数学与应用数学中英文外文翻译文献

中英文翻译UNITSOFMEASUREMENTANDFUNCTIONALFORM(VotingOutcomesandCampaignExpenditures)Inthevotingoutcomeequationin(2.28),R²=0.505.Thus,theshareofcampaignexpendituresexplainsjustover50percentofthevariationintheelectionoutcomesforthissample.ThisisafairlysizableportionTwoimportantissuesinappliedeconomicsare(1)understandinghowchangingtheunitsofmeasurementofthedependentand/orindependentvariablesaffectsOLSestimatesand(2)knowinghowtoincorporatepopularfunctionalformsusedineconomicsintoregressionanalysis.ThemathematicsneededforafullunderstandingoffunctionalformissuesisreviewedinAppendixA.TheEffectsofChangingUnitsofMeasurementonOLSStatisticsInExample2.3,wechosetomeasureannualsalaryinthousandsofdollars,andthereturnonequitywasmeasuredasapercent(ratherthanasadecimal).Itiscrucialtoknowhowsalaryandroearemeasuredinthisexampleinordertomakesenseoftheestimatesinequation(2.39).WemustalsoknowthatOLSestimateschangeinentirelyexpectedwayswhentheunitsofmeasurementofthedependentandindependentvariableschange.InExample2.3,supposethat,ratherthanmeasuringsalaryinthousandsofdollars,wemeasureitindollars.Letsalardolbesalaryindollars(salardol=845,761wouldbeinterpretedas$845,761.).Ofcourse,salardolhasasimplerelationshiptothesalarymeasuredinthousandsofdollars:salardol?1,000?salary.Wedonotneedtoactuallyruntheregressionofsalardolonroetoknowthattheestimatedequationis:salaˆrdol=963,191+18,501roe.Weobtaintheinterceptandslopein(2.40)simplybymultiplyingtheinterceptandtheslopein(2.39)by1,000.Thisgivesequations(2.39)and(2.40)thesameinterpretation.Lookingat(2.40),ifroe=0,thensalaˆrdol=963,191,sothepredictedsalaryis$963,191[thesamevalueweobtainedfromequation(2.39)].Furthermore,ifroeincreasesbyone,thenthepredictedsalaryincreasesby$18,501;again,thisiswhatweconcludedfromourearlieranalysisofequation(2.39).Generally,itiseasytofigureoutwhathappenstotheinterceptandslopeestimateswhenthedependentvariablechangesunitsofmeasurement.Ifthedependentvariableismultipliedbytheconstantc—whichmeanseachvalueinthesampleismultipliedbyc—thentheOLSinterceptandslopeestimatesarealsomultipliedbyc.(Thisassumesnothinghaschangedabouttheindependentvariable.)IntheCEOsalaryexample,c?1,000inmovingfromsalarytosalardol.Chapter2TheSimpleRegressionModelWecanalsousetheCEOsalaryexampletoseewhathappenswhenwechangetheunitsofmeasurementoftheindependentvariable.Defineroedec=roe/100tobethedecimalequivalentofroe;thus,roedec=0.23meansareturnonequityof23percent.Tofocusonchangingtheunitsofmeasurementoftheindependentvariable,wereturntoouroriginaldependentvariable,salary,whichismeasuredinthousandsofdollars.Whenweregresssalaryonroedec,weobtainsalˆary=963.191+1850.1roedec.Thecoefficientonroedecis100timesthecoefficientonroein(2.39).Thisisasitshouldbe.ChangingroebyonepercentagepointisequivalenttoΔroedec=0.01.From(2.41),ifΔroedec=0.01,thenΔsalˆary=1850.1(0.01)=18.501,whichiswhatisobtainedbyusing(2.39).Notethat,inmovingfrom(2.39)to(2.41),theindependentvariablewasdividedby100,andsotheOLSslopeestimatewasmultipliedby100,preservingtheinterpretationoftheequation.Generally,iftheindependentvariableisdividedormultipliedbysomenonzeroconstant,c,thentheOLSslopecoefficientisalsomultipliedordividedbycrespectively.Theintercepthasnotchangedin(2.41)becauseroedec=0stillcorrespondstoazeroreturnonequity.Ingeneral,changingtheunitsofmeasurementofonlytheindependentvariabledoesnotaffecttheintercept.Intheprevioussection,wedefinedR-squaredasagoodness-of-fitmeasureforOLSregression.WecanalsoaskwhathappenstoR2whentheunitofmeasurementofeithertheindependentorthedependentvariablechanges.Withoutdoinganyalgebra,weshouldknowtheresult:thegoodness-of-fitofthemodelshouldnotdependontheunitsofmeasurementofourvariables.Forexample,theamountofvariationinsalary,explainedbythereturnonequity,shouldnotdependonwhethersalaryismeasuredindollarsorinthousandsofdollarsoronwhetherreturnonequityisapercentoradecimal.Thisintuitioncanbeverifiedmathematically:usingthedefinitionofR2,itcanbeshownthatR2is,infact,invarianttochangesintheunitsofyorx.IncorporatingNonlinearitiesinSimpleRegressionSofarwehavefocusedonlinearrelationshipsbetweenthedependentandindependentvariables.AswementionedinChapter1,linearrelationshipsarenotnearlygeneralenoughforalleconomicapplications.Fortunately,itisrathereasytoincorporatemanynonlinearitiesintosimpleregressionanalysisbyappropriatelydefiningthedependentandindependentvariables.Herewewillcovertwopossibilitiesthatoftenappearinappliedwork.Inreadingappliedworkinthesocialsciences,youwilloftenencounterregressionequationswherethedependentvariableappearsinlogarithmicform.Whyisthisdone?Recallthewage-educationexample,whereweregressedhourlywageonyearsofeducation.Weobtainedaslopeestimateof0.54[seeequation(2.27)],whichmeansthateachadditionalyearofeducationispredictedtoincreasehourlywageby54cents.Becauseofthelinearnatureof(2.27),54centsistheincreaseforeitherthefirstyearofeducationorthetwentiethyear;thismaynotbereasonable.Suppose,instead,thatthepercentageincreaseinwageisthesamegivenonemoreyearofeducation.Model(2.27)doesnotimplyaconstantpercentageincrease:thepercentageincreasesdependsontheinitialwage.Amodelthatgives(approximately)aconstantpercentageeffectislog(wage)=β0+β1educ+u,(2.42)wherelog(.)denotesthenaturallogarithm.(SeeAppendixAforareviewoflogarithms.)Inparticular,ifΔu=0,then%Δwage=(100*β1)Δeduc.(2.43)Noticehowwemultiplyβ1by100togetthepercentagechangeinwagegivenoneadditionalyearofeducation.Sincethepercentagechangeinwageisthesameforeachadditionalyearofeducation,thechangeinwageforanextrayearofeducationincreasesaseducationincreases;inotherwords,(2.42)impliesanincreasingreturntoeducation.Byexponenttiating(2.42),wecanwritewage=exp(β0+β1educ+u).ThisequationisgraphedinFigure2.6,withu=0.Estimatingamodelsuchas(2.42)isstraightforwardwhenusingsimpleregression.Justdefinethedependentvariable,y,tobey=log(wage).Theindependentvariableisrepresentedbyx=educ.ThemechanicsofOLSarethesameasbefore:theinterceptandslopeestimatesaregivenbytheformulas(2.17)and(2.19).Inotherwords,weobtainβˆ0andβˆ1fromtheOLSregressionoflog(wage)oneduc.EXAMPLE2.10(ALogWageEquation)UsingthesamedataasinExample2.4,butusinglog(wage)asthedependentvariable,weobtainthefollowingrelationship:log(ˆwage)=0.584+0.083educ(2.44)n=526,R²=0.186.Thecoefficientoneduchasapercentageinterpretationwhenitismultipliedby100:wageincreasesby8.3percentforeveryadditionalyearofeducation.Thisiswhateconomistsmeanwhentheyrefertothe“returntoanotheryearofeducation.”Itisimportanttorememberthatthemainreasonforusingthelogofwagein(2.42)istoimposeaconstantpercentageeffectofeducationonwage.Onceequation(2.42)isobtained,thenaturallogofwageisrarelymentioned.Inparticular,itisnotcorrecttosaythatanotheryearofeducationincreaseslog(wage)by8.3%.Theinterceptin(2.42)isnotverymeaningful,asitgivesthepredictedlog(wage),wheneduc=0.TheR-squaredshowsthateducexplainsabout18.6percentofthevariationinlog(wage)(notwage).Finally,equation(2.44)mightnotcaptureallofthenon-linearityintherelationshipbetweenwageandschooling.Ifthereare“diplomaeffects,”thenthetwelfthyearofeducation—graduationfromhighschool—couldbeworthmuchmorethantheeleventhyear.WewilllearnhowtoallowforthiskindofnonlinearityinChapter7.Anotherimportantuseofthenaturallogisinobtainingaconstantelasticitymodel.EXAMPLE2.11(CEOSalaryandFirmSales)WecanestimateaconstantelasticitymodelrelatingCEOsalarytofirmsales.ThedatasetisthesameoneusedinExample2.3,exceptwenowrelatesalarytosales.Letsalesbeannualfirmsales,measuredinmillionsofdollars.Aconstantelasticitymodelislog(salary=β0+β1log(sales)+u,(2.45)whereβ1istheelasticityofsalarywithrespecttosales.Thismodelfallsunderthesimpleregressionmodelbydefiningthedependentvariabletobey=log(salary)andtheindependentvariabletobex=log(sales).EstimatingthisequationbyOLSgivesPart1RegressionAnalysiswithCross-SectionalDatalog(salˆary)=4.822?+0.257log(sales)(2.46)n=209,R²=0.211.Thecoefficientoflog(sales)istheestimatedelasticityofsalarywithrespecttosales.Itimpliesthata1percentincreaseinfirmsalesincreasesCEOsalarybyabout0.257percent—theusualinterpretationofanelasticity.Thetwofunctionalformscoveredinthissectionwilloftenariseintheremainderofthistext.Wehavecoveredmodelscontainingnaturallogarithmsherebecausetheyappearsofrequentlyinappliedwork.Theinterpretationofsuchmodelswillnotbemuchdifferentinthemultipleregressioncase.Itisalsousefultonotewhathappenstotheinterceptandslopeestimatesifwechangetheunitsofmeasurementofthedependentvariablewhenitappearsinlogarithmicform.Becausethechangetologarithmicformapproximatesaproportionatechange,itmakessensethatnothinghappenstotheslope.Wecanseethisbywritingtherescaledvariableasc1yiforeachobservationi.Theoriginalequationislog(yi)=β0+β1xi+ui.Ifweaddlog(c1)tobothsides,wegetlog(c1)+log(yi)+[log(c1)β0]+β1xi+ui,orlog(c1yi)?[log(c1)+β0]+β1xi+ui.(RememberthatthesumofthelogsisequaltothelogoftheirproductasshowninAppendixA.)Therefore,theslopeisstill?1,buttheinterceptisnowlog(c1)??0.Similarly,iftheindependentvariableislog(x),andwechangetheunitsofmeasurementofxbeforetakingthelog,thesloperemainsthesamebuttheinterceptdoesnotchange.YouwillbeaskedtoverifytheseclaimsinProblem2.9.Weendthissubsectionbysummarizingfourcombinationsoffunctionalformsavailablefromusingeithertheoriginalvariableoritsnaturallog.InTable2.3,xandystandforthevariablesintheiroriginalform.Themodelwithyasthedependentvariableandxastheindependentvariableiscalledthelevel-levelmodel,becauseeachvariableappearsinitslevelform.Themodelwithlog(y)asthedependentvariableandxastheindependentvariableiscalledthelog-levelmodel.Wewillnotexplicitlydiscussthelevel-logmodelhere,becauseitariseslessofteninpractice.Inanycase,wewillseeexamplesofthismodelinlaterchapters.Chapter2TheSimpleRegressionModelTable2.3ThelastcolumninTable2.3givestheinterpretationofβ1.Inthelog-levelmodel,100*β1issometimescalledthesemi-elasticityofywithrespecttox.AswementionedinExample2.11,inthelog-logmodel,β1istheelasticityofywithrespecttox.Table2.3warrantscarefulstudy,aswewillrefertoitoftenintheremainderofthetext.TheMeaningof“Linear”RegressionThesimpleregressionmodelthatwehavestudiedinthischapterisalsocalledthesimplelinearregressionmodel.Yet,aswehavejustseen,thegeneralmodelalsoallowsforcertainnonlinearrelationships.Sowhatdoes“linear”meanhere?Youcanseebylookingatequation(2.1)thaty=β0+β1x+u.Thekeyisthatthisequationislinearintheparameters,β0andβ1.Therearenorestrictionsonhowyandxrelatetotheoriginalexplainedandexplanatoryvariablesofinterest.AswesawinExamples2.7and2.8,yandxcanbenaturallogsofvariables,andthisisquitecommoninapplications.Butweneednotstopthere.Forexample,nothingpreventsusfromusingsimpleregressiontoestimateamodelsuchascons=β0+β1√inc+u,whereconsisannualconsumptionandincisannualincome.Whilethemechanicsofsimpleregressiondonotdependonhowyandxaredefined,theinterpretationofthecoefficientsdoesdependontheirdefinitions.Forsuccessfulempiricalwork,itismuchmoreimportanttobecomeproficientatinterpretingcoefficientsthantobecomeefficientatcomputingformulassuchas(2.19).WewillgetmuchmorepracticewithinterpretingtheestimatesinOLSregressionlineswhenwestudymultipleregression.Thereareplentyofmodelsthatcannotbecastasalinearregressionmodelbecausetheyarenotlinearintheirparameters;anexampleiscons=1/(β0+β1inc)+u.Estimationofsuchmodelstakesusintotherealmofthenonlinearregressionmodel,whichisbeyondthescopeofthistext.Formostapplications,choosingamodelthatcanbeputintothelinearregressionframeworkissufficient.EXPECTEDVALUESANDVARIANCESOFTHEOLSESTIMATORSInSection2.1,wedefinedthepopulationmodely=β0+β1x+u,andweclaimedthatthekeyassumptionforsimpleregressionanalysistobeusefulisthattheexpectedvalueofugivenanyvalueofxiszero.InSections2.2,2.3,and2.4,wediscussedthealgebraicpropertiesofOLSestimation.WenowreturntothepopulationmodelandstudythestatisticalpropertiesofOLS.Inotherwords,wenowviewβˆ0andβˆ1asestimatorsfortheparameters?0and?1thatappearinthepopulationmodel.Thismeansthatwewillstudypropertiesofthedistributionsof?ˆ0and?ˆ1overdifferentrandomsamplesfromthepopulation.(AppendixCcontainsdefinitionsofestimatorsandreviewssomeoftheirimportantproperties.)UnbiasednessofOLSWebeginbyestablishingtheunbiasednessofOLSunderasimplesetofassumptions.Forfuturereference,itisusefultonumbertheseassumptionsusingtheprefix“SLR”forsimplelinearregression.Thefirstassumptiondefinesthepopulationmodel.测量单位和函数形式在投票结果方程（2.28）中，R²=0.505。因此，竞选支出的份额解释只是在这样的选举结果的变化百分之50。这是一个相当大的部分在应用经济学的两个重要的问题是：（1）了解如何改变单位的依赖性和/或独立变量的OLS估计和测量的影响（2）知道如何将经济学中流行的功能形式为回归分析。一个完整的理解的功能形式问题所需要的数学在附录A了测量单位的OLS变化的影响统计在例2.3中，我们选择了测量数千美元的年薪，和净资产收益率是衡量百分之一（而不是一个小数点）。关键是要知道薪水和净资产收益率是衡量在这个例子中，为了使方程估计的意义（2.39）。我们还必须知道，OLS估计预期的方式改变时，完全的依赖和独立变量的测量单位的变化。在example2.3，假设，而不是千美元衡量的工资，我们衡量的美元。让salardol工资以美元（salardol=845761会被解释为845761美元。）。当然，salardol有数千美元的测量一个简单的关系：salardol工资？1000？工资。我们不需要实际运行的salardol对罗伊的回归知道估计方程为：萨拉ˆRDOL

