多元线性回归模型：假设检验.ppt

1 MultipleRegressionAnalysis y b0 b1x1 b2x2 bkxk u2 Inference 2 AssumptionsoftheClassicalLinearModel CLM Sofar weknowthatgiventheGauss Markovassumptions OLSisBLUE Inordertodoclassicalhypothesistesting weneedtoaddanotherassumption beyondtheGauss Markovassumptions Assumethatuisindependentofx1 x2 xkanduisnormallydistributedwithzeromeanandvariances2 u Normal 0 s2 3 CLMAssumptions cont UnderCLM OLSisnotonlyBLUE butistheminimumvarianceunbiasedestimatorWecansummarizethepopulationassumptionsofCLMasfollowsy x Normal b0 b1x1 bkxk s2 Whilefornowwejustassumenormality clearthatsometimesnotthecaseLargesampleswillletusdropnormality 4 x1 x2 Thehomoskedasticnormaldistributionwithasingleexplanatoryvariable E y x b0 b1x y f y x Normaldistributions 5 NormalSamplingDistributions 6 ThetTest 7 ThetTest cont KnowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistestsStartwithanullhypothesisForexample H0 bj 0Ifacceptnull thenacceptthatxjhasnoeffectony controllingforotherx s 8 ThetTest cont 9 tTest One SidedAlternatives Besidesournull H0 weneedanalternativehypothesis H1 andasignificancelevelH1maybeone sided ortwo sidedH1 bj 0andH1 bj 0areone sidedH1 bj 0isatwo sidedalternativeIfwewanttohaveonlya5 probabilityofrejectingH0ifitisreallytrue thenwesayoursignificancelevelis5 10 One SidedAlternatives cont Havingpickedasignificancelevel a welookupthe 1 a thpercentileinatdistributionwithn k 1dfandcallthisc thecriticalvalueWecanrejectthenullhypothesisifthetstatisticisgreaterthanthecriticalvalueIfthetstatisticislessthanthecriticalvaluethenwefailtorejectthenull 11 yi b0 b1xi1 bkxik uiH0 bj 0H1 bj 0 c 0 a 1 a One SidedAlternatives cont Failtoreject reject 12 Examples1 HourlyWageEquationH0 bexper 0H1 bexper 0 13 One sidedvsTwo sided Becausethetdistributionissymmetric testingH1 bjthan cthenwefailtorejectthenullForatwo sidedtest wesetthecriticalvaluebasedona 2andrejectH1 bj 0iftheabsolutevalueofthetstatistic c 14 yi b0 b1Xi1 bkXik uiH0 bj 0H1 bj 0 c 0 a 2 1 a c a 2 Two SidedAlternatives reject reject failtoreject 15 SummaryforH0 bj 0 Unlessotherwisestated thealternativeisassumedtobetwo sidedIfwerejectthenull wetypicallysay xjisstatisticallysignificantatthea level Ifwefailtorejectthenull wetypicallysay xjisstatisticallyinsignificantatthea level 16 Examples2 DeterminantsofCollegeGPAcolGPA collegeGPA greatpointaverage hsGPA highschoolGPAskipped averagenumbersofleturesmissedperweek 17 Testingotherhypotheses AmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0 bj ajInthiscase theappropriatetstatisticis 18 Examples3 CampusCrimeandEnrollmentH0 benroll 1H1 benroll 1 19 Examples4 HousingPricesandAirPollutionH0 blog nox 1H1 blog nox 1 20 ConfidenceIntervals Anotherwaytouseclassicalstatisticaltestingistoconstructaconfidenceintervalusingthesamecriticalvalueaswasusedforatwo sidedtestA 1 a confidenceintervalisdefinedas 21 Computingp valuesforttests Analternativetotheclassicalapproachistoask whatisthesmallestsignificancelevelatwhichthenullwouldberejected So computethetstatistic andthenlookupwhatpercentileitisintheappropriatetdistribution thisisthep valuep valueistheprobabilitywewouldobservethetstatisticwedid ifthenullweretrue 22 Mostcomputerpackageswillcomputethep valueforyou assumingatwo sidedtestIfyoureallywantaone sidedalternative