詹姆斯计量经济学第三版偶数题4-8答案

Chapter4LinearRegressionwithOneRegressor4.2.Thesamplesize200.n=Theestimatedregressionequationis2(215)9941(031)394081SER102WeightHeightR=.−.+..,=.,=..(a)Substituting70,65,and74Height=inchesintotheequation,thepredictedweightsare176.39,156.69,and192.15pounds.(b)39439415591.WeightHeight∆=.×∆=.×.=.(c)Wehavethefollowingrelations:1254and104536.incmlbkg=.=.Supposetheregressionequationincentimeter-kilogramunitsis01垐WeightHeightγγ=+.Thecoefficientsare0ˆ99410453645092;kgγ=−.×.=−.045361254ˆ39407036kgγ..=.×=.percm.The2Risunitfree,soitremainsat2081R=..Thestandarderroroftheregressionis1020453646267.SERkg=.×.=.4.4.(a)()(),fmfRRRRuβ−=−+sothatvar2()var()var()2fmfRRRRuββ−=×−++×cov(,).mfuRR−Butcov(,)0,mfuRR−=thus2var()var()var().fmfRRRRuβ−=×−+Withβ>1,var(R−Rf)>var(Rm−Rf),followsbecausevar(u)≥0.(b)Yes.Usingtheexpressionin(a)2var()var()(1)var()var(),fmfmfRRRRRRuβ−−−=−×−+whichwillbepositiveif2var()(1)var().mfuRRβ>−×−(c)7.3%3.5%3.8%.mfRR−=−=Thus,thepredictedreturnsare垐ˆ()3.5%3.8%fmfRRRRββ=+−=+×Wal-Mart:3.5%+0.3×3.8%=4.6%Kellogg:3.5%0.53.8%5.4%+×=WasteManagement:3.5%0.63.8%5.8%+×=Verizon:3.5%0.63.8%5.8%+×=Microsoft:3.5%1.03.8%7.3%+×=BestBuy:3.5%1.33.8%8.4%+×=BankofAmerica:3.5%+2.4×3.8%=11.9%4.6.Using(|)0,iiEuX=wehave010101(|)(|)(|)(|)iiiiiiiiiiEYXEXuXEXXEuXXββββββ=++=++=+.4.8.Theonlychangeisthatthemeanof0βˆisnowβ0+2.Aneasywaytoseethisisthisistowritetheregressionmodelas01(2)(2).iiiYXuββ=+++−Thenewregressionerroris(ui−2)andthenewinterceptis(β0+2).AlloftheassumptionsofKeyConcept4.3holdforthisregressionmodel.4.10.(a)E(ui|X=0)=0andE(ui|X=1)=0.(Xi,ui)arei.i.d.sothat(Xi,Yi)arei.i.d.(becauseYiisafunctionofXiandui).Xiisboundedandsohasfinitefourthmoment;thefourthmomentisnon-zerobecausePr(Xi=0)andPr(Xi=1)arebothnon-zerosothatXihasfinite,non-zerokurtosis.Followingcalculationslikethoseexercise2.13,uialsohasnonzerofinitefourthmoment.(b)var()0.2(10.2)0.16iX=×−=and0.2.Xµ=Also222var[()][()][()|0]Pr(0)[()|1]Pr(1)iXiiXiiXiiiiXiiiXuEXuEXuXXEXuXXµµµµ−=−=−=×=+−=×=wherethefirstequalityfollowsbecauseE[(Xi−µX)ui]=0,andthesecondequalityfollowsfromthelawofiteratedexpectations.2222[()|0]0.21,and[()|1](10.2)4.iXiiiXiiEXuXEXuXµµ−==×−==−×Puttingtheseresultstogether1222ˆ21(0.210.8)((10.2)40.2)121.250.16nnβσ××+−××==4.12.(a)Write22201111121221211垐?ˆ()()[()]()()ˆ().()nnniiiiiinniiiiniiiESSYYXYXXXXYYXXXXββββ=======−=+−=−∑−−=−=∑−∑∑∑∑Thisimplies21222211121112211111122()()()()()()()()()niiinnniiiiiiniiinnniiiinnXYXYXYXXYYESSRYYXXYYXXYYXXYYsrss=====−==−−∑−−==∑−∑−∑−∑−−=∑−∑−==(b)Thisfollowsfrompart(a)becauserXY=rYX.(c)BecauserXY=XYXYsss,rXYYXss=111222111()()()()(1)ˆ1()()(1)nniiiiiiXYnnXiiiiXXYYXXYYsnsXXXXnβ====−−−−−===−−−∑∑∑∑4.14.Because01垐YXββ=−,01ˆYXββ=+.Thesampleregressionlineis01ˆyxββ=+,sothatthesampleregressionlinepassesthrough(,XY).Chapter5RegressionwithaSingleRegressor:HypothesisTestsandConfidenceIntervals5.2.