=

963191

+

18501净资产收益率。我们得到的截距和斜率在（2.40）简单地乘以拦截和边坡（2.39）1000。这给出了方程（2.39）和（2.40）相同的解释。看（2.40），如果净资产收益率=

0，然后萨拉ˆRDOL

=

963191，所以预测的工资963191美元[我们得到的方程相同的值（2.39）]。此外，如果净资产收益率增加了一个，然后预测的工资增加18501美元；再次，这是我们从我们前面的分析方程（2.39）。一般来说，很容易找出发生了什么的截距和斜率的估计当因变量变化的测量单位。如果因变量乘以常数c-which意味着每个样品中的值乘以c-then

OLS截距和斜率的估计也乘以C（这是假设没有关于独立变量的变化。）在首席执行官工资的例子，C？1000从工资salardol。2章简单的回归模型我们还可以使用首席执行官工资的例子来看看会发生什么，当我们改变的独立变量的测量单位。定义roedec=净资产收益率/100是净资产收益率的十进制数；因此，roedec=0.23意味着股本回报率23%。专注于改变的独立变量的计量单位，我们回到我们原来的因变量，工资，这是在数千美元计算。当我们回归的工资roedec，我们获得萨尔ˆ进制=

963.191

+

1850.1

roedec。在roedec系数是100倍的系数对罗伊案（2.39）。这是应该的。一个百分点的变化的净资产收益率相当于Δroedec=0.01。从（2.41），如果Δroedec=0.01，然后Δ萨尔ˆ进制=1850.1（0.01）=18.501，这是通过使用（2.39）。请注意，从（2.39）至（2.41），独立变量除以100，并因此OLS斜率估计乘以100，保留方程的解释。一般来说，如果自变量除以或乘以某个非零常数，C，那么OLS斜率也乘以或除以c分别。拦截并没有改变（2.41）因为roedec=0对应于一零的股本回报率。一般来说，唯一的独立变量的变化不影响截距测量单位。在上一节中，我们定义了R2为善良的OLS回归拟合测量。我们也可以问发生什么，R2当单位是独立或依赖变量的变化测量。不做任何代数，我们应该知道结果：善良的模型拟合不应该依赖于我们的变量的测量单位。例如，变化的工资数额，由股本回报的解释，不应取决于工资是以美元或数千美元或是股本回报率是百分之一或一个十进制的测量。这种直觉可以验证数学：使用R2的定义，它可以表明，R2，事实上，在Y或X的单位不变将简单回归的非线性到目前为止我们侧重于依赖和独立的变量之间的线性关系。我们在1章中提到的，线性关系不几乎适合所有经济上的应用。幸运的是，它是将许多非线性的简单回归分析，通过适当定义的依赖和独立变量而容易。在这里，我们将两种可能性，应用工作中经常出现。在阅读中应用原文文献见估计一个模型如（2.42）是直接使用简单回归的时候。只是定义因变量，y，y=日志是（工资）。独立的变量是由X=教育代表。OLS的力学和以前一样：的截距和斜率的估计是由公式（2.17）和（2.19）。换句话说，我们获得了0和1βˆβˆ从日志OLS回归（工资）对教育。EX

M

PL

E2。1

0（L

O

G

W

G

E

q

u

T

O

N）使用相同的数据，如例2.4，但使用日志（工资）作为因变量，我们得到如下关系：日志（ˆ工资）=

0.584+

0.083教育（2.44）N=

526，R

=

0.186²。对教育的系数有一个解释当它乘以100：工资增加百分之8.3每增加一年的教育。这就是经济学家们所说的“回到一年的教育。“这是要记住，用工资的日志中的主要原因是重要的（2.42）工资实施教育的恒定比例的影响。一次方程（2.42）得到的工资，自然对数是很少提及。特别是，它是不正确的说，一年的教育可以增加日志（工资）8.3%。拦截（2.42）是非常有意义的，因为它给出了预测的日志（工资），当教育=0。R2表明教育解释变化在日志百分之18.6（工资）（而不是工资）。最后，方程（2.44）不可能捕捉到所有的非—在工资和教育之间的关系的线性度。如果有“文凭的影响，然后从高中毕业第十二年可以比第十一年前更值钱。我们将学习如何让这种在7章非线性。的自然对数的另一个重要用