BasicEconometrics

Notes

Econometricsisabranchofmathematicalstatisticsthatisconcernedwithapplyingstatisticstonon-experimentaldata.Usedfor;-Estimatingeconomicrelationships-Testingeconomictheories-Evaluatingpolicy-ForecastingmacrovariablesNon-experimentaldatareferstodatathatisnotaccumulatedthroughtheuseofcontrolledexperimentsonindividuals,firmsorsegmentsoftheeconomy.Non-experimentaldataisalsoreferredtoasobservationaldataorretrospectivedata.Experimentaldata,ontheotherhand,referstodatacollectedinlaboratoryenvironmentsinthenaturalsciences.Itishardtodeviseexperimentsineconomicsbecauseof;-Theyareoftenimpossibletoconduct-Theyareprohibitivelyexpensive-TheycanbemorallyrepugnanttoconductthekindsofexperimentsneededtoaddresseconomicissuesAnempiricalanalysisemploysdatatotestatheoryorestimatearelationship.4Stepsinconductinganempiricalanalysis.(S.E.E.F.)1.Specification(F.F.D.A.)Thisinvolvesspecifyingthemodel,whichisformulatingthequestionthatdescribesvariousrelationships.Aneconomicmodelconsistsofmathematicalequationsthatdescribevariousrelationships.Thisentailslookingat:-First-Thenatureofstochasticfunctions-Thereisanunobservable‘populationregressionfunction’(PRF),whichwetrytoapproximatewithour‘sampleregressionfunction’(SRF)-Thisgivesustheconceptofthe‘conditionalmean’,E(y|x)(E=expected)-Second-Thetypeofmathematicalfunctions-Themostcommonlyusedfunctionalformsarethefollowing;-Linearfunctions:yt=β0+β1xt-Quadraticfunctions:yt=β0+β1x2t-Powerfunctions:yt=β0(xt)β1-Exponentialfunctions:yt=β0eβ1t-Third–Thechoiceofvariables-Choosingthedataavailabilityanddataqualityused.-Fourth–Theassumptionsmadeabouttheerrorterms-Theerrorterm,ut,representsthosefactorsthatimpactonytwhicharenotincludedinthemodel-Wecanneverknowexactlywhattheimpactonytisofalltheothervariables-Becauseofthis,weneedtomakecertainassumptionsaboutthenatureofut-Thereare5keyassumptionsgenerallymadeabouttheerrorterms:1.Theerrortermsarerandomvariableswithanormaldistribution2.Theyhaveazeromean,thatis,E(ut)=0forallt=1,2,...,n3.Theyhavethesamevariance,thatis,Var(ut)=σ2forallt=1,2,...,n 4.Theyareindependentrandomvariables,thatis,Cov(ut,us)=0forallt,s=1,2,...,n

5.Theerrortermsandtheindependentvariablesthemselvesareindependentofeachother,thatis,Cov(ut,Xt)=02.Estimation-Themodelisestimatedhere,toestimateweuseordinaryleastsquaresmethod(OLS).Thisisbasedonminimisingthesumofthesquaredresidualsinthis

equation .OLSminimisesthefunctionwithrespecttobeta0andbeta1.(Anythingwithahatisestimatedorexpectedvalues)3.Evaluation-Duringthisstagewealwaysexaminetheestimatedresiduals.Ifourmodeliscorrectlyspecifiedthentheestimatedresiduals(e)arethesampleestimatesofthepopulationerrorterms(u).Wealsolookattheparameterssizeandsign(beta0and1)toseeiftheymakesense.-Oncewearesatisfiedthatitmakessenseweconductteststoseeifthereisasignificantrelationship.Individualtestsincludet-testsandcorrelationanalysisandtestthesignificanceofindividualvariables.JointteststestforoverallsignificanceandconsistofRsquared,RhatsquaredandANOVA.4.ForecastingWhenwetalkaboutforecasts,werefertoforecastsfromtwodifferentperiods:-Withinsampleforecasts,whichareforecastsgeneratedwithinthesampleperiodusedtoestimatethemodel-Outofsampleforecasts,whichareforecastsgeneratedforanyperiodthatwasnotusedinestimatingthemodelTherearealsotwotypesofforecasts,staticanddynamic:-Staticforecastsareobtainedbysubstitutingknownvaluesoftheindependentvariableintotheestimatedequation-Dynamicforecastsareobtainedbysubstitutingvaluesoftheindependentvariables,whicharenotactuallyknownbutaregeneratedbythemodelMostapplicationsareconcernedwithoutofsamplestaticforecastsToestimateamodelweneedtoconverttheeconomicmodelintoaestimableeconometricmodel:

Fromthis: Tothis:

Econometricmodelsmaybeusedforhypothesistesting.Forexample,theparameterB3representstheeffectoftrainingonthewage.Howlargeistheeffectandisitstatisticallydifferentfromzero?

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StructureofEconomicDataToundertakeaneconometricanalysis,dataisrequiredonthevariablesofinterest.-Thetypeofeconometricmethodsemployedwilldependonthenatureofthedataused.-Theuseofaninappropriateestimationmethodmayleadtomisleadingresults

• Therearefourdifferentkindsofeconomicdatasets• Cross-sectionaldata• Timeseriesdata• Pooledcrosssections• PanelorLongitudinaldata

Cross-sectionaldatasets:Thesedatasetsconsistofasampleofobservationsonindividuals,households,firms,cities,states,countries,orotherentitiesofinterest,measuredataparticularpointintime.(theaverageageexample).Cross-sectionalobservationsareassumedtohavebeenobtainedthroughpurerandomsamplingfromapopulation.Theyareassumedtobeindependent.Purerandomsamplingiswhereeachobservationisdrawnatrandomfromapopulation,sothateachindividualobservationisaslikelyasthenexttobedrawn,andthateachdrawisindependentofallotherdraws.Thisdataisusuallyemployedinfieldsofappliedmicroeconomics.Problemswiththisdataandpurerandomsampling:-Refusaltoparticipate(leadstosampleselectionbias)-Samplingcanleadtoclustering(wheresampledentitiesarelargelyrelativetothepopulation.-Differentvariablescancorrespondtodifferenttimeperiodsincross-sectionaldata.Example

Timeseriesdata:Thistypeofdatasetconsistsofobservationsonavariableorseveralvariables,measuredovertime.Timeseriesobservationsarenotgenerallyindependent.Theytendtobestronglyrelatedtotheirrecenthistories.Theyare‘seriallycorrelated’.Becauseofthisdependency,modificationstostandardestimation

techniqueshavebeendevelopedtotakeintoaccountthisdependency.Timeseriesdatatypicallydisplaytrendsandseasonalpatterns.Someofthemorecommonfrequenciesatwhichdataarecollectedaredaily,weekly,monthly,quarterly,andannualintervals.TimeseriesdataistypicallyemployedinfieldsofAppliedMacroeconomicsandFinance.Example:

Pooledcrosssections:(combinationofcross-sectionalandtimeseries)Thesedatasetshavebothacross-sectionalandtimeseriesdimension.Forinstance,supposethattwohouseholdsurveysaretakeninAustralia,onein2010andtheotherin2014.Combiningthetwoyearstoincreasethesamplesizecanformapooledcrosssection.Crosssectionsaredrawnindependentlyofeachother.Forinstance,anewrandomsampleofhouseholdswouldbetakenin2014toanswerthesamesurveyquestions.Pooledcrosssectionscanoftenbeusedtoevaluatepolicychanges.Forinstance,supposewewantedtoevaluatetheeffectofareductioninpropertytaxesthatoccurredin1994.Wewouldcollectarandomsampleofhousepricesfortheyear,1993.Anewrandomsampleofhousepriceswouldbecollectedfortheyear,1995.Wewouldthencomparetheeffectthatreductionsinpropertytaxeshavehadonhouseprices.Comparebeforeandafterthereformwasimplemented,where1993wasbeforethereformand1995isafterthereform.

Checkoutthesplitinyear’sform1993to1995.

