計量經(jīng)濟(jì)學(xué)導(dǎo)論第四版英文完整教學(xué)課件

計量經(jīng)濟(jì)學(xué)導(dǎo)論第四版英文完整教學(xué)課件Economics20-Prof.Anderson1Economics20-Prof.Anderson2WelcometoEconomics20WhatisEconometrics?Economics20-Prof.Anderson3WhystudyEconometrics?

Rareineconomics(andmanyotherareaswithoutlabs!)tohaveexperimentaldataNeedtousenonexperimental,orobservational,datatomakeinferencesImportanttobeabletoapplyeconomictheorytorealworlddataEconomics20-Prof.Anderson4WhystudyEconometrics?

AnempiricalanalysisusesdatatotestatheoryortoestimatearelationshipAformaleconomicmodelcanbetestedTheorymaybeambiguousastotheeffectofsomepolicychange–canuseeconometricstoevaluatetheprogramEconomics20-Prof.Anderson5TypesofData–CrossSectional

Cross-sectionaldataisarandomsampleEachobservationisanewindividual,firm,etc.withinformationatapointintimeIfthedataisnotarandomsample,wehaveasample-selectionproblemEconomics20-Prof.Anderson6TypesofData–Panel

Canpoolrandomcrosssectionsandtreatsimilartoanormalcrosssection.Willjustneedtoaccountfortimedifferences.Canfollowthesamerandomindividualobservationsovertime–knownaspaneldataorlongitudinaldataEconomics20-Prof.Anderson7TypesofData–TimeSeries

Timeseriesdatahasaseparateobservationforeachtimeperiod–e.g.stockpricesSincenotarandomsample,differentproblemstoconsiderTrendsandseasonalitywillbeimportantEconomics20-Prof.Anderson8TheQuestionofCausality

SimplyestablishingarelationshipbetweenvariablesisrarelysufficientWanttotheeffecttobeconsideredcausalIfwe’vetrulycontrolledforenoughothervariables,thentheestimatedceterisparibuseffectcanoftenbeconsideredtobecausalCanbedifficulttoestablishcausalityEconomics20-Prof.Anderson9Example:ReturnstoEducation

AmodelofhumancapitalinvestmentimpliesgettingmoreeducationshouldleadtohigherearningsInthesimplestcase,thisimpliesanequationlikeEconomics20-Prof.Anderson10Example:(continued)

Theestimateof

b1,

isthereturntoeducation,butcanitbeconsideredcausal?Whiletheerrorterm,u,includesotherfactorsaffectingearnings,wanttocontrolforasmuchaspossibleSomethingsarestillunobserved,whichcanbeproblematicEconomics20-Prof.Anderson11TheSimpleRegressionModel

y=b0+b1x+uEconomics20-Prof.Anderson12SomeTerminology

Inthesimplelinearregressionmodel,wherey=b0+b1x+u,wetypicallyrefertoyastheDependentVariable,orLeft-HandSideVariable,orExplainedVariable,orRegressandEconomics20-Prof.Anderson13SomeTerminology,cont.

Inthesimplelinearregressionofyonx,wetypicallyrefertoxastheIndependentVariable,orRight-HandSideVariable,orExplanatoryVariable,orRegressor,orCovariate,orControlVariablesEconomics20-Prof.Anderson14ASimpleAssumption

Theaveragevalueofu,theerrorterm,inthepopulationis0.Thatis,E(u)=0Thisisnotarestrictiveassumption,sincewecanalwaysuseb0

tonormalizeE(u)to0Economics20-Prof.Anderson15ZeroConditionalMean

WeneedtomakeacrucialassumptionabouthowuandxarerelatedWewantittobethecasethatknowingsomethingaboutxdoesnotgiveusanyinformationaboutu,sothattheyarecompletelyunrelated.Thatis,thatE(u|x)=E(u)=0,whichimpliesE(y|x)=b0+b1xEconomics20-Prof.Anderson16..x1x2E(y|x)asalinearfunctionofx,whereforanyx

thedistributionofyiscenteredaboutE(y|x)E(y|x)=b0+b1xyf(y)Economics20-Prof.Anderson17OrdinaryLeastSquares

