管理经济学（第14版） 课件 第4章　基本估计技术

Chapter04BasicEstimationTechniquesMANAGERIALECONOMICSFoundationsofBusinessAnalysisandStrategyFourteenthEditionCHRISTOPHERR.THOMAS©McGrawHillLLC.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGrawHillLLC.LearningOutcomes4.1Setupandinterpretsimplelinearregressionequations.4.2Estimateinterceptandslopeparametersofaregressionlineusingthemethodofleastsquares.4.3Determinestatisticalsignificanceusingeitherttestsorpvaluesassociatedwithparameterestimates.4.4Evaluatethe“fit”ofaregressionequationtothedatausingthestatisticandtestforstatisticalsignificanceofthewholeregressionequationusinganFtest.4.5Setupandinterpretmultipleregressionmodels.4.6Uselinearregressiontechniquestoestimatetheparametersoftwocommonnonlinearmodels:quadraticandlog‐linearregressionmodels.2BasicEstimationParametersThecoefficientsinanequationthatdeterminetheexactmathematicalrelationamongthevariables.ParameterestimationTheprocessoffindingestimatesofthenumericalvaluesoftheparametersofanequation.3RegressionAnalysisRegressionanalysisAstatisticaltechniqueforestimatingtheparametersofanequationandtestingforstatisticalsignificance.DependentvariableVariablewhosevariationistobeexplained.ExplanatoryvariablesVariablesthatarethoughttocausethedependentvariabletotakeondifferentvalues.4SimpleLinearRegressionTrueregressionlinerelatesdependentvariableY

tooneexplanatory(orindependent)variableXY=a+bXInterceptparameter(a)

givesvalueofY

whereregressionlinecrossesY-axis(valueofY

when

X

iszero).Slopeparameter(b)

givesthechangeinYassociatedwithaone-unitchangeinX:b=ΔY∕ΔX5AHypotheticalRegressionModelRegressionlineshowstheaverageorexpectedvalueofYforeachlevelofX.True(oractual)underlyingrelationbetweenYandXisunknowntotheresearcherbutistobediscoveredbyanalyzingthesampledata.RandomerrortermUnobservabletermaddedtoaregressionmodeltocapturetheeffectsofalltheminor,unpredictablefactorsthataffectYbutcannotreasonablybeincludedasexplanatoryvariables.6Figure4.1TheTrueRegressionLine:RelatingSalesandAdvertisingExpendituresAccessthetextalternativeforslideimages.7Table4.1TheImpactofRandomEffectsonJanuarySalesFirmAdvertisingexpenditureActualsalesExpectedsalesRandomeffectTampaTravelAgency$3,000$30,000$25,000$5,000BuccaneerTravelService3,00021,00025,000−4,000HappyGetawayTours3,00025,00025,00008DataTimeseriesAdatasetinwhichthedataforthedependentandexplanatoryvariablesarecollectedovertimeforaspecificfirm.Cross-sectionalAdatasetinwhichthedataforthedependentandexplanatoryvariablesarecollectedfrommanydifferentfirmsorindustriesatagivenpointintime.ScatterdiagramAgraphofthedatapointsinasample.9FittingaRegressionLineThepopulationregressionline

istheequationorlinerepresentingthetrue(oractual)underlyingrelationbetweenthedependentvariableandtheexplanatoryvariable(s).Y=a+bXThesampleregressionlineisanestimateofthetrue(orpopulation)regressionlineandrepresentsthelinethatbestfitsthescatterofdatapointsinthesample.wherearethefittedorpredictedvaluesofthetrue(population)parametersaandb,

andisthefittedorpredictedvalueofY.ThepredictedvalueofYisobtainedbysubstitutingavalueofXintothesampleregressionequation.10Table4.2SalesandAdvertisingExpendituresforaSampleofSevenTravelAgenciesFirmSalesAdvertisingexpenditureA$15,000$2,000B30,0002,000C30,0005,000D25,0003,000E55,0009,000F45,0008,000G60,0007,00011Figure4.2TheSampleRegressionLine:RelatingSalesandAdvertisingExpendituresAccessthetextalternativeforslideimages.12MethodofLeastSquaresMethodofleastsquaresisamethodofestimatingtheparametersofalinearregressionequationbyfindingthelinethatminimizesthesumofthesquareddistancesfromeachsampledatapointtothesampleregressionline.13ParameterEstimatesEstimatorsaretheformulasbywhichtheestimatesofparametersarecomputed.Parameterestimates

