Econometrics.docx
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Econometrics.docx
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Econometrics
1.ReviewofProbabilityandStatistics
1.1.RandomVariables
Xisarandomvariableifitrepresentsarandomdrawfromsomepopulation
adiscreterandomvariablecantakeononlyselectedvalues
acontinuousrandomvariablecantakeonanyvalueinarealinterval
associatedwitheachrandomvariableisaprobabilitydistribution
1.1.1.RandomVariables–Examples
theoutcomeofacointoss–adiscreterandomvariablewithP(Heads)=.5andP(Tails)=.5
theheightofaselectedstudent–acontinuousrandomvariabledrawnfromanapproximatelynormaldistribution
1.2.ExpectedValueofX–E(X)
TheexpectedvalueisreallyjustaprobabilityweightedaverageofX
E(X)isthemeanofthedistributionofX,denotedbymx
Letf(xi)betheprobabilitythatX=xi,then
1.3.VarianceofX–Var(X)
ThevarianceofXisameasureofthedispersionofthedistribution
Var(X)istheexpectedvalueofthesquareddeviationsfromthemean,so
ThesquarerootofVar(X)isthestandarddeviationofX
Var(X)canalternativelybewrittenintermsofaweightedsumofsquareddeviations,because
1.4.Covariance–Cov(X,Y)
CovariancebetweenXandYisameasureoftheassociationbetweentworandomvariables,X&Y,
Ifpositive,thenbothmoveupordowntogether
Ifnegative,thenifXishigh,Yislow,viceversa
1.5.CorrelationBetweenXandY
CovarianceisdependentupontheunitsofX&Y[Cov(aX,bY)=abCov(X,Y)]
Correlation,Corr(X,Y),scalescovariancebythestandarddeviationsofX&Ysothatitliesbetween1&–1
1.6.Extension
1.6.1.Correlation&Covariance
IfσX,Y=0(orequivalentlyσX,Y=0)thenXandYarelinearlyunrelated
IfρX,Y=1thenXandYaresaidtobeperfectlypositivelycorrelated
IfρX,Y=–1thenXandYaresaidtobeperfectlynegativelycorrelated
Corr(aX,bY)=Corr(X,Y)ifab>0
Corr(aX,bY)=–Corr(X,Y)ifab<0
1.6.2.Expectations
E(a)=a,Var(a)=0
E(μX)=μX,i.e.E(E(X))=E(X)
E(aX+b)=aE(X)+b
E(X+Y)=E(X)+E(Y)
E(X-Y)=E(X)-E(Y)
E(X-μX)=0orE(X-E(X))=0
E((aX)2)=a2E(X2)
1.6.3.Variance
Var(X)=E(X2)–mx2
Var(aX+b)=a2Var(X)
Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)
Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y)
Cov(X,Y)=E(XY)-μXμY
If(andonlyif)X,Yindependent,then
Var(X+Y)=Var(X)+Var(Y),E(XY)=E(X)E(Y)
1.7.TheNormalDistribution
Ageneralnormaldistribution,withmeanmandvariances2iswrittenasN(m,s2)
Ithasthefollowingprobabilitydensityfunction(pdf)
1.7.1.TheStandardNormal
Anyrandomvariablecanbe“standardized”bysubtractingthemean,μ,anddividingbythestandarddeviation,σ,soE(Z)=0,Var(Z)=1
Thus,thestandardnormal,N(0,1),haspdf
1.7.2.PropertiesoftheNormal
IfX~N(m,s2),thenaX+b~N(am+b,a2s2)
Alinearcombinationofindependent,identicallydistributed(iid)normalrandomvariableswillalsobenormallydistributed
IfY1,Y2,…Ynareiidand~N(m,s2),then
1.8.CumulativeDistributionFunction
Forapdf,f(x),wheref(x)isP(X=x),thecumulativedistributionfunction(cdf),F(x),isP(Xx);P(X>x)=1–F(x)=P(X<–x)
1.8.1.Forthestandardnormal,f(z),thecdfisF(z)=P(Z P(|Z|>a)=2P(Z>a)=2[1-F(a)] P(aZb)=F(b)–F(a) 1.9.TheChi-SquareDistribution SupposethatZi,i=1,…,nareiid~N(0,1),andX=(Zi2),then Xhasachi-squaredistributionwithndegreesoffreedom(df),thatis X~2n IfX~2n,thenE(X)=nandVar(X)=2n 1.10.Thetdistribution Ifarandomvariable,T,hasatdistributionwithndegreesoffreedom,thenitisdenotedasT~tn E(T)=0(forn>1)andVar(T)=n/(n-2)(forn>2) TisafunctionofZ~N(0,1)andX~2nasfollows: 1.11.TheFDistribution Ifarandomvariable,F,hasanFdistributionwith(k1,k2)df,thenitisdenotedasF~Fk1,k2 FisafunctionofX1~2k1andX2~2k2asfollows: 1.12.RandomSamplesandSampling ForarandomvariableY,repeateddrawsfromthesamepopulationcanbelabeledasY1,Y2,…,Yn Ifeverycombinationofnsamplepointshasanequalchanceofbeingselected,thisisarandomsample Arandomsampleisasetofindependent,identicallydistributed(i.i.d)randomvariables 1.13.EstimatorsandEstimates Typically,wecan’tobservethefullpopulation,sowemustmakeinferencesbasedonestimatesfromarandomsample Anestimatorisjustamathematicalformulaforestimatingapopulationparameterfromsampledata Anestimateistheactualnumbertheformulaproducesfromthesampledata 1.13.1.ExamplesofEstimators Supposewewanttoestimatethepopulationmean SupposeweusetheformulaforE(Y),butsubstitute1/nforf(yi)astheprobabilityweightsinceeachpointhasanequalchanceofbeingincludedinthesample,then Cancalculatethesampleaverageforoursample: 1.14.CriteriaforaGoodEstimator