(a)Theestimatedgendergapequals$2.12/hour.(b)Thenullandalternativehypothesesare010Hβ:=vs.110.Hβ:≠Thet-statisticis11ˆ02.125.89ˆ()036acttSEββ−===,.andthep-valueforthetestis-value2(||)2(5.89)2000000000actpt=Φ−=Φ−=×.=.(tofourdecimalplaces)Thep-valueislessthan0.01,sowecanrejectthenullhypothesisthatthereisnogendergapata1%significancelevel.(c)The95%confidenceintervalforthegendergap1βis{212196036},.±.×.thatis,11412.83.β.≤≤(d)Thesampleaveragewageofwomenis0ˆ1252/hour.$β=.Thesampleaveragewageofmenis01垐$12.52$2.12$14.64/hour.ββ+=+=(e)Thebinaryvariableregressionmodelrelatingwagestogendercanbewrittenaseither01iWageMaleuββ=++,or01iWageFemalevγγ=++.Inthefirstregressionequation,Maleequals1formenand0forwomen;0βisthepopulationmeanofwagesforwomenand01ββ+isthepopulationmeanofwagesformen.Inthesecondregressionequation,Femaleequals1forwomenand0formen;0γisthepopulationmeanofwagesformenand01γγ+isthepopulationmeanofwagesforwomen.Wehavethefollowingrelationshipforthecoefficientsinthetworegressionequations:001010γββγγβ=+,+=.Giventhecoefficientestimates0βˆand1βˆ,wehave0011001垐ˆ14.64垐垐212γββγγββ=+=,=−=−=−..Duetotherelationshipamongcoefficientestimates,foreachindividualobservation,theOLSresidualisthesameunderthetworegressionequations:.垐iiuv=Thusthesumofsquaredresiduals,21,ˆniiSSRu==∑isthesameunderthetworegressions.Thisimpliesthatboth1/21SSRSERn=−and21SSRRTSS=−areunchanged.Insummary,inregressingWageson,Femalewewillget14.64212WagesFemale=−.,20064.2RSER=.,=.5.4.(a)−5.38+1.76×16−$22.78perhour(b)Thewageisexpectedtoincreaseby1.76×2=$3.52perhour.(c)Theincreaseinwagesforcollegeeducationis14.β×Thus,thecounselor’sassertionisthat110/42.50.β==Thet-statisticforthisnullhypothesisis1.762.509.25,0.08actt−=−whichhasap-valueof0.00.Thus,thecounselor’sassertioncanberejectedatthe1%significancelevel.A95%confidenceforβ1×4is4×(1.76±1.96×0.08)or$6.41≤Gain≤$7.67.5.6.(a)Thequestionaskswhetherthevariabilityintestscoresinlargeclassesisthesameasthevariabilityinsmallclasses.Itishardtosay.Ontheonehand,teachersinsmallclassesmightablesospendmoretimebringingallofthestudentsalong,reducingthepoorperformanceofparticularlyunpreparedstudents.Ontheotherhand,mostofthevariabilityintestscoresmightbebeyondthecontroloftheteacher.(b)Theformulain5.3isvalidforheteroskesdasticityorhomoskedasticity;thusinferencesarevalidineithercase.5.8.(a)43.2±2.05×10.2or43.2±20.91,where2.05isthe5%two-sidedcriticalvaluefromthet28distribution.(b)Thet-statisticis61.5557.40.88,actt−==whichisless(inabsolutevalue)thanthecriticalvalueof20.5.Thus,thenullhypothesisisnotrejectedatthe5%level.