PanelorLongitudinalDataThistypeofdatasetconsistsofatimeseriesforeachcrosssectionalentity.Withpaneldata,thesamecross-sectionalunitsarefollowedovertime.Thisfeaturedistinguishesapaneldatasetfromthatofapooledcrosssection.Assuch,paneldatahaveacross-sectionalandatimeseriesdimension.Observingthesameentitiesovertimeleadstoseveraladvantagesovercross-sectionalorpooledcrosssectionaldata.Paneldataallowsustocontrolfortime-invariantunobservedcharacteristics.Paneldataallowsustomodellaggedresponses.Considerthefollowingexampleoncrimestatisticsfor150cities.Eachcityisobservedintwoyears,1986and1990.Anumberoftime-invariantunobservedcitycharacteristicsmaybemodelled.Theeffectofpoliceoncrimeratesmayexhibitatimelag,whichcanbemodelled.Example:2yeardataoncitycrimefor150cities.

CausalityandCeterisParibus:Ourgoalistoinferthatonevariablehasacausaleffectonanothervariable.Causalityiswhereachangeinonevariableleadstoachangeinanothervariable.Findinganassociationbetweentwoormorevariablesmaybesuggestivebutnotconclusiveunlesscausalityisestablished(achangemaybecoincidenceorafactorhasnotbeenaccountedforinthemodel).Weneedtoholdallotherfactorsconstantinordertoanalysetheeffectofoneparticularvariableonanother.Ifallotherfactorsarenotheldconstant,thenitisimpossibletoknowthecausaleffectofonevariableonanother.Someexampleswhereinferringcausalityarisesconsistof:-Measuringthereturnofeducation-Analysingtheeffectoflawenforcementoncrimelevels-AnalysingtheeffectoftheminimumwageonunemploymentThisisduetothefactthatitisdifficulttoisolatevariablesinarelationshipthatwillbeobviouslyaffectedbyothervariablesinasimplelinearregression.

CHAPTER2:TheSimpleRegressionModelDefinitions:Weareinterestedinexplainingyintermsofxorhowyvarieswithchangesinx.Inwritingamodelthatexplainsthistherearethreeissues:-First,sincethereisneveranexactrelationship,howdoweallowforotherfactorstoaffecty?-Second,whatisthefunctionalformbetweenyandx?-Third,howcanwebesurethatwearecapturingaceterisparibusrelationshipbetweenyandx?Someofthesecanbesolvedbywritingtheequationyt=β0+β1xt+uwhichisknownasthesimplelinearregression(SLR)orthetwovariableregressionmodelorabivariateregressionmodel.-Thevariablesyandxhaveseveraldifferentnamesthatareusedinterchangeably:-Theyvariableiscalledthedependentvariable,theexplainedvariable,theresponsevariable,thepredictedvariable,andtheregressand-Thexvariableiscalledtheindependentvariable,theexplanatoryvariable,thecontrolvariable,thepredictorvariable,andtheregressor-Thevariable,u,representsallfactorsotherthanxthataffecty.Itisknownastheerrorterm,thedisturbanceterm,thestochasticterm,andtherandomtermTherefore,ifthechangeinu=0thenthechangeinywill=thechangeinx.BorepresentstheinterceptandB1representstheslopeparameterinaSLR.UsingtheassumptionthatE(u)=0,andthatE(β0)=β0andE(β1)=β1,wecanobtainwhatisknownasthe‘populationregressionfunction’(PRF):E(y|x)=E(β0+β1x+u)E(y|x)=E(β0+β1x)+E(u)E(y|x)=E(β0)+E(β1x)+E(u)Therefore:E(y|x)=β0+β1x.ThismeansthelinearrelationshipofthePRFgivesusaoneunitincreasesinxchangestheexpectedvalueofybytheslopeamount.DerivingOLSestimatesAlinebestfitdrawniscalledthesampleregressionfunctionthatis

Foranyxivalue,thedifferencebetweentheactualvalueofyiandthevaluegivenbythesampleregressionfunctioniscalledtheresidual,ui,where:

Foracorrectlyspecifiedmodeltheresidualisthesampleestimateoftheerrorterm.Liketheerrorterm(u),itrepresentsthatpartofthevalueofthevariableythattheestimatedlinearmodelisunabletoexplain.TheOLSmethodstatesthatweshouldchoosetheSMFlinethathasthesmallestresiduals.Thelinethatminimisestheamountofvariationinythatcannotbe

i10i xˆˆy β+β=

)xˆˆ(yyyu i10iiii β+β−=−=

explainedbythemodel.Tomeasurethevariationwesumtheresiduals.Howevertheywouldequal0aretheyarenegativeandpositive.Toavoidthiswe

sumtheirsquaredvalues: TheOLSprocedureminimisesthefunctionwithrespecttotwounknownsBoandB1.Usingcalculusthenecessaryconditionsforlocalextremumare

Thisisnowexpandedtogive‘normalequations’

Andsolved:

Usinganexamplewherei=7wehave

Sothat:THESEFORMULASAREVERYIMPORTANT!

Andwegetaregressionfunctionof:

Weinterpretthisas:

§ Whenthetemperatureisequalto0oC,thelevelofmonthlysalesforFisher-Hausenis-$1,728.96

§ Anincreaseintemperatureby1oCleadstoanincreaseinmonthlysalesforFisher-Hausenby$466.33

§ Supposethatthetemperatureispredictedtobe20oCinAugust

WhatwouldbethelevelofsalesinAugustearnedbyFisher-Hausen?Ŝi=-1728.956+466.330(20)=$7597.64

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OLSSampledataproperties:

1.OLSresidualsiszerobecause 2.CovariancebetweenregressorsandOLSresidualsiszerobecause

3.ThepointxandyhatisalwaysontheestimatedOLSregressionline.Inotherwords,ifwetakethesampleregressionfunction,andpluginxhatforx,thenthepredictedvalueisyhat.Whenevaluatingthequalityofourestimatedmodelweusethevarianceoftheerrorterms.toobtainanunbiasedestimateofthevariancefortheerrortermweusethefollowingformula:

Wherekisthenumberofparametersestimated,excludingtheconstant.Anduhat=0

Standarderroris: Thedifferencebetweenyobservationandyestimateistotalvariance.

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7,000

8,000

9,000

10,000

11,000

12,000

13,000

14,000

15,000

16,000

20 22 24 26 28 30 32 34

Y

X

ExplainedvariationisbetweenthehorizontallineandtheendpointofOLSline.Theunexplainedvariationisthebracketedspacethatisuhat.Thereforetotalvariationisyobservation–yestimate=yobservation.Thisistruewhenwehavenocorrelationbetweenvariables.IfwecallthedistancebetweenY1andtheTotalVariation,thenthiscanbedividedintotwoparts:Thefirstpartisthedifferencebetweenthevaluepredictedbythemodelandthemeanvalue.-ThisiscalledtheExplainedVariationbecauseitisthecomponentofthevariationinYifrom,whichourregressionmodelcanexplain

Forobservation1,wehave:ExplainedVariation(ESS)= -ThesecondpartisthedifferencebetweentheactualvalueofY,andthevaluethatthemodelpredicts.ThisiscalledtheResidualorChanceVariationbecauseitisthecomponentofvariationthatthemodelisunabletoexplain:ResidualVariation(RSS)=

TotalvariationTSS=ESS+RSS

Oncethemodelhasbeenestimateditmustbefittedtodeterminethegoodnessofthisoverallfit,whichismeasuredbythecoefficientofdeterminationorrsquared.IfwedividebothsidesbytheTSS,weget:1=(ESS/TSS)+(RSS)/(TSS)