BasicideaofregressionistoestimatethepopulationparametersfromasampleLet{(xi,yi):i=1,…,n}denotearandomsampleofsizenfromthepopulationForeachobservationinthissample,itwillbethecasethat

yi=b0+b1xi+uiEconomics20-Prof.Anderson18....y4y1y2y3x1x2x3x4}}{{u1u2u3u4xyPopulationregressionline,sampledatapointsandtheassociatederrortermsE(y|x)=b0+b1xEconomics20-Prof.Anderson19DerivingOLSEstimates

ToderivetheOLSestimatesweneedtorealizethatourmainassumptionofE(u|x)=E(u)=0alsoimpliesthatCov(x,u)=E(xu)=0Why?RememberfrombasicprobabilitythatCov(X,Y)=E(XY)–E(X)E(Y)Economics20-Prof.Anderson20DerivingOLScontinued

Wecanwriteour2restrictionsjustintermsofx,y,b0andb1,sinceu=y–b0–b1xE(y–b0–b1x)=0E[x(y–b0–b1x)]=0ThesearecalledmomentrestrictionsEconomics20-Prof.Anderson21DerivingOLSusingM.O.M.

ThemethodofmomentsapproachtoestimationimpliesimposingthepopulationmomentrestrictionsonthesamplemomentsWhatdoesthismean?RecallthatforE(X),themeanofapopulationdistribution,asampleestimatorofE(X)issimplythearithmeticmeanofthesampleEconomics20-Prof.Anderson22MoreDerivationofOLS

WewanttochoosevaluesoftheparametersthatwillensurethatthesampleversionsofourmomentrestrictionsaretrueThesampleversionsareasfollows:Economics20-Prof.Anderson23MoreDerivationofOLSGiventhedefinitionofasamplemean,andpropertiesofsummation,wecanrewritethefirstconditionasfollowsEconomics20-Prof.Anderson24MoreDerivationofOLSEconomics20-Prof.Anderson25SotheOLSestimatedslopeisEconomics20-Prof.Anderson26SummaryofOLSslopeestimate

TheslopeestimateisthesamplecovariancebetweenxandydividedbythesamplevarianceofxIfxandyarepositivelycorrelated,theslopewillbepositiveIfxandyarenegativelycorrelated,theslopewillbenegativeOnlyneedxtovaryinoursampleEconomics20-Prof.Anderson27MoreOLS

Intuitively,OLSisfittingalinethroughthesamplepointssuchthatthesumofsquaredresidualsisassmallaspossible,hencethetermleastsquaresTheresidual,?,isanestimateoftheerrorterm,u,andisthedifferencebetweenthefittedline(sampleregressionfunction)andthesamplepointEconomics20-Prof.Anderson28....y4y1y2y3x1x2x3x4}}{{?1?2?3?4xySampleregressionline,sampledatapointsandtheassociatedestimatederrortermsEconomics20-Prof.Anderson29Alternateapproachtoderivation

Giventheintuitiveideaoffittingaline,wecansetupaformalminimizationproblemThatis,wewanttochooseourparameterssuchthatweminimizethefollowing:Economics20-Prof.Anderson30Alternateapproach,continued

Ifoneusescalculustosolvetheminimizationproblemforthetwoparametersyouobtainthefollowingfirstorderconditions,whicharethesameasweobtainedbefore,multipliedbynEconomics20-Prof.Anderson31AlgebraicPropertiesofOLS

