第二章 一元线性回归的检验和预测2016

CHAPTERTWOHYPOTHESISTESTINGANDPREDITIONOFTWO-VARIABLELINEARREGRESSIONMODEL1.ttestandconfidenceinterval2.prediction:pointpredictorintervalpredictor3.reportandanalysis4.caseHypothesistest:Isagivenobservationorfindingsufficientlyclosedtohypothesizedvaluesothatwecan’trejectthestatedhypothesis?There’retwoapproachestoproceedthehypothesistest:testofsignificanceandconfidenceinterval2.1testofsignificanceandconfidenceinterval2.1.1testofsignificanceConstitutingstandardizednormaldistributionButtoobtain,weneedUsually,trueisunknown,nowwesubstitutefor.Furthermorewesubstituteforandgetat-statistic1、Putforwardhypothesis:Nullhypothesis:H0：Alternativehypothesis:H1：2、Constitutet-statisticandcomputethevalueofT:3、Decisionrule:If|T|>ta/2(n-2)，rejecttheH0,,itmeansthattheexplanatoryvariableindeedinfectsdependentvariable.ThevariableXisstatisticallysignificant.Otherwise,acceptH0，itindicatesthatthereisn’tsignificantcorrelationbetweenexplanatoryanddependentvariablesupplement：decisionruleoft-testH0:H1:rejectH0双尾β1=β1*β1≠β1*|T|>tα/2右尾β1≤β1*β1＞β1*T>tα左尾β1≥β1*β1

＜β1*T<-tαα:levelofsignificance,fixed(1-α)%:confidencedf:degreeoffreedomCriticalvalueP-value:probabilityactualprobabilityofobtainingavalueofT-TESTasmuchasorgreaterthanobtainedinOLSestimationIffixedαisgreaterthanP,wemayrejectthenullhypothesis.2.1.2confidenceintervalForstatisticTfollowstdistribution：Givenαlevelofsignificance：IscalledtheconfidenceintervalofthetrueßgivenconfidenceorthelevelofsignificanceIfzerofallswithintheconfidenceinterval,weacceptthenullhypothesis.Otherwise,rejectnullhypothesis.Correspondingly,2.2theanalysisofvarianceSSD.F.ESS1RSSN-2TSSN-12.3predictionMeanpredictionIndividualpredictionWehavesampleregressionfunctionAfuturevalueofXisknownasX0，thenthefutureconditionalmeansofYcanbepredictedasfollowing:IsestimatorofthefutureconditionalmeansofYItiscalledpointpredictorAnotherquestionisintervalpredictor:whichistheconfidenceintervalofgivenconfidence2.3.1MEANPREDICTIONPredictionofconditionalmeansYongivenXαisthelevelofsignificance2.3.2individualpredictionPredictionofanindividualYvalueongivenX1、Thedistributionofresidual2.ConstitutetheT-statisticαisthelevelofsignificanceThepredictiveabilityfallsmarkedlyasX0departsprogressivelyfromthesamplemean!ConfidenceintervalformeansConfidenceintervalforindividualsConclusion:Thegreatern,themoreaccuratepredictionThecloserforXtosamplemeansX0,themoreaccuratepredictionThewidthofconfidencebandsissmallestwhenThewiderthesamplerangeofX,themoreaccurateprediction2.4normalitytestsThehypothesistestisbasedontheassumptionthatthedisturbancefollowsthenormaldistribution. But,infact,isthisassumptionsatisfied?WeusetheresidualtocheckitHistogramofResidualsNormalProbabilityPlotself-studyJBtest2.5reportandanalysisoftheresultofregression

Analysis:1、Arethesigns

ofestimatedregressioncoefficientaccordancewithourexpectationoreconomictheory?2、Istheslopecoefficientstatisticalsignificant?(statisticallysignificantlydifferentfromZero.)3、HowwelldoestheregressionmodelexplainthechangeorvariationofY?IfR2isgreatenough,wecansaythattheregressionmodelseemsquitegood.4、Isourmodelsatisfiedthenormalityassumption

？案例11977-1991，美国，讨论股票和黄金的保值作用，《货币战争》中的观点是否得到支持？

每盎司纽约黄金美元价格消费价格指数（CPI）纽约股票交易所指数(NYSE)147.9860.653.69193.4465.253.7307.6272.658.32612.5182.468.1459.6190.974.02376.0196.568.93423.8