《計量經(jīng)濟學(xué)導(dǎo)論》電子教案英文版

WelcometoEconomics20

WhatisEconometrics?

Economics20-Prof.Anderson

WhystudyEconometrics?

Rareineconomics(andmanyotherareaswithoutlabs!)tohaveexperimentaldata

Needtousenonexperimental,orobservational,datatomakeinferences

Importanttobeabletoapplyeconomictheorytorealworlddata

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WhystudyEconometrics?

Anempiricalanalysisusesdatatotestatheoryortoestimatearelationship

Aformaleconomicmodelcanbetested

Theorymaybeambiguousastotheeffectofsomepolicychange–canuseeconometricstoevaluatetheprogram

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TypesofData–CrossSectional

Cross-sectionaldataisarandomsample

Eachobservationisanewindividual,firm,etc.withinformationatapointintime

Ifthedataisnotarandomsample,wehaveasample-selectionproblem

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TypesofData–Panel

Canpoolrandomcrosssectionsandtreatsimilartoanormalcrosssection.Willjustneedtoaccountfortimedifferences.

Canfollowthesamerandomindividualobservationsovertime–knownaspaneldataorlongitudinaldata

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TypesofData–TimeSeries

Timeseriesdatahasaseparateobservationforeachtimeperiod–e.g.stockprices

Sincenotarandomsample,differentproblemstoconsider

Trendsandseasonalitywillbeimportant

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TheQuestionofCausality

Simplyestablishingarelationshipbetweenvariablesisrarelysufficient

Wanttotheeffecttobeconsideredcausal

Ifwe’vetrulycontrolledforenoughothervariables,thentheestimatedceterisparibuseffectcanoftenbeconsideredtobecausal

Canbedifficulttoestablishcausality

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Example:ReturnstoEducation

Amodelofhumancapitalinvestmentimpliesgettingmoreeducationshouldleadtohigherearnings

Inthesimplestcase,thisimpliesanequationlike

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Example:(continued)

Theestimateofb1,isthereturntoeducation,butcanitbeconsideredcausal?

Whiletheerrorterm,u,includesotherfactorsaffectingearnings,wanttocontrolforasmuchaspossible

Somethingsarestillunobserved,whichcanbeproblematic

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TheSimpleRegressionModel

y=b0+b1x+u

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SomeTerminology

Inthesimplelinearregressionmodel,wherey=b0+b1x+u,wetypicallyrefertoyasthe

DependentVariable,or

Left-HandSideVariable,or

ExplainedVariable,or

Regressand

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SomeTerminology,cont.

Inthesimplelinearregressionofyonx,wetypicallyrefertoxasthe

IndependentVariable,or

Right-HandSideVariable,or

ExplanatoryVariable,or

Regressor,or

Covariate,or

ControlVariables

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ASimpleAssumption

Theaveragevalueofu,theerrorterm,inthepopulationis0.Thatis,

E(u)=0

Thisisnotarestrictiveassumption,sincewecanalwaysuseb0tonormalizeE(u)to0

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ZeroConditionalMean

Weneedtomakeacrucialassumptionabouthowuandxarerelated

Wewantittobethecasethatknowingsomethingaboutxdoesnotgiveusanyinformationaboutu,sothattheyarecompletelyunrelated.Thatis,that

E(u|x)=E(u)=0,whichimplies

E(y|x)=b0+b1x

Economics20-Prof.Anderson

.

.

x1

x2

E(y|x)asalinearfunctionofx,whereforanyx

thedistributionofyiscenteredaboutE(y|x)

E(y|x)=b0+b1x

y

f(y)

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OrdinaryLeastSquares

Basicideaofregressionistoestimatethepopulationparametersfromasample

Let{(xi,yi):i=1,…,n}denotearandomsampleofsizenfromthepopulation

Foreachobservationinthissample,itwillbethecasethat

yi=b0+b1xi+ui

Economics20-Prof.Anderson

.

.

.