ThesumoftheOLSresidualsiszeroThus,thesampleaverageoftheOLSresidualsiszeroaswellThesamplecovariancebetweentheregressorsandtheOLSresidualsiszeroTheOLSregressionlinealwaysgoesthroughthemeanofthesampleEconomics20-Prof.Anderson32AlgebraicProperties(precise)Economics20-Prof.Anderson33MoreterminologyEconomics20-Prof.Anderson34ProofthatSST=SSE+SSREconomics20-Prof.Anderson35Goodness-of-Fit

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregressionR2=SSE/SST=1–SSR/SSTEconomics20-Prof.Anderson36UsingStataforOLSregressions

Nowthatwe’vederivedtheformulaforcalculatingtheOLSestimatesofourparameters,you’llbehappytoknowyoudon’thavetocomputethembyhandRegressionsinStataareverysimple,toruntheregressionofyonx,justtyperegyxEconomics20-Prof.Anderson37UnbiasednessofOLS

Assumethepopulationmodelislinearinparametersasy=b0+b1x+uAssumewecanusearandomsampleofsizen,{(xi,yi):i=1,2,…,n},fromthepopulationmodel.Thuswecanwritethesamplemodelyi=b0+b1xi+uiAssumeE(u|x)=0andthusE(ui|xi)=0

AssumethereisvariationinthexiEconomics20-Prof.Anderson38UnbiasednessofOLS(cont)

Inordertothinkaboutunbiasedness,weneedtorewriteourestimatorintermsofthepopulationparameterStartwithasimplerewriteoftheformulaasEconomics20-Prof.Anderson39UnbiasednessofOLS(cont)Economics20-Prof.Anderson40UnbiasednessofOLS(cont)Economics20-Prof.Anderson41UnbiasednessofOLS(cont)Economics20-Prof.Anderson42UnbiasednessSummary

TheOLSestimatesofb1andb0areunbiasedProofofunbiasednessdependsonour4assumptions–ifanyassumptionfails,thenOLSisnotnecessarilyunbiasedRememberunbiasednessisadescriptionoftheestimator–inagivensamplewemaybe“near”or“far”fromthetrueparameterEconomics20-Prof.Anderson43VarianceoftheOLSEstimators

NowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameterWanttothinkabouthowspreadoutthisdistributionisMucheasiertothinkaboutthisvarianceunderanadditionalassumption,soAssumeVar(u|x)=s2(Homoskedasticity)Economics20-Prof.Anderson44VarianceofOLS(cont)

Var(u|x)=E(u2|x)-[E(u|x)]2E(u|x)=0,sos2

=E(u2|x)=E(u2)=Var(u)Thuss2isalsotheunconditionalvariance,calledtheerrorvariance

s,thesquarerootoftheerrorvarianceiscalledthestandarddeviationoftheerrorCansay:E(y|x)=b0+b1xandVar(y|x)=s2Economics20-Prof.Anderson45..x1x2HomoskedasticCaseE(y|x)=b0+b1xyf(y|x)Economics20-Prof.Anderson46.x

x1x2yf(y|x)HeteroskedasticCasex3..E(y|x)=b0+b1xEconomics20-Prof.Anderson47VarianceofOLS(cont)Economics20-Prof.Anderson48VarianceofOLSSummary

Thelargertheerrorvariance,s2,thelargerthevarianceoftheslopeestimateThelargerthevariabilityinthexi,thesmallerthevarianceoftheslopeestimateAsaresult,alargersamplesizeshoulddecreasethevarianceoftheslopeestimateProblemthattheerrorvarianceisunknownEconomics20-Prof.Anderson49EstimatingtheErrorVariance

