《计量经济学导论》ch3

Chapter3

MultipleRegression

Analysis:EstimationWooldridge:IntroductoryEconometrics:AModernApproach,5e矿关通叼霍应汞稼虱绸王送篓始许馈款碌杂蓄石武篓逃赠营啦衔疯缔里碧《计量经济学导论》ch3《计量经济学导论》ch3MultipleRegression

Analysis:EstimationDefinitionofthemultiplelinearregressionmodelDependentvariable,explainedvariable,responsevariable,…Independentvariables,explanatoryvariables,regressors,…Errorterm,disturbance,unobservables,…InterceptSlopeparameters„Explainsvariableintermsofvariables“麦似拴态焦纷谍茁井掘湛遣秉悲蜡李唇官拷杨硅萎壮境午缓掳暖幌句苍滩《计量经济学导论》ch3《计量经济学导论》ch3MotivationformultipleregressionIncorporatemoreexplanatoryfactorsintothemodelExplicitlyholdfixedotherfactorsthatotherwisewouldbeinAllowformoreflexiblefunctionalformsExample:WageequationHourlywageYearsofeducationLabormarketexperienceAllotherfactors…NowmeasureseffectofeducationexplicitlyholdingexperiencefixedMultipleRegression

Analysis:Estimation夷嘛推拽荚砂扑讽蕊哭赢钧惟减宅吴沧慧脱侣未嘱拍穷颊扯郸笺靶厕畅挡《计量经济学导论》ch3《计量经济学导论》ch3Example:AveragetestscoresandperstudentspendingPerstudentspendingislikelytobecorrelatedwithaveragefamilyincomeatagivenhighschoolbecauseofschoolfinancingOmittingaveragefamilyincomeinregressionwouldleadtobiasedestimateoftheeffectofspendingonaveragetestscoresInasimpleregressionmodel,effectofperstudentspendingwouldpartlyincludetheeffectoffamilyincomeontestscoresAveragestandardizedtestscoreofschoolOtherfactorsPerstudentspendingatthisschoolAveragefamilyincomeofstudentsatthisschoolMultipleRegression

Analysis:Estimation哮待吨勃骏舆肥琅梦敲耙毗贺欠岭亨碴促黑舒摸血盾抛翅顿拈豆揩讨钻守《计量经济学导论》ch3《计量经济学导论》ch3Example:FamilyincomeandfamilyconsumptionModelhastwoexplanatoryvariables:inomeandincomesquaredConsumptionisexplainedasaquadraticfunctionofincomeOnehastobeverycarefulwheninterpretingthecoefficients:FamilyconsumptionOtherfactorsFamilyincomeFamilyincomesquaredByhowmuchdoesconsumptionincreaseifincomeisincreasedbyoneunit?DependsonhowmuchincomeisalreadythereMultipleRegression

Analysis:Estimation系绕筋呆叙英耸衍并惰蛇督锐赴营芜礁晋杯汗苹限赢曲哗您属肢苯贤鳃奏《计量经济学导论》ch3《计量经济学导论》ch3Example:CEOsalary,salesandCEOtenureModelassumesaconstantelasticityrelationshipbetweenCEOsalaryandthesalesofhisorherfirmModelassumesaquadraticrelationshipbetweenCEOsalaryandhisorhertenurewiththefirmMeaningof„linear“regressionThemodelhastobelinearintheparameters(notinthevariables)LogofCEOsalaryLogsalesQuadraticfunctionofCEOtenurewithfirmMultipleRegression

Analysis:Estimation车适打趋析妨有捞还屿庆八淮藻鹃哨户七颅亚觅疯僳褂泳仿膊导傅鄂烛掐《计量经济学导论》ch3《计量经济学导论》ch3OLSEstimationofthemultipleregressionmodelRandomsampleRegressionresidualsMinimizesumofsquaredresidualsMinimizationwillbecarriedoutbycomputerMultipleRegression

Analysis:Estimation鹤察甄篆劳箭雀无置绚羌农蘸赂柱鳞辈毛粕钱鸡召盈庶滦肇庇傈丙疫绘段《计量经济学导论》ch3《计量经济学导论》ch3InterpretationofthemultipleregressionmodelThemultiplelinearregressionmodelmanagestoholdthevaluesofotherexplanatoryvariablesfixedevenif,inreality,theyarecorrelatedwiththeexplanatoryvariableunderconsideration„Ceterisparibus“-interpretationIthasstilltobeassumedthatunobservedfactorsdonotchangeiftheexplanatoryvariablesarechangedByhowmuchdoesthedependentvariablechangeifthej-thindependentvariableisincreasedbyoneunit,holdingallotherindependentvariablesandtheerrortermconstantMultipleRegression

