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1、Chapter4Prediction,Goodness-of-fit,andModelingIssuesWalterR.PaczkowskiRutgersUniversity4.1LeastSquarePrediction4.2MeasuringGoodness-of-fit4.3ModelingIssues4.4PolynomialModels4.5Log-linearModels4.6Log-logModelsChapterContents4.1LeastSquaresPredictionTheabilitytopr
2、edictisimportantto:businesseconomistsandfinancialanalystswhoattempttoforecastthesalesandrevenuesofspecificfirmsgovernmentpolicymakerswhoattempttopredicttheratesofgrowthinnationalincome,inflation,investment,saving,socialinsuranceprogramexpenditures,andtaxrevenuesl
3、ocalbusinesseswhoneedtohavepredictionsofgrowthinneighborhoodpopulationsandincomesothattheymayexpandorcontracttheirprovisionofservicesAccuratepredictionsprovideabasisforbetterdecisionmakingineverytypeofplanningcontext4.1LeastSquaresPredictionInordertouseregression
4、analysisasabasisforprediction,wemustassumethaty0andx0arerelatedtooneanotherbythesameregressionmodelthatdescribesoursampleofdata,sothat,inparticular,SR1holdsfortheseobservationswheree0isarandomerror.Eq.4.14.1LeastSquaresPredictionThetaskofpredictingy0isrelatedtoth
5、eproblemofestimatingE(y0)=β1+β2x0AlthoughE(y0)=β1+β2x0isnotrandom,theoutcomey0israndomConsequently,aswewillsee,thereisadifferencebetweentheintervalestimateofE(y0)=β1+β2x0andthepredictionintervalfory04.1LeastSquaresPredictionTheleastsquarespredictorofy0comesfromth
6、efittedregressionlineEq.4.24.1LeastSquaresPredictionFigure4.1Apointprediction4.1LeastSquaresPredictionToevaluatehowwellthispredictorperforms,wedefinetheforecasterror,whichisanalogoustotheleastsquaresresidual:Wewouldliketheforecasterrortobesmall,implyingthatourfor
7、ecastisclosetothevaluewearepredictingEq.4.34.1LeastSquaresPredictionTakingtheexpectedvalueoff,wefindthatwhichmeans,onaverage,theforecasterroriszeroandisanunbiasedpredictorofy04.1LeastSquaresPredictionHowever,unbiasednessdoesnotnecessarilyimplythataparticularforec
8、astwillbeclosetotheactualvalueisthebestlinearunbiasedpredictor(BLUP)ofy0ifassumptionsSR1–SR5hold4.1LeastSquaresPredictionThevarianceoftheforecastisEq.4.44.1Lea