(c)Theonesided5%criticalvalueis1.70;tactislessthanthiscriticalvalue,sothatthenullhypothesisisnotrejectedatthe5%level.5.10.Letn0denotethenumberofobservationwithX=0andn1denotethenumberofobservationswithX=1;notethat11;niiXn==∑1;nXn=1111;niiiXYYn==∑22211()nniiiiXXXnX==−=−=∑∑21011111;nnnnnnnnn−=−=11001,niinYnYY=+=∑sothat0110nnYYYnn=+Fromtheleastsquaresformula11111221110011101000()()()ˆ()()/()nnniiiiiiiiinniiiiXXYYXYYXYYnXXXXnnnnnnnYYYYYYYnnnnβ=====∑−−∑−∑−===∑−∑−=−=−−=−and0101101011000垐()nnnnnYXYYYYYYnnnnββ+=−=+−−==5.12.Equation(4.22)gives()()02ˆ222var()where1iiXiiiiHuHXEXnEHβµσ=,=−.Usingthefactsthat(|)0iiEuX=andvar2(|)iiuuXσ=(homoskedasticity),wehave()()()222()()[(|)]000xxiiiiiiiiiiixiEHuEuXuEuEXEuXEXEXEXµµµ=−=−=−×=,and222222222222222[()]22|XiiiiiiXXiiiiiiiXXiiiiiiEHuEuXuEXEuXuXuEXEXEuEXEuXEXEXµµµµµ=−=−+=−+222222222222|21.XiiiXXXiuuuuiiiEXEuXEXEXEXEXµµµµσσσσ=−+=−Because()0,iiEHu=var2()[()]iiiiHuEHu=,so()2222var()[()]1XiiiiuiHuEHuEXµσ==−.Also()()()()()()()()22222222222222112121XXXiiiiiiiXXXiiiiEHEXEXXEXEXEXEXEXEXEXµµµµµµ=−=−+=−+=−.Thus()()()()()()022222ˆ222222222222221var()11()[]XuiiiuiXXiiiuiuXiXEXHunEHnnEXEXEXEXnnEXβµσσσµµσσσµ−===−−==.−5.14.(a)FromExercise(4.11),ˆiiaYβ=∑where21injjXiXa==∑.SincetheweightsdependonlyoniXbutnotoniY,βˆisalinearfunctionofY.(b)11121(|,,)ˆ(|,,)niiinnnijXEuXXEXXXβββ==∑=+=∑since1(|,,)0inEuXX=(c)2211122211var(|,,)ˆvar(|,,)niiinnnnijijXuXXXXXXσβ===∑==∑∑(d)Thisfollowstheproofintheappendix.Chapter6LinearRegressionwithMultipleRegressors6.4.(a)WorkersintheNortheastearn$0.69moreperhourthanworkersintheWest,onaverage,controllingforothervariablesintheregression.WorkersintheNortheastearn$0.60moreperhourthanworkersintheWest,onaverage,controllingforothervariablesintheregression.WorkersintheSouthearn$0.27lessthanworkersintheWest.(b)TheregressorWestisomittedtoavoidperfectmulticollinearity.IfWestisincluded,thentheinterceptcanbewrittenasaperfectlinearfunctionofthefourregionalregressors.(c)TheexpecteddifferenceinearningsbetweenJuanitaandJenniferis02706087.−.−.=−.6.6.(a)Thereareotherimportantdeterminantsofacountry’scrimerate,includingdemographiccharacteristicsofthepopulation.(b)Supposethatthecrimerateispositivelyaffectedbythefractionofyoungmalesinthepopulation,andthatcountieswithhighcrimeratestendtohiremorepolice.Inthiscase,thesizeofthepoliceforceislikelytobepositivelycorrelatedwiththefractionofyoungmalesinthepopulationleadingtoapositivevaluefortheomittedvariablebiassothat11ˆ.ββ>6.8.Omittedfromtheanalysisarereasonswhythesurveyrespondentssleptmoreorlessthanaverage.Peoplewithcertainchronicillnessesmightsleepmorethan8hourspernight.Peoplewithotherillnessesmightsleeplessthan5hours.Thisstudysaysnothingaboutthecausaleffectofsleeponmortality.6.10.