.

y4

y1

y2

y3

x1

x2

x3

x4

}

}

{

{

u1

u2

u3

u4

x

y

Populationregressionline,sampledatapoints

andtheassociatederrorterms

E(y|x)=b0+b1x

Economics20-Prof.Anderson

DerivingOLSEstimates

ToderivetheOLSestimatesweneedtorealizethatourmainassumptionofE(u|x)=E(u)=0alsoimpliesthat

Cov(x,u)=E(xu)=0

Why?RememberfrombasicprobabilitythatCov(X,Y)=E(XY)–E(X)E(Y)

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DerivingOLScontinued

Wecanwriteour2restrictionsjustintermsofx,y,b0andb1,sinceu=y–b0–b1x

E(y–b0–b1x)=0

E[x(y–b0–b1x)]=0

Thesearecalledmomentrestrictions

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DerivingOLSusingM.O.M.

Themethodofmomentsapproachtoestimationimpliesimposingthepopulationmomentrestrictionsonthesamplemoments

Whatdoesthismean?RecallthatforE(X),themeanofapopulationdistribution,asampleestimatorofE(X)issimplythearithmeticmeanofthesample

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MoreDerivationofOLS

Wewanttochoosevaluesoftheparametersthatwillensurethatthesampleversionsofourmomentrestrictionsaretrue

Thesampleversionsareasfollows:

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MoreDerivationofOLS

Giventhedefinitionofasamplemean,andpropertiesofsummation,wecanrewritethefirstconditionasfollows

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MoreDerivationofOLS

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SotheOLSestimatedslopeis

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SummaryofOLSslopeestimate

Theslopeestimateisthesamplecovariancebetweenxandydividedbythesamplevarianceofx

Ifxandyarepositivelycorrelated,theslopewillbepositive

Ifxandyarenegativelycorrelated,theslopewillbenegative

Onlyneedxtovaryinoursample

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MoreOLS

Intuitively,OLSisfittingalinethroughthesamplepointssuchthatthesumofsquaredresidualsisassmallaspossible,hencethetermleastsquares

Theresidual,??,isanestimateoftheerrorterm,u,andisthedifferencebetweenthefittedline(sampleregressionfunction)andthesamplepoint

Economics20-Prof.Anderson

.

.

.

.

y4

y1

y2

y3

x1

x2

x3

x4

}

}

{

{

??1

??2

??3

??4

x

y

Sampleregressionline,sampledatapoints

andtheassociatedestimatederrorterms

Economics20-Prof.Anderson

Alternateapproachtoderivation

Giventheintuitiveideaoffittingaline,wecansetupaformalminimizationproblem

Thatis,wewanttochooseourparameterssuchthatweminimizethefollowing:

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Alternateapproach,continued

Ifoneusescalculustosolvetheminimizationproblemforthetwoparametersyouobtainthefollowingfirstorderconditions,whicharethesameasweobtainedbefore,multipliedbyn

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AlgebraicPropertiesofOLS

ThesumoftheOLSresidualsiszero

Thus,thesampleaverageoftheOLSresidualsiszeroaswell

ThesamplecovariancebetweentheregressorsandtheOLSresidualsiszero

TheOLSregressionlinealwaysgoesthroughthemeanofthesample

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AlgebraicProperties(precise)

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Moreterminology

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ProofthatSST=SSE+SSR

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Goodness-of-Fit

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?

Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregression

R2=SSE/SST=1–SSR/SST

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UsingStataforOLSregressions

Nowthatwe’vederivedtheformulaforcalculatingtheOLSestimatesofourparameters,you’llbehappytoknowyoudon’thavetocomputethembyhand

RegressionsinStataareverysimple,toruntheregressionofyonx,justtype

regyx

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UnbiasednessofOLS

Assumethepopulationmodelislinearinparametersasy=b0+b1x+u

Assumewecanusearandomsampleofsizen,{(xi,yi):i=1,2,…,n},fromthepopulationmodel.Thuswecanwritethesamplemodelyi=b0+b1xi+ui

AssumeE(u|x)=0andthusE(ui|xi)=0

Assumethereisvariationinthexi

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UnbiasednessofOLS(cont)

Inordertothinkaboutunbiasedness,weneedtorewriteourestimatorintermsofthepopulationparameter

Startwithasimplerewriteoftheformulaas

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UnbiasednessofOLS(cont)

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UnbiasednessofOLS(cont)

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UnbiasednessofOLS(cont)

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UnbiasednessSummary

TheOLSestimatesofb1andb0areunbiased

Proofofunbiasednessdependsonour4assumptions–ifanyassumptionfails,thenOLSisnotnecessarilyunbiased

