专业-金融计量lecture02简单回归

Lecture02

Simplelinearmodel

Outline2.1Introducingsimplemodel2.2OLSestimation2.3PropertiesofOLSStatistics2.1IntroducingsimplelinearmodelSimpletaskineconometrics

“explainingyintermsofx,”orin“studyinghowyvarieswithchangesinx.”影响是否存在？如何影响？有多大影响？其实就是建立关系，甚至是因果关系三大任务都可以表示为这样：estimatingeconomicrelationshipsEffectofeducationonwages：whatisx,whatisy?EffectofspendingonpoliceoncrimeEffectofhoursspentstudyingonthefinalgradeattheexamSimpletaskineconometricstestingeconomictheoriesCAPM,whatisx,whatisy?andmoreevaluatinggovernmentandbusinesspolicy劳动培训是否能（在多大程度上）帮助失业的人找到工作，提高工资？Threeproblemswiththesimpletasksincethereisneveranexactrelationshipbetweentwovariables,howdoweallowforotherfactorstoaffecty?Wagedependsonmanyfactors,includingeducation不是充分必要的关系，不是确定性的关系whatisthefunctionalrelationshipbetweenyandx?Positivecorrelated,butlinearornonlinear?howcanwebesurewearecapturingaceterisparibusrelationshipbetweenyandx(ifthatisadesiredgoal)?AchievethegoalbyasimplelinearregressionAsimpleequationtoexplainyintermsofxiswhereuisarandomvariableuncorrelatedwithx在多元的情况下，controlvariableshavedifferentmeaningsfromtheotherterms

Threeproblemscanbesolvedinthisequationtohandletheproblem1：errortermThejointeffectofallotherfactorstohandleproblem2:linearrelationship:meaning:aone-unitchangeinxhasthesame

effectony,regardlessoftheinitialvalueofx.Istheeffectofeducationonwagelinear?10->11yrsvs.11->12yrs?Otherrelationship?3.Problem3,CeterisParibusassumptionpartialderivativesexplanation:Meaningoftheassumption->partialderivatives->inordertoletbetacapturetheeffect,partialderivativeofuoverxshouldbezero!Randomvariableexplanation:ifx,y,anduareallrandom,andxisindependentofu,onaverage,betawillcapturetheeffect.Infact,uncorrelatedisenoughTheuncorrelationassumptionisequivalenttoE(u|x)＝E(u)(2.6)economicinterpretationBecausethereexistsbeta0,wecanalwaysassumethatE(u)=0tohavepopulationregressionfunction(PRF)Model2.1isalsocalledlinearregressionmodel,regressionofyonx“回归”一词的由来回归分析最旱起源于生物学的研究，“回归”最初是遗传学中的一个名词。1889年英国的生物学家兼统计学家F.Gallton和他的朋友K.Pearson收集了上千个家庭成员的身高、臂长和腿长的记录。企图寻找出子女们身高与父母们身高之间关系的具体表现形式。假定线性模型X为父母的平均身高，Y为成年子女的平均身高。根据1078个家庭的调查结果估计系数得到：一个数值分析：平均174.23cm（？）,矮个父母（160cm）的孩子身高166.89cm；搞个父母（180cm）的孩子身高177.21cm。龙生龙，凤生凤，老鼠的儿子会打洞但并非高的越长越高，矮的越长越矮。父母身高增加一个单位（由于变异或其他原因），而Y仅增加0.516个单位）。高个子父母的子女身高有低于其父母身高的趋势，而矮个子父母的子女身高有高于其父母身高的趋势，结论：父母所生子女有回归于人类平均身高的趋势，故某人种的平均身高是相当稳定的。Galton把这种孩子的身高向中间值靠近的趋势称之为一种回归效应，而他发展的研究两个数值变量的方法称为回归分析。见1889年F.Gallton的论文《普用回归定律》回归的含义：任何变异的东西总有趋向于一般、平稳的势头。回归应该是mean-reverting回归定律的普适性穷不过三代，富不过三代好电影的续集JF上关于资本结构的获奖论文Whatifwehavetherelationbelow,stillmeanreverting?Eventhoughnotnecessarymeanreverting,itisstillcalledregressionmodel统计课上也讲回归分析,几点差别在统计中x一般是确定的，我们这里不要求，只要求x是外生的（x影响y，而不是y对x也有影响）统计中不讲为什么要求x确定（或外生）。2.2estimatingthesimplelinearmodelWhydoweneedtoestimateamodel?Fromdataandeconomictheory(intuition)tocausalityeffect,weneedtoknowtheparametersCase1:twoobservationsCaseII:multipleobservations,whichisthebest?DependsonthecriteriaTherearemanymethodstoestimateamodel.OLSisthesimplestone.Twostepstoderiveit:DefineobjectiveOptimizingtheobjectiveAnyestimationmethodinvolvesoptimizinganobjectiveMLE:log-likelihoodGMM:momentsdifference我们介绍最小二乘法（OLS），因为它直观（道理）、简单（计算）DerivingOLSestimatorDefineerror:Equation(2.20)isalsocalledsampleregressionfunction

DefineobjectiveWewanttochoosebeta_hattomakethesumofsquaredresiduals,assmallaspossible.简单和？绝对值之和？OptimizationtogettheformulaFOCSOCsolutionExample2.3:CEOsalaryandreturnonequity.OutsiderswanttoknowwhetherCEO’sarepaidbytheirperformanceTostudytherelationshipbetweenthefirmperformance(measuredasROE)andCEOcompensation(salary),wepostulatethesimplemodelData:CEOSAL1SAS:人工计算SAS：regcommandData:CEOSAL1SASInterprettheregressionexampleWhattheparameterestimatorsmean:beta0_hat,beta1_hatForecastingWhatisforecastinganyway?CausalityanalysisWhatdoweexpectforbeta1.Whatifbeta1_hat=0,<0,>>>0?Whatistheestimator?2.3PropertiesofOLSStatisticsAlgebraicPropertiesofOLSStatisticsWecanviewOLSasposingeachyiintotwopart拟合值及拟合误差Thesummationoferror=0Proof:FOC打猎Thesamplecovariancebetweentheregressorsandtheresiduals=0Proof:FOC22Goodness-of-FitWhyisitaproblem?Youmightthinkthatonewaytomeasurethefitofthemodelistoadduptheresiduals.However,bydefinitiontheresidualswillsumtozero.Analternativeistosquaretheresiduals,addthemup(givingtheRSS)andthentakethesquareroot.RootMSE=squarerootofRSS/n-kOnewaytointerprettherootMSEisthatitshowshowfarawayonaveragethemodelisfromexplainingyArelativeerrormeasure:R2Goodness-of-Fit:R2IntroducingSST,SSEandSSRItiseasytoshowthatIntroducingRsquare:TheR-squaredoftheregression,sometimescalledthecoefficientofdetermination,isdefinedasMeaningofR2theratiooftheexplainedvariationcomparedto