Lecture-16-Introduction-to-Asymptotics讲座16介绍的渐近性81课件

Lecture16IntroductiontoAsymptotics讲座16介绍的渐近性46、法律有权打破平静。——马·格林47、在一千磅法律里，没有一盎司仁爱。——英国48、法律一多，公正就少。——托·富勒49、犯罪总是以惩罚相补偿；只有处罚才能使犯罪得到偿还。——达雷尔50、弱者比强者更能得到法律的保护。——威·厄尔Lecture16IntroductiontoAsymptotics讲座16介绍的渐近性Lecture16IntroductiontoAsymptotics讲座16介绍的渐近性46、法律有权打破平静。——马·格林47、在一千磅法律里，没有一盎司仁爱。——英国48、法律一多，公正就少。——托·富勒49、犯罪总是以惩罚相补偿；只有处罚才能使犯罪得到偿还。——达雷尔50、弱者比强者更能得到法律的保护。——威·厄尔Lecture16:

IntroductiontoAsymptotics(Chapter12.1–12.3)Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AgendaTheAsymptoticPerspective(Chapter12.1)AsymptoticUnbiasedness(Chapter12.2)Consistency(Chapter12.2)ProbabilityLimits(Chapter12.2)ConsistencyofOLS(Chapter12.3)3Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Lecture16:

IntroductiontoAsymptotics(Chapter12.1–12.3)Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AgendaTheAsymptoticPerspective(Chapter12.1)AsymptoticUnbiasedness(Chapter12.2)Consistency(Chapter12.2)ProbabilityLimits(Chapter12.2)ConsistencyofOLS(Chapter12.3)3Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective

(Chapter12.1)Thislecturemarksamajorshiftinthemathematicalunderpinningsofthecourse.Sofar,wehavefocusedonfindingunbiasedandefficientestimators.Thatis,wewantestimatorsthatwillgive

youtherightansweronaverageoverallpossiblesamples,andthathavethelowestpossiblevariance.4Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)Sofar,wehavefocusedonfindingunbiasedandefficientestimators.Thislectureintroducesasetofnewcriteriaforjudgingestimators,basedonlarge-sample(asymptotic)properties.Insteadoflookingatthepropertiesoftheestimatoraveragingoverallpossiblesamples,welookatpropertiesasthesamplesizegetsvery,verylarge.5Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)Insteadoflookingatthepropertiesoftheestimatoraveragingoverallpossiblesamples,wewillstartlookingatpropertiesasthesamplesizegetsvery,verylarge.Whytheshift?Themathisgoingtobemuchmoreconvenient.Cross-samplepropertiesbecomeintractableaswerelaxthelast(andmostcrucial)Gauss–Markovassumption,thattheX

’sarefixedacrosssamples.6Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)Large-sample(asymptotic)propertiesaremathematicallytractable.Wejusthavetohopethatestimatorsthatweproveworkwellwitha

near-infinitenumberofobservationswillalsoworkwellwiththefinitedatasetsweactuallyobserve.7Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)OneuseofMonteCarlotechniquesis

tostudycomputationallythesmallsamplepropertiesofestimatorsthathavebeenderivedasymptotically.Estimatorsdesignedforlargesamplesdon’ttendtoworkwellinsmallsamples,butareappropriatefor“reasonablylarge”samples.8Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)Whataretheasymptoticproperties

ofinterest?Let’sstartwithsomeinformalexplanations.AsymptoticUnbiasedness:asthesamplesizegetsvery,verylarge,theestimatorbecomesunbiased(eventhoughtheremaybebiasesatsmallsamplesizes).9Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.Wecanseethispropertyinouroriginal

MonteCarlosimulationsofbg1-bg3.10Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithNormallyDistributedDisturbances(1of2)11Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithNormallyDistributedDisturbances(2of2)12Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.1