justdividethetwo sidedp valueby2Manysoftware suchasStataorEviewsprovidesthetstatistic p value and95 confidenceintervalforH0 bj 0foryou 23 TestingaLinearCombination Supposeinsteadoftestingwhetherb1isequaltoaconstant youwanttotestifitisequaltoanotherparameter thatisH0 b1 b2Usesamebasicprocedureforformingatstatistic 24 TestingLinearCombo cont 25 TestingaLinearCombo cont So touseformula needs12 whichstandardoutputdoesnothaveManypackageswillhaveanoptiontogetit orwilljustperformthetestforyouMoregenerally youcanalwaysrestatetheproblemtogetthetestyouwant 26 Examples5 SupposeyouareinterestedintheeffectofcampaignexpendituresonoutcomesModelisvoteA b0 b1log expendA b2log expendB b3prtystrA uH0 b1 b2 orH0 q1 b1 b2 0b1 q1 b2 sosubstituteinandrearrange voteA b0 q1log expendA b2log expendB expendA b3prtystrA u 27 Example cont Thisisthesamemodelasoriginally butnowyougetastandarderrorforb1 b2 q1directlyfromthebasicregressionAnylinearcombinationofparameterscouldbetestedinasimilarmannerOtherexamplesofhypothesesaboutasinglelinearcombinationofparameters b1 1 b2 b1 5b2 b1 1 2b2 etc 28 MultipleLinearRestrictions Everythingwe vedonesofarhasinvolvedtestingasinglelinearrestriction e g b1 0orb1 b2 However wemaywanttojointlytestmultiplehypothesesaboutourparametersAtypicalexampleistesting exclusionrestrictions wewanttoknowifagroupofparametersareallequaltozero 29 TestingExclusionRestrictions NowthenullhypothesismightbesomethinglikeH0 bk q 1 0 bk 0ThealternativeisjustH1 H0isnottrueCan tjustcheckeachtstatisticseparately becausewewanttoknowiftheqparametersarejointlysignificantatagivenlevel itispossiblefornonetobeindividuallysignificantatthatlevel 30 ExclusionRestrictions cont Todothetestweneedtoestimatethe restrictedmodel withoutxk q 1 xkincluded aswellasthe unrestrictedmodel withallx sincludedIntuitively wewanttoknowifthechangeinSSRisbigenoughtowarrantinclusionofxk q 1 xk 31 TheFstatistic TheFstatisticisalwayspositive sincetheSSRfromtherestrictedmodelcan tbelessthantheSSRfromtheunrestrictedEssentiallytheFstatisticismeasuringtherelativeincreaseinSSRwhenmovingfromtheunrestrictedtorestrictedmodelq numberofrestrictions ordfr dfurn k 1 dfur 32 TheFstatistic cont TodecideiftheincreaseinSSRwhenwemovetoarestrictedmodelis bigenough torejecttheexclusions weneedtoknowaboutthesamplingdistributionofourFstatNotsurprisingly F Fq n k 1 whereqisreferredtoasthenumeratordegreesoffreedomandn k 1asthedenominatordegreesoffreedom 33 0 c a 1 a f F F TheFstatistic cont reject failtoreject RejectH0atasignificancelevelifF c 34 Example MajorLeagueBaseballPlayers Salary 35 RelationshipbetweenFandtStat TheFstatisticisintendedtodetectwhetheranycombinationofasetofcoefficientsisdifferentfromzero Thettestisbestsuitedfortestingasinglehypothesis Groupabunchofinsignificantvarialbeswithasignificantvariable itispossibleconcludethattheentiresetofvariablesisjointlyinsignificant Often whenavariableisverystatisticallysignificantanditistestedjointlywithanothersetofvariables thesetwillbejointlysignificant 36 TheR2formoftheFstatistic BecausetheSSR smaybelargeandunwieldy analternativeformoftheformulaisusefulWeusethefactthatSSR SST 1 R2 foranyregression socansubstituteinforSSRuandSSRur 37 OverallSignificance AspecialcaseofexclusionrestrictionsistotestH0 b1 b2 bk 0SincetheR2fromamodelwithonlyan