Rememberunbiasednessisadescriptionoftheestimator–inagivensamplewemaybe“near”or“far”fromthetrueparameter

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VarianceoftheOLSEstimators

Nowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameter

Wanttothinkabouthowspreadoutthisdistributionis

Mucheasiertothinkaboutthisvarianceunderanadditionalassumption,so

AssumeVar(u|x)=s2(Homoskedasticity)

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VarianceofOLS(cont)

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

E(u|x)=0,sos2=E(u2|x)=E(u2)=Var(u)

Thuss2isalsotheunconditionalvariance,calledtheerrorvariance

s,thesquarerootoftheerrorvarianceiscalledthestandarddeviationoftheerror

Cansay:E(y|x)=b0+b1xandVar(y|x)=s2

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.

.

x1

x2

HomoskedasticCase

E(y|x)=b0+b1x

y

f(y|x)

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.

x

x1

x2

y

f(y|x)

HeteroskedasticCase

x3

.

.

E(y|x)=b0+b1x

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VarianceofOLS(cont)

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VarianceofOLSSummary

Thelargertheerrorvariance,s2,thelargerthevarianceoftheslopeestimate

Thelargerthevariabilityinthexi,thesmallerthevarianceoftheslopeestimate

Asaresult,alargersamplesizeshoulddecreasethevarianceoftheslopeestimate

Problemthattheerrorvarianceisunknown

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EstimatingtheErrorVariance

Wedon’tknowwhattheerrorvariance,s2,is,becausewedon’tobservetheerrors,ui

Whatweobservearetheresiduals,??i

Wecanusetheresidualstoformanestimateoftheerrorvariance

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ErrorVarianceEstimate(cont)

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ErrorVarianceEstimate(cont)

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MultipleRegressionAnalysis

y=b0+b1x1+b2x2+...bkxk+u

1.Estimation

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ParallelswithSimpleRegression

b0isstilltheintercept

b1tobkallcalledslopeparameters

uisstilltheerrorterm(ordisturbance)

Stillneedtomakeazeroconditionalmeanassumption,sonowassumethat

E(u|x1,x2,…,xk)=0

Stillminimizingthesumofsquaredresiduals,sohavek+1firstorderconditions

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InterpretingMultipleRegression

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A“PartiallingOut”Interpretation

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“PartiallingOut”continued

Previousequationimpliesthatregressingyonx1andx2givessameeffectofx1asregressingyonresidualsfromaregressionofx1onx2

Thismeansonlythepartofxi1thatisuncorrelatedwithxi2arebeingrelatedtoyisowe’reestimatingtheeffectofx1onyafterx2hasbeen“partialledout”

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SimplevsMultipleRegEstimate

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Goodness-of-Fit

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Goodness-of-Fit(continued)

Howdowethinkabouthowwelloursampleregressionlinefitsoursampledata?

Cancomputethefractionofthetotalsumofsquares(SST)thatisexplainedbythemodel,callthistheR-squaredofregression

R2=SSE/SST=1–SSR/SST

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Goodness-of-Fit(continued)

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MoreaboutR-squared

R2canneverdecreasewhenanotherindependentvariableisaddedtoaregression,andusuallywillincrease

BecauseR2willusuallyincreasewiththenumberofindependentvariables,itisnotagoodwaytocomparemodels

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AssumptionsforUnbiasedness

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

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

E(u|x1,x2,…xk)=0,implyingthatalloftheexplanatoryvariablesareexogenous

Noneofthex’sisconstant,andtherearenoexactlinearrelationshipsamongthem

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TooManyorTooFewVariables

Whathappensifweincludevariablesinourspecificationthatdon’tbelong?

Thereisnoeffectonourparameterestimate,andOLSremainsunbiased

Whatifweexcludeavariablefromourspecificationthatdoesbelong?

