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1、Ch.16:MonteCarloExperimentation9391.MonteCarloexperimentation1.1.IntroductionAttheoutset,itisusefultodistinguishMonteCarlomethodsfromdistributionsamplingeventhoughtheirapplicationineconometricsmayseemrathersimilar.Theformerisageneralapproachwherebymathematicalproblemso
2、fananalyti-calnaturewhichprovetechnicallyintractable(ortheirsolutioninvolvesprohibi-tivelyexpensivelabourcosts)canbesolvedbysubstitutinganequivalentstochasticproblemandsolvingthelatter.Incontrast,distributionsamplingisusedtoevaluatefeaturesofastatisticaldistributionbyr
3、epresentingitnumericallyanddrawingobservationsfromthatnumericaldistribution.ThislasthasbeenusedinstatisticsfromanearlydateandimportantexamplesofitsapplicationareStudent(1908),Yule(1926)andOrcuttandCochrane(1949)interalia.Thus,toinvestigatethedistributionofthemeanofrand
4、omsamplesofTobservationsfromadistributionwhichwasuniformbetweenzeroandunity,onecouldsimplydrawalargenumberofsamplesofthatsizefrom(say)asetofonemillionevenlyspacednumbersintheinterval[O,l]andplottheresultingdistribution.Suchaprocedure(thatis,numericallyrepresentingaknow
5、ndistributionandsamplingtherefrom)isinvariablypartofaMonteCarloexperiment[thenamederivingfromMetropolisandUlam(1949)]butoftenonlyasmallpart.ToillustrateaMonteCarloexperiment,considercalculating:[f(X)dx=1(say),(1)Joforacomplicatedfunctionf(x)whoseintegralisunknown.Intro
6、ducearandomvariableYE[a,b]withaknowndensityp(0)anddefine17=f(v)/p(v),then:(2)Thus,calculatingE(q)willalsoprovideZandasolutionisachievedbyestimatingE(q)[seeSobol(1974)],highlightingtheswitchfromtheinitialdeterministicproblem(evaluateI)tothestochasticequivalent(evaluatet
7、hemeanofarandomvariable).QuandtinChapter12ofthisHandbookdiscussesthenumericalevaluationofintegralsingeneral.Ratherclearly,distributionsamplingisinvolvedin(2),buttheexamplealsopointsupimportantaspectswhichwillbepresentinlaterproblems.Firstly,p(-)Ch.16:MonteCurloExperime
8、ntation941Asbefore,analyticalcalculationofE(k)ispresumedintractableforthepurposesoftheillustration[butsee,forexample,
