Nonparametric Policy Analysis - Harvard University
NonparametricPolicy AnalysisJAMESH. emeaneffectofa latedon a dependentvariable(Y). Thisis conventionallydoneusinga onof Y, givenpolicyandnonpolicyvariablesX, is X afterthepolicyinterventionlieswithinthesupportofX beforetheintervention,thenthisanalysiscan be cestimatoris developedforthemodelY, g(Xj) A'dj uj, whereg(-) is a continuousunknownfunctionof the continuousvariablesX, dj is an m-vectorof dummyof unknownvariables,A is an dujis an errortermwithE(uj Ixj, dj) 0. Theestimandeffects,is B Eg(Xj7) - Eg(Xj),whereX andX* (respectively)denotethevaluesofX beforeandafterthepolicyintervention.A nonparametricestimatorB,is proposed.The estimatoris ressionestimatesofE(Y I Xj*,dj)and E(Y I Xj, dj). To estimatetheseconditionalA is nparametricregressionsof Y and d on X. The consistencyand asymptoticof B, are studied.The ance,is examinedin a MonteCarloexperiment.In ator,relativeto theefficientparametricestimator,is foundto be hedimensionofX is largeandthesamplesizeis small,however,thenonparametricestimatorcan exhibitsubstantialbias.KEY WORDS: 1. INTRODUCTIONlatingtheindependentand dependentvariables.The esmodel thatA commoneconometricproblemis predictingtheav- timatoris developedfor a teobservaerageeffectofa proposedpolicyon medthattheregressionable, wherethe variabletypicallyis eitherdirectlyorcan be expressedas an unknownfunctiong(x),indirectlyrelatedto individualwelfare.For example,an olanalystmightbe elreflectsain housepricesresultingfromcleaningup a localhazard- workdaysresultingfromfromone observationalcella reductioninairpollution,orthechangeinaveragecon- whichg(x) linwhichg(x)sumptionfroma changein incometaxes.Moreresultingwouldbe specifiedas a finitelywe can thinkof a policyas transforminggenerally,some (as wellas umentsemiparaor all of theelementsof a k-dimensionalvectorof pendentvariablesfroman originalvalue,X, to X*. itswithcross-sectionaldatawhentheremightbe fewX on ecells.Thespecificationofg(x)imThe usualeconometricapproachto thisproblemis toposesnoparametricassumptionsconcerningthecontina ction.dependentvariables-forexample,a linearregression uouspartoftheregressionare fromthesame cell,Whenall of theobservationsfunction-andto estimatetheparametersof stimatingusing,say,leastsquaresor maximumlikelihood.The efof Y1,givenXi, posedpolicyarethenpredicted conditionalforeachobservationi odel.Unfortunately,al- yis dconcerningthepreciseparametricmodelto elaverageconsequencesofthisambiguitycanbe ectationsmetricmodelis misspecified,thecorrespondingbenefits timatesthisis matoris in subtractingeffects,This articleproposesa icyinterventionswhenTherehas been muchworkon hefunctionalformreIntroducedregression.by Nadaraya(1964) and istency* JamesH. Stock is anProfessor,JohnF. KennedySchool ofGovernment,HarvardUniversity,CambridgeMA 02138. This workwassupportedin part by National Science FoundationGrant SES-84-0879.The author thanks H. Bierens, J. Powell, D. Wise, two anonymousreferees,and an associate editor for helpfulsuggestionson an earlierdraft,and Rick Whiteand Robin Lumsdaine forresearchassistance.567? icanStatisticalAssociationJune1989,Vol.84, No.406,Theoryand Methods
568Journalofthe is EYi E[g(Xi)inderivingthe articularof theproposedestimatorare De- A 'di ui] E[g(Xi) A istencythismeanvalueis E*Yi E*[g(Xi) A'di],whereE*[.]vroye's(1978) and Bierens's(1983) uniformistakenoverX*. The estimandforkernelregression. renton Y ptoticthemeaneffectA varietynormality(at a rateslowerthann2ll, wheren is thesampleB E[g(Xi) A'di] - E*[g(Xi) A'di]size), werereviewedbyPrakasaRao (1983) and Bierenstechniquesnonparametricregression(1985). Alternative E[g(Xi)] - E*[g(Xi)],(2)includesplineregression(e.g., Wahba 1978), nearestthatthefunc- wherethesecondequalityobtainsby assumingneighborregression(e.g., Stone1977),and otsimplyas a conNotethat(1) shouldbe za (1983)].thatis