Wedon’tknowwhattheerrorvariance,s2,is,becausewedon’tobservetheerrors,uiWhatweobservearetheresiduals,?iWecanusetheresidualstoformanestimateoftheerrorvarianceEconomics20-Prof.Anderson50ErrorVarianceEstimate(cont)Economics20-Prof.Anderson51ErrorVarianceEstimate(cont)Economics20-Prof.Anderson52MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u1.EstimationEconomics20-Prof.Anderson53ParallelswithSimpleRegression

b0isstilltheintercept

b1tobkallcalledslopeparameters

uisstilltheerrorterm(ordisturbance)Stillneedtomakeazeroconditionalmeanassumption,sonowassumethatE(u|x1,x2,…,xk)=0Stillminimizingthesumofsquaredresiduals,sohavek+1firstorderconditionsEconomics20-Prof.Anderson54InterpretingMultipleRegressionEconomics20-Prof.Anderson55A“PartiallingOut”InterpretationEconomics20-Prof.Anderson56“PartiallingOut”continued

Previousequationimpliesthatregressingyonx1

andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimatingtheeffectofx1onyafterx2hasbeen“partialledout”Economics20-Prof.Anderson57SimplevsMultipleRegEstimateEconomics20-Prof.Anderson58Goodness-of-FitEconomics20-Prof.Anderson59Goodness-of-Fit(continued)

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregressionR2=SSE/SST=1–SSR/SSTEconomics20-Prof.Anderson60Goodness-of-Fit(continued)Economics20-Prof.Anderson61MoreaboutR-squared

R2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincreaseBecauseR2

willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytocomparemodelsEconomics20-Prof.Anderson62AssumptionsforUnbiasedness

Populationmodelislinearinparameters:y=b0+b1x1+b2x2+…+bkxk

+uWecanusearandomsampleofsizen,{(xi1,xi2,…,xik,yi):i=1,2,…,n},fromthepopulationmodel,sothatthesamplemodelisyi=b0+b1xi1+b2xi2+…+bkxik

+ui

E(u|x1,x2,…xk)=0,implyingthatalloftheexplanatoryvariablesareexogenousNoneofthex’sisconstant,andtherearenoexactlinearrelationshipsamongthemEconomics20-Prof.Anderson63TooManyorTooFewVariables

Whathappensifweincludevariablesinourspecificationthatdon’tbelong?Thereisnoeffectonourparameterestimate,andOLSremainsunbiasedWhatifweexcludeavariablefromourspecificationthatdoesbelong?OLSwillusuallybebiasedEconomics20-Prof.Anderson64OmittedVariableBiasEconomics20-Prof.Anderson65OmittedVariableBias(cont)Economics20-Prof.Anderson66OmittedVariableBias(cont)Economics20-Prof.Anderson67OmittedVariableBias(cont)Economics20-Prof.Anderson68SummaryofDirectionofBiasCorr(x1,x2)>0Corr(x1,x2)<0b2>0PositivebiasNegativebiasb2<0NegativebiasPositivebiasEconomics20-Prof.Anderson69OmittedVariableBiasSummary

Twocaseswherebiasisequaltozerob2=0,thatisx2doesn’treallybelonginmodelx1andx2areuncorrelatedinthesampleIfcorrelationbetweenx2,x1andx2,yisthesamedirection,biaswillbepositiveIfcorrelationbetweenx2,x1andx2,yistheoppositedirection,biaswillbenegativeEconomics20-Prof.Anderson70TheMoreGeneralCase

Technically,canonlysignthebiasforthemoregeneralcaseifalloftheincludedx’sareuncorrelatedTypically,then,weworkthroughthebiasassumingthex’sareuncorrelated,asausefulguideevenifthisassumptionisnotstrictlytrueEconomics20-Prof.Anderson71VarianceoftheOLSEstimators

NowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameterWanttothinkabouthowspreadoutthisdistributionisMucheasiertothinkaboutthisvarianceunderanadditionalassumption,soAssumeVar(u|x1,x2,…,xk)=s2(Homoskedasticity)Economics20-Prof.Anderson72VarianceofOLS(cont)

Letxstandfor(x1,x2,…xk)AssumingthatVar(u|x)=s2alsoimpliesthatVar(y|x)=s2

The4assumptionsforunbiasedness,plusthishomoskedasticityassumptionareknownastheGauss-MarkovassumptionsEconomics20-Prof.Anderson73VarianceofOLS(cont)Economics20-Prof.Anderson74ComponentsofOLSVariances

