Brooks 2002 Introductory Econometrics for Finance

‘IntroductoryEconometricsforFinance’©ChrisBrooks20021Chapter4Furtherissueswiththeclassicallinearregressionmodel‘IntroductoryEconometricsforFinance’©ChrisBrooks20022GoodnessofFitStatistics•Wewouldlikesomemeasureofhowwellourregressionmodelactuallyfitsthedata.•Wehavegoodnessoffitstatisticstotestthis:i.e.howwellthesampleregressionfunction(srf)fitsthedata.•ThemostcommongoodnessoffitstatisticisknownasR2.OnewaytodefineR2istosaythatitisthesquareofthecorrelationcoefficientbetweenyand.•Foranotherexplanation,recallthatwhatweareinterestedindoingisexplainingthevariabilityofyaboutitsmeanvalue,,i.e.thetotalsumofsquares,TSS:•WecansplittheTSSintotwoparts,thepartwhichwehaveexplained(knownastheexplainedsumofsquares,ESS)andthepartwhichwedidnotexplainusingthemodel(theRSS).$yttyyTSS2y‘IntroductoryEconometricsforFinance’©ChrisBrooks20023DefiningR2•Thatis,TSS=ESS+RSS•Ourgoodnessoffitstatisticis•ButsinceTSS=ESS+RSS,wecanalsowrite•R2mustalwaysliebetweenzeroandone.Tounderstandthis,considertwoextremesRSS=TSSi.e.ESS=0soR2=ESS/TSS=0ESS=TSSi.e.RSS=0soR2=ESS/TSS=1RESSTSS2RESSTSSTSSRSSTSSRSSTSS21ttttttuyyyy222ˆˆ‘IntroductoryEconometricsforFinance’©ChrisBrooks20024TheLimitCases:R2=0andR2=1tyytxtytx‘IntroductoryEconometricsforFinance’©ChrisBrooks20025ProblemswithR2asaGoodnessofFitMeasure•Thereareanumberofthem:1.R2isdefinedintermsofvariationaboutthemeanofysothatifamodelisreparameterised(rearranged)andthedependentvariablechanges,R2willchange.2.R2neverfallsifmoreregressorsareadded.totheregression,e.g.consider:Regression1:yt=1+2x2t+3x3t+utRegression2:y=1+2x2t+3x3t+4x4t+utR2willalwaysbeatleastashighforregression2relativetoregression1.3.R2quiteoftentakesonvaluesof0.9orhigherfortimeseriesregressions.‘IntroductoryEconometricsforFinance’©ChrisBrooks20026AdjustedR2•Inordertogetaroundtheseproblems,amodificationisoftenmadewhichtakesintoaccountthelossofdegreesoffreedomassociatedwithaddingextravariables.Thisisknownas,oradjustedR2:•Soifweaddanextraregressor,kincreasesandunlessR2increasesbyamorethanoffsettingamount,willactuallyfall.•Therearestillproblemswiththecriterion:1.A“soft”rule2.NodistributionfororR22R)1(1122RkTTR2R2R‘IntroductoryEconometricsforFinance’©ChrisBrooks20027ARegressionExample:HedonicHousePricingModels•Hedonicmodelsareusedtovaluerealassets,especiallyhousing,andviewtheassetasrepresentingabundleofcharacteristics.•DesRosiersandThérialt(1996)considertheeffectofvariousamenitiesonrentalvaluesforbuildingsandapartments5sub-marketsintheQuebecareaofCanada.•TherentalvalueinCanadianDollarspermonth(thedependentvariable)isafunctionof9to14variables(dependingontheareaunderconsideration).Thepaperemploys1990data,andfortheQuebecCityregion,thereare13,378observations,andthe12explanatoryvariablesare:LnAGE-logoftheapparentageofthepropertyNBROOMS-numberofbedroomsAREABYRM-areaperroom(insquaremetres)ELEVATOR-adummyvariable=1ifthebuildinghasanelevator;0otherwiseBASEMENT-adummyvariable=1iftheunitislocatedinabasement;0otherwise‘IntroductoryEconometricsforFinance’©ChrisBrooks20028HedonicHousePricingModels:VariableDefinitionsOUTPARK-numberofoutdoorparkingspacesINDPARK-numberofindoorparkingspacesNOLEASE-adummyvariable=1iftheunithasnoleaseattachedtoit;0otherwiseLnDISTCBD-logofthedistanceinkilometrestothecentralbusinessdistrictSINGLPAR-percentageofsingleparentfamiliesintheareawherethebuildingstandsDSHOPCNTR-distanceinkilometrestothenearestshoppingcentreVACDIFF1-vacancydifferencebetweenthebuildingandthecensusfigure•Examinethesignsandsizesofthecoefficients.–ThecoefficientestimatesthemselvesshowtheCanadiandollarrentalpricepermonthofeachfeatureofthedwelling.‘IntroductoryEconometricsforFinance’©ChrisBrooks20029HedonicHousePriceResultsDependentVariable:CanadianDollarsperMonthVariableCoefficientt-ratioApriorisignexpectedIntercept282.2156.09+LnAGE-53.10-59.71-NBROOMS48.47104.81+AREABYRM3.9729.99+ELEVATOR88.5145.04+BASEMENT-15.90-11.32-OUTPARK7.177.07+INDPARK73.7631.25+NOLEASE-16.99-7.62-LnDISTCBD5.844.60-SINGLPAR-4.27-38.88-DSHOPCNTR-10.04-5.97-VACDIFF10.295.98-Notes:AdjustedR2=0.65l;regressionF-statistic=2082.27.Source:DesRosiersandThérialt(1996).ReprintedwithpermissionoftheAmericanRealEstateSociety.‘IntroductoryEconometricsforFinance’©ChrisBrooks200210TestsofNon-nestedHypotheses•Allofthehypothesistestsconcludedthusfarhavebeeninthecontextof“nested”models.•Butwhatifwewantedtocomparebetweenthefollowingmodels?•WecoulduseR2oradjustedR2,butwhatifthenumberofexplanatoryvariablesweredifferentacrossthe2models?•Analternativeapproachisanencompassingtest,basedonexaminationofthehybridmodel:ttttttvxyuxy321221:2Model:1Modelttttwxxy33221:3Model‘IntroductoryEconometricsforFinance’©ChrisBrooks200211TestsofNon-nestedHypotheses(cont’d)•Thereare4possibleoutcomeswhenModel3isestimated:–2issignificantbut3isnot–3issignificantbut2isnot–2and3arebothstatisticallysignificant–Neither2nor3aresignificant•Problemswithencompassingapproach–Hybridmodelmaybemeaningless–Possiblehighcorrelationbetweenx2andx3.‘IntroductoryEconometricsforFinance’©ChrisBrook