(a)112122ˆ22,111uXXXnβσσρσ=−AssumeX1andX2areuncorrelated:1220XXρ=12ˆ1144001061410.001674006600βσ=−=⋅==(b)With12,0.5XXρ=12ˆ211440010.56114.00224000.756βσ=−==(c)ThestatementcorrectlysaysthatthelargeristhecorrelationbetweenX1andX2thelargeristhevarianceof1ˆ,βhowevertherecommendation“itisbesttoleaveX2outoftheregression”isincorrect.IfX2isadeterminantofY,thenleavingX2outoftheregressionwillleadtoomittedvariablebiasin1ˆ.βChapter7HypothesisTestsandConfidenceIntervalsinMultipleRegression7.1and7.2Regressor(1)(2)(3)College(X1)5.46**(0.21)5.48**(0.21)5.44**(0.21)Female(X2)−2.64**(0.20)−2.62**(0.20)−2.62**(0.20)Age(X3)0.29**(0.04)0.29**(0.04)Ntheast(X4)0.69*(0.30)Midwest(X5)0.60*(0.28)South(X6)−0.27(0.26)Intercept12.69**(0.14)4.40**(1.05)3.75**(1.06)(a)Thet-statisticis5.46/0.21=26.0,whichexceeds1.96inabsolutevalue.Thus,thecoefficientisstatisticallysignificantatthe5%level.The95%confidenceintervalis5.46±1.96×0.21.(b)t-statisticis−2.64/0.20=−13.2,and13.2>1.96,sothecoefficientisstatisticallysignificantatthe5%level.The95%confidenceintervalis−2.64±1.96×0.20.7.4.(a)TheF-statistictestingthecoefficientsontheregionalregressorsarezerois6.10.The1%criticalvalue(fromthe3,F∞distribution)is3.78.Because6.10>3.78,theregionaleffectsaresignificantatthe1%level.(b)TheexpecteddifferencebetweenJuanitaandMollyis(X6,Juanita−X6,Molly)×β6=β6.Thusa95%confidenceintervalis−0.27±1.96×0.26.(c)TheexpecteddifferencebetweenJuanitaandJenniferis(X5,Juanita−X5,Jennifer)×β5+(X6,Juanita−X6,Jennifer)×β6=−β5+β6.A95%confidenceintervalcouldbeconstructedusingthegeneralmethodsdiscussedinSection7.3.Inthiscase,aneasywaytodothisistoomitMidwestfromtheregressionandreplaceitwithX5=West.InthisnewregressionthecoefficientonSouthmeasuresthedifferenceinwagesbetweentheSouthandtheMidwest,anda95%confidenceintervalcanbecomputeddirectly.7.6.Inisolation,theseresultsdoimplygenderdiscrimination.Genderdiscriminationmeansthattwoworkers,identicalineverywaybutgender,arepaiddifferentwages.Thus,itisalsoimportanttocontrolforcharacteristicsoftheworkersthatmayaffecttheirproductivity(education,yearsofexperience,etc.)Ifthesecharacteristicsaresystematicallydifferentbetweenmenandwomen,thentheymayberesponsibleforthedifferenceinmeanwages.(Ifthisweretrue,itwouldraiseaninterestingandimportantquestionofwhywomentendtohavelesseducationorlessexperiencethanmen,butthatisaquestionaboutsomethingotherthangenderdiscriminationintheU.S.labormarket.)ThesearepotentiallyimportantomittedvariablesintheregressionthatwillleadtobiasintheOLScoefficientestimatorforFemale.Sincethesecharacteristicswerenotcontrolledforinthestatisticalanalysis,itisprematuretoreachaconclusionaboutgenderdiscrimination.7.8.