OLSwillusuallybebiased

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OmittedVariableBias

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OmittedVariableBias(cont)

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OmittedVariableBias(cont)

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OmittedVariableBias(cont)

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SummaryofDirectionofBias

Positivebias

Negativebias

b2<0

Negativebias

Positivebias

b2>0

Corr(x1,x2)<0

Corr(x1,x2)>0

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OmittedVariableBiasSummary

Twocaseswherebiasisequaltozero

b2=0,thatisx2doesn’treallybelonginmodel

x1andx2areuncorrelatedinthesample

Ifcorrelationbetweenx2,x1andx2,yisthesamedirection,biaswillbepositive

Ifcorrelationbetweenx2,x1andx2,yistheoppositedirection,biaswillbenegative

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TheMoreGeneralCase

Technically,canonlysignthebiasforthemoregeneralcaseifalloftheincludedx’sareuncorrelated

Typically,then,weworkthroughthebiasassumingthex’sareuncorrelated,asausefulguideevenifthisassumptionisnotstrictlytrue

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VarianceoftheOLSEstimators

Nowweknowthatthesamplingdistributionofourestimateiscenteredaroundthetrueparameter

Wanttothinkabouthowspreadoutthisdistributionis

Mucheasiertothinkaboutthisvarianceunderanadditionalassumption,so

AssumeVar(u|x1,x2,…,xk)=s2(Homoskedasticity)

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VarianceofOLS(cont)

Letxstandfor(x1,x2,…xk)

AssumingthatVar(u|x)=s2alsoimpliesthatVar(y|x)=s2

The4assumptionsforunbiasedness,plusthishomoskedasticityassumptionareknownastheGauss-Markovassumptions

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VarianceofOLS(cont)

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ComponentsofOLSVariances

Theerrorvariance:alargers2impliesalargervariancefortheOLSestimators

Thetotalsamplevariation:alargerSSTjimpliesasmallervariancefortheestimators

Linearrelationshipsamongtheindependentvariables:alargerRj2impliesalargervariancefortheestimators

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MisspecifiedModels

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MisspecifiedModels(cont)

Whilethevarianceoftheestimatorissmallerforthemisspecifiedmodel,unlessb2=0themisspecifiedmodelisbiased

Asthesamplesizegrows,thevarianceofeachestimatorshrinkstozero,makingthevariancedifferencelessimportant

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EstimatingtheErrorVariance

Wedon’tknowwhattheerrorvariance,s2,is,becausewedon’tobservetheerrors,ui

Whatweobservearetheresiduals,??i

Wecanusetheresidualstoformanestimateoftheerrorvariance

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ErrorVarianceEstimate(cont)

df=n–(k+1),ordf=n–k–1

df(i.e.degreesoffreedom)isthe(numberofobservations)–(numberofestimatedparameters)

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TheGauss-MarkovTheorem

Givenour5Gauss-MarkovAssumptionsitcanbeshownthatOLSis“BLUE”

Best

Linear

Unbiased

Estimator

Thus,iftheassumptionshold,useOLS

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MultipleRegressionAnalysis

y=b0+b1x1+b2x2+...bkxk+u

2.Inference

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AssumptionsoftheClassicalLinearModel(CLM)

Sofar,weknowthatgiventheGauss-Markovassumptions,OLSisBLUE,

Inordertodoclassicalhypothesistesting,weneedtoaddanotherassumption(beyondtheGauss-Markovassumptions)

Assumethatuisindependentofx1,x2,…,xkanduisnormallydistributedwithzeromeanandvariances2:u~Normal(0,s2)

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CLMAssumptions(cont)

UnderCLM,OLSisnotonlyBLUE,butistheminimumvarianceunbiasedestimator

WecansummarizethepopulationassumptionsofCLMasfollows

y|x~Normal(b0+b1x1+…+bkxk,s2)

Whilefornowwejustassumenormality,clearthatsometimesnotthecase

Largesampleswillletusdropnormality

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.

.

x1

x2

Thehomoskedasticnormaldistributionwith

asingleexplanatoryvariable

E(y|x)=b0+b1x

y

f(y|x)

Normal

distributions

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NormalSamplingDistributions

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ThetTest

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ThetTest(cont)

Knowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistests

Startwithanullhypothesis

Forexample,H0:bj=0

Ifacceptnull,thenacceptthatxjhasnoeffectony,controllingforotherx’s

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ThetTest(cont)

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tTest:One-SidedAlternatives

Besidesournull,H0,weneedanalternativehypothesis,H1,andasignificancelevel

H1maybeone-sided,ortwo-sided

H1:bj>0andH1:bj<0areone-sided

H1:bj??0isatwo-sidedalternative

Ifwewanttohaveonlya5%probabilityofrejectingH0ifitisreallytrue,thenwesayoursignificancelevelis5%

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One-SidedAlternatives(cont)

