古扎拉蒂计量经济学第四版讲义Ch9-Model-Specification

1ModelSpecificationandDiagnosticTesting1.Introductionmodelspecificationerrormodelspecificationbias123456modelevaluationModelSelectionCriteria123456Typesofspecificationerrorsyand/orcommittheerrorsofmeasurementbiasmultiplicativelyadditivelyTheassumptionoftheCLRMthattheeconometricmodeliscorrectlyspecifiedhastwomeanings.One,therearenoequationspecificationerrors,andtwo,therearenomodelspecificationerrors.equationspecificationerrorsModelmodelspecificationerrorsModelmis-specificationerrors2GDP2.Consequencesofmodelspecificationerrors(theformertwotypes)1underfittingamodel2overfittingamodelUnderfittingamodel(Omittingarelevantvariable)Supposethetruemodelis12233iiiiyxxβββε=+++13.3.1b1b2Butforsomereasonwefitthefollowingmodel:122iiiyxvαα=++13.3.1a1a2x31x2x3230r≠a1a2()11Eaβ≠()22Eaβ≠2x2x3a1a232σ4a2b25612()22332Eabββ=+13.3.332bx3x2332bβ00a2332bβ0a2overestimate2βpositivebias3Theestimateof2σis2/RSSdfσ=.RSSdfRSSdf34()()2222variaxxσ=−∑13.3.4()()()()2222222232var1iibVIFxxrxxσσ==−−−∑∑13.3.5()2varb()2vara()2varb()2vara0223r1()2vara()2varba2b2tradeoffOverfittingamodel(Inclusionofanirrelevantvariable)Supposethetruemodelis122iiiyxββε=++13.3.6b1b2Butforsomereasonwefitthefollowingmodel:12233iiiiyxaxvαα=+++13.3.7a1a21OLS()()()112233,,0EaEaEaβββ====13A.222σ34aab4()()2222varibxxσ=−∑13.3.8()()()2222223var1iaxxrσ=−−∑13.3.9()()()22223var1var1abr=−13.3.10422301r≤≤()()22varvarab≥a13.Testsofspecificationerrors(theformerthreetypes)DetectingthepresenceofunnecessaryvariablestFpuristapproachPuristapproachanddataminingTestsforomittedvariablesandincorrectfunctionalformExaminationofresidualsvisualdiagnosticyxyxxTheDubin-WatsondstatisticyxDW0.716,1,038,2.70modifieddtestDWTousetheDurbin-Watsontestfordetectingmodelspecificationerrors,weproceedasfollows:1.Fromtheassumedmodel,obtaintheOLSresiduals.2.Ifitisbelievedthattheassumedmodelismis-specifiedbecauseitexcludesarelevant5explanatoryvariable,say,Zfromthemodel,ordertheresidualsobtainedinstep1accordingtoincreasingvaluesofx.NotethattheZvariablecouldbeoneofthexvariablesincludeintheassumedmodeloritcouldbesomefunctionofthatvariablesuchasx2orx3.3.Computethedstatisticfromtheresidualsthusorderedbytheusualdformula,namely,()21221ntttntteede−==−=∑∑12.6.54.FromtheDurbin-Watsontables,iftheestimateddvalueissignificant,thenonecanacceptthehypothesisofmodelmis-specification.Ifthatturnsouttobethecase,theremedialmeasureswillnaturallysuggestthemselves.Ramsey’sRESETtestRamseyhasproposedageneraltestofspecificationerrorcalledRESET(regressionspecificationerrortest).Herewewillillustrateonlythesimplestversionofthetest.12iiiyxααε=++13.4.6yxyyTheideabehindRESET:Althoughie∑andiiey∑arenecessarilyzero,theresidualsinthisfigureshowapatterninwhichtheirmeanchangessystematicallywithiy.Thiswouldsuggestthatifweintroduceiyinsomeformasregressorsin13.4.6,itshouldincreaseR2.AndiftheincreaseinR2isstatisticallysignificant(onthebasisoftheFtest),itwouldsuggestthatthelinearfunction13.4.6wasmis-specified.ThestepsofRESETtest:1.Fromthechosenmodel,e.g.,13.4.6,obtainedtheestimatediy.2.Rerun13.4.6introducingiyinsomeformasanadditionalregressor(s).yiyThus,werun231234iiiiiyxyyuββββ=++++13.4.763.LettheR2obtainedfrom13.4.7be2newRandthatobtainedfrom13.4.6be2oldR.ThenwecanusetheFtest()()()222/1/newoldnewRRnumberofnewregressorsFRnnumberofparametersinthenewmodel−=−−8.5.1tofindoutiftheincreaseinR2from13.4.7isstatisticallysignificant.4.IfthecomputedFvalueissignificant,say,atthe5percentlevel,onecanacceptthehypothesisthatthemodel13.4.6ismis-specified.RESETtestLagrangeMultiplier(LM)testforaddingvariablesThisisanalternativetoRamsey’sRESETtest.restrictedversionTherestrictedregressionassumesthatthecoefficientsofthesquaredandcubedoutputtermsareequaltozero.LMtest1.Estimatetherestrictedregression,i.e.,linearmodel,byOLSandobtaintheresiduals,ie.2.Ifinfacttheunrestrictedregression,i.e.,cubicmodel,isthetrueregression,theresidualsobtainedfromstep1shouldberelatedtothesquaredandcubedterms,thatis,2ixand3ix.3.Thissuggeststhatweregresstheieobtainedinstep1onalltheregressors(includingthoseintherestrictedregression),whichinthepresentcasemeans231234iiiiiexxxvαααα=++++14.4.11wherevisanerrortermwiththeusualproperties.4.Forlarge-samplesize,Englehasshownthatn(thesamplesize)timestheR2estimatedfromthe(auxiliary)regression13.4.11followsthechi-squaredistributionwithdfequaltothenumberofrestrictionsimposedbytherestrictedregression,twointhepresentexamplesincetheterm2ixand3ixaredroppedfromthemodel.Symbolically,wewrite22numberofrestrictionsasynRχ∼13.4.125.Ifthechi-squarevalueobtainedfrom13.4.12exceedsthecriticalchi-squarevalueatthechosenlevelofsignificance,werejecttherestrictedregression.Otherwise,wedonotrejectit.75.Incorrectspecificationofthestochasticerrortermiiiyxβε=13.2.8iiiyxαε=+13.2.9lniεOLS13.2.813.2.913.A.4()2ln0,iNεσ∼()222/2,1ilognormaleeeσσσε−∼13.6.1()2/2Eeσαβ=13.6.2αβ7.ModelSelectionCriteriaInthissectionwediscussseveralcriteriathathavebeenusedtochooseamongcompetingmodelsand/ortocomparemodelsforforecastingpurposes.Herewedistinguishbetweenin-sampleforecastingandout-of-sampleforecasting.In