MBA统计学 28页

RelationshipamongvariablesFunctionCorrelation(statisticalrelationship)YdependsonX,butisn’tmerelydeterminedbyX.Example:price—demandforproducttemperature—demandforair-conditioningRegression—Accordingtoobservantdata,establishregressionmodelandmakestatisticalreferenceonvariableshavingstatisticalrelationship.Chapter10Regression1\nWhatdoesregressiondo?Solvethefollowingproblems:Determinewhethertherehasstatisticalrelationshipamongvariables,ifhas,showtheformula.Forecastthevalueofanothervariableaccordingtoonevariableoragroupofvariables.2\nLinearRegressionAssumptionsNormalityEveryvalueofX,YfollowsthenormaldistributionTheerrorprobabilityfollowsthenormaldistributionHomoscedasticity(ConstantVariance)IndependenceofErrorsLinearity3\nExample:X-price，Y-demandfortheproductWehavedata:1.Scatterplot2.Regressionequation(OrdinaryLeastSquareEstimation)3.CorrelationcoefficientrTestingtheregressionmodel4.Forecasting5.RegressioncanbelinearitiedSimpleLinearRegressionX(Yuan)708090100110Y(thousand)11.2511.2811.6511.7012.144\nLinearRegressionModelVariablesconsistofalinearfunction.YXiii01SlopeY-InterceptIndependent(Explanatory)VariableDependent(Response)VariableRandomError5\nPopulationLinearRegressionModeli=randomerrorXYXiX01YXiii01ObservedValueObservedValueY6\nSampleLinearRegressionModelei=randomerrorYXYbbXeiii01^YbbXii01UnsampledObservedValueSampledObservedValue7\nOrdinaryLeastSquaresTheleastsquaresmethodprovidesanestimatedregressionequationthatminimizesthesumofsquareddeviationsbetweentheobservedvaluesofthedependentvariableyiandtheestimatedvaluesofthedependentvariable.e2YXe1e3e4YbbXeiii01^YbbXii01OLSMineeeeeii2112223242PredictedValue8\nCoefficient&EquationsYbXbXYnXYXnXbYbXiiiiiniin011122101SampleregressionequationSlopefortheestimatedregressionequationInterceptfortheestimatedregressionequationb9\nEvaluatingtheModelTestCoefficientofDeterminationandStandardDeviationofEstimationResidualAnalysisTestCoefficientsofSignificance^YbbXii0110\nMeasuresofVariationinRegression1.TotalSumofSquares(SST)MeasurethevariationbetweentheobservedvalueYiandthemeanY.2.ExplainedVariation(SSR)VariationcausedbytherelationshipbetweenXandY.3.UnexplainedVariation(SSE)Variationcausedbyotherfactors.11\nVariationMeasuresYXYXiSST(Yi-Y)2SSE(Yi-Yi)2^SSR(Yi-Y)2^Yi^YbbXii0112\nCoefficientofDetermination0r21rbYbXYnYYnYiiiininiin201211212ExplainedvariationTotalvariationSSRSSTAmeasureofthegoodnessoffitoftheestimatedregressionequation.Itcanbeinterpretedastheproportionofthevariationinthedependentvariableythatisexplainedbytheestimatedregressionequation.13\nCorrelationCoefficientAnumericalmeasureoflinearassociationbetweentwovariablesthattakesvaluesbetween–1and+1.Valuesnear+1indicateastrongpositivelinearrelationship,valuesnear–1indicateastrongnegativelinearrelationship,andvaluesnearzeroindicatelackofalinearrelationship.14\nCoefficientsofDetermination(r2)andCorrelation(r)r2=1,r2=0,YYi=b0+b1XiX^YYi=b0+b1XiX^YYi=b0+b1XiX^YYi=b0+b1XiX^r=+1r=-1r=+0.9r=015\nTestofSlopeCoefficientforSignificance1.TestsaLinearRelationshipBetweenX&Y2.HypothesesH0:1=0(NoLinearRelationship)H1:10(LinearRelationship)3.TestStatistic16\nExampleTestofSlopeCoefficientH0:1=0H1:10.05df5-2=3Criticalvalue:Statistic:Determine:Conclusion:tbSb1110700019153655...Rejectat=0.05Thereisevidenceofarelationship.t03.1824-3.1824.025RejectReject.02517\nMultipleRegressionModelThereexistslinearrelationshipamongandependentvariableandtwoormorethantwoindependentvariables.YXXXiiiPPii01122slopeofpopulationinterceptofpopulationYrandomerrorDependentVariableIndependentVariables18\nExample:NewYorkTimesYouworkintheadvertisementdepartmentofNewYorkTimes(NYT).Youwillfindtowhatextentdoadssize(squareinch)andpublishingvolume(thousand)influencetheresponsetoads(hundred).Youhavecollectedthefollowingdata:responsesizevolume112488131357264410619\nExample(NYT)ComputerOutputParameterEstimatesParameterStandardTforH0:VariableDFEstimateErrorParam=0Prob>|T|INTERCEP10.06400.25990.2460.8214ADSIZE10.20490.05883.6560.0399CIRC10.28050.06864.0890.0264b2b0bPb120\nInterpretationofCoefficients1.Slope(b1)Ifthepublishingvolumeremainsunchanged,whenadssizeincreasesonesquareinch,theresponseisexpectedtoincrease0.2049hundredtimes.2.Slope(b2)Ifadssizeremainsunchanged,whenpublishingvolumeincreasesonethousand,theresponseisexpectedtoin-crease0.2805hundredtimes.21\nEvaluatingtheModel1.Howdoesthemodeldescribetherelationshipamongvariables?2.Closenessof‘BestFit’3.Assumptionsmet4.Significanceofestimates5.Correlationamongvariables6.Outliers(unusualobservations)22\nTestingOverallSignificanceTestwhetherthereislinearrelationshipbetweenYandalltheindependentvariables.2.UseFstatistic.HypothesisH0:1=2=...=P=0ThereisnolinearrelationshipbetweenYandindependentvariables.H1:Atleastthereisacoefficientisn’tequalto0.AtleastthereisanindependentvariableinfluencesY23\nTestingOverallSignificanceComputerOutputAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValueProb>FModel29.24974.624955.4400.0043Error30.25030.0834CTotal59.5000Pn-P-1n-1MSR/MSEpValue24\nTransformationsinRegressionModelsNon-linearmodelsthatcanbetransformedintolinearmodels(convenienttocarryoutOLS).DataTransformationMultiplicativeModelExampleYXXYXXiiiiiiii0120112212lnlnlnlnln25\nSquare-RootTransformationYXXiiii011221>01<0YX126\nLogarithmicTransformationYXXiiii01122lnln1>01<0YX127\nExponentialTransformationYiieXXii011221>01<0YX128