Havingpickedasignificancelevel,a,welookupthe(1–a)thpercentileinatdistributionwithn–k–1dfandcallthisc,thecriticalvalue

Wecanrejectthenullhypothesisifthetstatisticisgreaterthanthecriticalvalue

Ifthetstatisticislessthanthecriticalvaluethenwefailtorejectthenull

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yi=b0+b1xi1+…+bkxik+ui

H0:bj=0H1:bj>0

c

0

a

(1-a)

One-SidedAlternatives(cont)

Failtoreject

reject

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One-sidedvsTwo-sided

Becausethetdistributionissymmetric,testingH1:bj<0isstraightforward.Thecriticalvalueisjustthenegativeofbefore

Wecanrejectthenullifthetstatistic<–c,andifthetstatistic>than–cthenwefailtorejectthenull

Foratwo-sidedtest,wesetthecriticalvaluebasedona/2andrejectH1:bj??0iftheabsolutevalueofthetstatistic>c

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yi=b0+b1Xi1+…+bkXik+ui

H0:bj=0H1:bj>0

c

0

a/2

(1-a)

-c

a/2

Two-SidedAlternatives

reject

reject

failtoreject

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SummaryforH0:bj=0

Unlessotherwisestated,thealternativeisassumedtobetwo-sided

Ifwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthea%level”

Ifwefailtorejectthenull,wetypicallysay“xjisstatisticallyinsignificantatthea%level”

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Testingotherhypotheses

AmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0:bj=aj

Inthiscase,theappropriatetstatisticis

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ConfidenceIntervals

Anotherwaytouseclassicalstatisticaltestingistoconstructaconfidenceintervalusingthesamecriticalvalueaswasusedforatwo-sidedtest

A(1-a)%confidenceintervalisdefinedas

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Computingp-valuesforttests

Analternativetotheclassicalapproachistoask,“whatisthesmallestsignificancelevelatwhichthenullwouldberejected?”

So,computethetstatistic,andthenlookupwhatpercentileitisintheappropriatetdistribution–thisisthep-value

p-valueistheprobabilitywewouldobservethetstatisticwedid,ifthenullweretrue

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Stataandp-values,ttests,etc.

Mostcomputerpackageswillcomputethep-valueforyou,assumingatwo-sidedtest

Ifyoureallywantaone-sidedalternative,justdividethetwo-sidedp-valueby2

Stataprovidesthetstatistic,p-value,and95%confidenceintervalforH0:bj=0foryou,incolumnslabeled“t”,“P>|t|”and“[95%Conf.Interval]”,respectively

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TestingaLinearCombination

Supposeinsteadoftestingwhetherb1isequaltoaconstant,youwanttotestifitisequaltoanotherparameter,thatisH0:b1=b2

Usesamebasicprocedureforformingatstatistic

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TestingLinearCombo(cont)

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TestingaLinearCombo(cont)

So,touseformula,needs12,whichstandardoutputdoesnothave

Manypackageswillhaveanoptiontogetit,orwilljustperformthetestforyou

InStata,afterregyx1x2…xkyouwouldtypetestx1=x2togetap-valueforthetest

Moregenerally,youcanalwaysrestatetheproblemtogetthetestyouwant

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Example:

Supposeyouareinterestedintheeffectofcampaignexpendituresonoutcomes

ModelisvoteA=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+u

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Example(cont):

Thisisthesamemodelasoriginally,butnowyougetastandarderrorforb1–b2=q1directlyfromthebasicregression

Anylinearcombinationofparameterscouldbetestedinasimilarmanner

Otherexamplesofhypothesesaboutasinglelinearcombinationofparameters:

b1=1+b2;b1=5b2;b1=-1/2b2;etc

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MultipleLinearRestrictions

Everythingwe’vedonesofarhasinvolvedtestingasinglelinearrestriction,(e.g.b1=0orb1=b2)

However,wemaywanttojointlytestmultiplehypothesesaboutourparameters

Atypicalexampleistesting“exclusionrestrictions”–wewanttoknowifagroupofparametersareallequaltozero

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TestingExclusionRestrictions

NowthenullhypothesismightbesomethinglikeH0:bk-q+1=0,...,bk=0

ThealternativeisjustH1:H0isnottrue

Can’tjustcheckeachtstatisticseparately,becausewewanttoknowiftheqparametersarejointlysignificantatagivenlevel–itispossiblefornonetobeindividuallysignificantatthatlevel

