测量单位和函数形式外文翻译.docx

测量单位和函数形式外文翻译.docx

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测量单位和函数形式外文翻译

河南科技学院

2014届本科毕业论文（设计）

英文文献及翻译

UNITSOFMEASUREMENTANDFUNCTIONALFORM

学生姓名：

所在院系：

数学科学学院

所学专业：

数学与应用数学

导师姓名：

完成时间：

2013年12月25日

UNITSOFMEASUREMENTANDFUNCTIONALFORM

（VotingOutcomesandCampaignExpenditures）

Inthevotingoutcomeequationin（2.28）,R²=0.505.Thus,theshareofcampaignexpendituresexplainsjustover50percentofthevariationintheelectionoutcomesforthissample.Thisisafairlysizableportion

Twoimportantissuesinappliedeconomicsare

（1）understandinghowchangingtheunitsofmeasurementofthedependentand/orindependentvariablesaffectsOLSestimatesand

（2）knowinghowtoincorporatepopularfunctionalformsusedineconomicsintoregressionanalysis.ThemathematicsneededforafullunderstandingoffunctionalformissuesisreviewedinAppendixA.

TheEffectsofChangingUnitsofMeasurementonOLS

Statistics

InExample2.3,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）simplybymultiplyingtheinterceptandthe

slopein（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,ifroe

increasesbyone,thenthepredictedsalaryincreasesby$18,501;again,thisiswhatwe

concludedfromourearlieranalysisofequation（2.39）.

Generally,itiseasytofigureoutwhathappenstotheinterceptandslopeestimateswhenthedependentvariablechangesunitsofmeasurement.Ifthedependentvariableismultipliedbytheconstantc—whichmeanseachvalueinthesampleismultipliedbyc—thentheOLSinterceptandslopeestimatesarealsomultipliedbyc.（Thisassumesnothinghaschangedabouttheindependentvariable.）IntheCEOsalaryexample,c?

1,000inmovingfromsalarytosalardol.

Chapter2

TheSimpleRegressionModel

WecanalsousetheCEOsalaryexampletoseewhathappenswhenwechangetheunitsofmeasurementoftheindependentvariable.Defineroedec=roe/100tobethedecimalequivalentofroe;thus,roedec=0.23meansareturnonequityof23percent.Tofocusonchangingtheunitsofmeasurementoftheindependentvariable,wereturntoouroriginaldependentvariable,salary,whichismeasuredinthousandsofdollars.Whenweregresssalaryon

roedec,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）,theindependent

variablewasdividedby100,andsotheOLSslopeestimatewasmultipliedby100,preservingtheinterpretationoftheequation.Generally,iftheindependentvariableisdividedormultipliedbysomenonzeroconstant,c,thentheOLSslopecoefficientisalsomultipliedordividedbycrespectively.

Theintercepthasnotchangedin（2.41）becauseroedec=0stillcorrespondstoazeroreturnonequity.Ingeneral,changingtheunitsofmeasurementofonlytheindependentvariabledoesnotaffecttheintercept.

Intheprevioussection,wedefinedR-squaredasagoodness-of-fitmeasureforOLSregression.WecanalsoaskwhathappenstoR2whentheunitofmeasurementofeithertheindependentorthedependentvariablechanges.Withoutdoinganyalgebra,weshouldknowtheresult:

thegoodness-of-fitofthemodelshouldnotdependon

theunitsofmeasurementofourvariables.Forexample,theamountofvariationinsalary,explainedbythereturnonequity,shouldnotdependonwhethersalaryismeasuredindollarsorinthousandsofdollarsoronwhetherreturnonequityisapercentoradecimal.Thisintuitioncanbeverifiedmathematically:

usingthedefinitionofR2,itcanbeshownthatR2is,infact,invarianttochangesintheunitsofyorx.

IncorporatingNonlinearitiesinSimpleRegression

Sofarwehavefocusedonlinearrelationshipsbetweenthedependentandindependentvariables.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）.Thisequation

isgraphedinFigure2.6,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）.EstimatingthisequationbyOLSgives

Part1

RegressionAnalysiswithCross-SectionalData

log（salˆary）=4.822?

+0.257log（sales）

（2.46）

n=209,R²=0.211.

Thecoefficientoflog（sales）istheestimatedelasticityofsalarywithrespecttosales.Itimpliesthata1percentincreaseinfirmsalesincreasesCEOsalarybyabout0.257percent—theusualinterpretationofanelasticity.

Thetwofunctionalformscoveredinthissectionwilloftenariseintheremainderofthistext.Wehavecoveredmodelscontainingnaturallogarithmsherebecausetheyappearsofrequentlyin