TheDistributions

ofg1,g2,and

g3forSeveral

SampleSizeswithNormallyDistributedDisturbancesTheAsymptoticPerspective(cont.)Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.WecanseethispropertyinouroriginalMonteCarlosimulationsofbg1-bg3.Whatifwerepeatthesimulations,

butdrawefromaskewed,non-

Normaldistribution?14Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(1of2)15Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(2of2)16Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2

TheDistributions

ofg1,g2,andg3

forSeveralSampleSizeswithSkewed,DiscreteDisturbancesTheAsymptoticPerspectiveTheLawofLargeNumbers:

undersuitable(andeasilyattained)conditions,thesamplemeanisaconsistentestimatorofthecorrespondingpopulationmean.18Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)AsymptoticNormality:asthesamplesizegrowsvery,verylarge,theestimatorfollowstheNormaldistribution(eventhoughitmayfollowadifferentdistributioninsmallersamples).19Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)Aswesawinthepreviousfigures,thedistributionofbg1,bg2,andbg3appearstobeNormalevenatsmallsamplesizeswhenMoresurprisingly,thedistributionofbg1,bg2,andbg3alsoappearstobeNormalatlargersamplesizesevenwheneisdistributedquitenon-Normally.20Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(1of2)21Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2TheDistributionsofg1,g2,

andg3forSeveralSampleSizeswithSkewed,DiscreteDisturbances(2of2)22Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.2

TheDistributions

ofg1,g2,andg3

forSeveralSampleSizeswithSkewed,DiscreteDisturbancesTheAsymptoticPerspectiveTheCentralLimitTheorem:

undersuitable(andoftenreasonable)conditions,thedistributionofasample

meantendstobeapproximatelyNormallydistributedinlargesamples.TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.24Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F,andChi-squareddistributions)forconstructingteststatisticswhensamplesizesarelarge.Werarelyknowforsurethedistributionof

,soitisre-assuringtoknowourestimatorswilltypicallybeapproximatelyNormalin

largesamples,regardlessofitssmall-

sampledistribution.25Copyright©2006PearsonAddison-Wesley.Allrightsreserved.TheAsymptoticPerspective(cont.)KeyTermssofar:AsymptoticAsymptoticUnbiasednessConsistencyTheLawofLargeNumbersAsymptoticNormalityTheCentralLimitTheorem26Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(Chapter12.2)Let’slookatsomeofthesepropertiesmoreclosely,startingwithasymptoticunbiasedness.Formanybiasedestimators,thebiasshrinkssmallerandsmallerasthesamplesizegrows.Asthesamplesizegrowsinfinitelylarge,thebiasshrinkstozero.Suchestimatorsareasymptoticallyunbiased.27Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.)Forexample,wehavelearnedthatwemustmakea“degreesoffreedom”correctiontoourestimatedstandarderrors.28Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.)Weknowthats2isanunbiasedestimateof

s2.Butwhatifweneglectedthedegreesoffreedomcorrection?29Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.)30Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.)31Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AsymptoticUnbiasedness(cont.)n30.3333100.80001000.98001,0000.998010,0000.999832Copyright©2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstandingWhichofthefollowingareasymptoticallyunbiasedestimatorsofbundertheGauss–Markovassumptions?33Copyright©2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)34Copyright©2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)35Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyWewouldlovetobeguaranteedthatourestimatorwillbeexactlyright,oratleastvery,veryclosetoexactlyright.Certaintyistoomuchtoaskforinastochasticworld.However,ifourestimatorisconsistent,wecanbe“almostalwaysalmostright”invery,verylargesamples.36Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Consistency(cont.)Moreprecisely:anestimatorisconsistentif,providedwegetthesamplesizehighenough,theprobabilitycanbeascloseto1aswelikethatourestimateisasclosetobeingrightaswelike.Withfinitesamplesizes,thereisalwayssomeprobabilitythataconsistentestimatorisverywrong.Butasthesamplesizegrows,thatprobabilitybecomesvery,verysmall.37Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Figure12.3TheCollapseofaConsistentEstimator’sDistributionasnGrows38Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyWhatdoesittaketogetconsistency?Oneoften-encounteredwayistohave