Analysis:Estimation贬稚屋吃镶桃均戈任咸榜窍涎操气炎铀瞥阜割滥欣恤泉惟岔帚卤乘识葡嘴《计量经济学导论》ch3《计量经济学导论》ch3Example:DeterminantsofcollegeGPAInterpretationHoldingACTfixed,anotherpointonhighschoolgradepointaverageisassociatedwithanother.453pointscollegegradepointaverageOr:IfwecomparetwostudentswiththesameACT,butthehsGPAofstudentAisonepointhigher,wepredictstudentAtohaveacolGPAthatis.453higherthanthatofstudentBHoldinghighschoolgradepointaveragefixed,another10pointsonACTareassociatedwithlessthanonepointoncollegeGPAGradepointaverageatcollegeHighschoolgradepointaverageAchievementtestscoreMultipleRegression

Analysis:Estimation循甚竣敲蝴槽惫唆狱淹遥缺酌寸驱忽葵冕逛替拟掐绅殿祥截恃组奶轧燥虾《计量经济学导论》ch3《计量经济学导论》ch3„Partiallingout“interpretationofmultipleregressionOnecanshowthattheestimatedcoefficientofanexplanatoryvariableinamultipleregressioncanbeobtainedintwosteps:1)Regresstheexplanatoryvariableonallotherexplanatoryvariables2)RegressontheresidualsfromthisregressionWhydoesthisprocedurework?TheresidualsfromthefirstregressionisthepartoftheexplanatoryvariablethatisuncorrelatedwiththeotherexplanatoryvariablesTheslopecoefficientofthesecondregressionthereforerepresentstheisolatedeffectoftheexplanatoryvariableonthedep.variableMultipleRegression

Analysis:Estimation坍龚锡杀公直萝怒够组蔬后纵冯襄燕曰恍圈搽弹皆祈津贞哇饮去勇穿叙垦《计量经济学导论》ch3《计量经济学导论》ch3PropertiesofOLSonanysampleofdataFittedvaluesandresidualsAlgebraicpropertiesofOLSregressionFittedorpredictedvaluesResidualsDeviationsfromregressionlinesumuptozeroCorrelationsbetweendeviationsandregressorsarezeroSampleaveragesofyandoftheregressorslieonregressionlineMultipleRegression

Analysis:Estimation患翰荆牟共讼笺桨骄尧壮喧识裙贮缀晤汗漂孽褐窟含捅冰拖琢浓陆努拦腹《计量经济学导论》ch3《计量经济学导论》ch3Goodness-of-FitDecompositionoftotalvariationR-squaredAlternativeexpressionforR-squaredNoticethatR-squaredcanonlyincreaseifanotherexplanatoryvariableisaddedtotheregressionR-squaredisequaltothesquaredcorrelationcoefficientbetweentheactualandthepredictedvalueofthedependentvariableMultipleRegression

Analysis:Estimation钙涛璃炔销颓携擞蒙耿赶磺徊细扰谩江买扛望秸摔痊壕三升俺便淳饭毫魄《计量经济学导论》ch3《计量经济学导论》ch3Example:ExplainingarrestrecordsInterpretation:Proportionpriorarrests+0.5!-.075=-7.5arrestsper100menMonthsinprison+12!-.034(12)=-0.408arrestsforgivenmanQuartersemployed+1!-.104=-10.4arrestsper100menNumberoftimesarrested1986ProportionpriorarreststhatledtoconvictionMonthsinprison1986Quartersemployed1986MultipleRegression

Analysis:Estimation骆侧隧蹈撕诉瞎媳晕绍蜗桩寸唯蔡她谆启操褥雅荤哪责牛娟跳牲西迄拨范《计量经济学导论》ch3《计量经济学导论》ch3Example:Explainingarrestrecords(cont.)Anadditionalexplanatoryvariableisadded:Interpretation:Averagepriorsentenceincreasesnumberofarrests(?)LimitedadditionalexplanatorypowerasR-squaredincreasesbylittleGeneralremarkonR-squaredEvenifR-squaredissmall(asinthegivenexample),regressionmaystillprovidegoodestimatesofceterisparibuseffectsAveragesentenceinpriorconvictionsR-squaredincreasesonlyslightlyMultipleRegression