inasastructuralmodelbutditionalexpectation,conis pplitheoreticalvenienceand ion(2)valid,g(x)cable to thebenefits-estimationproblem.Thebeforeand afterthe policyis implemented.modelandtheproposed differThe g(x)mightchangepolicyresultsare presentedin Section2. Asymptoticestimatorofinreactionsare statedin changes,perhapsbecauseofsophisticatedthe behaviorof the policyquestion,andis shownto be consistent,Section3. The ncenteredto as the "Lucas critique").Thisto (thisis ty,however,this limitingdistributionat n"2, fasterthan is oposedThisresultcomestimators.exhibitedby tingaverageofRobinson's(1988)n '2 ratefortheestimatorplementsthe parametricpart of this semiparametricregression changein astedisposalsite.Let Yof thisestimatorare up a contaminatedmodel.In Section4, thepropertiesXa vectorof housingrepresenthousingprice,representconina nceto theincluding(for example)of edictionwasteandsite,g(x) be a hedonicobservationsis smalland thenumberofregressorslarge, ebiasoftheproposedestimatorcanbe severe.Thisbias gpricesgivenmetropolitananditssourcesare discussedinSection4. lication,housingin Section5.summarizedcleaningup a givenhazardous-wastesite,X* denotestheafterthecleanup,andB is theaveragecleanup2. THEMODELAND THEPROPOSEDESTIMATOR attributesbenefit,measuredin termsof increasedhousingprices.2.1 TheModeltheoreticalworkon theuseTherehas beenconsiderableto edtobe generatedbya non- ofhedonichousing-pricesuchas cleaningup a hazardouslinearversionof theusuallinearmodel,withm dummy ofpolicyinterventions,are made on the waste site; for example,see Harrisonand Rubinfeldassumptionsvariables.No parametricfunctionthecontinuousvariablesto (1978), Polinskyand Shavell (1975, 1976, 1978), worksuggeststhedependentvariable.Thedataconsistofn observations Scotchmerpriceequationisfromm 1 observationalcellson thedependentvariable, of an pendentYi, politan(althoughthecellXi, and a vectorofzerosand ones,di,indicatingneednotbesmallforiteachistobe validhouse);unlikelyis drawn.Omittingone lcontrog(x)unchanged(ifm. ametricpolicyanalyses,anditis maintainedY1 g(XI) A'di ui,i 1, . . ., n, (1) in thenonparametrictreatmenthereas well.efvectorof cell-specificwhereA is them-dimensional2.2 The Proposed Estimatorfects.It is assumedthat(Xi, di, YI) are iid,E(ui IXi, di)The proposedestimator 0, and E(u2 I Xi, di) o2(X,, di), wheretherearesidestepstheproblemofspecconstantsU2 and i2 suchthat0 a'2 a2(X, d) a2 a functionalformforg(x) throughifyingtheuseofkernelThe basicidea oftheestimatoroXforall (x, d) in thesupportof (Xi, di). The function nonparametricregression.g(x) is assumedto be continuousinx.is a simpleone: Fortheithobservation,obtainconsistentI considertheproblemofestimatingtheaveragechange nonparametricestimatesof E(Y Xi) and E(Y X*).froma shiftin the Repeat thisforeach observation,in thedependentvariableY resultingi 1, . . . , n. ThedistributionofX. Let X* denotethevectorofpolicyand nscontrolvariablesaftertheintervention,andletH(x) and at Xi* and Xi providesan estimateof the effectof theH*(x) denotethe marginaldistributionsof Xi anldXi*. proposedpolicyshifton Yi foreach i. The estimatorof
Stock: NonparametricPolicy Analysis569themeanbenefitsis theaverageofeach oftheseindivid- anduallyestimatedbenefits.nnWhenthereare no dummyvariables,theestimatorinw-((Xi-x)lbn)f2n(X) w((Xi - x)lbn)divolvesdirectaveragesof nonparametrici lestimatesofthei lregressionfunction.Let thekernelweightfunctionw(t)(9)be a densityon Rk (technicalconditionsare givenin Sec.3), b, be thekernelbandwidthandg&(x)de- Letqiand(i otethekernelestimatorofg(x):sions, ngn (x) / n w((Xi- x)Ibbn)Y,i lw((Xi-Vi x)/bbn).Yi(10)fM(XM)andi l(3)(i di- f2n(Xi)(11)Underweak conditionson the densities,if bn- 0 and A is thenestimatedbytheOLS regressionof'i ontoQi:nbn-? cc, then gn(x) is a consistentestimatorof g(x)(Spiegelmanand Sacks 1980).The proposedestimatoristhesampleanalogof(2), (3) evaluatedat