(a)UsingtheexpressionsforR2and2,Ralgebrashowsthat2222111(1),so1(1).11nnkRRRRnkn−−−=−−=−−−−−242011Column1:1(10.049)0.0514201R−−=−−=−242021Column2:1(10.424)0.4274201R−−=−−=−242031Column3:1(10.773)0.7754201R−−=−−=−242031Column4:1(10.626)0.6294201R−−=−−=−242041Column5:1(10.773)0.7754201R−−=−−=−(b)034134:0:,0HHββββ==≠≠Unrestrictedregression(Column5):2011223344unrestricted,0.775YXXXXRβββββ=++++=Restrictedregression(Column2):201122restricted,0.427YXXRβββ=++=22unrestrictedrestrictedHomoskedasticityOnlyunrestricted2unrestrictedunrestricted()/,420,4,2(1)/(1)(0.7750.427)/20.348/20.174322.22(10.775)/(42041)(0.225)/4150.00054RRqFnkqRnk−====−−−−====−−−5%CriticalvalueformF2,00=4.61;FHomoskedasticityOnly=F2,00soHoisrejectedatthe5%level.(c)t3=−13.921andt4=0.814,q=2;|t3|>c(Wherec=2.807,the1%BenferronicriticalvaluefromTable7.3).Thusthenullhypothesisisrejectedatthe1%level.(d)−1.01±2.58×0.277.10.Because222restrictedunrestrictedunrestrictedrestricted1,SSRSSRSSRRRRTSSTSS−=−−=and2unrestrictedunrestricted1.SSRRTSS−=Thusrestrictedunrestrictedunrestricted22unrestrictedrestricted2unrestrictedunrestrictedunrestrictedrestrictedunrestrictedunrestrictedun/()/(1)/(1)/(1)()//(SSRSSRTSSSSRTSSqRRqFRnknkSSRSSRqSSRnk−−==−−−−−−=−restricted1)−Chapter8NonlinearRegressionFunctions8.2.(a)Accordingtotheregressionresultsincolumn(1),thehousepriceisexpectedtoincreaseby21%(=100%×0.00042×500)withanadditional500squarefeetandotherfactorsheldconstant.The95%confidenceintervalforthepercentagechangeis100%×500×(0.00042±1.96×0.000038)=[17.276%to24.724%].(b)Becausetheregressionsincolumns(1)and(2)havethesamedependentvariable,2Rcanbeusedtocomparethefitofthesetworegressions.Thelog-logregressionincolumn(2)hasthehigher2,Rsoitisbettersouseln(Size)toexplainhouseprices.(c)Thehousepriceisexpectedtoincreaseby7.1%(=100%×0.071×1).The95%confidenceintervalforthiseffectis100%×(0.071±1.96×0.034)=[0.436%to13.764%].(d)Thehousepriceisexpectedtoincreaseby0.36%(100%×0.0036×1=0.36%)withanadditionalbedroomwhileotherfactorsareheldconstant.Theeffectisnotstatisticallysignificantata5%significancelevel:0.0036||0.097301.96.0.037t==<Notethatthiscoefficientmeasurestheeffectofanadditionalbedroomholdingthesizeofthehouseconstant.Thus,itmeasurestheeffectofconvertingexistingspace(from,sayafamilyroom)intoabedroom.(e)Thequadratictermln(Size)2isnotimportant.Thecoefficientestimateisnotstatisticallysignificantata5%significancelevel:0.0078||0.055711.96.0.14t==<(f)Thehousepriceisexpectedtoincreaseby7.1%(=100%×0.071×1)whenaswimmingpoolisaddedtoahousewithoutaviewandotherfactorsareheldconstant.Thehousepriceisexpectedtoincreaseby7.32%(=100%×(0.071×1+0.0022×1))whenaswimmingpoolisaddedtoahousewithaviewandotherfactorsareheldconstant.Thedifferenceintheexpectedpercentagechangeinpriceis0.22%.Thedifferenceisnotstatisticallysignificantata5%significancelevel:0.0022||0.0221.96.0.10t==<8.4.