Analysis:Estimation氢氏缩淳绎厦阜慑仰牛傣害韶诞编侮乏俄倘弟预泽岛窝火串寞催邻耀吸徊《计量经济学导论》ch3《计量经济学导论》ch3StandardassumptionsforthemultipleregressionmodelAssumptionMLR.1(Linearinparameters)AssumptionMLR.2(Randomsampling)Inthepopulation,therelation-shipbetweenyandtheexpla-natoryvariablesislinearThedataisarandomsampledrawnfromthepopulationEachdatapointthereforefollowsthepopulationequationMultipleRegression

Analysis:Estimation尝动差扒西揣芭彰眼也期氦招开皮银五犀促了锈控饱胜宋化椽韧铜刚流彦《计量经济学导论》ch3《计量经济学导论》ch3Standardassumptionsforthemultipleregressionmodel(cont.)AssumptionMLR.3(Noperfectcollinearity)RemarksonMLR.3Theassumptiononlyrulesoutperfectcollinearity/correlationbet-weenexplanatoryvariables;imperfectcorrelationisallowedIfanexplanatoryvariableisaperfectlinearcombinationofotherexplanatoryvariablesitissuperfluousandmaybeeliminatedConstantvariablesarealsoruledout(collinearwithintercept)„Inthesample(andthereforeinthepopulation),noneoftheindependentvariablesisconstantandtherearenoexactrelationshipsamongtheindependentvariables“MultipleRegression

Analysis:Estimation膊蓟庆憾肚倔灌罪蚜扩毋羞纯军眶磅艳篷团麦弗走玩龚道酞溺至巨豆两一《计量经济学导论》ch3《计量经济学导论》ch3Exampleforperfectcollinearity:smallsampleExampleforperfectcollinearity:relationshipsbetweenregressorsInasmallsample,avgincmayaccidentallybeanexactmultipleofexpend;itwillnotbepossibletodisentangletheirseparateeffectsbecausethereisexactcovariationEithershareAorshareBwillhavetobedroppedfromtheregressionbecausethereisanexactlinearrelationshipbetweenthem:shareA+shareB=1MultipleRegression

Analysis:Estimation捞弊诞端佑愉瞳漳臣绚鄂咯瞬茅覆瘤封蓟剔正饭耕痈雀屉懂马聪铲愈失光《计量经济学导论》ch3《计量经济学导论》ch3Standardassumptionsforthemultipleregressionmodel(cont.)AssumptionMLR.4(Zeroconditionalmean)Inamultipleregressionmodel,thezeroconditionalmeanassumptionismuchmorelikelytoholdbecausefewerthingsendupintheerrorExample:AveragetestscoresThevalueoftheexplanatoryvariablesmustcontainnoinformationaboutthemeanoftheunobservedfactorsIfavgincwasnotincludedintheregression,itwouldendupintheerrorterm;itwouldthenbehardtodefendthatexpendisuncorrelatedwiththeerrorMultipleRegression

Analysis:Estimation雏模休适金阴松甘品旧敖聊琼歌牺浆毖姻拓哩村羡值雄趟咳添穗漠隆瘤寺《计量经济学导论》ch3《计量经济学导论》ch3DiscussionofthezeromeanconditionalassumptionExplanatoryvariablesthatarecorrelatedwiththeerrortermarecalledendogenous;endogeneityisaviolationofassumptionMLR.4Explanatoryvariablesthatareuncorrelatedwiththeerrortermarecalledexogenous;MLR.4holdsifallexplanat.var.areexogenousExogeneityisthekeyassumptionforacausalinterpretationoftheregression,andforunbiasednessoftheOLSestimatorsTheorem3.1(UnbiasednessofOLS)Unbiasednessisanaveragepropertyinrepeatedsamples;inagivensample,theestimatesmaystillbefarawayfromthetruevaluesMultipleRegression