areobtainedbysubstitutingsampledataintoestimators(theyarethevaluesofaandbthatminimizethesumofsquaredresiduals).TheresidualisthedifferencebetweentheactualandfittedvaluesofY:ThefittedorpredictedvalueofYassociatedwithaparticularvalueofX,whichisobtainedbysubstitutingthatvalueofXintothesampleregressionequation.14Table4.3ExamplesofPrintoutsforRegressionAnalysis(PanelA:“Generic”Style)DEPENDENTVARIABLE:SR-SQUAREF-RATIOP-VALUEONFOBSERVATIONS:

70.765216.300.0100VARIABLEPARAMETERESTIMATESTANDARDERRORT-RATIOP-VALUEINTERCEPTA11573.04.971917150.831.231541.624.040.16650.010015Table4.3ExamplesofPrintoutsforRegressionAnalysis(PanelB:MicrosoftExcel)1ABCDEFG1SUMMARYOUTPUT23RegressionStatisticsRegressionStatistics4MultipleR0.87485RSquare0.76526AdjustedRSquare0.71837StandardError8782.64388Observations7916Table4.3ExamplesofPrintoutsforRegressionAnalysis(PanelB:MicrosoftExcel)2ABCDEFG10ANOVA11dfSSMSFSignificanceF12Regression11257182986125718298616.300.010013Residual5385674157.377134831.4614Total616428571431516CoefficientsStandardErrortStatP-valueLower95%Upper95%17Intercept11573.07150.831.620.1665−6808.722229954.789618A4.971911.231544.040.01001.80618.137717StatisticalSignificanceStatisticalsignificanceThereissufficientevidencefromthesampletoindicatethatthetruevalueofthecoefficientisnotzero.HypothesistestingAstatisticaltechniqueformakingaprobabilisticstatementaboutthetruevalueofaparameter.18Figure4.3RelativeFrequencyDistributionforbwhenb=5Accessthetextalternativeforslideimages.19UnbiasedEstimatorsTheestimatesdonotgenerallyequalthetruevaluesofaandb.•arerandomvariablescomputedusingdatafromarandomsample.Anestimatorisunbiased

ifitproducesestimatesofaparameterthatare,onaverage,equaltothetruevalueoftheparameter.20TestingforStatisticalSignificanceMustdetermineifthereissufficientstatisticalevidencetoindicatethatYistrulyrelatedtoX(thatis,b

≠0).Evenifb

=0,itispossiblethatthesamplewillproduceanestimatethatisdifferentfromzero.Testforstatisticalsignificanceusingttestsorpvalues.21LevelofConfidenceandLevelofSignificanceLevelofsignificanceistheprobabilityoffindingaparameterestimatetobestatisticallydifferentfromzerowhen,infact,itiszero.TypeIerroriswhenaparameterestimateisfoundtobestatisticallysignificantwhenitisnot.Levelofconfidence

istheprobabilityofcorrectlyfailingtorejectthetruehypothesisthatb=0;equalto:1−levelofsignificance=levelofconfidence22At

TestAttest

isastatisticaltestusedtotestthehypothesisthatthetruevalueofaparameterisequaltozero(b=0)tratioiscomputedaswhereisthestandarderroroftheestimateAtstatisticisthenumericalvalueofthetratio.23PerformingatTestUseattabletochoosecriticaltvaluewithn

−kdegreesoffreedomforthechosenlevelofsignificance.Thecriticalvalueoft

isthevaluethatthetstatisticmustexceedinordertorejectthehypothesisthatb=0.Degreesoffreedomarethenumberofobservationsinthesampleminusthenumberofparametersbeingestimatedbytheregressionanalysis(n−k).Iftheabsolutevalueoftstatisticisgreaterthanthecriticalt,theparameterestimateisstatisticallysignificantatthegivenlevelofsignificance.24Usingp