Theerrorvariance:alargers2impliesalargervariancefortheOLSestimatorsThetotalsamplevariation:alargerSSTjimpliesasmallervariancefortheestimatorsLinearrelationshipsamongtheindependentvariables:alargerRj2impliesalargervariancefortheestimatorsEconomics20-Prof.Anderson75MisspecifiedModelsEconomics20-Prof.Anderson76MisspecifiedModels(cont)

Whilethevarianceoftheestimatorissmallerforthemisspecifiedmodel,unlessb2=0themisspecifiedmodelisbiasedAsthesamplesizegrows,thevarianceofeachestimatorshrinkstozero,makingthevariancedifferencelessimportantEconomics20-Prof.Anderson77EstimatingtheErrorVariance

Wedon’tknowwhattheerrorvariance,s2,is,becausewedon’tobservetheerrors,uiWhatweobservearetheresiduals,?iWecanusetheresidualstoformanestimateoftheerrorvarianceEconomics20-Prof.Anderson78ErrorVarianceEstimate(cont)

df=n–(k+1),ordf=n–k–1df(i.e.degreesoffreedom)isthe(numberofobservations)–(numberofestimatedparameters)Economics20-Prof.Anderson79TheGauss-MarkovTheorem

Givenour5Gauss-MarkovAssumptionsitcanbeshownthatOLSis“BLUE”BestLinearUnbiasedEstimatorThus,iftheassumptionshold,useOLSEconomics20-Prof.Anderson80MultipleRegressionAnalysis

y=b0+b1x1+b2x2+...bkxk+u2.InferenceEconomics20-Prof.Anderson81AssumptionsoftheClassicalLinearModel(CLM)

Sofar,weknowthatgiventheGauss-Markovassumptions,OLSisBLUE,Inordertodoclassicalhypothesistesting,weneedtoaddanotherassumption(beyondtheGauss-Markovassumptions)Assumethatuisindependentofx1,x2,…,xkanduisnormallydistributedwithzeromeanandvariances2:u~Normal(0,s2)Economics20-Prof.Anderson82CLMAssumptions(cont)

UnderCLM,OLSisnotonlyBLUE,butistheminimumvarianceunbiasedestimatorWecansummarizethepopulationassumptionsofCLMasfollows

y|x~Normal(b0+b1x1+…+bkxk,s2)Whilefornowwejustassumenormality,clearthatsometimesnotthecaseLargesampleswillletusdropnormalityEconomics20-Prof.Anderson83..x1x2ThehomoskedasticnormaldistributionwithasingleexplanatoryvariableE(y|x)=b0+b1xyf(y|x)NormaldistributionsEconomics20-Prof.Anderson84NormalSamplingDistributionsEconomics20-Prof.Anderson85ThetTestEconomics20-Prof.Anderson86ThetTest(cont)

KnowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistestsStartwithanullhypothesisForexample,H0:bj=0Ifacceptnull,thenacceptthatxjhasnoeffectony,controllingforotherx’sEconomics20-Prof.Anderson87ThetTest(cont)Economics20-Prof.Anderson88tTest:One-SidedAlternatives

Besidesournull,H0,weneedanalternativehypothesis,H1,andasignificancelevelH1maybeone-sided,ortwo-sidedH1:bj>0andH1:bj<0areone-sidedH1:bj

0isatwo-sidedalternativeIfwewanttohaveonlya5%probabilityofrejectingH0ifitisreallytrue,thenwesayoursignificancelevelis5%Economics20-Prof.Anderson89One-SidedAlternatives(cont)

Havingpickedasignificancelevel,a,welookupthe(1–a)thpercentileinatdistributionwithn–k–1dfandcallthisc,thecriticalvalueWecanrejectthenullhypothesisifthetstatisticisgreaterthanthecriticalvalueIfthetstatisticislessthanthecriticalvaluethenwefailtorejectthenullEconomics20-Prof.Anderson90yi=b0+b1xi1+…