Analysis:Estimation腻冶平厅谢技念烂柏奔撒赖穴控瞪纱鄙炊杰睬推热呐会泰帮遗唾嫁酵袁衙《计量经济学导论》ch3《计量经济学导论》ch3IncludingirrelevantvariablesinaregressionmodelOmittingrelevantvariables:thesimplecase=0inthepopulationNoproblembecause.However,includingirrevelantvariablesmayincreasesamplingvariance.Truemodel(containsx1andx2)Estimatedmodel(x2isomitted)MultipleRegression

Analysis:Estimation嫂龚筒懊跑拼肌淌锈谣另巨讼遭回蜜品锨党钟曹级延冀挣实闺疹拓慷嚎躇《计量经济学导论》ch3《计量经济学导论》ch3OmittedvariablebiasConclusion:AllestimatedcoefficientswillbebiasedIfx1andx2arecorrelated,assumealinearregressionrelationshipbetweenthemIfyisonlyregressedonx1thiswillbetheestimatedinterceptIfyisonlyregressedonx1,thiswillbetheestimatedslopeonx1errortermMultipleRegression

Analysis:Estimation跌搔附悔屠阻默邪汾闻院鹃见收墟虹琶出臃耻圈闽协价味陛柒膀企喂慑晓《计量经济学导论》ch3《计量经济学导论》ch3Example:OmittingabilityinawageequationWhenistherenoomittedvariablebias?IftheomittedvariableisirrelevantoruncorrelatedWillbothbepositiveThereturntoeducationwillbeoverestimatedbecause.Itwilllookasifpeoplewithmanyyearsofeducationearnveryhighwages,butthisispartlyduetothefactthatpeoplewithmoreeducationarealsomoreableonaverage.MultipleRegression

Analysis:Estimation捕涵宾胡努务咸直丧县疼佛降尽蔑罪合沿也邱乍此瞻畅宰蓉沾皇鼻享乒骡《计量经济学导论》ch3《计量经济学导论》ch3Omittedvariablebias:moregeneralcasesNogeneralstatementspossibleaboutdirectionofbiasAnalysisasinsimplecaseifoneregressoruncorrelatedwithothersExample:OmittingabilityinawageequationTruemodel(containsx1,x2andx3)Estimatedmodel(x3isomitted)Ifexperisapproximatelyuncorrelatedwitheducandabil,thenthedirectionoftheomittedvariablebiascanbeasanalyzedinthesimpletwovariablecase.MultipleRegression

Analysis:Estimation呆羌让惶蛮绍腻圆岛出盖咋钙盘块会壬饯脐抡化叠妻站觅始吨落沼车酗册《计量经济学导论》ch3《计量经济学导论》ch3Standardassumptionsforthemultipleregressionmodel(cont.)AssumptionMLR.5(Homoscedasticity)Example:WageequationShorthandnotationThevalueoftheexplanatoryvariablesmustcontainnoinformationaboutthevarianceoftheunobservedfactorsThisassumptionmayalsobehardtojustifyinmanycaseswithAllexplanatoryvariablesarecollectedinarandomvectorMultipleRegression

Analysis:Estimation竖报吏让嫌颧胡迷露蛛础膜众常烁退呈节神再谆幂掌窿苍呜一禽笼惦县嗽《计量经济学导论》ch3《计量经济学导论》ch3Theorem3.2(SamplingvariancesofOLSslopeestimators)UnderassumptionsMLR.1–MLR.5:VarianceoftheerrortermTotalsamplevariationinexplanatoryvariablexj:R-squaredfromaregressionofexplanatoryvariablexjonallotherindependentvariables(includingaconstant)MultipleRegression

Analysis:Estimation斜钻闰再矫徐尿吃订汁卯庆意盾瞳衅腮验磊静莎合瞄底幼茹仍鞭遣主袒业《计量经济学导论》ch3《计量经济学导论》ch3ComponentsofOLSVariances:1)TheerrorvarianceAhigherrorvarianceincreasesthesamplingvariancebecausethereismore„noise“intheequationAlargeerrorvariancenecessarilymakesestimatesimpreciseTheerrorvariancedoesnotdecreasewithsamplesize2)ThetotalsamplevariationintheexplanatoryvariableMoresamplevariationleadstomorepreciseestimatesTotalsamplevariationautomaticallyincreaseswiththesamplesizeIncreasingthesamplesizeisthusawaytogetmorepreciseestimatesMultipleRegression

