專業(yè)-金融計(jì)量lecture02簡單回歸

Lecture02

Simplelinearmodel

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

“explainingyintermsofx,”orin“studyinghowyvarieswithchangesinx.”影響是否存在？如何影響？有多大影響？其實(shí)就是建立關(guān)系，甚至是因果關(guān)系三大任務(wù)都可以表示為這樣：estimatingeconomicrelationshipsEffectofeducationonwages：whatisx,whatisy?EffectofspendingonpoliceoncrimeEffectofhoursspentstudyingonthefinalgradeattheexamSimpletaskineconometricstestingeconomictheoriesCAPM,whatisx,whatisy?andmoreevaluatinggovernmentandbusinesspolicy勞動(dòng)培訓(xùn)是否能（在多大程度上）幫助失業(yè)的人找到工作，提高工資？Threeproblemswiththesimpletasksincethereisneveranexactrelationshipbetweentwovariables,howdoweallowforotherfactorstoaffecty?Wagedependsonmanyfactors,includingeducation不是充分必要的關(guān)系，不是確定性的關(guān)系whatisthefunctionalrelationshipbetweenyandx?Positivecorrelated,butlinearornonlinear?howcanwebesurewearecapturingaceterisparibusrelationshipbetweenyandx(ifthatisadesiredgoal)?AchievethegoalbyasimplelinearregressionAsimpleequationtoexplainyintermsofxiswhereuisarandomvariableuncorrelatedwithx在多元的情況下，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“回歸”一詞的由來回歸分析最旱起源于生物學(xué)的研究，“回歸”最初是遺傳學(xué)中的一個(gè)名詞。1889年英國的生物學(xué)家兼統(tǒng)計(jì)學(xué)家F.Gallton和他的朋友K.Pearson收集了上千個(gè)家庭成員的身高、臂長和腿長的記錄。企圖尋找出子女們身高與父母們身高之間關(guān)系的具體表現(xiàn)形式。假定線性模型X為父母的平均身高，Y為成年子女的平均身高。根據(jù)1078個(gè)家庭的調(diào)查結(jié)果估計(jì)系數(shù)得到：一個(gè)數(shù)值分析：平均174.23cm（？）,矮個(gè)父母（160cm）的孩子身高166.89cm；搞個(gè)父母（180cm）的孩子身高177.21cm。龍生龍，鳳生鳳，老鼠的兒子會(huì)打洞但并非高的越長越高，矮的越長越矮。父母身高增加一個(gè)單位（由于變異或其他原因），而Y僅增加0.516個(gè)單位）。高個(gè)子父母的子女身高有低于其父母身高的趨勢，而矮個(gè)子父母的子女身高有高于其父母身高的趨勢，結(jié)論：父母所生子女有回歸于人類平均身高的趨勢，故某人種的平均身高是相當(dāng)穩(wěn)定的。Galton把這種孩子的身高向中間值靠近的趨勢稱之為一種回歸效應(yīng)，而他發(fā)展的研究兩個(gè)數(shù)值變量的方法稱為回歸分析。見1889年F.Gallton的論文《普用回歸定律》回歸的含義：任何變異的東西總有趨向于一般、平穩(wěn)的勢頭?；貧w應(yīng)該是mean-reverting回歸定律的普適性窮不過三代，富不過三代好電影的續(xù)集JF上關(guān)于資本結(jié)構(gòu)的獲獎(jiǎng)?wù)撐腤hatifwehavetherelationbelow,stillmeanreverting?Eventhoughnotnecessarymeanreverting,itisstillcalledregressionmodel統(tǒng)計(jì)課上也講回歸分析,幾點(diǎn)差別在統(tǒng)計(jì)中x一般是確定的，我們這里不要求，只要求x是外生的（x影響y，而不是y對(duì)x也有影響）統(tǒng)計(jì)中不講為什么要求x確定（或外生）。2.2estimatingthesimplelinearmodelWhydoweneedtoestimateamodel?Fromdataandeconomictheory(intuition)tocausalityeffect,weneedtoknowtheparametersCase1:twoobservationsCaseII:multipleobservations,whichisthebest?DependsonthecriteriaTherearemanymethodstoestimateamodel.OLSisthesimplestone.Twostepstoderiveit:DefineobjectiveOptimizingtheobjectiveAnyestimationmethodinvolvesoptimizinganobjectiveMLE:log-likelihoodGMM:momentsdifference我們介紹最小二乘法（OLS），因?yàn)樗庇^（道理）、簡單（計(jì)算）DerivingOLSestimatorDefineerror:Equation(2.20)isalsocalledsampleregressionfunction

DefineobjectiveWewanttochoosebeta_hattomakethesumofsquaredresiduals,assmallaspossible.簡單和？絕對(duì)值之和？OptimizationtogettheformulaFOCSOCsolutionExample2.3:CEOsalaryandreturnonequity.OutsiderswanttoknowwhetherCEO’sarepaidbytheirperformanceTostudytherelationshipbetweenthefirmperformance(measuredasROE)andCEOcompensation(salary),wepostulatethesimplemodelData:CEOSAL1SAS:人工計(jì)算SAS：regcommandData:CEOSAL1SASInterprettheregressionexampleWhattheparameterestimatorsmean:beta0_hat,beta1_hatForecastingWhatisforecastinganyway?CausalityanalysisWhatdoweexpectforbeta1.Whatifbeta1_hat=0,<0,>>>0?Whatistheestimator?2.3PropertiesofOLSStatisticsAlgebraicPropertiesofOLSStatisticsWecanviewOLSasposingeachyiintotwopart擬合值及擬合誤差Thesummationoferror=0Proof:FOC打獵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