Asymptoticproperties(forlargesamples): Consistency 1.14.1.UnbiasednessofEstimator Wantyourestimatortoberight,onaverage Wesayanestimator,W,ofaPopulationParameter,q,isunbiasedifE(W)=E(q) Proof: SampleMeanisUnbiased 1.14.2.EfficiencyofEstimator Wantyourestimatortobeclosertothetruth,onaverage,thananyotherestimator Wesayanestimator,W,isefficientifVar(W) 1.14.3.MeanSquareError(MSE)ofEstimator Whatifcan’tfindanunbiasedestimator? DefinemeansquareerrorasE[(W-q)2] Gettradeoffbetweenunbiasednessandefficiency,sinceMSE=variance+bias2 Forourexample,thatmeansminimizing 1.14.4.ConsistencyofEstimator Asymptoticproperties,thatis,whathappensasthesamplesizegoestoinfinity? WantdistributionofWtoconvergetoq,i.e.lim[p(W)]=q Forourexample,thatmeanswewant Anunbiasedestimatorisnotnecessarilyconsistent–supposechooseY1asestimateofmY,sinceE(Y1)=μY,thenplim(Y1)μY Anunbiasedestimator,W,isconsistentifVar(W)0asn LawofLargeNumbersreferstotheconsistencyofsampleaverageasestimatorforμ,thatis,tothefactthat: 1.15.CentralLimitTheorem AsymptoticNormalityimpliesthatP(Z Thecentrallimittheoremstatesthatthestandardizedaverageofanypopulationwithmeanmandvarianceσ2isasymptotically~N(0,1),or 1.16.EstimateofPopulationVariance WehaveagoodestimateofμY,wouldlikeagoodestimateofσ2Y Canusethesamplevariancegivenbelow–notedivisionbyn-1,notn,sincemeanisestimatedtoo, Ifknowmcanusen 1.17.EstimatorsasRandomVariables Eachofoursamplestatistics(e.g.thesamplemean,samplevariance,etc.)isarandomvariable-Why? Eachtimewepullarandomsample,we’llgetdifferentsamplestatistics Ifwepulllotsandlotsofsamples,we’llgetadistributionofsamplestatistics 2.Introduction 2.1.WhystudyEconometrics? Rareineconomics(andmanyotherareaswithoutlabs! )tohaveexperimentaldata Needtousenonexperimental,orobservational,datatomakeinferences Importanttobeabletoapplyeconomictheorytorealworlddata Anempiricalanalysisusesdatatotestatheoryortoestimatearelationship Aformaleconomicmodelcanbetested Theorymaybeambiguousastotheeffectofsomepolicychange–canuseeconometricstoevaluatetheprogram. Whatiseconometricsandwhatdoeconometriciansdo? Basically,theytrytoanswerquestionsasdiverseasmakingforecast,assessthesafetyofnuclearpowerplants,testtheoriestomakeextraprofitsonthestockexchangeorevaluatetheefficiencyofpolicies.SoyoumayneedeconometricsnotonlytogetyourMasterbutalsoinvariousjobs(industry,banking,consultancy,academia…). Comparedwithmathematicalstatistics,thedifferenceisthattheeconometriciansrelymostlyonobservationaldataratherthanexperimentaldata.Thiscreatesspecificproblemstheoreticallybutalsoempirically.Soforthiscourse,wewillputtheemphasisonhowtoconductanempiricaleconometricanalysis,andthenintroducethenecessarytheory. Becauseweusedatatoanswerquantitativequestions,ouranswerswillvaryifweuseadifferentsetofdata.Thus,notonlyshouldweprovideananswerbutalsoameasureofhowprecisethisansweris.(UnlikeintheHitchhikerguidetotheGalaxy,weretheanswerisalways42! ). Generally,aneconometricanalysisisconductedtotestsomehypotheses,soafterpresentingthesimplemodel,wewillmoveontothebasicoftesting. Dataspecificitymeansthatwewillhavetodepartfromthesimplemodelinmorethanoneway,whichiswhatthevariousremainingchaptersareallabout.For,Malinvaud(1966)-‘Theartoftheeconometricianconsistsinfindingthesetofassumptionswhicharebothsufficientlyspecificandsufficientlyrealistictoallowhimtakethebestpossibleadvantageofthedataavailabletohim’Hendrysaysnottoconfuseeconometricswitheconomic-tricksoreconomystics! ! ! Whilstexperimentaldatawillgreatlysimplifyourtasks,suchdataarerarelyavailableinsocialsciences(ethicreason).Inthiscourse,wewillbeconcernedwithtwotypesofdata: crosssection,whentypicallyalarge(random)sampleofthepopulationofinterestissurveyedatonepointintimeandtimeseries,wheretheevolutionofafewvariablesovertimeisrecorded.Crosssectioncanalsobepooledacrosstime,inordertoincreasethesamplesizeandmoreimportantlytoassesshowakeyrelationshipchangesovertime(forexampleafterapolicychange).Crosssectiondataistypicallyusedinmicro-economicanalysiswhiletimeseriesaremoreassociatedwithmacro-analysisandfinance.Theanalysisoftimeseriesiscomplicatedbyissuesofseasonality,trendandpersistence,butwewillseehowtodealwiththeseissues. Othertypesofdataexist,suchaspanel(longitudinaldata),whereobservati
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