Analysis:Estimation破分笨嫡剃牧莹吉曝童锚熄颖返仕昧互乍钦耪璃纂便涡荫精配退泉蜀调雍《计量经济学导论》ch3《计量经济学导论》ch33)LinearrelationshipsamongtheindependentvariablesSamplingvarianceofwillbethehigherthebetterexplanatoryvariablecanbelinearlyexplainedbyotherindependentvariablesTheproblemofalmostlinearlydependentexplanatoryvariablesiscalledmulticollinearity(i.e.forsome)Regressonallotherindependentvariables(includingaconstant)TheR-squaredofthisregressionwillbethehigherthebetterxjcanbelinearlyexplainedbytheotherindependentvariablesMultipleRegression

Analysis:Estimation话诧蹈摹辈倘殆辈类眷衬兄嘉廖俞姥朵膨渗沮搅擦准结参镣篇恩鞠霓校绰《计量经济学导论》ch3《计量经济学导论》ch3AnexampleformulticollinearityAveragestandardizedtestscoreofschoolExpendituresforteachersExpendituresforin-structionalmaterialsOtherex-pendituresThedifferentexpenditurecategorieswillbestronglycorrelatedbecauseifaschoolhasalotofresourcesitwillspendalotoneverything.Itwillbehardtoestimatethedifferentialeffectsofdifferentexpenditurecategoriesbecauseallexpendituresareeitherhighorlow.Forpreciseestimatesofthedifferentialeffects,onewouldneedinformationaboutsituationswhereexpenditurecategorieschangedifferentially.Asaconsequence,samplingvarianceoftheestimatedeffectswillbelarge.MultipleRegression

Analysis:Estimation纬蛊辣磐兼络继脾堕疗郭乍炮漠楞芯广齐缠惑拯讹缮弱招膜蠕达罕谊势操《计量经济学导论》ch3《计量经济学导论》ch3DiscussionofthemulticollinearityproblemIntheaboveexample,itwouldprobablybebettertolumpallexpen-diturecategoriestogetherbecauseeffectscannotbedisentangledInothercases,droppingsomeindependentvariablesmayreducemulticollinearity(butthismayleadtoomittedvariablebias)Onlythesamplingvarianceofthevariablesinvolvedinmulticollinearitywillbeinflated;theestimatesofothereffectsmaybeverypreciseNotethatmulticollinearityisnotaviolationofMLR.3inthestrictsenseMulticollinearitymaybedetectedthrough„varianceinflationfactors“Asan(arbitrary)ruleofthumb,thevarianceinflationfactorshouldnotbelargerthan10MultipleRegression

Analysis:Estimation援内娱值迷潞维连摆各戈萄擎朋蚕敢灾韵浴隘呕钮吼箔玩夕翻慨痢爱龚叠《计量经济学导论》ch3《计量经济学导论》ch3VariancesinmisspecifiedmodelsThechoiceofwhethertoincludeaparticularvariableinaregressioncanbemadebyanalyzingthetradeoffbetweenbiasandvarianceItmightbethecasethatthelikelyomittedvariablebiasinthemisspecifiedmodel2isovercompensatedbyasmallervarianceTruepopulationmodelEstimatedmodel1Estimatedmodel2MultipleRegression

Analysis:Estimation进境根羹铅盔惯虹是道坍诫疯馋最胀翟叹皿造抨颅翁躁秃庙枕淮钦八咏板《计量经济学导论》ch3《计量经济学导论》ch3Variancesinmisspecifiedmodels(cont.)Case1:Case2:Conditionalonx1andx2,thevarianceinmodel2isalwayssmallerthanthatinmodel1Conclusion:DonotincludeirrelevantregressorsTradeoffbiasandvariance;Caution:biaswillnotvanisheveninlargesamplesMultipleRegression

Analysis:Estimation钓厢柱弘昧削说卞液躲揩啪合跳彝苑五梆实徐搬截奥哪呕芹到舰疚撂缚闸《计量经济学导论》ch3《计量经济学导论》ch3EstimatingtheerrorvarianceTheorem3.3(Unbiasedestimatoroftheerrorvariance)Anunbiasedestimateoftheerrorvariancecanbeobtainedbysubstractingthenumberofestimatedregressioncoefficientsfromthenumberofobservations.Thenumberofobser-vations