+bkxik+uiH0:bj=0H1:bj>0c0a(1-a)One-SidedAlternatives(cont)FailtorejectrejectEconomics20-Prof.Anderson91One-sidedvsTwo-sided

Becausethetdistributionissymmetric,testingH1:bj<0isstraightforward.ThecriticalvalueisjustthenegativeofbeforeWecanrejectthenullifthetstatistic<–c,andifthetstatistic>than–cthenwefailtorejectthenullForatwo-sidedtest,wesetthecriticalvaluebasedona/2andrejectH1:bj

0iftheabsolutevalueofthetstatistic>cEconomics20-Prof.Anderson92yi=b0+b1Xi1+…

+bkXik+uiH0:bj=0H1:bj>0c0a/2(1-a)-ca/2Two-SidedAlternativesrejectrejectfailtorejectEconomics20-Prof.Anderson93SummaryforH0:bj=0

Unlessotherwisestated,thealternativeisassumedtobetwo-sidedIfwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthea%level”Ifwefailtorejectthenull,wetypicallysay“xjisstatisticallyinsignificantatthea%level”Economics20-Prof.Anderson94TestingotherhypothesesAmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0:bj=aj

Inthiscase,theappropriatetstatisticisEconomics20-Prof.Anderson95ConfidenceIntervals

Anotherwaytouseclassicalstatisticaltestingistoconstructaconfidenceintervalusingthesamecriticalvalueaswasusedforatwo-sidedtestA(1-a)%confidenceintervalisdefinedasEconomics20-Prof.Anderson96Computingp-valuesforttests

Analternativetotheclassicalapproachistoask,“whatisthesmallestsignificancelevelatwhichthenullwouldberejected?”So,computethetstatistic,andthenlookupwhatpercentileitisintheappropriatetdistribution–thisisthep-value

p-valueistheprobabilitywewouldobservethetstatisticwedid,ifthenullweretrueEconomics20-Prof.Anderson97Stataandp-values,ttests,etc.

Mostcomputerpackageswillcomputethep-valueforyou,assumingatwo-sidedtestIfyoureallywantaone-sidedalternative,justdividethetwo-sidedp-valueby2Stataprovidesthetstatistic,p-value,and95%confidenceintervalforH0:bj=0foryou,incolumnslabeled“t”,“P>|t|”and“[95%Conf.Interval]”,respectivelyEconomics20-Prof.Anderson98TestingaLinearCombination

Supposeinsteadoftestingwhetherb1isequaltoaconstant,youwanttotestifitisequaltoanotherparameter,thatisH0:b1=b2UsesamebasicprocedureforformingatstatisticEconomics20-Prof.Anderson99TestingLinearCombo(cont)Economics20-Prof.Anderson100TestingaLinearCombo(cont)

So,touseformula,needs12,whichstandardoutputdoesnothaveManypackageswillhaveanoptiontogetit,orwilljustperformthetestforyouInStata,afterregyx1x2…xkyouwouldtypetestx1=x2togetap-valueforthetestMoregenerally,youcanalwaysrestatetheproblemtogetthetestyouwantEconomics20-Prof.Anderson101Example:

SupposeyouareinterestedintheeffectofcampaignexpendituresonoutcomesModelisvoteA=b0+b1log(expendA)+b2log(expendB)+b3prtystrA+u

H0:b1=-b2,orH0:q1=b1+b2=0

b1=q1–b2,sosubstituteinandrearrange

voteA=b0+q1log(expendA)+b2log(expendB-expendA)+b3prtystrA+uEconomics20-Prof.Anderson102Example(cont):