Rsquared: ThevalueofR2liesbetween0and1,thatis0≤R2≤1IfESS=TSS,thenR2=1Thisimpliesaperfectfit.R2isameasureofthegoodnessoffitoftheestimatedregressionmodeltothesampledata,andthebetterthisfitis,thecloserisR2toavalueof1IftheESS=0,thenR2=0-ThissuggeststhattheregressionmodeldoesnotprovidebetterestimatesforYi.-Moreimportantly,afindingthatR2=0suggeststhatalinearrelationshipdoesnotexistbetweenYiandtheregressors(xvariables).-Thisimpliesthatanon-linearrelationshipexistsbetweenYiandtheregressors.-WhethertheobtainedR2ishighenoughdependsontheactualproblemstudiedandthetypeofdatathathasbeenused.-Inthecaseofcross-sectionaldata,R2≥0.5isusuallyconsideredtobeagoodenoughfit.Itislowerbecauseitdoesnottakeintoaccountthenatureoffluctuationsoveratimeseries;ratheritisasnapshotintime.ThisiswhytimeseriesdatarequiresahigherRsquared.-Intime-seriesdata,R2≥0.9isusuallyconsideredtobeagoodenoughfit.

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ThenumberofregressorsdoesnotaffecttheTSS;itisfixedforagivensample.TheRSSdecreases,ordoesnotincrease,asmoreandmoreregressorsareincludedinthemodel.R2isanincreasingfunctionofk,oratleast,isnon-decreasingink.Thatis,R2onlyincreaseswhentheadditionofanotherindependentvariablehelpsexplainthevariationinthedependentvariable.ThevalueofR2isconstrainedbythesamplesize.But,kcannotbeincreasedindefinitely,becausedf=n-(k+1)≥0TheOLSestimatorsdonotexistifdf<0,andthelargerthedf,themorereliabletheestimatedmodel.Aquestionthatiscommonlyaskedis:‘shouldweincludeaconstant’?Ifwedon’t,thenweareforcingthemodeltopassthroughtheorigina.k.a.zeroonboththeyandx-axis(inthebottomleftmostcorner).Ineffect,weareimposingtherestrictionthatβ0=0whichtheoretically,thismaybeincorrect.Forinstance,inestimatingaconsumptionfunction,imposingβ0maybeinvalid,becauseconsumptionexpenditureisstillundertakenthroughsavingsandothermeasuresofwealth,andnotjustdirectlythroughincome(peoplestillconsumewheretheirincomeiszero).Weshouldalmostalwaysincludeβ0inourestimatedmodel.Evenifβ0fromtheestimatedmodelisveryclosetozeroinmagnitude,andisstatisticallyinsignificant,itisgoodtoleaveitin.Ifwedon’t,weareforcingthemodeltogothroughtheorigin.Donotexcludeβ0fromtheestimatedmodelunlessyouhavesomestrongtheoreticalreasontodosowhichisrare.Chapter3:MultipleRegression,estimationAsimplelinearregressionmodelisquitelimitedinhowwetrytoexplainonlyonevariableintermsofanother.Incontrast,amultiplelinearregressionmodeldealswithseveralexplanatoryvariables.Weavoidtheproblemofomittedvariablebiasandincludefactorsthatwouldobviouslybeincludedintheerrorterm.Weinterpretthevariablesthesameasasimplelinearregression.Tohelpuschoosewhatvariablestoincludeinourmodelweuseacorrelationmatrix.

Doesthelittleprepresentsigma?Thismeansthatsigma,iscloseto+1thereisastrongpositiverelationship.Closeto-1meansthereisastrongnegativerelationship.+1or-1beingtheperfectrelationship.0meansthatxandyareindependent.Ifwesquaresigma,thisgivesustheproportionofthevarianceinonevariablethatcanbedirectlyrelatedtochangesintheother.