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ExclusionRestrictions(cont)

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

Intuitively,wewanttoknowifthechangeinSSRisbigenoughtowarrantinclusionofxk-q+1,,…,xk

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TheFstatistic

TheFstatisticisalwayspositive,sincetheSSRfromtherestrictedmodelcan’tbelessthantheSSRfromtheunrestricted

EssentiallytheFstatisticismeasuringtherelativeincreaseinSSRwhenmovingfromtheunrestrictedtorestrictedmodel

q=numberofrestrictions,ordfr–dfur

n–k–1=dfur

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TheFstatistic(cont)

TodecideiftheincreaseinSSRwhenwemovetoarestrictedmodelis“bigenough”torejecttheexclusions,weneedtoknowaboutthesamplingdistributionofourFstat

Notsurprisingly,F~Fq,n-k-1,whereqisreferredtoasthenumeratordegreesoffreedomandn–k–1asthedenominatordegreesoffreedom

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0

c

a

(1-a)

f(F)

F

TheFstatistic(cont)

reject

failtoreject

RejectH0ata

significancelevel

ifF>c

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TheR2formoftheFstatistic

BecausetheSSR’smaybelargeandunwieldy,analternativeformoftheformulaisuseful

WeusethefactthatSSR=SST(1–R2)foranyregression,socansubstituteinforSSRuandSSRur

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OverallSignificance

AspecialcaseofexclusionrestrictionsistotestH0:b1=b2=…=bk=0

SincetheR2fromamodelwithonlyaninterceptwillbezero,theFstatisticissimply

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GeneralLinearRestrictions

ThebasicformoftheFstatisticwillworkforanysetoflinearrestrictions

Firstestimatetheunrestrictedmodelandthenestimatetherestrictedmodel

Ineachcase,makenoteoftheSSR

Imposingtherestrictionscanbetricky–willlikelyhavetoredefinevariablesagain

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Example:

Usesamevotingmodelasbefore

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

nownullisH0:b1=1,b3=0

Substitutingintherestrictions:voteA=b0+log(expendA)+b2log(expendB)+u,so

UsevoteA-log(expendA)=b0+b2log(expendB)+uasrestrictedmodel

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FStatisticSummary

Justaswithtstatistics,p-valuescanbecalculatedbylookingupthepercentileintheappropriateFdistribution

Statawilldothisbyentering:displayfprob(q,n–k–1,F),wheretheappropriatevaluesofF,q,andn–k–1areused

Ifonlyoneexclusionisbeingtested,thenF=t2,andthep-valueswillbethesame

Economics20-Prof.Anderson

MultipleRegressionAnalysis

y=b0+b1x1+b2x2+...bkxk+u

3.AsymptoticProperties

Economics20-Prof.Anderson

Consistency

UndertheGauss-MarkovassumptionsOLSisBLUE,butinothercasesitwon’talwaysbepossibletofindunbiasedestimators

Inthosecases,wemaysettleforestimatorsthatareconsistent,meaningasn??∞,thedistributionoftheestimatorcollapsestotheparametervalue

Economics20-Prof.Anderson

SamplingDistributionsasn??

b1

n1

n2

n3

n1<n2<n3

Economics20-Prof.Anderson

ConsistencyofOLS

UndertheGauss-Markovassumptions,theOLSestimatorisconsistent(andunbiased)

Consistencycanbeprovedforthesimpleregressioncaseinamannersimilartotheproofofunbiasedness

Willneedtotakeprobabilitylimit(plim)toestablishconsistency

Economics20-Prof.Anderson

ProvingConsistency

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AWeakerAssumption

Forunbiasedness,weassumedazeroconditionalmean–E(u|x1,x2,…,xk)=0

Forconsistency,wecanhavetheweakerassumptionofzeromeanandzerocorrelation–E(u)=0andCov(xj,u)=0,forj=1,2,…,k

Withoutthisassumption,OLSwillbebiasedandinconsistent!

Economics20-Prof.Anderson

DerivingtheInconsistency

Justaswecouldderivetheomittedvariablebiasearlier,nowwewanttothinkabouttheinconsistency,orasymptoticbias,inthiscase

Economics20-Prof.Anderson

AsymptoticBias(cont)

So,thinkingaboutthedirectionoftheasymptoticbiasi