all samplepoints:((It is shownin thenextsectionthatAnis consistentforA.The secondstep involvesobtaininga consistentnonj 1estimatorof g(x), giventhisconsistentparametricestiwheregn(Xj*)and gn(Xj) are the kernelestimatorsof matorofA. Suchan estimatorcan be obtainedbynotingbothconstructedusing(3). that(1) and (7) implythatE(Y Ix) g(x) E(d Ix)'Ag(Xj) andg(Xj), respectively,[Note that one could alternativelyestimateB by Bn g(x) f2(x)'A,whereas(6) statesthatE(Y I x) n-lyn[g(X) - Yj]. I focuson Bn althoughthe is similar.]g(x) f,(x) - theproblemconsiderably,sinceitis nownecessarytoestimatethe Althoughg(x) cannotbe estimateddirectly,each comnuisanceparameterA in (1) as wellas themeanbenefits. ponenton the rightside of (13) can. Accordingly,g(x)To motivatetheproposedtechnique,recalltheordinary can be estimatedbyleastsquares(OLS) estimatorofA wheng(x) is ation,gn(x) fl,(x) - f2n(x)'An.(14)Bn nn- I[gn(XJ*)- gn(Xj)],(4)(1) isThe thirdstepincomputingthebenefitsestimatoris toevaluategn(x)at each samplevalueofX andX*. Whenas longas thepolicydoesnotchangefiandA in(5) canbe estimatedTheparametersusingOLS therearecelleffects,thecellinwhichtheobservationis locatedtheestimatorwithbothX andD as heOLS estimatorAlternatively,ofA can be writtenas stillhastheform(4), withtheregressionreplacingthesimplerone in (3).A (D'MxD) - ID'Mx Y, whereMx I - X(X'X) 1X'.thevariousexpressionsCombiningforB. andgj(x),theThatis,A can be computedbyregressingtheresidualsofisa regressionofY onX againsttheresidualsofa regression proposedestimatorofD on X.nWheng(x) is unknown,thereisnoclearwaytoestimatey Y(Xj)(Y - dJAn),n(15)Bnj 1g andA simultaneously.Instead,A, g, andB areestimatedinthreesteps.The firststepis theestimationofA, which whereis motivatedby analogyto theOLS estimator:EstimatethecelleffectsusingOLS byregressingtheresidualsfrom yn(X) An*(x) -An(a kernelnonparametricregressionof Y on X, againstthen/nresidualsfroma kernelregressionof d on X. Let f1(x) U.Y Xfl DA(5)and f2(x) (respectively)denote the conditionalexpecta-A*()tionsof Y andd, givenx, lestimators:f,(x) E(YIx),f2(x) w((X1IE(d x),nf1,,(x) -i l(6)(7)/nx)Ib,,)Y/i lw((X, - x)Ib,), Ei lw((x-X*)lbn)InInn' 1An(x) Ei ln)j 1w((x - Xi)Ibn)/w((Xj-X*)lb)Jw((Xi - Xi)Ibn)1 j lwhereAnis givenin (12). NotethatboththeOLS andthenonparametricbenefitsestimatorsare linearin the dependentvariable,withweightsthatdependsolelyon {Xi,,(8) X*', d,}.
Journalofthe American StatisticalAssociation,June1989570to itspointwiselimitprobability3. CONSISTENCYAND ASYMPTOTICNORMALITYpointin S, it convergesat a rateslowerthann"l2.Thisdifficultyalso hecell-ef- thenonparametricbenefitsestimator:Ifg(x) is n. BothAn thenEto B at a rateslowerthan[BnI{Xi, dj}]convergesandBnare consistent.In addition,whencenteredaround n.itsexpectationconditionalon {Xi, di}(i 1, . . ., n), estimatoris asymptoticallynormal.benefitsmeanof Bn to B, whencenteredat E[Bn I {XiI di}] theThefollowingaremadeconcerningthedis- estimatorassumptionsnormaland convergesto itsis asymptoticallyofX and X* and theconditionaltributionsexpectations limitingdistributionat the rate n"2.E(Y I X) and E(d I X).Theorem3. SupposethatAssumptions1 and 2 hold,Assumption1. (a) H(x) and H*(X) (respectively)( 0 usdensitiesH A suchh(x) and h*(x). In addition,thatEluj12 A Xoforall j. Then,n"12(BnandH* havea commoncompactsupport, and3 h1andE [BnI{Xi, dl}])4 N(O, V), whereV E [a2(X, d)(y(X)h2suchthat0 h,' h(x),h*(x) c h2 X forall x E- R'M- l[d - f2(X)])2]. In addition,Vn n1( yn(Xj)inx uni(b) f1(x)and f2(x)are boundedand continuous- RAMn-'zni)2u21AX7,V, where7rnj w((X1 formlyovert. (c) 0 f [h*(x)lh(x)- 112dH(x) oo.X bniEi 1 w((Xj - Xi)lbn)and unj Yj -gn (Xj)The assumptionthatX and X* havethesamesupportis A'dj.Ifa2(X, d) U2 forall (x, d), thenV a2[f (h*(x)lnotinnocuous:It restrictsthepolicyexperimentsthatcan h(x) - 1)2 dH(x) R'M-'R]. The proofis givenin thebe consideredto