• |ρ|>0.8–strongcorrelation,withρ2≈2/3• |ρ|<0.5–weakcorrelation,withρ2=¼• Inbetween–moderatecorrelation

Thisisacorrelationmatrix

StatisticalSignificance:Probabilityvaluelessthan0.05(p-value)T-testforsignificanceLEARNTHISFORMULA

LEARNWHATLEVELLOGLEVELLEVELANDLOGLOGRELATIONSHIPSARE.Mustknowtheseformulas:PRF=

SRF=

OLS= Whenworkingwithamultipleregressionthet-statis

Multicollinearityiswhenseveralindependentvariablesarelinearlyrelated.Thiscanalterresultsdrastically.2typescanoccurthatincludeexactandnear.Iftwoormoreindependentvariableshavealinearrelationshipbetweenthem,thenwehaveperfectorexactmulticollinearity.ThereisnouniquesolutiontothenormalequationsobtainedbyOLS.Whenthereisalinearrelationshipamongthevariables,thevariance-covariancevaluesbecomeundefined.Itisnotpossibletoestimatetheregressioncoefficientsinthecaseofexactmulticollinearity.Mostcomputerprogramswillhighlightthisbydisplayinganerrormessage,suchas‘matrixsingular’,‘nearsingularmatrix’,or‘exactcollinearityencountered’Inmostpracticalsituations,thereisaclosebutnotexactrelationshipbetweentheXvariablesinthemodel.Inthisinstance,weobtainactualparameterestimates.Estimationpackagesdon’ttellusthatwehaveahighlycollinearrelationshipamongtheindependentvariables.Whentheseestimateshaveverylargevariances,itmeansthattheirvaluesvarywidelyfromonesampletoanother.Thismeansthatwhenanewsetofobservationsbecomesavailableattheendofthemonthortheendofthequarter,whenweusethesenewvaluestore-estimateourmodel,wemayobtainestimatesthatdiffersignificantlyfromtheoneswearenowusing.Ageneralruleofthumbthat’susedtoindicatewhetherornotmulticollinearityisaproblemisifR21▪≥0.90,whichimpliesthattheVIF≥10.

Whywedon’tignoremulticollinearity?

First,iftwoormoreexplanatoryvariablesareexactlylinearlyrelated,thenthemodelcannotbeestimated.Second,ifsomeexplanatoryvariablesarenearlylinearlyrelated,thentheOLSestimatorsarestillBLUE,buthaveverylargevariancesandcovariance’s,makingpreciseestimationdifficult.

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Third,nearmulticollinearityincreasesthestandarderrorsoftheregressioncoefficientsandreducesthet-statistics,makingcoefficientslesssignificantandpossiblyeveninsignificant.Fourth,thecovariancebetweentheregressioncoefficientsofapairofhighlycorrelatedvariableswillbeveryhigh,inabsoluteterms,makingitdifficulttointerpretindividualcoefficients.

MCisaproblemwiththedatanotthemodel.Itcomesupwithahighrsquaredandlowtstatvalues.Highvaluesforcorrelationcoefficientsbetweentheindependentvariables.Thisisdifferenttoahighcorrelationcoefficientwiththedependentandindependentvariables.Regressioncoefficientsappeartobesensitivetomodelspecification.Notethatmulticollinearitymaystillbepresentevenifapairwisecorrelationcoefficientmaybequitelow.Thisisbecause3ormorevariablesmaybelinearlyrelated;yetpairwisecorrelationcoefficientsmaynotbehigh.Insuchinstances,multicollinearityisobservedwhenregressioncoefficientsaredrasticallyaltered,evenpossiblyreversingsigns,whenvariablesareaddedordropped.TosolveMC:-Eliminatevariables(butthiscanleadtoomittedvariablebias)-IncreasesamplesizeWhenyouhaveamultipleregressionmodel,addinganewindependentvariable,nomatterhowirrelevantorridiculousitis,eitherleavestheR2thesame,orincreasesitsvalue.Thus,weareunabletouseR2tocomparetwomodelswithdifferentnumbersofindependentvariables.WemustthenusetheadjustedRsquared.

Thedegreesoffreedomforthesumsofsquaresare:

TSS:n-1 ESS:(k+1)-1 RSS:n–(k+1)

and,dfTSS=dfESS+dfRSS,wherekrepresentsthenumberofregressors,and(k+1)isthenumberofparameterstobeestimated,whichincludestheconstantterm.

Althoughahigherisusuallypreferredtoalowerone,oneshouldneverattempttomaximisethevalueoftheadjustedR2attheexpenseofeconomictheory.

)R-1(1-k-n

1-n-1

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