8.-Regression-Model-Specification

RegressionModelSpecicationEC320ProfessorJeremyPiger1/21RegressionModelSpecicationReadings:DoughertyChapter6ModelSpecicationLeavingVariables\OutThatShouldBe\InLeavingVariables\InThatShouldBe\OutTestingLinearRestrictions2/21ModelSpecicationTothispoint,wehavefocusedonestimationofgivenregressionmodels.However,oftenwewillbeinterestedinexplainingadependentvariable,butwewon'tknowexactlywhattheregressionmodelshouldlooklike.Aprimarysourceofsuchuncertaintyiswithregardstowhichexplanatoryvariablesshouldbeincludedintheregressionmodel.Decidingwhichexplanatoryvariablesshouldbeintheregressionispartofmodelbuildingormodelspecication.Insome(rare)cases,economictheorywilltellusexactlywhichexplanatoryvariablesshouldbeincludedinthemodel.However,therewillusuallybeuncertaintyregardingthischoice.3/21ModelSpecicationTheneedtospecifyaregressionmodelcreatesmanyquestionsandnewtoolstoanswerthem.Herewewillbegintodiscusstheseissues.Inparticular,wewillfocusontheconsequencesofmodelmisspecication,meaningwegotthemodelwrong.Remember,therstclassicalassumptionforthelinearregressionmodelisthatthemodelweareestimatingisthe\truemodel.Ifourmodelismisspecied,thenitisnotthetruemodel.Thus,weareestimatingamodelforwhichoneoftheclassicalassumptionshasbeenbroken.Whataretheconsequencesofthis?4/21ModelSpecicationWewillexploretheconsequencesoftwodierentwaysinwhichyoucanmisspecifyamodel.Intherst,weleaveanexplanatoryvariableoutofaregressionthatshouldbeintheregression.Thatis,weomitarelevantvariable.Inthesecond,weleaveanexplanatoryvariableinaregressionthatdoesn'tbelongintheregression.5/21LeavingaVariable\Outthatshouldbe\InWewilldemonstratetheeectsofomittingarelevantvariablewiththefollowingexample:Supposethetruelinearregressionmodelis:Yi=1+2X2i+3X3i+uiHowever,whenwespecifyourregressionmodel,supposewedon'trecognizethatX3belongsintheregression,andinsteadestimatethefollowingwrongregression:Yi=1+2X2i+ui6/21LeavingaVariable\Outthatshouldbe\InDenotetheOLSestimatorof2fromthewrongregressionasb2.Itcanbeshownthattheexpectedvalueofb2is:E(b2)=2+3nPi=1(X2iX2)(X3iX3)nPi=1(X2iX2)2Thisisequivalentto:E(b2)=2+3rx2x3Awhere,A0andrx2x3isthesamplecorrelationcoecientbetweenX2andX3.7/21LeavingaVariable\Outthatshouldbe\InFromthepreviousslide:E(b2)=2+3rx2x3AThisequationsaysthat:Ingeneral,b2isbiased.E(b2)6=2exceptincertainspecialcircumstances.ThereasonforthebiasisthatX2willbepickingupsomeoftheeectofX3.Thesizeofthebiasisgivenby3rx2x3AThedirectionofthebiasdependsonthesignsof3andrx2x3ThebiasthatexistsintheOLSestimatorbecauseoftheomissionofrelevantvariablesiscalledomittedvariablesbias.8/21LeavingaVariable\Outthatshouldbe\InExample:EngelCurvesConsiderthefollowinglogarithmicregressionmodelforestimatingtheincomeelasticityofdemandforfoodandnon-alcoholicbeveragesconsumedathome:log(fdho)=1+2log(expend)+3log(size)+uiWewouldthinkthat3ispositive.Also,thesamplecorrelationcoecientbetweenlog(expend)andlog(size)ispositive.Thus,ifwemistakenlyestimate:log(fdho)=1+2log(expend)+uitheOLSestimateb2willbeabiasedestimatorof2.Thedirectionofthebiasispositive(b2isbiasedupward).9/21LeavingaVariable\Inthatshouldbe\OutWewillnowexploretheconsequencesofincludingexplanatoryvariablesintheregressionthatshouldn'tbethere.Again,consideranexample:Supposethetruelinearregressionmodelis:Yi=1+2X2i+uiHowever,whenwespecifyourregressionmodel,supposewealsoincludeX3intheregression,andinsteadestimatethefollowingwrongregression:Yi=1+2X2i+3X3i+uiImportant:Notethatthewrongmodelisthetruemodelwhen3=0.Thus,thetruevalueof3iszero.10/21LeavingaVariable\Inthatshouldbe\OutResult:TheOLSestimatesfromthewrongmodelb1,b2,andb3,areunbiasedestimatorsofthetrueregressionmodelparameters,1,2and3=0.Thus,whenweestimateamodelthatincludesunnecessaryvariables,theOLSestimatorsarestillunbiased.However,theOLSestimatorsobtainedfromthewrongmodelwillnotbeasecientastheOLSestimatorsobtainedfromthetruemodel.Thatis,theOLSestimatorsfromthewrongmodelwillhaveahighervariance.11/21LeavingaVariable\Inthatshouldbe\OutThereasonforthelossofeciencyissimple:Remember,whenexplanatoryvariablesinaregressionmodelarecorrelated,thiswillincreasethevarianceoftheOLSestimator.Thisisbecauseitbecomesmorediculttoseparateouttheeectsofonevariablefromtheother.Thus,iftwovariablesarecorrelated,andonedoesnotbelongintheregression,thenitwouldbebettertoleavethisvariableoutoftheregression.Leavingitoutwillyieldlowervariance(moreecient)estimates.12/21TestingMoreComplicatedHypothesesWeconcludetheclasswithadiscussionofhowtotestsomemorecomplicatednullhypotheses.Inparticular,considerthemultiplelinearregressionmodel:Yi=1+2X2i+3X3i+:::+kXki+uiWehavefocusedontestinghypothesesofthefollowingtype:H0:2=02(useat-test)or:H0:2=3=0(useaF-test)13/21TestingLinearRestrictionsThenullhypotheseswehavetestedsofarareexamplesofnullhypothesesthatimposelinearrestrictionsonthedata.Alinearrestrictionisarestrictionthatcanbeexpressedusingalinearequation:2=c+a33+a44+::::+akkWewillnowconsiderhowtotestanullhypothesisthatimposesagenerallinearrestriction.14/21Test