Thisisthesamemodelasoriginally,butnowyougetastandarderrorforb1–b2=q1directlyfromthebasicregressionAnylinearcombinationofparameterscouldbetestedinasimilarmannerOtherexamplesofhypothesesaboutasinglelinearcombinationofparameters:b1=1+b2;b1=5b2;b1=-1/2b2;etcEconomics20-Prof.Anderson103MultipleLinearRestrictions

Everythingwe’vedonesofarhasinvolvedtestingasinglelinearrestriction,(e.g.b1=0orb1=b2)However,wemaywanttojointlytestmultiplehypothesesaboutourparametersAtypicalexampleistesting“exclusionrestrictions”–wewanttoknowifagroupofparametersareallequaltozeroEconomics20-Prof.Anderson104TestingExclusionRestrictions

NowthenullhypothesismightbesomethinglikeH0:bk-q+1=0,...,bk=0ThealternativeisjustH1:H0isnottrueCan’tjustcheckeachtstatisticseparately,becausewewanttoknowiftheqparametersarejointlysignificantatagivenlevel–itispossiblefornonetobeindividuallysignificantatthatlevelEconomics20-Prof.Anderson105ExclusionRestrictions(cont)

Todothetestweneedtoestimatethe“restrictedmodel”withoutxk-q+1,,…,xkincluded,aswellasthe“unrestrictedmodel”withallx’sincludedIntuitively,wewanttoknowifthechangeinSSRisbigenoughtowarrantinclusionofxk-q+1,,…,xk

Economics20-Prof.Anderson106TheFstatistic

TheFstatisticisalwayspositive,sincetheSSRfromtherestrictedmodelcan’tbelessthantheSSRfromtheunrestrictedEssentiallytheFstatisticismeasuringtherelativeincreaseinSSRwhenmovingfromtheunrestrictedtorestrictedmodel

q=numberofrestrictions,ordfr–dfur

n–k–1=dfurEconomics20-Prof.Anderson107TheFstatistic(cont)

TodecideiftheincreaseinSSRwhenwemovetoarestrictedmodelis“bigenough”torejecttheexclusions,weneedtoknowaboutthesamplingdistributionofourFstatNotsurprisingly,F~Fq,n-k-1,whereqisreferredtoasthenumeratordegreesoffreedomandn–k–1asthedenominatordegreesoffreedomEconomics20-Prof.Anderson1080ca(1-a)f(F)FTheFstatistic(cont)rejectfailtorejectRejectH0atasignificancelevelifF>cEconomics20-Prof.Anderson109TheR2formoftheFstatistic

BecausetheSSR’smaybelargeandunwieldy,analternativeformoftheformulaisusefulWeusethefactthatSSR=SST(1–R2)foranyregression,socansubstituteinforSSRuandSSRurEconomics20-Prof.Anderson110OverallSignificance

AspecialcaseofexclusionrestrictionsistotestH0:b1=b2=…=bk=0SincetheR2fromamodelwithonlyaninterceptwillbezero,theFstatisticissimplyEconomics20-Prof.Anderson111GeneralLinearRestrictions

ThebasicformoftheFstatisticwillworkforanysetoflinearrestrictionsFirstestimatetheunrestrictedmodelandthenestimatetherestrictedmodelIneachcase,makenoteoftheSSRImposingtherestrictionscanbetricky–willlikelyhavetoredefinevariablesagainEconomics20-Prof.Anderson112Example:

UsesamevotingmodelasbeforeModelisvoteA=b0+b1log(expendA)+b2log(expendB)+b3prtystrA+unownullisH0:b1=1,b3=0Substitutingintherestrictions:voteA=b0+log(expendA)+b2log(expendB)+u,soUsevoteA-log(expendA)=b0+b2log(expendB)+uasrestrictedmodelEconomics20-Prof.Anderson113FStatisticSummary

Justaswithtstatistics,p-valuescanbecalculatedbylookingupthepercentileintheappropriateFdistributionStatawilldothisbyentering:displayfprob(q,n–k–1,F),wheretheappropriatevaluesofF,q,andn