ValuesApvaluegivestheexactlevelofsignificanceforateststatistic,whichistheprobabilityoffindingsignificancewhennoneexists.Treatasstatisticallysignificantonlythoseparameterestimateswithpvaluessmallerthanthemaximumacceptablesignificancelevel.25CoefficientofDetermination(R2)measuresthefractionoftotalvariationinthedependentvariable(Y)thatisexplainedbytheregressionequation(orexplainedbythevariationinX).Rangesfrom0to1.HighindicatesYandXarehighlycorrelatedbutdoesnotprovethatYandXarecausallyrelated.26Figure4.4HighandLowCorrelationAccessthetextalternativeforslideimages.27FStatisticTheFstatisticisusedtotestforsignificanceoftheoverallregressionequation.CompareFstatistictocriticalFvaluefromtheFtable.Twodegreesoffreedom,n−kandk−1.Specifylevelofsignificance.IfFstatisticexceedsthecriticalF,theregressionequationoverallisstatisticallysignificantatthespecifiedlevelofsignificance.28MultipleRegressionRegressionmodelsthatusemorethanoneexplanatoryvariabletoexplainthevariationinthedependentvariable.Coefficientforeachexplanatoryvariablemeasuresthechangeinthedependentvariableassociatedwithaone-unitchangeinthatexplanatoryvariable,holdingallothervariablesconstant.29QuadraticRegressionModelsAquadraticregressionmodelisanonlinearregressionmodel:UsewhencurvefittingscatterplotisU-shapedor∩-shaped.ForlineartransformationcomputenewvariableEstimateY=a+bX+cZ.30Figure4.5AQuadraticRegressionEquationAccessthetextalternativeforslideimages.31Log-LinearRegressionModelsAnonlinearregressionmodeloftheform:Transformbytakingnaturallogarithms:bandcareelasticities.32Figure4.6ALog-LinearRegressionEquation(PanelA)Accessthetextalternativeforslideimages.33Figure4.6ALog-LinearRegressionEquation(PanelB)Accessthetextalternativeforslideimages.34Summary1AsimplelinearregressionmodelrelatesadependentvariableY

toasingleexplanatoryvariableX.

Theregressionequationiscorrectlyinterpretedasprovidingtheaveragevalue(expectedvalue)ofY

foragivenvalueofX.Parameterestimatesareobtainedbychoosingvaluesofa

andb

thatcreatethebestfittinglinethatpassesthroughthescatterdiagramofthesampledatapoints.Iftheabsolutevalueofthetratioisgreater(less)thanthecriticaltvalue,thenis(isnot)statisticallysignificant.Exactlevelofsignificanceassociatedwithatstatisticisitspvalue.AhighindicatesY

andX

arehighlycorrelatedandthedatatightlyfitthesampleregressionline.35Summary2IftheFstatisticexceedsthecriticalFvalue,theregressionequationisstatisticallysignificant.Inmultipleregression,thecoefficientsmeasurethechangeinY

associatedwithaone-unitchangeinthatexplanatoryvariable.QuadraticregressionmodelsareappropriatewhenthecurvefittingthescatterplotisU-shapedor∩-shapedLog-linearregressionmodelsareappropriatewhentherelationisinmultiplicativeexponentialformTheequationistransformedbytakingnaturallogarithms.36EndofMainContent©McGrawHillLLC.Allrightsreserved.NoreproductionordistributionwithoutthepriorwrittenconsentofMcGrawHillLLC.AccessibilityContent:TextAlternativesforImages38Figure4.1TheTrueRegressionLine:RelatingSalesandAdvertisingExpenditures

-TextAlternativeReturntoparent-slidecontainingimages.Theverticalaxis(S)islabeledmonthlysalesindollarsrangingfrom0to55,000.Thehorizontalaxis(A)islabeledmonthlyadvertisingexpendituresrangingfrom0to9,000.AtrueregressionlinewithapositiveslopelabeledSequals10,000plus5Astartsat(0,10,000)andpassesthroughthepointsat(3,000,25,000),(4,000,30,000),and(9,000,55,000).Solidhorizontallinesfromtheverticalaxisanddashedverticallinesfromthehorizontalaxisconnecttheplots.Arightangleisformedbetweenthepoints(3,000,25,000)and(4,000,30,000).Returntoparent-slidecontainingimages.39Figure4.2TheSampleRegressionLine:RelatingSalesandAdvertisingExpenditures