anasymptoticallyunbiasedestimatorwhosevarianceshrinkstozeroasthesample

sizegrows.Inlargesamples,theestimatoronaverage

isright.Inlargesamples,nosingleestimateislikelytobeveryfarfromtheestimator’saverage.39Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Consistency(cont.)Apathologicalexampleofconsistencywithoutasymptoticunbiasedness:40Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Consistency(cont.)41Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(Chapter12.2)Howcanwedetermineifanestimatorisconsistentornot?Weneedanewmathematicaltool,theprobabilitylimit(orplim).Theplimisconceptuallymorecomplicatedthanexpectations,butinpracticeitiseasiertoworkwith.42Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)Arandomvariablebconvergesinprobabilitytoaconstantvaluecif,asthesamplesizegrowsverylarge,theprobabilityapproaches1thatbtakesonavalueveryclosetoc.Wecallctheprobabilitylimitofb.plim(b)=c43Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)44Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)45Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)46Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ProbabilityLimits(cont.)47Copyright©2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding48Copyright©2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)49Copyright©2006PearsonAddison-Wesley.Allrightsreserved.CheckingUnderstanding(cont.)50Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AlgebraofProbabilityLimits51Copyright©2006PearsonAddison-Wesley.Allrightsreserved.AlgebraofProbabilityLimits(cont.)52Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(Chapter12.3)Howdoweshowthatanestimator

isconsistent?Let’sstartwiththeGauss–Markovassumptions,andastraightlinethroughtheorigin.53Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)54Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)First,notethatweknowbg4isunbiased,

soE(bg4)=b.Thus,plim(bg4)=b

iftheVar(bg4)

approaches0asngrows.Var(bg4)approaches0ifSXi2approaches

∞(ands2isfinite).55Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)Var(bg4)approaches0ifSXi2

approaches∞(ands

2isfinite).Theseassumptionswilltypicallybemet,butwedoneedtoaugmentourGauss–Markovassumptionstoruleouttheunusualcases.56Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)Nowlet’saddanintercepttothemodel:57Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)Wecouldusethesamemethodweusedforbg4:showthatOLSisunbiasedandthatthevarianceshrinksto0as

ngrowslarger.Instead,we’regoingtousea

differentmethod.58Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)59Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)60Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)WenowneedtoaddanassumptiontotheGauss–MarkovDGP,toguaranteethatWeneedtoassumethat,asngrowslarge,61Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)62Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)63Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)64Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)65Copyright©2006PearsonAddison-Wesley.Allrightsreserved.ConsistencyofOLS(cont.)66Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)Thislectureintroducesasetofnewcriteriaforjudgingestimators,basedonlarge-sample(asymptotic)properties.Insteadoflookingatthepropertiesoftheestimatoraveragingoverallpossiblesamples,welookatpropertiesasthesamplesizegetsvery,verylarge.67Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)Whataretheasymptoticproperties

ofinterest?AsymptoticUnbiasedness:asthesamplesizegetsvery,verylarge,theestimatorbecomesunbiased(eventhoughtheremaybebiasesatsmallsamplesizes).68Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)Whataretheasymptoticproperties

ofinterest?Consistency:asthesamplesizegetsvery,verylarge,theestimateisvery,verylikelytobevery,veryclosetotherightanswer.69Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)TheLawofLargeNumbers:undersuitable(andeasilyattained)conditions,thesamplemeanisaconsistentestimatorofthecorrespondingpopulationmean.70Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)Whataretheasymptoticproperties

ofinterest?AsymptoticNormality:asthesamplesizegrowsvery,verylarge,theestimatorfollowstheNormaldistribution(eventhoughitmayfollowadifferentdistributioninsmallersamples).71Copyright©2006PearsonAddison-Wesley.Allrightsreserved.Review(cont.)TheCentralLimitTheorem:

undersuitable(andreasonablyeasytoattain)conditions,thedistributionofasamplemeantendstobeapproximatelyNormallydistributedinlargesamples.TheCentralLimitTheoremjustifiesourrelianceontheNormaldistribution(anditsoff-spring,thet,F