onesforwhichtherealreadyexistssome Appendix.experiencein thedata. Thisis a consequenceof theinofkernelregressionto extrapolation.applicability4. MONTECARLO RESULTSThe kernelw(u) is assumedto ithonepolicyvariable,2. w(u) is a n 9Tk withan absolutelycharac- and fromzero to two controlvariables.The data wereintegrablegeneratedbya linearversionof (1):teristicfunction.1/2kDefineYii 1 E Xrij Aj uij,nRn n-1 E Yn(Xj)djr 1(16) 1, ., m. Each observationon Xrijwas drawnfroma uniformindependentlydistributionon theunitinR E(d Ix) dH*(x) E(d I x) dH (x),terval.The errorsuj weredrawnfroman NI(O, .25) disntribution.The simulatedshiftin thepolicyvariablewasMn n-1E jXli X1j for all i and j. Under these assumptions,thej 1truevalueof B is -.1667. Whentheobservationsweredrawnfrommorethanone cell,[n/(m1)]observations M E[(di E(di IXi))(di E(di I Xi))'],weredrawnfromthefirstm cellsand ll,where[.] denotes the greatestlesser integer.The estimatorwasy(x) h*(x)/h(x)- 1.computedusinga multivariateGaussiankernel,wheretheThe firsttworesultsare thatthecell-effectsestimator samplecovariancematrixofX wasusedas thecovarianceestimatorAn and thebenefitsBnare utedusing wherebisa(bln1/2)l/k,parametervariedacrosssim1. IfAssumptionsTheorem1 and2 hold,bn- 0, and bnulations.nbnk oo, thenMn P M andAn A A.This studyexaminesBn and two varianceestimatorsTheorem2. Underthe conditionsof Theorem1, Bn suggestedforVngivenin Theorem3.bytheexpressionsA sioncanunderstatetheregressionvariancein atively0Theratesofconvergenceto wereestimatedusingtherems1 and 2 are op-j"kernelregressions,Unj- lowerrateis one of Bierens's(1983) conditionsmo.Unj Yi - [fIn(j)(XI) (dj - f2n(j)(Xj))'An, (17)fortheuniformconsistencyof kernelregression,a resultwhereusedto proveTheorems1 and 2.The consistencyof B,,arisesfromtheuniformconsisw((X, - Xj)Ibn)Y i/ w((X1- formcon- fl()x) ri?ji?jsistencyof the kernelestimatorsof f1(x)and tentateach(18)-*fj 1fforj
Stock: NonparametricPolicy j)lbn)(19)(e.g., see Li 1984; Devroyeand Penrod1984; Marron1985;Rice nsideredareVI, nn(20)ECniUnii landnV2n C)(n(21)Uniwherecni Yn(Xi) - RnMnwere17ni.The simulationscomputedusing100drawsforn 20, 40, and60, and50drawsforn 100.Selectedsimulationresultsare presentedin Table 1.Severalfeaturesare apparentfromtheseresults.Evensectionwithn 40, thebias discussedin theprecedingcan be smallwhenk 1. As k increases,however,theestimatoris increasinglybiasedtoward0. Withlargek,thisbiascan be substantial,evenforlargen. The biasoftheestimatoralso growsas thenumberofcellsincreases,thisdeteriorationdoes notseemto be as imporalthoughtantas allsamplesizes;Theeffectofmis mostpronouncedforexample,forn 60 andk 3 themeanofBnchangesas m increases.The twovarianceestimatorsonlyslightlyhave similarmeans,particularlyas n increases.In genvarianceand typerallyfallwithin10% ofthesimulationicallyare conservative(as is expectedusingthe drop-jregressionresiduals).As
in the nonparametric treatment here as well. 2.2 The Proposed Estimator The proposed estimator sidesteps the problem of spec- ifying a functional form for g(x) through the use of kernel nonparametric regression. The basic idea of the estimator is a simple one: For the ith observation, obtain consistent
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Recent developments in nonparametric methods offer powerful tools to tackle the inconsistency problem of earlier specification tests. To obtain a consistent test, we may estimate the infinite-dimensional alternative or true model by nonparametric methods and compare the nonparametric model with the para-
Nonparametric Tests Nonparametric tests are useful when normality or the CLT can not be used. Nonparametric tests base inference on the sign or rank of the data as opposed to the actual data values. When normality can be assumed, nonparametr ic tests are less efficient than the
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