(a)With2yearsofexperience,theman’sexpectedAHEis2ln()(0.103216)(0.4510)(0.0134016)(0.01342)0.0002112)(0.0950)(0.0920)(0.0231)1.5033.159AHE=×−×+××+×−×−×−×−×+=With3yearsofexperience,theman’sexpectedAHEis2ln()(0.103216)(0.4510)(0.0134016)(0.01343)0.0002113)(0.0950)(0.0920)(0.0231)1.5033.172AHE=×−×+××+×−×−×−×−×+=Difference=3.172−3.159=0.013(or1.3%)(b)With10yearsofexperience,theman’sexpectedAHEis2ln()(0.103216)(0.4510)(0.0134016)(0.013410)0.00021110)(0.0950)(0.0920)(0.0231)1.5033.253AHE=×−×+××+×−×−×−×−×+=With11yearsofexperience,theman’sexpectedAHEis2ln()(0.103216)(0.4510)(0.0134016)(0.013411)0.00021111)(0.0950)(0.0920)(0.0231)1.5033.263AHE=×−×+××+×−×−×−×−×+=Difference3.263−0.010(or1.0%)(c)Theregressioninnonlinearinexperience(itincludesPotentialexperience2).(d)Letβ1denotethecoefficientonPotentialExperienceandβ2denotethecoefficienton(PotentialExperience)2.Then,followingthediscussionintheparagraphsjustaboutequation(8.7)inthetext,theexpectedchangeinpart(a)isgivenbyβ1+5β2andtheexpectedchangein(b)isgivenbyβ1+21β2.Thedifferencebetweenthese,say(b)−(a),is16β2.Becausetheestimatedvalueofβ2issignificantatthe5%level(thet-statisticfor2βˆis−0.000211/0.000023=−9.2),thedifferencebetweentheeffectsin(a)and(b)(=16β2)issignificantatthe5%level.(e)No.Thiswouldaffectthelevelofln(AHE),butnotthechangeassociatedwithanotheryearofexperience.(f)IncludeinteractiontermsFemale×PotentialexperienceandFemale×(Potentialexperience)2.8.6.(a)(i)Thereareseveralwaystodothis.Hereisone.Createanindicatorvariable,sayDV1,thatequalsoneif%Eligibleisgreaterthan20%andlessthan50%.Createanotherindicator,sayDV2,thatequalsoneif%Eligibleisgreaterthan50%.Runtheregression:0123%1%2%otherregressorsTestScoreEligibleDVEligibleDVEligibleββββ=++×+×+Thecoefficientβ1showsthemarginaleffectof%EligibleonTestScoresforvaluesof%Eligible<20%,β1+β2showsthemarginaleffectforvaluesof%Eligiblebetween20%and50%andβ1+β3showsthemarginaleffectforvaluesof%Eligiblegreaterthan50%.(ii)Thelinearmodelimpliesthatβ2=β3=0,whichcanbetestedusinganF-test.(b)(i)Thereareseveralwaystodothis,perhapstheeasiestistoincludeaninteractiontermSTR×ln(Income)totheregressionincolumn(7).(ii)Estimatetheregressioninpart(b.i)andtestthenullhypothesisthatthecoefficientontheinteractiontermisequaltozero.8.8(a)and(b)(c)(d)(e)8.10.(a)1121211312(,)(,),YfXXXfXXXXXββ∆=+∆−=∆+∆×so1321.YXXββ∆=+∆(b)122122312(,)(,),YfXXXfXXXXXββ∆=+∆−=∆+×∆2so2312.YXXββ∆=+∆(c)1122120111222311220112231213212312312(,)(,)()()()()()()().YfXXXXfXXXXXXXXXXXXXXXXXXXXβββββββββββββ∆=+∆+∆−=++∆++∆++∆+∆−+++=+∆++∆+∆∆8.12.(a)BecauseofrandomassignmentwithinthegroupofreturningstudentsE(X1i|ui)=0in“γ-regression,”sothat1ˆ&ga