75%.4343045.450000390%.4481351.4500911Variance95%.4497166.450096Skewness99%.4500911.4501086Kurtosis********************************************Endofthealgorithmtoestimatethegpscore********************************************
368EstimatingtheGPSandthedose–responsefunction
******************************************************************************Thesetofthepotentialtreatmentvaluesisdividedinto3intervalsThevaluesofthegpscoreevaluatedattherepresentativepointofeachtreatmentintervalaredividedinto5intervals
*****************************************************************************************************************************************SummarystatisticsofthedistributionoftheGPSevaluatedattherepresentativepointofeachtreatmentinterval
***********************************************************
VariableObsMeanStd.Dev.MinMax
gps_1Variablegps_2Variable
gps_3
237Obs237Obs237
.262852
Mean.4178101
Mean.1814998
.0956436Std.Dev..0373217Std.Dev..088236
.0583948
Min.2433839
Min.0181741
.4486237
Max.4501224
Max.4141454
******************************************************************************Testthattheconditionalmeanofthepre-treatmentvariablesgiventhe
generalizedpropensityscoreisnotdifferentbetweenunitswhobelongtoaparticulartreatmentintervalandunitswhobelongtoallothertreatmentintervals
******************************************************************************TreatmentIntervalNo1-[1.139000058174133,22.98200035095215]
MeanStandardDifferenceDeviationt-value
agew
maleownhsowncolltixbotworkthenyearwyearm1yearm2yearm3yearm4yearm5yearm6
-.25322.04799.15044.20765.33298.00154.00156.33117.908721.2445.74998.962991.4414
1.814.04246.156.23738.48448.05608.191351.90521.77191.67561.56251.71751.7774
-.139591.1304.96433.87476.68729.0275.00813.17382.51284.74274.47999.5607.81098
M.BiaandA.Mattei
TreatmentIntervalNo2-[23.08799934387207,79.11299896240234]
MeanStandardDifferenceDeviationt-value
agew
maleownhsowncolltixbotworkthenyearwyearm1yearm2yearm3yearm4yearm5yearm6
-.13308-.03419-.2294-.20996-.26933.03013-.32817.51467.23703.41572.46856-.00903-.33587Mean
Difference-1.7504-.04742.34062.23199-.03159-.07006.3672-.63678-.83409-1.2074-1.351-1.6137-2.2111
1.8294.0657.13927.21228.43812.05266.170081.77411.70381.66561.5711.62421.6445StandardDeviation2.3202.06211.1914.28116.56716.07448.226131.94281.83561.73221.59821.87921.8615
-.07275-.52041-1.6471-.98908-.61474.57227-1.9295.2901.13912.24959.29826-.00556-.20423
369
TreatmentIntervalNo3-[82.98699951171875,484.7900085449219]
t-value-.75444-.763421.7796.82512-.0557-.940691.6238-.32777-.45441-.69707-.84534-.8587-1.1878
agewmaleownhsowncolltixbotworkthen
yearwyearm1yearm2yearm3yearm4yearm5yearm6
Accordingtoastandardtwo-sidedt-test:
ModerateevidenceagainstthebalancingpropertyThebalancingpropertyissatisfiedatlevel0.05
Thisoutputisthemostdetailedwecanhavebecausewespeci?edthedetailoption.Whenthisoptionisnotspeci?ed,someinformationisomittedfromtheoutput.Specif-ically,theresultsofthegoodness-of-?ttestfornormality,thesummarystatisticsofthedistributionoftheGPSevaluatedattherepresentativepointofeachtreatmentinterval,andtheresultsofthebalancingtestwithineachtreatmentintervalarenotshown.Insuchacase,theprogramprovidesonlyshortsentencesinformingtheuserwhetherthenormaldistributionmodelandthebalancingpropertyarestatisticallysatis?ed.