-TextAlternativeReturntoparent-slidecontainingimages.Theverticalaxis(S)islabeledsalesindollarsrangingfrom0to70,000inincrementsof10,000.Thehorizontalaxis(A)islabeledadvertisingexpendituresindollarsrangingfrom0to10,000inincrementsof2,000.Sevenpointsareplottedat(2,000,15,000),(2,000,30,000),(3,000,25,000),(5,000,30,000),(7,000,60,000),(8,000,45,000),and(9,000,56,000).Alinewithapositiveslopelabeledasasampleregressionline,Scapequals11,573plus4.9719Astartsat(0,11,000)andpassesthroughthepointat(9,000,56,000).Apointat(7,000,46,376)islabeledScapsubscriptiequals46,376.Apointat(7,000,60,000)islabeledSsubscriptiequals60,000.ThedistancebetweenScapsubscriptiandSsubscriptiislabeledesubscripti.Notethatthevaluesareapproximate.Returntoparent-slidecontainingimages.40Figure4.3RelativeFrequencyDistributionforbwhenb=5

-TextAlternativeReturntoparent-slidecontainingimages.Theverticalaxisislabeledastherelativefrequencyofbcaprangingfrom0to1.Thehorizontalaxisislabeledleast-squaresestimateofb(bcap)rangingfrom0to10inincrementsof1.Abell-shapedcurvestartsnearzero,concavedown,graduallyrisestopeakat5,thenconcaveup,andgraduallyfallsnear10.Averticaldashedlineat5denotesthepeakofthecurve.Returntoparent-slidecontainingimages.41Figure4.4HighandLowCorrelation

-TextAlternativeReturntoparent-slidecontainingimages.PanelA:TheverticalaxisislabeledYandthehorizontalaxisislabeledX.Alinefromlefttorightwithanegativeslopepassesthroughninescatteredpointsthatareplottedaroundthelinelinearly.Fourpointsareplottedabovethelineandfivepointsareplottedbelowtheline.PanelB:TheverticalaxisislabeledYandthehorizontalaxisislabeledX.Alinefromlefttorightwithapositiveslopepassesthrough12pointsthatareplottedaroundthelinelinearly.Sevenpointsareplottedabovethelineandfivepointsareplottedbelowtheline.Returntoparent-slidecontainingimages.42Figure4.5AQuadraticRegressionEquation

-TextAlternativeReturntoparent-slidecontainingimages.Theverticalaxis(Y)rangesfrom0to120inincrementsof10.Thehorizontalaxis(X)rangesfrom0to20inincrementsof5.12pointsareplottedat(3,83),(3,107),(4,61),(5,76),(6,68),(8,30),(10,57),(12,40),(14,81),(15,68),(17,102)and(18,110).Anupwardopenparaboliccurve,labeledYcapequals140.08minus19.51Xplus1.01Xsquared,startsat(2,109)passesthrough(10,40),andendsat(18,112).Atablewiththreecolumnsandtwelverowsisshownattherightofthegraph.ThefirstcolumnislabeledYandentriesbelowitare83,107,61,76,68,30,57,40,81,68,102,and110.ThesecondcolumnislabeledXandentriesbelowitare3,3,4,5,6,8,10,12,14,15,17,and18.ThethirdcolumnislabeledZandentriesbelowitare9,9,16,25,36,64,100,144,196,225,289,and324.Notethatthevaluesareapproximate.Returntoparent-slidecontainingimages.43Figure4.6ALog-LinearRegressionEquation(PanelA)

-TextAlternativeReturntoparent-slidecontainingimages.Theverticalaxis(Y)rangesfrom0to3000inincrementsof1000.Thehorizontalaxis(X)rangesfrom0to200inincrementsof50.AconcaveupdecreasingcurvelabeledYcapequals63,575Xtothepowernegative0.96,startsat(40,